Admissible Non-Injective Transitions as the Primitive of Physical Description

The programme's thesis is that quantum mechanics is the effective theory of physically real unresolved admissible structures and their resolution. This paper proposes axioms for that thesis on a single primitive: the local structure of admissible transitions between observable states. The four axioms are local projective admissibility (A1), structural non-injectivity (A2), admissibility as non-premature selection (A3), and projection locking (A4). On non-injective transitions, 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. We define the proto-state as the unresolved configuration shared between two successive states and derive from A1 and A3 its structural indeterminacy among the admissible directions; its physical reality is an interpretive reading. A3 preserves these open directions but does not, by itself, forbid an erasure of their relative phases. Given a representation dictionary for the fibre, A1 and A3 impose irreducibility only under an additional reading, not derived here, on which a generator-stable decomposition counts as premature selection. For a carrier of dimension greater than one generated by a conjugate pair, irreducibility implies non-commutation. It does not make the commutator central or select a finite Heisenberg group, as an explicit $\mathfrak{S}_3$ countermodel shows: the Heisenberg carrier of the spectral programme is supplied, and distinct from its Weil representation. The discreteness of quantum transitions is postulated by A4; on the supplied shell realisation, a realised Born–Infeld saturation depth is an integer, and its existence is not established. Complex carriers are supplied, the Born weight is an additional rule, and Hilbert structure is open.