This repository contains the source of the Q2 Cosmochrony paper Quantum Structure beyond Spin-1/2: Co-Admissible Sectors and the Emergence of SU(2) as a Fixed Point.

This work extends the quantum reconstruction programme beyond the minimal $SU(2)$ parity sector established in Q1.

It addresses the following question:

Is the quantum structure derived from admissibility specific to the spin-$\tfrac{1}{2}$ sector, or does it extend to other admissible representation sectors — and if so, how is the physically realised sector selected?

Quick Summary

Q2 shows that the answer is deeper than expected.

Quantum structure is not unique to a single sector.

It emerges across a co-admissible family, but with a structural selection mechanism.

More precisely:

• the spin-$\tfrac{1}{2}$ and spin-$\tfrac{3}{2}$ sectors are exactly co-admissible on $2I$
• both sectors independently reproduce:
  • phase coherence
  • singlet correlator
  • Tsirelson bound
  • Born rule
• the difference lies only in normalisation and representation dimension
• admissibility refinement selects $SU(2)$ as a stable fixed point

Thus:

• quantum mechanics is not tied to a specific representation
• it is the effective theory of a co-admissible family
• $SU(2)$ emerges as the unique stable limit under admissibility flow

Context

Q1 established that:

• admissibility implies phase coherence
• the admissible fibre carries complex amplitudes
• the singlet correlator $E(\hat{a}, \hat{b}) = -\hat{a}\cdot\hat{b}$ is derived
• the Tsirelson bound follows structurally
• the Born rule is uniquely fixed by coherence and correlators :contentReference[oaicite:0]{index=0}

However:

• the derivation was restricted to the spin-$\tfrac{1}{2}$ (SU(2)) parity sector
• it remained unclear whether this sector is uniquely selected
• the role of higher admissible sectors was unknown

This defines the scope of Q2.

Core Result

The paper establishes that:

The spin-$\tfrac{1}{2}$ and spin-$\tfrac{3}{2}$ sectors are co-admissible on the binary icosahedral group $2I$, and both reproduce the full quantum structure from admissibility alone.

Moreover:

$SU(2)$ emerges as the unique stable fixed point of this co-admissible family under admissibility refinement.

Main Structural Results

1. Exact co-admissibility

Result. The spectral data satisfy: $\lambda_{1/2} = \lambda_{3/2},$

Thus:

• both sectors share the same admissibility window
• neither is preferred at the discrete level
• the degeneracy is algebraically forced, not numerical :contentReference[oaicite:1]{index=1}

2. Phase coherence beyond spin-$\tfrac{1}{2}$

Result. The admissibility argument of Q1 extends unchanged:

• coherence is preserved by non-premature selection (A3)
• conjugate structures remain BI-indiscernible

Thus:

• phase coherence is sector-independent

3. Modified normalisation via Casimir

Result. In the spin-$\tfrac{3}{2}$ sector:

• the normalisation $E(\hat{a}, \hat{a}) = -1$ is replaced
• the Casimir operator provides the correct scaling

Thus:

• the structure is preserved
• only the representation-dependent normalisation changes

4. Singlet correlator in higher sector

Result. A singlet-type correlator is derived:

• still bilinear in observables
• still constrained by symmetry and isotropy
• consistent with admissibility and coherence

Thus:

• correlation structure is robust across sectors

5. Tsirelson bound generalisation

Result. A Tsirelson-type bound still holds:

• derived from fibre geometry
• independent of quantum postulates

Thus:

• non-classical correlations are structural, not sector-specific

6. Born rule persistence

Result. The Born rule is derived again:

• from coherence + correlator compatibility
• without invoking Gleason or external axioms

Thus:

• probability remains a geometric consequence of admissibility

7. Admissibility flow and sector selection

Result. In the LPS limit:

• the degeneracy lifts
• spin-$\tfrac{1}{2}$ becomes strictly dominant

Thus:

• $SU(2)$ is not imposed
• it is selected as a stable fixed point

Foundational Chain

The extension is fully internal:

Admissibility (A1–A3)
$\to$ phase coherence (Q1)
$\to$ quantum structure (Q1)
$\to$ co-admissible sectors on $2I$ (Q2)
$\to$ admissibility flow
$\to$ $SU(2)$ as fixed point

No additional quantum postulate is introduced.

