Specular Bit Architecture (SBA) for Reliability Protocols Across Photonic and Neuromorphic Hardware: A Formal Mathematical Framework

This paper presents the Specular Bit Architecture (SBA) as a substrate-agnostic formal informational layer for reliability-aware hardware, with emphasis on photonic, neuromorphic, and hybrid accelerator settings under pre-registered evaluation protocols. SBA encodes signed information with specular carriers (without an explicit sign bit), uses Fibonacci positional weights for exact local rewrites, and defines deterministic sanitization and reconstruction with bounded propagation. The framework provides: (1) a signed-digit model with Zeckendorf-style uniqueness per channel; (2) exact rewrite rules derived from Fibonacci identities; (3) synchronous read-compute-write semantics with deterministic local arbitration (Protocol C); (4) a direct canonical unranking map that recovers the unique shifted-Zeckendorf support associated with an integer label without simulating intermediate rounds; (5) an exact width-5 combinatorial count for canonical patterns and ordered center-overflow pairs together with exhaustive local enumeration yielding rule-specific empirical certificates and extraction of rule-, policy- and configuration-specific bounded-propagation constants ; and (6) an informational mass-energy bookkeeping model with an optional discrete-continuous geometric extension for propagation-resource estimation, including an exact closed-form asymmetric combinatorial load density for the Pascal-triangle support graph. The same formal layer now also makes explicit the rewrite-induced directional topological bias (+1)+(-2)/2=-1/2≠0 and exact integer-lattice reachability under exact arithmetic, for which the precision-induced maximal Lyapunov exponent satisfies λ_max = 0. Formal informational results are separated from hardware-modeling claims: any substrate mapping beyond the formal layer is treated as calibration-bound and deployment-specific, with explicit comparability and uncertainty protocols. Parameters such as m₀, c, and c₁ are modeling constants or calibration-dependent proxies and require independent calibration for physical use ; the optional Φ-ratio concurrency normalization is retained only as an exploratory calibration-layer ansatz, not as a theorem-level derivation. Under the stated synchronous and local-arbitration assumptions, the resulting primitives provide strict locality, deterministic normalization, and bounded error propagation across heterogeneous substrates, and they admit a continuum-limit bridge for propagation-resource estimation in the modeling layer under explicit calibration assumptions. A companion open-source proofbench (MIT-licensed) provides reproducible computational artifacts for independent audit of configuration-specific certificates, including measured-front causal-pyramid diagnostics, manuscript-ready Pascal-triangle summary tables, bounded-propagation certificates, and hardware-oriented benchmark exports under explicit separation between formal, modeling, and hypothesis-layer claims. The present revision also records: a model-layer Shannon-entropy ceiling for ternary readout uncertainty; a model-layer weighted-Landauer-erasure refinement that adds Fibonacci-weighted excess dissipation above the universal Landauer floor under substrate calibration ; an exact Lucas-theorem characterization of triadic self-similar zero-void structure in the ternary-reduced Pascal load field ; a selection-rule result showing that exact local rewrite preservation forces the weight law into the recurrence with unique expanding eigenvalue ; a discrete impulse-response interpretation in which sanitization plus Protocol C exactly damps local noncanonical conflict modes ; and a guarded evolution-to-index construction for declared canonical state families and a three-stage microstate/update/canonical projection discipline. No theorem in this manuscript, by itself, establishes a validated claim about substrate physics or deployment-level reliability ; all substrate interpretations remain calibration-bound and require independent experimental validation. Index Terms: neuromorphic computing, photonic computing, reliability-aware architectures, Specular Bit Architecture, bounded propagation, signed-digit arithmetic, Fibonacci positional coding