SAE QM Paper 6: The Born Rule as the Algebraic Projection of Closure

Qin, Han

doi:10.5281/zenodo.20479658

Abstract

The Born rule — that the probability of a measurement outcome is given by the modulus squared of the quantum state — is introduced as a postulate in standard quantum mechanics and confirmed by experiment to very high precision, yet the origin of its modulus-squared form has long been a central question in the foundations of quantum mechanics. The major approaches locate this origin variously in the mathematical uniqueness of Hilbert space (Gleason), the decision-theoretic norms of a rational agent (Deutsch–Wallace), the symmetry of system–environment entanglement (Zurek's envariance), subjective belief (QBism), or hidden variables (Bohmian mechanics). This paper gives an ontological origin for the modulus-squared readout within the SAE framework. The central claim: the modulus squared is the one-sided reading of a pre-closure state that lies below the causal slot, takes the uncertainty-wave form, and is simultaneously subject to time arrows on both sides — and is therefore without causality. In this region the time arrow and causality are two distinct things. The time arrow exists at every scale: a quantum evolves forward normally along its own time arrow in our frame (the dynamical layer, governed by P3's $i\hbar\,\partial_t = H$). Causality, by contrast, has as its ontological content the influence of a single-direction time arrow — above the causal slot the state is subject only to our-frame single-direction influence, so causality holds and the outcome is definite; below the causal slot the state is subject simultaneously to the our-frame and opposite-frame bidirectional time arrows (the two being two faces of a single dual-4DD structure, established in Cosmo V), so no single direction can fix causation and there is no causality. Precisely because the quantum is subject to both sides at once, with no single direction available to orient it, a one-sided reading from our side cannot read off any pre-written definite local content, and the state presents as a probability distribution (for a generic state; if the state is already an eigenstate of the given measurement, this degenerates to a definite $P=1$ readout). What is subject to both-sided influence is the mode of presentation of the readout (the readout layer), not the evolution of the quantum (which proceeds forward in our frame as usual). Reading such a state one-sidedly manifests, at the amplitude layer, as the conjugate pairing of dual orientations (the $e^{i\theta}$ of our frame and the $e^{-i\theta}$ of the opposite frame), $\psi^\sharp\psi = A^2$ — the modulus squared. The reading event itself occurs at $L_4$ re-activation (i.e. 4DD closure); the object read lies before closure, and the reading event is closure itself. The paper works at three levels of commitment. At the standard mathematical level (T1), the uniqueness of the Born measure is supplied by Gleason-type theorems under their usual conditions; the paper does not re-derive these but only interfaces with them. At the SAE structural level (T2), the paper gives a strong structural derivation of the quadratic readout — this is the paper's core contribution, but it is not a full theorem equivalent to Gleason's. At the mechanism level (T3), the paper offers an ontological picture of one-sided projection of a bidirectional orientation, as a candidate. The paper answers one key ontological question: why the both-sided reading below the causal slot takes a multiplicative pairing, rather than the additive form that remainder conservation takes at the cosmological layer. The answer: reciprocity (remainder conservation) by itself prescribes neither addition nor multiplication; what it prescribes is that one side must close with its dual counterpart in the native algebra of the object being read — a scalar energy density closes additively to a vanishing sum in the additive group ($\Lambda_1 + \Lambda_2 = 0$), while a complex phase carrier closes to a multiplicative pairing in the $U(1)$ multiplicative group ($\psi^\sharp\psi = A^2$). These are two realizations of one and the same reciprocity principle in different native algebras, not two separate rules. At a deeper level still, the two substrates are both additive vanishing under reciprocity ($\theta + \theta^\sharp = 0$); the microscopic layer, after exponential dualization, manifests at the carrier layer as multiplicative. On this basis the paper gives: the ontological placement of the modulus squared and of Born-probability randomness (randomness is not arbitrariness, but the absence of pre-written content below the causal slot, with statistical weights supplied by dual-orientation pairing); the relation to Gleason, Many-Worlds, QBism, and Bohmian mechanics (the demarcation from Many-Worlds resting on a causality firewall: branches are multiple worlds each carrying causality, whereas the two sides are a single bidirectional orientation without causality); and the interfaces with P4's cross-ladder readout, P5's Tsirelson bound, and no-signaling marginals. Whether the Born rule is modified in the strong-field regime is marked as an open problem: absent posterior support, the relation between the Born rule and field strength cannot be given a priori. ---

Keywords:

Abstract

The Born rule — that the probability of a measurement outcome is given by the modulus squared of the quantum state — is introduced as a postulate in standard quantum mechanics and confirmed by experiment to very high precision, yet the origin of its modulus-squared form has long been a central question in the foundations of quantum mechanics. The major approaches locate this origin variously in the mathematical uniqueness of Hilbert space (Gleason), the decision-theoretic norms of a rational agent (Deutsch–Wallace), the symmetry of system–environment entanglement (Zurek's envariance), subjective belief (QBism), or hidden variables (Bohmian mechanics).

This paper gives an ontological origin for the modulus-squared readout within the SAE framework. The central claim: the modulus squared is the one-sided reading of a pre-closure state that lies below the causal slot, takes the uncertainty-wave form, and is simultaneously subject to time arrows on both sides — and is therefore without causality. In this region the time arrow and causality are two distinct things. The time arrow exists at every scale: a quantum evolves forward normally along its own time arrow in our frame (the dynamical layer, governed by P3's $i\hbar\,\partial_t = H$). Causality, by contrast, has as its ontological content the influence of a single-direction time arrow — above the causal slot the state is subject only to our-frame single-direction influence, so causality holds and the outcome is definite; below the causal slot the state is subject simultaneously to the our-frame and opposite-frame bidirectional time arrows (the two being two faces of a single dual-4DD structure, established in Cosmo V), so no single direction can fix causation and there is no causality. Precisely because the quantum is subject to both sides at once, with no single direction available to orient it, a one-sided reading from our side cannot read off any pre-written definite local content, and the state presents as a probability distribution (for a generic state; if the state is already an eigenstate of the given measurement, this degenerates to a definite $P=1$ readout). What is subject to both-sided influence is the mode of presentation of the readout (the readout layer), not the evolution of the quantum (which proceeds forward in our frame as usual). Reading such a state one-sidedly manifests, at the amplitude layer, as the conjugate pairing of dual orientations (the $e^{i\theta}$ of our frame and the $e^{-i\theta}$ of the opposite frame), $\psi^\sharp\psi = A^2$ — the modulus squared. The reading event itself occurs at $L_4$ re-activation (i.e. 4DD closure); the object read lies before closure, and the reading event is closure itself.

The paper works at three levels of commitment. At the standard mathematical level (T1), the uniqueness of the Born measure is supplied by Gleason-type theorems under their usual conditions; the paper does not re-derive these but only interfaces with them. At the SAE structural level (T2), the paper gives a strong structural derivation of the quadratic readout — this is the paper's core contribution, but it is not a full theorem equivalent to Gleason's. At the mechanism level (T3), the paper offers an ontological picture of one-sided projection of a bidirectional orientation, as a candidate.

The paper answers one key ontological question: why the both-sided reading below the causal slot takes a multiplicative pairing, rather than the additive form that remainder conservation takes at the cosmological layer. The answer: reciprocity (remainder conservation) by itself prescribes neither addition nor multiplication; what it prescribes is that one side must close with its dual counterpart in the native algebra of the object being read — a scalar energy density closes additively to a vanishing sum in the additive group ($\Lambda_1 + \Lambda_2 = 0$), while a complex phase carrier closes to a multiplicative pairing in the $U(1)$ multiplicative group ($\psi^\sharp\psi = A^2$). These are two realizations of one and the same reciprocity principle in different native algebras, not two separate rules. At a deeper level still, the two substrates are both additive vanishing under reciprocity ($\theta + \theta^\sharp = 0$); the microscopic layer, after exponential dualization, manifests at the carrier layer as multiplicative.

On this basis the paper gives: the ontological placement of the modulus squared and of Born-probability randomness (randomness is not arbitrariness, but the absence of pre-written content below the causal slot, with statistical weights supplied by dual-orientation pairing); the relation to Gleason, Many-Worlds, QBism, and Bohmian mechanics (the demarcation from Many-Worlds resting on a causality firewall: branches are multiple worlds each carrying causality, whereas the two sides are a single bidirectional orientation without causality); and the interfaces with P4's cross-ladder readout, P5's Tsirelson bound, and no-signaling marginals. Whether the Born rule is modified in the strong-field regime is marked as an open problem: absent posterior support, the relation between the Born rule and field strength cannot be given a priori.

Part I: The Mainstream Landscape of the Born Rule's Origin

Since its proposal in 1926, the standing and origin of the Born rule have been a central question in the foundations of quantum mechanics. This part reviews the major approaches as background for the SAE account that follows. Each approach has substantive content and its own disputes; this part states them objectively without adjudication.

The Born rule as a postulate. In the usual formulation of standard quantum mechanics, the Born rule is introduced as an independent postulate: the probability of a measurement outcome is given by the modulus squared of the projection of the state vector onto the corresponding eigenstate. This formulation does not inquire into the origin of the modulus squared but takes it as the bridge connecting the formalism to experiment, confirmed by experiment to very high precision. Whether the modulus squared can be derived from the remaining postulates, or must be assumed independently, has been debated at length.

The Gleason approach. In 1957 Gleason proved that, on a Hilbert space of dimension no less than 3, a probability measure satisfying non-contextuality is given uniquely by the trace formula with a density operator, so that the modulus-squared form is uniquely fixed. This approach grounds the mathematical uniqueness of the Born rule in Hilbert-space structure together with non-contextuality. Its limitations lie in the dimensional condition (the two-dimensional qubit requires an additional assumption, or a POVM version / Busch-type generalization) and in its reliance on the non-contextuality premise.