Mathematical Role of Q2

Q2 provides:

• extension of Q1 beyond minimal sector
• proof of co-admissibility
• representation-theoretic generalisation of quantum structure
• structural selection mechanism for $SU(2)$

More precisely, it:

• removes the apparent arbitrariness of the spin-$\tfrac{1}{2}$ choice
• embeds quantum mechanics in a family of admissible sectors
• explains why $SU(2)$ is realised physically

Epistemic Structure

Established input

• admissibility axioms (A1–A4)
• parity fibre (O18)
• projection locking (O22)
• quaternionic minimality (O23)
• Q1 quantum reconstruction

New results

• co-admissibility of spin-$\tfrac{1}{2}$ and spin-$\tfrac{3}{2}$
• extension of coherence and correlators
• Casimir-based normalisation
• sector-independent Born rule
• admissibility flow
• emergence of $SU(2)$ as fixed point

Remaining open problems

• existence of additional co-admissible sectors
• extension to larger symmetry groups (e.g. $SU(3)$)
• multipartite mixing between sectors
• full Hilbert space reconstruction

Interpretation of the Result

The conceptual shift is:

• previous view: $SU(2)$ is the quantum symmetry
• Q2: $SU(2)$ is the stable endpoint of an admissibility flow

Thus:

• quantum mechanics is not tied to a fixed algebra
• it emerges from a selection process over admissible sectors
• representation theory becomes dynamical (in the admissibility sense)

Structural Role in the Programme

Q2 completes the first layer of the quantum bridge:

• Foundation: admissibility axioms
• HeisenbergStructure: algebra of the fibre
• Q1: quantum structure in minimal sector
• Q2: extension and selection of sectors

Thus:

• the existence of quantum structure is established (Q1)
• its universality and selection are established (Q2)

What Q2 Adds

• co-admissible sector framework
• extension beyond spin-$\tfrac{1}{2}$
• structural robustness of quantum laws
• admissibility flow concept
• derivation of $SU(2)$ as fixed point

Outcome

The quantum layer is now:

• derived (Q1)
• representation-independent (Q2)
• structurally selected (Q2)

Quantum mechanics is:

• not postulated
• not tied to a single representation
• the effective theory of admissible coherent fibres

Residual Open Problems

Higher symmetry groups

Determine whether admissibility admits fixed points beyond $SU(2)$.

Sector hierarchy

Understand the full structure of co-admissible families.

Multipartite structure

Extend the framework to entangled systems mixing sectors.

Continuum limit

Clarify the emergence of continuous Hilbert space.

Status

The programme now establishes:

• admissibility as primitive
• Heisenberg structure of the fibre
• emergence of quantum structure (Q1)
• universality and selection of sectors (Q2)

The remaining problem is:

• full generalisation beyond the parity sector and $SU(2)$ fixed point

Repository Structure

paper/
├── out/      # Compiled Q2 PDF
├── tex/      # LaTeX sources
└── README.md

Citation

If you reference this work, please cite:

J. Beau: Quantum Structure beyond Spin-$\tfrac{1}{2}$: Co-Admissible Sectors and the Emergence of $SU(2)$ as a Fixed Point Zenodo, 2026.

Acknowledgements

Portions of the conceptual synthesis, structural organisation, and editorial refinement benefited from iterative interactions with large language models used as analytical assistants.

All theoretical results, computations, and interpretations remain the sole responsibility of the author.

Contributions

This repository is intended as a research reference.

Critical feedback, independent verification, and further analysis of:

• co-admissible sectors
• admissibility flow
• representation selection
• higher symmetry emergence
• quantum structure beyond SU(2)

are welcome.

Please open an issue to discuss conceptual points, technical details, or possible extensions.