cert_inhabited
plain-language theorem explainer
The second RS forcing-chain certificate is inhabited: a single record packs diagonal vanishing of domain cost, its nonnegativity on positive arguments, and positivity of the canonical threshold. Anyone assembling the eight-tick structural package cites this existence fact. The proof is a one-line term that exhibits the prebuilt certificate witness.
Claim. The type of certificates packing (i) $\mathrm{domainCost}(r,r)=0$ for all $r\neq 0$, (ii) $\mathrm{domainCost}(m,e)\ge 0$ whenever $m>0$ and $e>0$, and (iii) strict positivity of the canonical threshold, is nonempty.
background
Module 2 of the RS forcing chain records the eight-tick structural layer: period $2^D=2^3=8$ states per recognition epoch, forced once spatial dimension is $D=3$ (landmark T7/T8). Status is a structural theorem with no sorry and no extra axioms.
The certificate structure bundles three elementary cost facts used throughout that layer. Domain cost is the local cost functional on measure/expectation pairs; the first field asserts it vanishes on the diagonal for nonzero reals, the second that it is nonnegative when both arguments are positive. The third field asserts that the module's canonical threshold (the cutoff used to gate recognition events in this layer) is strictly positive.
Upstream, those three properties are already proved as named lemmas and assembled into a concrete certificate value; the present declaration only records that the certificate type is inhabited.
proof idea
One-line term proof. The ambient module already constructs a concrete value cert : RSForcingChain002Cert by packing the three proved fields (diagonal vanishing, nonnegativity, threshold positivity). The theorem is simply the introduction rule for Nonempty, written as the anonymous constructor ⟨cert⟩. No further rewriting or case analysis occurs.
why it matters
Gives a single inhabited certificate object for Foundation Module 2, so downstream forcing-chain consumers can depend on one Nonempty fact rather than three separate lemmas. It sits inside the eight-tick octave package (period $2^3=8$, forced by $D=3$), which is landmark T7 of the T0–T8 forcing chain. No further used-by edges are recorded yet; the declaration is the module-level existence seal that later chain assemblers are expected to import when they require the Module-2 cost/threshold package as a unit.
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