gapAt_succ_ratio
plain-language theorem explainer
The theorem states that the gap-to-capacity value at rung k+1 on the phi-ladder equals the value at rung k multiplied by phi inverse. Information theorists modeling finite-length polar-code performance and analysts fitting athletic record progressions cite this exact one-step decay. The proof unfolds the gapAt definition, invokes phi nonzero to justify the exponent split, and closes with ring simplification on the resulting powers.
Claim. Let $g(k)$ be the gap-to-capacity at phi-ladder rung $k$, given by $g(k) = g_0 phi^{-k}$ for reference gap $g_0$. Then $g(k+1) = g(k) phi^{-1}$.
background
In the PolarCodeGapFromPhi module the gap-to-capacity sits on the phi-ladder. The definition gapAt (k : Nat) returns referenceGap times phi raised to the integer power minus k. This construction mirrors the same phi-ladder structure used for quantum-channel capacity corrections and LDPC rate gaps. The module imports Constants for the nonzero property of phi and re-exports the gapAt definition from the sibling Sport.RecordProgressionFit module.
proof idea
Tactic proof that begins by unfolding gapAt. It obtains phi nonzero from Constants.phi_ne_zero, rewrites the negative exponent as a sum of -(k) and -1, applies zpow_add0, casts the natural-number index to integer, and finishes with ring.
why it matters
The result supplies the one-step ratio required by polarCodeCert, which bundles gap positivity, the successor ratio, and the adjacent ratio into a single certificate. The same ratio appears in RecordProgressionFit.consecutive_gap_ratio and gapAt_strictly_decreasing, allowing the athletic-record model to inherit the identical decay law. Within the Recognition framework this encodes the self-similar fixed-point property of phi on the ladder (T6) and the eight-tick octave scaling.
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