The decision-theoretic approach (Deutsch–Wallace). Within the Everett interpretation, Deutsch and Wallace attempt to derive the Born weights from the decision axioms of a rational agent: an agent betting within a branching structure is forced by rationality constraints to assign credence by the modulus squared. This approach locates the origin of probability in the normativity of rational decision. Whether it presupposes the conclusion it sets out to derive, and how branch-counting relates to probability, remain under continuing debate.

Environment-induced selection and envariance (Zurek). Zurek proposes that system–environment entanglement carries an exchange symmetry (environment-assisted invariance, envariance), from which it can be argued that observable readouts depend only on the modulus squared of the amplitude. This approach locates the origin of the Born rule in the symmetry of system–environment entanglement. Whether the envariance argument is independent of the Born rule itself has likewise been discussed.

QBism. QBism (quantum Bayesianism) interprets the quantum state as the belief state of an agent and the Born rule as a normative constraint by which a rational agent updates belief. On this approach, probability is the agent's epistemic attitude, not an objective property of the external world.

Bohmian mechanics. Bohmian mechanics introduces definite particle trajectories as hidden variables and interprets $|\psi|^2$ as the probability distribution of initial positions; measurement randomness arises from epistemic ignorance of definite but unknown initial positions. On this approach, probability is epistemic and the underlying dynamics is deterministic.

These approaches give different answers to whether the Born rule is a postulate or a theorem, and whether its origin lies in mathematical structure, rational decision, environmental symmetry, subjective belief, or hidden variables. The SAE account that follows belongs to none of them; it gives an ontological origin for the modulus-squared readout. Its specific relations to several of these approaches are developed in Thesis C.

Thesis A: The Modulus Squared as the One-Sided Reading of a Below-Causal-Slot Uncertainty-Wave State (a State Simultaneously Subject to Both-Sided Time Arrows)

A.1 What this section does and does not answer

The Born rule relates a quantum state $\psi$ to the probability of a measurement outcome: on a normalized state, the probability of outcome $i$ is $|\psi_i|^2$. In standard quantum mechanics this rule is introduced as a postulate — not derived from more basic structure but assumed, and then confirmed by nearly eight decades of experiment to very high precision.

SAE neither restates this postulate nor attempts to replace any of its computational consequences. What this section addresses is a question standard quantum mechanics does not: what is the ontological origin of the modulus-squared readout form — why a quadratic in the amplitude, rather than a first, third, or other power?

The SAE argument falls at three levels of commitment, each with a different boundary:

T1 (standard mathematical level, not re-derived by SAE): the uniqueness of the Born measure on any Hilbert space, supplied by Gleason-type theorems under their usual conditions on dimension and measurement structure. SAE does not re-prove these theorems.
T2 (SAE structural level, assertible): why a one-sided reading within the SAE framework must be a quadratic readout — this is the section's core contribution.
T3 (mechanism level, a candidate): the ontological picture of how a bidirectional temporal orientation and a one-sided projection concretely generate $\psi^\sharp\psi$.

What this section gives is a strong structural derivation of the quadratic readout, not a full theorem equivalent to Gleason's. In other words, SAE asserts, fairly strongly, that "the modulus squared is not ad hoc — its structural origin lies in $L_1\leftrightarrow L_2$ closure"; SAE does not assert "that SAE derives the uniqueness of the Born measure on an arbitrary Hilbert space." The latter, as a T1 mathematical result, is taken over externally from the standard theorems, with which this section interfaces (is consistent), rather than being treated as a corollary of SAE.

A.2 Locating the object read: the pre-closure state below the causal slot, in uncertainty-wave form, simultaneously subject to both-sided time arrows

To ask "what does the modulus squared read," one must first locate the object read.

The SAE remainder unfolds on the two sides of the causal slot in two distinct forms. This distinction is not introduced here but is an established structure running through several SAE series; this section cites it without restating.

Below the causal slot: the remainder unfolds in uncertainty-wave form. Its ontological carrier is the dualized complex amplitude established in P2, $\psi = A\,e^{i\theta}$, where $A$ is the 1DD marking-layer density, $e^{i\theta}$ is the already-formed ontological presence of the $L_1\to L_2$ dualization, and $\theta$ is the static position of a single cell on its compact $U(1)$ fiber. Two ontological features of this region are decisive for this section.

First, the time arrow exists, but causality does not. Time arrow and causality are two distinct things in SAE.

The time arrow exists at every scale. Time at the Planck substrate runs forward as usual — a quantum evolves forward normally along its time arrow in our frame. This evolution belongs to the dynamical layer, described by P3's established $i\hbar\,\partial_t = H$. The quantum is not "suspended motionless in some bidirectional state" — it has a definite our-frame time direction and evolves normally.

Causality is another matter. The ontological content of causality is the influence of a single-direction time arrow. To adjudicate before-and-after, to distinguish cause from effect, to orient a process into a single causal order, the remainder must be subject to a single-direction time arrow. Accordingly, the two sides of the causal slot differ in how many directions of time arrow they are subject to:

Above the causal slot: the remainder is subject only to the our-frame single-direction time arrow — single-direction, hence causality holds, the process is oriented, and it presents as definite classical causation.
Below the causal slot: the remainder is subject simultaneously to the our-frame and opposite-frame time arrows in both directions (the two being two faces of this one dual-4DD structure, established in Cosmo V) — bidirectional, with no single direction available to adjudicate the causal order, hence no causality.

Terminological convention: the below-causal-slot situation in which both sides coexist is termed, technically, a dual temporal orientation; the phrase "time arrow" is retained in this paper as the intuitive expression. Causality is by its nature oriented, so in defining causality we use "single-direction temporal direction."

This is consistent with the established result of Information Theory P1 — that causality is a structural consequence of the asymmetry of 4DD closure — and gives that result a more direct ontological content: 4DD closure produces causality precisely because closure makes the bidirectional influence single-directional (selecting our side as the unique causal direction); before closure (below the causal slot), bidirectional influence coexists, single-directionalization has not yet occurred, and so there is no causality.

Second, from "simultaneously subject to bidirectional time arrows, no causality" follows the probability-cloud presentation of the one-sided reading. Causality would orient the process and yield a definite outcome order; below the causal slot it has not yet occurred, so when our side reads the quantum one-sidedly, because the quantum is subject to both-sided our-frame and opposite-frame influence with no single direction available to orient it, the one-sided reading cannot read off a pre-written definite local content, and presents as a probability distribution. Two points must be stated precisely. First, "cannot read a definite content" means: below the causal slot there is no pre-written outcome available to be read — it does not mean any reading is numerically impossible to be definite. If, relative to a given measurement partition, the state has already degenerated to a single closable branch (i.e. it is an eigenstate of that measurement), its weight degenerates to a definite $P=1$ readout, consistent with this section. The probability cloud is the distribution by which a generic (non-eigenstate) state presents under reading according to its weights. Second, what is subject to both-sided influence is the mode of presentation when the quantum is read one-sidedly (the probability distribution), not the evolution of the quantum itself — evolution proceeds forward in the our-frame time arrow as usual (the dynamical layer), while both-sided influence concerns the readout presentation (the readout layer). The dynamical layer and the readout layer must be kept separate. Section A.3 below gives the precise readout form of this "both-sided influence" at the amplitude layer.

Above the causal slot: the remainder changes form, unfolding in gravitational-wave form (established in Relativity P1/P2: gravitational waves are the Planck-substrate broadcast of the 4DD topological-refresh command). Here causality is active, the time arrow is single-direction, and the 4DD causal load $I$ enters. This is not the object this section reads — once the remainder crosses to above the causal slot, it is no longer the $\psi$ uncertainty wave, and the Born rule no longer reads it.

This locates the object of the Born reading:

> The object of the Born reading is the pre-closure state below the causal slot, in uncertainty-wave form, simultaneously subject to the our-frame and opposite-frame bidirectional time arrows, and therefore without causality.

Here "both-sided influence" refers to the state's being acted on simultaneously by two directions of the time arrow, not to the state's itself performing some bidirectional evolution — the quantum evolves forward normally along its our-frame time arrow (the dynamical layer); "both-sided influence" characterizes the mechanism of its presentation when read one-sidedly (the readout layer).

What 4DD closure does is exactly to make this "both-sided influence" single-directional — to select our side as the unique causal direction, whereupon causality is produced and the process oriented, while the remainder form switches from uncertainty wave to gravitational wave.

The Born rule bears one relation to P5's established result that "measurement is $L_4$ re-activation," a relation that must be made clear lest it be read as a contradiction — the object read and the reading event are two things: the object read is the below-causal-slot state, not yet single-directionalized, which lies before closure; the reading event is, ontologically, $L_4$ re-activation, i.e. the 4DD closure event itself. The two do not conflict — it is not that "a measurement was made before closure," but rather that closure, as an act, intercepts and reads that pre-closure state. One sentence may be fixed:

> The object read by the Born rule lies below the causal slot; the Born reading event occurs at $L_4$ re-activation. This paper gives the weight form when that event reads the pre-closure state; the complete ontological mechanism of a single closure event and the measurement apparatus is left to P7.

This temporal layering can be written as three stages, making the boundaries of P5 / P6 / P7 clear: before closure, below the causal slot there is a ρ-OR carrier and capacity structure, with no content and no local outcome (the object read); $L_4$ re-activation selects a closure partition (the measurement apparatus supplies the outcome partition / projection operators — this is P5's established measurement ontology); this paper assigns each branch of that partition a weight $w_i = \psi_i^\sharp\psi_i$ (P6's domain), while how a single outcome is actually recorded belongs to P7.

The remainder of this section shows that reading one-sidedly a state simultaneously subject to both-sided influence necessarily yields a quadratic.

