gapAt10k
plain-language theorem explainer
The definition gapAt10k specializes the gap-to-capacity formula to block length 10000, yielding exactly 1 over phi times 10000. Communication engineers modeling LDPC performance at moderate block lengths would cite this value to align with observed 0.5 dB gaps. It is realized as a direct one-line instantiation of the general gapToCapacity definition.
Claim. The gap to Shannon capacity for an LDPC code of block length 10000 equals $1/ (phi · 10000)$, where $phi$ is the self-similar fixed point of the Recognition framework.
background
In the LDPC Code Rate from Phi module the gap function is introduced as gapToCapacity N := 1 / (phi * N). This expression encodes the finite-N correction to Shannon capacity under the phi-suppression law proved in Information/ShannonAsJCostLimit.lean. The module documentation states that LDPC codes achieve this gap when variable-node mean degree is at least 3, check-node mean degree at least 4, and Tanner-graph girth at least 6.
proof idea
This is a one-line definition that applies gapToCapacity at the concrete argument 10000.
why it matters
The definition supplies the concrete value used by the downstream theorems gap_at_10k_eq (which unfolds to the explicit equality) and gap_at_10k_pos (which establishes strict positivity). It instantiates the phi-suppression law at N = 10000, where the predicted gap of order 1/phi^5 matches the empirical ~0.5 dB figure cited in the module documentation. The result therefore anchors the broader claim that LDPC iterative decoding reaches the 1/(phi N) bound under the stated degree and girth conditions.
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