polarCodeCert
plain-language theorem explainer
polarCodeCert constructs a certificate asserting that the polar-code gap to capacity lies on the phi-ladder and shrinks by the factor phi inverse at each adjacent level. Information theorists comparing finite-blocklength codes to Shannon capacity under Recognition Science scaling would reference this when aligning polar codes with LDPC rate gaps. The definition is a direct structure instantiation that supplies the three required fields from the supporting gapAt theorems.
Claim. Let $g(k)$ denote the gap-to-capacity at phi-ladder level $k$. The certificate asserts that $g(k)>0$ for all natural $k$, that $g(k+1)=g(k)phi^{-1}$ for all $k$, and that $g(k+1)/g(k)=phi^{-1}$ for all $k$.
background
In the PolarCodeGapFromPhi module the gap function is defined as referenceGap scaled by negative integer powers of phi. The PolarCodeCert structure collects three properties: strict positivity of the gap, the exact multiplicative recurrence by phi inverse, and the adjacent ratio equality. This construction mirrors the LDPC gap treatment in the sibling LDPCCodeRateFromPhi module and the quantum-channel-capacity correction noted in the module doc. The local setting states that polar codes achieve capacity with gap decaying as O(2^{-N^{0.5}}) in block length N, yet the RS model places the finite-length gap on the phi-ladder with adjacent ratios exactly 1/phi.
proof idea
The definition is a direct structure constructor for PolarCodeCert. It assigns the gap_pos field to the theorem gapAt_pos, the one_step_ratio field to gapAt_succ_ratio, and the adjacent_ratio field to gapAt_adjacent_ratio. The three upstream theorems already supply the required statements, so no additional tactics or reductions occur.
why it matters
This definition completes the polar-code gap certificate inside the Information domain, parallel to the LDPC gap_pos result. It supports the claim that finite-length corrections for polar codes obey the same phi-scaling as other RS information quantities. The module doc explicitly links the construction to Arıkan 2009 and the shared 1/phi structure with the quantum-channel-capacity correction. No downstream uses appear yet, leaving open its integration into a unified information bound.
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