IndisputableMonolith.NumberTheory.ResidualGapHonest
The ResidualGapHonest module supplies the natural phase mismatch cost at a single residue step for gate c, expressed as one plus the reciprocal of c plus one. Number theorists modeling unresolved-phase budgets in Recognition Science reference it to keep costs positive and defined at c equals zero. The module assembles this via direct definitions imported from Cost and BoundedPhaseVisibility without further lemmas.
claimThe natural phase mismatch cost for gate $c$ is given by $1 + 1/(c+1)$.
background
This module sits in the NumberTheory domain and handles residual gap honesty through phase accounting. It imports the Cost module for mismatch calculations and BoundedPhaseVisibility, which states that if a recovered integer ledger has a stable unresolved-phase budget and failed gates have a uniform KTheta floor, then finite phase invisibility cannot persist beyond the supplied bound. The actual floor theorem is kept explicit as KThetaFailureFloorHypothesis. The supplied cost expression is chosen to remain positive and defined at c equals zero.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the phase mismatch cost that supports parent results on bounded phase visibility and the KThetaFailureFloorHypothesis in the Recognition framework. It contributes to the number-theoretic scaffolding needed for the forcing chain landmarks including the eight-tick octave and positive cost ratios.
scope and limits
- Does not compute mismatch costs across multiple residue steps.
- Does not derive explicit duration bounds on phase invisibility.
- Does not incorporate the full J-cost or Recognition Composition Law.