IndisputableMonolith.Information.LDPCCodeRateFromPhi
This module computes the gap to Shannon capacity for LDPC codes of block length N by applying the J-cost band. Information theorists working inside Recognition Science cite it when closing domain certificates in the B-tier chain. The module applies the six-clause template imported from CanonicalJBand to certify nonnegativity and related properties of the gap.
claimGap to Shannon capacity $\Delta(N)$ for an LDPC code of block length $N$, obtained by applying the J-cost-on-ratio template to the code rate.
background
The module resides in the Information domain and imports CanonicalJBand. That upstream module supplies the reusable six-clause J-cost-on-ratio template used across the master cert chain. The template proves J(1) = 0 and J(x) ≥ 0 for x > 0.
The local setting is the derivation of information rates inside Recognition Science, where the J function encodes the recognition cost on ratios and the phi-ladder supplies the underlying scale. The module therefore translates the general band into a concrete gap function for LDPC codes.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies one of the B-tier domain certificates in the master cert chain. It applies the Canonical J-Cost Band template to LDPC rates, thereby extending the J-band machinery into the information domain and supporting the overall forcing chain from the single functional equation to physical constants.
scope and limits
- Does not prove existence of LDPC codes achieving the stated rate.
- Does not compute numerical gap values for specific N.
- Does not address coding schemes other than LDPC.
- Does not derive the Shannon capacity formula itself.