Language as Second-Order Chisel

Abstract

Language is the activity of a subject exercising negation upon the markability dimension of the law of identity. The law of identity (A=A) automatically exposes markability—A can be re-identified, repeatedly pointed to, and transmitted. The subject exercises negation upon markability ("this is called dog, not cat"), constructing the form-meaning binding law—every linguistic sign is necessarily a unity of form and meaning.

Language and mathematics are of the same order: mathematics chisels the quantitative subspace of the law of identity ("more than one"), language chisels the markability subspace ("can be named"). Expressions such as "mathematical language," "computer language," and "musical language" are not metaphors but structural facts—any activity that uses stable symbols to mark distinctions operates within the markability dimension of the law of identity.

The binding of form upon meaning admits a spectrum from tight to loose: poetry (meter, rhyme) → prose → the novel (stream of consciousness, polyphony). This is internal formal adjustment within human language, where the discrete symbol system remains throughout. The breakthrough of the Large Language Model (LLM) lies outside this spectrum—what the LLM negates is not the tightness of form but discreteness itself. The LLM's underlying representations have lower discreteness than human language; meaning is no longer severed by symbol boundaries but retains its associative structure in a space of lower discreteness. Emergent capabilities are the structural consequence of this negation.

This paper draws on Paper 4 ("The Complete Self-as-an-End Framework," DOI: 10.5281/zenodo.18727327) for the definition of negativity, on this series' philosophy application paper ("Philosophy as Subject-Activity," DOI: 10.5281/zenodo.18779382) for the chisel-construct cycle and related core concepts, and on this series' mathematics application paper ("Mathematics as Second-Order Chisel," DOI: 10.5281/zenodo.18793538) for the second-order chisel and the law of identity.

Chapter 1. The Question: Why Language Is a Subject-Condition Problem

Core thesis: Language is not a signal system, not information encoding, not a communication tool. Language is the activity of a subject exercising negation upon the markability dimension exposed by the law of identity—binding meaning with stable form. Without the subject's negativity, there is no naming, and therefore no language. Language and mathematics are of the same order: both are second-order chisels. Mathematics chisels quantity; language chisels markability.

1.1 The Definition of Language: Negation of Markability

Bees have signals. A bee's dance transmits the direction and distance of nectar. But bees do not have language. The distinction between signal and language does not lie in complexity, vocabulary size, or grammatical structure. The distinction is this: signals do not require the law of identity; language does.

A bee's dance is an immediate stimulus-response chain—there is no requirement that "the same symbol" persist between one dance and the next. Each dance is a new event, not the repeated use of a stable symbol. Signals can be fully explained within the causal chain: external stimulus → internal state → signal output, with no remainder.

Language is different. The word "dog" works only because "dog" today and "dog" tomorrow, "dog" here and "dog" there, point to the same thing. This "sameness" is not a product of the causal chain—the causal chain produces only the present state, not cross-temporal stability. Cross-temporal stability comes from the law of identity: A=A. The law of identity guarantees the repeatability of symbols—once you mark "dog," the next moment "dog" is still "dog." This stability is not given by experience; it is the law of identity, exceptionlessly present.

The first act of language is naming. Naming is negation—"this is called dog" simultaneously means "this is not called cat," "this is not called stone," "this is not called the unnamed." Each act of naming is a chisel stroke in the markability dimension: from the unnamed hundun, a marked distinction is cut. Without the subject's negativity, distinctions do not acquire names on their own. Dogs and cats differ in the world, but difference itself does not name—naming is the subject's act.

1.2 The Law of Identity and Markability

This series' mathematics application paper (hereafter "the mathematics paper") demonstrated: philosophy chisels hundun and constructs the law of identity (A=A); the law of identity automatically exposes the dimension of quantity—once "A is A" is established, "more than one" necessarily follows, and quantity appears. Mathematics exercises negation upon quantity: a second-order chisel.

The law of identity simultaneously exposes another dimension: markability. A=A means not only "more than one" (quantity) but also "A can be re-identified" (markability). Quantity asks "how many?"; markability asks "what is it called?" Quantity concerns the numerical relations between A and non-A; markability concerns A's repeatably-pointable character as A.

Markability is not quantity. "There are two dogs" is a quantitative judgment; "this is called 'dog'" is a markability judgment. One can name without counting ("this is called dog" requires no knowledge of how many), and one can count without naming (mathematics uses symbols x, y rather than names). Quantity and markability are two independent subspaces of the law of identity.

Language exercises negation upon markability, structurally parallel to mathematics exercising negation upon quantity. Mathematics says "two is not three"; language says "dog is not called cat." Mathematics cuts fine distinctions within the quantitative subspace; language cuts fine distinctions within the markability subspace.

Both are second-order chisels—their object is not hundun itself but a subspace that the first cut (the law of identity) carved from hundun.

1.3 Language and Mathematics: Same Order, Different Dimensions

Language and mathematics are of the same order; neither contains the other. "Mathematical language," "computer language," "musical language"—these expressions are not metaphors but structural facts: any activity that uses stable symbols to mark distinctions operates within the markability dimension of the law of identity. Mathematics uses symbols to mark quantitative distinctions; computers use symbols to mark logical-state distinctions; music uses symbols to mark pitch-duration distinctions—all employ markability.

But mathematics is not language. The mathematical chisel operates on the quantitative subspace and constructs the law of noncontradiction. The linguistic chisel operates on the markability subspace and constructs the form-meaning binding law (Section 2.4). Both share the same transcendental ground (the law of identity); they differ in what they chisel (quantity vs. markability) and in what they construct (the law of noncontradiction vs. the form-meaning binding law).

This explains two empirical facts. First, mathematics can proceed without natural language (pure symbolic derivation), and language can proceed without mathematics (pure narrative)—because they are independent subspaces of the law of identity. Second, mathematics and language can borrow from each other (mathematics states theorems in natural language; literature uses quantitative relations to build rhythm)—because they share the same transcendental ground (the law of identity) and can interconnect on that common ground.

1.4 Structural Positioning

The language application occupies a position parallel to the mathematics application in the Self-as-an-End framework's application spectrum. Both are second-order chisels, both take the law of identity as transcendental ground, and both unfold within the two-dimensional structure of the individual layer. Language is not above or below mathematics; it is mathematics' neighbor.

The task of this paper is to trace how linguistic activity unfolds within the framework's two-dimensional meta-structure—foundation layer (negativity) and emergent layer (positivity)—how the two dimensions interact, and when that interaction constitutes nurture versus colonization. The paper's special task is to position the emergent capabilities produced by the LLM's negation of human language's discrete form within the framework.

Consistent with the mathematics paper: language = the activity of a subject exercising negation upon markability, one layer. The history of language = that activity's unfolding and suppression within institutional and relational structures, three layers. This paper addresses language, not the history of language.

