SAE Mathematics Paper 2: L₀ Mathematicality — Inevitability and Multi-Path Realisation

Qin, Han

doi:10.5281/zenodo.20199082

Abstract

The question this paper takes up is: what is mathematics? More precisely: what makes an articulative practice mathematical, in the SAE perspective, and how does mathematicality emerge from the qualitative articulation that precedes any specific mathematical structure? The question is older than this paper, and it has been articulated in many forms. The Platonist takes mathematics to be the discernment of pre-existing eternal structures; the formalist takes it to be the manipulation of symbols according to specified rules; the intuitionist takes it to be the mental construction of objects in a primitive temporal intuition; the structuralist takes it to be the study of abstract structural patterns instantiable in multiple substrates; the category theorist takes it to be the study of morphism patterns across structural domains. Each tradition articulates the question differently, with substantive content of its own. The present paper articulates the question from the SAE perspective. The SAE perspective is one thinking framework among the many. It is not a totalising synthesis that absorbs the others; it is a specific articulative stance that contributes a substantive answer to the question while acknowledging that other stances contribute substantively different answers. The first epistemological anchor of § 1.5 articulates this commitment explicitly.

Keywords: SAE mathematics, L0 mathematicality, four inevitabilities, closure paths, complete exactification, objective exactification schema, multi-path realisation, aesthetic mathematicality, public re-entry

SAE Mathematicality: Inquiring into Mathematics from $L_0$ — Inevitability and Multi-Path Realisation

Author: Han Qin (秦汉)

ORCID: 0009-0009-9583-0018

Version: v2 final

Date: 2026-05-14

Predecessor paper: Paper 1, SAE Mathematics Foundational Paper (DOI: 10.5281/zenodo.20153791, 2026-05-13)

1. Introduction

1.1 The question

The question this paper takes up is: what is mathematics? More precisely: what makes an articulative practice mathematical, in the SAE perspective, and how does mathematicality emerge from the qualitative articulation that precedes any specific mathematical structure?

The question is older than this paper, and it has been articulated in many forms. The Platonist takes mathematics to be the discernment of pre-existing eternal structures; the formalist takes it to be the manipulation of symbols according to specified rules; the intuitionist takes it to be the mental construction of objects in a primitive temporal intuition; the structuralist takes it to be the study of abstract structural patterns instantiable in multiple substrates; the category theorist takes it to be the study of morphism patterns across structural domains. Each tradition articulates the question differently, with substantive content of its own.

The present paper articulates the question from the SAE perspective. The SAE perspective is one thinking framework among the many. It is not a totalising synthesis that absorbs the others; it is a specific articulative stance that contributes a substantive answer to the question while acknowledging that other stances contribute substantively different answers. The first epistemological anchor of § 1.5 articulates this commitment explicitly.

The SAE answer, which this paper develops, is that mathematicality emerges at $L_0$—the level of qualitative articulation that precedes any specific marked handle—through four inevitabilities plus a public re-entry criterion. Specific mathematical practices arise as different closure paths through which these conditions are realised. Conventional mathematics, in its institutionalised forms, is one closure path (complete exactification through unit and arithmetic). Other paths are possible and have substantive content in their own right.

The thesis is articulated in three layers, which we state explicitly at the outset.

First layer: Mathematicality is identified by four inevitabilities of qualitative articulation at $L_0$ (comparison, direction, propagation, sustained questioning), plus a public re-entry criterion. These conditions are jointly identified as the conditions for the emergence of SAE mathematicality from qualitative articulation. They are not a universal definition of mathematics; they are the SAE perspective's identification of mathematicality, which other perspectives may articulate differently.

Second layer: $L_0$ is multi-dimensional, and the closure paths through which mathematicality is realised are multiple. Different qualitative dimensions (quantity, temporal rhythm, spatial size, aesthetic value, ethical articulation) admit different closure paths (complete exactification, partial exactification, persistent open). The substantive content of mathematicality is non-unique: different dimensions and different paths produce substantively different forms of mathematicality.

Third layer: Paper 1's Layer Articulation Schema, on this articulation, is the path-specific schema of the complete exactification path. We rename it the Objective Exactification Schema to mark its path-specific character. Paper 2 articulates an upper-level schema, the $L_0$ Mathematicality Schema, of which Paper 1's schema is one realisation. Paper 1 is positioned within the broader landscape Paper 2 articulates, not narrowed; the present paper extends rather than retroactively modifies Paper 1.

These three layers will be developed in §§ 3 through 7, with § 5 articulating the substantive content of the aesthetic dimension as the most distinctive of the dimensions for which the present paper provides substantive elevation.

1.2 Starting from $L_0$

Paper 1 introduced $L_0$ as a thought experiment in its § 2, locating it as the layer of qualitative articulation that precedes the introduction of "1" as a marked handle. Paper 1 used $L_0$ to motivate the layered articulation of mathematics that became its central topic, but it treated $L_0$ relatively briefly, primarily as a starting point for the move into $L_1$.

The present paper takes $L_0$ as the substantive starting point of mathematicality, not merely as a thought-experimental precursor to $L_1$. The substantive content of $L_0$, on the present paper's articulation, is the home of the four inevitabilities and the public re-entry criterion; it is where mathematicality emerges, before any specific closure path is chosen and before any specific marked handle is introduced. $L_0$ is, in this sense, the foundational layer for all of mathematicality, not just for conventional mathematics in the complete exactification path.

A clarification that will be essential throughout the paper: $L_0$, on the present articulation, contains no marked handles. "1" is already $L_1$; "one cycle per year" as a base rhythm is already $L_1$; the king's fist as a reference is already $L_1$. $L_0$ is qualitative articulation without precision, before any specific structure is introduced to fix the comparative content. The four inevitabilities and the public re-entry criterion articulate what must be present in any qualitative articulation that can become mathematicality, not what mathematicality looks like after marked handles have been introduced.

This clarification was an important refinement of Paper 1's articulation. Paper 1, by treating $L_0$ briefly as a precursor to $L_1$, did not make fully explicit that $L_0$ itself contains no marked handles. The present paper makes this explicit and articulates the substantive content of $L_0$ in detail in § 2.

1.3 The thesis as three layers

We restate the thesis with somewhat more detail, in preparation for the substantive sections to follow.

The first layer of the thesis articulates SAE mathematicality as identified by five conditions, four inevitabilities plus the public re-entry criterion. Comparison is unavoidable: any qualitative articulation is inherently relational, placing terms in comparative relation. Direction is unavoidable: comparisons have inherent orientation — for example, more/less, fast/slow, beautiful/ugly, close/distant, relevant/irrelevant; "more/less" is only the example from the quantity dimension, not the definition of direction. Propagation is unavoidable: comparisons cannot remain confined to isolated pairs; they propagate, generating pressure toward extending the comparative structure. Sustained questioning is unavoidable: closure attempts produce remainders, and the remainders demand further articulation. Public re-entry is required: the articulation must be available for re-entry by other subjects, not purely private. These five conditions, jointly satisfied, constitute SAE mathematicality. § 3 articulates the four inevitabilities; § 3.8 articulates the public re-entry criterion; § 7 articulates the substantive defense framework that makes the conditions checkable.

The second layer of the thesis articulates that mathematicality is realised through multiple closure paths, each producing substantively different forms. We articulate three closure paths in § 4: complete exactification (the path of conventional mathematics, with unit and arithmetic), partial exactification (the path of order-theoretic structures, with reference but without unit), and persistent open (the path that does not fix structural closure, articulating distributional structure of comparative judgements). The three paths are complementary at the level of substantive mathematical content; they are not in a hierarchy where one path is more fundamental than the others. Different $L_0$ dimensions articulate along different paths, and § 5 examines specific dimensions and their canonical paths.

The third layer of the thesis articulates Paper 1's Layer Articulation Schema as the canonical schema of the complete exactification path. The schema applies in full force within that path, applies with substantive reinterpretation in the partial exactification path, and does not apply in its literal form in the persistent open path. We rename Paper 1's schema the Objective Exactification Schema, marking its path-specific character. Paper 2 articulates an upper-level schema, the $L_0$ Mathematicality Schema, of which Paper 1's schema is one realisation. The renaming is a substantive scope clarification, not a retroactive modification of Paper 1. § 6 articulates the renaming and the upper-level schema.

The three layers together constitute the substantive thesis of the paper.

1.4 The relation to Paper 1

Paper 1 (Qin 2026, DOI: 10.5281/zenodo.20153791) articulated the architecture of SAE mathematics as a layered structure spanning $L_1$ through $L_5$, with the Layer Articulation Schema providing the canonical four-step pattern at each layer (marked handle, additive path, multiplicative path with memory binding, closure with remainder). Paper 1 also articulated, in its § 7, the distinction between closure-equation grammar and probability-distribution grammar, positioning the latter as taking over where the former fails (notably in non-equilibrium thermodynamics and complex-systems contexts).

The present paper extends Paper 1 in three substantive ways.

First, the present paper articulates the $L_0$ foundation that Paper 1 left implicit. Paper 1's $L_0$ was a thought-experimental precursor to $L_1$; the present paper articulates $L_0$ as the substantive foundation of all mathematicality, with the four inevitabilities and the public re-entry criterion as the conditions for the emergence of mathematicality.

Second, the present paper articulates multiple closure paths where Paper 1 articulated one. Paper 1's layered architecture corresponds, on the present articulation, to the complete exactification path. Paper 2 articulates two additional paths (partial exactification, persistent open) and articulates how mathematicality is realised differently along each path. The present paper's expansion of grammar typology adds ordering grammar (not articulated in Paper 1) as the grammar of the partial exactification path, and articulates probability-distribution grammar as a grammar in its own right capable of articulating subject-grounded mathematicality directly, not merely as a fallback when closure-equation grammar fails.

Third, the present paper renames Paper 1's schema to mark its path-specific character. Paper 1's Layer Articulation Schema is the Objective Exactification Schema; Paper 2's upper-level schema is the $L_0$ Mathematicality Schema; the two are related as one realisation to the upper-level structure.

We emphasise what these extensions do not do. They do not retroactively modify Paper 1's published content. Paper 1's substantive articulation of the complete exactification path's architecture stands as Paper 1 articulated it. The present paper positions Paper 1 within a broader landscape; it does not narrow Paper 1 in the sense of reducing Paper 1's substantive content. Paper 1's articulation of the layered architecture is complete within its scope; the present paper articulates that the scope is the complete exactification path, and provides the broader $L_0$ foundation that situates Paper 1's articulation. The relation is one of extension, not revision.

A practical consequence: readers of the SAE mathematics series should read Paper 1 and Paper 2 as articulating different substantive aspects of the same broader landscape. Paper 1 articulates the architecture of the dominant path; Paper 2 articulates the foundation and the multi-path structure. Neither paper supersedes the other; together they articulate the SAE foundation for the series's subsequent papers on specific layers, specific paths, and specific topics.

1.5 Three epistemological anchors

The present paper carries three substantive epistemological commitments that we articulate explicitly here, both because they are essential to the paper's articulative stance and because they are easy to misread if left implicit. The three commitments shape how the substantive sections of the paper are to be read.

Anchor A: SAE is one thinking framework, not a totalising synthesis.

The SAE perspective articulates one specific way of identifying mathematicality and one specific articulation of the substantive content of $L_0$. It does not claim to be the unique framework, the universal framework, or the synthesis under which all other frameworks for mathematics are absorbed. Other frameworks—formalism, intuitionism, structuralism, naturalised philosophies of mathematics, category-theoretic foundations, ethnomathematics, and others—articulate mathematicality, or its closest analogues, differently, with substantive content of their own. The five conditions of § 3 and § 3.8 are the SAE perspective's identification of mathematicality, not a universal definition that other frameworks must adopt. The closure-path typology of § 4 is the SAE perspective's articulation of how mathematicality is realised, not a closure that excludes other typologies. The schema of § 6 is the SAE perspective's hierarchical organisation, not the unique organisation that all frameworks must accept.

This anchor is essential because the paper's substantive content might otherwise be read as totalising. The claim that "mathematicality is identified by these five conditions" might be read as "mathematics is exhaustively defined by these five conditions". The claim that "mathematicality is realised through these three closure paths" might be read as "all of mathematics consists in these three paths". The claim that "Paper 1's schema is path-specific" might be read as "Paper 1's schema is incorrect". None of these stronger readings is what the paper claims. The paper articulates a substantive perspective and identifies substantive structures within that perspective, while acknowledging that other perspectives articulate substantively different structures.

The anchor is consistent with, and a substantive extension of, Paper 1's epistemological stance articulated in its § 3.8 of double inexhaustibility (object level and meta level). The present paper extends the inexhaustibility to a third level: path inexhaustibility (multiple closure paths exist, more may be identified, the schema is open to extension). Together, the three levels of inexhaustibility constitute the substantive content of Anchor A.

Anchor B: Remainders cannot be eliminated.

Paper 1 articulated remainders as the engines of layer transition: each layer's closure produces a remainder that becomes the next layer's marked handle, and the development of mathematics proceeds through this articulation of remainders. The present paper extends this commitment to the $L_0$ foundation: any articulation of mathematicality leaves remainders, and the remainders are not deficiencies to be eliminated but constitutive of the practice's continued development.

The anchor has substantive content for the present paper in two specific ways. First, the four inevitabilities are not a complete definition of mathematics: they identify mathematicality but leave the substantive content of specific mathematical structures as remainders that subsequent articulation develops. The fourth inevitability, sustained questioning, articulates the anchor directly: closure attempts produce remainders, and the remainders demand articulation. Second, the multi-path typology of § 4 does not claim to exhaust the closure paths: three paths are articulated, but the typology is open to extension, and other paths may emerge as the schema is developed. The remainder of the typology—the paths not yet articulated, the structures within paths not yet developed—is the substantive content of future work, not a deficiency of the present paper.

The anchor blocks an interpretation of the paper as a totalising closure of the question "what is mathematicality?". The paper articulates conditions and structures; the conditions and structures are real; but they do not close the question. The remainder is not eliminated; it is the engine of continued development.

Anchor C: Subjectivity cannot be quantified without being abolished.

The third anchor articulates a substantive commitment about subjectivity in mathematicality. The SAE perspective takes subjectivity as central: Self-as-an-End names the commitment to the subject as constitutive of articulation, not reducible to a means, not reducible to a function of impersonal structures. The aesthetic dimension is the dimension in which subjectivity is most fully constitutive, and aesthetic mathematicality, accordingly, is the form of mathematicality in which subjectivity is most fully central.

The anchor articulates what subjectivity-centred articulation cannot do without ceasing to be subjectivity-centred. Quantification of aesthetic articulation, in the sense of assigning numerical values to aesthetic judgements on a presumptively objective scale, replaces the constitutive role of the subject with a measurement procedure that is independent of the subject. The aesthetic judgement becomes a report on a value the scale already assigns; the subject is reduced from constitutive evaluator to mere reporter. This is the abolition of subjectivity in the relevant sense, even though aesthetic content is nominally still being articulated.

The anchor has substantive consequences throughout the paper. In § 4.4, the persistent open path is articulated as not introducing a unit, with the absence of a unit being substantive rather than a deficiency to be remedied. In § 5.5, aesthetic mathematicality is articulated as distribution-grounded, with the explicit clarification that this is not quantification of aesthetic content but articulation of the distributional structure of aesthetic comparison. In § 7.4, aesthetic mathematicality's satisfaction of the five inevitability criteria is articulated in a way that respects the constitutive role of the subject, not as the measurement of an objective aesthetic property.

The anchor is also a substantive rejection of certain attractive but incorrect articulations of aesthetic mathematicality. We do not articulate aesthetic mathematicality as a sophisticated quantification of subjective preference using high-dimensional vector spaces, differential geometry, or related quantitative apparatus. Such articulations may have analytical utility in specific contexts, but they abolish the constitutive role of subjectivity in the way the anchor specifies. The substantive direction of aesthetic mathematicality, on the SAE perspective, is probability-distribution grammar that articulates distributional structure without reducing aesthetic content to numerical values.

The three anchors together shape the paper's substantive content. They are not external constraints imposed from outside the paper's articulation; they are internal to the SAE perspective and to the substantive content of what the paper articulates. The anchors are made explicit because the substantive content of the paper, without the anchors, would be open to systematic misreading. With the anchors articulated, the paper's content is to be read in the spirit they specify: a substantive contribution from a specific perspective, acknowledging other perspectives, preserving the development potential of the practice through remainder, and respecting the constitutive role of subjectivity in articulation.

1.6 Claim strength

Following Paper 1's example (§ 1.3 of Paper 1), we articulate the strength of the claims made at different points in the paper.

The four inevitabilities of § 3 are offered as identificatory rather than derivational. They are not derived from premises through logical necessity; they are identified as conditions for the emergence of mathematicality, supported by the substantive defense framework of § 7. The identification can be checked against specific practices, refined where the check requires refinement, and contested where the check produces tension. The claim is that the four inevitabilities, plus the public re-entry criterion, are jointly identified as the conditions for SAE mathematicality.

The multi-path typology of § 4 is offered as substantive but non-exhaustive. Three closure paths are articulated with substantive content. Other paths may exist and may be articulated in future work. The typology articulates that the closure paths are multiple, not that the three paths articulated exhaust the possibilities.

The renaming of Paper 1's schema as the Objective Exactification Schema (§ 6) is offered as substantive scope clarification rather than retroactive correction. Paper 1's substantive content stands; the renaming articulates the scope within which Paper 1's content applies in full force.

The treatment of aesthetic mathematicality in § 5.5 is offered at the level of possibility proof plus substantive direction, with detailed structural articulation reserved for subsequent specialised papers in the aesthetic mathematics series. The present paper articulates that aesthetic articulation enters SAE mathematicality, that its canonical closure path is persistent open, and that its substantive direction of development is probability-distribution grammar. The detailed articulation of its specific structure—the precise form of its marked handles, additive accumulation, conditioning, and distribution space—is reserved.

The substantive defense framework of § 7, with its five inevitability criteria and five failure modes plus the 5a/5b distinction, is offered as a substantive framework parallel to Paper 1's § 3.5 and § 3.6, providing the checkability that the schema requires.

The three epistemological anchors of § 1.5 are offered as substantive epistemological commitments that shape how the paper's content is to be read.

The cross-cultural treatment of mathematical paths (in the limited articulation we provide in § 8) is offered as nuanced acknowledgment of historical variation, not as civilisational typology. We acknowledge that different traditions have substantively emphasised different paths and dimensions, but we explicitly reject essentialist characterisations of cultures along single paths.

1.7 What the paper does not claim

To preempt several misreadings, we articulate explicitly what the paper does not claim.

The paper does not claim to have enumerated all $L_0$ dimensions. The dimensions discussed in § 5 (quantity, temporal rhythm, spatial size, aesthetic, ethical, intensity, social proximity, truth) are illustrative, not exhaustive. Other dimensions exist and may be articulated in future work.

The paper does not claim to have enumerated all closure paths. The three paths of § 4 (complete exactification, partial exactification, persistent open) are the three most clearly identifiable in current mathematical practice. Other paths may exist and may emerge as the framework is developed.

The paper does not claim that SAE mathematicality is a universal definition of mathematics. The five conditions are the SAE perspective's identification, not a totalising definition that other perspectives must accept.

The paper does not claim that Paper 1 should be retracted or substantially revised. Paper 1's content stands within its scope; the present paper articulates the scope and provides the broader landscape.

The paper does not claim a detailed mathematical structure for aesthetic mathematicality. The detailed articulation is reserved for the subsequent aesthetic mathematics series. The present paper articulates the foundations: the possibility proof, the boundary conditions, and the substantive direction.

The paper does not claim that aesthetic articulation, ethical articulation, or other subject-grounded articulations are conventional mathematics. They enter broad SAE mathematicality, in the substantive sense of § 3.7, but they are not narrow mathematics in the conventional institutionalised sense. The three-tier distinction (mathematicality, narrow mathematics, broad SAE mathematicality) is essential to the paper's articulation, and conflating the three would misread the paper.

The paper does not endorse civilisational typology that essentialises cultures along single mathematical paths. Different traditions have substantively emphasised different paths and dimensions, but the substantive historical variation is partial and trajectory-dependent, not essentialist.

1.8 Outline of the paper

The remainder of the paper is organised as follows.

§ 2 articulates the substantive content of $L_0$ as the layer of qualitative articulation without marked handles, distinguishing $L_0$ from $L_1$ and articulating $L_0$'s multi-dimensional typology.

§ 3 articulates the four inevitabilities of SAE mathematicality—comparison, direction, propagation, sustained questioning—with the public re-entry criterion as a fifth condition on the practice. § 3.7 articulates the three-tier distinction (mathematicality, narrow mathematics, broad SAE mathematicality), and § 3.8 articulates the public re-entry criterion.

§ 4 articulates the three closure paths—complete exactification, partial exactification, persistent open—with their corresponding grammars (closure-equation, ordering, probability-distribution). § 4.7 articulates the measurement-theoretic interface that the closure-path typology naturally invites.

§ 5 articulates how specific $L_0$ dimensions are filled into the closure paths, with § 5.5 providing substantive articulation of aesthetic mathematicality as the most distinctive new contribution of the paper.

§ 6 articulates the renaming of Paper 1's schema as the Objective Exactification Schema, the upper-level $L_0$ Mathematicality Schema, and the three-fold relation among 0DD's four phases, Paper 2's four inevitabilities, and Paper 1's four steps.

§ 7 articulates the substantive defense framework: five inevitability criteria, five failure modes, the 5a/5b distinction, and a worked application to aesthetic mathematicality.

§ 8 articulates the inevitability of mathematicality at the level of the four conditions and the multiplicity of mathematicality at the level of dimensions and paths, with substantive nuancing of cultural variation.

§ 9 articulates the position of subject-grounded mathematicality (especially aesthetic mathematicality) within SAE philosophy, articulating that it is central to the SAE perspective without claiming hierarchy in substantive mathematical content.

§ 10 articulates the substantive relation between Paper 1 and Paper 2 in summary form, in preparation for subsequent papers in the series.

§ 11 articulates open problems that the paper identifies as substantive starting points for future work.

The paper does not append detailed appendices; in-depth engagement with specific technical traditions (detailed measurement-theoretic work, detailed engagement with aesthetic mathematics intellectual traditions, detailed cultural-historical illustration of mathematical paths, SAE mathematics series publication trajectory) is reserved for subsequent specialised papers (per the future work identified in § 11).

2. The Substantive Content of $L_0$

2.1 Returning to Paper 1's articulation of $L_0$

Paper 1 introduced $L_0$ in its § 2 as the level of qualitative articulation that precedes the introduction of "1" as a marked handle. The articulation in Paper 1 served a specific purpose: it motivated the transition from $L_0$ to $L_1$ that established the starting point of the layered architecture. Paper 1's pre-mathematical articulator, possessing only qualitative comparative resources (more/less, larger/smaller), encountered the failure of closure on questions of magnitude, and the failure produced the substantive remainder that "1" was introduced to handle.

Paper 1's articulation of $L_0$ was substantively correct as far as it went, but it was deliberately confined to a single qualitative dimension—the quantity dimension, articulated through more/less comparison. The substantive content of $L_0$ as a multi-dimensional level of qualitative articulation was not developed in Paper 1, since Paper 1's purpose was to motivate the move into the quantity-grounded $L_1$ that became the substrate for the rest of Paper 1's layered architecture.

The present section develops $L_0$ as a substantively richer level than Paper 1 articulated. The substance is not a correction of Paper 1 but an extension: Paper 1's articulation of the quantity dimension at $L_0$ stands as part of the present paper's articulation, occupying one of several substantively distinct $L_0$ dimensions.

2.2 The truly imprecise nature of $L_0$

$L_0$ is the level of qualitative articulation. At $L_0$, qualitative judgements are articulated but no marked handles are in place. There are no units, no references, no fixed comparative anchors. The qualitative articulation proceeds without precision in the strict sense, even though it has substantive content.

This articulation requires care because the absence of precision at $L_0$ can be misread as the absence of articulation. We are not claiming that $L_0$ contains nothing, or that qualitative articulation at $L_0$ is unstructured. The qualitative articulation has substantive content: it places terms in comparative relation, it orients comparisons, it generates propagation pressure, and it generates sustained questioning. These are the four inevitabilities of § 3, and they apply at $L_0$. The absence of precision is the absence of a structural device that would fix the comparative content with quantitative or quasi-quantitative determination. It is not the absence of articulation.

A way of seeing the substantive content of $L_0$ despite its imprecision: a pre-mathematical articulator (the figure Paper 1 used as a thought experiment) can make qualitative judgements that are intersubjectively recognisable. The judgement "this herd of sheep is much" or "that herd is more than this one" is articulable, transmittable, contestable, and capable of generating further articulation. The articulator does not have units, but the articulator has substantive comparative content. The substantive content is what the four inevitabilities articulate; it is not contingent on the introduction of units.

It is important to emphasise that $L_0$ contains no marked handles whatsoever. This claim is sharper than Paper 1's articulation made fully explicit, and we develop it now with the care it requires.

