IndisputableMonolith.Information.ChannelCapacity2FromJCost
Packages the RS J-cost into a domain cost and a strictly positive canonical threshold, then exposes an inhabited certificate that the associated channel capacity equals 2. Information-side RS developments cite the certificate and the nonnegativity lemmas rather than rebuilding the cost bridge. Structure is definitional wrapping of Cost/Constants plus elementary positivity and inhabitation facts.
claimDefines a domain cost $C$ induced by the J-cost, a canonical threshold $\theta>0$, and an inhabited certificate asserting that the corresponding channel capacity equals $2$.
background
Recognition Science measures mismatch by the unique cost $J(x)=\frac{x+x^{-1}}{2}-1$ (equivalently $\cosh(\log x)-1$), forced at T5 of the unified chain. The Cost import supplies that functional; Constants supplies the RS-native tick. Channel-capacity language in the Information domain needs a domain-level cost, a positive readout threshold, and a package that downstream proofs can inhabit rather than reconstruct.
This module is that package. It lifts $J$ to a domain cost with an evaluation identity and a nonnegativity lemma, fixes a canonical positive threshold, and records a certificate type whose inhabitant asserts capacity $2$. The value $2$ is the capacity claim being certified, not a Shannon derivation from scratch.
proof idea
Definition-and-certificate module, not a deep derivation. domainCost wraps the imported J-cost on the working domain; domainCost_at_eq and domainCost_nonneg discharge evaluation and $C\ge 0$. canonicalThreshold names the cutoff and canonicalThreshold_pos proves it is strictly positive. ChannelCap2Cert is the certificate structure; cert and cert_inhabited supply a concrete inhabitant so callers obtain capacity-$2$ as a closed package.
why it matters in Recognition Science
Moves T5 J-uniqueness from the cost layer into Information as a reusable capacity-$2$ certificate. No downstream edges are recorded yet; the inhabited cert is the intended import surface for later channel or coding arguments in the monolith. Ties RS cost geometry to capacity language without re-proving J-uniqueness. The capacity value $2$ sits near the binary structure that feeds the eight-tick octave (T7), but this module only certifies the capacity claim, it does not identify it with the octave step.
scope and limits
- Does not derive classical Shannon capacity outside the RS J-cost model.
- Does not prove the threshold is the unique possible cutoff, only that the canonical choice is positive.
- Does not identify capacity 2 with the T7 eight-tick octave inside this module.
- Does not bound alpha, masses, or other RS constants.
- Does not supply an operational coding theorem or asymptotic rate region.