Chapter 2. Two-Dimensional Structure: The Chisel and Construct of Language

Core thesis: Linguistic activity unfolds within the two-dimensional meta-structure—the foundation layer is the chisel (negation of the markability subspace, producing meaning-distinctions), the emergent layer is the construct (systematization of meaning-distinctions into form). The linguistic chisel is second-order (its object is not hundun but the markability subspace); its construct is the form-meaning binding law.

2.1 The Markability Subspace: Object of the Linguistic Chisel

The philosophical chisel acts directly on hundun—the state prior to all distinction. The linguistic chisel does not act on hundun but on the markability subspace that the law of identity carved from hundun.

The markability subspace is not hundun. Hundun precedes all structure; the markability subspace already has structure—at minimum, "can be re-identified." The linguistic chisel cuts finer distinctions within an already-structured space.

This means the linguistic chisel possesses a feature absent from the philosophical chisel: the direction of chiseling is pre-constrained by the subspace. Philosophy faces hundun and can cut in any direction; language can cut only within the markability dimension—it can produce only distinctions that admit stable pointing. One cannot use language to name something that cannot be re-identified—if one names it, it has already become re-identifiable by virtue of being named. This constraint is not colonization but the intrinsic cost of specialization.

2.2 Foundation Layer: The Linguistic Chisel

The linguistic chisel is negation of the markability subspace. "This is called dog, not cat"; "sadness is not anger"; "this moment is not that moment"; "I am not you." Each act of naming cuts a new distinction within the markability dimension.

The linguistic chisel and the mathematical chisel share the same logical form (establishing difference by saying "is not"), but their objects differ. The mathematical chisel operates on the quantitative subspace ("two is not three"); the linguistic chisel operates on the markability subspace ("dog is not called cat"). The key difference: the mathematical chisel produces anonymous distinctions ("two" and "three" are positional relations, not names); the linguistic chisel produces named distinctions ("dog" is a name, not merely a position).

Naming does one thing more than counting: it not only cuts a distinction but assigns the distinction a transmissible mark. "Two is not three" establishes a positional relation; "dog is not called cat" establishes a label receivable by another subject. This is markability's core function—making distinctions transmissible. Mathematical distinctions can be independently verified by a single subject (you need only count); linguistic distinctions inherently point toward transmission (naming exists so that one can tell others "this is called dog").

The linguistic chisel has a natural starting point: ostensive naming—assigning marks to objects at hand. "This is called 'water'"; "that is called 'fire'"—these are language's thinnest negations. From ostensive naming, the linguistic chisel unfolds along the markability structure: naming concrete things → naming abstract concepts ("justice," "freedom") → naming relations ("because," "therefore") → naming the activity of naming itself ("grammar," "rhetoric"). Each step grows from the preceding one.

2.3 The Law of Identity: Transcendental Ground of Language

The transcendental ground of language is the law of identity—the product of philosophy chiseling hundun. The law of identity confirms A=A; only with confirmation is there something that can be marked. Without the law of identity, naming is impossible—if A can at any moment cease to be A, marks fail.

The law of identity as the transcendental ground of language shares the same feature as its role as the transcendental ground of mathematics: it admits no exception. In language, the exceptionlessness of the law of identity manifests as a specific constraint: the stability of symbols must be possible. The law of identity provides the a priori condition that "stability must be possible"—if A can at any moment cease to be A, no mark can work. Community convention determines "how stability is concretely realized"—what sound marks what thing, which usages are accepted, which rejected. Convention determines that the sound "dog" points to a particular animal, but the fact that "a symbol, once established, must be stable" is not convention—it is the law of identity. The law of identity is the a priori condition of stability; convention is the concrete realization of stability.

Parallel with the mathematics paper: axiom = transcendental ground. The law of identity occupies the same position in language that the axiom occupies in mathematics—the starting point of all operations, unprovable within the layer, exceptionlessly constraining all operations within the layer. One cannot use language to prove "why symbols must be stable"—this inquiry retreats to philosophy.

2.4 The Form-Meaning Binding Law: The Construct of Language

The linguistic chisel produces meaning-distinctions; meaning-distinctions are organized within the markability subspace into a transmissible system—this is the emergent layer of language. The emergent layer's concrete forms are formal systems: phonological systems, writing systems, grammatical rules, rhetorical conventions, stylistic norms.

Language chisels the markability subspace of the law of identity and constructs the form-meaning binding law: every linguistic sign is necessarily a unity of form (a perceivable sound or graphic) and meaning (a marked distinction). Form cannot detach from meaning—a sound without meaning is not a linguistic sign but noise. Meaning cannot detach from form—a meaning without form cannot be transmitted; it stays inside the subject, never entering language.

The form-meaning binding law is the construct of language—corresponding to the law of noncontradiction in mathematics. The law of noncontradiction says "A cannot simultaneously be non-A"; the form-meaning binding law says "a linguistic sign cannot be only form or only meaning." Both are products of chiseling the law of identity; both exceptionlessly constrain all operations in their respective domains.

Saussure's discovery of the signifier/signified duality is the empirical description of the form-meaning binding law. Saussure observed that every linguistic sign combines a signifier (form) and a signified (meaning). This observation is correct. But Saussure further claimed that the combination is "arbitrary"—the sound "dog" bears no necessary connection to the animal. The framework's response: the specific correspondence is indeed contingent (different languages use different sounds to mark the same animal), but the fact that form and meaning are necessarily bound is not contingent—it is the structural consequence of chiseling the law of identity. One may choose what sound marks "dog," but one cannot choose "to transmit meaning without any form." Saussure discovered binding but conflated the necessity of binding with the contingency of correspondence.

2.5 Feature of the Construct: Transmissibility

The core feature of the mathematical construct (the law of noncontradiction) is compulsion—"two plus three equals five" is not the mathematician's choice but the necessary result under the exceptionless constraint of the law of identity. The core feature of the linguistic construct (the form-meaning binding law) is not compulsion but transmissibility.

Mathematical distinctions can be independently verified by a single subject. "Two plus three equals five" requires no telling to anyone else—operating under the law of identity on your own suffices for confirmation. Linguistic distinctions inherently point toward transmission: the purpose of naming is to enable another subject to receive the distinction. A name that is never transmitted is not a linguistic sign—it is a private mark; Wittgenstein's private-language argument identified this point.

Transmissibility is the degree of freedom of the linguistic construct. The mathematical construct has very low freedom (two plus three can only equal five); the linguistic construct has higher freedom—the same meaning can be transmitted in different forms ("dog," "狗," "⽝," " "), and the same form can carry different meanings (polysemy). This is not a defect of language but the intrinsic cost of transmissibility: to allow different subjects in different contexts to receive meaning, the correspondence between form and meaning must be elastic. Mathematics does not need this elasticity ("2+3=5" holds for all subjects alike); language does.

