IndisputableMonolith.Information.DataCompression3FromJCost
Package linking the RS J-cost to a three-fold data-compression certificate. It defines a domain cost pulled back from J, a positive canonical threshold, and an inhabited DataCompr3Cert record. Information theorists working in the RS stack cite it when they need the compression factor three as a proved numeric bound rather than an ansatz. The argument is definitional plus elementary nonnegativity and positivity lemmas over the Cost module.
claimFrom the RS cost $J(x)=(x+x^{-1})/2-1$, the module introduces a domain cost $C$, proves $C\ge 0$, fixes a canonical threshold $\theta>0$, and packages an inhabited certificate that the effective data-compression factor is $3$.
background
Recognition Science measures mismatch by the unique cost $J$ forced at T5: $J(x)=\cosh(\log x)-1$, equivalently $(x+x^{-1})/2-1$. The Cost import supplies that functional and its elementary calculus (nonnegativity, minimum at $1$, composition law). Constants supplies the RS tick $\tau_0$ used when costs are read as information per tick.
This module sits in the Information domain. It lifts $J$ to a domain-level cost (a real-valued functional on the objects being compressed) and isolates a single positive threshold that marks when compression by factor three becomes admissible. Sibling names indicate equality-at-evaluation lemmas, nonnegativity, and a certificate record DataCompr3Cert whose inhabitation is the module's main deliverable.
No external analytic number theory is required: everything is real analysis on $J$ plus bookkeeping of the threshold.
proof idea
Definition module with short supporting lemmas, not a deep derivation. domainCost is introduced as the pullback of $J$ (or a monotone transform of it) to the data domain; domainCost_at_eq and domainCost_nonneg discharge evaluation and sign. canonicalThreshold is a closed-form positive real; positivity is a one-line arithmetic check. The certificate record bundles these facts; cert_inhabited builds a concrete inhabitant, so downstream code can assume the factor-three bound without re-proving the inequalities.
why it matters in Recognition Science
In the RS forcing chain, $J$ is unique (T5) and the eight-tick octave (T7) already forces discrete period $2^3=8$. A compression factor of three is the natural information-theoretic counterpart that pairs with spatial dimension $D=3$ (T8) and with ternary structure in coding bounds. This module turns that numerology into a named, inhabited certificate consumable by later Information lemmas.
No downstream edges are recorded yet (used_by empty), so the module is a leaf exporter: it freezes the factor-three claim for any parent that needs a compression certificate tied to $J$ rather than to Shannon entropy alone. It does not itself touch $\alpha$, mass rungs, or the Berry threshold; those remain in other domains.
scope and limits
- Does not derive Shannon capacity or classical rate-distortion theorems.
- Does not prove uniqueness of the factor three among all possible thresholds.
- Does not connect the certificate to alpha, mass ladders, or gravity constants.
- Does not supply an algorithmic compressor; only a numeric certificate.
- Does not claim empirical compression ratios on real data sets.