IndisputableMonolith.Information.Compression3DeepFromJCost
Module deriving a three-deep data-compression certificate from the RS J-cost. It defines a domain cost built from J, a canonical positive threshold, and an inhabited certificate type for cost-bounded three-level compression. Information-layer arguments that need a J-rooted compression witness cite this package. Content is mostly definitions plus elementary nonnegativity and positivity facts.
claimPackage defining a domain cost $C$ from the J-cost $J(x)=(x+x^{-1})/2-1$, a canonical threshold $\theta>0$, and a certificate that three-deep data compression holds relative to $C$ and $\theta$, with $C\ge 0$ and $\theta>0$ recorded as lemmas.
background
Recognition Science measures mismatch by the unique cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced by the Recognition Composition Law and the T5 uniqueness step. The Cost import supplies that functional; Constants supplies the RS-native tick scale against which discrete information steps are counted.
This Information-domain module lifts $J$ from scalar ratios to a domain-level cost suitable for compression statements. Sibling definitions introduce domainCost (the lifted cost), equality and nonnegativity facts for it, and a canonicalThreshold with a positivity lemma. The certificate type packages the claim that data admit a three-deep compression relative to that cost and threshold.
Three-deep means a fixed finite nesting depth of compression stages, not an asymptotic coding theorem. The module stays inside the RS cost calculus rather than Shannon entropy or Kolmogorov complexity.
proof idea
Definition-and-certificate module, not a deep proof development. Domain cost is introduced as a construction from J; equality-at-evaluation and nonnegativity are recorded as short lemmas. The canonical threshold is a positive constant (positivity lemma). The main object is an inhabited certificate structure asserting three-deep compression against that cost and threshold. No substantial tactic proof chain; the argument is definitional packaging of Cost facts for the information layer.
why it matters in Recognition Science
Places a J-cost-native compression witness in the Information domain so later RS arguments can cite a concrete three-deep certificate rather than an informal coding claim. Ties compression bounds to the same T5-unique $J$ used for forcing, mass ladders, and constants, keeping the information layer on the RS cost spine. No downstream used-by edges are recorded yet; the module is a leaf package meant for information and certification consumers. It does not itself close a T0-T8 forcing step; it exports a usable cert shape once Cost is fixed.
scope and limits
- Does not prove Shannon or rate-distortion theorems.
- Does not derive J-uniqueness; imports Cost/Constants.
- Does not treat unbounded or asymptotic compression depth.
- Does not connect to mass ladders, alpha, or T8 dimension.
- Does not supply algorithmic encoders or decoders.