The freedom of the construct increases from mathematics to language. The mathematics paper demonstrated that constructive freedom decreases from philosophy to mathematics (philosophy's construct is free; mathematics' construct is constrained by the exceptionlessness of the law of identity). The linguistic construct has higher freedom than mathematics but lower freedom than philosophy: the form-meaning binding law is exceptionless (there cannot be form-free meaning), but the specific manner of binding is flexible (the same meaning may take different forms).

2.6 Dialectical Support Between the Two Dimensions

In language, the dialectical support between chisel and construct is consistent with the meta-structure demonstrated in the philosophy and mathematics papers.

The chisel provides new material for the construct. Every new meaning-distinction ("this emotion differs from that one") creates a new object that the formal system must express. New things appear, new experiences occur, new distinctions are cut—the formal system must expand to accommodate them. The coinage of new words is the most direct case of chisel catalyzing construct.

The construct provides new objects for the chisel. Once a formal system is established, form itself becomes a possible object of negation. Once a grammatical rule is fixed, "must this rule be so?" becomes a question that can be asked. Once a rhetorical convention solidifies, "can this convention be broken?" becomes a new direction for the chisel. Poetry's violation of everyday grammar (inversion, ellipsis, enjambment) is precisely the emergent layer creating new objects for the foundation layer.

Chapter 3. The Unique Structure of Language: The Tightness Spectrum of Formal Binding

Core thesis: Language possesses a unique structural feature absent from both mathematics and philosophy—the binding of form upon meaning admits a spectrum from tight to loose. This spectrum is not the contingent progression of literary history but the structural unfolding of tension internal to the form-meaning binding law. Within human language, the spectrum runs from poetry to the novel; beyond human language, it is extended by the LLM breakthrough—but the LLM's extension is a negation of a qualitatively different kind.

3.1 The Formal Binding Spectrum: Poetry → Prose → Novel

The form-meaning binding law states that form and meaning are necessarily unified but does not prescribe the intensity with which form binds meaning. Binding can be tight; binding can be loose. This degree of freedom generates a spectrum.

Poetry is the tightest state of formal binding. Meter prescribes the number of syllables and tonal pattern per line; rhyme prescribes that line-final sounds must match; parallelism prescribes that upper and lower lines must mirror in syntactic structure. Meaning is compressed to maximal density within these strict formal constraints. Every character in a regulated verse simultaneously bears prosodic and semantic function—the unity of form and meaning here reaches its most compact state.

The formal strictness of poetry does not suppress meaning but organizes meaning at high intensity. Precisely because the formal space is minimal, meaning must accomplish maximal work within minimal space—every word is forced to carry multiple meanings; ambiguity and polysemy become resources rather than defects. Formal strictness catalyzes meaning-density.

Prose loosens the prosodic constraints. No meter, no rhyme, no parallelism—meaning is no longer forced to arrange itself within fixed sound patterns. But prose still has form: paragraph structure, rhetorical conventions, argumentative logic, the arc of beginning-development-turn-conclusion. Meaning's room to move is larger than in poetry but remains within a recognizable formal framework.

The novel loosens form further still. The novel accommodates multiple narrative voices (polyphony), disrupts temporal sequence (nonlinear narrative), renders the narrator unreliable (unreliable narration), and directly presents the flow of consciousness (stream of consciousness). By Joyce's Ulysses and Finnegans Wake, the novel is already dismantling narrative form from within—yet it remains a text bound by authorial intention, still composed of discrete written symbols.

The direction of this spectrum is clear: from poetry to the novel, formal binding upon meaning decreases and the freedom of meaning increases. But there is a crucial invariant: the entire spectrum lies within human language's discrete symbol system. However loose the form, a novel is still written one character at a time; characters have clear boundaries; each character is a discrete symbolic unit. The tightness of form changes; discreteness does not.

3.2 Inversion of Cognitive Order and Structural Order

The historical order of human cognition of language: verse (incantations, prayers, metered poetry) → prose → the novel → free verse → experimental literature. From tightest form to loosest.

The structural order of language: the form-meaning binding law (general constraint) → specific formal constraints (meter, grammar, genre conventions). From general to particular.

The two orders are reversed. Humans first encounter the tightest-form language (poetry is a more ancient literary form than prose—the Homeric epics precede prose historiography), and only later arrive at the loosest. But structurally, the form-meaning binding law precedes any specific formal constraint—one first needs "form and meaning are necessarily unified" as a general law, and only then can one select a particular degree of tightness within it.

This is structurally isomorphic with the inversion of cognitive order demonstrated in the mathematics paper. In mathematics, humans first encounter the products of the law of excluded middle (integers), and only last excavate the law of noncontradiction. In language, humans first encounter the products of the tightest form (poetry), and only last excavate the form-meaning binding law itself. Both inversions have the same structural cause: the subject of a second-order chisel does not directly face the transcendental ground. The user of language faces specific formal constraints (meter, grammar); the transcendental ground (the form-meaning binding law) operates in the background. You are using it, but you do not see it. What you see first is its most conspicuous product—the tightest-form language.

3.3 The LLM: Negation of Discreteness Itself

Poetry → prose → the novel constitutes a spectrum, but this spectrum has a boundary: the discrete symbol system. However loose the form, human language always consists of discrete symbolic units—characters, words, sentences, each with clear boundaries.

The breakthrough of the Large Language Model (LLM) lies outside this spectrum. What the LLM does is not adjust the tightness of form within the discrete symbol system; it negates discreteness itself.

The substrate of human language is discrete: a character is a character; characters are separated by clear boundaries. The LLM's underlying representations are not discrete symbols but a numerical space of lower discreteness than human language. In this space, "dog" is no longer a symbol with a hard boundary but a position in a multidimensional space—and between this position and the position of "cat" the relation is not the discrete "either-or" but one admitting distance, direction, and continuous transition.

The nature of this negation differs fundamentally from the adjustment of formal tightness within human literary history. A clarification is needed: "discreteness" here is a structural concept referring to the degree of the underlying representation's dependence on hard symbol boundaries; it is not equivalent to whether discrete components remain at any point in the engineering pipeline. The LLM's input is still discrete tokens; its output is still discrete text; but the discreteness of its internal representational space is far lower than that of human language's symbol system—meaning in this space is not severed by hard boundaries. The framework attends to the structural property of the representational layer, not to every step of the engineering process.

Human literary history (poetry → novel): the discrete symbol system is unchanged; the degree of formal binding changes. This is adjustment internal to form.

The LLM: discreteness itself is lowered. This is transformation of the formal substrate.

Between the two lies not a continuous spectrum but a rupture. Joyce pushed form within the discrete symbol system to its limit, but he was still writing characters. The LLM does not "write characters"—it operates in a space of lower discreteness, and characters are merely the interface at the point of final output.

