The A|Ω⟩ Formalism: Consolidated Master Reference

We collect, in a single self-contained reference, the operator-algebraic formalism underlying the A|Ω⟩ programme. The common substrate is a Krein space K = HΦ ⊗ HΨ carrying a fundamental symmetry J: a positive-definite physical sector HΨ and an indefinite-metric conformal/phenomenal ghost sector HΦ. Part I stablishes this absolute mathematical architecture. Part II introduces the Batalin-Vilkovisky (BV) Quantum Master Equation and wheeled operad geometries to govern the closure of causal cells, establishing the topological Hermite-Biehler positivity of the de Branges kernel. Part III derives three superficially distinct phenomena directly from this foundation: (i) an indefinite-metric, BRST-quantized model of Euclidean quantum gravity whose physical cohomology is positive; (ii) a spectral realization in which the regularized vacuum partition function reproduces the completed Riemann ξ-function and the nontrivial zeros are realized as the spectrum of a Berry–Keating dilatation generator; and (iii) a phenomenal–physical correspondence predicting a falsifiable π/2 phase-offset in sensorimotor neural oscillations. Part IV provides the unifying dictionary and an honest status map of the programme’s analytic results.