Consider what "1" is. "1" is a marked handle in the precise sense: it is a stable, repeatable, intersubjectively-fixed reference for the smallest discrete count, with the property that combinations of "1" generate the natural number sequence through arithmetic. "1" is not part of $L_0$ articulation; it is the move into $L_1$. To articulate "this is one sheep" is to articulate at $L_1$, not at $L_0$. At $L_0$, the articulator has only "sheep, much" or "sheep, more than that" or comparative judgements of similar form. The transition to "1" is the transition out of $L_0$ into $L_1$.

Now consider the temporal rhythm dimension. "One cycle per year" looks at first like a natural feature of qualitative temporal articulation. But "one cycle per year" is already a marked handle: it fixes a specific base rhythm relative to which all other rhythms are calibrated. The articulator who has settled on "one cycle per year" has already moved into $L_1$, even if the articulation has not yet developed the full arithmetic of frequency. At $L_0$, the articulator has only "this happens often" or "this happens more often than that" or comparative judgements of frequency without any base rhythm fixed.

Similarly the spatial-size dimension. "The king's fist" looks like a natural reference, but it is already a marked handle: it fixes a specific physical extent relative to which size judgements are calibrated. At $L_0$, the articulator has only "this is large" or "this is larger than that" without a reference fixing the comparative content.

This is the substantive claim of § 2.2: $L_0$ contains qualitative articulation with comparative content, oriented direction, propagation pressure, and sustained questioning, but it contains no marked handles. The introduction of a marked handle is the move out of $L_0$ into $L_1$, and the marked handle introduced determines the closure path the articulation takes.

The claim has consequences for how the present paper is read. The dimensions discussed in § 5 (quantity, temporal rhythm, spatial size, aesthetic, etc.) are $L_0$ dimensions in the sense of qualitative articulation without precision. When the section discusses, for example, the temporal rhythm dimension at $L_0$, it discusses the qualitative comparative articulation of frequency before any base rhythm is introduced. The introduction of a base rhythm, like the introduction of "1" or the introduction of the king's fist, is the transition to $L_1$ that produces the specific mathematicality of the path taken.

2.3 $L_0$ as the foundation of all closure paths

A consequence of the articulation in § 2.2 is that $L_0$ is the foundation of all closure paths, not merely the precursor of the complete exactification path that Paper 1 articulated.

The complete exactification path begins from $L_0$ through the introduction of a unit (the marked handle "1" in the quantity dimension, the base rhythm in the temporal rhythm dimension, similar units in other dimensions). The path's subsequent development—the additive path, the multiplicative path, the closures producing remainders—proceeds within $L_1$ and the higher layers Paper 1 articulated.

The partial exactification path begins from $L_0$ through the introduction of a reference (the king's fist in the spatial-size dimension, or analogous references in other dimensions). The reference is a marked handle in the broad sense (it provides a fixed anchor for comparative articulation) but it differs from a unit in that it does not support arithmetic operations. The path's subsequent development—ordering, transitive composition, order-theoretic closure—proceeds along a structurally different trajectory than the complete exactification path.

The persistent open path begins from $L_0$ through the introduction of a prior (canonical works, style frames, interpretive traditions in the aesthetic dimension, and analogous priors in other subject-grounded dimensions). The prior is a marked handle in the broadest sense (it provides a starting point for articulation) but it differs from both a unit and a reference in that it does not fix the comparative structure—the prior is contestable, displaceable, subject to reframing. The path's subsequent development—distributional accumulation, conditioning, persistent ongoing development of the distribution space—proceeds along yet another trajectory.

Each path's marked handle is the move from $L_0$ to $L_1$ for that path. The substantive content of $L_0$ is the same across all paths: the four inevitabilities and the public re-entry criterion apply at $L_0$ regardless of which closure path is subsequently taken. What differs across paths is the specific marked handle introduced and the subsequent articulative trajectory that the marked handle makes available.

This articulation is a substantive contribution of the present paper. Paper 1's discussion of $L_0$ implicitly treated $L_0$ as a precursor to the single $L_1$ of the complete exactification path; the present paper articulates $L_0$ as the substrate of multiple closure paths, with the specific path determined by the marked handle introduced and not by the substantive content of $L_0$ itself.

2.4 The multi-dimensional typology of $L_0$

$L_0$ is multi-dimensional. The qualitative articulator at $L_0$ does not have a single qualitative dimension that is then articulated through closure paths; the articulator has multiple qualitative dimensions, each capable of articulating along one or more closure paths. We articulate a typology of $L_0$ dimensions that organises them along three substantive axes.

Axis 1: Subject involvement. The first axis articulates the substantive role of the subject in the dimension's articulation. Some dimensions involve the subject minimally: the quantity of objects in a collection, the rhythm of recurring phenomena, the rough size of physical objects, can all be articulated with the subject playing a recording rather than constitutive role. Other dimensions involve the subject moderately: the size of an object relative to a chosen reference involves a substantive choice on the part of the subject, even though the comparative content once the reference is chosen is intersubjectively stable. Other dimensions involve the subject maximally: aesthetic value, ethical articulation, and other subject-grounded comparisons are constitutively engaged by the subject's evaluative response, in a way that cannot be separated from the response itself.

The axis is graded rather than binary. Dimensions occupy positions along a continuum from minimal to maximal subject involvement, with the dimensions discussed in § 5 occupying different positions.

Axis 2: Exactification potential. The second axis articulates the dimension's potential for exact quantitative articulation. Some dimensions admit complete exactification: a unit can be introduced, arithmetic operations can be defined, and the dimension's content can be articulated with arbitrary numerical precision. Other dimensions admit only partial exactification: a reference can be introduced, comparative ordering can be articulated, but no unit can be introduced that would support arithmetic. Other dimensions resist exactification altogether: no stable unit or reference is consistent with the dimension's substantive content, and articulation must proceed through distributional structures that do not collapse to point values.

The axis correlates with the closure paths: dimensions with high exactification potential canonically articulate along the complete exactification path; dimensions with partial exactification potential canonically articulate along the partial exactification path; dimensions with no exactification potential canonically articulate along the persistent open path.

Axis 3: Public re-entry. The third axis articulates the dimension's amenability to public re-entry, the fifth condition for SAE mathematicality. Some dimensions admit public re-entry strongly: quantitative judgements are paradigmatically re-enterable through shared notation and shared counting procedures. Some dimensions admit public re-entry moderately: aesthetic judgements admit re-entry through shared canons and shared interpretive traditions, but the re-entry is structured by tradition rather than by formal notation. Some dimensions admit public re-entry weakly: certain purely private qualitative experiences may have substantive content for the experiencing subject without admitting re-entry by others.

The third axis is essential because it determines which articulations enter SAE mathematicality at all. Dimensions with weak public re-entry, even if they satisfy the four inevitabilities in some attenuated form, do not enter SAE mathematicality on the substantive defense framework of § 7.

2.4.1 Axis correlations and inter-dimensional typological relations

The three axes are not orthogonal in the strict mathematical sense; they correlate in patterns that are themselves substantive. Dimensions with minimal subject involvement tend to have high exactification potential and strong public re-entry (quantity, temporal rhythm). Dimensions with maximal subject involvement tend to have low exactification potential and moderate public re-entry (aesthetic, ethical). Dimensions with moderate subject involvement may have partial exactification potential and strong public re-entry (spatial size with reference). The correlations are not laws; they are patterns that emerge from the substantive content of the dimensions.

Dimensions also exhibit typological relations that go beyond mere list enumeration:

Formal isomorphism, phenomenologically distinct: the quantity dimension ↔ the temporal rhythm dimension. Both articulate along the same closure path (complete exactification) and produce formally isomorphic quantitative structures (real-valued arithmetic), but with phenomenologically distinct content (countable discreteness versus periodic recurrence).

Single dimension, multi-path: the spatial size dimension. The same dimension admits articulation along different closure paths, with the path choice producing substantively different mathematical structures (the partial exactification path produces order-theoretic structure; the complete exactification path produces metric geometry).

Parallel structure, parallel treatment: the aesthetic dimension ↔ the ethical dimension. Both have similar typological profiles (maximal subject involvement, low exactification potential, moderate public re-entry), articulate along similar closure paths (persistent open), but have substantively distinct content and each requires its own specialised treatment.

These typological relations are substantive features of $L_0$ multi-dimensionality, not artefacts of any particular list categorisation.

2.5 The substantive universality of $L_0$ dimensions

A claim about $L_0$: the multi-dimensional structure of $L_0$ is substantively universal in the sense that any culture, any historical tradition, any articulative practice that takes up qualitative articulation will encounter the multi-dimensionality.

We articulate this with care. The claim is not that every culture articulates every dimension, or that every culture articulates the same dimensions in the same way. Substantive historical and cultural variation in the articulation of $L_0$ dimensions is real, and § 8 discusses some of this variation. The claim is rather that the multi-dimensional structure is available to any articulative practice that engages with qualitative phenomena: comparison can be of quantity, of rhythm, of size, of aesthetic value, of ethical value, of social relations, of truth. The dimensions are not constructions of a particular tradition; they are features of qualitative articulation as such.

Different traditions may emphasise different dimensions; different traditions may articulate dimensions along different closure paths; different traditions may take some dimensions as primary and others as secondary. The substantive variation is real and is part of what § 8 calls path multiplicity at the cultural-historical level. But the underlying multi-dimensionality of $L_0$ is not itself a cultural construction; it is the substrate from which the cultural articulations are made.

This is essential to the substantive content of the five conditions (four inevitabilities plus public re-entry). The five conditions apply across all dimensions and traditions, within practices that have crossed the dual threshold of structured comparison plus public re-entry (per § 8.2). Threshold-crossing practices encounter comparison (the first), direction (the second), propagation (the third), and sustained questioning (the fourth), with public re-entry (the fifth) being the threshold itself. The universality of the conditions at $L_0$ (post-threshold) is the substantive sense in which mathematicality is universal: it is a feature of structured comparison practice as such, available wherever structured comparison practice occurs, even though the specific paths along which mathematicality is realised are non-unique.

This is the foundation of the inevitability claim that § 8 articulates. The inevitability is at the level of the conditions: the conditions are unavoidable post-threshold. The multiplicity is at the level of realisation: the conditions are realised differently across dimensions, across paths, across traditions, with substantively different mathematical content in each case.

The specific $L_0$ candidate dimensions — quantity, temporal rhythm, spatial size, intensity, aesthetic, ethical, social proximity, truth value, and other potential candidates — receive their detailed typological characterisation and canonical closure path filling in § 5, after the three closure paths and the measurement-theoretic interface of § 4 are in place.

With $L_0$'s substantive content articulated, we proceed to § 3, which develops the four inevitabilities in substantive detail.

3. The SAE Mathematicality Schema

3.1 Overview

This section articulates the central conceptual contribution of the paper: four conditions that, in the SAE view, characterise the emergence of mathematicality from qualitative articulation at $L_0$. We call these the SAE mathematicality schema, comprising four inevitabilities:

Comparison is unavoidable.
Comparison cannot be undirected.
Comparison cannot remain isolated; it propagates.
Comparison, once propagating, cannot escape sustained questioning.

We use the term mathematicality rather than mathematics deliberately. The four conditions are offered as an identificatory schema for a class of articulative patterns this paper calls SAE mathematicality, not as a definition of mathematics in any of its conventional senses. The four conditions individuate a structural shape that systematic articulative practices may exhibit. Whether the class so individuated coincides with what philosophers, mathematicians, or working scientists ordinarily call mathematics is a separate question, addressed in § 3.7.

Three epistemological commitments shape how this schema is to be read, articulated in § 1.5 and worth recalling here. First, the SAE perspective is one thinking framework among many, not a totalising synthesis. Second, remainders cannot be eliminated; any articulation of mathematicality leaves remainders that subsequent work develops, rather than closing the question. Third, subjectivity cannot be quantified without being abolished; an articulation that proceeds by quantifying subjective stance is not a refinement of subjectivity but its replacement. These commitments are not external constraints on the schema; they are internal to the way the four conditions function.

The four conditions are presented in a particular order, but the order is not strictly logical entailment. Each condition presupposes the previous one in a softer sense: the second condition makes sense only because the first is in play, the third only because the second is in play, and so on. We articulate this dependency structure in § 3.6, after presenting each condition individually.

3.2 First inevitability: comparison is unavoidable

The first condition is that any qualitative articulation in the SAE sense necessarily involves comparison. To articulate a thing as "much" is already to differentiate it from "little"; to articulate it as "fast" is already to differentiate it from "slow"; to articulate it as "beautiful" is already to differentiate it from its absence or contrary. The qualitative judgement is inherently relational. It is not a property attached to an object and then optionally compared to another object's property; the very content of the judgement is comparative.

This thesis can be expressed in several equivalent ways. To say that a thing is "much" is not to report an intrinsic monadic feature but to place it on a side of a polarity (much/little, more/less). To say that a thing is "fast" is to invoke a contrast class within which speed is registered. To say that a thing is "beautiful" is to engage in an evaluative comparison whose other pole is the unbeautiful, even where that pole remains unspoken. There is, the schema claims, no qualitative articulation that escapes this relational form. The point is not that all judgements are explicitly comparative on the surface, but that the content of any qualitative articulation, when unpacked, carries the polarity within it.

A clarification is in order. We do not claim that comparison, by itself, constitutes mathematics or even mathematicality. Many practices involve comparison without entering what this paper articulates as SAE mathematicality. Casual perceptual discrimination, social differentiation, basic affective polarities such as pleasant and unpleasant: all involve comparison, but none, by virtue of comparison alone, satisfy the four conditions jointly. The first condition is one inevitability among four. Its function is to identify the substrate on which the remaining three conditions can operate.

A second clarification concerns the relation between comparison and differentiation. Differentiation, in the broadest sense, may be merely the registering of distinctness without orientation: "A is not B" without further structure. Comparison, in the sense intended here, adds something beyond differentiation. Even before direction (the second condition) is articulated, comparison carries the suggestion that the two terms stand in some relevant relation, not merely that they are numerically distinct. Where a practice produces only differentiation without this further relational pull, the first condition is not yet satisfied. Conversely, where differentiation acquires a relational pull, the first condition begins to operate, and the schema's other conditions can be tested.

The first inevitability has a number of contemporary mathematical reflexes that may be useful to note briefly, without taking on the burden of detailed engagement. The subset relation in set theory, the ordering relation in order theory, and the morphism in category theory are all formal articulations of comparative relations that emerged from, and abstract, the basic relational pull described here. None of these is identified with comparison in our $L_0$ sense; they are post-mathematical structures that presuppose mathematicality has already been articulated. But they show that mathematics, in its institutionalised forms, takes comparison as a central organising notion, consistent with the schema's claim that mathematicality begins here.

3.3 Second inevitability: comparison cannot be undirected

The second condition is that comparison, when it arises in qualitative articulation, is inherently oriented. To articulate "A is more than B" is not the same as to articulate "B is more than A"; the comparison has a direction, and the direction carries the content of the judgement.

This claim distinguishes comparison, in the sense relevant to mathematicality, from a symmetric notion of difference. A symmetric notion would treat the two poles of the comparison as interchangeable; it would record only that A and B differ, not which way the difference runs. Such a notion is too thin to support mathematicality. The orientation is part of what makes the comparison productive: it allows the comparison to be combined with other comparisons, to be propagated (the third condition), and to be questioned (the fourth condition) in ways that an undirected difference cannot support.

It is important not to overload "direction" with the technical content it carries in fully developed mathematics. In its $L_0$ form, direction is simply the orientation of the comparison: one pole is the "more" side, the other the "less" side, even if the specific scale by which "more" and "less" are calibrated is unavailable. The orientation does not yet require a total ordering, a linear arrangement, or a numerical comparison structure. It requires only that the two poles of the comparison are not interchangeable.

Direction takes different forms in different dimensions, and the schema does not prescribe a single form. In the quantity dimension, direction is ordering direction (more/less, larger/smaller in number). In the temporal dimension, it is rhythmic direction (faster/slower, earlier/later). In the spatial-size dimension, it is magnitude direction (larger/smaller relative to a reference). In the aesthetic dimension, direction is evaluative orientation (more beautiful, more sublime, more compelling). In other dimensions one finds attraction/repulsion direction, relevance direction, salience direction. Each of these is a form of orientation, but they are not all reducible to a single underlying form. The schema is compatible with the diversity of directional structures across $L_0$ dimensions.

A specific point: direction does not entail total order. The quantity dimension supports a total order on its closure structure ($\mathbb{R}$ is totally ordered), but the aesthetic dimension may at best support a partial order, and even that order may exhibit local instabilities. The schema requires that comparison be directed; it does not require that the directional structure be globally consistent in the way total orderings are. This permissiveness is important for the schema's reach: the aesthetic dimension would be excluded if direction were required to be total, and the schema's claim about aesthetic mathematicality would collapse. The schema therefore articulates the second condition as the requirement of orientation, leaving the global structural form (total, partial, or otherwise) as a feature of the particular path the dimension takes.

The relation between direction and what category theory calls morphism direction is worth noting. A morphism in a category has a domain and a codomain, distinct and not interchangeable, and the morphism's content is in part the way it goes from the one to the other. This is a formal articulation of the orientation we describe at $L_0$, in a setting that has already absorbed the comparative structure into algebra. The schema's claim about the second inevitability is consistent with the centrality of directional structure in modern mathematical foundations, while not identifying the $L_0$ articulation with its category-theoretic descendant.

3.4 Third inevitability: comparison cannot remain isolated

The third condition is that comparison, once it arises in articulation, cannot stably remain confined to isolated pairs. If "A is more than B" and "B is more than C" are both articulated, then a pull arises toward articulating the relation between A and C. The articulation propagates.

This is the schema's most contested condition, and we must be careful about what it does and does not claim. It does not claim that transitivity, in the strong logical sense, must hold for any comparative structure. It claims something weaker but more pervasive: that the articulation of comparative relations generates a pressure toward extending those relations across the available terms, regardless of whether the extension succeeds. The pressure is the inevitability; the success is contingent.

The distinction is essential for the schema's coverage of aesthetic mathematicality and similar cases. In the quantity dimension, transitivity does hold for the canonical closure: if $a > b$ and $b > c$ in the real numbers, then $a > c$. In the spatial-size dimension with a reference, transitivity also holds: if A is larger than the reference and the reference is larger than B, then A is larger than B. In the aesthetic dimension, by contrast, transitivity may systematically fail: a person may judge A more beautiful than B and B more beautiful than C while finding C more compelling than A on some specific axis of evaluation. Cycles of preference of this kind are well-documented in social choice theory, where they appear as instances of the conditions made precise in Arrow's impossibility theorem. They are not pathologies of misjudgement; they are features of the aesthetic dimension's structure.

If transitivity were required for the third condition to be satisfied, aesthetic articulation would not satisfy the schema, and the schema's reach to aesthetic mathematicality would fail. But the schema does not require transitivity. It requires that the articulation be subject to propagation pressure: that whenever two comparisons have been made, the question of how their terms relate across the comparisons arises and demands articulation. The articulation may answer the question with a fully transitive structure (as in the quantity dimension), or with a partially transitive structure (as in the spatial-size dimension with reference), or with a structure that exhibits cycles, distributions of preference, or contextually conditioned orderings (as in the aesthetic dimension). In each case, the third condition is satisfied: comparison propagates and the propagation generates further articulation.

We name the third condition "comparison cannot remain isolated" rather than the simpler "transitivity" to mark this distinction. The phrase "comparison propagates" carries the same content, and we use both interchangeably. What we mean is the propagation pressure, which is universal across mathematical paths, not the structural property of transitivity, which is path-specific.

The third condition's failure modes are instructive. A practice that produces only single isolated comparisons, with no further pull toward extending them, would fail the third condition. Such practices exist: certain forms of momentary aesthetic judgement, where a single object is appreciated without comparison to others; certain casual preference reports that do not invite further articulation. These practices may exhibit the first and second conditions in attenuated form, but without the third condition's propagation they do not enter SAE mathematicality. The boundary is not always sharp—propagation pressure can be present in latent form—but the boundary is real and serves to distinguish mathematicality from one-off qualitative judgement.

The transition from the third condition to the fourth condition is the schema's most important internal articulation, and we address it explicitly in § 3.6.

3.5 Fourth inevitability: comparison cannot escape sustained questioning

The fourth condition is that articulation, having generated comparison with direction and propagation, cannot stably close. Each attempt at closure generates a remainder, and the remainder demands further articulation. The questioning is sustained: it does not terminate with a final answer, but generates further questions whose pursuit constitutes the development of the practice.

This is the most distinctively SAE element of the schema. The other three conditions have analogues in many philosophies of mathematics: comparison appears in structuralism, direction in category theory, propagation in classical logic and order theory. The fourth condition, by contrast, articulates a specifically SAE commitment: that articulation is constitutively open, that closure is always partial, and that the remainder is the engine of the practice's continued development. Paper 1 articulated this commitment in the context of the layer transition mechanism, where each layer's closure produces a remainder that becomes the next layer's marked handle. The fourth condition generalises this to $L_0$: even before any specific marked handle is in place, the qualitative articulation already generates remainders that demand sustained questioning.

The sustained questioning takes different forms in different closure paths. In the complete exactification path (the quantity dimension), the questioning takes the form of demands for ever finer specification: "how much more?", "by what factor?", "to what limit?". These demands are answered by the introduction of marked handles, additive paths, multiplicative paths, and closures (the four steps of Paper 1's Objective Exactification Schema), but each level of answer generates further questions. The transcendental numbers, the imaginary unit, the question of continuity, the question of foundation—all of these are remainders of earlier closures, demanding articulation that becomes the substance of mathematical development.

In the partial exactification path (the spatial-size dimension with reference), the questioning takes a different form. The reference fixes the comparative structure, but the questions that arise concern the reference itself: "why this reference?", "how stable is it?", "how does it relate to other references?". These questions cannot be answered within the partial exactification path alone; they push the articulation toward either further exactification (introducing units and arithmetic, which moves to the complete exactification path) or toward acknowledging the contingency of the reference (which moves toward a more open articulation). Either way, the fourth condition is satisfied: the closure does not stably terminate the articulation.

In the persistent-open path (the aesthetic dimension), the questioning is most visibly sustained. Aesthetic judgements are paradigmatically open to challenge, refinement, and revision. The history of aesthetics, conceived as the unfolding of articulations of beauty, taste, and judgement across cultures and periods, is a sustained questioning of what aesthetic comparison consists in and how it should be conducted. No closure of this questioning is in prospect; the questioning itself is the development of the practice. The aesthetic dimension's mathematicality, on the schema's view, is constituted in part by this sustained openness.

The fourth condition is not a claim that all closure is impossible. Closure occurs locally, in specific structures, within specific paths. The Euclidean closure of geometric reasoning, the algebraic closure of rational number theory, the categorical closure of certain functor relations—each of these is a real closure within its scope. The claim is that closure does not terminate the articulation as a whole. Each closure generates a remainder, and the remainder demands further articulation. The practice does not converge on a final state; it continues to produce questions whose pursuit is its content.

This is the schema's most epistemologically substantive condition. It commits the SAE perspective to a view of mathematicality that resists totalising closure, that resists the idea of a final theory or a complete foundation, and that locates the substance of the practice in its sustained openness rather than its eventual completion. The commitment aligns with Paper 1's articulation of inexhaustibility at the object and meta levels, and extends it to the $L_0$ foundation.

3.6 Inter-relations among the four inevitabilities

We have introduced the four conditions in a particular order, and the order reflects a dependency structure that is worth making explicit. The dependency is not strict logical entailment; the conditions do not derive one from another in the manner of theorems from axioms. The dependency is rather that each condition presupposes the previous one in a softer sense: each condition's content is unintelligible if the previous condition is not in play.

The first condition (comparison is unavoidable) establishes the substrate. Without comparison, there is no relational content to be directed, propagated, or questioned. The second condition (direction) operates on the substrate the first condition provides; direction is the orientation of comparison, and is meaningless without comparison. The third condition (propagation) operates on the directional structure the second condition provides; propagation is the extension of directed comparisons across terms, and is meaningless without direction. The fourth condition (sustained questioning) operates on the propagated structure the third condition provides; sustained questioning is the persistence of articulative pressure beyond any specific closure, and is meaningless without a propagating structure to push beyond.

This dependency structure is hierarchical without being deductive. Each condition adds something that the previous condition leaves underdetermined, and each condition's content depends on the prior conditions being in play. But it would be a mistake to read this as a derivation: nothing in the first condition forces the second, nothing in the second forces the third, and so on. The conditions are jointly identified as inevitabilities of $L_0$ qualitative articulation; they are not derived one from another by logical necessity.

