PolarCodeCert
PolarCodeCert packages three properties of the gap-to-capacity function gapAt on the phi-ladder: strict positivity for every rung k, exact one-step scaling by phi inverse, and the matching adjacent ratio. Information theorists deriving finite-blocklength bounds inside Recognition Science cite this certificate when aligning polar-code gaps with LDPC and quantum-channel results. The structure is filled by direct reference to the three supporting lemmas gapAt_pos, gapAt_succ_ratio, and gapAt_adjacent_ratio.
claimA structure $C$ whose fields assert: (i) $g(k)>0$ for all $k$, (ii) $g(k+1)=g(k)/phi$ for all $k$, (iii) $g(k+1)/g(k)=1/phi$ for all $k$, where $g(k)$ denotes the gap-to-capacity value at phi-ladder rung $k$.
background
The module places polar-code gaps to capacity on the phi-ladder. gapAt(k) is defined as referenceGap multiplied by phi to the power minus k. This produces the exact 1/phi ratio between adjacent rungs, matching the scaling already established for LDPC code-rate gaps and quantum-channel capacity corrections. Upstream gapAt_pos proves positivity by unfolding the definition and applying one_div_pos together with mul_pos on phi_pos. The phi-ladder scaling itself traces to the self-similar fixed point forced in the T0-T8 chain.
proof idea
This is a structure definition. Its three fields are populated verbatim by the lemmas gapAt_pos, gapAt_succ_ratio, and gapAt_adjacent_ratio. No additional tactics or reductions occur inside the declaration.
why it matters in Recognition Science
The certificate is instantiated inside polarCodeCert to supply the gap properties required for polar codes. It completes the uniform treatment of finite-length gaps across polar, LDPC, and quantum channels, all reducing to the same 1/phi ratio on the phi-ladder. The construction sits inside the Information domain and inherits the phi scaling that originates from the Recognition Composition Law and the T5 J-uniqueness step.
scope and limits
- Does not derive the numerical value of referenceGap.
- Does not recover the asymptotic O(2^{-N^{0.5}}) decay in block length.
- Does not incorporate error probability or explicit block-length dependence beyond rung index k.
- Does not prove that 1/phi is the unique possible scaling factor.
formal statement (Lean)
51structure PolarCodeCert where
52 gap_pos : ∀ k, 0 < gapAt k
53 one_step_ratio : ∀ k, gapAt (k + 1) = gapAt k * phi⁻¹
54 adjacent_ratio : ∀ k, gapAt (k + 1) / gapAt k = phi⁻¹
55
56/-- Polar-code gap-to-capacity certificate. -/