IndisputableMonolith.Information.ErrorCorrectionCodesFromJCost
The module defines error correction code families derived from the J-cost in Recognition Science. It establishes that the decoding threshold gap 1-r on the φ-ladder shrinks for deeper families. Information theorists working in the RS framework would cite these constructions when linking recognition composition to code rates. The module consists of definitions for ECCFamily together with lemmas proving the gap is positive and strictly decreasing.
claimThe module introduces the family of error-correcting codes ECCFamily parameterized by depth on the φ-ladder, together with the decoding threshold gap function satisfying thresholdGap(r) = 1 - r > 0 and strictly decreasing for deeper families.
background
Recognition Science derives all physics from the single functional equation whose information measures are governed by the J-cost J(x) = (x + x^{-1})/2 - 1. This module sits in the information domain and imports the fundamental RS time quantum τ₀ = 1 tick from the Constants module. It introduces ECCFamily as a parameterized collection of codes whose rates sit on the φ-ladder and defines thresholdGap to quantify the residual gap 1-r, with sibling lemmas establishing its positivity and monotonicity.
proof idea
This is a definition module that first declares ECCFamily, eccFamily_count and thresholdGap. It then supplies the supporting lemmas thresholdGap_pos and thresholdGap_strictDecr by direct appeal to the ordering properties of the φ-ladder.
why it matters in Recognition Science
The module supplies the concrete code families needed to embed error correction inside Recognition Science information theory. It fills the step that converts J-cost into controllable decoding thresholds on the φ-ladder, thereby preparing the ground for later integration with the eight-tick octave and the alpha band. No immediate parent theorems are listed in the used-by graph, yet the construction directly supports any downstream claim that invokes error-corrected information measures.
scope and limits
- Does not compute explicit code rates or distances for individual rungs.
- Does not prove that the codes achieve capacity or meet the Shannon bound.
- Does not address quantum error-correction variants or physical implementations.
- Does not connect the gap to the mass formula or Berry creation threshold.