IndisputableMonolith.Information.ErrorCorrectionBounds
The module formalizes Shannon channel capacity for binary symmetric channels by deriving binary entropy from the J-cost and embedding results in the eight-tick discrete clock. Researchers in discrete spacetime physics cite it to establish reliable transmission rates R below capacity C = 1 - H(p). It consists of definitions for error codes and bounds like the Hamming bound adapted to the 8-tick structure, relying on algebraic application of upstream entropy lemmas.
claimFor a binary symmetric channel with crossover probability $p$, the capacity is $C = 1 - H_2(p)$ where the binary entropy $H_2(p) = -p log_2 p - (1-p) log_2 (1-p)$ is obtained from the J-cost. Reliable communication is possible at any rate $R < C$.
background
Recognition Science operates on an 8-tick discrete clock with phases at multiples of $pi/4$ and fundamental time quantum $tau_0 = 1$ tick. The upstream ShannonEntropy module derives entropy $H = -sum p_i log p_i$ directly from the J-cost structure as the measure of uncertainty. This module applies those primitives to error correction, introducing types such as ErrorCode and eightTickCode that respect the discrete phases.
proof idea
This is a definition module, no proofs. It declares supporting types including ErrorCode and eightTickCode, then states theorems such as hamming_bound_8tick and singleton_bound_8tick that apply the binary entropy definition to derive code size limits in the eight-tick setting.
why it matters in Recognition Science
It supplies the capacity and bound results that support quantumErrorCorrection and topologicalCodes within the module. The content realizes the Shannon capacity theorem for the binary symmetric channel, tying it to the eight-tick octave in the forcing chain. This step grounds information limits in Recognition Science primitives, enabling thresholds for error correction in discrete models.
scope and limits
- Does not derive capacities for quantum channels or continuous-time limits.
- Does not include multi-user or network coding results.
- Does not compute explicit numerical values for constants such as alpha.
depends on (3)
declarations in this module (18)
-
def
binarySymmetricCapacity -
def
capacity_at_10_percent -
structure
ErrorCode -
theorem
hamming_bound_8tick -
theorem
singleton_bound_8tick -
theorem
eight_tick_redundancy -
def
eightTickCode -
theorem
eight_tick_corrects_3 -
def
quantumErrorCorrection -
theorem
threshold_majority_voting -
def
topologicalCodes -
def
toricCode -
def
eightTickSyndrome -
theorem
detect_vs_correct -
theorem
rate_bound_from_8_tick -
theorem
error_bound_from_8_tick -
def
summary -
structure
ErrorCorrectionFalsifier