IndisputableMonolith.Information.ErrorCorrection3DeepFromJCost
Module packaging a certificate that three-deep error correction is forced by the J-cost geometry of Recognition Science. It defines a domain cost pulled back from J, a positive canonical threshold, and an inhabited ErrCorr3DeepCert record. Information theorists working in RS cite it when linking channel depth to the unique cost functional. The argument is definitional plus nonnegativity and positivity lemmas, not a deep derivation.
claimFrom the unique cost $J(x)=(x+x^{-1})/2-1$ one obtains a domain cost $C$, a canonical threshold $\theta>0$, and an inhabited certificate that error correction of depth three is admissible relative to $(C,\theta)$ in RS-native units.
background
Recognition Science fixes a unique nonnegative cost $J$ on the positive reals by the Recognition Composition Law and the normalization $J(1)=0$, $J''(1)=1$. Explicitly $J(x)=\cosh(\log x)-1$. The Cost import supplies that functional; Constants supplies the RS tick $\tau_0=1$.
This module works in the Information domain. It pulls $J$ back to a domain cost on whatever discrete or continuous alphabet is under study, records that the cost is nonnegative, and isolates a strictly positive canonical threshold against which decoding depth is measured. The three-deep certificate is the structured witness that depth-three correction sits above that threshold.
No external channel model is assumed beyond the cost geometry; the module is local to the J-derived numerics.
proof idea
Definition module with short supporting lemmas. domainCost is introduced as the J-pullback; domainCost_at_eq and domainCost_nonneg discharge equality and nonnegativity by reduction to properties of $J$. canonicalThreshold is a positive constant (canonicalThreshold_pos). ErrCorr3DeepCert is a structure bundling cost, threshold, and the depth-three claim; cert and cert_inhabited supply a concrete inhabited instance. No multi-step tactic proof; the content is the certificate type and its witness.
why it matters in Recognition Science
Places three-deep error correction inside the J-cost forcing chain rather than as an external coding-theory postulate. Downstream consumers (none linked in the current graph) would use ErrCorr3DeepCert when assembling information-capacity or ledger-consistency arguments that need a fixed correction depth. The depth three is the natural discrete counterpart of the spatial dimension $D=3$ forced at T8 and of the eight-tick octave at T7, keeping the information layer aligned with the same self-similar scale $\phi$ that fixes the rest of the RS constants. The module closes a scaffolding gap between pure Cost and applied Information certificates.
scope and limits
- Does not derive a Shannon capacity or explicit code constructions.
- Does not prove uniqueness of depth three beyond the certificate witness.
- Does not connect to physical noise models outside J-cost geometry.
- Does not depend on or export mass-ladder or alpha-band results.
- Does not claim empirical BER bounds; only the abstract threshold certificate.