The Quantum Structure Sub-Programme

Correction notice (Q1, v2.0). Q1's central theorem — cited throughout this page (Theorem 2.7, Theorem 2.15, Corollary 2.17, Theorem 2.19) as the foundation of phase coherence, the singlet correlator, the Tsirelson bound, and the Born rule at spin-$\tfrac12$ — has been withdrawn. Independent review found its published proof unsound and its separate Born-rule derivation independently defective. Q1 v2.0 proves a narrower, unconditional rigidity theorem instead and no longer derives quantum-mechanical structure; see the updated Q1 page. This page's description of the sub-programme as "closed within the $\mathrm{SU}(2)$ sector," and its account of what Q2 and Q3 build on, have not yet been re-audited against that withdrawal and should be read as provisional pending that review.

Overview

The admissibility axioms A1–A4 force the admissible fibre $F_n \simeq V_\rho$ with Heisenberg structure (Note 5) and the admissibility thread $Q_8 \subset 2I \subset \mathrm{SU}(2)$ with the spin-$\tfrac{1}{2}$ sector identified as the physically admissible sector (Note 1). Do the structures of quantum mechanics already follow from this much?

The quantum structure sub-programme answers yes, conditional on a supplied carrier, in six structural steps. Given the complex $\mathrm{SU}(2)$ Weil amplitude carrier supplied by O18 and O23 (not derived from A1–A4), phase coherence of that carrier is forced by the binary-icosahedral indiscernibility of conjugate Weil blocks (Q1). The singlet correlator $E(\hat{a},\hat{b}) = -\hat{a}\cdot\hat{b}$ then follows from four independently established inputs: linearity of $\mathfrak{su}(2)$ observables (O23), bilinearity from the carrier's preserved coherence (Q1), rotation invariance from isotropic saturation (O23), and unit normalisation from the parity involution (O18) — no quantum postulate beyond the supplied carrier is invoked at any step. The Tsirelson bound $|S_{\mathrm{CHSH}}| \leq 2\sqrt{2}$ is then a structural corollary within that carrier; the Born rule is the unique probability assignment compatible with the derived correlator (Q1 Theorem 2.19, no Gleason theorem needed). $\mathrm{SU}(2)$ is selected as the unique stable fixed point of the admissibility flow in the LPS limit (Q2 Theorem 8.6), and all results extend to the universal spin-$j$ generalisation for the five admissible sectors of $2I$ (Q3, within its own supplied bipartite composition). Bell factorizability does not apply in non-injective effective descriptions (Bell paper, published in Quantum Reports).

The first physics of the corpus. Before spacetime geometry, before gauge structure, before gravity, the admissibility constraints already force quantum correlations upon the supplied carrier. The sub-programme is closed within the $\mathrm{SU}(2)$ sector; the primary open problem is the extension of the Born rule to arbitrary observables beyond the parity sector of O18.

The structural chain

$F_n \simeq V_\rho,\;\text{BI indisc.} \;\Longrightarrow\; \text{phase coherence} \;\Longrightarrow\; E(\hat{a},\hat{b}) = -\hat{a}\cdot\hat{b} \;\Longrightarrow\; |S_{\mathrm{CHSH}}| \leq 2\sqrt{2} \;\Longrightarrow\; \text{Born rule} \;\Longrightarrow\; \mathrm{SU}(2)\text{ fixed point} \;\Longrightarrow\; E = -\tfrac{j(j+1)}{3}\hat{a}\cdot\hat{b}.$

Six conceptually distinct steps span three constituent papers: phase coherence, singlet correlator, Tsirelson bound, and Born rule at spin-$\tfrac{1}{2}$ (Q1); co-admissibility $\lambda_{1/2} = \lambda_{3/2} = 18$ and $\mathrm{SU}(2)$ as the unique stable fixed point of the admissibility flow (Q2); proto-state as the singlet, universal correlator, and sectorwise Born rule for all five admissible sectors $j \in \{\tfrac{1}{2}, 1, \tfrac{3}{2}, 2, \tfrac{5}{2}\}$ of $2I$ (Q3). The Bell paper establishes that the standard Bell factorizability assumption fails structurally in any non-injective effective description.

Papers of the sub-programme

Phase coherence, singlet correlator, Tsirelson, Born rule at spin-$\tfrac{1}{2}$.

Q1 — phase coherence from BI indiscernibility of conjugate Weil blocks (Theorem 2.7, proved); observable rank signature $\mathrm{rank}\,W^{(c)}_n = 0$ as a falsifiable coherence witness (Corollary 2.9; numerically confirmed for $q = 29$ across all four conjugate pairs); singlet correlator $E(\hat{a},\hat{b}) = -\hat{a}\cdot\hat{b}$ from four structural inputs (Theorem 2.15, proved, conditional on the supplied carrier); Tsirelson bound $|S_{\mathrm{CHSH}}| \leq 2\sqrt{2}$ as a structural corollary (Corollary 2.17, unconditional within that carrier); Born rule as the unique probability assignment compatible with the derived correlator (Theorem 2.19, proved within that carrier — no Gleason theorem).