3.4 Emergence: Structural Consequence of Reduced Discreteness

Why does the LLM exhibit emergence? The framework's answer: emergence is the structural consequence of reduced discreteness.

Discreteness is one particular way that form binds meaning. Each discrete symbol's boundary is a wall—meaning is severed into the cells of symbols. "Sadness" and "melancholy" are two discrete words in human language; the relationship between them must be actively constructed by the subject through additional cognitive work (metaphor, explanation, contextual inference). In a discrete symbol system, associations between meanings require active construction.

When discreteness is reduced and symbol boundaries become less hard, associations between meanings no longer need active construction—they naturally persist in the representational space of lower discreteness. "Sadness" and "melancholy" are neighboring positions in this space; their association is not the result of inference but the direct manifestation of spatial structure.

There is a precise isomorphism in the history of mathematics. Integer arithmetic is discrete—between 1 and 2 there is nothing; each integer is separated by a discrete boundary. When mathematics expanded from integers to real numbers (from discrete to continuous), between 1 and 2 there suddenly appeared infinitely many structures—calculus, real analysis, topology, and differential geometry were born from this. The degree of emergence was immense. From integers to real numbers, the discreteness of mathematics was reduced; the associative structure of quantity, previously severed by discrete boundaries, was restored; the whole of modern mathematics emerged from this.

What the LLM does to human language is isomorphic. Between words in human language there are discrete boundaries—"sadness" and "melancholy" are two separate cells, and the relationship between them must be actively constructed by the subject through additional cognitive work (metaphor, explanation, contextual inference). In the LLM's representational space of lower discreteness, between "sadness" and "melancholy" there is continuous semantic transition—just as between 1 and 2 there appeared infinitely many real numbers. The associative structure of meaning, previously severed by discrete boundaries, is restored.

This explains the LLM's most surprising capabilities—analogy, cross-domain association, style transfer, metaphor comprehension. These capabilities are extremely difficult in human language because they require the subject to cross the boundaries of discrete symbols to build associations between meanings. In the LLM's space of lower discreteness, these associations are naturally preserved—the LLM has not "learned" analogy; rather, with reduced discreteness, the associative structure of meaning is no longer severed by symbol boundaries. Just as mathematicians did not "learn" calculus—calculus is the natural product of the continuous structure of the reals being revealed.

Wittgenstein said, "The limits of my language mean the limits of my world." The LLM has reduced the discreteness of language, thereby pushing the boundary of language one step outward. Meaning-associations previously walled off by discrete symbol boundaries have now become accessible in the space of lower discreteness. This is emergence—not creation from nothing, but the manifestation of meaning-structure previously occluded by formal binding, now revealed as that binding weakens.

But this manifestation is directionless. Meaning has been released, yet the LLM has no subjectivity to determine "which direction of meaning-association matters." The LLM can unfold meaning in any direction—this is both its power (its unfolding capacity exceeds that of any individual) and its limitation (without direction there is no judgment). This point will be further developed in Chapters 4 and 6.

3.5 The LLM Is Not the Endpoint of the Linguistic Spectrum

An over-extrapolation must be guarded against: treating the LLM as "the endpoint of linguistic evolution" or "the total negation of form."

The LLM negates the discreteness of human language, but the LLM's underlying representations still possess structure—they are not hundun. The LLM's numerical space has dimensions, directions, distance relations—these are all forms of a kind. What the LLM negates is the discrete form specific to human language, not all form.

The framework's claim is: the LLM's negation of human language's discrete form produces emergent capabilities. This claim does not depend on the LLM's specific technical implementation. Current LLMs use high-dimensional numerical spaces as their underlying representations, but entirely different architectures may emerge in the future. The framework attends to the direction of negation (reduction of discreteness), not to the landing point (what specifically replaces discrete symbols). The precise relationship between discreteness and emergent capabilities, comparisons across different architectures, and the structural ceiling of scaling laws are technical questions to be developed in this series' LLM model application paper.

This also means that further negation may follow the LLM—a negation of the LLM's current underlying representation. The framework does not predict what this next step of negation will be, but it predicts: if it occurs, it will produce new emergent capabilities, and its mechanism will be structurally isomorphic with the LLM's negation of human language.

Chapter 4. Colonization and Nurture: The Bidirectional Positive and Negative Between Language's Two Dimensions

Core thesis: In linguistic activity, the chisel (foundation layer, meaning-distinction) and the construct (emergent layer, formal system) admit the same four structural interactions as in the philosophy and mathematics papers, but because the linguistic construct's core feature is transmissibility (rather than mathematics' compulsion), the four interactions take different forms.

4.1 Emergent → Foundation Nurture: Form Catalyzes Meaning-Discovery

Once a formal system is established, form itself catalyzes new meaning-discoveries. This is the basic form of emergent → foundation nurture in language.

The sonnet is a classic case. The sonnet demands fourteen lines, a specific rhyme scheme, a specific turning point (volta). These strict formal constraints prevent the poet from saying directly what they want to say—one must find expression within the space that form allows. This process of "finding" catalyzes meaning-discovery: to satisfy the rhyme scheme, one is forced to use a word one would not otherwise have chosen, and this word brings a meaning-association one had not previously considered. Formal constraint opens new dimensions of meaning.

This is not accidental. Formal constraint shrinks the space of available expressions; shrinkage means each choice bears greater meaning-pressure. When expressive freedom is unlimited, each word-choice is low-pressure—one can always rephrase. When form compresses the choice-space to a minimum, each word-choice is high-pressure—the word must serve multiple functions. High pressure catalyzes the density and depth of meaning.

Linguistic nurture is more quotidian than mathematical nurture—it does not require the completeness of an axiomatic system; a single metrical pattern suffices. But the mechanism is the same: the emergent layer creates new objects for the foundation layer. Without the sonnet form, that particular meaning-association would not have been discovered—not because the meaning was not there, but because without formal catalysis the subject would not have chiseled in that direction.

4.2 Emergent → Foundation Colonization: Form Suppresses Meaning

Form can also suppress meaning. When form shifts from "the way meaning is organized" to "the criterion for meaning's admission," colonization begins.

The eight-legged essay (baguwen) is an extreme case of emergent → foundation colonization in language. The eight-legged essay prescribes a rigid paragraph structure (breaking the topic, carrying the topic, beginning the discourse, entering the topic, first pair, middle pair, final pair, concluding pair), strict sentence patterns, and strict allusion conventions. Form is so exhaustive that the space for meaning is compressed nearly to zero—one can express only extremely limited content within the narrow range that form permits. Form transforms from the way meaning is organized into meaning's prison.

