gapToCapacity
plain-language theorem explainer
The definition supplies the gap to Shannon capacity for an LDPC code of block length N as 1 over phi times N. Information theorists bounding finite-blocklength LDPC performance in 5G and storage systems would cite this expression when quantifying overhead. It is introduced as a direct algebraic formula with no reduction steps or lemmas.
Claim. The gap to Shannon capacity for an LDPC code of block length $N$ is given by $g(N) = 1/ (phi N)$, where $phi$ is the golden-ratio fixed point.
background
The module treats the finite-N correction to Shannon capacity for LDPC codes as phi-suppressed. Low-density parity-check codes approach capacity when variable-node degree is at least 3, check-node degree at least 4, and Tanner-graph girth at least 6. The gap expression follows directly from the Recognition Science claim that the correction term equals 1 over phi N.
proof idea
This definition is a one-line algebraic expression that substitutes 1 over (phi times N) for the gap function.
why it matters
The definition anchors the LDPC rate certification by supplying the explicit phi-suppression law g(N) = 1/(phi N). It is invoked by gap_pos, gap_decreasing, gap_doubling_halves, gap_times_N_invariant, gapAt10k, and the LDPCRateCert structure. The construction realizes the B16 deepening on LDPC codes, consistent with phi-ladder scaling and the eight-tick octave from the core forcing chain.
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