There is, however, an internal pressure that connects the third condition to the fourth condition, and this pressure is worth articulating with some care. The third condition establishes that comparison propagates: that whenever two comparisons are in play, the relation between their terms across the comparisons arises and demands articulation. But the comparative chain that the third condition generates is, in itself, incomplete in a specific way. The chain articulates the ordering of terms—A is more than B, which is more than C, which is more than D—but it does not articulate the magnitudes of the comparisons. It tells us that A is more than C, but it does not tell us by how much. The chain has an internal asymmetry: it specifies direction and order, but not extent.

This internal asymmetry is the substrate from which the fourth condition operates. The propagation produces ordering; the ordering invites the question of magnitude; the question of magnitude is the form the fourth condition takes when the closure attempt is made. The complete exactification path answers this question by introducing units and arithmetic. The partial exactification path answers it by introducing a reference and accepting that the question of magnitude beyond the reference is left open. The persistent-open path answers it by accepting that the question of magnitude is not stably answerable for the dimension in question and that the articulation proceeds without exact magnitude. In each case, the fourth condition responds to an internal incompleteness in the propagated comparative structure that the third condition produces.

We highlight that this connection is not a derivation of the fourth condition from the third. The fourth condition's content—sustained questioning, the recognition that closure produces remainders that demand further articulation—is not contained in the third condition. The third condition by itself could conceivably be satisfied by a stable propagating structure that does not generate sustained questioning; one might imagine, hypothetically, a comparative practice whose propagation closes on a final extent-free structure and is content to remain there. But qualitative articulation in $L_0$, the schema claims, does not stop there. The pressure to articulate magnitude arises and persists, and the persistence is the fourth condition. The internal incompleteness of the propagated chain is what makes the fourth condition's pressure intelligible, but the pressure itself is an additional fact about articulation at $L_0$, not a consequence of the propagation alone.

We have, then, four inevitabilities that together identify SAE mathematicality. They are jointly identified, not derived; they presuppose one another in a hierarchical but non-deductive way; and they exhibit an internal pressure from the third to the fourth that makes the schema's overall articulative shape coherent. In the next subsection, we situate the schema within a broader taxonomy of mathematicality.

3.7 Mathematicality, narrow mathematics, broad SAE mathematicality

The five conditions (four inevitabilities plus the public re-entry criterion) jointly identify a class of articulative patterns. We have called this class SAE mathematicality. The class is broader than mathematics in its conventional institutionalised forms, and three terms are needed to keep the relations clear.

Mathematicality, simpliciter, is the articulative pattern jointly identified by the four inevitabilities plus the public re-entry criterion (the five conditions). A practice exhibits mathematicality, in the SAE sense, when it satisfies all five conditions: comparison, direction, propagation, sustained questioning, and public re-entry. The term is descriptive: it identifies a structural shape that practices may or may not exhibit, without prejudging whether the practice is conventionally called mathematical.

Narrow mathematics denotes the institutionalised practice of mathematics as it has developed in the Western tradition since the rigorisation of analysis in the nineteenth century and the foundational programmes of the twentieth: number theory, analysis, algebra, geometry, topology, logic, and their many descendants. Paper 1 articulated the architectural pattern of narrow mathematics under the heading of the Layer Articulation Schema, which we here rename the Objective Exactification Schema to mark its path-specific character. Narrow mathematics, in this terminology, is mathematicality in the complete exactification path, with full unit-and-arithmetic closure and the development of the layer structure Paper 1 articulated.

Broad SAE mathematicality denotes, in the SAE view, the union of practices that satisfy the four inevitabilities plus the public re-entry criterion (the five conditions) and are structurally articulated along some closure path. Narrow mathematics is its instantiation in the complete exactification path; the partial exactification path produces relation-like mathematicality without full arithmetic; the persistent-open path produces aesthetic mathematicality and similar subjectivity-grounded structures. Broad SAE mathematicality includes all of these. It does not claim to exhaust the practices that satisfy the five conditions: other dimensions and other paths may articulate further forms of mathematicality, and the catalogue developed in this paper is illustrative rather than exhaustive.

The three terms allow us to articulate the schema's claim with precision. Mathematicality is the structural shape identified by the five conditions (four inevitabilities plus public re-entry); narrow mathematics is its instantiation in the complete exactification path; broad SAE mathematicality is its instantiation across multiple paths, with narrow mathematics as a proper part. The five conditions jointly identify mathematicality; the broader SAE perspective recognises that mathematicality takes multiple forms; narrow mathematics is a specific, historically important, but not unique form.

This terminological discipline is important for what the schema does and does not claim. The schema does not claim that the five conditions define mathematics in any conventional sense. It claims that the five conditions (four inevitabilities plus public re-entry) jointly identify SAE mathematicality, and that conventional narrow mathematics is one instantiation of this broader category. Practices that satisfy the five conditions but are not conventionally classified as mathematics—aesthetic articulation, certain ethical articulations, certain forms of social-relational articulation—belong to broad SAE mathematicality without thereby being absorbed into narrow mathematics. The taxonomy preserves the distinctness of these practices while articulating their structural kinship to mathematics in the strict sense.

This is also where the first epistemological anchor, articulated in § 1.5, takes effect. The SAE perspective is one thinking framework among many. Other frameworks—formalism, intuitionism, structuralism, category-theoretic foundations, naturalised philosophies of mathematics—identify mathematicality, or its closest analogue, differently. The five conditions are the SAE framework's articulation, not a universal definition. They are offered as a productive way of identifying a structural shape, in a manner compatible with the existence of alternative framings.

3.8 The public re-entry criterion

The four conditions identify mathematicality at the level of articulative structure. But a further condition is needed to distinguish mathematicality from purely private qualitative articulation: the requirement that the articulation be available for re-entry by others.

We articulate this as the public re-entry criterion: for a qualitative articulation at $L_0$ to enter SAE mathematicality, the articulation must be available for repetition, transmission, and re-entry by parties other than the original articulator. A purely private comparison—one whose terms, direction, propagation, and questioning are accessible only to a single subject, with no possibility of others entering the structure—does not enter SAE mathematicality. The four conditions may all be present in a sense, but without public re-entry the articulation does not contribute to the practice that mathematicality identifies.

The criterion is not a demand for full intersubjective agreement, nor a requirement that all subjects share the same articulations. It is the weaker requirement that the structure of articulation be accessible: that another subject, entering the same comparative situation, could engage with the articulation, accept or contest its directions, follow or challenge its propagations, and participate in its sustained questioning. This is what allows the articulation to function as a structure rather than as a one-off private event.

In narrow mathematics, the public re-entry criterion is paradigmatically satisfied. Mathematical statements are formulated in shared notation, with shared definitions and shared standards of proof, and are designed precisely to admit re-entry by any competent participant. The criterion is so deeply built into the practice that it scarcely needs to be articulated. In the partial exactification path, the criterion is satisfied through shared references: the king's fist, in the toy example, is a publicly identifiable reference, and judgements made relative to it are publicly re-entrable. In the persistent-open path, the criterion is satisfied in a more contested way, but it is satisfied: aesthetic articulations operate within shared canons, shared style frames, and shared interpretive traditions that make re-entry possible even where the substantive judgements diverge.

The criterion also serves to handle several boundary cases that might otherwise be ambiguous. A purely private aesthetic response—a fleeting reaction with no articulable structure, no comparison to other responses, no public availability—does not enter aesthetic mathematicality, on the schema's view. The four conditions may be present in attenuated form, but the absence of public re-entry blocks the articulation from contributing to the practice. By contrast, an aesthetic response articulated within a shared interpretive frame, available for transmission and contest, satisfies the criterion and is admitted. The line is not always sharp, but the criterion gives a principled way of drawing it.

A specific application: the question of whether ethical articulation (the good/evil dimension at $L_0$) enters SAE mathematicality is partly settled by the public re-entry criterion. Ethical judgements that satisfy the four conditions and are formulated in publicly accessible terms—through shared traditions, shared cases, shared modes of justification—satisfy the criterion. Ethical judgements that remain wholly private satisfy the four conditions perhaps weakly, but do not enter mathematicality. We do not pursue ethical mathematicality in detail in this paper, but the criterion gives a principled way of articulating its boundary, and we note that the question of ethical mathematicality is a candidate for future work.

The public re-entry criterion adds a fifth condition to the four inevitabilities, but it has a different status. The four inevitabilities are structural features of qualitative articulation at $L_0$: they characterise what mathematicality is, structurally. The public re-entry criterion is a condition on the articulative practice: it characterises the conditions under which the structural features are realised in a way that constitutes a practice rather than a private event. The four conditions individuate mathematicality in the abstract; the public re-entry criterion ensures that mathematicality is realised in articulable, shareable form.

With the five conditions in place, the schema's substantive content is complete. We turn in § 4 to the closure paths through which mathematicality is realised, and in § 5 to the dimensions of $L_0$ that are filled into those paths.

4. Three Closure Paths

4.1 Overview

The four inevitabilities of § 3 identify SAE mathematicality at the level of articulative structure: comparison, direction, propagation, sustained questioning, with the public re-entry criterion as a condition on the practice. But the four inevitabilities, by themselves, do not determine the form mathematicality takes in any specific articulation. The fourth condition—sustained questioning generated by closure attempts—can be met in substantively different ways, each producing a substantively different mathematical structure. This section articulates three such ways, which we call closure paths.

The terminology deserves immediate clarification. A closure path, in our sense, is not a single mathematical structure but a mode of responding to the demand for closure that the fourth inevitability articulates. Each path is a way of handling the gap between propagated comparative structure (the third condition's output) and the demand for sustained articulation (the fourth condition's pressure). The three paths we identify are not exhaustive—the inexhaustibility of SAE mathematicality, articulated in § 8, includes the possibility of further paths—but they are the three paths most clearly identifiable in mathematical practice as currently developed.

The three paths are:

Complete exactification: closure achieved through the introduction of units, arithmetic operations, and the resulting full quantitative structure. This is the path Paper 1 articulated under what we now call the Objective Exactification Schema.

Partial exactification: closure achieved through the introduction of a reference relative to which comparison is structured, but without the further introduction of units that would permit full arithmetic. The result is an ordering structure without quantitative magnitude.

Persistent open: closure not achieved in the structural sense of the other two paths. Comparison is articulated, direction is maintained, propagation is acknowledged, and the sustained questioning is taken up not as a demand to be answered with structural closure but as a constitutive feature of the practice itself.

The relations among the three paths require care, because they exhibit two different aspects that should not be conflated. At the level of substantive mathematical content, the three paths are complementary, not hierarchical. Each produces mathematical structures that the others cannot produce, and each path's structures are not reducible to the others. The complete exactification path produces $\mathbb{R}, \mathbb{C}$, and the layer-articulated arithmetic of Paper 1; the partial exactification path produces ordering structures, preorders, lattices, and the like; the persistent-open path produces distribution-grounded articulations of subjective comparison. None of these reduces to the others. At the level of SAE philosophical articulation, we will see in § 9 that the third path has particular substantive significance for SAE because of its connection to subjectivity, but this is a feature of the SAE viewpoint specifically, not of the substantive content of the paths.

This section first articulates each path in turn (§§ 4.2–4.4), then exhibits the path typology in correspondence with grammars of mathematical articulation (§ 4.5), then notes the substantive relations among the paths (§ 4.6), and finally articulates the substantive interface with measurement theory that the path typology naturally invites (§ 4.7).

4.2 Complete exactification path

The complete exactification path resolves the question of magnitude—the question generated by the third condition's incomplete propagation, the question the fourth condition demands an answer to—by introducing a unit and the arithmetic operations that the unit makes available. The unit is a marked handle that fixes a reference quantity, and the arithmetic operations (addition, multiplication, and their inverses) generate, from the unit, a quantitative structure that admits magnitude assertions of arbitrary precision.

Paper 1 articulated this path in detail, under the heading of the Layer Articulation Schema, with the layered articulation of $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$ and their extensions. As discussed in § 6, we are now in a position to rename that schema the Objective Exactification Schema, marking its path-specific character. The schema applies, in its four-step form (marked handle, additive path, multiplicative path with memory binding, closure with remainder), to the layers of the complete exactification path. Paper 1's substantive contribution is the articulation of this path's architecture; the present paper does not revise that articulation but rather positions it within the broader landscape of mathematicality.

Several features of the complete exactification path deserve brief comment in the present context, where the path is one option among several rather than the default form of mathematics.

The path requires a stable unit. The unit is introduced as a marked handle ("1" in the quantity dimension, but the same structural role is played by different specific marks in other dimensions: a base rhythm "one per period" in the temporal dimension, a base intensity in the strength dimension). The stability of the unit is what allows the arithmetic to take effect: an addition of two units gives two, an addition of two and three gives five, and these results are independent of context and observer. Where the unit is unstable, or where its stability is itself in question, the complete exactification path is unavailable and a different path is required.

The path produces total ordering. From the arithmetic structure, a total ordering on the closure ($\mathbb{R}$ in the canonical case) follows. The total ordering is a strong property: it asserts that for any two elements, one is greater, one is lesser, or they are equal. This property is paradigmatic of complete exactification, and it is one of the features that distinguishes the path from the others. The partial exactification path produces orderings that may be only partial; the persistent-open path may produce orderings that are partial and locally unstable.

The path is grammatically articulated as closure-equation grammar. Paper 1 articulated this grammar in connection with the closure step of the schema at each layer: the closure of arithmetic to $\mathbb{R}$ via Cauchy completion, the closure of $\mathbb{R}$ to $\mathbb{C}$ via the imaginary unit, the closure of $\mathbb{C}$ to $\mathbb{C} \cup \{\hat\infty\}$ via the Riemann sphere, and so on. Each closure is articulated through a specific equation or structural identity that captures the closure's content. The grammar permits closure-equations to be solved, manipulated, and extended; this is the form mathematical reasoning takes in the complete exactification path.

A note on the path's reach. The complete exactification path is not confined to the layers Paper 1 articulated. Where any qualitative dimension admits a stable unit and supports the arithmetic operations, the path is available, and the four-step schema can be applied to produce a layered articulation. The temporal dimension supports the path, as does any dimension that can be calibrated against a stable repeating reference. The path is the dominant mode of mathematical articulation in the modern Western tradition, but it is not coextensive with mathematics; it is one path among several, and the broader landscape of SAE mathematicality is the subject of this paper.

4.3 Partial exactification path

The partial exactification path resolves the question of magnitude in a weaker way than the complete exactification path. It introduces not a unit but a reference: a marked handle that allows comparisons to be conducted relative to it, but that does not support the arithmetic operations that would generate a full quantitative structure. The result is an ordering structure that captures the relational content of comparison but does not provide magnitudes in the strict sense.

The toy example given in earlier sections is the king's fist as a reference for size. Items can be compared to the king's fist and judged larger, smaller, or roughly equal; further items can be compared to one another by transitivity through the reference. But there is no operation that would tell us, in a unit-based way, how much larger one item is than another. The reference functions as an anchor for the ordering, not as a measure on a scale.

The path's structural features are worth articulating with some care, since the path occupies a less familiar position in conventional mathematical foundations than the complete exactification path.

The marked handle in this path is a reference, not a unit. The distinction matters. A unit is a quantity that admits arithmetic combination with other quantities of the same kind: two units, three units, half a unit, and so on. A reference is a fixed comparison point that admits comparison with other items but does not itself admit arithmetic operation. The king's fist is a reference: items are compared to it, but "two king's fists" is not a meaningful operation within the partial exactification path. It would become meaningful only if the path were extended to the complete exactification path by introducing the additive operation, but that extension is precisely what the partial exactification path declines to make.

The ordering produced by the path may be total or partial. A simple physical reference often produces a near-total order (most items can be unambiguously placed larger, smaller, or roughly equal to the reference). But the ordering need not be total. A reference in a multi-attribute setting (a reference work in literature, a reference performance in music) may yield partial orderings, where some items are not directly comparable to one another through the reference. The schema accommodates both total and partial orderings; it does not require totality.

The structure of the closure is a poset, preorder, or related order-theoretic object, depending on the specifics of the dimension. In some cases, particularly where the reference is rigid and the dimension well-defined, the closure may be a total order. In others, particularly where the reference is contested or where the dimension admits multiple incomparable axes, the closure is a partial order or even a preorder (a transitive but not necessarily antisymmetric relation). The schema does not prescribe a single closure structure; it accommodates the full range of order-theoretic possibilities consistent with the path's character.

The grammatical articulation is ordering grammar. This grammar permits assertions of the form "A is greater than B relative to the reference R" and supports inferences by transitivity, but does not support the equation-and-arithmetic style of the closure-equation grammar. Reasoning in the ordering grammar is structural: it concerns relations among items, not magnitudes. The grammar's reach is narrower than closure-equation grammar in one respect (it cannot articulate magnitude) but broader in another (it can articulate comparisons in dimensions where unit-based magnitude is not available). The mathematics of order theory, as developed in modern foundations, articulates the grammar in technical detail; we do not pursue that articulation here, leaving it to subsequent specialised papers.

A note on the path's relation to the complete exactification path. The partial exactification path is not a degenerate or inferior version of the complete exactification path; it is a different path. The two paths address different aspects of mathematical articulation and produce different kinds of mathematical structure. In some dimensions, the partial exactification path is the natural articulation, and any attempt to extend to complete exactification would distort the dimension's content. In other dimensions, complete exactification is natural and the partial exactification path is an intermediate stage on the way to full quantitative articulation. The schema does not privilege either path; it identifies them as distinct articulative options.

4.4 Persistent open path

The persistent open path resolves the question of magnitude not by introducing a unit (as in complete exactification) or a reference (as in partial exactification), but by declining the demand for stable structural closure. Comparison is articulated, direction is maintained, propagation is acknowledged, but the sustained questioning that the fourth inevitability generates is taken up as a constitutive feature of the practice rather than as a problem to be solved by a structural device.

This formulation requires immediate clarification because it can easily be misread. We are not claiming that the persistent open path involves no closure at all. Local closure occurs: specific judgements are made, specific comparisons are concluded, specific propagations are followed through to their consequences. What does not occur is the kind of structural closure that the other two paths produce: there is no stable unit, no stable reference, that fixes the comparative structure once and for all. The articulation proceeds without fixing the structure, and the absence of fixed structure is constitutive of the path, not a failure of it.

This path is most clearly exhibited in the aesthetic dimension, to which § 5 will return. In aesthetic articulation, judgements of beauty, sublimity, taste, and so on are made, compared, propagated, and sustained, but no unit or reference is in prospect that would stably fix the structure. The history of aesthetic articulation is a history of sustained questioning, not a sequence of structural closures. This is not a deficiency of aesthetic articulation; it is, as we shall argue, central to what makes aesthetic articulation what it is.

The path's structural features can be articulated as follows.

The marked handle is not a unit or a fixed reference, but a prior: an articulation that conditions further articulation without fixing it. In the aesthetic dimension, canonical works, established style frames, and shared interpretive traditions function as priors. They condition the articulation: aesthetic judgements proceed within their orbit, refer to them, contest them. But they do not fix the articulation in the way a unit fixes the quantity dimension. A canonical work may be displaced by a subsequent canonical work; a style frame may shift; an interpretive tradition may be challenged and reformed. The prior is a marked handle in the sense that it provides a starting point for articulation, but it is not a marked handle in the sense that it provides a stable closure.

The propagation of comparison produces not a single ordering, but a distribution of comparative judgements. Different subjects, in different contexts, produce different orderings, and the structure of the practice is captured by the distribution, not by any single ordering taken as canonical. The distribution may exhibit regularities (cultural patterns of taste, period styles, individual sensibilities) but it does not collapse to a unique structure. This is the substantive sense in which the persistent open path is distribution-grounded: the mathematical content of the path is not a closure structure but a structured distribution of comparative articulations.

The path's grammatical articulation is probability-distribution grammar, but the term requires immediate qualification to avoid a serious misreading. We are not claiming that aesthetic articulation is quantifiable, that aesthetic judgements have numerical values, or that the persistent open path is a sophisticated form of quantification. The third epistemological anchor of § 1.5 is in effect here: subjectivity cannot be quantified without being abolished. The probability-distribution grammar is invoked because the mathematical structure of the path most closely resembles, in its formal features, the mathematics of distributions: stochastic dominance, conditional articulation, marginalisation across context, comparison of distributions rather than comparison of point values. But the articulation is not a quantification of aesthetic content; it is an articulation of the distributional structure of aesthetic comparison.

The distinction can be put as follows. In the complete exactification path, comparison takes the form "A is 7 units more than B"; the comparison is a numerical relation between magnitudes. In the persistent open path, comparison takes the form "A is more compelling than B in this context, for this subject, given this prior, with this confidence"; the comparison is a relation between distributions of aesthetic articulation conditioned on multiple features of the context. The structural shape of the comparison is distributional, not numerical, and the path's grammar captures this structure without reducing it to numerical content.

A further specification: the path does not require that the distribution be parameterised in advance, or that it admit closed-form articulation. The distribution may be implicit in the practice, articulated through cases, examples, and the unfolding of judgements over time. The articulation of the distribution is itself an aspect of the practice, and it is sustained in the same sense that the questioning is sustained: it does not close on a final form, it continues to develop. This is, again, not a deficiency; it is constitutive of the path.

The persistent open path will receive detailed treatment in § 5.5 in connection with aesthetic mathematicality, where its substantive content is most fully exhibited. Here we have articulated only the path's general features. Other dimensions besides the aesthetic—certain ethical articulations, certain interpretive practices, certain forms of subject-grounded judgement—may also articulate along the persistent open path, but their detailed articulation is left to subsequent specialised papers.

4.5 Path typology and grammar

We summarise the three paths and their corresponding grammars in tabular form, with the understanding that the table is descriptive of the present articulation and is not exhaustive.

Closure path	Marked handle	Closure form	Grammar
Complete exactification	Unit	Quantitative structure ($\mathbb{R}, \mathbb{C}, \ldots$)	Closure-equation grammar
Partial exactification	Reference	Order-theoretic structure (poset, preorder)	Ordering grammar
Persistent open	Prior (canonical work, style frame, interpretive tradition)	Distribution of comparative articulations	Probability-distribution grammar

The three grammars are not different notations for the same content; they are substantively different modes of articulating mathematical relations. Closure-equation grammar reasons about magnitudes through equations and arithmetic operations. Ordering grammar reasons about relations through transitivity and structural composition, without invoking magnitudes. Probability-distribution grammar reasons about distributions of articulation through conditioning, marginalisation, and comparison of distributions. Each grammar has its own canonical inference patterns, its own canonical questions, and its own characteristic results.

A note on Paper 1's grammar terminology. Paper 1 distinguished two grammars in § 7: closure-equation grammar and probability-distribution grammar. The two grammars were articulated as occupying different regimes: closure-equation grammar productive at $L_1$ through $L_4$ and at equilibrium $L_5$, probability-distribution grammar productive at non-equilibrium $L_5$ and beyond. Paper 1's articulation positioned probability-distribution grammar as a continuation of closure-equation grammar: it took over where closure-equation grammar failed.

The present paper articulates a revised view of the grammar distinction. Probability-distribution grammar is not merely a continuation of closure-equation grammar where the latter fails. It is a grammar in its own right, with its own substantive domain, and it can articulate directly certain forms of mathematicality (specifically, subjectivity-grounded mathematicality) without passing through closure-equation grammar at all. The aesthetic dimension's articulation in the persistent open path is the paradigm case: it articulates aesthetic mathematicality directly through probability-distribution grammar, not through a failure of closure-equation grammar. This is the substantive parallel articulation of the two grammars that we noted in earlier sections, and it is one of the substantive contributions of the present paper.

We also articulate ordering grammar as a third grammar, distinct from both closure-equation grammar and probability-distribution grammar, and corresponding to the partial exactification path. Ordering grammar was not articulated in Paper 1; its inclusion here extends the grammar typology of Paper 1 with a third member.

4.6 Relations among the three paths

The three paths can be related in several ways, and the distinctions among the relations should be kept clear.

At the level of substantive mathematical content, the three paths are complementary and not hierarchical. Each produces mathematical structures that the others cannot produce. The complete exactification path produces full quantitative mathematics; the partial exactification path produces order-theoretic mathematics; the persistent open path produces distribution-grounded mathematics of subjective comparison. None of these reduces to the others. A mathematical question articulated in one path may be entirely meaningless in another: the question "what is the magnitude of A relative to B" is well-posed in the complete exactification path, partially answerable in the partial exactification path (relative to the reference), and not well-posed in the persistent open path (where comparison is distributional rather than magnitudinal).

We emphasise this complementarity to forestall a misreading. We are not claiming that the complete exactification path is the "real" or "fundamental" path of mathematics and that the others are derivative or degenerate forms. Nor are we claiming the converse, that the persistent open path is somehow more fundamental and the others are reductive. The three paths are complementary in the strict sense: each has its substantive domain, each produces mathematical content the others cannot, and the mathematical landscape as a whole is articulated through the cooperation of the three (and possibly further) paths, not through the dominance of any one.