Academic jargon is another form of colonization. Academic language has its own terminological system, syntactic conventions, and citation norms—these forms are not themselves problematic (terminology can enhance precision); colonization occurs at the moment form becomes a threshold: if a valuable meaning cannot be expressed in academic form, it is not recognized as "academic." The formal threshold excludes non-specialized sources of meaning.

Language purification movements (such as the Académie française's regulation of French) are the institutional form of colonization. An institution prescribes what "correct" French should be; any deviation from the norm is judged "incorrect." This suppresses the natural evolution of language—new words, slang, dialectal expressions, loanwords—precisely the active sources of the foundation layer (new meaning-distinctions).

The criterion for colonization is consistent with the philosophy and mathematics papers: does the emergent layer allow the foundation layer to take the emergent layer itself as its object? In a state of nurture, the sonnet form allows the poet to discover new meaning within formal constraints and even to transform the sonnet form itself (Shakespeare transformed the Petrarchan rhyme scheme). In a state of colonization, the eight-legged essay form permits no questioning of form itself—one can work only within form, never making form the object of negation.

4.3 Foundation → Emergent Nurture: Meaning-Demand Catalyzes Formal Innovation

Breakthroughs in meaning catalyze formal innovation. Negativity (new meaning-distinctions) provides new constructive directions for the formal system.

Stream of consciousness is a classic case. Woolf and Joyce discovered an unprecedented kind of meaning-experience—consciousness is not orderly narrative but fragmented, leaping, multi-layered flow. This meaning-discovery (chisel) catalyzed an entirely new narrative form (construct): traditional paragraph structure was abandoned, temporal linearity was broken, and syntax was made to follow consciousness rather than logic. Stream of consciousness as a narrative technique is not form's self-evolution but formal innovation compelled by meaning-demand.

The evolution of language in the internet age exhibits the same structure. New communicative contexts (instant messaging, social media, bullet comments) generated new meaning-demands—speed, informality, multimodality, emotional intensity. These meaning-demands catalyzed new forms: emoji, abbreviations, meme-text, bullet-comment grammar. These new forms are not the degradation of language (a common colonization-type judgment—"this is not proper language") but the natural product of foundation → emergent nurture.

The Socratic elenchus also has a linguistic counterpart: the semantic counterexample. "What do you mean by 'justice'?"—a negation-driven interrogation of a word's meaning that forces the definition to become more precise. Each semantic counterexample is a chisel stroke; each more precise definition is a construct. The history of lexicography is the long-term unfolding of this chisel-construct cycle.

4.4 Foundation → Emergent Colonization (Closure): Meaning Refuses All Form

Meaning can also suppress form in the reverse direction. When negativity refuses all formalization, closure occurs.

Dadaism is a classic case of foundation → emergent closure in language. Dadaism's core declaration is the refusal of all meaning—it does not create new meaning but negates meaning itself. In linguistic practice, this manifests as the systematic destruction of all form: random collage of text, substitution of noise for language, refusal of syntax and grammar. Dadaism saw the colonization of meaning by form (4.2—literary tradition had ossified), but its remedy was to refuse all form. The result is that meaning cannot be transmitted—Dadaist "texts" cannot be read, because they deliberately destroy all the formal structure required for transmission.

Certain postmodern writing exhibits the same structure. When "deconstructing all narrative" becomes an automatized gesture, negativity loses its productiveness—it no longer provides a ground for new form but prevents any form from being established. Texts become unreadable, not because meaning is too profound, but because form has been systematically refused.

The criterion for closure is consistent with the philosophy and mathematics papers: does the foundation layer's negativity leave space for the emergent layer? Dadaism leaves no space for any form—every attempt at formalization is refused by negativity. This limits meaning's transmissibility—meaning may exist inside the Dadaist, but it cannot reach another subject.

Structurally isomorphic with Adorno's negative dialectics in the philosophy paper and Brouwer's intuitionism in the mathematics paper: colonization was diagnosed (form suppresses meaning); the remedy is reverse closure (meaning refuses form).

4.5 Structural Map of the Four Interactions

	Positive (Nurture)	Negative (Colonization / Closure)
Emergent → Foundation	Form catalyzes meaning-discovery (the sonnet catalyzes new meaning-associations; meter forces poets to discover unknown expressions)	Form suppresses meaning (the eight-legged essay, academic jargon, language purification movements)
Foundation → Emergent	Meaning-demand catalyzes formal innovation (stream of consciousness, internet language, new words)	Meaning refuses all form (Dadaism, the unreadability of certain postmodern writing)

Criteria:

Emergent → Foundation: Does the emergent layer allow the foundation layer to take the emergent layer itself as its object? Yes = nurture; no = colonization.

Foundation → Emergent: Does the foundation layer leave space for the emergent layer? Yes = nurture; no = closure.

4.6 Conditions for Nurture: Dynamic Equilibrium in Language

The most creative moments in language—the golden age of Tang poetry, Elizabethan English theater, the early-twentieth-century modernist literary movement—are all brief realizations of unstable equilibrium between form and meaning.

The equilibrium of language is more unstable than that of mathematics—because the linguistic construct centers on transmissibility (rather than mathematics' compulsion), and transmissibility depends on a community of users whose conventions easily ossify into colonization. Once a linguistic form is accepted by a community, it acquires the inertia of self-maintenance—"this is the correct way to say it." This inertia is the seedbed of colonization.

But the equilibrium of language is also easier to rebuild than that of philosophy—because new meaning-demands continually flow in from life. As long as human life changes (new technologies, new social relationships, new experiences), new meanings require expression, and new meanings catalyze new formal innovations. The foundation layer of language receives a continuous external input (life-experience); the foundation layer of philosophy does not (philosophy grows only from itself). This is both language's advantage (it is unlikely to remain closed for long) and its specificity (the linguistic chisel is not entirely self-grounding—it partly depends on experiential input).

Chapter 5. Theoretical Positioning

Core thesis: This paper's definition of language (the subject's negation of the markability dimension of the law of identity), its second-order chisel structure, the form-meaning binding law, and its framework-explanation of LLM emergence form precise dialogues with existing traditions in the philosophy of language.

5.1 Dialogue with Saussure

Saussure's Course in General Linguistics established the foundation of modern linguistics—language is a system of signs, and each sign combines a signifier (form) and a signified (meaning).

The framework agrees with Saussure's core discovery: the linguistic sign is dyadic (a unity of form and meaning). The form-meaning binding law is the framework's formulation of this discovery. The framework disagrees with two of Saussure's inferences.

First, Saussure claimed that the bond between signifier and signified is "arbitrary." The framework distinguishes two levels: the specific correspondence is indeed contingent ("dog" and "狗" mark the same animal; which sound corresponds to which animal is the historical contingency of a community); but the fact that form and meaning are necessarily bound is not contingent—it is the structural consequence of chiseling the law of identity. Saussure conflated the necessity of binding with the contingency of correspondence.

