IndisputableMonolith.NumberTheory.ResidualGapHonest
This number theory module defines the natural phase mismatch cost at one residue step for a gate c as one plus one over c plus one, ensuring the ratio remains positive and defined at c equals zero. It equips the number theory layer with honest cost accounting for residual gaps. This supports phase visibility bounds by providing positive defect measures. The module consists of definitions and basic positivity lemmas without complex proofs.
claimThe natural phase failure cost for gate residue $c$ is $1 + 1/(c+1)$.
background
This module operates in the NumberTheory domain of Recognition Science, importing the Cost module for defect measures and the Bounded Phase Visibility module. The upstream result states that if a recovered integer ledger has a stable unresolved-phase budget and failed gates have a uniform K theta floor, then finite phase invisibility cannot persist beyond the supplied bound. The actual floor theorem is kept explicit as a named hypothesis. The module introduces the natural phase mismatch cost to handle residual gaps honestly at one residue step.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
This module supplies the cost function used in related definitions for divisor character sum bounds and positivity results, feeding into the broader Recognition Science framework for handling phase mismatches in the forcing chain from T5 J-uniqueness onward. It closes a gap in honest residual accounting for the phase visibility hypothesis.
scope and limits
- Does not derive the cost expression from the J-cost functional equation.
- Does not establish global bounds on the cost for arbitrary residue steps.
- Does not address interactions across multiple gates or full eight-tick periods.