IndisputableMonolith.Information.ErrorCorrectionCodesFromJCost
This module constructs families of error-correcting codes whose decoding thresholds derive from the J-cost on the phi-ladder. Researchers deriving information bounds in Recognition Science cite it for explicit gap formulas. The module supplies definitions for ECCFamily and thresholdGap together with count and monotonicity lemmas.
claimFor an ECCFamily $f$ indexed by rung on the phi-ladder the decoding threshold gap equals $1-r(f)$, where deeper families produce strictly smaller gaps.
background
Recognition Science obtains all constants from the forcing chain T0-T8 with J-cost $J(x)=(x+x^{-1})/2-1$. The imported Constants module fixes the RS time quantum as tau0=1 tick. This module introduces ECCFamily as code families parameterized by phi-ladder position and defines thresholdGap as the quantity 1-r measuring the gap to the Shannon limit.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
The module supplies the ECCFamily and thresholdGap objects that support information-theoretic constructions inside the Recognition Science framework. It fills the gap between the J-cost definition and explicit error-correction thresholds on the phi-ladder.
scope and limits
- Does not exhibit concrete codewords or encoding maps.
- Does not derive numerical values for any specific rung.
- Does not address physical realizability of the codes.
- Does not connect the gap to the alpha band or G constant.