IndisputableMonolith.Gravity.RS_GRV_Structural_003
Structural certificate package for Recognition Science gravity item 003: a nonnegative domain cost functional, its evaluation identity, and a strictly positive canonical threshold. Gravity and RS-constants workers cite it when wiring cost bounds into GRV structural claims. The module is mostly definitions plus elementary nonnegativity and positivity lemmas, closed by an inhabited certificate record.
claimThe module introduces a domain cost $C_{\mathrm{dom}}$, proves $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at specified points, fixes a canonical threshold $\theta>0$, and packages these facts as an inhabited structural certificate $\mathrm{RSGRVStructural003}$.
background
Recognition Science gravity work builds on the global cost calculus from the Cost import and the RS-native constants (including the time quantum $\tau_0=1$ tick). The cost side supplies the J-type recognition cost and related nonnegativity infrastructure; gravity modules specialize that cost to geometric or field domains.
Here the specialization is a domain cost $C_{\mathrm{dom}}$ together with a canonical numerical threshold $\theta$. The siblings domainCost_nonneg and canonicalThreshold_pos record the two sign facts needed downstream: the cost never goes negative, and the threshold is strictly positive. An equality lemma pins the cost's value at designated evaluation points so later GRV arguments can substitute without re-deriving the functional.
The module sits in the Gravity domain and does not itself derive Einstein equations or mass ladders; it only freezes the structural cost/threshold interface used by later GRV certificates.
proof idea
Definition-heavy module. domainCost and canonicalThreshold are introduced as defs; domainCost_at_eq is an evaluation identity; domainCost_nonneg and canonicalThreshold_pos are short nonnegativity/positivity proofs that lean on the imported Cost nonnegativity facts and the Constants layer. The certificate record RSGRVStructural003Cert bundles those lemmas; cert and cert_inhabited witness that the record is realizable. No deep tactic scripts: elementary rewriting and sign lemmas.
why it matters in Recognition Science
Gives Gravity a named, inhabitable structural certificate (item 003) so later GRV developments can depend on a single package rather than ad-hoc cost and threshold hypotheses. Feeds any parent that imports this module for domain-cost bounds or threshold comparisons; the supplied used-by list is currently empty, so this is a leaf certificate awaiting wiring into higher GRV theorems.
In the broader RS chain it sits downstream of the Cost calculus and Constants ($\tau_0$, and ultimately the $\phi$-ladder and J-uniqueness from the forcing chain), and upstream of any gravity claim that needs a nonnegative domain cost and a positive cutoff. It does not close dynamical field equations; it only locks the structural interface those equations will cite.
scope and limits
- Does not derive Einstein field equations or Newtonian limits.
- Does not fix numerical values of G, c, or alpha beyond imported Constants.
- Does not prove uniqueness of the domain cost functional.
- Does not connect the threshold to observational GR data.
- Does not discharge dynamical or cosmological GRV claims.