StructMaterialsM73Cert
plain-language theorem explainer
Packages the three structural axioms for the Materials-domain certificate at recognition rung 73: domain cost vanishes on the diagonal, is nonnegative for positive mass and energy arguments, and the canonical threshold is strictly positive. Materials and RS-certificate consumers cite it as the type of a filled certificate. It is a pure structure definition; inhabitation is discharged elsewhere by the concrete `cert` value.
Claim. A Materials rung-73 structural certificate is a record of three facts: (i) for every nonzero real $r$, the domain cost at equal arguments satisfies $C(r,r)=0$; (ii) for all positive reals $m,e>0$, the domain cost obeys $C(m,e)\ge 0$; (iii) the canonical threshold $T$ satisfies $T>0$.
background
This module states a structural Recognition Science certificate for the Materials domain at recognition rung 73 (Plan v7, 120th pass). Status is structural theorem: zero sorry, zero axiom. The certificate is the data type that later gets inhabited by an explicit witness.
Domain cost is the Materials-specialized cost functional (built from the RS J-cost). Upstream, ObserverForcing records that every recognition event has nonnegative cost via Jcost_nonneg on a positive state. The same nonnegativity pattern is re-specialized here to positive mass and energy arguments. The diagonal vanishing condition encodes that equal arguments sit at the J-cost minimum (identity ratio). Canonical threshold is the positive cutoff used by the Materials rung prediction.
Sibling lemmas in the module (domainCost_at_eq, domainCost_nonneg, canonicalThreshold_pos) are the concrete proofs that will fill the three fields.
proof idea
No proof body: this is a structure (record type) with three propositional fields. Inhabitation is not claimed here. Downstream, the noncomputable definition cert fills the fields by assigning domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos respectively; cert_inhabited then wraps that witness as Nonempty.
why it matters
Gives the typed interface for the Materials structural certificate at rung 73. Downstream cert is the explicit inhabitant and cert_inhabited proves the type is nonempty, closing the structural theorem for this module. In the RS forcing picture, nonnegative cost and diagonal vanishing are the Materials-domain shadow of the global J-cost minimum (T5 J-uniqueness / RCL identity at ratio 1). The positive threshold is the gate used by rung-level Materials predictions on the phi-ladder. The declaration itself does not compute masses or material constants; it only freezes the certificate shape so later passes can cite a single inhabited record.
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