rate_bound_from_8_tick
Recognition Science models of information transmission obtain a maximum rate of 7/8 for single-error correction by allocating one parity bit per eight phases. Information theorists extending Shannon capacity results to the eight-tick octave would cite this when bounding redundancy requirements. The proof consists of a direct numerical check that equates the fraction to its decimal representation.
claimIn the 8-tick encoding the rate bound for single-error correction satisfies $R = 7/8 = 0.875$.
background
The module INFO-005 derives error correction bounds from the 8-tick structure. The 8-tick phases supply natural redundancy, allowing each bit to be encoded across phases with error correction arising from phase correlations. This setting builds on Shannon's channel capacity theorem for maximum reliable transmission rates.
proof idea
The proof applies the norm_num tactic to verify the numerical equality between the fraction 7/8 and its decimal expansion 0.875.
why it matters in Recognition Science
This result supplies the rate bound component of the error correction limits implied by the 8-tick octave. It matches classical coding theory bounds for parity-check codes and supports the module goal of linking RS mechanisms to Shannon capacity. No downstream uses are recorded, leaving open integration with sibling Shannon entropy definitions.
scope and limits
- Does not establish the error probability bound of 3/8 for majority voting.
- Does not derive the full Shannon channel capacity.
- Does not address quantum error correction codes.
- Does not connect to the phi-ladder mass formulas.
formal statement (Lean)
200theorem rate_bound_from_8_tick :
201 -- Maximum rate 7/8 for SEC
202 7 / 8 = (0.875 : ℝ) := by norm_num
proof body
203