cert
plain-language theorem explainer
Packages a structural certificate for the astrophysics domain at recognition rung 41: domain cost vanishes on the diagonal, is nonnegative for positive arguments, and the canonical threshold is positive. Anyone citing the mod-41 astrophysics structural prediction uses this bundle. The definition is a pure record assembly of three already-proved field lemmas.
Claim. There is a structural certificate for astrophysics at recognition rung 41: for every nonzero real $r$, the domain cost satisfies $C(r,r)=0$; for all positive reals $m,e$, one has $C(m,e)\ge 0$; and the canonical threshold $T$ obeys $T>0$.
background
The module states a structural Recognition Science prediction for the astrophysics domain at recognition rung 41 (Plan v7, 120th pass), with status structural theorem: zero sorry, zero axiom.
The certificate structure collects three elementary cost properties. Domain cost is the local cost functional on pairs of positive reals used in this astrophysics slice; the diagonal identity $C(r,r)=0$ for $r\neq 0$ is the zero-defect fixed point. Nonnegativity for positive mass/energy-like arguments mirrors the global fact that recognition cost is nonnegative (upstream: cost of any recognition event is nonnegative, via $J$-cost nonnegativity). The canonical threshold is the positive cutoff against which domain cost is compared in the structural claim.
Sibling lemmas domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos discharge the three fields; this definition only names the inhabited certificate.
proof idea
One-line structure inhabitant: fill cost_at_eq by the sibling diagonal identity for domain cost, cost_nonneg by the sibling nonnegativity lemma for domain cost on positive arguments, and threshold_pos by positivity of the canonical threshold. No extra algebra; pure record assembly of three prior proofs.
why it matters
Gives a single named certificate object for the rung-41 astrophysics structural layer so downstream consumers can depend on one term rather than three separate lemmas. Fits the RS pattern of packaging domain cost identities (diagonal zero, nonnegativity) plus a positive threshold as the minimal structural interface before quantitative mass or scaling claims. Sits under the forcing-chain cost infrastructure ($J$-cost nonnegativity from observer forcing) and the phi-ladder rung indexing used for domain placement. No used_by edges are recorded yet; the companion inhabitedness fact is the immediate consumer pattern. Does not itself force $D=3$, the eight-tick octave, or the alpha band; those live upstream in the unified forcing chain.
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