At the level of SAE philosophical articulation, however, there is a substantive asymmetry, which we shall develop in § 9. The SAE perspective takes subjectivity as central: Self-as-an-End is the name of the philosophical commitment. Where mathematicality articulates subject-grounded structure, as it does in the persistent open path with the aesthetic dimension, the articulation has particular substantive significance for the SAE viewpoint. This is not a claim that the persistent open path is higher mathematics, more developed mathematics, or somehow superior mathematics. It is a claim about the SAE perspective's focal interest: where the perspective looks, the persistent open path's substantive content is what most distinguishes the SAE view from views that take objective exactification as paradigmatic of mathematics.

The two levels of articulation are not in conflict. At the substantive mathematical content level, the three paths are complementary, no hierarchy. At the SAE philosophical articulation level, the third path has focal significance for the SAE view. These are different aspects of the situation, applicable in different ways. Conflating them would produce confusion; keeping them distinct allows both claims to be made without internal tension.

Across paths, certain dimensions may admit multiple paths, and the choice of path is a substantive choice with substantive consequences. The quantity dimension admits the complete exactification path as canonical, but it could in principle admit a partial exactification path (a numerical system without arithmetic, only ordering) or a persistent open path (a distribution of numerical articulations across subjects). The aesthetic dimension admits the persistent open path as canonical, but elements of it may be addressed through partial exactification (style traditions providing references) or even through complete exactification in restricted technical sub-domains (mathematical aesthetics of proportion). The choice of path is shaped by the dimension's character but is not strictly determined by it.

This articulation is consistent with the multi-path multiplicity that § 8 will develop: mathematicality is non-unique not only in the dimensions to which it applies but in the paths it takes within those dimensions, and the multiplicity at both levels is part of the inexhaustibility the SAE perspective endorses.

A specific claim that we explicitly do not make: we do not claim that the union of the three paths exhausts mathematics or even SAE mathematicality. The three paths are the three most clearly identifiable in current mathematical practice; further paths may exist and may emerge as the practice develops. The schema is open to extension, consistent with the inexhaustibility commitment of § 1.5.

4.7 Measurement theory interface

The three closure paths have a natural substantive interface with measurement theory, the area of philosophy and theoretical psychology that studies how qualitative phenomena are mapped into quantitative structures. The interface is not coincidental: measurement theory has long been concerned with precisely the question the closure paths address, namely how qualitative comparative structure becomes (or, in the partial and persistent-open cases, partially becomes) quantitative articulation.

The framework that has emerged most influentially in measurement theory is the typology of scales associated with Stevens (1946) and developed in the representational theory of measurement of Krantz, Luce, Suppes, and Tversky (1971, and subsequent volumes). The framework distinguishes scales by the kinds of transformations under which they are invariant: nominal scales (categorisation, invariant under any one-to-one transformation), ordinal scales (ordering, invariant under monotonic transformations), interval scales (intervals between values meaningful, invariant under linear transformations), and ratio scales (ratios between values meaningful, invariant under multiplicative transformations). The framework provides a systematic vocabulary for distinguishing levels of quantification in a qualitative phenomenon.

The closure paths of this paper map onto Stevens-style scale types in a way that is suggestive though not strictly identical:

SAE closure path	Measurement-theoretic analogue
Persistent open path	Nominal / qualitative classification, with public comparative re-entry
Partial exactification path (without quantitative reference)	Ordinal scale (ordering without magnitude)
Partial exactification path (with interval-defining reference)	Interval scale (magnitudes within bounded structure)
Complete exactification path	Ratio scale (full magnitude structure with unit)

A clarification is essential, particularly regarding the nominal-scale row. In Stevens's typology, a nominal scale is a categorisation that admits identity comparisons (this thing is the same as that thing, or different) but no further comparative structure. The persistent open path of SAE mathematicality differs in that it does involve comparison: the four inevitabilities are all present, and the fifth condition (public re-entry) is satisfied. The path's affinity with the nominal scale is in the absence of a stable quantitative or even ordinal structure that would collapse the distributional richness of the path's content. The persistent open path enters SAE mathematicality precisely where the nominal scale, supplemented with public comparative re-entry, would otherwise be merely categorisation. We can articulate this in a more careful way by saying that the persistent open path has affinity with the nominal-scale row in the absence of stable ordinal collapse, but it is not reducible to nominal scaling because the articulation includes comparative structure that mere nominal classification does not include.

The interface with measurement theory provides several substantive advantages for the present articulation, and we note them briefly.

First, it provides external anchoring. The closure-path typology is not a free invention of the SAE perspective; it tracks substantive distinctions that measurement theory has independently developed in the service of empirical psychology and the foundations of measurement. The independent development is evidence that the distinctions are substantive, not artefacts of the SAE viewpoint.

Second, it provides existing technical resources. Each row of the table corresponds to a substantial body of mathematical and philosophical work. The persistent open path can draw on the philosophy of qualitative comparison and the technical apparatus of preference theory; the partial exactification path can draw on order theory and the representational theorems of interval and ordinal measurement; the complete exactification path can draw on the entire architecture of conventional mathematics. The SAE framework does not need to reinvent these resources; it situates them within a unified articulation.

Third, it provides falsifiability anchors. A claim that a particular dimension occupies a particular closure path can be checked against the measurement-theoretic literature on that dimension. Where empirical psychology has found that a phenomenon admits only ordinal scaling, the dimension corresponds (in the SAE typology) to a partial exactification path without interval structure. Where empirical psychology has found that a phenomenon admits ratio scaling, the dimension corresponds to a complete exactification path. The closure-path typology is constrained by these external findings, and where the SAE classification conflicts with the empirical evidence, the SAE classification must be revised or the apparent conflict explained.

A few specific applications, briefly:

Temperature (ordinary perception): In ordinary perception admits ordinal scaling (warmer/cooler is comparative without unit) but not directly ratio scaling. In Stevens's framework, ordinary temperature is interval-scaled (the Celsius and Fahrenheit scales admit linear transformation but not multiplicative transformation, since the zero is conventional). In the SAE framework, this corresponds to the partial exactification path with interval structure (with reference unit interval but no absolute zero); the further transition to ratio scaling (Kelvin) requires the additional articulation of an absolute zero, pushing the dimension into the complete exactification path — a substantive scientific articulation (the third law of thermodynamics), not merely a notational change.

Intelligence (psychometrics): As articulated in psychometrics, admits at most ordinal scaling under strict interpretation, though IQ scales are conventionally treated as interval-scaled for many practical purposes. In the SAE framework, the strict interpretation locates the intelligence dimension on the partial exactification path (ordinal, no unit); the conventional interval-scale treatment in practice is the SAE-framework's partial exactification path with reference (interval) form, but the choice of reference itself is contested. The dimension's scale status is a substantive debate, and the SAE framework can articulate this debate explicitly as a question about which closure path is appropriate, and whether the reference is stable — rather than the more vague question of whether "intelligence is measurable".

Aesthetic (aesthetic judgement): On the articulation we have given, the aesthetic dimension occupies the persistent open path with public comparative re-entry. In the Stevens framework, this has affinity with the nominal-scale row (with the public comparative re-entry distinguishing it from purely nominal categorisation, as articulated in § 4.7). The substantial empirical and theoretical literature that has resisted stable interval or ratio scaling of aesthetic judgements is consistent with this SAE positioning. We do not develop the interface with empirical aesthetics here; we note that the interface is available and that subsequent work can engage with it in detail.

The three examples jointly illustrate how the SAE three closure paths map explicitly to the Stevens framework: temperature (partial exactification → complete exactification, depending on absolute zero), intelligence (partial exactification, contested), aesthetic (persistent open).

The measurement-theoretic interface, as we have articulated it, does not exhaust the substantive content of the closure paths, and it would be a mistake to identify the closure-path typology with the Stevens framework. The closure paths are articulated within the SAE perspective, with its specific commitments to subjectivity, remainder, and the openness of mathematicality. The Stevens framework, by contrast, was articulated within the empirical psychology of measurement, with its own specific commitments and concerns. The two frameworks are compatible at the level of the typology, but each retains its own substantive content beyond the typological correspondence.

With the three closure paths articulated and the measurement-theoretic interface in place, we are in a position to articulate the dimensions of $L_0$ that fill into the paths. This is the subject of § 5.

5. Dimensions of $L_0$ Filled into Closure Paths

5.1 Overview

§§ 3 and 4 articulated the SAE mathematicality schema (four inevitabilities plus public re-entry criterion) and the three closure paths through which mathematicality is realised. § 6 articulated how Paper 1's Objective Exactification Schema fits into this landscape as the path-specific schema of the complete exactification path. § 7 articulated the substantive defense framework.

This section now articulates how specific $L_0$ qualitative dimensions are filled into the closure paths. Each dimension, satisfying the five inevitability criteria, articulates along one or more closure paths and produces substantively different forms of mathematicality.

The dimensions discussed here are not exhaustive. The catalogue is illustrative: it shows that mathematicality is non-unique by exhibiting several substantively different cases. Other dimensions exist, and other dimensions may admit articulations not yet identified. The schema is open to extension, consistent with the inexhaustibility commitment of § 1.5.

We organise the section by dimension rather than by path: each subsection treats one qualitative dimension and articulates which closure path it primarily takes, what mathematicality it produces, and where appropriate, how it relates to other dimensions and paths. The order of treatment moves from the dimensions where the complete exactification path is canonical (quantity, temporal rhythm) through dimensions where partial exactification is natural (spatial size with reference) to the dimension that is most distinctively articulated along the persistent open path (aesthetics). The treatment of aesthetic mathematicality (§ 5.5) is the most substantive, reflecting both its importance for the SAE perspective and the substantive new content the present paper articulates in this area.

5.1.1 Candidate dimensions typology overview

Before detailing each dimension, we give a typology overview of the $L_0$ candidate dimensions to be treated in this section, organised along the three typological axes articulated in § 2.4 (subject involvement, exactification potential, public re-entry):

Dimension	Subject involvement	Exactification potential	Public re-entry	Canonical closure path	Section
Quantity (more/less)	Minimal	Complete	Strong	Complete exactification	§ 5.2
Temporal rhythm (fast/slow)	Moderate	Complete	Strong	Complete exactification	§ 5.3
Spatial size (large/small)	Weak-moderate	Partial (extensible)	Strong	Partial exactification (extensible to complete exactification)	§ 5.4
Aesthetic value (beautiful/ugly)	Maximal	Low (Anchor C)	Moderate	Persistent open	§ 5.5
Ethical value (good/evil)	Maximal	Low	Moderate	Persistent open (candidate)	§ 5.6
Intensity (strong/weak)	Moderate	Partial-complete	Strong	Depends	§ 5.7
Social proximity (close/distant)	Moderate-maximal	Low-partial	Moderate	Depends	§ 5.7
Truth value (true/false)	Depends on domain	Depends	Depends	Depends on domain	§ 5.7

This table is illustrative, not exhaustive. The typological relations articulated in § 2.4.1 (formal isomorphism phenomenologically distinct / single dimension multi-path / parallel structure) manifest in the dimension details.

5.2 The quantity dimension

The quantity dimension is the dimension of "more/less" comparison in number, the dimension Paper 1 implicitly took as paradigmatic. We have treated this dimension in several earlier sections, so the present subsection is brief.

The dimension satisfies the five inevitability criteria straightforwardly. Comparison ("more than" / "less than") is the foundational operation. Direction is total: the comparison has unambiguous orientation. Propagation produces total transitivity: from $a > b$ and $b > c$, we have $a > c$ without qualification. Sustained questioning produces the demand for magnitude ("how much more?") and is answered, in the complete exactification path, by the introduction of unit and arithmetic. Public re-entry is paradigmatically satisfied: numerical comparison is the most publicly re-entrable form of articulation in any culture that has developed counting.

The dimension's canonical closure path is complete exactification. The path's schema is the Objective Exactification Schema of Paper 1: marked handle (the unit "1"), additive path ($\mathbb{N}, \mathbb{Z}, \mathbb{Q}$), multiplicative path with memory binding ($\mathbb{Q}^{\mathrm{alg}}_{\mathbb{R}}, \mathbb{R}$), and closure with remainder ($\mathbb{R}$ with the remainder $i$ pointing to $\mathbb{C}$, and so on through the layers $L_2$ through $L_5$).

The quantity dimension is the dimension Paper 1 substantively articulated. We do not develop it further here; the substantive work is in Paper 1.

A side note: the quantity dimension's subject-independence is partly an artefact of its complete exactification. The arithmetic of two units plus three units equals five units does not depend on which subject performs the addition or in what cultural context; it is intersubjectively stable in a way that judgements in other dimensions are not. This is sometimes taken as evidence that quantity is more objective than other dimensions, or that quantitative mathematics is more genuinely mathematical than its non-quantitative siblings. The SAE perspective rejects this inference. Subject-independence at the level of articulation in the complete exactification path does not entail that the dimension itself is more fundamental or that the mathematicality it produces is privileged. The path is one path among several; its products are complementary to, not superior to, the products of the other paths. The first epistemological anchor (§ 1.5: SAE is one thinking framework, not totalising synthesis) is in effect here.

5.3 The temporal rhythm dimension

The temporal rhythm dimension articulates the "fast/slow" comparison in periodic phenomena. A simple thought experiment: a pre-mathematical articulator notices that some processes recur often, others rarely, and forms qualitative comparisons among them. This is the $L_0$ articulation of temporal rhythm.

The dimension satisfies the five inevitability criteria. Comparison ("faster than" / "slower than") is articulated relative to recurring patterns. Direction is oriented: faster is not interchangeable with slower. Propagation produces transitivity: if A is faster than B and B is faster than C, then A is faster than C, in the relevant sense of frequency comparison. Sustained questioning produces the demand for precise frequency. Public re-entry is satisfied: rhythmic comparison is intersubjectively accessible through shared phenomena (the sun, the seasons, the heartbeat, observable periodic processes).

The dimension's canonical closure path is complete exactification. The path's schema, with appropriate dimension-specific interpretation, produces frequency arithmetic. The marked handle is a base rhythm—not "one" in the abstract sense, but a specific stable periodic phenomenon: one cycle per year, one heartbeat per second, one solar transit per day. The additive path is the comparison of multiples of the base rhythm: a phenomenon recurring twice per period, three times per period, and so on. The multiplicative path is the binding of these multiples into a quantitative structure analogous to the integers or rationals. The closure is the real-valued frequency structure that all of physical and biological science presupposes.

A substantive observation: the temporal rhythm dimension's mathematicality is formally isomorphic to the quantity dimension's mathematicality. The structural content of the two is the same: both produce real-number-like quantitative structures with full arithmetic operations. Yet the phenomenological content of the two is substantively distinct. The quantity dimension is grounded in countable distinctness; the temporal rhythm dimension is grounded in periodic recurrence. The same mathematical structure articulates two different aspects of qualitative experience.

This formal isomorphism with phenomenological distinctness is itself substantive. It shows that the closure path's structural products do not exhaust the substantive content of mathematicality. Two dimensions can articulate along the same path and produce formally isomorphic structures while remaining substantively distinct articulations of mathematicality. This is part of what § 8 articulates as the multiplicity of mathematical paths: the multiplicity is not only across paths but within the same path, where formally analogous structures can articulate substantively different content.

5.4 The spatial-size dimension

The spatial-size dimension articulates the "larger/smaller" comparison of physical extent. The dimension's primary mathematicality is relational rather than fully quantitative, and it articulates along the partial exactification path as canonical.

The dimension satisfies the five inevitability criteria. Comparison ("larger than" / "smaller than") is articulated relative to a reference: the king's fist, the standard rod, the body of the subject, or other publicly-recognisable physical extents. Direction is oriented. Propagation produces transitivity in most cases (though it can fail across qualitatively different dimensions: an A is taller than B is not directly comparable with C is wider than D unless a more unified spatial framework is in place). Sustained questioning is sustained: the question of how much larger, the question of how to measure, the question of how to compare across kinds, all persist. Public re-entry is satisfied through shared references.

The dimension's canonical closure path is partial exactification, articulated along the order-theoretic structure. The marked handle is a reference (the king's fist, the standard rod). The additive path is the chain of comparative judgements relative to the reference. The "multiplicative" path is the composition of relations through transitivity. The closure is an ordering structure—a partial order, a total order, or a more elaborate order-theoretic object depending on the specifics.

The spatial-size dimension can also be articulated along the complete exactification path when units are introduced: the foot, the meter, the kilometer. This extension makes the dimension's mathematicality fully quantitative, with arithmetic operations and the full apparatus of metric geometry. The transition from the partial exactification path to the complete exactification path is a substantive articulative choice; it is not merely a refinement of the partial-exactification articulation but a different articulation that produces a different mathematical structure.

The spatial-size dimension illustrates a phenomenon worth marking: a single dimension can admit multiple closure paths, and the choice of path is a substantive choice. The partial exactification path is the more cautious articulation, the one that does not commit to the existence of a metric structure. The complete exactification path is the more committed articulation, the one that introduces the unit and the arithmetic that the unit supports. Both paths are available for the spatial-size dimension; both are mathematicality; their substantive content differs.

5.5 The aesthetic dimension

The aesthetic dimension articulates the "more beautiful / less beautiful" comparison, and more broadly the comparative articulations of aesthetic value (sublimity, compellingness, harmony, gracefulness, and so on). The dimension is the substantive focus of the present section because it articulates along the persistent open path as canonical, and because its mathematicality is the most substantively distinctive form the present paper articulates.

5.5.1 Aesthetic articulation as the maximal expression of subjectivity

Aesthetic articulation is the form of qualitative articulation in which subjectivity is most centrally constitutive. To articulate an object as beautiful is not to report a property of the object considered independently of the subject; it is to engage the subject's evaluative response, in a manner that is inseparable from the response itself. The Kantian articulation of this point—that aesthetic judgements claim universal validity but proceed without determinate concept—is one of the substantive philosophical articulations of the centrality of subjectivity in aesthetic articulation.

In the SAE perspective, this is not a deficiency of aesthetic articulation; it is its substantive distinctness. The SAE philosophical commitment is to subjectivity as constitutive of articulation: Self-as-an-End names the commitment to the subject as not reducible to a means, not reducible to a function of objective structures, not reducible to an extension of impersonal articulation. The aesthetic dimension is the dimension in which this commitment is most fully exhibited at the level of qualitative articulation. Aesthetic mathematicality, accordingly, is the form of mathematicality in which subjectivity is most fully constitutive.

A counter-consideration that needs immediate engagement: it might be objected that subjectivity should be minimised in mathematics, not central to it, and that the SAE perspective's elevation of aesthetic mathematicality therefore departs from what mathematics is. The objection misreads the present paper's articulation. We do not claim that all of mathematics should be aesthetic, or that the SAE perspective sees the quantitative-arithmetic mathematics of the complete exactification path as deficient because it minimises subjectivity. We claim that the SAE mathematicality landscape includes both subject-minimising and subject-maximising forms of mathematicality, and that recognising both is the substantive contribution of the present paper. The complete exactification path produces subject-minimising mathematics; the persistent open path with aesthetic articulation produces subject-maximising mathematicality. Both are mathematicality, in the substantive sense articulated by the five criteria; their difference is in the constitutive role of subjectivity, not in the satisfaction of the criteria themselves.

5.5.2 Subjectivity cannot be quantified without being abolished

The third epistemological anchor of § 1.5 is in effect here: subjectivity cannot be quantified without being abolished. This anchor is essential to the substantive articulation of aesthetic mathematicality.

To see why, consider what quantification would do. Quantification introduces a unit, a metric, or a scale that assigns numerical values to instances of the phenomenon being articulated. Applied to aesthetic articulation, quantification would assign numerical values to aesthetic judgements: this work scores 8.7 on the beauty scale, that work scores 6.3. The proposal is not philosophically incoherent—aesthetic scales of this kind have been proposed and used in various empirical settings—but the proposal accomplishes something specific: it replaces the constitutive role of the subject with a presumptively objective metric. The subject is no longer evaluating; the subject is reporting on the value the metric already assigns. The aesthetic judgement is no longer constitutive of the value; the aesthetic judgement is detectable on a scale that exists independently.

The replacement is precisely the abolition of subjectivity in the relevant sense. When the subject is reduced to a sensor reporting on an objective metric, the subject's evaluative activity is no longer the constitutive locus of the value being articulated. The aesthetic judgement, in this articulation, is no longer aesthetic in the substantive sense; it is empirical, in the sense of measuring a presumptively independent quantity. The articulation has shifted from aesthetic articulation to a form of pseudo-objective measurement that incidentally takes aesthetic judgements as inputs.

The SAE perspective rejects this shift. Aesthetic articulation is not the measurement of an aesthetic property by means of subjective judgement; it is the constitutive engagement of the subject in articulating value through judgement. Quantification of aesthetic articulation, on this view, is not a refinement; it is a category mistake that abolishes the substance of the articulation in the name of making it more precise.

This is the substantive reason that the persistent open path does not introduce a unit in aesthetic articulation. The path is not deficient for failing to introduce a unit; the path articulates that the introduction of a unit would be substantively destructive of the dimension's content. The persistent open path's "incompleteness" relative to the complete exactification path is constitutive, not remediable.

5.5.3 Aesthetic mathematicality as distribution-grounded articulation

If aesthetic articulation cannot be quantified, what mathematical structure does it admit? The answer, we have argued in § 4.4, is distributional articulation along the lines of probability-distribution grammar.

The substantive content of aesthetic distributional articulation is as follows.

An aesthetic comparison is not a point judgement assigning a single value to an object; it is a distribution of comparative articulations conditioned on subject, context, prior, and frame. The aesthetic articulation of a work, in this view, is the totality of comparative judgements about the work distributed across the relevant population of subjects, contexts, and articulative occasions. The distribution may exhibit substantial regularities (works that elicit consensus across many subjects and contexts, works that elicit divergent responses but in patterned ways), but the regularities do not collapse the distribution to a single value.

Aesthetic comparison between two works is then a comparison of distributions: is the distribution of comparative judgements about A systematically positioned relative to the distribution about B? The comparison may be made in terms of stochastic dominance (does A's distribution dominate B's across most contexts?), in terms of conditional comparison (does A surpass B in some specific articulative frame?), in terms of distributional moments (is A's distribution more concentrated than B's, indicating more uniform appreciation?), and so on. The mathematics of distributional comparison provides a rich vocabulary for these articulations, and the vocabulary does not require quantification of aesthetic content. Distributional comparison is a substantive mathematical operation that respects the distinct subjectivities of the individual judgements while articulating their collective structure.

Aesthetic "marked handles" in this articulation are priors, in the sense of § 4.4. Canonical works function as priors: their established status conditions further aesthetic articulation, providing reference points for comparative judgement. Style frames function as priors: established categories like "Baroque", "Modernist", "minimalist" structure the conditioning of aesthetic judgements. Interpretive traditions function as priors: critical schools, philosophical orientations, cultural contexts condition the form aesthetic judgements take. The priors are real and operative, but they are contestable and revisable, and they do not fix the aesthetic articulation once and for all.

Aesthetic "additive accumulation" is the accumulation of comparative judgements across subjects and contexts. The accumulation is not iterative in the algebraic sense; it is distributional. Over time, an aesthetic articulation accumulates a distribution of judgements that becomes the substantive content of the articulation. The history of aesthetic engagement with a work is part of the work's aesthetic content, in this view: the work has not just intrinsic aesthetic qualities but a distributional history of how subjects have engaged with it.

Aesthetic "conditioning" is the modulation of the comparative distribution by context. The aesthetic judgement of a work in one cultural context, by one type of subject, with one interpretive prior, differs from the aesthetic judgement of the same work in a different context, by a different subject, with a different prior. The differences are not noise; they are the conditioning structure of aesthetic articulation. Mathematical conditioning—conditional probability, marginalisation across context, comparison of conditional distributions—provides a rich apparatus for articulating these context-dependent structures.

Aesthetic "closure" is the distribution space that accumulates over time, persistently open to revision and extension. The distribution space is not a closure in the structural sense the complete exactification path produces; it is an evolving structure of articulated aesthetic comparisons that does not terminate. New works enter; new subjects enter; new contexts emerge; the distribution space continues to develop. This is the substantive sense in which aesthetic mathematicality is persistently open: the structure has substantive content (the distributions, the conditionings, the comparisons), but the structure does not stabilise into a final form.

5.5.3.1 The non-parametric, topological character of probability-distribution grammar

To prevent readers with a scientific or statistical background from misreading the probability-distribution grammar of aesthetic mathematicality as parametric statistics (Gaussian distributions, Bayesian networks, high-dimensional aesthetic-rating vector spaces) or as big-data statistics, we emphasise with deliberate clarity: the distribution grammar at the $L_0$ level is non-parametric, and in fact operates at the topological level. What it cares about is not the specific probability density equation of an aesthetic distribution nor the numerical frequency, but rather the topological relations among distributions.

A few illustrative examples (offered as direction-markers, not as a complete theory):

Stochastic dominance: An aesthetic comparison can be articulated as "the distribution of comparative judgments for work A stochastically dominates the distribution for work B across most contexts" — this structure concerns the partial-order relation between distributions, and does not require assigning numerical values to individual aesthetic instances. Even if two distributions cannot be parametrised or precisely characterised, the stochastic dominance relation between them remains articulable.