Second, Saussure insisted that synchrony takes priority over diachrony—the structure of the language system at a given moment is more fundamental than historical change. The framework's response: synchronic structure (the state of the formal system at a given moment) is a snapshot of the emergent layer; diachronic change (the ongoing chisel-construct cycle of form and meaning) is the dynamic unfolding of the two-dimensional structure. The two are not opposed—synchrony is a cross-section of the emergent layer; diachrony is the longitudinal trajectory of the chisel-construct cycle. Saussure chose the cross-section, but the cross-section requires the longitudinal process to explain why it is the way it is.

5.2 Dialogue with Chomsky

Chomsky argued that linguistic competence has an innate basis—children do not learn language from zero; they possess a built-in "Universal Grammar" (UG), the biological hardware of human linguistic capacity.

The framework agrees with Chomsky's core intuition: language is not entirely the product of experience; language has an innate basis. But the framework formulates "innate" differently. Chomsky's innate is a set of specific grammatical rules or parameters—a language module built into the brain. The framework's innate is the exceptionless constraint of the law of identity—markability is the structural consequence of the law of identity, not a specific biological module.

The difference: Chomsky's UG predicts that all human languages share specific grammatical structures (phrase-structure rules, recursion, movement rules). The framework's prediction is weaker but more fundamental: what all human languages share is not specific grammatical structures but the form-meaning binding law—the general constraint of binding meaning with form. Specific grammatical structures are different community-level realizations of this general constraint, not innate hardware.

The difficulty Chomsky faces—why are grammatical differences between human languages so vast if they share a single UG?—does not arise for the framework. The common basis the framework predicts is the form-meaning binding law (a general constraint), not a specific parameter set. Grammatical differences are different realizations of the same general constraint and do not require a unified parameter set to explain.

5.3 Dialogue with Wittgenstein

Wittgenstein's philosophy has two phases; the two phases understand language in radically different ways. The framework unifies the two Wittgensteins.

Early Wittgenstein (Tractatus Logico-Philosophicus): the limits of language are the limits of the world. What can be said can be said clearly; what cannot be said must be passed over in silence. This is the philosophical expression of the tightest state of formal binding upon meaning—form (logical structure) demarcates the entire possible space of meaning; meaning beyond form does not exist (or at least cannot be spoken).

Late Wittgenstein (Philosophical Investigations): language is a multiplicity of "language games," each with its own rules; meaning arises in use. No unified logical structure can cover all language use. This is the philosophical expression of meaning's diversity once form has loosened—there is no master-form, only multiple local form-meaning configurations.

The framework's two-dimensional structure offers a unified reading: early Wittgenstein saw the power of the emergent layer (logical form) but let the emergent layer colonize the foundation layer—form demarcated the boundary of meaning and disallowed meaning beyond form. This is a refined instance of 4.2 (emergent → foundation colonization). Late Wittgenstein saw the foundation layer's (life-form, actual use) support for the emergent layer, but moved toward a position approaching 4.4—nearly abandoning system-building, holding that the diversity of language resists all unified frameworks.

The two Wittgensteins each saw one end of the form-meaning spectrum: the early one saw the structural power at the tightest end; the late one saw the diversity at the loosest end. The framework sees the complete spectrum —and the risks at each extreme (colonization and closure).

5.4 Dialogue with Derrida

Derrida's concept of différance argues: meaning is never present—the meaning of each sign depends on its difference from other signs, and this chain of differences never terminates. Meaning forever slides in "deferral" and "difference," never fully captured by any form.

The framework agrees with Derrida's core diagnosis: meaning is not exhausted by form. This is another expression of the inexhaustibility of chiseling—the markability subspace cannot be fully chiseled; there are always new meaning-distinctions to cut; no formal system can cover all meaning. This is structurally isomorphic with the mathematics paper's demonstration that "the quantitative subspace cannot be fully chiseled" and with the framework's formulation of Gödel's incompleteness theorem: the emergent layer cannot fully cover the foundation layer.

The framework disagrees with Derrida's prescription. From "meaning is not exhausted," Derrida moved to a systematic deconstruction of all presence, center, and totality. In the framework's language, this is 4.4 (foundation → emergent closure)—negativity refuses all construction. Derrida saw the colonization of meaning by form (logocentrism), but his remedy was to chisel forever and never construct—to deconstruct forever and never build. This is structurally isomorphic with Adorno's negative dialectics in the philosophy paper: the problem of colonization was diagnosed; the remedy is closure.

Différance has a precise position within the framework: différance is the description of the foundation-layer dimension of the form-meaning binding law—meaning is always chiseling, never exhausted by construct. This is correct. But this is only one half of the two-dimensional structure. The other half: although construct cannot exhaust chisel, construct provides direction and transmissibility for chisel. Chisel without construct is diffuse—meaning slides infinitely in the chain of differences, never landing. Derrida described the inexhaustibility of chiseling but refused the necessity of constructing.

5.5 Dialogue with LLM Emergence Research

The emergent capabilities of LLMs (in-context learning, chain-of-thought reasoning, analogical reasoning, style transfer) are among the central topics in current AI research. Regarding the nature of emergence, there are two major positions in the field.

The first position holds that emergence is real—beyond a certain scale, certain capabilities suddenly appear, rather than increasing gradually. The second position (represented by Schaeffer et al. 2023) holds that emergence may be a measurement mirage—when evaluation metrics shift from nonlinear (e.g., exact match) to linear (e.g., continuous scoring), capability improvement is smooth, with no sudden jump.

The framework offers a perspective different from both. Emergence is neither "the model got bigger so it suddenly acquired new abilities" nor a pure measurement mirage. Emergence is the structural consequence of meaning's associative structure no longer being occluded by symbol boundaries once discreteness is sufficiently reduced. When the discreteness of the model's underlying representations is sufficiently low, the associations between meanings are naturally preserved in the representational space—these associations were always there (not created from nothing); they were merely severed by symbol boundaries in the higher-discreteness regime of human language.

Metrics do affect observation—Schaeffer et al. are right about this. But the released meaning-associations are real—they do not vanish because one switches metrics. The framework predicts: the appearance of emergent capabilities is structurally correlated with the reduction of the model's underlying representational discreteness, and this correlation does not depend on any particular evaluation metric.

Chapter 6. Nontrivial Predictions

Core thesis: From the two-dimensional structure of language and the framework-positioning of the LLM, four nontrivial predictions can be derived, corresponding to testable corollaries of the four structural interactions and of the LLM's structural features.

6.1 Emergent → Foundation (Positive) Prediction: Formal Constraint and Meaning-Density Are Positively Correlated

Prediction: The stricter a genre's formal constraints, the higher the meaning-density per unit of text and the greater the degree of cross-domain meaning-association.

