The Cosmochrony Research Programme: A Structural Roadmap and Paper Inventory

The Cosmochrony programme develops a pre-geometric framework in which physical structure — spacetime, quantum mechanics, gauge symmetry, and Standard Model observables — arises from a single primitive: the local structure of admissible non-injective transitions between observable states. The corpus comprises 110 papers across three theory branches. Branch I (Foundation track) builds the axiomatic base: four axioms suffice to derive the Heisenberg group ${\mathrm{Heis}}_3(\mathbb{Z}/q\mathbb{Z})$ and its Weil representation as theorems, without any background substrate or relaxation mechanism. Branch II (O-series) develops the spectral admissibility sub-programme, deriving and numerically validating the transfer chain $c_{\mathrm{BI}} \to \delta_{\mathrm{pair}} \to \beta^* \approx 0.126$ connecting the BI saturation constant to the charged-lepton mass hierarchy. Branch III (Q-series and companion papers) derives quantum mechanics, spacetime geometry, gauge structure, and further physical observables from Branch I axioms and Branch II spectral data. Paper Q3 completes the ${\mathrm{SU}}(2)$ quantum sector: by combining BI indiscernibility with Schur's lemma on the Clebsch–Gordan decomposition of $\chi_{2j+1}\otimes\chi_{2j+1}$, the universal singlet correlator $E(\hat{a},\hat{b}) = -j(j+1)/3\cdot(\hat{a}\cdot\hat{b})$ and the Born rule are derived for all five admissible sectors of $2I$, closing the general-$j$ extension that was open in Q2. The geometric emergence sub-programme (Q5a–Q10, U1, W1, H2) closes the effective spacetime identification: under the Q5a continuum-limit hypotheses, the lifting hypothesis [H-lift] is proved (Q9), asymptotic $\mathfrak{su}(2)$-isotropy $A_H \to 2$ is proved (Q10, U1), the admissibility weight convergence [H-w] is proved (W1), yielding the effective Lorentzian co-metric $g^{\mu\nu} = \mathrm{diag}(-2, 2, 2, 2) \propto \eta^{\mu\nu}$ ($A_\tau = 2$, Q11 ). Q5a v2.0 establishes that [H1] is not needed in full generality. Q5a-O2 proves spectral atomicity of the admissible sector (each pair spans three pure Fourier modes) and closes hypotheses [H-$\mathcal{E}$1] and [C] without Nash inequalities. Hypothesis [H2] (strong convergence of the rescaled Weil generators) is proved (H2 paper), closing all hypotheses of the Q5a large-$q$ convergence theorem unconditionally on the admissible sector. O31 develops the structural framework for the SU(3) identification: the metaplectic intertwining of ${\mathrm{Heis}}_3(\mathbb{Z}/q\mathbb{Z})$ forces triplet co-admissibility unconditionally for the colour-adapted Cayley graph, and the single-frequency BI fingerprint mechanism (v1.5) makes [H-color] exact pointwise on the standard graph at finite $q$, so SU(3) and the Standard Model composite $G_{\mathrm{SM}} = {\mathrm{SU}}(3)\times{\mathrm{SU}}(2)\times\mathrm{U}(1)$ are identified unconditionally for the standard graph. O33 gives an exact arithmetic description of the O12 three-character filtration, derives its Mackey–Hecke multiplicity doublets and Harper split, and proves a parity-closure obstruction protecting the odd Weyl-symbol sector. Its stabiliser analysis now includes an orbit-divisibility sieve, uniform exact $C_4$ and $C_6$ exceptional families, and (v1.1) an additive rigidity theorem proving $A_K=\{\pm1\}$ whenever every run and gap of the mode set is longer than the run count, together with a third uniform near-saturation $C_4$ family; completeness of the exceptional orders remains open only in the residual degenerate boundary regime. Version 1.4 describes that regime mechanically — the depth-zero square extends to an infinite uniform Gaussian-box $C_4$ family, and an exhaustive audit through $q=151$ classifies all $75$ exceptional sets into three refined mechanisms with no order beyond six — and proves that the porous depths of a uniform block converge to an explicit limiting point process with rational count law and mean $28/15$. Version 1.5 fixes the exact stabiliser value inside two of those mechanisms: a fourth-moment argument proves that every proper Gaussian box, direct or wrapped, has stabiliser exactly $C_4$, and every free four-point complement orbit has the same exact stabiliser; completeness of the three refined mechanisms, and a uniform identification of the top autocorrelation level of the Eisenstein $C_6$ sets, remain open. Its mixing criterion is closed (v1.3): a Galois separation theorem removes the cyclotomic phase determinant for every generic character, and the boundary-sum zeros are completely classified into antipodal-pair, cube-triple, and double-pair mechanisms on at most four hole pairs. This paper provides a complete inventory, a logical dependency map, and a structured account of what is proved, structural, heuristic, or open. It is intended both as an entry point for external readers and as an internal navigation reference.