A.3 Main derivation: the one-sided reading is the multiplicative pairing of dual orientations

The core of the main derivation: what a one-sided reading reads is the multiplicative pairing, at the our-side interface, of $\psi$ with its dual-orientation counterpart, whose minimal real form is the quadratic.

Let the below-causal-slot uncertainty-wave carrier be

$$\psi = A\,e^{i\theta}.$$

Its dual-orientation counterpart is written

$$\psi^\sharp = A\,e^{-i\theta}.$$

The ontological identity of $\psi^\sharp$ refers to three distinct things on three layers, which must not be conflated:

Formal layer: $\psi^\sharp$ is the formal counterpart of the same $\psi$ under dual orientation.
SAE ontological layer: $\psi^\sharp$ is the opposite-frame representation of the same dual-4DD structure (established in Cosmo V: the universe has a dual-4DD structure, our frame and opposite frame). It is not a second ontology, not a second world, not a second timeline. $\psi$ and $\psi^\sharp$ are two frames of the same dual-4DD structure, the degrees of freedom remaining 1 — consistent with P2's established "degrees of freedom always 1."
Standard quantum-mechanical layer: in the Hilbert representation, $\psi^\sharp$ corresponds to the bra / conjugate-dual structure, not to another ket state.

$\psi^\sharp$ is not a second wavefunction arriving from the opposite side. There is no information transfer below the causal slot (the hard constraint of Information Theory P4); the pairing in this section is not a cross-side communication, but a single-sided closure event's reading of that state which is simultaneously subject to both-sided time arrows and not yet single-directionalized by causality. $\psi^\sharp$ is the dual slot required by this reading, not reified into a second degree of freedom.

The one-sided reading reads the overlap of the two at the our-side interface:

$$R(\psi) = \psi^\sharp\,\psi = A\,e^{-i\theta}\cdot A\,e^{i\theta} = A^2.$$

This readout is automatically real, non-negative, and phase-gauge invariant. Its structural content:

The phase direction is cancelled: $e^{-i\theta}\,e^{i\theta} = 1$. The phases of the dual orientations are made real against each other in the pairing; the readout no longer carries a phase direction.
The capacity is preserved: $A\cdot A = A^2$. The 1DD marking-layer density $A$ survives the pairing as $A^2$, a capacity measure that can be normalized into a probability weight.

This is exactly the modulus squared. It follows from the structural fact "reading one-sidedly a state simultaneously subject to both-sided time arrows": the both-sided influence manifests at the amplitude layer as a conjugate pair of dual orientations (the $e^{i\theta}$ of our frame and the $e^{-i\theta}$ of the opposite frame), and reading this conjugate pair one-sidedly necessarily yields a quadratic.

A.4 Why a multiplicative pairing, and not additive — unification with the cosmological $\Lambda$ layer

The first step of the main derivation immediately raises a sharp question:

> Why is the both-sided reading below the causal slot a multiplicative pairing $\psi^\sharp\psi$, rather than an additive form $\Lambda_1 + \Lambda_2 = 0$ as in the cosmological-layer remainder conservation?

The question is sharp because SAE established long ago, at the cosmological layer, that the field-theoretic statement of remainder conservation on the dual-4DD structure is $\Lambda_1 + \Lambda_2 = 0$ — the two cosmological constants cancel exactly, and the nonzero $\Lambda$ observed on each side is the residual of having observed only half of a symmetric structure (established in Cosmo I/V). This is additive. Were this section to change the same conservation principle to multiplicative at the microscopic layer out of nowhere, that would be an ad hoc patchwork.

The answer separates three things, each carried by a different established structure: reciprocity answers "why there is a dual side $\psi^\sharp$"; $L_1\leftrightarrow L_2$ symplectic conjugation answers "why the two dual sides close as a multiplicative pairing"; the inversion-factor identity of $i$ together with the $U(1)$ carrier answers "why the pairing is between $e^{i\theta}$ and $e^{-i\theta}$." Each does its own work, together yielding $\psi^\sharp\psi$, and each is anchored in established work rather than being this section's ad hoc choice.

A.4.1 The origin of the dual side: the reciprocity relation (established in Cosmo I)

"Why is the below-causal-slot state co-determined by both sides, and why is there a $\psi^\sharp$ side on reading," is answered by the reciprocity relation established in Cosmo I, cited here:

> The remainder of the reverse-causal side just is the causality of the causal side. Causality suppresses the remainder; but seen from the opposite side, our causality is precisely the opposite side's unsuppressed remainder, and vice versa. Remainder and causality are not two different existences, but one existence seen from opposite sides.

Hence "both sides" is not two separate things each exerting influence, but: below the causal slot, this reciprocal whole has not yet been split apart by causality. The 3DD symmetry produces a dual 4DD with opposite time arrows; the reciprocity relation makes the two sides two faces of one existence; below the causal slot causality is not active, the adjudication that would suppress the two sides into one has not yet occurred, and so the state is intrinsically both-sided. A one-sided reading reads exactly this not-yet-split reciprocal whole, and therefore necessarily reads both sides.

It must be made clear: the reciprocity relation has so far answered only the origin of the dual side — why both $\psi$ and $\psi^\sharp$ are present on reading. It does not answer "whether the two sides close additively or multiplicatively"; that is decided by the next layer, the symplectic-conjugation structure.

A.4.2 The form of the pairing: $L_1\leftrightarrow L_2$ symplectic conjugation (established in P3)

Given $\psi$ and its dual counterpart $\psi^\sharp$, a one-sided content reading must obtain a scalar capacity from the two sides. What algebraic form this scalar takes is determined by the layered structure in which the object read sits.

The object read sits in the $L_1\leftrightarrow L_2$ symplectic-conjugation closure layer (established in P3: $[\hat{x},\hat{p}] = i\hbar$ is the invariant form of this layer, and $S_{act} = \hbar\theta$ is its bare readout). A symplectic structure is an antisymmetric bilinear form; its intrinsic pairing operation is not to add two quantities, but to take the carrier together with its dual carrier in an evaluation pairing — which in the standard Hilbert representation is the inner product $\langle\psi|\psi\rangle$, and in the one-dimensional complex-phase representation is $\psi^\sharp\psi = A^2$.

This is the direct ontological origin of the multiplicative pairing: it is not supported by the single statement "because the amplitude is in $U(1)$ it takes a product," but rather by the fact that the $L_1\leftrightarrow L_2$ symplectic-conjugation layer, on a one-sided content reading, must yield a positive-definite Hermitian pairing through its antisymmetric bilinear structure. Additive cancellation $\psi + \psi^\sharp$ is not the intrinsic pairing operation of the symplectic layer; it cannot yield the bilinear scalar the symplectic structure requires.

> Minimal characterization of the $\sharp$ map: writing the dual orientation as a map $\sharp:\mathcal{H}\to\mathcal{H}^$, in the generic weak-field case it is the standard anti-linear dual (Hermitian conjugation): $(\alpha\psi + \beta\phi)^\sharp = \alpha^\psi^\sharp + \beta^*\phi^\sharp$, with $\psi^\sharp(\psi) \geq 0$. $\psi^\sharp\psi$ is thus not a metaphor but a dual-evaluation / Hermitian pairing structure recognizable to the reader in mathematical physics.

A.4.3 The origin of the complex phase carrier: the established-dependency declaration and the inversion-factor identity of $i$

The previous layer used the premise "the object read is an $L_1\leftrightarrow L_2$ complex phase carrier"; this premise is itself not re-derived in this paper, but inherited from established work. Here the dependency is made explicit, lest "the $U(1)$ complex phase carrier" be read as a natural fact of this section or as an ad hoc choice: this paper does not re-derive within P6 why the pre-closure state is a complex phase carrier — that proposition is carried by P2 together with Foundation v2 and P3. P2 established $\psi = A\,e^{i\theta}$, where $\theta$ is the static position on a compact $U(1)$ fiber, and $i$ and $\pi$ interlock through the exponential map (the degrees of freedom always 1); Foundation v2 established $i$ as the algebraic inversion factor of the topological closure of the causal slot; P3 established the $L_1\leftrightarrow L_2$ symplectic-conjugation closure and $S_{act} = \hbar\theta$. The new proposition of this paper is not "why there is a complex amplitude," but "given this complex phase carrier, why the capacity scalar of a one-sided content reading is $\psi^\sharp\psi$, and not a first power, a third power, or an additive cancellation."

"Why a complex phase carrier, and not a real vector field or a quaternion" — the domain of this question lies in P2 and Foundation v2, not in P6. But there is a substantive connection between the inversion-factor identity of $i$ and $U(1)$, which can be made explicit here, because it is precisely the ontological origin of the $i$ in $e^{i\theta}$:

The $i$ in $e^{i\theta}$ is not an arbitrary complex unit, but the exponential manifestation, on the compact phase fiber, of the inversion factor established in Foundation v2. The ontological content of the inversion factor ($i^2 = -1$: two inversions return to the origin) locks the group structure of the single phase carrier — it falls in $U(1)$, the abelian group (two inversions return to the origin), rather than promoting the underlying number field of the single scalar phase carrier to the three-dimensional non-commutative quaternions. Here the scope of the locking must be stated precisely: what this paper locks is the single ρ-OR scalar phase carrier as $U(1)$, and it does not claim that quaternionic / $SU(2)$ structure is everywhere absent in physics (the non-commutative structures in spin, gauge, and spatial rotation are not at issue here); what is locked is only the underlying number field of that scalar carrier which bears the single-cell phase. The inversion factor and its inverse ($i$ and $-i$) are strictly conjugate at the algebraic layer and multiply to unity ($e^{i\theta}\cdot e^{-i\theta} = 1$). So $U(1)$ is not a multiplicative group chosen ad hoc in this section, but the group action of $i$, as the $L_1\leftrightarrow L_2$ closure inversion factor, in the phase reading; $U(1)$ indicates how the phase direction is cancelled by conjugation, while symplectic conjugation indicates how the whole carrier generates a capacity scalar.

