IndisputableMonolith.Information.ErrorCorrection3DeepFromJCost
Module packaging a three-deep error-correction certificate built from the Recognition Science J-cost. It defines a domain cost, proves nonnegativity and an evaluation identity, fixes a positive canonical threshold, and exposes an inhabited certificate record. RS information theorists cite the certificate when wiring J-cost bounds into discrete correction depth. The content is mostly definitional with short positivity and nonnegativity lemmas.
claimFrom the J-cost $J(x)=(x+x^{-1})/2-1$ one obtains a domain cost $C$, a canonical threshold $\theta>0$, and an inhabited certificate asserting that three-deep error correction is available when the domain cost stays controlled by $\theta$.
background
Recognition Science measures mismatch with the unique cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced at T5 of the unified forcing chain and obeying the Recognition Composition Law. The Cost import supplies that functional; Constants supplies the RS tick $\tau_0=1$ as the native time quantum.
This Information-domain module turns that scalar cost into a domain-level cost suitable for discrete correction arguments. Sibling definitions introduce domainCost (with an evaluation identity and a nonnegativity lemma), a strictly positive canonicalThreshold, and the certificate type ErrCorr3DeepCert together with an inhabited instance. The local setting is certificate packaging rather than a full coding theorem: the objects name the cost bound that licenses depth-three correction in the RS stack.
proof idea
Definition module with short supporting lemmas, not a long derivation. Domain cost is defined from J; an evaluation identity and nonnegativity are recorded directly from J's known properties. The canonical threshold is a positive constant (positivity lemma). The certificate record bundles those facts; inhabitance is a constructor application assembling the nonnegativity and threshold data. No deep tactic script: algebraic inheritance from Cost plus record packaging.
why it matters in Recognition Science
Gives the Information layer a named, inhabited bridge from J-cost to three-deep error correction, so later RS arguments can cite a single certificate instead of re-opening Cost. Ties to T5 J-uniqueness and the RCL: the same forced cost that organizes the forcing chain here bounds discrete correction depth. Downstream use edges are not yet recorded on this page; the natural consumers are channel or ledger modules that need a depth-three correction hypothesis discharged by a cost threshold. Closes a packaging gap between Cost and Information without claiming a Shannon-style capacity theorem.
scope and limits
- Does not prove a Shannon capacity or asymptotic coding theorem.
- Does not derive J-uniqueness; imports Cost and assumes T5-level J.
- Does not treat correction depths other than three.
- Does not specify a concrete noisy channel model or decoder.
- Does not yet show recorded downstream consumers on this page.