cert
plain-language theorem explainer
Packages the three structural properties of the materials-domain cost at recognition rung 83 into one certificate: diagonal vanishing, non-negativity off the identity, and a strictly positive canonical threshold. Materials or RS auditors cite it as the inhabited structural certificate for mod-83. Construction is a pure structure fill from three already-proved sibling lemmas.
Claim. There is a structural materials certificate at recognition rung 83: for the materials domain cost $C$, one has $C(r,r)=0$ whenever $r\neq 0$; $C(m,e)\ge 0$ whenever $m>0$ and $e>0$; and the canonical threshold $\tau$ satisfies $\tau>0$.
background
The module states a structural Recognition Science prediction for the Materials domain at recognition rung 83 (Plan v7, 120th pass), with status structural theorem: zero sorry, zero axiom. The certificate type bundles three Prop fields that any such structural claim must discharge.
Domain cost is the materials specialization of the RS cost functional (imported from Cost and Constants). The first field requires the cost to vanish on the diagonal (measured equals expected, nonzero scale). The second requires non-negativity for positive measured and expected values. The third requires the module's canonical threshold to be strictly positive.
Upstream, ObserverForcing records that every recognition-event cost is non-negative via $J$-cost non-negativity at positive state. That global fact is the conceptual ancestor of the domain-level non-negativity field wired here.
proof idea
Pure structure inhabitation. The three fields of StructMaterialsM83Cert are filled by the sibling lemmas domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos respectively. No extra tactics or algebraic work: the definition is a one-shot record constructor.
why it matters
This is the concrete certificate object for the materials structural theorem at rung 83. Downstream use is not yet wired in-tree (used_by empty), but the sibling cert_inhabited and any later materials forcing or threshold comparison will consume it as the single inhabited proof package.
It sits in the materials branch of RS structural predictions, resting on the same $J$-cost non-negativity that appears in ObserverForcing and ultimately on the T5 uniqueness of $J(x)=(x+x^{-1})/2-1$. It does not itself touch the eight-tick octave, $D=3$, or the alpha band; those enter only if a later materials mass or gap formula cites this cert.
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