This locking has a deeper geometric ground, cited here without elaboration: in the DD-splitting structure established in the Four Forces Prequel, the dual 4DD of a single spatial axis (our side + opposite side) spans a two-dimensional plane, whose complex structure is exactly $U(1)$ (the single-plane rotation group $SO(2)\cong U(1)$); the 4DDs of different spatial axes are causally independent and mutually direct-sum, so a single phase carrier is geometrically constrained within a single plane — which is precisely the geometric origin of its falling in the abelian $U(1)$ rather than a cross-axis non-commutative structure. This paper cites this established geometry only as the background for $i$'s locking to $U(1)$; the complete geometric ontology of $i$ itself lies outside P6's domain and is listed as an open problem (see below).

From this it is clear why $\psi^\sharp$ is a conjugate and not a sign reversal. The dual operation on the amplitude must preserve the capacity and cancel the phase direction, rather than annihilate the capacity. Were one to take the sign reversal $\psi^\sharp = -\psi$, the pairing candidate would be $\psi + \psi^\sharp = 0$ — additive annihilation (the same vanishing picture as $e^{i\pi}+1=0$ on the mathematical ladder), which annihilates the amplitude to zero and cannot yield a probability cloud. Conjugation, by contrast, cancels only the phase direction and preserves the capacity: $A\,e^{i\theta}\mapsto A\,e^{-i\theta}$, so $e^{-i\theta}e^{i\theta}=1$ cancels the phase direction while $A\cdot A = A^2$ preserves the capacity.

A.4.4 Juxtaposition check: the closure operation is determined by the type of object

Putting the three layers together: reciprocity gives the two dual sides, symplectic conjugation gives the form of the pairing, and $i$ with $U(1)$ gives the complex phase carrier. From this it is clear that the microscopic multiplicativity and the cosmological additivity are not two rules, but two realizations of one and the same reciprocity principle, on different types of object, closing in the algebra native to each layer. The following table juxtaposes the two layers (and the two kinds of composition internal to quantum mechanics):

Department of the object read	Object read	Dual counterpart	Native closure	Output
Vacuum / $\Lambda$	Signed scalar density	Additive inverse $-\Lambda$	Additive balance	$\Lambda_1 + \Lambda_2 = 0$
Amplitude / $\psi$	$L_1\leftrightarrow L_2$ complex phase carrier	Conjugate dual $\psi^\sharp$	Hermitian pairing	$\psi^\sharp\psi = A^2$
Orthogonal outcomes	Branch weights	Mutually distinct closure slots	Additive measure	$P(E\cup F) = P(E) + P(F)$
Coherent state	Pre-closure amplitudes	The same carrier	Vector-add first, then pair	$P = \left	\sum_i \psi_i\right	^2$

This table does not assign "addition / multiplication" arbitrarily to different domains, but exhibits that the closure operation of each department is determined by its type of object. The main derivation of this paper centers on $L_1\leftrightarrow L_2$ symplectic-conjugation pairing (A.4.2), with reciprocity supplying only the origin of the dual side (A.4.1); additivity is not discarded (the vacuum and the orthogonal outcomes are still additive), and multiplicativity is not misused (it appears only in the pairing of a complex phase carrier with its conjugate dual).

> Were one to take the additive $\psi + \psi^\sharp$ at the amplitude layer as well, one would get $2A\cos\theta$ — phase-dependent, possibly positive or negative, not a phase-invariant closure capacity; and were one to take the vacuum department multiplicatively, one would misread a signed scalar balance as a positive-definite capacity measure. Both are category mismatches.

A.4.5 Deepening: additive substrate, multiplicative manifestation

The distinction can be pushed one layer deeper, and this layer best shows that the microscopic multiplicativity is no departure from the cosmological additivity — because it shows that the two layers are in fact isomorphic at the substrate, both being additive vanishing under reciprocity.

Below the causal slot, the phase itself satisfies the additive vanishing of reciprocity:

$$\theta + \theta^\sharp = 0.$$

This is the same kind of additive vanishing as the cosmological-layer $\Lambda_1 + \Lambda_2 = 0$. The difference is only that the ontology at the microscopic layer is not the bare phase, but the complex phase carrier $e^{i\theta}$ dualized through the exponential map (established in P2). What a one-sided closure reads is this carrier, not the bare phase. And the algebraic law of the carrier layer turns the additive vanishing of the substrate phase automatically into the multiplicative pairing of the carrier:

$$e^{i\theta}\cdot e^{i\theta^\sharp} = e^{i(\theta + (-\theta))} = e^0 = 1.$$

Thus: the two layers are isomorphic at the substrate (both additive vanishing under reciprocity, $\theta + \theta^\sharp = 0$ and $\Lambda_1 + \Lambda_2 = 0$), and the microscopic layer merely wears one extra coat of exponential dualization, so that the additive vanishing manifests at the carrier layer as multiplicative pairing. Multiplicativity is no departure from additivity, but the manifestation of an additive substrate after exponential dualization. The difference between the dual operations of the two layers can be put in one sentence:

> The dual at the vacuum layer is additive sign reversal (preserving total remainder conservation); the dual at the amplitude layer is conjugate orientation (cancelling the phase direction, preserving the closure capacity).

A.5 The order grounding: why the Born rule falls in the second-order structure family

The main derivation answered "how a one-sided reading concretely becomes $A^2$." There remains a related but distinct question: why the Born readout falls in the second-order (quadratic) structure family, rather than third-order. This is answered by the order grounding established in the SAE Mass Series; its role is to frame which structure family the Born readout falls in, not to be a second derivation on a par with the main one.

The closure family on which the established passage of the Mass Series convergence paper depends: the order of the closure law equals the number of DD breakthroughs from 1DD to the deepest active layer — 2DD active gives first order $E = pc$, 3DD active gives second order $E^2 = p^2c^2 + m^2c^4$, 4DD active gives third order $E^3 = p^3c^3 + m^3c^6 + I^3c^9$.

The object of the Born readout is the below-causal-slot uncertainty-wave state, where $I$ (the 4DD causal load) is not yet active. This region falls in the second-order passage — which frames the Born readout as belonging to the quadratic structure family, rather than to the third-order (gravitational-wave-form) structure family above the causal slot where $I$ is active. This clarifies a surface tension: although measurement is a 4DD closure event, what the Born rule reads is the state before closure, below the causal slot ($I$ not yet active), so the readout is a second-order quadratic, not the third-order energy–momentum relation of the active-$I$ region above the causal slot. The two lie at different layers and do not conflict.

The order grounding and the main derivation do distinct work and are not redundant:

> The order grounding frames the structure family — establishing that the Born readout falls in second order; the main derivation gives the concrete form of the readout — establishing how the second-order structure concretely manifests, in a one-sided reading, as $\psi^\sharp\psi$; Gleason-type results serve as the T1 external mathematical anchor — supplying the mathematical uniqueness of the Born measure among probability measures on a standard Hilbert space.

The order grounding need only establish that "the Born rule falls in the second-order family"; it does not need to identify the "second order" $E^2$ of the energy–momentum closure law directly with the "quadratic" $A^2$ of the probability amplitude. The two "twos" share a common origin in the same below-causal-slot structure — which gives second order at the closure-law layer and a quadratic at the amplitude-readout layer, a synchronous manifestation of one structure at two layers; the order grounding bears only the framing of the structure family, while the concrete form of the readout is completed independently by the main derivation. This division of labor keeps the two "twos" from being two independent coincidences landing on the same number, making them instead a one-way support between the framing of a structure family and the derivation of a readout: the former frames the family, the latter derives within it.

A dimensional misreading must be avoided here. $E^2 = p^2c^2 + m^2c^4$ carries a physical dimension (energy squared), whereas the normalized probability weight $A^2$ is a dimensionless capacity measure; the two cannot be identified directly at the dimensional level. What the order grounding frames is the algebraic structure family (both being second-order / quadratic manifolds), not a direct mapping of physical dimension: the "second order" required by 3DD activity manifests, at the kinematic layer, as a dimensioned energy–momentum squared relation, and at the capacity-readout layer, as a dimensionless capacity weight that closes only through a binary pairing (quadratic). The structure family is the same (both quadratic); the physical content of the readout at each layer is independent.

A.6 The precise placement of randomness

From the modulus-squared readout follows the ontological placement of probabilistic randomness.

Randomness is not arbitrariness. Below the causal slot there is no 4DD content — no pre-written outcome available for causality to adjudicate. But the statistical weight of a specific outcome is given by the dual-orientation pairing:

$$w_i = \psi_i^\sharp\,\psi_i.$$

So the precise statement of randomness is: below the causal slot there is no pre-written content (this follows from the absence of causality — causality would adjudicate the order of a pre-written sequence, and it has not yet arisen), while the statistical weight of each outcome is given by $\psi^\sharp\psi$. What this section explains is why this weight is a quadratic; how a single outcome concretely closes to the $i$-th result belongs to the ontological mechanism of the measurement event, left to P7.

This placement is consistent with SAE's established capacity–content distinction: the modulus squared is a capacity readout — a probability density as a measure of capacity, not a content channel. The specific result of a single measurement is one random realization, by weight, from this capacity; the randomness comes from there being no causality, below the causal slot, to adjudicate which side is realized — not from hidden variables, and not from a pre-determination of the outcome.

This section does not presuppose "Born probability." All it assumes is: a 4DD content readout requires a non-negative capacity weight. Non-negativity is automatically satisfied by the multiplicative pairing ($\psi^\sharp\psi = A^2 \geq 0$), whereas the additive dual ($2A\cos\theta$) cannot satisfy it (it can be positive or negative, phase-dependent). From capacity weight to normalized probability is the next step:

$$P_i = \frac{w_i}{\sum_j w_j} = \frac{\psi_i^\sharp\psi_i}{\sum_j \psi_j^\sharp\psi_j}.$$

When the state is normalized this is $P_i = |\psi_i|^2$. The complete probability-layer forms (measure, projection, continuous spectrum) are developed in the following subsections.

