RSGRVStructural002Cert
plain-language theorem explainer
Certificate bundle for Gravity structural module 2: the domain cost vanishes on equal positive arguments, stays non-negative for positive mass/energy pairs, and the canonical threshold is strictly positive. Gravity and RS-cost auditors cite it as the typed interface that `cert` inhabits. As a structure definition there is no proof body; inhabitance is discharged downstream by the sibling lemmas.
Claim. A certificate is a triple of properties: (i) for every real $r \neq 0$, the domain cost of the pair $(r,r)$ is zero; (ii) for all $m,e > 0$, the domain cost of $(m,e)$ is non-negative; (iii) the canonical threshold is strictly positive.
background
This module is Gravity RS Structural Module 2. It records the golden-ratio recognition cost: the RS J-cost attains $J(\varphi)=\varphi-3/2\approx 0.11803$ at the self-similar fixed point $\varphi$. Status is structural (zero sorry, zero axiom).
The domain cost is the local cost functional on pairs of positive reals (mass/energy-style arguments) used in this gravity layer. The canonical threshold is the positive cutoff against which that cost is compared. Upstream, ObserverForcing already has non-negativity of recognition-event cost via $J$-cost non-negativity on positive states; the present fields specialize that discipline to the gravity domain-cost interface.
The structure does not prove the three facts. It only names the Prop bundle that a later cert value must satisfy.
proof idea
No proof body: this is a structure definition packing three fields. Inhabitance is not argued here. Downstream, cert fills the fields by the sibling lemmas domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos, and cert_inhabited wraps that value as Nonempty.
why it matters
Gives the typed certificate that Gravity structural module 2 must discharge before claiming a clean J-cost minimum story at $\varphi$. Downstream cert is the concrete inhabitant and cert_inhabited records non-emptiness, so later gravity lemmas can depend on a single named bundle rather than three loose hypotheses.
In the Recognition framework this sits under the gravity domain and ties to T5/T6 landmarks: J-uniqueness and $\varphi$ as the self-similar fixed point, with the module headline $J(\varphi)=\varphi-3/2$. It does not itself force $\varphi$ or derive GR; it only standardizes the cost-and-threshold interface those structural claims use.
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