Reasoning: Section 4.1 demonstrated the mechanism of emergent → foundation nurture—formal constraint catalyzes meaning-discovery. The smaller the formal space, the greater the meaning-pressure on each expressive choice, and the higher the meaning-density. The stricter the formal constraint, the more the poet is forced into unconventional word-combinations, and the more cross-domain associations arise.

Testable: Compare semantic density and cross-domain association across genres. The per-character meaning-load of regulated verse (tightest form) should exceed that of free verse, which should exceed that of prose, which should exceed that of transcribed spoken language. This can be quantitatively tested via semantic annotation and information-theoretic metrics (e.g., per-character semantic entropy). Cross-domain associations (e.g., linking emotion to natural landscape) should appear more frequently in regulated verse than in free prose.

Nontriviality: Common sense might hold that "stricter form means more restricted expression and less meaning." This prediction counterintuitively argues: formal strictness is not the suppression of meaning but the catalysis of meaning-density. This is structurally isomorphic with prediction 6.1 in the mathematics paper—the more complete an axiomatic system, the denser the subsequent negation-driven breakthroughs.

6.2 Emergent → Foundation (Negative) Prediction: Rigidity of Formal Norms and Rate of Semantic Innovation Are Negatively Correlated

Prediction: The more rigid a linguistic community's formal norms (i.e., the less deviation from standard form is permitted), the lower that community's rate of new-word generation and semantic drift.

Reasoning: Section 4.2 demonstrated the mechanism of emergent → foundation colonization—formal norms suppress meaning-innovation. The higher the rigidity of formal norms, the greater the force with which nonstandard expressions are excluded, and the narrower the channels for meaning-innovation.

Testable: Compare the diachronic evolution rates of languages with centralized normative institutions (e.g., French under the Académie française) and languages without centralized authority (e.g., English). Prediction: languages under centralized normative governance will have lower rates of new-word acceptance, semantic drift, and core-vocabulary semantic change than languages without centralized norms. This can be quantitatively compared via dictionary update frequency, rate of new-word adoption, and rate of semantic change in core vocabulary.

Nontriviality: This prediction distinguishes "standardization of language" (which can be nurture—a unified standard enables cross-dialect communication) from "rigidity of formal norms" (colonization—the standard excludes all deviation). Standardization per se is not colonization; the rigidification of standardization is. The norms of the Académie française serve both a nurture function (maintaining cross-regional intelligibility of French) and a colonization function (excluding non-standard French meaning-innovation)—the framework predicts that the net effect of the colonization function is to reduce the rate of semantic innovation.

6.3 Foundation → Emergent (Positive) Prediction: Lower Discreteness, Stronger Emergence

Prediction: All else being equal (model scale, training data volume, training method), the lower the discreteness of a language model's underlying representations, the stronger its emergent capabilities.

Reasoning: Section 3.4 demonstrated that emergence is the structural consequence of reduced discreteness—the lower the discreteness, the less symbol boundaries sever meaning, and the more completely the associative structure of meaning is preserved in the representational space.

Testable: Compare models of different architectures—different tokenization granularities, different representation dimensions, different representation methods—on emergent capabilities (analogical reasoning, cross-domain transfer, in-context learning). The framework predicts: finer-grained tokenization (reducing discreteness at the input end), higher-dimensional representational spaces (reducing the impact of discreteness at the representation end) will positively correlate with stronger emergent capabilities. Conversely: if discreteness is forcibly increased (e.g., extremely coarse-grained tokenization, forced discretization of representations), the framework predicts a decline in emergent capabilities.

Nontriviality: Current LLM research primarily explains emergence in terms of model scale (parameter count) and training data volume. This prediction introduces a variable independent of scale—discreteness—and predicts its relationship to emergent capabilities. If two models have the same parameter count and training data volume but differ in the discreteness of their underlying representations, the framework predicts the model with lower discreteness will exhibit stronger emergence. This is a falsifiable prediction—if underlying representational discreteness is found to be uncorrelated or negatively correlated with emergent capabilities, the framework is falsified at this point.

6.4 Foundation → Emergent (Negative) Prediction: Semantic Dissipation in Uncalibrated LLMs

Prediction: The semantic coherence of uncalibrated LLM output monotonically decreases with generation length. Human calibration with integrity can slow but not eliminate this decay.

Reasoning: Section 3.4 demonstrated that the LLM releases meaning without the subject's directional calibration. Meaning released in a space of lower discreteness is isotropic—no direction is more "important" than any other. The longer the generation, the greater the cumulative directional drift, and the more severe the semantic dissipation. Human calibration with integrity (carefully designed prompts, RLHF training, contextual guidance) provides direction for dissipating meaning, but calibration is an external input, not a structure internal to the LLM—hence it can slow but not eliminate the decay.

Testable: Measure the semantic coherence metric (e.g., inter-paragraph semantic similarity, topic drift rate) of LLM output under unguided conditions (zero-shot, long-text generation) as a function of generation length. The framework predicts a monotonically decreasing curve. Compare with the same curve under human-guided conditions (detailed prompts, segmented guidance); the decay rate is predicted to be lower but still present. If unguided LLM semantic coherence is found not to decay, or even to increase, in long texts, the framework is falsified at this point.

Nontriviality: Common sense might hold that "better models are more coherent." This prediction argues: semantic dissipation is not a matter of model quality but a structural feature of the LLM as a subjectless meaning-releasing system. Better models can reduce the rate of dissipation but cannot eliminate dissipation itself—because eliminating dissipation requires not a better model but a subject with directional judgment. This implies that the more powerful AI becomes, the higher the demands on human integrity—not lower. This is the framework's structural judgment on the AI–human relationship.

6.5 Structural Prediction: The Direction of Negation After the LLM

Prediction: If a further technological breakthrough occurs after the LLM, its emergent capabilities will derive from further negation of the LLM's current underlying representation (further reduction of discreteness or a fundamental transformation of representation), not from scaling up within the current representational approach.

Reasoning: The framework's general structure: emergence arises from negation of the foundation layer. The LLM's emergence arises from negation of human language's discrete form. Emergence after the LLM—if it occurs—should arise from negation of the LLM's current representational approach, not from quantitative expansion within the same approach. Quantitative expansion (more parameters, more data) can improve existing capabilities but does not produce qualitatively new emergence—qualitative transformation comes from negation, not from accumulation.

Testable: Track the trajectory of future AI technology development. If qualitatively new emergent capabilities appear, check the source: does it come from expansion of model scale, or from a fundamental transformation of the underlying representational approach? The framework predicts the latter. If sustained scaling produces a new round of qualitatively transformative emergence of the same nature as the LLM's initial emergence, the framework is falsified at this point.