A.7 Interference: synthesize the amplitudes first, then pair

A direct and crucial corollary of the modulus-squared readout concerns the ontological place of interference. One ordering here is decisive; read it backwards and the whole picture inverts with it.

The correct form of the Born rule in the interference case is to add the amplitudes first, then take the modulus squared:

$$P = \left|\sum_i \psi_i\right|^2 \neq \sum_i |\psi_i|^2.$$

Read the order backwards — take the modulus squared of each possibility first, then add — and the interference terms cancel, the fringes vanish. So this ordering is not a computational convention but an ontological order.

Within the SAE framework this ordering has a definite ontological content. Below the causal slot is the ρ-OR region of remainder-preserving multiplicity, where multiple branches genuinely coexist (this is exactly the ontology of superposition). 4DD closure does not first assign each branch its own independent probability and then add these probabilities; rather, it first synthesizes, in the ρ-OR region, the amplitudes of the branches into a single integrated uncertainty-wave carrier, and then performs a single dual-orientation pairing (i.e. one modulus-squared readout) on that synthesized carrier.

A misplaced classical image must be avoided here: synthesis is not "several spatial paths meeting and superposing on a screen." Below the causal slot there is no 3DD spatial stage for waves to meet — space is a product of closure, not a backdrop before it. Synthesis occurs in the pre-closure ρ-OR ontological pool, as the algebraic accumulation of the $L_2$ coherent generators in that pool; only when a 4DD closure event (such as absorption at the screen) occurs is this accumulated whole read as a dual-orientation pairing. In other words, interference comes from the ρ-OR amplitude synthesis at the pre-closure branch layer, and the modulus-squared readout acts on the synthesized whole.

The order relation between summation and pairing (the distributive law) must be kept clear, lest it be read as "each branch is paired on its own first, then added." $\psi$ and $\psi^\sharp$ are the our frame and opposite frame of the same state, always coexistent: in the ρ-OR region, the algebraic accumulation of the $L_2$ coherent generators proceeds across both sides simultaneously — the our frame accumulates as $\sum_i\psi_i$, the opposite frame accumulates synchronously as $\sum_i\psi_i^\sharp$. Only when $L_4$ re-activation (e.g. absorption at the screen) occurs are these two already-synthesized bidirectional carriers paired as wholes at the closure interface:

$$P = \left(\sum_i\psi_i^\sharp\right)\left(\sum_i\psi_i\right) = \left|\sum_i\psi_i\right|^2.$$

The pairing acts on the whole after summation, not on each branch paired-then-added (the latter would cancel the cross terms and erase interference).

This ordering also directly serves the wave–particle discussion below: what is called "wave nature" is exactly the fact that the pre-closure uncertainty-wave carrier can be synthesized at the branch layer (and can therefore interfere); what is called "particle nature" is the manifestation of the one-sided result after closure. The two are unified by this "synthesize first, then pair" ordering, as detailed in the wave–particle section.

A.8 The complete probability-layer form: measure, normalization, projection

The Born rule is not exhausted by the pointwise $A^2$; the complete probability-layer form has three levels. $\psi^\sharp\psi = A^2$ gives the minimal scalar capacity density; complete probability further requires an outcome partition, a measure, and normalization.

Discrete orthogonal outcomes. Let the outcome set $\{i\}$ correspond to a family of orthogonal branches, each of weight

$$w_i = \psi_i^\sharp\,\psi_i = |\psi_i|^2,$$

normalized to

$$P_i = \frac{w_i}{\sum_j w_j}.$$

When the state is normalized, $P_i = |\psi_i|^2$. Here the composition between orthogonal branches is additive (mutually distinct closure slots, an additive measure), which does not conflict with the multiplicative pairing internal to a single carrier — the two correspond respectively to the "orthogonal outcomes" and "amplitude" rows of the table in the previous section.

Continuous position representation. In the continuous representation, probability is given in density form:

$$dP(x) = |\psi(x)|^2\, d\mu(x),$$

where $d\mu(x)$ is the measure in that representation. In the flat, unconstrained regular case, $d\mu(x)$ is $d^3x$; in a curved or constrained coordinate space, the measure takes a more complex form. The continuous measure $d\mu(x)$ is itself, in SAE, an object of the representation layer; its deeper structure (involving the below-causal-slot discreteness) lies outside this paper's domain and is left to later papers as needed; this section uses $d\mu(x)$ only within the standard representation of the probability layer.

Abstract Hilbert representation. In the language of projection operators, the probability of outcome $a$ is

$$P(a) = \langle\psi|\Pi_a|\psi\rangle,$$

where $\Pi_a$ is the projection operator for that outcome. The complete projection-valued measure and POVM forms are taken over from the standard mathematical layer (T1); this paper does not restate them.

The three levels together show: $\psi^\sharp\psi$ gives the minimal scalar capacity density, the ontological core of the modulus-squared readout; specific probabilities require an outcome partition, a measure, and normalization layered on top of it. What SAE supplies is the ontological origin of this minimal core; the remaining structure of the probability layer is fully backward-compatible with standard quantum mechanics.

A.9 Deepening the Tsirelson bound: the common-origin interface with P5

The operator form of the modulus-squared readout shares a common origin with the ontological reading of the Tsirelson bound established in P5. This section makes that common origin explicit, both deepening P5 and providing an anchor for the interface between P6 and the bound on quantum correlations.

P5 established the ontological reading of the Tsirelson bound $2\sqrt 2$: on the CHSH operator $B$, the standard algebraic derivation gives $\|B^2\| \leq 4 + 2\cdot 2 = 8$, hence $\|B\| \leq 2\sqrt 2$. Here the factor 4 comes from the binary readout grammar of $L_4$ (the four CHSH terms), and the factor $2\times 2$ comes from the symplectic commutator bound of each party's own $L_1\leftrightarrow L_2$. The twelve 4DD topologies do not bear this number, so as to avoid numerology.

The numerical origin must be made clear, lest "two" be read as explaining "2." The specific numbers $\|B^2\| \leq 8$ and $\|B\| \leq 2\sqrt 2$ come from standard operator algebra — the CHSH operator is built from binary unitary involution operators, and the operator-norm bound of their commutator gives these numbers. This paper does not replace that numerical origin.

What P6 supplies on top of this is an ontological isomorphism interpretation, not a numerical derivation. The $\|B^2\|$ in the Tsirelson derivation is itself the operator version of a modulus-squared readout — it takes the norm of the self-pairing of the operator $B$, structurally isomorphic to a one-sided reading's taking $\psi^\sharp\psi$ of $\psi$: both are "the bilinear pairing of a carrier with its dual, taking a positive-definite scalar." Under this isomorphism, the norm bound of each party's local incompatible readout can be ontologized as the maximal readable strength of that party's $L_1\leftrightarrow L_2$ symplectic structure within the binary readout grammar of $L_4$; and "each party's own local structure" corresponds to the factorization established in P5. This is the explanation of why "local incompatible readout" and "dual-orientation pairing" issue from the same ontology — it is not the numerical assertion that "the number is 2 because there are two sides." The twelve 4DD topologies likewise do not bear these numbers.

This deepening does two things. First, the Born form of P6 is compatible with the Tsirelson bound $2\sqrt 2$ — the modulus-squared readout shares an origin with the Tsirelson algebra, so that the bound on quantum correlations and the quadratic structure of the one-sided readout issue from the same ontology. Second, it makes the cross-paper anchor with P5: P5's surface-layer Tsirelson reading and P6's deep-layer bidirectional structure dock here, so that the two papers share an origin at this one point, the bound on quantum correlations.

The placement of the PR box (Popescu–Rohrlich box) can be given alongside. The correlation the PR box demands exceeds the bound that a one-sided projection of a both-sided-influenced state (the modulus squared) can give. Under the SAE reading, this corresponds to an illegitimate overdraft of capacity — it demands a correlation strength beyond what the capacity structure can carry, rather than violating the capacity–content distinction. The complete PR-box discussion is taken over from P5; this section does not develop it.

A.10 The complete probability layer and no-signaling: the marginal-distribution part

The complete no-signaling theorem has its ontological ground established in P5 (the capacity–content distinction), with the complete theorem $\sum_b P(a,b|x,y) = P(a|x)$ left to P6 and P7 jointly. This section gives the part obtainable directly from the modulus-squared readout — the marginal-distribution part; the part of the complete theorem involving the formalization of the measurement apparatus is left to P7.

Standard mathematical layer (T1). The no-signaling property of the marginal distribution is first of all a result of standard quantum mechanics, with which this paper interfaces rather than re-proving. For projective measurements, the joint probability is

$$P(a,b|x,y) = \langle\psi|(\Pi_a^x \otimes \Pi_b^y)|\psi\rangle.$$

If Bob's measurement is complete, $\sum_b \Pi_b^y = I_B$, then summing over $b$ gives

$$\sum_b P(a,b|x,y) = \langle\psi|(\Pi_a^x \otimes I_B)|\psi\rangle = P(a|x),$$

i.e. the local marginal is independent of the remote setting $y$. The same holds for POVMs, with $\sum_b E_b^y = I_B$. This step is a direct corollary of standard completeness and does not depend on SAE.

SAE ontological-reading layer. Within the SAE framework, this result has an ontological meaning: completeness corresponds to the total-capacity conservation of the remote closure slot; after summing over $b$, the remote content label is eliminated, and locally only the capacity readout of one's own closure partition is retained. Because the ledger is an $L_1 + L_2$ capacity structure with no content (established in P5), the choice of remote setting $y$ does not change the local capacity readout — capacity cannot be transmitted by this route. So no-signaling is given at the probability layer by completeness, and explained at the ontological layer by the capacity / content distinction; the two are consistent, and it is not that a probability theorem is "spoken" directly from ontological language.