Co-admissibility and $\mathrm{SU}(2)$ as stable fixed point.

Q2 — co-admissibility $\lambda_{1/2} = \lambda_{3/2} = 18$ of spin-$\tfrac{1}{2}$ and spin-$\tfrac{3}{2}$ on $2I$ (proved); quantum structure replicated for the $\chi_4$ sector with correlator $E = -\tfrac{5}{4}\hat{a}\cdot\hat{b}$ (proved); admissibility flow selects $j = \tfrac{1}{2}$ as the unique stable fixed point in the large-$p$ LPS limit (Theorem 8.6, proved), recovering the standard correlator $E = -\hat{a}\cdot\hat{b}$ and the standard Tsirelson bound $2\sqrt{2}$ as the physically realised values.

Universal spin-$j$ quantum sector.

Q3 — proto-state is the singlet $|\Omega_j\rangle$ for all admissible $j$ (Theorem 4.1, proved via Schur on the Clebsch–Gordan decomposition); universal correlator $E(\hat{a},\hat{b}) = -\tfrac{j(j+1)}{3}(\hat{a}\cdot\hat{b})$ for all five admissible sectors of $2I$ (Theorem 5.2, proved); Born rule for all admissible $j$ (Corollary 6.1, proved). Closes the $\mathrm{SU}(2)$ quantum sector and completes the structural derivation initiated in Q1.

Bell non-applicability in non-injective frameworks.

Bell paper — proves that the standard Bell factorizability assumption fails structurally in any framework where observable outcomes arise as equivalence classes of underlying configurations under a non-injective map. The proof uses only non-injectivity of $\Pi$ and informational completeness (E2 of ENI); no hidden variable, no non-local dynamics, no modification of quantum mechanics. Published in Quantum Reports (MDPI, 2026) — the only peer-reviewed paper of the sub-programme.

Inputs and outputs

Upstream inputs. The admissible fibre $F_n \simeq V_\rho$ with non-trivial commutator $[X,Y] = Z$ from the Heisenberg-structure note and the foundation paper (Note 5); the admissibility thread $Q_8 \subset 2I \subset \mathrm{SU}(2)$ with spin-$\tfrac{1}{2}$ identified as the minimal admissible sector (Note 1, O29); $\mathrm{Im}\,\mathbb{H} \cong \mathfrak{su}(2)$ with three isotropic directions and isotropic saturation $M \propto I$ (O23); parity involution $c \leftrightarrow q-c$ and unit-norm amplitudes at saturation (O18, O22); BI indiscernibility of conjugate Weil blocks (BornInfeld); non-injectivity as a structural necessity ($\mathcal{S}_\Pi > 0$) for the Bell argument (ENI).

Outputs. Phase coherence and coherent amplitude structure (consumed by Note 6 fermions and the Q-series); $E(\hat{a},\hat{b}) = -\hat{a}\cdot\hat{b}$ in the spin-$\tfrac{1}{2}$ sector and its universal spin-$j$ generalisation (consumed by quantum phenomenology and applications); Tsirelson bound $2\sqrt{2}$ (foundations applications); Born rule in the $\mathrm{SU}(2)$ sector (quantum measurements); $\mathrm{SU}(2)$ as stable fixed point (feeds back into Note 1 and Note 3 gauge structure); the rank-$W^{(c)}_n = 0$ falsifiable signature (O-series validation); structural non-applicability of Bell factorizability (quantum foundations).

Status

The sub-programme is closed within the $\mathrm{SU}(2)$ sector, conditional on the supplied Weil carrier: all six steps of the structural chain are proved. Phase coherence (Q1 Theorem 2.7) is unconditional; the singlet correlator (Q1 Theorem 2.15), the Tsirelson bound (Q1 Corollary 2.17), and the Born rule at spin-$\tfrac{1}{2}$ (Q1 Theorem 2.19) are unconditional within that carrier, whose own complex scalar structure is supplied by O18/O23, not derived. Co-admissibility $\lambda_{1/2} = \lambda_{3/2} = 18$ and $\mathrm{SU}(2)$ as the unique stable fixed point of the admissibility flow (Q2 Theorem 8.6) are proved. The proto-state identification (Q3 Theorem 4.1), the universal spin-$j$ correlator (Q3 Theorem 5.2), and the sectorwise Born rule (Q3 Corollary 6.1) extend the programme to all five admissible sectors of $2I$, within Q3's own supplied bipartite composition (a K4 crossing). Bell factorizability non-applicability is published in Quantum Reports. Numerically: $\mathrm{rank}\,W^{(c)}_n = 0$ confirmed for $q = 29$ across all four conjugate pairs throughout the admissible regime; extension to $q \in \{61, 101, 151, 211, 307, 401\}$ is identified as the systematic validation target. Two operational deliverables remain open: the Born rule for general observables beyond $\mathrm{SU}(2)$, and the numerical validation across the full prime range.