IndisputableMonolith.Gravity.RS_GRV_Structural_006
Structural certificate module for RS gravity claim 006: a non-negative domain cost built from the Recognition J-cost, a positive canonical threshold, and an inhabited certificate packaging those facts. Gravity auditors cite it when wiring cost-threshold comparisons into GRV structural lemmas. The file is mostly definitions plus elementary non-negativity and positivity proofs from the Cost and Constants imports.
claimDefine a domain cost $C$ from the Recognition cost $J$, prove $C \ge 0$ and an evaluation identity, fix a canonical threshold $\theta > 0$, and package these into an inhabited structural certificate $\mathrm{RSGRVStructural006Cert}$.
background
Recognition Science measures mismatch with the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the T5 step of the unified forcing chain and obeying the Recognition Composition Law. The Cost import supplies that functional; Constants supplies the RS-native units and fixed points (including $\phi$) used elsewhere in the gravity stack.
This module sits in the Gravity domain and introduces a domain-level cost built from $J$, together with a canonical positive threshold against which that cost is compared. The certificate type bundles the cost, its non-negativity, the threshold, and positivity into a single inhabited record used by structural GRV arguments.
No curved-spacetime dynamics are defined here; the setting is purely the algebraic cost-threshold layer that later gravity lemmas can assume without re-proving elementary inequalities.
proof idea
Definition-heavy module. domainCost is introduced from the imported J-cost; domainCost_nonneg and domainCost_at_eq are short inequality and evaluation lemmas. canonicalThreshold is a fixed positive real; canonicalThreshold_pos records positivity. RSGRVStructural006Cert (with cert and cert_inhabited) is a structure packing those facts so downstream gravity code can take a single inhabitant rather than reassemble hypotheses. No deep tactic scripts; proofs are elementary applications of Cost non-negativity and constant positivity.
why it matters in Recognition Science
Gives the Gravity stack a reusable structural certificate (claim 006) so cost-versus-threshold comparisons do not scatter ad-hoc lemmas. Downstream used_by edges are empty in the current graph, so this file is a leaf certificate provider rather than a step inside a larger proved theorem. It anchors the RS pattern that gravity structural claims should cite an inhabited cert recording $J$-derived non-negative cost and a positive canonical cutoff, consistent with the broader forcing-chain discipline (T5 J-uniqueness feeding cost-based physics). Auditors use it when checking that GRV structural hypotheses are discharged by named, non-sorry objects rather than free-floating inequalities.
scope and limits
- Does not derive Einstein equations or any curved-spacetime field equation.
- Does not fix numerical GRV observables or match solar-system data.
- Does not prove uniqueness of the canonical threshold beyond positivity.
- Does not connect domainCost to the mass ladder or eight-tick octave.
- Does not supply dynamics; only cost, threshold, and an inhabited cert record.