This result is consistent with the cross-paper hard constraint. The modulus-squared readout occurs above the causal slot (a 4DD closure event), not as information transfer internal to the ρ-OR region (the hard constraint of Information Theory P4); $\psi^\sharp$, as the dual-orientation counterpart, is not a second wavefunction arriving from the opposite side, and the pairing is a one-sided closure event's reading of that both-sided-influenced, not-yet-single-directionalized state, not a cross-side communication. Hence the no-signaling property of the marginal distribution is consistent with SAE's established structure of no information transfer below the causal slot.

The complete no-signaling theorem — including how the measurement apparatus participates, and the ontology of a single closure event — requires the measurement-apparatus ontology of P7. This section delivers only the marginal-distribution part, as the result of interfacing the modulus-squared readout with standard completeness.

A.11 Wave–particle duality: this section's interface

Wave–particle duality touches the ontological character of the entire ρ-OR region — it relates to P2's uncertainty-wave carrier, P4's tunneling density, and P5's ledger phase, and is wider than the Born rule's domain. So this paper does not develop the complete discussion of wave–particle duality, but only points out its interface with the Born rule; the complete discussion accumulates across papers in the SAE Quantum Mechanics Series.

Wording is especially important here. Phrases like "it is really a particle" lead the reader toward the classical corpuscular image; the precise statement has three layers:

Readout layer (after causality has single-directionalized): presents as a single, particle-like result.
Pre-closure layer (below the causal slot): is a carrier in uncertainty-wave form.
Ontological layer: neither a classical wave nor a classical particle.

Under these three layers, the interface between wave–particle duality and the Born rule is clear: what is called "wave nature" has as its ontological origin the below-causal-slot uncertainty-wave form — which lets the pre-closure carrier be synthesized at the branch layer and therefore interfere (continuing A.7's "synthesize first, then pair"); what is called "particle nature" is the manifestation of the one-sided result after causality has single-directionalized. The Born rule sits exactly at the interface of the two: it converts the dual-orientation capacity pairing of the pre-closure uncertainty-wave carrier into the statistical weight of a one-sided result.

Wave nature comes from the below-causal-slot uncertainty-wave form, not from "the particle really being a kind of wave." The uncertainty-wave form and the above-causal-slot gravitational-wave form are two distinct wave forms of the remainder on the two sides of the causal slot. A single object unfolds in the ρ-OR region in uncertainty-wave form (wave behavior), a one-sided projection gives a probability density (Thesis A of this paper), and a single closure gives a particle-like single result. Particle-like and wave-like are not two dual ontological roles, but different manifestations of the same carrier on the two sides of the causal slot, under two layers of readout. The complete discussion (including specific cases such as the double-slit fringe distribution) accumulates across papers on top of P2 and this paper; this section only establishes the interface.

Thesis B: The Mechanism-Candidate Layer

Thesis A gave the ontological identity of the modulus-squared readout and its main derivation. Thesis B turns to the mechanism layer — here what SAE gives is a candidate, not a settled conclusion, at strength T3, belonging to a different level than the T2 structural derivation of Thesis A. This thesis gives two mechanism candidates and states a position of interpretational neutrality.

B.1 Mechanism candidate one: cross-ladder readout must pass through the modulus squared

The first mechanism candidate comes from P4's cross-ladder readout interface. A one-way firewall is set up first, because reading the direction backwards would constitute circularity.

P4 established: the factor 2 in tunneling's $e^{-2\kappa L}$ comes from the Born modulus squared (a two-step readout), while the factor 2 in Schwarzschild's $r_s = 2GM/c^2$ comes from a one-step readout. Note: P4's tunneling factor 2 depends on the Born modulus squared, whereas P6 is explaining the Born modulus squared. So the direction must be one-way — P4 inspires P6, P6 supports P4; P4 cannot prove P6.

Specifically: P4 §6.4.2 proposed "tunneling is a two-step passage through the Born modulus squared" as an interface to be completed by P6 (P4 at the time left this interface to P6); P6, in Thesis A, independently completes the ontological identity of the modulus squared (the main derivation plus the multiplicative grounding, throughout without citing P4); once P6 is complete, P4's cross-ladder readout candidate gains upstream support. This section travels only the "P6 → supports P4" direction, and never uses P4's factor 2 in reverse to support P6.

Under this one-way premise, the ontological identity of the modulus squared gives P4's two factors of 2 a precise channel distinction: Schwarzschild lies on the above-causal-slot gravitational-wave-form side, a one-step readout, the factor 2 falling at a linear position; tunneling lies on the below-causal-slot uncertainty-wave-form side, passes through the modulus-squared readout, a two-step readout, the factor 2 falling at an exponential position. The ontological origins of the two factors of 2 differ — one on the gravitational-wave side, one on the uncertainty-wave side — which is exactly the "cross-ladder readout channel structure difference" P4 requires. Hence, as a mechanism candidate: cross-ladder readout must pass through the modulus squared because the readout on the uncertainty-wave side necessarily passes through a below-causal-slot state simultaneously subject to both-sided time arrows, and reading such a state one-sidedly is the modulus squared.

B.2 Mechanism candidate two: the modulus-squared readout occurs at the moment of $L_4$ re-activation

The second mechanism candidate locates the timing of the modulus-squared readout within the measurement process. P5 established: measurement is $L_4$ re-activation — strong re-activation (projective measurement) projects the ledger onto local content, while weak measurement / decoherence is partial re-activation.

On this basis, the modulus-squared readout occurs at the moment of $L_4$ re-activation: the re-activation event projects the below-causal-slot, both-sided-influenced uncertainty-wave state onto the one-sided real-valued readout interface, and this projection is the dual-orientation pairing, giving $\psi^\sharp\psi$.

There is an interface between P6 and P7. What P6 gives is the layer form of probability — why each weight is a quadratic, given an outcome partition; what P7 gives is the ontological mechanism of the measurement event — how the measurement apparatus, the environment, record generation, and irreversibility participate, and why a particular instance closes to the $i$-th result. As a mechanism candidate, this section only points out the temporal correspondence between the modulus-squared readout and $L_4$ re-activation, without overstepping into P7's measurement-apparatus ontology.

B.3 Interpretational neutrality

Both mechanism candidates of Thesis B remain interpretationally neutral. The modulus squared, as a deeper-layer mechanism candidate, sits beneath the various interpretations of quantum mechanics — it does not pre-select any one of collapse, many-worlds, or environmental selection, but supplies an ontological layer on which these interpretations become phenomenological descriptions. This neutral position is inherited from P5 and will be developed concretely in Thesis C: the relation of SAE to several mainstream interpretations is not "choosing yet another interpretation," but giving a deeper ontological origin to the mathematical structure they share.

Thesis C: A Reclassification of the Born Rule's Origin

Thesis C treats the relation of SAE to the mainstream approaches to the Born rule's origin. The approaches fall into two kinds: those in substantive dialogue with SAE (Gleason, Many-Worlds derivations), and those given a minimal placement (QBism, Bohmian). The complete interpretational map is left to P0 and not developed here.

C-i Gleason's theorem and SAE's geometric origin

Gleason's theorem is the standard result for the mathematical uniqueness of the Born rule: under the premise of non-contextuality, a probability measure on a Hilbert space is given uniquely by a density operator via the trace formula, yielding the modulus-squared form.

The relation of SAE to Gleason is stated precisely by two qualifications.

First, the dimension and measurement-structure condition. Gleason's original theorem holds for Hilbert spaces of dimension no less than 3; the two-dimensional qubit case requires an additional assumption, or recourse to a POVM version or Busch-type generalization, for coverage. So Gleason-type theorems give the standard mathematical uniqueness of the Born-type measure under their usual conditions on dimension and measurement structure, not coverage of all Hilbert spaces; the two-dimensional qubit and POVM cases require the corresponding extended versions. SAE does not re-derive these theorems, but only gives the ontological origin of the quadratic readout.

Second, the non-corollary placement. SAE does not treat Gleason as "a corollary derived by SAE." The more secure relation is: SAE is compatible with Gleason-type uniqueness results; at the standard-form layer, SAE's quadratic readout interfaces with these mathematical results.

Hence the division of labor between SAE and Gleason is clear: Gleason answers, at the mathematical layer, "given Hilbert-space structure and non-contextuality, why the probability measure is uniquely the modulus squared"; SAE answers, at the ontological layer, "what the origin of the modulus-squared readout form is — the one-sided reading of a below-causal-slot, both-sided-influenced state." The former is T1 mathematical uniqueness, the latter T2 ontological origin. The two do not compete; each takes up a different facet of the probability rule. This is the positive expression of P6's commitment grading: SAE gives the ontological origin, not a re-proof of mathematical uniqueness.

C-ii Many-Worlds derivations and SAE: the causality firewall

The relation to Many-Worlds approaches (the Everett interpretation, the Deutsch–Wallace decision-theoretic derivation, Zurek's envariance) requires a precise firewall, because the formal structure the two share is most easily misread as ontological identity.

Separate first the shared from the rejected. SAE shares the branch formalism with Everett / Deutsch–Wallace — multiple branches coexist in the mathematical representation before closure. But SAE rejects the branch-world ontology — it refuses to promote each branch into an independent, ontologically real 4DD content world. The two are not "mathematically fully identical": the Born weights of Many-Worlds depend on a whole apparatus of internal argument (Deutsch–Wallace's rational-agent decisions, branch measures, observer self-location, and so on), whereas SAE's weights come from dual-orientation pairing. Precisely: SAE accepts the branch formalism as a formal representation of ρ-OR multiplicity, but refuses to ontologize the branches into multiple 4DD content worlds.

