COSMOLOGY OF THE FINITE REGISTRY

This paper presents the Octomorphic Registry, a finite-state geometric substrate consisting of exactly 84 admissible address states, from which atomic structure, chemical valence, lattice rigidity, and macroscopic physical constraints are derived algorithmically. The work converts the Octomorphic Framework from a conceptual proposal into a fully reproducible engineering artifact. All claims are enforced through deterministic construction rules, explicit gate predicates, and executable census algorithms. No phenomenological fitting, probabilistic assumptions, or narrative explanations are used. Core Contributions Finite Registry Construction A canonical derivation of an 84-seat address space generated from Fano-plane incidence geometry augmented by a two-bit lift. The registry is fixed in size and topology. Seat Ledger vs. Motif Census (Architectural Separation) A strict distinction is enforced between: the Registry Seat Ledger (R₈₄) — the immutable hardware address map, and the Triangle / Square Census — admissible runtime motifs discovered under gate constraints. This separation eliminates category errors between address capacity and structural occupancy. Gate-Based Admissibility All edges and cycles are filtered through: a Phase Strain gate (|ΔL| ≤ 0.25), and a Triality Balance gate (TB ≥ 0.60). Structures that fail these gates are explicitly forbidden, not approximated. Carbon Singularity and Valence Derivation The registry census proves that only one block (Block 5) contains: a self-stabilizing core triangle, and exactly four non-overlapping square docking ports, derived via an exact Maximal Independent Set computation. This yields tetravalence (methane) as a geometric necessity, not a chemical assumption. Lattice Rigidity and Diamond Structure Covalent bonding is defined as face-to-face port fusion, not point adjacency. Iterated fusion of Block-5 units yields a tetrahedral, rotation-locked lattice corresponding to diamond, deriving hardness as a topological inevitability. Finite Cosmology Space, motion, inertia, drag, and heat are derived as consequences of adjacency rewriting in a finite registry executing at a global update cadence. Time is defined as update ordering; heat as irreversible processing loss. Reproducibility The paper includes a Reproducibility Pack with: deterministic registry construction code, exact motif census and MIS solvers, full execution transcripts, validation ledgers mapping structural predictions to empirical constraints. All artifacts are frozen, normative, and independently executable. The framework introduces no new free parameters beyond explicitly marked open conjectures. Scope This work is intended as an engineering blueprint, not a review or interpretation. It targets researchers in mathematical physics, discrete geometry, theoretical chemistry, and foundational cosmology interested in finite-state formulations of physical law. License Released under the Octomorphic Open License v1.1.