Quantum Mechanics from Records: Three Consistency Postulates and One Kernel Recover the Dirac–von Neumann Formalism, with the Projection Postulate as a Theorem

We give an axiom set for quantum mechanics in which measurement is derived rather than posited. The primitives are consistency constraints on observer-indexed information states: three postulates — determinacy indexed to an information state, informational blanks identified with superpositions, and a two-generation consistency-pruning rule whose ∃∀ form is selected by elimination within a stated class — together with one quantitative axiom, a path-pair weighting kernel over the powerset of alternatives. The main result is a recovery theorem. Under the assignment of a cluster of alternatives to the vector $\sum_p e^{iS[p]/\hbar}\,|x(p)\rangle$ over the readout configurations, the axiom set recovers the Dirac–von Neumann formalism axiom by axiom: states are rays, superposition is the union of clusters in the interacting regime, observables are the partitions that records induce, the Born rule is the weight of a record-restricted cluster, the projection postulate is pruning followed by renormalization, composites are products of path spaces, evolution is the kernel’s composition law, and no-signalling is an algebraic identity. Six of the seven axioms are theorems of the set; the seventh, the Born rule, is absorbed into the one weight law, the amplitudes being recovered from cluster weights by polarization up to a global phase and complex conjugation. No measurement, projection, or observables axiom remains. The theorem is exhibited three times: a classification theorem sorting every candidate superposition under an arbitrary detector configuration into three bins fixed by the records, which relocates the quantum–classical boundary from scale to information; a two-tag interferometry configuration displaying both composition regimes in one experiment and closing on the complementarity relations for reasons that are theorems; and a reductio showing the resulting realism surviving Bell’s theorem, which prices hidden valuations and finds none here to price. The paper is a re-axiomatization, not an operational reconstruction: the kernel is posited, and its content — continuity, local tomography, the quadratic form — is what the reconstruction programs seek from operational axioms. Every assumption is inventoried and the open constructions are named.