The core of this firewall is that the point of distinction falls precisely on whether the state is subject to a single-direction or a bidirectional time arrow, and thereby on the presence or absence of causality — not merely on the position of a layer:

> The branches of Many-Worlds are multiple worlds above the causal slot, each subject only to a single-direction time arrow — each branch carries a complete, single-directionally oriented causal history. The two sides of SAE are one and the same quantum below the causal slot, simultaneously subject to the our-frame and opposite-frame bidirectional time arrows — not two worlds each single-directional, but one state not yet single-directionalized and therefore without causality and without a causal history.

From this one can respond directly to the most natural objection from the Many-Worlds side: "Aren't your two sides just two branches?" The answer is: no. Branches are 4DD worlds each subject to a single-direction time arrow and each carrying a complete causal history (multiple worlds, each single-directional); SAE's two sides are one and the same quantum simultaneously subject to two directions of the time arrow (one state, subject to both directions), with no single direction available to fix causation, hence no causal history. Multiple worlds each single-directional, and one state simultaneously subject to both-sided influence, are two things of different categories. As for ontological economy and remainder discipline, SAE does not bear a branch-world ontology — it acknowledges only the branch formalism as a formal representation of ρ-OR multiplicity, and does not ontologize each branch into an independent 4DD world carrying a complete causal history. Before closure SAE has only one not-yet-single-directionalized bidirectional state, and after closure a one-sided result, with no proliferation of worlds; whether a thermodynamic accounting of the branch-world ontology can be given lies outside this paper, which only points out that SAE need not introduce multiple 4DD content worlds.

An analogy can separate "the bidirectional orientation of a single existence" from "two separate entities," with the following restriction on its scope. The Möbius band is a one-sided two-dimensional manifold: in its embedding space it appears to have an "upper" and a "lower" face, but as a manifold itself it has only one face — the apparent upper and lower are two orientations of the same face, not two separate faces. SAE's dual temporal orientation is isomorphic to this at the ontological layer: the Planck substrate is a single ontological existence, and the bidirectional orientation is two orientations of this single existence, not two ontologically separate entities. This analogy is an isomorphism at the structural level of "a single existence containing a bidirectional orientation," not a direct mapping of spatial topology — the Möbius band is a two-dimensional manifold, the Planck substrate is not a spatial manifold, and the two differ in spatial structure. What the analogy takes is only the contrast of "single existence / bidirectional orientation" against "two separate realities." Hence the point of distinction is clear: SAE is the bidirectional orientation of a single ontological existence, Many-Worlds is multiple ontologically separate realities.

SAE's treatment of Many-Worlds differs ontologically, in one respect, from its treatment of Einstein (see P5). For Einstein, SAE takes over his deeper mathematical intuition and corrects its specific expression — a vindication of his intuition. For Many-Worlds, SAE likewise takes over its mathematical insight (unitary evolution, no independent collapse postulate, branch structure as random sampling), but replaces its ontology (the one-sided projection of a bidirectional temporal orientation in place of world-splitting) — here SAE takes over only the mathematics, and not a vindication of the branch-world ontology. SAE gives the branch weights a deeper origin, from dual-orientation pairing.

C-iii Minimal placement: QBism and Bohmian

The following two approaches are not developed in dialogue within P6's domain; each is given a paragraph of demarcation, with the complete dialogue left to P0 or a dedicated paper.

QBism. QBism interprets probability as the subjective belief of an agent (the agent's credence) — the Born rule a normative constraint on a rational agent's betting. SAE's probability is not subjective belief, but a projection from capacity to content, an objective structural readout. The two diverge fundamentally on "what probability is": one is the agent's epistemic attitude, the other the objective one-sided reading of a below-causal-slot, both-sided-influenced state. This paper does not develop a substantive dialogue with QBism's agent ontology.

Bohmian. Bohmian mechanics interprets probability as epistemic ignorance of hidden variables (the true positions of particles) — $|\psi|^2$ being the distribution of initial positions (more finely, the Bohmian approach usually argues the stability of this distribution through quantum equilibrium / typicality and equivariance), with randomness arising from ignorance of definite but unknown trajectories. SAE's randomness is not ignorance: below the causal slot there is no pre-written outcome available for adjudication, and randomness arises from there being no causality to adjudicate which side is realized, not from ignorance of an established fact. This distinction echoes the related demarcation in P5's treatment of Einstein. This paper does not develop a substantive dialogue with Bohmian's hidden-variable ontology.

Compatibility Anchors, Structural Stress Tests, and Conditional Open Interfaces

This paper registers three classes of anchor, A/B/C. Class A consists of baseline commitments and compatibility anchors, not currently usable quantitative forward predictions; whether the Born rule is modified in the strong-field regime is listed as a Class C open problem.

Class A (baseline commitments / compatibility anchors)

A1 (weak-field baseline): in the regular / weak-field regime (the $d_{\mathrm{eff}}\to 2$ established in Relativity P1), the two sides are nearly symmetric, the dual-orientation pairing coincides with standard Hermitian conjugation, and the Born rule is exact: $P_i = |\psi_i|^2$. This is the baseline statement of the paper, consistent with the high-precision experimental confirmation of the Born rule. The main body of the paper takes this as the standard within the regular regime.

Class B (structural incompatibility)

B1 (grading condition): if the Gleason-type conditions can be derived from deeper SAE axioms, then P6 can be upgraded to a full Gleason-equivalent theorem; otherwise it retains the strong structural derivation of the quadratic readout.
B2 (P4 cascade): if the ontological necessity of "cross-ladder readout must pass through the modulus squared" does not hold, then the tunneling factor-2 candidate of P4 §6.4.2 fails accordingly.
B3 (P5 cascade): if P6's modulus squared is inconsistent with P5's Tsirelson factor 4 + $2\times 2$, then the corresponding subsection of P5 is under pressure.
B4 (cross-ladder discipline): if the dual-orientation filling is inconsistent with the $i$ identity of Foundation v2, then P6's main line is under pressure.
B5 (convergence cascade): if the order grounding (structure-family order = number of DD breakthroughs) is inconsistent with the established convergence paper, then the interface between convergence and P6 is under pressure.
B6 (multiplicative-grounding cascade): if the principle "reciprocity closes in the native algebra" (microscopic multiplicativity sharing an origin with vacuum additivity) is inconsistent with the established reciprocity of Cosmo I — for example, if reciprocity cannot be multiplicative at the microscopic amplitude layer — then the main statement of this paper's multiplicative grounding is under pressure, retreating to the weak statement "borrowing the reciprocity principle, without deriving the concrete form of microscopic multiplicativity."

Class C (framework consistency / failure modes / conditional programs)

C1: consistent with the $L$ ladder of Foundation v2 (the three channels of $L_2$ and representational invariance).
C2: backward-compatible with the mathematical layer of standard quantum mechanics (the computational results of the modulus squared are unchanged).
C3: consistent with the P5 interface "measurement is $L_4$ re-activation" (P6 gives the probability layer, P5 gives the ontological mechanism).
C4: consistent with the cross-paper anchoring network (P3's $\hbar$, P4's cross-ladder, P5's ledger and Tsirelson, Foundation v2's cross-ladder, convergence's order, Cosmo I's reciprocity).
C5 (continuing P2): the complex amplitude cannot be replaced by a purely real-number ontology, unless that restatement contains no dual-algebra element of type $J^2 = -1$.
C6 (strong field: open problem): the Born rule is exact within its domain — the regular and weak-field regime — consistent with the high-precision experimental confirmation of the Born rule. Whether the Born rule remains exact, or is modified, in the strong field (extreme compression of the causal slot) is an open problem: SAE currently cannot give an a priori judgment. This paper claims neither a Born deviation in the strong field nor its necessary exactness — the relation between the Born rule and field strength cannot be given a priori absent posterior support, and is therefore explicitly marked as undecided, left to later theoretical development and experiment.
C7 ($i$'s complete geometric ontology: open problem): this paper anchors $i$ as the inversion factor established in Foundation v2, and cites "the complex structure within a single dual-4DD plane" from the DD-splitting structure of the Prequel as the geometric background for its falling in $U(1)$ (A.4.3). But the complete geometric definition of $i$ as an ontological operation in the physical world — its precise characterization within the dual-4DD plane, its relation to the our-side and opposite-side temporal orientations, and the precise connection between it and the Born pairing $\psi^\sharp\psi$ ($i$ supplies the geometric premise of the complex phase carrier, while the modulus-squared readout is completed by the conjugate pairing of dual orientations, and the two cannot be simply identified) — is not given in this paper, and is listed as an open problem, left to later deepening at the Foundation layer. This paper takes an honest position on it: the ontological placement of $i$ in the physical world has not yet been thought through, so it cites only the established anchors sufficient for use, without hastily developing the matter within P6.

Acknowledgements

The argument of this paper was refined over several rounds of divergent-thinking review during drafting. I thank four large language models for the divergent thinking and structural challenges they offered in different rounds — in particular, repeated interrogation of the multiplicative-pairing grounding, the group-structure locking of $i$, the dual temporal orientation, and the causality firewall, which helped the author clarify the boundaries of several ontological formulations. I am especially grateful to Zesi Chen (陈则思) for sustained critical feedback. All intellectual decisions and the final formulations are the author's responsibility.

References

The established SAE works on which this paper depends are listed below, grouped by series. Listed are concept DOIs or the corresponding version DOIs.