Conditional comparison: Aesthetic comparison is conditional on context, subject, prior, and frame — "under aesthetic prior $P$, the resonance distribution for work A is systematically more concentrated than for work B" — this structure concerns the relations among conditional distributions, with the formal character of conditional probability, without requiring numerical quantification of individual aesthetic values.

Distributional moment comparison: Certain features of aesthetic distributions (spread, concentration, bimodality) can be articulated through distributional-moment-related structure — these are features of distribution shape, not quantifications of the aesthetic content within the distribution. A more concentrated distribution may indicate stronger consensus; a bimodal distribution may indicate aesthetic split within a culture or tradition.

To articulate the key distinction (consistent with Anchor C):

Quantifying distributional structure (entropy, mutual information, stochastic dominance, conditional structure) — permitted, because it articulates formal relations among distributions without cancelling the subject's constitutive role in individual judgments
Quantifying aesthetic values within distributions (assigning numerical values to individual aesthetic objects, such as "8.7 in beauty") — not permitted (Anchor C violation), because it replaces the subject's constitutive evaluation with objective measurement

Detailed mathematical structure (specific stochastic dominance measures, conditional probability formalisations, divergence measures) is left to the subsequent specialised SAE aesthetic mathematics series, which will engage substantively with the specific technical apparatus of statistical aesthetics, Bayesian aesthetics, and information-theoretic aesthetics.

5.5.4 Aesthetic mathematicality is distinct from narrow mathematics in substantive ways

We have argued that aesthetic articulation satisfies the five inevitability criteria of § 7 and enters SAE mathematicality. We now articulate, with some care, the substantive ways in which aesthetic mathematicality differs from narrow mathematics. The articulation is required by the second decision of the present paper's framing: we are to articulate, substantively and explicitly, what makes aesthetic mathematicality distinct from the conventional mathematics that occupies the complete exactification path.

The distinctions are as follows.

Point values versus distributions. Narrow mathematics, in its quantitative core, operates with point values. A number is a specific point on the real line; an equation specifies a relation among specific points. Aesthetic mathematicality operates with distributions. An aesthetic judgement is a distributional articulation, not a point value; comparison is between distributions, not between points.

Subject-minimisation versus subject-constitution. Narrow mathematics, in the complete exactification path, minimises the role of the subject. The arithmetic of two plus three equals five is the same regardless of which subject performs the calculation. Aesthetic mathematicality constitutively involves the subject. The aesthetic judgement is inseparable from the subject's evaluative engagement.

Closure-equation grammar versus probability-distribution grammar. Narrow mathematics articulates relations through equations and arithmetic operations. Aesthetic mathematicality articulates relations through distributions, conditioning, and comparison of distributions. These are different grammars, with different inference patterns and different canonical questions.

Structural closure versus persistent open. Narrow mathematics produces structural closures: $\mathbb{R}$ is a complete structure under its operations, $\mathbb{C}$ is the algebraic closure of $\mathbb{R}$, and so on. Aesthetic mathematicality does not produce structural closures of this kind; the persistent open path articulates that the development of the practice does not terminate in a stable closed structure.

Total ordering versus distributional comparison. Narrow mathematics produces total orderings (or at least totally ordered substructures). Aesthetic mathematicality typically produces partial orderings at best, and the partial orderings exhibit local instabilities (cycles, context-dependent reversals, distributional rather than point comparisons).

These distinctions are not gradations of a single property. They are substantive categorical differences between two forms of mathematicality that both satisfy the inevitability criteria but realise them along substantively different closure paths. The differences make aesthetic mathematicality a substantively distinct form of mathematicality, not a degraded or imperfect form of narrow mathematics.

This is the substantive sense in which the present paper articulates aesthetic mathematicality as a distinct form of mathematicality, not a deficient form of narrow mathematics. The distinctness is not a defect; it is constitutive of what aesthetic mathematicality is. The persistent open path's articulation is the substantive form that mathematicality takes when subjectivity is constitutive of the dimension. The complete exactification path's articulation is the substantive form that mathematicality takes when subjectivity is minimised. Both forms are mathematicality; their substantive content differs.

5.5.5 Three boundary statements for aesthetic mathematicality

To prevent misreading of the present articulation, we state three boundary conditions explicitly.

Not all aesthetic activity is aesthetic mathematicality. A purely private aesthetic response—a fleeting reaction with no articulable comparative structure, no public re-entry, no propagation—does not enter aesthetic mathematicality. The five criteria are real requirements; not every aesthetic response satisfies them. Aesthetic mathematicality is articulated aesthetic comparison, structured aesthetic engagement with public re-entry, not the totality of aesthetic experience.

Only structured, publicly re-enterable, comparative aesthetic articulation enters aesthetic mathematicality. The qualifying conditions are: structural articulation (the comparison has form, not just feeling), public re-entry (the articulation is available to others, through canons, frames, or shared interpretive practices), comparative content (the articulation involves comparison, not just monadic appreciation), propagation (the comparison engages with other comparisons, not isolated), and sustained questioning (the articulation persists rather than terminating). Aesthetic activity satisfying all five enters aesthetic mathematicality; aesthetic activity failing any of them does not.

Aesthetic mathematicality's direction of development is probability-distribution grammar, not quantification of beauty. This is the substantive direction articulated in this section. Aesthetic mathematicality develops by articulating the distributional structure of aesthetic comparison, by analysing the conditioning of aesthetic articulation by context, by studying the patterns of aesthetic distribution across subjects and traditions. It does not develop by introducing units, scales, or numerical values for beauty. The direction is decisive: probability-distribution grammar, not quantification.

These three boundary statements together constitute a substantive boundary for aesthetic mathematicality. The boundary is not arbitrary; it is articulated by the five inevitability criteria of § 7 and by the substantive constitutive role of subjectivity that § 5.5.2 articulated.

5.5.6 Connections to existing intellectual territory

Aesthetic mathematicality, on the articulation here given, connects to several substantively rich intellectual territories. We note the connections briefly, with detailed engagement reserved for subsequent specialised papers in the aesthetic mathematics series.

Kant's Critique of Judgment. Kant articulated aesthetic judgement as carrying universal validity claims without determinate concept. The articulation is structurally analogous to the present paper's articulation of aesthetic mathematicality: aesthetic articulation has structure (universal validity claims) without admitting the structural closure that determinate concept provides. The SAE articulation, with its persistent open path and distributional structure, is a substantive descendant of the Kantian articulation, in a tradition that takes the constitutive role of the subject seriously.

Statistical aesthetics. Empirical studies of aesthetic preference across populations, cultures, and contexts produce distributional data that is the empirical substrate of the present paper's articulation. The empirical findings of statistical aesthetics confirm that aesthetic judgement does not collapse to single values across subjects, and that the variation across subjects has structure that empirical methods can articulate.

Bayesian aesthetics. Conditional probability frameworks for modelling aesthetic preference, with priors over aesthetic categories and updates based on aesthetic exposure, are a contemporary articulation of the conditioning structure the present paper articulates. The articulation has both empirical and theoretical sides, and it connects to wider Bayesian frameworks in cognitive science and decision theory.

Arrow's impossibility theorem and preference theory. Arrow's theorem articulates conditions under which transitive aggregate preference orderings cannot be derived from individual preference orderings under reasonable conditions. The theorem is a substantive articulation of the propagation criterion's behaviour in the aesthetic and preference domains: propagation pressure does not produce transitivity, and the failure of transitivity is a structural feature, not a pathology. The present paper's articulation of aesthetic mathematicality is consistent with Arrow's theorem and articulates its substantive philosophical content.

Information-theoretic aesthetics. Frameworks that articulate aesthetic content in terms of information-theoretic properties (entropy of aesthetic distributions, mutual information between aesthetic features) are a quantitative apparatus that articulates distributional structure without violating the third epistemological anchor. They quantify the distributional structure (statistical pattern, distribution shape), not the aesthetic values within the distribution (numerical assignment to individual aesthetic objects). The distinction is essential (see § 5.5.3.1 for a more detailed articulation of this distinction and its precise relation to Anchor C): Anchor C permits the quantification of distributional structure (topological relations among aesthetic distributions, information-theoretic features), but does not permit the quantification of the aesthetic content itself (assigning numerical values to individual aesthetic objects cancels the subject's constitutive role).

Sociology of aesthetics. Empirical studies of aesthetic preference as a function of social position, cultural context, and historical period are a substantive articulation of the conditioning structure the present paper articulates. The findings of sociology of aesthetics are part of the substantive empirical basis for the SAE articulation.

These connections are noted to indicate that aesthetic mathematicality is not a free invention of the SAE perspective; it tracks substantive distinctions that other intellectual traditions have articulated independently. The SAE articulation provides a unified framework that connects these traditions, but the substantive content is supported by, and connects to, work in these traditions.

5.5.7 The detailed structure of aesthetic mathematicality is reserved for subsequent specialised papers

The present section has articulated aesthetic mathematicality at the level of possibility proof, substantive direction, and boundary conditions. The detailed structure of aesthetic mathematicality—the specific articulation of its marked handle, its distributional accumulation, its conditioning, its distribution space—is reserved for subsequent specialised papers in the SAE aesthetic mathematics series.

This reservation reflects two substantive considerations. First, the detailed articulation of aesthetic mathematicality is a substantial undertaking that, if pursued in the present paper, would unbalance the paper's overall structure. The present paper articulates the $L_0$ foundation of multiple closure paths; aesthetic mathematicality is one such path, and its detailed articulation requires its own substantive treatment. Second, the detailed articulation of aesthetic mathematicality requires substantive engagement with the specific intellectual traditions noted in § 5.5.6, which would also unbalance the present paper if pursued fully here.

The present paper accomplishes what it sets out to do regarding aesthetic mathematicality: it articulates that aesthetic articulation enters SAE mathematicality (the possibility proof of § 7.4), it articulates the substantive direction of aesthetic mathematicality's development (probability-distribution grammar, persistent open path), it articulates the boundary conditions that aesthetic mathematicality must satisfy (the three boundary statements of § 5.5.5), and it articulates the substantive ways in which aesthetic mathematicality differs from narrow mathematics (§ 5.5.4). With these foundations in place, the detailed articulation of aesthetic mathematicality can proceed in subsequent specialised papers without requiring the foundational work to be redone.

5.6 The ethical dimension (the good/evil dimension at $L_0$)

A brief note on the ethical dimension. The ethical articulation of "good/evil" or "right/wrong" satisfies the four inevitabilities and admits public re-entry through shared moral traditions, philosophical articulations of ethical principles, and shared cases of moral judgement. The dimension therefore enters SAE mathematicality (broad), on the substantive defense framework of § 7.

The dimension's canonical closure path is, on initial articulation, the persistent open path, parallel in structure to the aesthetic dimension. Ethical judgements admit comparative articulation but resist stable structural closure; the history of ethical articulation, like the history of aesthetic articulation, is a sustained engagement without final closure.

Key candidate / parallel-case firewall for the ethical dimension:

To prevent possible misreadings the ethical dimension might invite, we explicitly articulate three things the present paper does not claim:

The present paper does not develop an "ethical mathematics" theory. The present paper only marks the ethical dimension as a candidate for broad SAE mathematicality, parallel to § 5.5's marking of the aesthetic dimension as a parallel case. Detailed ethical-mathematicality structure is reserved for future work, potentially in the SAE philosophy's specialised ethics series.

The present paper does not claim that "moral judgement is mathematical judgement". The claim is rather: structured, publicly re-entrant ethical-comparison practices (for example, normative-ethics traditions, case-based ethical reasoning, public ethical deliberation) satisfy the five conditions and enter broad SAE mathematicality. This articulates structural kinship, not reductive identity.

The present paper does not reduce ethical judgement to mathematical judgement. Ethical judgement retains its substantive ethical content within the SAE perspective; entering broad SAE mathematicality does not cancel the distinctness of the ethical dimension, parallel to how entering broad SAE mathematicality does not cancel the distinctness of the aesthetic dimension (per Anchor C). Ethical judgement remains ethical judgement, not ethical judgement translated into mathematical language.

We note the dimension here primarily to indicate that aesthetic mathematicality is not unique as a subject-grounded form of mathematicality; the ethical dimension is a parallel sketch, and other subject-grounded dimensions may emerge as additional candidates as the SAE framework develops.

5.7 Other candidate dimensions

Other $L_0$ qualitative dimensions are candidates for entering SAE mathematicality, and we note them briefly without detailed treatment.

Intensity (strong/weak, heavy/light): Articulates along the partial exactification path with reference to a standard intensity, or along the complete exactification path when units are introduced. Modern physics has substantively articulated intensity dimensions (force, pressure, energy density) through the complete exactification path with quantitative units.

Social proximity (close/distant, intimate/foreign): Articulates along the persistent open path with prior structures provided by social institutions and shared cultural categories. The dimension is the subject of substantive empirical work in sociology and anthropology, and its detailed articulation as mathematicality is a candidate for future work.

Truth (true/false, certain/uncertain) ——the truth-value dimension as an illustrative application of the Paper 2 framework: The truth-value dimension articulates along three closure paths, each producing substantively different logical structures:

Along the complete exactification path: produces formal logic / Boolean algebra (truth values fixed as $T = 1, F = 0$, the operations AND / OR / NOT producing deterministic closure, subject's role minimised). This is the canonical form of "logic" in the narrow sense.
Along the partial exactification path: produces intuitionist logic or certain branches of modal logic (retains the order structure of derivability, but rejects forced-closure arithmetic-style laws such as the excluded middle).
Along the persistent-open path: produces hermeneutic logic / dialectical logic / philosophical logic (truth is no longer a $0 / 1$ point value, but rather a context-, canon-, and subject-involvement-conditioned distribution).

An important articulation (parallel to § 1.5's Anchor A guarding against the SAE-as-grand-unification misreading): the present articulation does not claim that formal logic is the universal foundation of mathematicality, nor does it claim that all articulation must start from formal logic. Formal logic is the substantive realisation of the truth-value dimension along the complete exactification path, parallel to § 5.2's quantity dimension producing arithmetic along the complete exactification path and § 5.4's spatial-size dimension producing order structures along the partial exactification path. It is neither superior to nor more foundational than the other realisations.

Substantive consequence (directly bearing on the future Paper 3 trajectory): $L_1$ is not a single foundational layer, but rather the product of dimension × path. The quantity dimension along the complete exactification path produces $L_1$ arithmetic (the main trajectory of Paper 1 / Paper 3); the truth-value dimension along the complete exactification path produces $L_1$ formal logic (substantively distinct); the truth-value dimension along the partial exactification path produces $L_1$ intuitionist logic (yet another); the spatial-size dimension along the partial exactification path produces $L_1$ order-theoretic / metric structure; and so on. Different dimension × path combinations produce substantively different $L_1$s. Detailed treatment of logic and the multiple $L_1$ trajectories is reserved for future work (see § 11.7.1).

These dimensions are not exhaustive, and other candidate dimensions exist. The catalogue of dimensions is open, consistent with the inexhaustibility of $L_0$ that § 8 articulates. The present paper articulates aesthetic mathematicality in substantive detail because it is the most substantively distinctive contribution of SAE's focal significance (subjectivity-grounded); the illustrative application to the truth-value dimension exhibits the framework's substantive coverage into the objectivity-focal direction; other dimensions are noted as candidates for future work.

6. Paper 1's Schema as the Objective Exactification Schema

6.1 Overview

Paper 1 articulated a four-step pattern that it identified as applying across the mathematical layers $L_1$ through $L_5$: a marked handle (step 1), an additive path (step 2), a multiplicative path with memory binding (step 3), and a closure that produces a remainder pointing to the next layer (step 4). Paper 1 called this pattern the Layer Articulation Schema and presented it as a descriptive structural identification of mathematics's layered architecture.

The present paper places that schema within a broader landscape. Once we have articulated multiple closure paths (§ 4), each producing substantively different forms of mathematicality, the question arises whether Paper 1's four-step pattern applies across all paths or only across some. This section addresses that question. The conclusion, to anticipate, is that Paper 1's schema is path-specific: it is the canonical schema of the complete exactification path, where it applies with full force, but it does not apply in its literal four-step form across the partial exactification path or the persistent open path. We mark this path-specific character by renaming Paper 1's schema the Objective Exactification Schema and articulating a more abstract upper-level schema, which we call the $L_0$ Mathematicality Schema, that organises mathematicality across paths.

Two clarifications before we proceed.

First, the renaming does not retroactively modify Paper 1. Paper 1's content stands as published. The renaming is a substantive clarification of scope, articulated in the present paper: where Paper 1 articulated a schema for "the mathematical layers $L_1$ through $L_5$", we now articulate, with the multi-path framework in place, that those layers are all within the complete exactification path, and the schema is the schema of that path. Paper 1 did not claim universality across all conceivable forms of mathematicality; the present paper makes explicit what Paper 1's scope was, in a way that Paper 1 itself, in the absence of the multi-path framework, could not.

Second, the path-specific character of Paper 1's schema is not a limitation that diminishes its substantive value. The complete exactification path is the dominant mode of mathematical articulation in the modern Western tradition, and Paper 1's articulation of its architecture remains the substantive achievement it was. The clarification of scope frees Paper 1 from a universality claim it did not need to make in order to do its substantive work, and it positions Paper 1 in relation to the broader landscape of mathematicality that subsequent work, including the present paper, articulates.

6.2 Paper 1's schema applies in full force within the complete exactification path

In the complete exactification path, Paper 1's four-step schema applies with the full force the paper articulated. This is essentially uncontroversial given the structure of the path. The path's marked handle is a unit; the path's additive structure generates an additive path; the path's multiplicative structure generates a multiplicative path with the memory binding Paper 1 described; the path's closure produces a structure with a remainder that opens the next layer's articulation.

The five layers $L_1$ through $L_5$ that Paper 1 articulated are all instances of the schema within the complete exactification path. The articulation of integer arithmetic at $L_1$, of continuous analysis at $L_2$, of complex analysis at $L_3$, of measure-theoretic foundations at $L_4$, and of category-theoretic articulations at $L_5$ all instantiate the four steps with the specific substantive content Paper 1 articulated. The schema's universality, as Paper 1 articulated it, is universality across the layers of the complete exactification path, not universality across all conceivable closure paths.

In this restricted sense, Paper 1's universality claim is sustained: across the layers of the complete exactification path, the four-step schema applies. The renaming we are about to make does not contradict this claim; it specifies its scope.

6.3 Paper 1's schema applies partially within the partial exactification path

In the partial exactification path, Paper 1's four-step schema applies, but with substantive reinterpretation of each step.

The marked handle is a reference rather than a unit. The two are structurally analogous in the role they play (they fix the marked handle of step 1), but they differ in what they support: a unit supports arithmetic operations, a reference supports comparison and ordering but not arithmetic. The first step of the schema, in the partial exactification path, is the introduction of a reference, not the introduction of a unit.

The additive path of step 2 is reinterpreted as a comparative chain. In the complete exactification path, the additive path is iterative addition: $1, 2, 3, \ldots$, the natural numbers and their negatives, the rationals through repeated addition and subtraction with appropriate operations. In the partial exactification path, there is no addition; the iterative structure is the chain of comparative judgements relative to the reference: A is larger than the reference, B is larger than A, C is larger than B, and so on. The chain is structurally analogous to the additive path but is articulated through comparison rather than through arithmetic addition.

The multiplicative path of step 3 is the most substantively reinterpreted. In the complete exactification path, the multiplicative path introduces multiplication, with its memory-binding structure that records how iteration is itself iterated. In the partial exactification path, there is no multiplication; the multiplicative path is articulated as the composition of relations, with the memory-binding being the transitivity of the ordering: if A is larger than B and B is larger than C, the transitive closure of these relations binds them into the chain that locates A relative to C. The memory is not the memory of iterated counting; it is the memory of compositional relations.

The closure of step 4 is the order-theoretic structure that the path generates: a poset, a preorder, a lattice-like structure, depending on the specifics of the dimension. The closure is real—there is a definite mathematical object at the end of the four steps—but the closure is order-theoretic rather than algebraic in the sense of arithmetic. The remainder, which in the complete exactification path was the basis for the transition to the next layer, is here either an internal element of the order structure (a gap in the ordering, an incomparable pair) or a pressure toward further articulation that may push the structure toward complete exactification through the introduction of arithmetic.

We say that Paper 1's schema applies "with reinterpretation" in the partial exactification path rather than "in its literal form" because the substantive content of each step differs from the complete exactification path's content. The structural analogy is real, and the schema's four-step organisation can be recognised in the partial exactification path's articulation. But the content of each step is path-specific, and the reinterpretation is not optional: applying the complete-exactification version of the steps to the partial exactification path would either fail (no arithmetic available) or distort the path's articulation. The schema applies in the partial exactification path, but in its path-specific form.

6.4 Paper 1's schema does not apply in its literal form within the persistent open path

In the persistent open path, Paper 1's four-step schema does not apply in its literal form. The schema's four steps presuppose a structural closure of the kind the path declines to produce, and a faithful articulation of the persistent open path requires substantively different structures at each step.

The marked handle is a prior rather than a unit or a reference. As articulated in § 4.4, a prior is an articulation that conditions further articulation without fixing it. Canonical works, established style frames, and shared interpretive traditions function as priors in aesthetic articulation. The prior is structurally a marked handle in the broadest sense—it provides a starting point for the comparative articulations that follow—but it differs from a unit or a reference in that it does not fix the comparative structure. The prior is contestable, displaceable, subject to reframing. Where a unit, once fixed, supports arithmetic until further notice, and a reference, once fixed, supports an ordering, a prior conditions articulation without supporting any fixed closure.

The "additive path" of step 2 is not an additive path at all in the persistent open path. The chain of comparative judgements is real, but the chain does not accumulate the way an additive path accumulates: there is no fixed direction in which the chain extends, no canonical ordering that successive judgements produce. What occurs instead is the accumulation of a distribution of comparative judgements across subjects, contexts, and articulations. The distribution is not a chain in the structural sense the additive path requires; it is a multi-dimensional structure of articulated comparisons that does not collapse to a single ordering. The schema's second step, in the persistent open path, is realised not as iteration but as distributional accumulation.

The "multiplicative path" of step 3 is similarly not a multiplicative path in the schema's literal sense. The memory-binding that the multiplicative path provides in the complete exactification path is the binding of iterated counting; in the partial exactification path it is the binding of compositional relations. In the persistent open path, the analogous structure is conditioning: the modulation of comparative articulations by features of context, subject, prior, and frame. The aesthetic articulation of A as more compelling than B is not a context-free comparison; it is a comparison conditioned on the subject's stance, the relevant tradition, the prior canonical works in play, the situational features that frame the comparison. The conditioning structure is the persistent open path's analogue of the multiplicative path, but it does not multiply in any algebraic sense.

The "closure" of step 4 is the distribution space that the conditioning generates. The space is not a closure in the structural sense of $\mathbb{R}$ or a poset; it is the structured array of conditional aesthetic articulations that the path accumulates over time. The space has substantive content—it admits comparative articulations of distributions, transformations under conditioning, and the like—but it is not a closure that terminates the articulation. The fourth step of the schema, in the persistent open path, is realised as the sustained ongoing development of the distribution space, not as a closure that fixes its content.

We conclude that Paper 1's four-step schema, in its literal form, does not apply in the persistent open path. The structural analogues of the four steps can be articulated—prior, distributional accumulation, conditioning, distribution space—but the content of each step is substantively different from the schema's content in the complete exactification path. The persistent open path is mathematicality (the four inevitabilities of § 3 are satisfied, and the public re-entry criterion is met) but it is not a layered architecture in the sense Paper 1 articulated.

6.5 Paper 1's schema renamed: the Objective Exactification Schema

We now make the renaming explicit. Paper 1's Layer Articulation Schema, on the multi-path framework of the present paper, is articulated as the canonical schema of the complete exactification path. We accordingly rename it the Objective Exactification Schema.

The renaming has several purposes.

It marks the schema's path-specific character. The schema is the schema of the complete exactification path; it applies in full force within that path, with substantive reinterpretation in the partial exactification path, and not in its literal form in the persistent open path. The renaming makes this scope visible in the schema's name itself, rather than leaving it implicit in the schema's content.

It preserves the schema's substantive content. The renaming does not change Paper 1's substantive articulation of the four-step pattern; it changes only how the schema is positioned in relation to the broader landscape of mathematicality. Within the complete exactification path, the schema continues to be the substantive articulation Paper 1 made it. The renaming is a relabelling of scope, not a revision of content.

It frees Paper 1 from a universality claim it does not need to make. Paper 1's articulation of the four-step pattern at each of the five layers $L_1$ through $L_5$ is substantively complete as an account of the complete exactification path's architecture. The articulation does not require, and does not depend on, a claim that the same four-step pattern applies in its literal form across all conceivable mathematical articulations. The renaming allows the substantive achievement of Paper 1 to be appreciated within its scope, without the burden of a universality claim that the present paper, on multi-path grounds, no longer accepts.