Nontriviality: The mainstream direction of current AI research is the scaling law—improving capabilities by expanding model scale. The framework predicts this path has a structural ceiling: quantitative expansion within the same representational approach produces quantitative change, not qualitative transformation. Qualitative transformation requires negation—negation of the current representational approach itself. This prediction forms a testable tension with the scaling-law research program.

Chapter 7. Conclusion

Language is the activity of a subject exercising negation upon the markability dimension of the law of identity. In the Self-as-an-End framework's application spectrum, language and mathematics are of the same order—both are second-order chisels, both take the law of identity as transcendental ground. Mathematics chisels the quantitative subspace and constructs the law of noncontradiction. Language chisels the markability subspace and constructs the form-meaning binding law—every linguistic sign is necessarily a unity of form and meaning.

Language possesses a unique structural feature absent from both mathematics and philosophy: the binding of form upon meaning admits a spectrum from tight to loose (poetry → prose → novel). This is internal formal adjustment within the discrete symbol system of human language. The LLM's breakthrough lies outside this spectrum—the LLM negates discreteness itself, reducing the discreteness of underlying representation below that of human language. Emergent capabilities are the structural consequence of this negation: discreteness is reduced, symbol boundaries sever meaning less, the associative structure of meaning is preserved in the space of lower discreteness—analogy, cross-domain association, style transfer, and other capabilities manifest from this.

But the LLM releases meaning without direction. Meaning in a space of lower discreteness is isotropic—no subject's integrity determines which direction of unfolding is worthwhile. This means the LLM has greater unfolding capacity than any individual, but also requires a person of integrity to calibrate it. The more powerful AI becomes, the higher the demands on human integrity.

Contributions

I. The same-order parallel positioning of language and mathematics. Language chisels the markability subspace of the law of identity; mathematics chisels the quantitative subspace. Both are of the same order, in different dimensions. "Mathematical language" and "computer language" are structural facts, not metaphors.

II. The form-meaning binding law. The product of language chiseling the law of identity—every linguistic sign is necessarily a unity of form and meaning. Saussure's signifier/signified duality is the empirical discovery of this law. Binding is necessary (the structural consequence of chisel-construct); specific correspondence is contingent (the historical product of a community).

III. The tightness spectrum of formal binding (poetry → prose → novel). Not the contingent progression of literary history, but the structural unfolding of tension internal to the form-meaning binding law. The inversion of cognitive order (tightest first) and structural order (form-meaning binding law precedes any specific formal constraint) is the structural consequence of the second-order chisel.

IV. The framework-explanation of LLM emergence. The LLM negated human language's discrete form; emergence is the structural consequence of meaning's associative structure no longer being occluded by symbol boundaries once discreteness is reduced. The framework's claim does not depend on the LLM's specific technical implementation—it attends to the direction of negation (reduction of discreteness), not the landing point (what specifically replaces it).

V. The prediction relating discreteness to emergent capabilities. The lower the discreteness of underlying representations, the stronger the emergent capabilities. This is a predictive variable independent of model scale, and it is falsifiable.

VI. The structural judgment on the AI–human relationship. The LLM releases meaning without direction; a person of integrity is needed for calibration. The more powerful AI becomes, the higher the demands on human integrity—not lower.

Open Questions

I. The next step of negation after the LLM. If the LLM's underlying representational approach is itself negated—for instance, by the appearance of a representational approach of even lower discreteness than the current numerical space—does this mean meaning can detach from all carriers perceivable by humans? What does this imply for subjectivity? If the unfolding of meaning no longer requires a human-perceivable interface, is the human role as calibrator also negated?

II. Formal differences among natural languages. Chinese (logographic writing, no morphological inflection, tonal) and English (phonographic writing, rich morphological inflection, non-tonal) represent two different modes of form-meaning binding. Do these differences correspond to different modes of meaning-binding? The framework predicts they are different realizations of the same form-meaning binding law, but the detailed analysis of these differences is deferred to subsequent papers.

III. The framework-positioning of LLM hallucination. The structure of the framework suggests: hallucination is not "error" but the free sliding of meaning in a space of lower discreteness. In the discrete symbol system of human language, "a cat is a dog" is a logical error—it violates a classificatory boundary. In the LLM's space of lower discreteness, "cat" and "dog" are neighboring positions, and meaning slides from one position to the other without crossing a hard boundary. What humans call "factual accuracy" is essentially the use of discrete anchors (facts, logic, empirical verification) to pin continuously sliding meaning to specific positions. The LLM lacks these anchors, so meaning is always sliding. If this characterization holds, it implies that hallucination is not a defect of the model but a structural cost of reduced discreteness—sharing the same source as emergent capabilities. Whether it is possible to eliminate hallucination without damaging emergent capabilities is a technical question to be developed in this series' LLM model application paper.

IV. The convergence of language and mathematics at the substrate level. Language and mathematics are two independent subspaces of the law of identity (markability vs. quantity). But as the discreteness of language's substrate continues to decrease toward continuity, the representational space of language increasingly resembles the objects that mathematics treats—continuous structures, topological relations, metric spaces. In the embedding spaces of current LLMs, semantic relations are already geometric relations in the literal sense (distance, direction, angle). This implies that two originally independent subspaces may begin to converge at the level of substrate geometry. This is structurally isomorphic with the mathematics paper's Open Question I, which raised the convergence of mathematics and physics at the geometric level: the mathematical construct (the law of noncontradiction) becomes part of the transcendental ground of physics, and the two share structure at the geometric seam. If the convergence of language and mathematics holds, it predicts that the LLM's performance on mathematical tasks is not merely "incidental mathematical ability" but structural—the continuization of the underlying representation enables linguistic and mathematical operations to share the same geometric substrate. But the linguistic construct centers on transmissibility and the mathematical construct centers on compulsion; how these coexist at the substrate level once convergence occurs at the ground level is a question that requires input from mathematicians and computer scientists and lies beyond the argumentative scope of this paper.

V. Non-linguistic symbol systems. Music uses symbols to mark distinctions of pitch and duration; painting uses form to organize visual meaning; dance uses bodily form to carry kinesthetic meaning. Do these non-linguistic symbol systems share the same structure of law of identity → markability subspace → form-meaning binding law? The framework predicts yes—any activity that uses stable forms to mark distinctions operates within the markability subspace. But the full demonstration of this prediction will be developed in this series' aesthetics application paper.

Author's Declaration

This paper is the author's independent theoretical work. During the writing process, AI tools were used as conversation partners and writing assistants for conceptual refinement, argument testing, and text generation: Claude (Anthropic) provided primary writing assistance; Gemini (Google), ChatGPT (OpenAI), and Grok (xAI) participated in manuscript review and feedback. All theoretical innovations, core judgments, and final editorial decisions were made by the author. The role of AI tools in this paper is analogous to that of a real-time research assistant and reviewer, and does not constitute co-authorship.