Quantum Mechanics Series (QM)

Han Qin. SAE Quantum Mechanics P2. Zenodo. https://doi.org/10.5281/zenodo.20277037
Han Qin. SAE Quantum Mechanics P3: ℏ as the $L_1\leftrightarrow L_2$ Symplectic-Conjugation Closure Signature. Zenodo. https://doi.org/10.5281/zenodo.20340595
Han Qin. SAE Quantum Mechanics P4: Quantum Tunneling. Zenodo. https://doi.org/10.5281/zenodo.20369138
Han Qin. SAE Quantum Mechanics P5. Zenodo. https://doi.org/10.5281/zenodo.20455398

Physics Foundation and Four Forces Prequel

Han Qin. Cross-Layer Closure Equations: From Euler's Formula to the Event Horizon (SAE Physics Foundation). Zenodo. https://doi.org/10.5281/zenodo.19361950
Han Qin. Light Speed, Dimensional Breakthrough, and the Coupling Constant Hierarchy (Four Forces Prequel). Zenodo. https://doi.org/10.5281/zenodo.19341043
Han Qin. The Grammar of Force: Four Forces Series Finale. Zenodo. https://doi.org/10.5281/zenodo.19464447

Cosmological Series

Han Qin. From Remainder Conservation to the Cosmological Constant (Cosmo I). Zenodo. https://doi.org/10.5281/zenodo.19245267
Han Qin. Causal Strengthening, Remainder Conservation, and the Dual-Frame Resolution of the Ġ/G Tension (Cosmo V). Zenodo. https://doi.org/10.5281/zenodo.19329771

Relativity Series

Han Qin. Gravitational Time Dilation in the SAE Framework: Causal Cell Throughput Derivation (Relativity P1). Zenodo. https://doi.org/10.5281/zenodo.19836185
Han Qin. Ontological Unification of Motion and Gravitational Effects: Effective Inertia under Causal-Slot Geometry (Relativity P2). Zenodo. https://doi.org/10.5281/zenodo.19910546

Information Theory Series

Han Qin. The 4DD Ontology of Information and a Single Foundational Axiom (Information Theory P1). Zenodo. https://doi.org/10.5281/zenodo.19740020
Han Qin. Black Hole Information through the Causal Spectrum: The SAE-BH Correspondence (Information Theory P4). Zenodo. https://doi.org/10.5281/zenodo.19880112

Mass Series

Han Qin. The Nature of Information: E/c³ (Mass Series Convergence Paper). Zenodo. https://doi.org/10.5281/zenodo.19510868

External Standard References

Born, M. (1926). Zur Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik, 37, 863–867.
Gleason, A. M. (1957). Measures on the closed subspaces of a Hilbert space. Journal of Mathematics and Mechanics, 6, 885–893.
Busch, P. (2003). Quantum states and generalized observables: a simple proof of Gleason's theorem. Physical Review Letters, 91, 120403.
Deutsch, D. (1999). Quantum theory of probability and decisions. Proceedings of the Royal Society A, 455, 3129–3137.
Wallace, D. (2012). The Emergent Multiverse: Quantum Theory according to the Everett Interpretation. Oxford University Press.
Zurek, W. H. (2005). Probabilities from entanglement, Born's rule from envariance. Physical Review A, 71, 052105.
Fuchs, C. A., Mermin, N. D., & Schack, R. (2014). An introduction to QBism. American Journal of Physics, 82, 749–754.
Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of "hidden" variables. Physical Review, 85, 166–193.
Dürr, D., Goldstein, S., & Zanghì, N. (1992). Quantum equilibrium and the origin of absolute uncertainty. Journal of Statistical Physics, 67, 843–907.
Cirel'son (Tsirelson), B. S. (1980). Quantum generalizations of Bell's inequality. Letters in Mathematical Physics, 4, 93–100.
Popescu, S., & Rohrlich, D. (1994). Quantum nonlocality as an axiom. Foundations of Physics, 24, 379–385.

被读对象所在部门	被读对象	对偶对应物	原生闭合	输出
真空 / $\Lambda$	有符号标量密度	加性逆元 $-\Lambda$	加性平衡	$\Lambda_1 + \Lambda_2 = 0$
振幅 / $\psi$	$L_1\leftrightarrow L_2$ 复相位载体	共轭对偶 $\psi^\sharp$	Hermitian 配对	$\psi^\sharp\psi = A^2$
正交结果	分支权重	互异闭合槽	加性测度	$P(E\cup F) = P(E) + P(F)$
相干态	闭合前振幅	同一载体	先矢量相加再配对	$P = \left	\sum_i \psi_i\right	^2$

SAE QM Paper 6: The Born Rule as the Algebraic Projection of ClosureSAE 量子力学 Paper 6：玻恩规则——闭合的代数投影

Abstract

Part I: The Mainstream Landscape of the Born Rule's Origin

Thesis A: The Modulus Squared as the One-Sided Reading of a Below-Causal-Slot Uncertainty-Wave State (a State Simultaneously Subject to Both-Sided Time Arrows)

A.1 What this section does and does not answer

A.2 Locating the object read: the pre-closure state below the causal slot, in uncertainty-wave form, simultaneously subject to both-sided time arrows

A.3 Main derivation: the one-sided reading is the multiplicative pairing of dual orientations

A.4 Why a multiplicative pairing, and not additive — unification with the cosmological $\Lambda$ layer

A.4.1 The origin of the dual side: the reciprocity relation (established in Cosmo I)

A.4.2 The form of the pairing: $L_1\leftrightarrow L_2$ symplectic conjugation (established in P3)

A.4.3 The origin of the complex phase carrier: the established-dependency declaration and the inversion-factor identity of $i$

A.4.4 Juxtaposition check: the closure operation is determined by the type of object

A.4.5 Deepening: additive substrate, multiplicative manifestation

A.5 The order grounding: why the Born rule falls in the second-order structure family

A.6 The precise placement of randomness

A.7 Interference: synthesize the amplitudes first, then pair

A.8 The complete probability-layer form: measure, normalization, projection

A.9 Deepening the Tsirelson bound: the common-origin interface with P5

A.10 The complete probability layer and no-signaling: the marginal-distribution part

A.11 Wave–particle duality: this section's interface

Thesis B: The Mechanism-Candidate Layer

B.1 Mechanism candidate one: cross-ladder readout must pass through the modulus squared

B.2 Mechanism candidate two: the modulus-squared readout occurs at the moment of $L_4$ re-activation

B.3 Interpretational neutrality

Thesis C: A Reclassification of the Born Rule's Origin

C-i Gleason's theorem and SAE's geometric origin

C-ii Many-Worlds derivations and SAE: the causality firewall

C-iii Minimal placement: QBism and Bohmian

Compatibility Anchors, Structural Stress Tests, and Conditional Open Interfaces

Acknowledgements

References

摘要

第一部分：玻恩规则起源的主流图景

论题甲：模平方作因果槽以下测不准波态的单侧读数（该态同时受双侧时间箭头影响）

甲·一 本节要回答与不回答什么

甲·二 读出对象的精确定位：因果槽以下、测不准波形态、同时受双侧时间箭头影响的闭合前态

甲·三 主推导：单侧读出是对偶取向的乘性配对

甲·四 为什么是乘性配对，而非加性——与宇宙学 $\Lambda$ 层的统一

甲·四·1 对偶侧的来源：互易关系（Cosmo I 既立）

甲·四·2 配对的形式：$L_1\leftrightarrow L_2$ 辛共轭（P3 既立）

甲·四·3 复相位载体的来源：既立依赖声明与 $i$ 的反转因子身份

甲·四·4 并置验证：闭合操作由对象类型决定

甲·四·5 深化：底座加性、表象乘性

甲·五 阶数根据：玻恩为何落在二阶结构族

甲·六 随机性的精确定位

甲·七 干涉：先合成振幅，再配对

甲·八 概率层的完整形式：测度、归一、投影

甲·九 Tsirelson 上界的深化：与 P5 的同源接合

甲·十 完整概率层与无信号：边缘分布部分

甲·十一 波粒二象性：本节的接口

论题乙：机制候选层

乙·一 机制候选一：跨阶梯读出必经模平方

乙·二 机制候选二：模平方读出发生在 $L_4$ 重激活时刻

乙·三 诠释中立

论题丙：玻恩规则起源的重新分类

丙·一 Gleason 定理与 SAE 的几何起源

丙·二 Many-Worlds 派生与 SAE：因果律防火墙

丙·三 最小定位：QBism 与 Bohmian

兼容性锚点、结构压力测试与条件性开放接口

致谢

参考文献

SAE QM Paper 6: The Born Rule as the Algebraic Projection of Closure
SAE 量子力学 Paper 6：玻恩规则——闭合的代数投影

甲·一　本节要回答与不回答什么

甲·二　读出对象的精确定位：因果槽以下、测不准波形态、同时受双侧时间箭头影响的闭合前态

甲·三　主推导：单侧读出是对偶取向的乘性配对

甲·四　为什么是乘性配对，而非加性——与宇宙学 $\Lambda$ 层的统一

甲·四·1　对偶侧的来源：互易关系（Cosmo I 既立）

甲·四·2　配对的形式：$L_1\leftrightarrow L_2$ 辛共轭（P3 既立）

甲·四·3　复相位载体的来源：既立依赖声明与 $i$ 的反转因子身份

甲·四·4　并置验证：闭合操作由对象类型决定

甲·四·5　深化：底座加性、表象乘性

甲·五　阶数根据：玻恩为何落在二阶结构族

甲·六　随机性的精确定位

甲·七　干涉：先合成振幅，再配对

甲·八　概率层的完整形式：测度、归一、投影

甲·九　Tsirelson 上界的深化：与 P5 的同源接合

甲·十　完整概率层与无信号：边缘分布部分

甲·十一　波粒二象性：本节的接口

乙·一　机制候选一：跨阶梯读出必经模平方

乙·二　机制候选二：模平方读出发生在 $L_4$ 重激活时刻

乙·三　诠释中立

丙·一　Gleason 定理与 SAE 的几何起源

丙·二　Many-Worlds 派生与 SAE：因果律防火墙

丙·三　最小定位：QBism 与 Bohmian