RSGRVStructural005Cert
plain-language theorem explainer
Certificate bundle for Gravity structural module 5: the domain cost vanishes on the diagonal, stays non-negative for positive mass/energy arguments, and the canonical threshold is strictly positive. Gravity and RS-GRV auditors cite it as the typed interface that a concrete witness must inhabit. The structure itself is pure data; inhabitance is discharged by the sibling lemmas on domainCost and the threshold.
Claim. A certificate consists of three facts: (i) for every $r \neq 0$, the domain cost satisfies $C(r,r)=0$; (ii) for all $m,e>0$, $C(m,e)\ge 0$; (iii) the canonical threshold $T$ obeys $T>0$.
background
Module 5 sits in the Gravity RS structural series. The module doc fixes the eight-tick setting: one full traversal of the binary recognition lattice has period $2^D=8$, matching the T7 octave forced in the unified chain.
The domain cost $C(m,e)$ is the local cost functional on positive mass/energy-type arguments used by this gravity layer. The diagonal identity $C(r,r)=0$ says equal arguments carry zero excess cost. Non-negativity for $m,e>0$ is the gravity-side shadow of the foundation fact that every recognition event has non-negative cost (ObserverForcing: "The cost of any recognition event is non-negative," via $J$-cost non-negativity). The canonical threshold is the positive cutoff against which structural comparisons in this module are made.
The certificate is the Prop-bundle that packages exactly those three obligations; it does not itself compute masses or curvatures.
proof idea
No proof body: this is a structure declaration. The three fields are named hypotheses (diagonal vanishing of domainCost, non-negativity of domainCost on positive arguments, positivity of canonicalThreshold). Downstream, cert fills them by the sibling lemmas domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos; cert_inhabited then wraps that witness as Nonempty.
why it matters
In the RS gravity stack this certificate is the typed gate for structural module 5. Downstream cert and cert_inhabited show the interface is realized with zero sorry, matching the module claim "STRUCTURAL THEOREM (0 sorry, 0 axiom)."
It anchors the eight-tick gravity layer (period $2^D=8$) to the same non-negative cost discipline used in ObserverForcing and the $J$-cost calculus (T5 uniqueness of $J$). Without a positive threshold and a non-negative diagonal-vanishing domain cost, later GRV structural comparisons would lack a well-posed baseline. The declaration does not close dynamics or phenomenology; it only freezes the structural hypotheses those results may assume.
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