IndisputableMonolith.Information.ErrorCorrectionBounds
This module assembles error-correction bounds for binary symmetric channels inside the Recognition Science eight-tick clock by importing Shannon entropy derived from J-cost. Information theorists working on discrete physical channels would cite the capacity formula and the adapted Hamming and singleton bounds. The module is a collection of definitions and statements that restate classical results in RS-native units without new derivations.
claimFor a binary symmetric channel with crossover probability $p$, the capacity is $C=1-H(p)$ where $H(p)=-p\log_2 p-(1-p)\log_2(1-p)$ is the binary entropy; reliable transmission is possible at any rate $R<C$. The module also states the Hamming bound and singleton bound specialized to eight-tick codes.
background
The module operates in the Information domain and imports three upstream modules. Constants supplies the fundamental time quantum $\tau_0=1$ tick. EightTick defines the discrete 8-tick cycle with phases $0,\pi/4,\pi/2,3\pi/4,\pi,5\pi/4,3\pi/2,7\pi/4$. ShannonEntropy states that Shannon entropy $H=-\sum p_i\log p_i$ arises directly from the J-cost structure of Recognition Science.
The supplied doc-comment reproduces Shannon's channel-capacity theorem for the binary symmetric channel and notes that any rate below capacity permits reliable transmission. Sibling declarations inside the module (binarySymmetricCapacity, hamming_bound_8tick, eight_tick_redundancy, etc.) translate these classical statements into the eight-tick setting.
proof idea
This is a definition module, no proofs. It simply records the classical capacity formula, the binary-symmetric-channel specialization, and the standard Hamming and singleton bounds after substituting the eight-tick period and J-cost entropy.
why it matters in Recognition Science
The module supplies the information-theoretic layer required by the eight-tick octave (T7) before error-correction arguments can be applied to physical codes. It feeds the sibling declarations that define eightTickCode, topologicalCodes and toricCode, thereby linking Shannon capacity to the discrete clock structure. No downstream theorems are recorded yet; the module therefore closes the interface between the ShannonEntropy import and concrete coding bounds.
scope and limits
- Does not derive the capacity theorem from J-cost axioms.
- Does not treat continuous or non-binary channels.
- Does not construct explicit codes, only bounds.
- Does not address quantum error correction beyond naming the type.
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