Gap Bridge Theory: Structured Correspondences Across Domains

Gap Bridge Theory is a working theory of structured correspondences: maps that let you carry a piece of reasoning from one mathematical domain into another without losing the identity of the object you started with. When a correspondence preserves the right invariants (cardinality of a residue alphabet, size of an analytic error, injectivity of a folding map), conclusions transport. We develop three instances. Bridge A equates the multiplicative structure of coprime residue alphabets with an additive rate channel C(n)=log2φ(n) — the noiseless alphabet-size rate of (ℤ/nℤ)×. Bridge B converts Basel-type analytic tails into a precision-bit budget via B(ε)=−log2ε. Bridge C identifies coprimality with alias-free folding of residue channels (CRT injectivity; lcm collisions when moduli share factors). Classical number theory and elementary information counting supply the ingredients; Lean 4 / Mathlib locks the statements we rely on. Beyond the three bridges, this manuscript develops the theory as a usable architecture: discussion of when each bridge applies, what transport buys over ordinary citation, honest limits, and the value of formalization; then concrete applications — modulus choice, alphabet sizing, bit-budget conversion, protocol checklists, teaching moves, Lean reading practice, and anti-patterns. Laboratory numerical corroboration on finite grids supports the formalized statements. Zenodo deposit note: Published as v2 (version field 2.0.0) on the original record family of doi:10.5281/zenodo.18750234 (concept doi:10.5281/zenodo.18011947). Manuscript content stamped 2.2.0. This version deposits the PDF manuscript only. Resource type corrected to Preprint; access open. Calm lineage: earlier Novel Bridges packaging (18750234 / open siblings 18750166, 18750117) developed the same bridges; the present deposit shares Gap Bridge Theory as a theory-first exposition.