cert
plain-language theorem explainer
Packages three already-proved facts into the structural certificate for Gravity module 8: the domain cost vanishes on equal arguments, is nonnegative for positive mass and energy, and the canonical threshold is positive. Anyone citing the RS rung-spacing structural theorem (adjacent rungs separated by φ) uses this bundle. The definition is a pure structure inhabitant wiring three local lemmas.
Claim. There is a structural certificate asserting: (i) for every nonzero real $r$, the domain cost of $(r,r)$ is zero; (ii) for all positive reals $m,e$, the domain cost of $(m,e)$ is nonnegative; (iii) the canonical threshold is strictly positive.
background
Gravity RS Structural Module 8 records the rung-spacing law of Recognition Science: adjacent mass/energy rungs differ by the golden ratio $\phi \approx 1.618$. The module is marked structural (zero sorry, zero axiom).
The domain cost is the local cost functional on pairs of positive reals (mass and energy scales). It is built from the Recognition J-cost $J(x)=(x+x^{-1})/2-1$, which is nonnegative and vanishes only at the identity $x=1$. Upstream, cost_nonneg in ObserverForcing states that every recognition event has nonnegative cost via $J$-cost nonnegativity.
The certificate structure collects three Prop fields: diagonal vanishing of domain cost, nonnegativity on the positive quadrant, and positivity of the canonical threshold used to separate rungs.
proof idea
One-line structure inhabitant. The three fields are filled by the sibling lemmas domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos respectively. No new reasoning occurs here; the definition only packages those proofs into RSGRVStructural008Cert.
why it matters
This certificate is the exportable witness that Module 8's structural claims hold. It sits under the RS gravity ladder, where mass scales live on the $\phi$-rung lattice (primer mass formula: yardstick times $\phi^{\mathrm{rung}-8+\mathrm{gap}(Z)}$). Adjacent-rung factor $\phi$ is the T6 self-similar fixed point of the forcing chain.
No downstream consumers are recorded yet (used_by empty), so the immediate role is to close the module's structural interface and feed later gravity or mass-ladder theorems that demand a single named certificate rather than three separate lemmas. It does not itself derive $\phi$ or the eight-tick octave; those live upstream in the UnifiedForcingChain.
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