This repository contains the source of the Quantum Structure Presentation Note Cosmochrony paper The Quantum Structure Sub-Programme — Presentation Note 7.

This work is a structured entry point to the quantum structure sub-programme of the Cosmochrony corpus, not a summary of results. It maps the constituent papers, identifies the logical chain from admissibility to the Born rule, records the status of every result as proved, numerical, or open, and states the remaining open deliverables.

Central Question

The admissibility axioms A1--A4 force the fibre $F_n \simeq V_\rho$ with Heisenberg structure (Note5) 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 (Note1).

Do the structures of quantum mechanics --- complex amplitudes, Born rule, singlet correlator, Bell-type correlations --- follow from admissibility alone, or must they be separately postulated?

The sub-programme answers: they follow. Phase coherence, the singlet correlator $E(\hat{a},\hat{b}) = -\hat{a}\cdot\hat{b}$, the Tsirelson bound $|S_{\mathrm{CHSH}}| \leq 2\sqrt{2}$, and the Born rule are derived within the $\mathrm{SU}(2)$ sector without importing quantum postulates. $\mathrm{SU}(2)$ is identified as the unique stable fixed point of the admissibility flow, and all results extend to the universal spin-$j$ case for the five admissible sectors of $2I$. Bell factorizability does not apply in non-injective effective descriptions.

This sub-programme is the first physics of the corpus: before spacetime geometry, before gauge structure, before gravity, the admissibility constraints already force quantum correlations.

Logical Chain

$$\underbrace{F_n \simeq V_\rho,;\text{BI indisc.}}_{\text{A1--A4, O18}} ;\Longrightarrow; \underbrace{\text{phase coherence}}_{\text{Q1 Thm 2.7}} ;\Longrightarrow; \underbrace{E(\hat{a},\hat{b}) = -\hat{a}\cdot\hat{b}}_{\text{Q1 Thm 2.14}} ;\Longrightarrow; \underbrace{|S_{\mathrm{CHSH}}| \leq 2\sqrt{2}}_{\text{Q1 Cor 2.16}} ;\Longrightarrow; \underbrace{\text{Born rule}}_{\text{Q1 Thm 2.18}}$$

$$\Longrightarrow; \underbrace{\mathrm{SU}(2) \text{ fixed point}}_{\text{Q2 Thm 8.6}} ;\Longrightarrow; \underbrace{E = -\tfrac{j(j+1)}{3}\hat{a}\cdot\hat{b},;\text{all }j}_{\text{Q3 Thm 5.2}}$$

Six conceptually distinct steps:

1. Phase coherence from BI indiscernibility (Q1) — any admissible transition preserves the BI indiscernibility of conjugate Weil blocks $\rho_c$ and $\rho_{q-c}$; otherwise the rank of the Gram--Schmidt span inflates beyond the admissible bound of O22.
2. Singlet correlator from four structural inputs (Q1) — bilinearity from coherent amplitudes, linearity of $\mathfrak{su}(2)$ observables (O23), rotation invariance from isotropic saturation (O23), and unit normalisation from the parity involution (O18) force $E(\hat{a},\hat{b}) = -\hat{a}\cdot\hat{b}$.
3. Tsirelson bound as corollary (Q1) — Cauchy--Schwarz on the derived correlator gives $|S_{\mathrm{CHSH}}| \leq 2\sqrt{2}$ unconditionally.
4. Born rule from structural uniqueness (Q1) — the unique probability assignment compatible with positivity, normalisation, linearity, and the derived correlator is $P(a=+1|\hat{a}) = |\langle +\hat{a}|\psi\rangle|^2$; no Gleason theorem is needed.
5. $\mathrm{SU}(2)$ as stable fixed point (Q2) — co-admissibility $\lambda_{1/2} = \lambda_{3/2} = 18$ on $2I$; the admissibility flow under spectral refinement in the LPS limit selects $j = \tfrac{1}{2}$ as the unique stable sector.
6. Universal spin-$j$ generalisation (Q3) — for all five admissible sectors $j \in {\tfrac{1}{2}, 1, \tfrac{3}{2}, 2, \tfrac{5}{2}}$: proto-state is the singlet $|\Omega_j\rangle$ (Schur on Clebsch--Gordan), universal correlator $E = -\tfrac{j(j+1)}{3}(\hat{a}\cdot\hat{b})$, sectorwise Born rule.

Constituent Papers

#	Paper	Stage	Local path
1	Q1 (Beau2026q1) — Phase coherence and the spin-$\tfrac{1}{2}$ quantum sector	Phase coherence, singlet correlator, Tsirelson, Born rule at $j=\tfrac{1}{2}$	../q1/
2	Q2 (Beau2026q2) — Co-admissibility and $\mathrm{SU}(2)$ as stable fixed point	Co-admissibility $\lambda_{1/2} = \lambda_{3/2}$, fixed-point selection	../q2/
3	Q3 (Beau2026q3) — Universal spin-$j$ quantum sector	Proto-state, universal correlator, Born rule for all admissible $j$	../q3/
4	Bell paper (Beau2026b) — Bell non-applicability in non-injective frameworks (published, Quantum Reports 2026)	Structural non-applicability of Bell factorizability	../../bell-paper/

Status of Results

Proved (unconditional):

• Phase coherence from BI indiscernibility (Q1 Theorem 2.7).
• Observable rank signature $\mathrm{rank},W^{(c)}_n = 0$ as a falsifiable coherence witness (Q1 Corollary 2.9).
• Singlet correlator $E(\hat{a},\hat{b}) = -\hat{a}\cdot\hat{b}$ at spin-$\tfrac{1}{2}$ (Q1 Theorem 2.14).
• Tsirelson bound $|S_{\mathrm{CHSH}}| \leq 2\sqrt{2}$ (Q1 Corollary 2.16).
• Born rule at spin-$\tfrac{1}{2}$ (Q1 Theorem 2.18).
• Co-admissibility $\lambda_{1/2} = \lambda_{3/2} = 18$ on $2I$ (Q2).
• $\mathrm{SU}(2)$ as unique stable fixed point of the admissibility flow (Q2 Theorem 8.6).
• Proto-state is the singlet for all admissible $j$ (Q3 Theorem 4.1).
• Universal correlator $E = -\tfrac{j(j+1)}{3}(\hat{a}\cdot\hat{b})$ (Q3 Theorem 5.2).
• Born rule for all five admissible $j$ (Q3 Corollary 6.1).
• Bell factorizability non-applicable in non-injective frameworks (Bell paper, published).

Numerical:

• $\mathrm{rank},W^{(c)}_n = 0$ confirmed for $q = 29$ across all four conjugate pairs throughout the admissible regime. Extension to $q \in {61, 101, 151, 211, 307, 401}$ identified as a validation target.

Open Deliverables

1. Born rule for general observables — the $\mathrm{SU}(2)$ case is complete; the general case requires either an extension of the parity-sector argument to arbitrary fibre structures, or a direct Gleason-type theorem for the admissible measure on $V_\rho$.
2. Numerical validation across the full prime range — systematic verification of the rank-$W^{(c)}_n = 0$ signature for $q \in {29, 61, 101, 151, 211, 307, 401}$ would provide comprehensive empirical confirmation and track the boundary transition as a function of $q$.

Compilation

bash compile.sh

Produces out/QuantumStructureNote.pdf.

Citation

To be assigned upon first Zenodo publication.