IndisputableMonolith.Information.Compression3DeepFromJCost
Module packaging a certificate that three-deep data compression is forced by the RS J-cost on positive reals. It defines a domain cost, a canonical positive threshold, and an inhabited DataCompr3DeepCert record. Information theorists in the RS stack cite it when linking compression depth to the cost functional rather than to ad-hoc coding assumptions. The argument is definitional plus nonnegativity and positivity lemmas imported from Cost and Constants.
claimOn the positive reals, the domain cost induced by the J-cost $J(x)=(x+x^{-1})/2-1$ is nonnegative. There is a canonical positive threshold such that three-deep data compression is certified whenever the cost lies below that threshold; the certificate type is inhabited.
background
Recognition Science measures mismatch by the unique symmetric cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced at T5 of the unified forcing chain. The Cost import supplies that functional and its elementary inequalities; Constants supplies the RS-native tick scale used when thresholds are stated in discrete time.
This module sits in the Information domain. It lifts $J$ to a domain-level cost (how expensive it is to keep a configuration distinguishable across a domain) and isolates a single positive canonical threshold against which compression depth is judged. Three-deep compression means the data can be folded through three successive recognition layers without residual cost above threshold.
Sibling definitions name the pieces: domain cost and its value at equality, nonnegativity of that cost, the canonical threshold and its positivity, and the certificate record DataCompr3DeepCert together with an inhabited instance.
proof idea
Definition module with supporting lemmas, not a single deep proof. domainCost is introduced from J; domainCost_at_eq and domainCost_nonneg record the equality case and nonnegativity inherited from Cost. canonicalThreshold is a closed-form positive constant (canonicalThreshold_pos). DataCompr3DeepCert packages the compression claim; cert and cert_inhabited discharge inhabitation so downstream code can assume the certificate exists without constructing it by hand.
why it matters in Recognition Science
Links the T5 J-uniqueness landmark to a concrete information-theoretic claim: three-deep compression is not an extra coding hypothesis but a consequence of the same cost that forces $\varphi$, the eight-tick octave, and $D=3$. No downstream used_by edges are recorded yet; the module is a leaf certificate intended for the broader Information stack (compression bounds, channel capacity in RS units, and any ledger that treats depth-three folding as free once cost is subthreshold). It closes a scaffolding gap between pure Cost lemmas and usable compression certificates.
scope and limits
- Does not derive compression schemes or explicit codes; only a cost-threshold certificate.
- Does not treat depths other than three or non-J cost functionals.
- Does not prove uniqueness of the canonical threshold among all possible cutoffs.
- Does not connect to mass-ladder or alpha-band numerics.
- Does not supply runtime or bit-complexity bounds outside RS-native units.