What question did this study set out to answer?

To demonstrate that quantum-mechanical phase coherence derives from projective admissibility rather than serving as an independent axiom.

June 4, 2026Open Access

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

Key Points

To demonstrate that quantum-mechanical phase coherence derives from projective admissibility rather than serving as an independent axiom.
Establish theorems about admissible transitions and phase coherence structure in quantum mechanics.
Exhibit a constructive separating functional confirmed to be BI-admissible.
Conduct numerical validation for specific parameters across conjugate pairs.
Proved that admissible transitions preserve the indiscernibility of conjugate Weil blocks with $ ho_c$ and $ ho_{q-c}$ maintaining phase coherence.
Confirmed that rank $W_n^{(c)} = 0$ persists, establishing representational indistinguishability of conjugate blocks.
Derived quantitative relations such as the Tsirelson bound and Born probability rule without invoking quantum postulates.

Abstract

We establish that quantum-mechanical phase coherence is not an independent postulate but a structural consequence of projective admissibility within the Cosmochrony framework. The central result (Theorem thm: admissibility-phase-coherence) shows that any admissible transition preserves the Born–Infeld indiscernibility of conjugate Weil blocks c and ₐ-₂ on the Heisenberg Cayley graph Cay (Heis₃ (Z/qZ), Sq), and thereby maintains the metaplectic phase coherence of the admissible fibre. The proof rests on three independently established results: the minimal parity fibre structure of the non-injective projection (O18), the dynamical identity of conjugate Weil blocks (O17), and the discrete admissibility filter of projection locking (O22). A constructive separating functional Rₙ^ (c) is exhibited and proved BI-admissible. Numerical validation for q=29 across four conjugate pairs confirms that rank\, Wₙ^ (c) = 0 exactly throughout the admissible regime, establishing structural indistinguishability of conjugate blocks as a representation-theoretic fact rather than a dynamical coincidence. As a structural consequence, the admissible fibre carries complex amplitude structure with interference, constituting the natural precursor of an effective Hilbert sector — without quantum mechanics as a postulate. The singlet correlator E (a, b) = -a is then derived from admissibility, the parity involution (O18), and the isotropic SU (2) structure (O23), without invoking any quantum postulate. As a consequence, the Tsirelson bound |S₂₇ₒ₇| 22 follows unconditionally from the geometry of Im\, H under admissible projection. Finally, the Born probability rule P (a=+1\, |\, a) = |+a||² is derived as the unique probability assignment compatible with the coherent fibre structure, the SU (2) observable algebra, and the two-point correlator law. No quantum postulate is invoked at any step of the derivation.

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