Admissible Non-Injective Transitions as the Primitive of Physical Description

Abstract

Quantum mechanics is the effective theory of physically real unresolved admissible structures and their resolution. This paper establishes the minimal axiomatic foundation from which this claim follows: physical structure is derived from a single primitive, the local structure of admissible transitions between observable states. Four axioms govern this structure: local projective admissibility (A1), structural non-injectivity (A2), admissibility as non-premature selection (A3), and projection locking (A4). Along the transitions A2 asserts to be generically non-injective, A1 and A2 give immediate-image information loss and structural fluctuations; neither permanent non-recovery nor the arrow of time follows, since non-invertible steps admit cycles. From A3, we derive the proto-state as a physically real unresolved configuration shared between two successive states, and show that admissibility forces the retention of phase coherence. Under an explicit representation-theoretic reading of the admissible fibre, A3 forbids invariant factorisation and hence imposes irreducibility. For a carrier of dimension greater than one generated by a conjugate pair, irreducibility implies non-commutation. It does not, however, make the commutator central or select a finite Heisenberg group: an explicit $\mathfrak{S}_3$ countermodel satisfies the extracted carrier contract and violates that conclusion. The Heisenberg carrier used by the spectral programme therefore remains a supplied realisation, and must be distinguished from its associated Weil representation. From A4, we derive the discrete character of quantum transitions as a consequence of the interaction between continuous Born–Infeld saturation and the discrete shell structure of the observable space. The foundation is stated with the point at which it stops short of carrier selection.