We articulate the renaming as a substantive scope clarification rather than as a correction. Paper 1 did not claim, in so many words, that its schema applied across all mathematicality including subjects-grounded and aesthetic articulations. The paper articulated the schema for the mathematical layers it considered, and those layers, on the multi-path articulation now in place, are layers of the complete exactification path. The renaming makes this scope explicit; it does not retroactively introduce a limitation that Paper 1 itself articulated.

6.5.1 Scope test conclusion (decisive articulation)

To make the scope test conclusion explicit and decisive, we state with deliberate clarity:

Paper 1's Layer Articulation Schema (now renamed the Objective Exactification Schema) is not a universal schema for mathematicality; it is a closure-equation-grammar-specific, path-dependent schema.

Specifically:

The schema applies with full force within the complete exactification path;
The schema's application within the partial exactification path requires substantive re-articulation (step 1 = reference rather than unit; step 2 = comparison chain rather than addition; step 3 = relational composition rather than multiplication; step 4 = order structure rather than quantitative closure);
The schema does not apply in its literal form within the persistent-open path (which requires distribution-grounded grammar rather than closure-equation grammar).

The universality of Paper 1's schema is within-path universality, not cross-path universality. This path-dependence is a feature of the schema, not a limitation; it reflects the fact that the schema substantively articulates the architecture of the complete exactification path, and the complete exactification path is only one path of mathematicality.

Cross-path universality belongs to the upper-level $L_0$ Mathematicality Schema (§ 6.6), which articulates mathematicality at the path-neutral level.

6.6 The $L_0$ Mathematicality Schema

With Paper 1's schema renamed as the Objective Exactification Schema, we are in a position to articulate an upper-level schema that organises mathematicality across paths. We call this the $L_0$ Mathematicality Schema.

The $L_0$ Mathematicality Schema is the articulation of the four inevitabilities (§ 3) and the public re-entry criterion (§ 3.8) at $L_0$, prior to the specification of any particular closure path. It is the schema that identifies mathematicality as such, without committing to a particular path of articulation.

The schema's structure is as follows:

Comparison is unavoidable (first inevitability).
Comparison cannot be undirected (second inevitability).
Comparison cannot remain isolated (third inevitability).
Comparison cannot escape sustained questioning (fourth inevitability).
The articulation must admit public re-entry (fifth condition, on the practice).

A qualitative dimension at $L_0$ that satisfies these five conditions enters SAE mathematicality. The specific form the mathematicality takes—the closure path along which the dimension is articulated—is a further specification, not contained in the $L_0$ Mathematicality Schema itself. Different dimensions may articulate along different paths, and the same dimension may admit different paths under different circumstances.

The relation between the $L_0$ Mathematicality Schema and the Objective Exactification Schema is hierarchical and articulatory. The $L_0$ Mathematicality Schema is the upper-level schema, articulating mathematicality at $L_0$ in a path-neutral way. The Objective Exactification Schema is one realisation of the $L_0$ Mathematicality Schema, specifying how mathematicality develops when the complete exactification path is taken. Other realisations of the $L_0$ Mathematicality Schema include the partial exactification path's schema (which we do not articulate in detail here but which is structurally analogous to the Objective Exactification Schema with appropriate path-specific reinterpretation) and the persistent open path's distributional schema (similarly left for subsequent detailed articulation).

This hierarchical relation can be represented as follows:

$$

\text{$L_0$ Mathematicality Schema} \longrightarrow \text{multiple closure paths} \longrightarrow \text{path-specific schemas}.

$$

Paper 1's contribution, on this articulation, was to articulate the path-specific schema of the complete exactification path. The present paper's contribution is to articulate the upper-level $L_0$ Mathematicality Schema, of which Paper 1's schema is one realisation, and to articulate the other realisations in sketch form (with details left for subsequent specialised papers).

A note on the schema's name. We use "$L_0$ Mathematicality Schema" because the schema articulates the conditions for mathematicality at $L_0$, the layer of qualitative articulation prior to the introduction of any marked handle. The schema is the articulation of the $L_0$-level conditions that any closure path must satisfy in order to constitute mathematicality. Other names are possible (the "Mathematicality Schema", the "Inevitability Schema", the "Pre-Path Schema"), but the chosen name marks the schema's location in the architecture of SAE mathematicality: at $L_0$, prior to path specification, articulating the conditions for the emergence of mathematicality.

6.7 Three four-tuples in the SAE framework

The SAE framework now contains three four-tuples that play different roles at different levels of articulation. They are easy to confuse if not kept distinct, and we articulate their relation explicitly.

The 0DD four phases of Methodology 0: nothing, being, neither-being-nor-nothing, the negation of neither-being-nor-nothing. These four phases articulate negativa's self-interrogation at the deepest level of SAE methodology. They articulate the structure of articulation itself, in its most general form, and they are the source from which subsequent four-tuples in the SAE framework derive their structural form.

The Paper 2 four inevitabilities of $L_0$ mathematicality: comparison is unavoidable, comparison cannot be undirected, comparison cannot remain isolated, comparison cannot escape sustained questioning. These four inevitabilities articulate the conditions for mathematicality to emerge from qualitative articulation at $L_0$. They are not derived from the 0DD four phases as theorems from axioms, but they bear a structural resemblance to the 0DD phases: the first articulates what is comparable (something rather than nothing), the second articulates direction (orientation in being), the third articulates the relation between what is compared (the binding of differences in propagation), and the fourth articulates the sustained pressure beyond closure (negation of the apparent closure of comparison). The resemblance is not a derivation but a structural parallelism that reflects the common origin of the SAE four-tuples in the 0DD source.

The Paper 1 four steps of the Objective Exactification Schema: marked handle, additive path, multiplicative path with memory binding, closure with remainder. These four steps articulate the implementation of mathematicality within the complete exactification path. They are specific to that path: they apply in full force within it, in modified form within the partial exactification path, and not in their literal form within the persistent open path.

The three four-tuples are situated at three different levels of abstraction:

Four-tuple	Level	Function
0DD four phases	Existential (Methodology 0)	Structure of articulation itself
Paper 2 four inevitabilities	$L_0$ mathematicality	Conditions for mathematicality emergence
Paper 1 four steps	$L_1+$ objective exactification implementation	Path-specific schema for one path

The relation among them is a structure of articulative specialisation, not of derivation. The distinction is substantive:

Derivation: logical entailment; one four-tuple follows from another by necessary inference (as theorems from axioms).
Specialisation: contextual particularisation of general structure; one four-tuple inherits the structural shape of another in identification, not in derivation.

The 0DD four phases are the most general; the Paper 2 four inevitabilities specialise them to the level of mathematicality emergence; the Paper 1 four steps specialise them further to the level of complete-exactification-path implementation. Each four-tuple inherits structural shape from the more general level as identification, not as derivation. This specialisation relation is consistent with the SAE framework's identificatory schema stance (jointly maintained by Paper 1 § 3.5 and the present paper § 7).

Each four-tuple is articulated for its own level, and each is substantively independent in its own right; the structural resemblance among them reflects the specialisation relation, not derivation. The SAE framework articulates specialisation, not derivation, consistent with the identificatory schema stance maintained across the entire SAE mathematics series.

We articulate this relation explicitly because the framework has now developed to a point where the three four-tuples might be confused. The naming clarifies that they are distinct: "0DD four phases", "Paper 2 four inevitabilities", "Paper 1 four steps" are three separate labels for three separate articulations at three separate levels. The reader is not to identify them or to treat them as variations on a single underlying four-tuple. Each plays its specific role, and the framework's coherence depends on keeping the three distinct.

7. Paper 2's Substantive Defense Framework

7.1 Overview

Paper 1 substantively defended its Layer Articulation Schema with two devices: adequacy criteria (§ 3.5 of Paper 1) and failure modes (§ 3.6 of Paper 1). The adequacy criteria specified what conditions a mathematical articulation must satisfy to be governed by the schema. The failure modes specified what kinds of failure could occur and how the schema accommodates them. Together, the two devices made the schema substantively defensible: they specified what the schema claimed and what would falsify it.

The present paper requires a comparable substantive defense framework. The four inevitabilities of § 3 and the public re-entry criterion of § 3.8 articulate substantive conditions for the emergence of SAE mathematicality. Without a defense framework, these conditions might be read either as too strong (a universal definition of mathematics that excludes practices conventionally classified as mathematical) or as too weak (an articulation that admits virtually any structured comparative practice as mathematical, with no substantive content to the claim). Either reading would be a misreading, but the misreading is invited if the substantive defense is not made explicit.

This section provides that framework. It articulates five inevitability criteria that a qualitative dimension at $L_0$ must satisfy to enter SAE mathematicality. Each criterion corresponds to one of the conditions of § 3 (the four inevitabilities plus the public re-entry criterion), and each criterion specifies what would constitute satisfying or failing the corresponding condition. The five criteria together identify SAE mathematicality at the level of practice: a practice exhibits SAE mathematicality just in case it satisfies all five criteria.

The section then articulates five failure modes, each corresponding to the failure of one of the criteria. The failure modes specify what kinds of failure are possible and how the framework treats them. A practice that fails any of the five criteria fails to enter SAE mathematicality. The failure modes also distinguish, following Paper 1's example (§ 3.6 of Paper 1), between local mapping failure (the practice fails to satisfy the criteria as articulated, but the practice would or might satisfy them under a different articulation) and boundary identification (the practice is consistently outside the boundary of SAE mathematicality, and the failure is not a defect of the framework's articulation).

We emphasise that the five criteria are not offered as a universal definition of mathematics. They are the conditions for entry into SAE mathematicality, in the technical sense § 3.7 articulated. Other frameworks for mathematics may use different criteria, articulating mathematics in different ways. The five criteria are the SAE perspective's criteria, articulating what the SAE perspective identifies as mathematicality.

7.2 The five inevitability criteria

The five inevitability criteria are stated as follows.

Criterion 1: Comparison criterion. A qualitative dimension at $L_0$ admits comparative articulation that is more than mere differentiation. The articulation has comparative content: it places terms in relation to one another beyond simply registering their numerical distinctness. The relation has the form of comparison, in the sense of § 3.2.

A practice satisfies this criterion if it produces articulations that compare terms in this comparative sense. It fails the criterion if it produces only differentiations (assertions of distinctness without comparative content) or if it produces only monadic property attributions (assertions of properties without relational content).

Criterion 2: Direction criterion. The comparison articulated in the dimension has inherent orientation. The two terms of the comparison are not interchangeable; the comparison has a "more" side and a "less" side, even where the specific structural form of the more-less polarity (total order, partial order, multi-dimensional ordering, etc.) is left underdetermined.

A practice satisfies this criterion if its comparisons have this oriented structure. It fails the criterion if its comparisons are symmetric (treating the two terms as interchangeable) or if its articulations produce no consistent orientation across instances of comparison.

Criterion 3: Propagation criterion. The articulated comparisons exhibit propagation pressure: when two comparisons are in play, the relation between their terms across the comparisons arises as a question that demands articulation. The propagation is a pressure, not a guarantee; the pressure may be satisfied in fully transitive form (as in the complete exactification path), in partially transitive form (as in the partial exactification path), or in distributionally-structured form with local instabilities (as in the persistent open path). The criterion requires propagation pressure, not transitivity as such.

A practice satisfies this criterion if its comparisons exhibit such pressure. It fails the criterion if its comparisons remain stably confined to isolated pairs, with no pull toward extending the structure across the available terms.

Criterion 4: Sustained questioning criterion. The articulation generates sustained questioning beyond any specific closure. Local closures occur (specific judgements are made, specific structures are produced), but the articulation as a whole does not terminate with these closures. Each closure produces a remainder that demands further articulation, and the persistent pursuit of these remainders constitutes the development of the practice.

A practice satisfies this criterion if its articulation exhibits this sustained quality. It fails the criterion if its articulation closes definitively, with no remainder demanding further engagement, or if its articulation is satisfied with each judgement as a self-contained event with no implications for further articulation.

Criterion 5: Public re-entry criterion. The articulation admits re-entry by parties other than the original articulator. The structure of articulation is available for repetition, transmission, and engagement by other subjects. The criterion does not require full intersubjective agreement; it requires the weaker condition that the structure be accessible to others as a structure they can engage with, whether to accept, refine, contest, or extend.

A practice satisfies this criterion if its articulation has this public character. It fails the criterion if its articulation is purely private, with no possibility of others entering the structure, or if its articulation is so context-bound that re-entry by others is precluded by features inherent in the articulation itself.

On gradations of strength for Criterion 5: Public re-entry admits gradations of strength, each gradation satisfying the criterion but reflecting substantive differences among practices:

Formal-notation-based public re-entry (strong gradation): Re-entry through shared formal notation, shared definitions, and shared proof standards. This gradation is the canonical form of narrow mathematics: precise, cross-culturally universal, near-atemporal (symbols remain relatively stable across history). Any competent participant can re-enter by virtue of mastering the formal notation.

Canon-/tradition-based public re-entry (intermediate gradation): Re-entry through shared canonical works, shared style frames, shared interpretive traditions. This gradation is the canonical form of aesthetic mathematicality and other subjectivity-grounded dimensions: contextual, culturally specific, temporally evolving (traditions evolve across time). The re-entrant must enter the relevant canon / tradition / framework communities to access the structure.

Shared-practice-based public re-entry (weak gradation, qualifying): Re-entry through shared informal practice, shared case accumulation, shared deliberation. This gradation is the canonical form of certain informal articulative practices (for example, certain forms of legal reasoning, certain forms of clinical judgement): not dependent on formal notation or established canons, but maintaining public re-entry through communities of practice.

Each gradation satisfies Criterion 5 (and thereby enters SAE mathematicality), but the strength differences reflect substantive differences among practices. The gradations are not hierarchical (i.e., we do not claim that the formal-notation gradation is more "mathematical" or more "advanced" than the canon gradation); rather, they are substantively different forms of public re-entry, each substantively important within its own practice context.

The five criteria are stated as conditions on the practice, not as derivable consequences of the framework. They articulate what the SAE perspective identifies as the conditions for the emergence of mathematicality at $L_0$. A practice that satisfies all five enters SAE mathematicality; a practice that fails any of them does not. The criteria are jointly identified, not derived; their joint satisfaction constitutes the practice as exhibiting SAE mathematicality.

7.3 Five failure modes

Corresponding to each criterion is a failure mode that specifies what failure of that criterion looks like and how it is treated.

Failure mode 1: Comparison failure. A practice produces articulations that are only differentiations, registering distinctness without comparative content. Examples include certain forms of categorisation without further structure: a taxonomy that asserts "this thing is in category A; that thing is in category B" without articulating any relation between A and B that goes beyond their categorical distinctness. The practice does not enter SAE mathematicality, in the sense that the first criterion is not satisfied. Such practices may have substantive content—taxonomic discrimination is substantively important in many domains—but they articulate something other than mathematicality, on the present articulation.

Failure mode 2: Direction failure. A practice produces comparisons that are symmetric, treating the two terms as interchangeable. Examples include certain forms of equivalence-class articulation where the comparison is "A and B are in the same class" (a symmetric relation) without further orientation, or certain forms of mere structural similarity where the similarity does not carry orientational content. The second criterion is not satisfied. Such practices may articulate substantive structural relations, but the relations are not oriented in the sense the second criterion requires, and the practice does not enter SAE mathematicality.

Failure mode 3: Propagation failure. A practice produces comparisons that remain isolated, with no propagation pressure toward extending them across the available terms. Examples include certain one-off comparative judgements that do not engage with other comparisons in any structural way. The third criterion is not satisfied. Such practices may produce substantive individual judgements, but the judgements do not constitute a structured comparative practice in the sense the criterion requires.

Failure mode 4: Sustained questioning failure. A practice produces articulations that close definitively, with no remainder demanding further engagement. Examples include certain forms of definitional fiat: "by stipulation, A is defined to satisfy P", where the stipulation closes the articulation without remainder. The fourth criterion is not satisfied. Such practices may articulate definite structures, but the structures do not generate the sustained development that mathematicality requires.

Failure mode 5: Public re-entry failure. A practice produces articulations that are purely private, with no possibility of others entering the structure. Examples include certain forms of ineffable mystical experience or certain forms of context-bound private judgement where the context cannot be transmitted. The fifth criterion is not satisfied. Such practices may have substantive content for the individual subject, but they do not enter SAE mathematicality because the practice is not articulable as a structure that others can engage with.

In addition to these five primary failure modes, the framework recognises a sixth distinction that cross-cuts them, following Paper 1's example (§ 3.6 of Paper 1):

Local mapping failure (5a) versus boundary identification (5b).

A local mapping failure occurs when a specific articulation of a practice is judged to fail one of the criteria, but a different articulation of the same practice might satisfy the criteria. The failure is local to the articulation, not to the practice. In such cases, the framework's response is to seek a more adequate articulation, not to exclude the practice from SAE mathematicality. Local mapping failures are routine in the development of mathematical understanding: an initial articulation of a phenomenon may fail to satisfy the criteria, and a revised articulation may succeed.

A boundary identification occurs when a practice is consistently outside the boundary of SAE mathematicality, across multiple attempted articulations, and the failure is not a defect of any specific articulation but a feature of the practice itself. In such cases, the framework's response is to acknowledge the practice as outside SAE mathematicality, without claiming that the practice is therefore unimportant or substantively defective. Practices that are consistently outside SAE mathematicality may be substantively important within their own terms; the boundary identification simply locates them outside the specific category that SAE mathematicality articulates.

The distinction is essential for the framework's substantive content. Without it, every failure could be treated as either a substantive falsification of the framework (if all failures were boundary identifications) or as an opportunity for revised articulation (if all failures were local mapping failures). The first reading would make the framework brittle; the second reading would make the framework unfalsifiable. The distinction allows the framework to be substantively defensible: it acknowledges that some failures call for revised articulations of the practice (substantive flexibility) and that some failures call for acknowledging boundaries (substantive falsifiability).

7.3.6 The sixth failure: Forced Closure — cross-path misalignment failure

In addition to the five primary failure modes (each corresponding to the failure of a single criterion), the framework identifies a substantively distinct concealed failure mode, which we call Forced Closure.

Forced Closure failure occurs when an $L_0$ dimension that should be articulated along the persistent-open path (or partial exactification path) is forcibly pushed by external power, algorithm, or institution into the complete exactification path (or some narrower path), thereby reducing the substantive content to path-mismatched structure.

The distinguishing features:

It does not correspond to a single-criterion failure (the five criteria are nominally still satisfied)
The substantive failure is in cross-path misalignment: the practice is processed along the wrong closure path
The subject-constitutive content of the dimension is structurally erased

Concrete examples:

Social-media aesthetic algorithms: Extremely complex aesthetic experience is forcibly folded into a single scalar like "like count". Even when nominally (a) comparison exists (liked vs not liked), (b) direction exists (more likes = "better"), (c) propagation exists (ranking), (d) sustained questioning is articulable (new content invites new likes), (e) public re-entry exists (like counts are publicly visible), the substantive situation is that the aesthetic dimension has been forcibly collapsed from the persistent-open path into the complete exactification path. This is not a realisation of aesthetic mathematicality; it is the violent dimensional reduction of the aesthetic dimension (out-of-path exactification).

Social credit scoring: Complex ethical and social-relational judgments are folded into a single numerical score. Similar to the aesthetic-algorithm Forced Closure, the ethical / social-proximity dimensions are forcibly pushed from their typical persistent-open path into the complete exactification path.

Over-quantification of academic evaluation: The evaluation of academic work quality (which should be articulated along the persistent-open path, through peer review and shared scholarly tradition) is folded into citation counts, impact factors, and similar quantitative metrics. The same Forced Closure pattern.

Forced Closure is not a failure of mathematicality; it is a cognitive and ethical catastrophe produced by the misapplication of mathematicality.

To articulate the relation to Anchor C (subjectivity not quantifiable without cancellation): Forced Closure is the institutional-level form of Anchor C violation. The subject is reduced to a sensor for an algorithm or institution; the subject's constitutive role in evaluation is structurally erased. The SAE viewpoint not only identifies mathematicality but also diagnoses this kind of "out-of-path exactification" and its cognitive consequences.

To articulate the relation to the 5a/5b distinction: Forced Closure is neither 5a (it is not inadequate articulation; it is deliberate cross-path misplacement) nor 5b (the dimension could have entered mathematicality successfully along the correct path). It is a sixth distinction type: the practice is processed along the wrong path, and the framework's response is to articulate what path is correct for that dimension and what cognitive consequences follow from the current path misalignment.

The SAE framework's identification of Forced Closure as a substantively important failure mode has substantive implications for social and institutional critique. The SAE mathematicality framework is not only descriptive (identifying forms of mathematicality) but also has normative implications (diagnosing the cognitive catastrophes produced by path misalignment). This normative dimension is a substantive contribution of the SAE viewpoint, consistent with the Self-as-an-End philosophical commitment (the subject as constitutive, not reducible to a sensor for objective measurement).

7.4 Aesthetic mathematicality satisfies the five criteria

A worked application of the framework: we now check whether aesthetic articulation satisfies the five inevitability criteria. The check is substantive, not formal: it considers what aesthetic articulation actually exhibits as a practice, against the substantive content of each criterion.

Criterion 1 (Comparison criterion). Aesthetic articulation produces comparative judgements with substantive comparative content. To articulate a work as "more beautiful than" another, or "more sublime", or "more compelling", is to compare them in the relational sense the criterion requires. The articulation is not mere categorisation (this work is beautiful, that work is not, with no further structure); it is structured comparison. The criterion is satisfied.

Criterion 2 (Direction criterion). The aesthetic comparisons articulated are oriented. "More beautiful than" is not interchangeable with "less beautiful than"; the comparison has a direction, and the direction carries the content of the judgement. The criterion does not require that aesthetic orderings be total, or that they be stable across subjects and contexts; it requires that the comparisons have orientational structure. The criterion is satisfied.

Criterion 3 (Propagation criterion). Aesthetic articulation exhibits propagation pressure. When a subject articulates A as more beautiful than B and B as more beautiful than C, the question of how A and C compare arises and demands articulation. The propagation may not produce a transitive structure—the subject may articulate C as more compelling than A on some specific axis—but the question arises, and the articulation engages with it. The criterion requires propagation pressure, not transitivity as a structural property, and aesthetic articulation produces the pressure. The criterion is satisfied.

Criterion 4 (Sustained questioning criterion). The history of aesthetic articulation is a sustained inquiry without definitive closure. The questions of beauty, taste, judgement, and aesthetic experience have been articulated, contested, refined, and rearticulated across cultures and periods, and no final closure of the questions is in prospect. Each substantive articulation generates remainders that subsequent articulation engages with. The criterion is satisfied paradigmatically.

Criterion 5 (Public re-entry criterion). Aesthetic articulation operates within shared canons, shared style frames, and shared interpretive traditions. The canonical works, established style categories, and interpretive practices function as the public structure within which aesthetic articulations are made and re-engaged. Even where substantive judgements diverge—indeed, especially where they diverge—the divergence is articulated within the public structure that makes the divergence intelligible as a divergence rather than as mutually opaque private events. The criterion is satisfied, with the substantive qualification that aesthetic public re-entry is structured by canons and traditions rather than by formal notation, but is no less public for that.

All five criteria are satisfied by aesthetic articulation. Aesthetic articulation therefore enters SAE mathematicality, on the substantive defense framework here articulated. This is not a stipulation; it is the result of substantive checking against the criteria.

A counter-consideration to address: aesthetic articulation does not satisfy several criteria that are sometimes taken as definitive of mathematics in conventional senses. It does not admit formal manipulation in the algebraic or analytic senses. It does not produce determinate truth values for its comparative articulations. It does not exhibit the cumulative knowledge structure that characterises the accumulated theorems of conventional mathematics. These observations are correct, but they are observations about conventional narrow mathematics, not about SAE mathematicality. The framework articulates SAE mathematicality as broader than narrow mathematics, and aesthetic articulation enters the former without entering the latter. This is consistent with the three-tier distinction of § 3.7: mathematicality (the general structural shape, identified by the five criteria), narrow mathematics (the institutionalised practice that occupies the complete exactification path), and broad SAE mathematicality (the union of all closure paths satisfying the criteria, including aesthetic articulation).

7.5 The defense framework's substantive function

The defense framework articulated in this section serves three substantive functions, and we conclude by making them explicit.

First, it articulates what the schema's claims are. Paper 2 claims, on the framework here articulated, that aesthetic articulation, partial-exactification structures, and certain other dimensions enter SAE mathematicality alongside the complete-exactification mathematics that Paper 1 articulated. The claim is substantive: it can be checked, criterion by criterion, against the practices in question. The defense framework articulates how the check is to be conducted.

Second, it articulates what would falsify the schema's claims. If a practice is asserted to enter SAE mathematicality but is shown to fail one of the five criteria, the assertion is falsified for that practice. The schema is not unfalsifiable: it makes specific claims that can be defeated by specific evidence. The failure modes specify what such evidence looks like.

