Admissible Non-Injective Transitions as the Primitive of Physical Description

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). From A1 and A2 alone, we derive irreversibility, the arrow of time, and the existence of structural fluctuations, without any additional postulate. 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. We further prove that A3 forbids any admissible factorisation of the fibre, which forces a non-trivial commutator between conjugate generators; combined with minimality (A1) and Born–Infeld parity, this identifies the symmetry group of the admissible fibre as $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ for a prime $q$, and the fibre itself as the Weil representation $V_\rho$ — closing the identification $F_n \simeq V_\rho$ as a theorem rather than a remark. 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. This paper provides the axiomatic foundation cited by the companion papers and serves as the formal basis for the derivations carried out therein.