IndisputableMonolith.Foundation.RecognitionScienceSummary3
Summary module packaging a domain-level cost functional, its nonnegativity and evaluation identities, a positive canonical threshold, and an inhabited RSSummary3 certificate. Foundation authors cite it when they need a single import that re-exports Cost/Constants facts in certificate form. Structure is definitional plus short positivity and equality lemmas, not a deep derivation.
claimThe module introduces a domain cost $C$, proves $C \ge 0$ and an evaluation identity at distinguished points, fixes a canonical threshold $\theta > 0$, and packages these into an inhabited certificate $\mathrm{RSSummary3Cert}$.
background
Recognition Science builds physics from a unique cost $J$ forced by the Recognition Composition Law, with $J(x)=(x+x^{-1})/2-1$. The Cost import supplies that functional and its elementary inequalities; Constants supplies the RS-native tick $\tau_0=1$.
This file sits in the Foundation layer as a thin summary layer. It lifts the abstract cost to a named domain cost, records nonnegativity and a pointwise evaluation lemma, and introduces a positive canonical threshold used as a comparison scale in later forcing or certification steps.
The certificate bundle $\mathrm{RSSummary3Cert}$ is the module's public face: an inhabited record that downstream code can require instead of re-proving the same elementary facts.
proof idea
Definition module with short supporting lemmas. domainCost is introduced by definition from the Cost layer; domainCost_nonneg and domainCost_at_eq are direct appeals to Cost nonnegativity and evaluation. canonicalThreshold is a positive constant (canonicalThreshold_pos). RSSummary3Cert packages the above; cert and cert_inhabited discharge inhabitation by assembling those lemmas. No multi-step forcing argument lives here.
why it matters in Recognition Science
Gives Foundation consumers a single certificate-shaped import for domain cost, threshold positivity, and the Cost/Constants baseline without threading raw Cost lemmas. Used_by is empty in the graph snapshot, so it presently acts as a leaf summary rather than a proved parent of T5–T8. It still aligns with the forcing-chain scaffolding: J-cost uniqueness (T5) and the tick/octave bookkeeping that later force $\varphi$, eight-tick period, and $D=3$. Closes a packaging gap so later summary or audit modules can demand RSSummary3Cert rather than ad-hoc hypotheses.
scope and limits
- Does not prove J-uniqueness or the Recognition Composition Law.
- Does not derive phi, eight-tick structure, or D=3.
- Does not supply mass-ladder or alpha-band numerics.
- Does not claim downstream consumers beyond the empty used_by list.
- Does not replace the full Cost module; only re-exports a summary slice.