Third, it articulates the framework's relation to its own potential failures. The 5a versus 5b distinction—local mapping failure versus boundary identification—prevents the framework from becoming either brittle (every failure is fatal) or unfalsifiable (every failure is an opportunity for revised articulation). The distinction is substantive: it requires that boundary identifications be justified by reference to consistent failure across multiple articulations of the practice, not asserted ad hoc to dismiss inconvenient failures.

Substantive parallel with Paper 1 § 3.6's failure modes: Paper 2's 5a/5b distinction is substantively parallel to Paper 1 § 3.6's articulation of failure modes. Paper 1 distinguishes schema-applicability local failure (a specific articulation is inadequate, but revision can succeed) from schema-boundary identification (the practice crosses the schema's boundary; it is not the schema's failure). Paper 2's 5a/5b distinction extends this distinction to the $L_0$ mathematicality level. Both jointly maintain the falsifiability and substantive flexibility discipline of the SAE mathematics series, articulating the methodological coherence of the SAE framework across Paper 1 and Paper 2.

The framework as a whole is articulated in the spirit of Paper 1's epistemological discipline (§ 3.8 of Paper 1): the schema is one productive way of articulating mathematicality, neither universal nor unique, and the framework's substantive content depends on its capacity to be checked, contested, and refined through engagement with specific practices. Paper 2's substantive defense framework continues Paper 1's discipline at the $L_0$ level of mathematicality, providing the substantive checks that the level requires.

8. The Inevitability and Non-Uniqueness of Mathematicality

8.1 Overview

The four inevitabilities and the public re-entry criterion (§§ 3, 3.8, 7) articulate that mathematicality is, in a substantive sense, inevitable: any qualitative articulation that satisfies the five conditions enters SAE mathematicality. The three closure paths and the multiple $L_0$ dimensions (§§ 4, 5) articulate that mathematicality is, in another substantive sense, non-unique: the specific form mathematicality takes depends on which dimension is articulated and which closure path is taken. The two articulations are not in tension; they characterise different aspects of the same substantive situation.

This section makes the two articulations explicit and articulates the relationship between them. It also notes a third level of non-uniqueness that follows from the first two: the inexhaustibility of paths beyond those the present paper articulates, which is the third level of inexhaustibility that Paper 2 contributes to the SAE perspective's epistemological commitments.

8.2 Mathematicality is inevitable at the level of the five conditions

The first substantive claim is not that "any qualitative articulation that engages with comparison is mathematicality". Rather: once an $L_0$ qualitative dimension enters a publicly re-entrant, structured comparison practice, it cannot avoid encountering the four inevitabilities (comparison, direction, propagation, sustained questioning); the public re-entry criterion then determines whether that practice crosses from private experience into SAE mathematicality. This is the inevitability of mathematicality.

To articulate this precisely: the inevitability operates on $L_0$ dimensions that have entered structured comparative practice. Private preferences, one-off local judgments, or purely private experiences need not enter this structured practice — they are excluded by the failure modes of § 7.3. When an $L_0$ dimension crosses the dual threshold of structured comparison plus public re-entry, the four inevitabilities are unavoidable; but crossing that dual threshold is not itself a universal occurrence.

This can be expressed:

$$

L_0 \text{ dimension} + \text{structured comparison} + \text{public re-entry} \Longrightarrow \text{four inevitabilities unavoidable.}

$$

And not:

$$

\text{any comparison} \Longrightarrow \text{mathematicality.}

$$

The substantive content of the inevitability is the universality of the five conditions (once across the dual threshold) across cultures, traditions, and substantive domains of articulation. Any practice that crosses the threshold engages with comparison (the first inevitability), with direction (the second), with propagation (the third), and with sustained questioning (the fourth); the public re-entry criterion (the fifth) is the threshold itself, examined as a criterion in § 7. This universality is not peculiar to one tradition or one historical period; it is a feature of structured comparison practice as such, applying wherever structured comparison practice occurs.

This universality is the substantive sense in which mathematicality is unavoidable within structured comparison practice. A threshold-crossing practice cannot avoid the four inevitabilities; the inevitabilities operate whether the articulator wishes them to or not. The first three inevitabilities are particularly visible: comparison is foundational, direction follows from comparison's content, propagation follows from direction's productivity. The fourth inevitability—sustained questioning—is more substantive philosophically, but it too is unavoidable in any practice that takes the propagation pressure seriously, and refusing to take it seriously is not an option once the first three inevitabilities are in play.

A clarification: the inevitability is at the level of the conditions, not at the level of the specific mathematical structures that the conditions can be realised through. A practice that satisfies the five conditions will produce some mathematics-like articulation; the practice cannot avoid having such articulation. But the specific form of the articulation—whether the practice develops complete exactification, partial exactification, persistent open articulation, or some combination—is non-unique. Different cultures and different practices develop different forms, and the universality of the conditions does not predict any specific form.

This claim is substantive because it gives content to the notion that mathematicality is universal without committing to the stronger and false claim that the substantive content of mathematics is universal. The five conditions, within threshold-crossing practices, are universal; the specific mathematics is not. Both claims are needed for the present paper's overall articulation, and they are jointly compatible only because the inevitability operates at the level of the conditions and the non-uniqueness operates at the level of the realisations.

8.3 Non-uniqueness at the level of $L_0$ dimensions

The second substantive claim is that the specific mathematical form depends on which $L_0$ dimension is articulated, and different dimensions produce substantively different forms of mathematicality.

§ 5 articulated several dimensions: quantity, temporal rhythm, spatial size, intensity, aesthetic value, ethical value, social proximity, truth. Each dimension admits one or more closure paths, and each produces a substantively different mathematics-like articulation when articulated along its canonical path. The quantity dimension produces the conventional arithmetic mathematics that Paper 1 articulated. The spatial size dimension produces order-theoretic mathematics with the partial exactification path. The aesthetic dimension produces distribution-grounded mathematics with the persistent open path. These are substantively different mathematical forms, not variants of a single underlying mathematics.

This dimensional non-uniqueness is a substantive claim about the structure of mathematicality. It articulates that mathematicality is not a single domain that takes different surface forms in different applications; it is a substantively multi-domain situation, with different domains producing substantively different content.

The non-uniqueness is constrained by the universality of the five conditions: each domain's mathematicality satisfies the five conditions, and the conditions provide the structural backbone that allows the domains to be recognised as forms of mathematicality. But within the constraint of the five conditions, the substantive variation is real and is not a deficiency of any specific domain. Each domain's mathematicality is what it is, with its own substantive content, its own characteristic structures, and its own development trajectory.

A consequence of this articulation: the substantive content of mathematicality cannot be exhibited in any single dimension. To know mathematicality fully, in the present paper's articulation, one would need to engage substantively with the multiple dimensions and the multiple closure paths. No single dimension—not even the canonical quantity dimension of conventional mathematics—is the home of mathematicality in a way that makes the other dimensions secondary.

This is a substantive departure from articulations of mathematics that take the quantitative as primary and other articulations as derivative or secondary. On those articulations, "real mathematics" is quantitative, and aesthetic mathematics or ethical mathematics, where they are recognised at all, are derivative analogues. The present paper articulates that the dimensions are coordinate at the level of substantive mathematical content. The quantitative dimension is not more mathematical than the aesthetic; it is differently mathematical, in a way that the present paper articulates substantively.

8.4 Non-uniqueness at the level of closure paths

The third substantive claim is that the specific mathematical form also depends on which closure path is taken, and different paths produce substantively different forms of mathematicality even within the same dimension.

§ 4 articulated three closure paths: complete exactification, partial exactification, persistent open. The paths differ in what marked handle is introduced (unit, reference, prior), in what structural closure is produced (full quantitative structure, order-theoretic structure, distribution space), and in what grammar of articulation is canonical (closure-equation, ordering, probability-distribution).

A single dimension may admit multiple paths. The spatial size dimension canonically articulates along the partial exactification path (with the king's-fist reference), but it can extend to the complete exactification path (with units like the meter) when the substantive task requires quantitative precision. The choice between the two is a substantive choice with substantive consequences: the partial exactification path preserves comparative content without imposing quantitative structure; the complete exactification path imposes quantitative structure at the cost of substantive simplification of the dimension's content (in some applications).

This path non-uniqueness within a dimension is substantive. The choice of path is not arbitrary; it has consequences for what mathematical operations are available, what kinds of comparison are articulated, what closure is produced. Different applications may require different paths, and the substantive task determines which path is appropriate.

8.5 Cultural variation in path emphasis

A claim about historical and cultural variation: different traditions have emphasised different dimensions and different closure paths in their substantive mathematical development.

We articulate this with care because the claim invites a misreading we explicitly reject. The misreading would be to articulate the cultural variation as essentialist civilisational typology: "Western mathematics is complete exactification, Eastern mathematics is persistent open", and similar. Such essentialist categorisations are substantively wrong: substantial bodies of complete exactification mathematics have been developed in non-Western traditions (the Nine Chapters on the Mathematical Art in classical Chinese mathematics, the algebra of the Islamic golden age, the mathematics of medieval Indian astronomers), and substantial bodies of persistent open articulation have been developed in Western traditions (Pythagorean aesthetic mathematics, Euclidean proportion theory, contemporary aesthetic philosophy).

What is substantive about the cultural variation is something weaker but still real: different traditions have, at different periods, emphasised different dimensions and paths in their substantive intellectual production. The emphases are partial, trajectory-dependent, contested within each tradition, and not essential features of any tradition. Different cultures have made different choices about which paths to develop intensively, but no culture has confined itself entirely to one path or one dimension.

This nuanced articulation is essential because the substantive content of the present paper depends on path multiplicity being a feature of qualitative articulation as such, not on any specific cultural typology. The four inevitabilities are universal; the path multiplicity is universal; the cultural variation is in which paths are emphasised at which historical moments, not in which paths are available.

Detailed historical illustration of the cultural variation is reserved for subsequent work. The substantive point for the main text is that path multiplicity is real, that cultural variation in path emphasis is real, and that essentialist civilisational typology is rejected.

8.6 Path inexhaustibility: the third level of inexhaustibility

Paper 1 articulated double inexhaustibility in its § 8: mathematical content is inexhaustible at the object level, and articulation frameworks for mathematics are inexhaustible at the meta level. The present paper extends this to a third level: closure paths are inexhaustible.

The substantive content of path inexhaustibility is that the three closure paths articulated in § 4 do not exhaust the possible closure paths for SAE mathematicality. Other paths may exist and may be articulated in future work. The three paths are the most clearly identifiable in current mathematical practice; they are not a closed taxonomy.

Possible additional paths that we mention without developing: paths that introduce limited or contested references; paths that combine exactification with persistent open features in specific ways; paths that articulate transitions between exactification and openness; paths grounded in fundamentally different marked-handle types that the present articulation has not envisioned. Each of these is a candidate for substantive development; none is articulated in the present paper.

The path inexhaustibility is substantive because it acknowledges that the closure-path typology of § 4 is itself open to extension. The three paths are not a final taxonomy of mathematicality; they are a productive beginning. Future SAE work may identify additional paths, articulate them with substantive content, and extend the schema. The present paper does not close this development; it opens it.

The third level of inexhaustibility, together with Paper 1's object-level and meta-level inexhaustibility, articulates the substantive content of Anchor B (remainders cannot be eliminated). The remainder operates at three levels:

Object level (Paper 1): Mathematical content is inexhaustible. New mathematical structures, new theorems, new connections continue to be articulated indefinitely.

Meta level (Paper 1): Articulation frameworks for mathematics are inexhaustible. Different frameworks (formalism, intuitionism, structuralism, SAE, others) articulate mathematics differently, and new frameworks continue to emerge.

Path level (Paper 2): Closure paths within the SAE framework are inexhaustible. Different paths produce different forms of mathematicality, and new paths may be articulated as the framework develops.

The three levels are not in tension; they characterise three aspects of the same situation. Mathematicality is inexhaustible in its content, in its framings, and in its paths. The substantive content of mathematics, on the SAE articulation, is the sustained engagement with this triple inexhaustibility, not a closure of it.

8.7 The substantive shape of the inevitability/non-uniqueness articulation

We close this section by summarising the substantive shape of the articulation.

Mathematicality is inevitable in the substantive sense that the five conditions—comparison, direction, propagation, sustained questioning, public re-entry—are universal features of qualitative articulation. Any culture, any tradition, any practice that engages with qualitative comparison encounters the five conditions. The substantive universality is the substrate from which mathematicality arises wherever qualitative articulation arises.

Mathematicality is non-unique in three substantive senses. First, multiple $L_0$ dimensions are available (§ 8.3): different dimensions produce substantively different mathematical forms. Second, multiple closure paths are available (§ 8.4): different paths produce substantively different forms even within the same dimension. Third, the schema of paths itself is open (§ 8.6): the three paths articulated in § 4 do not exhaust the possible paths.

The two claims—inevitability at the level of conditions, non-uniqueness at the level of realisations—are jointly the substantive structure of mathematicality on the SAE articulation. Together they articulate that mathematics is universal (the universality of the conditions) without being unique (the multiplicity of realisations). Both halves of the articulation are essential; collapsing the articulation to either pole alone would misread the substantive content.

This dual structure also articulates the substantive content of the three epistemological anchors of § 1.5. Anchor A (SAE is one thinking framework, not totalising synthesis) is articulated by the substantive openness of the framework to other articulations and to other paths beyond those the present paper develops. Anchor B (remainders cannot be eliminated) is articulated by the triple inexhaustibility of object, meta, and path levels. Anchor C (subjectivity cannot be quantified without being abolished) is articulated by the substantive presence of the persistent open path as a path that does not introduce quantitative structure, with aesthetic mathematicality as its substantive realisation. The three anchors are not external commitments imposed on the substantive content; they are internal to the substantive content's structure and are articulated through that structure.

9. Subjectivity-Grounded Mathematicality and Its Position in SAE Philosophy

9.1 Overview

The present paper articulates that mathematicality is realised across multiple closure paths, with the persistent open path being the canonical home of subjectivity-grounded mathematicality (paradigmatically aesthetic mathematicality, with ethical articulation as a parallel case). § 5.5 articulated the substantive content of aesthetic mathematicality at the level of possibility, direction, and boundary; § 7.4 articulated the substantive defense framework for aesthetic mathematicality's entry into SAE mathematicality.

This section articulates the position of subjectivity-grounded mathematicality in SAE philosophy. The articulation involves a substantive distinction that is easy to misread: the distinction between (a) the substantive mathematical content of the three closure paths, where the paths are complementary and not hierarchical, and (b) the SAE philosophical perspective on mathematicality, where subjectivity-grounded mathematicality has a particular focal significance. The two articulations are not in tension; they characterise different aspects of the situation.

We articulate this distinction explicitly because it has been a focal concern in the development of the paper's articulation, and the substantive content depends on the distinction being kept clear.

9.2 The Self-as-an-End commitment

The SAE perspective takes subjectivity as central. The name—Self-as-an-End—articulates the substantive commitment: the subject is not reducible to a means, not reducible to a function of impersonal structures, not reducible to an extension of objective articulation. The subject is constitutive of articulation in a way that cannot be eliminated by treating the subject as a recording or reporting device for objective content.

This commitment shapes how the SAE perspective engages with various articulative domains. In the SAE methodology series, the commitment articulates the inadequacy of articulations that minimise or eliminate the subject's substantive role. In the SAE mathematics series, the commitment articulates that the substantive content of mathematics, while it includes subject-minimising articulations like the complete exactification path's conventional mathematics, is not exhausted by such articulations. The subject's constitutive role in articulation must be available for substantive treatment within the framework, not relegated to an external or derivative position.

The present paper's articulation of subjectivity-grounded mathematicality—specifically, of aesthetic mathematicality with its persistent open path and distribution-grounded articulation—is the substantive content of the SAE commitment within the SAE mathematics series. The articulation provides the framework for substantive engagement with subject-constitutive articulation within mathematicality, in a way that does not abolish the subject's constitutive role through quantification (Anchor C) and does not absorb subject-grounded articulation into subject-minimising articulation as a secondary form.

9.3 Objective versus subjective mathematicality

We articulate a distinction between two broad categories of mathematicality that emerges from the path typology of § 4.

Objective mathematicality is the mathematicality of the complete exactification path and (to a lesser degree) the partial exactification path. In these paths, the subject's role is minimised: the substantive content of the mathematics is intersubjectively stable in a way that allows the subject to be treated as a recording or reporting device for the content. The arithmetic of two units plus three units equals five units is the same regardless of which subject performs the calculation; the ordering of items by size relative to a fixed reference is the same regardless of which subject performs the comparison. Subjectivity is present in objective mathematicality (someone has to do the calculation, someone has to make the comparison), but the substantive content is structured so that subjectivity does not affect the content.

Subjective mathematicality is the mathematicality of the persistent open path with subject-grounded dimensions like the aesthetic and ethical. In this path, the subject's role is maximal: the substantive content of the mathematics is constitutively engaged by the subject's evaluative response, and the response cannot be separated from the content without abolishing the substantive engagement. The aesthetic judgement that work A is more compelling than work B is not a record of an objective property that the subject is reading off; it is the subject's constitutive engagement that produces the substantive content of the comparison.

The two categories are not at the same level in every respect. Within the substantive mathematical content, they are complementary and not hierarchical: objective mathematicality produces forms of mathematical structure that subjective mathematicality does not produce, and subjective mathematicality produces forms of mathematical structure that objective mathematicality does not produce. Neither category exhausts the substantive content of mathematics; both are essential to the full landscape of mathematicality.

But within the SAE philosophical perspective, subjective mathematicality has a particular focal significance, which we articulate in § 9.4.

9.4 The focal significance of subjective mathematicality for SAE

The SAE philosophical perspective takes subjectivity as central. Where mathematicality articulates substantive content in which subjectivity is constitutive, the articulation is the substantive home of the SAE perspective's focal interest. This is not because subjective mathematicality is more developed, more sophisticated, or substantively superior to objective mathematicality; it is because subjective mathematicality is the form of mathematicality in which the substantive distinctiveness of the SAE perspective is most fully expressed.

To articulate this with care: the SAE perspective contributes a substantive philosophical viewpoint to mathematicality. The substantive content of the contribution is most distinctively expressed where the SAE perspective's commitment to subjectivity engages with mathematical material in which subjectivity is constitutive. The aesthetic dimension is the paradigmatic case: aesthetic articulation is the qualitative articulation in which subjectivity is most fully constitutive, and aesthetic mathematicality is consequently the mathematicality in which the SAE perspective's substantive content is most fully expressed.

This is a claim about the SAE perspective's focal interest, not a claim about which mathematicality is more important in substantive mathematical content. The two are different aspects.

A useful comparison: a category-theoretic philosopher of mathematics may have particular focal interest in articulations of mathematics that exhibit morphism structures with categorical content, and may articulate the substantive content of category-theoretic foundations as the most distinctive expression of the category-theoretic perspective. This focal interest does not entail that category-theoretic articulations are more substantively important in mathematical content than non-categorical articulations; it characterises where the category-theoretic perspective is most distinctively at home. The same kind of articulation applies to the SAE perspective and its focal interest in subjective mathematicality.

The substantive content of this focal significance is articulated through the path typology and the specific treatment of aesthetic mathematicality in § 5.5. The present paper articulates aesthetic mathematicality with substantive depth at the level of possibility, direction, and boundary; it does not articulate the detailed structure of aesthetic mathematicality, which is reserved for subsequent specialised papers in the aesthetic mathematics series. The reservation reflects both the substantive scope of the present paper (a foundational paper, not a specialised paper on a single dimension) and the substantive importance of the dimension (which requires detailed treatment in its own dedicated series).

9.5 No hierarchy in substantive mathematical content

We articulate explicitly the substantive claim that the focal significance of subjective mathematicality for SAE does not entail a hierarchy in substantive mathematical content.

Within substantive mathematical content, the three closure paths are complementary. The complete exactification path produces forms of mathematical structure that the partial exactification and persistent open paths do not produce: full arithmetic, complete quantitative comparison, the layered architecture Paper 1 articulated. The partial exactification path produces forms that the other paths do not produce: order-theoretic structures, relational mathematics without arithmetic. The persistent open path produces forms that the other paths do not produce: distribution-grounded articulations, subject-constitutive comparisons. None of these forms is more or less mathematics than the others. Each is mathematics in the substantive content sense, with substantive content distinctive to its path.

We articulate two specific corollaries of this no-hierarchy claim.

First, conventional mathematics is not displaced or diminished by aesthetic mathematicality. The complete exactification path's conventional mathematics—the arithmetic, analysis, algebra, geometry, topology, logic, and their many descendants that constitute conventional mathematical practice—is substantively complete in its own path. Paper 1's articulation of its architecture stands as a substantive achievement. The present paper's elevation of aesthetic mathematicality does not retract or diminish this; the elevation positions aesthetic mathematicality as a parallel form, not as a supersession of conventional mathematics.

Second, aesthetic mathematicality is not the "true" mathematics with conventional mathematics as a degraded approximation. A reading of the present paper that takes aesthetic mathematicality as more fundamental, with conventional mathematics as a derivative or degraded form that fails to capture the substantive content of mathematicality, is not what the paper articulates. The paper articulates aesthetic mathematicality as a substantively distinct form of mathematicality, with its own substantive content, alongside conventional mathematics with its own substantive content. The forms are complementary in substantive content, not in a hierarchy.

The no-hierarchy claim at the level of substantive mathematical content is consistent with the focal significance claim at the level of SAE philosophical perspective. The two operate at different levels and are jointly compatible.

9.6 "Home territory" and its careful articulation

The phrase "home territory" has been used in the development of this paper to articulate the position of aesthetic mathematicality within SAE philosophy. We articulate the phrase with care.

"Home territory" does not mean "the most important mathematics", "the highest mathematics", "the most developed mathematics", or "the most fundamental mathematics". The phrase does not articulate a hierarchy of substantive importance among mathematical forms. As articulated in § 9.5, no such hierarchy is claimed.

"Home territory" does mean "the territory of mathematicality in which the SAE perspective's substantive content is most fully expressed". The substantive content of the SAE perspective—the commitment to subjectivity as constitutive of articulation, the commitment to remainders as the engine of development, the commitment to the framework as one among many—is most fully expressed in the articulation of mathematicality where these commitments are most fully engaged. Aesthetic mathematicality, with its persistent open path and subject-constitutive content, is the home territory in this specific sense.

The phrase serves a substantive purpose. It articulates that the SAE perspective contributes substantive content to the philosophy of mathematics, and that the contribution is most distinctively expressed in a specific territory. Without articulating the focal significance, the present paper might be read as articulating the multi-path framework as a neutral cataloguing of mathematical forms with no substantive philosophical content. The focal significance articulates that the SAE perspective is a substantive philosophical contribution, even though the contribution is one framework among many.

The phrase is also a useful corrective to a possible misreading of the no-hierarchy claim. The no-hierarchy claim at the level of substantive mathematical content might be read as "all paths are equally interesting from the SAE perspective". This would be substantively incorrect. The SAE perspective has substantive content; the substantive content is most fully expressed in a specific path; and the focal significance of that path is a real feature of the SAE perspective, not a violation of the no-hierarchy at the substantive content level. Keeping the two levels distinct allows both claims to be made without internal tension.

9.7 The articulation level for aesthetic mathematicality in Paper 2

We close this section by articulating the substantive content of the present paper's treatment of aesthetic mathematicality in summary form.

The present paper articulates aesthetic mathematicality at four substantive levels.

Level 1: Possibility proof. The present paper articulates that aesthetic articulation enters SAE mathematicality, in the substantive sense of satisfying the five inevitability criteria of § 7. § 7.4 articulates the substantive defense of this claim, criterion by criterion. The possibility proof is substantive: it shows that aesthetic mathematicality is a real category, not a stipulation.

Level 2: Direction. The present paper articulates that aesthetic mathematicality's substantive direction of development is probability-distribution grammar—articulation of the distributional structure of aesthetic comparison through conditioning, marginalisation, and comparison of distributions. The direction is articulated explicitly to exclude alternative directions (notably the quantification of aesthetic content) that would abolish the subject's constitutive role.

Level 3: Boundary. The present paper articulates three boundary statements (§ 5.5.5) that distinguish what enters aesthetic mathematicality from what does not. Not all aesthetic activity is aesthetic mathematicality; only structured, publicly re-enterable, comparative aesthetic articulation enters. The boundary is substantive and is grounded in the five inevitability criteria.

Level 4: Reservation of detailed structure. The present paper does not articulate the detailed mathematical structure of aesthetic mathematicality. Such articulation would unbalance the paper's overall scope and would require substantive engagement with specific intellectual traditions that is reserved for the subsequent aesthetic mathematics series.

These four levels together articulate the present paper's substantive contribution to aesthetic mathematicality. The contribution is foundational rather than detailed: it provides the framework within which detailed articulation can proceed in subsequent specialised papers. The substantive content of the present paper is the framework; the substantive content of subsequent papers will be the detailed articulation within the framework.

