IndisputableMonolith.Information.LDPCCodeRateFromJCost
Defines an LDPC-style certificate whose rate bound is read off the Recognition Science J-cost, together with a domain cost, its equilibrium value, nonnegativity, and a positive canonical threshold. Information and coding results inside the RS stack would cite the certificate and threshold. The module is largely definitional: identities at equilibrium, positivity lemmas, and an inhabited certificate record.
claimFrom the J-cost $J(x)=(x+x^{-1})/2-1$, the module defines a domain cost $C$, proves $C\ge 0$ and the equilibrium identity for $C$, introduces a positive canonical threshold $t_*>0$, and packages an LDPC certificate type with a canonical inhabited instance witnessing a J-derived code-rate bound.
background
Recognition Science forces a unique symmetric cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) via the Recognition Composition Law and the T5 uniqueness step. The Cost import supplies that functional; Constants supplies the RS-native tick scale used elsewhere in the stack.
This Information module treats coding rate as a cost-limited resource. A domain cost is built from $J$, so equilibrium configurations (where the argument of $J$ is $1$) sit at zero excess cost, matching the usual $J(1)=0$ normalization. Nonnegativity of the domain cost is the coding-side shadow of $J\ge 0$.
A canonical threshold is cut from that cost landscape and shown positive. The LDPC certificate is a small Prop/structure bundle asserting that a low-density parity-check style rate meets the J-derived bound; an inhabited instance supplies a default witness for downstream use.
proof idea
Definition-heavy module, not a single deep theorem. Domain cost is introduced from $J$; equilibrium and nonnegativity are short algebraic consequences of the standard $J$ identities ($J(1)=0$, $J\ge 0$). The canonical threshold is a closed-form cut from that cost, with positivity a one-line arithmetic check. The LDPC certificate is a structure (or bundled Prop) plus a default inhabited instance; no heavy tactic proof is required beyond assembling those pieces.
why it matters in Recognition Science
Places LDPC-style rate bounds on the same J-cost footing as the rest of Recognition Science, so coding constraints speak the same language as the forcing chain (T5 J-uniqueness and the RCL). No downstream consumers are wired in the graph yet (used_by is empty); the module is a leaf that supplies domainCost, the threshold lemmas, and LDPCCert/cert for later information-theoretic or channel-capacity arguments in the monolith. It does not itself close a T0–T8 step; it exports cost-native certificates those steps can cite when rates appear.
scope and limits
- Does not prove a Shannon or Gallager LDPC capacity theorem in classical information theory.
- Does not derive numerical code rates or construct explicit parity-check matrices.
- Does not connect the certificate to alpha, mass rungs, or the eight-tick octave.
- Does not claim the canonical threshold is optimal outside the J-cost normalization.
- Does not supply downstream consumers; the use graph is currently empty.