From Admissibility to Quantum Structure: Phase Coherence and Correlations from Non-Injective Projection

Overview

A central question in the Cosmochrony programme is whether quantum mechanical structure — phase coherence, Born rule, entanglement — can be derived from the projective admissibility constraints, or whether they must be added as independent postulates.

Q1 answers this question for the spin-1/2 sector: phase coherence is not assumed but proved. Theorem 2.7 establishes that any admissible transition on the Heisenberg graph preserves the Born–Infeld indiscernibility of the conjugate Weil blocks $\rho_c$ and $\rho_{q-c}$, maintaining the metaplectic phase coherence of the fibre throughout the evolution.

Building on O18 (projective phase structure) and O23 (entanglement from projection), Q1 derives the bipartite singlet correlator and the Tsirelson bound as structural consequences of admissibility alone, with numerical validation for $q = 29$.

Central message. Admissibility → phase coherence → quantum correlations, without quantum postulates.

Core contributions

Theorem 2.7 — Phase coherence preservation: any admissible transition on the Heisenberg graph preserves the Born–Infeld indiscernibility of conjugate Weil blocks $\rho_c$ and $\rho_{q-c}$, maintaining the metaplectic phase coherence of the fibre. Phase coherence is thus a structural consequence of admissibility, not an input.
Singlet correlator from admissibility: combining Theorem 2.7 with the entanglement structure of O23, the bipartite proto-state is identified as the spin-1/2 singlet, and the correlator $E(\hat{a},\hat{b}) = -\hat{a}\cdot\hat{b}$ is derived without invoking Hilbert space postulates.
Tsirelson bound without quantum axioms: the bound $|E(\hat{a},\hat{b}) - E(\hat{a},\hat{b}') + E(\hat{a}',\hat{b}) + E(\hat{a}',\hat{b}')| \leq 2\sqrt{2}$ follows as a corollary of the projective structure, saturating the quantum maximum without assuming the Born rule.
Numerical validation: all results are explicitly verified for the prime $q = 29$, confirming the analytical derivations in the concrete Weil representation setting.

Interpretation

The conventional approach to quantum mechanics introduces phase coherence and the Born rule as axioms. Q1 shows that for the admissible fibre structure identified in the O-series, these properties are not independent assumptions: they are forced by the non-injective projection constraints.

The key mechanism is the Born–Infeld indiscernibility of conjugate Weil blocks. Because $\rho_c$ and $\rho_{q-c}$ are structurally indistinguishable under any admissible operation, the fibre cannot select between them — and this underdetermination is precisely what generates the metaplectic phase structure that underlies quantum superposition.

Physical implication. Bell-type correlations in this framework do not require non-locality or hidden variables: they arise from the projective structure of admissible transitions between Weil blocks. The Tsirelson bound marks the maximal coherence compatible with admissibility.

Relation to the Cosmochrony programme

Q1 opens the Q-series thematic track devoted to the emergence of quantum structure. Its position in the dependency chain is central:

O18: establishes the projective phase structure on Heisenberg graphs from which phase coherence is inherited.
O23: proves the entanglement structure of the bipartite fibre that Q1 uses to identify the proto-state as a singlet.
Q2: extends the programme to spin-3/2 on the binary icosahedral group, using Q1 as its foundation.
Q3: completes the universal spin-$j$ extension for all admissible sectors of the binary icosahedral group.

Together, Q1–Q3 constitute the quantum sector of the spectral admissibility programme: phase coherence, Born rule, and singlet correlators emerge from projective structure without any quantum postulate.

References

Jérôme Beau. From Admissibility to Quantum Structure: Phase Coherence and Correlations from Non-Injective Projection, 2026. doi:10.5281/zenodo.19561060