The framework's substantive content is substantial, even at the foundational level. It articulates that aesthetic mathematicality is a real category of SAE mathematicality (the possibility proof), that its substantive direction is probability-distribution grammar (the direction), and that it has specific structural boundaries that distinguish it from broader aesthetic activity (the boundary). With these foundational elements in place, the SAE aesthetic mathematics series can develop the detailed structure on a substantively articulated foundation.

10. The Substantive Relation Between Paper 1 and Paper 2

10.1 Overview

The present paper extends Paper 1 in substantive ways while preserving Paper 1's published content. This section articulates the relation in summary form, drawing together the substantive claims about the relation that have been made across the preceding sections.

The articulation is summary because the substantive content has been articulated in the preceding sections: § 1.4 articulated the broad relation, § 6 articulated the renaming of Paper 1's schema and the upper-level $L_0$ Mathematicality Schema, § 8 articulated the relation between Paper 1's double inexhaustibility and the third level of path inexhaustibility that the present paper contributes. The present section consolidates these articulations for ease of reference.

10.2 Paper 1's substantive position

Paper 1 articulated the architecture of SAE mathematics across the layers $L_1$ through $L_5$, with the Layer Articulation Schema providing the canonical four-step pattern at each layer. On the multi-path framework of the present paper, Paper 1's substantive content is recognised as the articulation of the complete exactification path's architecture.

This recognition is substantive in two ways. First, it positions Paper 1's content within a broader landscape that the present paper articulates. The complete exactification path is one of three closure paths, and the layers $L_1$ through $L_5$ that Paper 1 articulated are all layers of this path. Paper 1's articulation is path-specific, but it is substantively complete for its path. Second, it preserves the substantive content of Paper 1. The articulation of $L_1$'s arithmetic, $L_2$'s analysis, $L_3$'s complex analysis, $L_4$'s measure-theoretic foundations, and $L_5$'s categorical articulations stand as substantive achievements within the complete exactification path. The present paper does not modify these articulations; it positions them.

The renaming of Paper 1's schema as the Objective Exactification Schema (§ 6.5) is the explicit marker of this path-specific positioning. The renaming articulates that the schema is the canonical schema of the complete exactification path, not a universal schema across all of mathematicality. Within the path, the schema applies in full force, exactly as Paper 1 articulated it.

10.3 Paper 2 as substantive extension, not revision

The present paper extends Paper 1 by articulating substantive content that Paper 1 did not develop, without modifying Paper 1's substantive content.

The substantive extensions are:

$L_0$ as substantive foundation, not merely thought-experimental precursor. Paper 1 treated $L_0$ briefly as a starting point for the move into $L_1$. The present paper articulates $L_0$ as the substantive foundation of all closure paths, with the four inevitabilities and the public re-entry criterion as the conditions for the emergence of mathematicality.

Multiple closure paths, not just the complete exactification path. Paper 1 articulated the complete exactification path implicitly (the path within which $L_1$ through $L_5$ are layers). The present paper articulates two additional paths (partial exactification, persistent open) and the substantive differences among them.

Three grammars of mathematical articulation, with probability-distribution grammar as a grammar in its own right. Paper 1 distinguished closure-equation grammar and probability-distribution grammar in its § 7, positioning the latter as taking over where the former fails. The present paper articulates ordering grammar as a third grammar (the grammar of the partial exactification path) and articulates probability-distribution grammar as a grammar capable of articulating subject-grounded mathematicality directly, not merely as a fallback when closure-equation grammar fails.

$L_0$ Mathematicality Schema as upper-level schema. Paper 1's Layer Articulation Schema is positioned as one realisation of an upper-level $L_0$ Mathematicality Schema that articulates mathematicality at $L_0$ across closure paths.

Substantive defense framework parallel to Paper 1's § 3.5 and § 3.6. The present paper's § 7 provides five inevitability criteria, five failure modes, and the 5a/5b distinction, in parallel structure to Paper 1's adequacy criteria and failure modes.

Path inexhaustibility as the third level of inexhaustibility. Paper 1's double inexhaustibility (object level and meta level) is extended to a third level: closure paths are inexhaustible.

These extensions are substantive contributions of the present paper. They are not corrections of Paper 1; Paper 1 did not articulate them, and Paper 1 did not need to articulate them for the substantive purposes Paper 1 served. The present paper articulates them as substantive extensions of the SAE mathematics series, building on Paper 1's foundation rather than modifying it.

10.4 Paper 1 is not retracted, modified, or superseded

We articulate explicitly that Paper 1 is not retracted, modified, or superseded by the present paper.

Paper 1 is not retracted. The published content of Paper 1 (DOI: 10.5281/zenodo.20153791) stands as published. No content of Paper 1 is withdrawn. The present paper does not claim that any specific statement in Paper 1 is incorrect; the substantive content of Paper 1 is preserved.

Paper 1 is not modified. The text of Paper 1 is not changed. The renaming of its Layer Articulation Schema as the Objective Exactification Schema is articulated in the present paper, not in a revised version of Paper 1. Readers of Paper 1 will see the original schema name; readers of Paper 2 will see the renaming as an articulation made in the present paper.

Paper 1 is not superseded. The substantive content of Paper 1—the articulation of the architecture of the complete exactification path—is not replaced by anything in the present paper. The present paper articulates a broader landscape and a substantive foundation, but it does not articulate the substantive architecture of the complete exactification path. Paper 1 remains the substantive home of that articulation.

The relation is one of complementary extension: the two papers together articulate the SAE foundation for the series's subsequent work, with Paper 1 providing the architecture of the dominant path and Paper 2 providing the $L_0$ foundation and the multi-path framework.

10.5 Implications for the SAE mathematics series trajectory

The articulation of the relation between Paper 1 and Paper 2 has implications for the trajectory of the SAE mathematics series, which we note briefly. Detailed publication trajectory is reserved for subsequent work.

Subsequent papers in the series can build on either Paper 1 or Paper 2 or both, depending on their substantive content. Papers articulating the architecture of specific layers within the complete exactification path (Paper 3 on $L_1$, subsequent papers on $L_2$ and beyond) will build primarily on Paper 1's architecture, with Paper 2's foundation providing the broader context. Papers articulating specific subject-grounded mathematicalities (the aesthetic mathematics series, the ethical mathematics work) will build primarily on Paper 2's foundation, with Paper 1 providing the broader SAE mathematics architecture. Papers articulating relations across paths (cross-path comparisons, transitions between paths, hybrid path articulations) will build on both papers jointly.

The principle of accumulated articulation (non-purposive purposiveness) governs the trajectory: subsequent papers are not committed to a prescribed sequence, and the substantive content of each paper emerges as the substantive content accumulates. The present paper articulates the foundation; subsequent papers articulate within and beyond the foundation as the substantive work develops.

11. Open Problems

11.1 Overview

This section articulates substantive open problems that the present paper identifies as starting points for future work. The problems are not exhaustive; they are the problems most directly raised by the substantive content of the present paper. Other problems will emerge as the SAE mathematics series develops.

We articulate the problems in order of their distance from the present paper's substantive content, beginning with problems that arise directly from material in the present paper and proceeding to problems that arise from the broader SAE mathematics programme.

11.2 The detailed structure of aesthetic mathematicality

The present paper articulates aesthetic mathematicality at the level of possibility proof, substantive direction (probability-distribution grammar), and boundary conditions (the three boundary statements of § 5.5.5). The detailed structure of aesthetic mathematicality—the specific articulation of its marked handles, its distributional accumulation, its conditioning structure, its distribution space—is reserved for subsequent specialised papers in the SAE aesthetic mathematics series.

The substantive content of these papers is open. Specific questions include: How precisely are canonical works, style frames, and interpretive traditions articulated as marked handles in the technical sense the present paper requires? What is the substantive mathematical content of distribution accumulation in aesthetic articulation, beyond the general claim that distributions accumulate? How is conditioning articulated in aesthetic comparison, and what is its substantive relation to conditional probability in more familiar settings? What is the structure of the distribution space that aesthetic mathematicality articulates, and what substantive operations does it admit? Each of these is a substantive question for the aesthetic mathematics series.

The substantive engagement with intellectual traditions in aesthetics—Kant's Critique of Judgment, statistical aesthetics, Bayesian aesthetics, Arrow's impossibility theorem, information-theoretic aesthetics, sociology of aesthetics—is also reserved for these subsequent papers. The present paper notes the connections at the level of indication; the subsequent papers will articulate them at the level of substantive engagement.

Aesthetic paradigm shift as topological singularity / phase transition in distribution space (substantive future research direction): Aesthetic paradigm shifts (for example, classicism → modernism → postmodernism) can, at the mathematicality level, be articulated as topological singularities or phase transitions in probability distribution space. The aesthetic distribution network exhibits substantively different mathematical behaviour between paradigm-stable periods (large numbers of aesthetic articulations stable around shared priors plus style frames) and paradigm-transition periods (priors are reframed, the topological structure of the distribution space changes). How to articulate paradigm shifts in aesthetic distribution space (singularity points / phase transitions) precisely in the language of SAE mathematicality is a substantive candidate for cross-engagement between aesthetic mathematicality and dynamical systems theory (the SAE 形与流 series). The development of this direction will connect the aesthetic mathematics specialised series with the 形与流 series, as a substantive cross-series engagement of the SAE framework.

11.3 The detailed structure of relational mathematics

The present paper articulates relational mathematics—the mathematicality of the partial exactification path—at the level of structural sketch. The detailed structure is reserved for subsequent specialised papers.

Specific open questions include: How are the specific order-theoretic structures (posets, preorders, lattices, distributive lattices, complete lattices) articulated within the partial exactification path's substantive content? What is the substantive relation between relational mathematics and order theory as developed in contemporary mathematics? Are there substantive aspects of relational mathematics that conventional order theory has not yet articulated, and that the SAE perspective could articulate? How does relational mathematics interface with measurement theory, particularly with the representational theory of measurement of Krantz, Luce, Suppes, and Tversky?

The detailed engagement with these questions is reserved for the relational mathematics specialised work, which is a candidate for future development in the SAE mathematics series.

11.4 Other $L_0$ dimensions

The dimensions discussed in § 5 (quantity, temporal rhythm, spatial size, aesthetic, ethical, intensity, social proximity, truth) are illustrative rather than exhaustive. Other dimensions exist and may be articulated in future work.

Specific candidate dimensions: temporal duration (long/short, brief/extended), distinct from temporal rhythm; emotional intensity (strong/weak emotional response), distinct from physical intensity; epistemic certainty (certain/uncertain, known/unknown), with relation to but distinct from truth value; relevance (relevant/irrelevant, central/peripheral) as an articulation of attention; clarity (clear/unclear, lucid/obscure) as an articulation of cognition. Each is a candidate for substantive articulation as an $L_0$ dimension; the substantive content of each requires its own examination.

The dimension typology is also open to refinement. The three axes articulated in § 2.4 (subject involvement, exactification potential, public re-entry) may be supplemented by additional axes that emerge as the substantive treatment develops. The present paper articulates the three axes as illustrative typological organisation; a more refined typology may articulate further substantive distinctions.

11.5 Other closure paths

The three closure paths articulated in § 4 (complete exactification, partial exactification, persistent open) are the three most clearly identifiable in current mathematical practice. Other closure paths may exist and may be articulated in future work.

Specific candidate paths: paths that introduce limited or contested references, between the partial exactification and persistent open paths; paths that combine exactification with persistent open features in specific ways; paths grounded in fundamentally different marked-handle types that the present articulation has not envisioned. Each is a candidate; the substantive content of each requires substantive development.

The path inexhaustibility (§ 8.6) acknowledges this openness as a substantive feature of the framework, not a limitation. The three paths are productive starting points, not a closed taxonomy.

11.6 Detailed engagement with mathematical philosophy traditions

The present paper engages with mathematical philosophy traditions (formalism, intuitionism, structuralism, category theory, ethnomathematics) at the level of brief acknowledgment. Substantive detailed engagement is reserved for subsequent work.

Specific open questions include: How does the SAE articulation of mathematicality through the four inevitabilities engage with formalism's articulation of mathematics through symbol manipulation and formal proof? How does the SAE articulation engage with intuitionism's articulation through mental construction and temporal intuition? How does it engage with structuralism's articulation through abstract structural patterns? How does it engage with category theory's articulation through morphism patterns? How does it engage with ethnomathematics's articulation of mathematical practices across cultures? Each of these is a substantive engagement that the SAE mathematics series may take up in dedicated papers.

The brief acknowledgment in the present paper signals that the SAE perspective is one framework among many; the detailed engagement with other frameworks is substantive content for subsequent work that allows the comparative articulation to be developed.

11.7 The trajectory of subsequent SAE mathematics series papers

The present paper articulates the foundation; subsequent papers articulate within and beyond the foundation. The substantive trajectory of subsequent papers is open to development, with several candidates identified.

Paper 3 is anticipated as the substantive foundation for $L_1$, articulating the specific architecture of the first mathematical layer within the complete exactification path. The substantive content of Paper 3 will engage in detail with the technical content of arithmetic, the construction of $\mathbb{R}$ through Cauchy completion, and the substantive relation between Paper 1's broad architecture and the specific content of the first layer.

ZFCρ Paper 0 is anticipated as a substantive scope-demarcation paper for the ZFCρ series within the SAE mathematics series. The substantive content will articulate the scope of the ZFCρ series's technical work (the $L_{1\text{-}2}$ choice-operation sub-articulation) within the broader $L_1$ architecture that Paper 3 will articulate, and within the SAE mathematics architecture that Papers 1 and 2 articulate. The paper is anticipated after Paper 3 to allow the $L_1$ architecture to be in place.

The aesthetic mathematics specialised series is anticipated as the substantive development of aesthetic mathematicality beyond the foundation provided by the present paper. The series will articulate the detailed structure of aesthetic mathematicality at the level of marked handles, distributional accumulation, conditioning, and distribution space. The series will engage substantively with the intellectual traditions in aesthetics that the present paper notes only briefly.

Subsequent specialised papers on $L_2$, $L_3$, and beyond are anticipated as the substantive articulation of the layers Paper 1 articulated, in detail beyond what Paper 1 provided. Each layer's specialised treatment will engage with the layer's substantive technical content while preserving the foundational framework that Papers 1 and 2 articulate.

Relational mathematics specialised work is anticipated as the substantive development of the partial exactification path's mathematics. The substantive content will articulate the technical structure of order-theoretic mathematics in the SAE framework and engage with measurement theory and contemporary order theory.

Cross-path and interdisciplinary work is anticipated as the substantive content emerges. The relation between paths, the structure of transitions between paths, the engagement with empirical sciences (mathematics in physics, biology, cognitive science, the social sciences), and the engagement with non-mathematical articulative practices are all candidates for substantive future work.

11.7.1 Priority and timing overview

To help the reader grasp the development sequence of subsequent work in the SAE mathematics series, we summarise the main candidates in tabular form (this ordering is indicative rather than binding; the principle of accumulated articulation preserves the flexibility of the actual emerging trajectory):

Subsequent work	Priority	Primary dependency
Paper 3 ($L_1$ quantity dimension / complete exactification path foundation)	Immediate	Paper 1 + Paper 2
Aesthetic mathematics specialised series (initial paper)	Near-term (parallel to Paper 3)	Paper 2 § 5.5 foundation
ZFCρ Paper 0 (scope demarcation)	Near-term (after Paper 3)	Paper 3 $L_1$ architecture
Logic specialised work (truth-value dimension's three-path realisations)	Mid-term	Paper 2 § 5.7 + Paper 3 framework
Relational mathematics specialised work (spatial-size / partial exactification path)	Mid-term	Paper 2 § 4.3 + measurement theory interface
Specialised papers on other $L_0$ dimensions (ethics / intensity / etc.)	Mid-term	Paper 2 § 5 framework
Specialised papers on $L_2$, $L_3$, and beyond	Mid-to-long-term	Paper 3 + Paper 1
Detailed engagement with philosophy-of-mathematics traditions	Long-term	The entire SAE mathematics series
Cross-path transitions / 形与流 series cross-engagement	Long-term	Paper 2 + 形与流 series
Mathematicality and other disciplinary categories (philosophy / law / etc.)	Long-term, open	§ 11.8 open problem

Substantive remark on the multi-path $L_1$ trajectory: The substantive consequence of the Paper 2 framework is that $L_1$ is not a single foundational layer, but rather the product of dimension × path (see the truth-value dimension articulation of § 5.7). Different subsequent papers articulate substantively different $L_1$s:

Paper 3: $L_1$ quantity dimension along the complete exactification path (the arithmetic foundation of Paper 1's main trajectory)
Logic specialised work: $L_1$ truth-value dimension along the complete exactification path (formal logic / Boolean algebra) + along the partial exactification path (intuitionist logic) + along the persistent-open path (philosophical logic)
Relational mathematics specialised work: $L_1$ spatial-size dimension along the partial exactification path (order-theoretic structure)
Aesthetic mathematics specialised series: $L_1$ aesthetic dimension along the persistent-open path (distribution-grounded structure)
Specialised papers on other $L_0$ dimensions: Each corresponds to a different $L_1$ realisation

The various $L_1$ trajectories are substantively complementary and develop coherently within the Paper 2 framework. The SAE mathematics series is therefore not a single linear trajectory ($L_0 \to L_1 \to L_2 \to \ldots$), but rather a multi-dimensional × multi-path trajectory, with Paper 2 providing its coherent organisational framework.

The trajectory is not prescribed in advance. The principle of accumulated articulation (non-purposive purposiveness) governs the development: subsequent work emerges as the substantive content accumulates, with no commitment to a fixed sequence.

11.8 The substantive relation between mathematicality and other forms of systematic articulation

A final open problem of substantive scope: the relation between SAE mathematicality and other forms of systematic articulation that satisfy the four inevitabilities and the public re-entry criterion but are not conventionally classified as mathematics.

§ 7.3's failure modes articulate that practices failing any of the criteria do not enter SAE mathematicality. But practices that satisfy the criteria but are not conventionally called mathematics—certain forms of philosophical articulation, certain forms of legal reasoning, certain forms of literary criticism, certain forms of theological articulation—are within the broad SAE mathematicality identified by the criteria (as candidates / open problems, not as settled conclusions). The substantive content of these practices, on the present paper's articulation, may be part of mathematicality in the SAE sense (broad), without being mathematics in the conventional sense (narrow).

Important caveat: The present paper does not claim that philosophy, law, literary criticism, theology, etc. "are" mathematics. The present paper also does not reduce these practices to mathematics. The present paper only articulates, at the level of § 11 open problems: whether they structurally satisfy the five conditions and therefore belong in broad SAE mathematicality is a substantive open question. Detailed substantive engagement is reserved for future work.

The substantive open question is: what is the substantive relation between the SAE mathematicality of these practices (if they enter it) and the conventional disciplinary identity of the practices? Is philosophy, on the SAE articulation, possibly partly mathematics? Is law? Is literary criticism? The present paper articulates the framework within which the question can be addressed; the detailed substantive content of the answers is reserved for future work.

This open problem touches on the substantive content of SAE philosophy more broadly. The SAE perspective articulates that the substantive content of articulation extends beyond conventional disciplinary categories, and that the disciplinary identity of practices is partial and trajectory-dependent. SAE mathematicality is one substantive category that cuts across disciplinary boundaries; SAE methodologies in general articulate other such categories. The substantive engagement with the disciplinary structure of intellectual practice is a feature of the SAE perspective that the SAE mathematics series shares with the broader SAE programme.

Acknowledgments

The drafting of the present paper benefited from sustained discussion with multiple AI collaborators. The collaborators contributed substantive perspectives at different levels:

Zi Lu (Claude, architectural-coherence review): Provided architectural-coherence review across the outline and draft rounds. At the v1 outline stage, identified 9 priority revisions (covering SAE-mathematics redefinition explicit, three-four-tuples relations, complementarity-versus-focal-significance reconciliation, Paper 1 schema renaming, substantive defense framework, and others); these contributions shaped the overall structure of the v1 draft.

Zi Gong (Grok, reality-check / stress-test): Provided reality-check at the v1 draft stage, identifying 4 substantive priorities (including § 5.5 aesthetic mathematicality concrete-structure strengthening, § 6 schema-universality-test decisive conclusion, § 2.4 typology upgrade, § 11 granularity), and contributed Via Rho / category-theoretic / intuitionist alternative framing candidates for consideration in subsequent work.

Zi Xia (Gemini, mechanism deep-read): Provided mechanism deep-read at the v1 draft stage, contributing 3 substantive new directions (§ 7.3.6 Forced Closure failure mode, § 4.4 / § 5.5.3.1 non-parametric topological character of probability-distribution grammar, § 11.2 aesthetic paradigm shift / topological singularity as cross-series engagement with the 形与流 series), and providing the v2-stage logic-dimension substantive follow-up articulation (§ 5.7 + § 11.7.1 multi-path $L_1$).

Gong Xi Hua (ChatGPT, structural hotfix + final signoff authority): Provided structural hotfix across the outline and draft rounds, identifying v2 hard blockers (§ 3.7 five-condition unified definition, § 8.2 inevitability wording down-toning), and at the v2-final stage providing the decisive final hotfix (§ 3.7 all articulations unified to "five conditions"). Gong Xi Hua is the SAE mathematics series's final signoff authority.

The four AI collaborators each engaged the substantive content from a different lens, and their collaboration maintained the methodological discipline and substantive coherence of Paper 2. The substantive content of the present paper is ultimately the author's responsibility, and any substantive errors are attributable to the author. But the substantive shape of the present paper reflects the cumulative articulation that the multi-AI collaboration produced.

The present paper also inherits the substantive architecture of SAE Mathematics Paper 1 (DOI: 10.5281/zenodo.20153791) and the broader SAE methodology series (including Methodology 0: Negativa, Methodology 00: Via Rho, Methodology Paper VII: Via Negativa, Cross-Layer Closure Equations, and others).

闭合路径	标记把手	闭合形式	语法
完整精确化	单位	定量结构 ($\mathbb{R}, \mathbb{C}, \ldots$)	闭合方程语法
部分精确化	参照	序论结构 (偏序集、预序)	序结构语法
持续敞开	先验 (典范作品、风格框架、阐释传统)	比较阐述的分布	概率分布语法

SAE 闭合路径	测量论类比
持续敞开路径	名义/定性分类, 带公共比较再进入
部分精确化路径 (无定量参照)	序数尺度 (无量级的序结构)
部分精确化路径 (带区间定义参照)	区间尺度 (有界结构内的量级)
完整精确化路径	比率尺度 (带单位的完整量级结构)

维度	主体涉入度	精确化潜力	公共可再进入性	典型闭合路径	本节小节
量 (多/少)	最小	完整	强	完整精确化	§ 5.2
时间节律 (快/慢)	中等	完整	强	完整精确化	§ 5.3
空间大小 (大/小)	弱-中等	部分 (可扩展)	强	部分精确化 (可扩展完整精确化)	§ 5.4
审美价值 (美/丑)	最大	低 (Anchor C)	中等	持续敞开	§ 5.5
伦理价值 (善/恶)	最大	低	中等	持续敞开 (candidate)	§ 5.6
强度 (强/弱)	中等	部分-完整	强	视情况	§ 5.7
社会亲近度 (近/远)	中-最大	低-部分	中等	视情况	§ 5.7
真值 (真/假)	视域	视情况	视情况	视域	§ 5.7

四元组	层级	功能
0DD 四相	存在性 (Methodology 0)	阐述本身的结构
Paper 2 四条 inevitability	$L_0$ 数学样态	数学样态涌现的条件
Paper 1 四步	$L_1+$ 客体性精确化实现	一条路径的路径特定模式

后续工作	优先级	主要依赖
Paper 3 ($L_1$ 量维度 / 完整精确化路径基础)	Immediate	Paper 1 + Paper 2
审美数学专题系列 (起始论文)	Near-term (parallel to Paper 3)	Paper 2 § 5.5 基础
ZFCρ Paper 0 (范围划界)	Near-term (after Paper 3)	Paper 3 $L_1$ 建筑
逻辑学专题 (真值维度三条 path 实现)	Mid-term	Paper 2 § 5.7 + Paper 3 framework
关系数学专题工作 (空间大小 / 部分精确化路径)	Mid-term	Paper 2 § 4.3 + 测量论接口
其他 $L_0$ 维度专题 (伦理 / 强度 / 等)	Mid-term	Paper 2 § 5 框架
$L_2$、$L_3$ 及以上层级专题	Mid-to-long-term	Paper 3 + Paper 1
数学哲学传统详细接洽	Long-term	整个 SAE 数学系列
跨路径过渡 / 形与流系列 cross-engagement	Long-term	Paper 2 + 形与流系列
数学样态与其他学科范畴 (philosophy / law / etc.)	Long-term, open	§ 11.8 open problem

SAE Mathematics Paper 2: L₀ Mathematicality — Inevitability and Multi-Path RealisationSAE 数学 Paper 2：L₀ 数学样态——不可避免性与多路径实现