IndisputableMonolith.Gravity.RecognitionGeodesic3_FromJCost
Module packaging a 3D recognition-geodesic cost built from the RS J-functional, with a nonnegativity lemma, a positive canonical threshold, and an inhabited deep certificate. Gravity and ledger workers cite it when they need a J-derived domain cost on spatial triples rather than a free metric ansatz. The argument is definitional plus elementary positivity from the Cost layer.
claimOn spatial triples one defines a domain cost $C$ from the RS $J$-cost, proves $C \ge 0$ and $C$ agrees with a pointwise evaluation, fixes a positive canonical threshold $\theta_*>0$, and packages an inhabited deep certificate $\mathsf{RecogGeodesic3DeepCert}$ asserting the 3D recognition-geodesic claims derived from $J$.
background
Recognition Science forces the cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) by the T5 uniqueness step of the forcing chain, subject to the Recognition Composition Law. The Cost import supplies that $J$ and its elementary inequalities; Constants supplies the RS tick $\tau_0=1$ used to normalize time units.
This Gravity module lifts that scalar cost to a domain cost on 3D recognition configurations (the spatial dimension forced at T8). The siblings name the pieces: a domain cost functional, its agreement with pointwise evaluation, nonnegativity, a positive canonical threshold, and a deep certificate type with an inhabited witness.
The local setting is not a free Riemannian geodesic equation. It is a ledger-side cost on recognition paths whose minimizers are the RS stand-in for geodesics in three spatial dimensions.
proof idea
Definition-heavy module. domainCost is introduced from the imported J-cost; domainCost_at_eq records agreement with the pointwise formula; domainCost_nonneg is the nonnegativity inheritance from Cost. canonicalThreshold and canonicalThreshold_pos fix a strictly positive cutoff. RecogGeodesic3DeepCert, cert, and cert_inhabited package those facts into an inhabited certificate rather than a long tactic proof. No deep analytic PDE work lives here; positivity and packaging do.
why it matters in Recognition Science
Gives Gravity a J-native 3D geodesic cost instead of an external metric postulate, tying the spatial side to T5 (J-uniqueness) and T8 ($D=3$). Downstream used_by edges are empty in the graph snapshot, so this module is a leaf certificate source: other gravity or ILG developments can import the inhabited RecogGeodesic3DeepCert rather than re-derive nonnegativity and the threshold. It closes the FromJCost path for recognition geodesics in three dimensions inside the monolith.
scope and limits
- Does not derive Einstein field equations or a full spacetime metric from J.
- Does not treat dimensions other than the forced $D=3$ spatial setting.
- Does not prove dynamical geodesic completeness or global minimizing paths.
- Does not fix numerical GR observables beyond the abstract positive threshold.
- Does not replace the Cost-layer proof of J-nonnegativity; it inherits it.