cert
plain-language theorem explainer
Packages three already-proved structural facts (diagonal vanishing of the domain cost, nonnegativity for positive mass/energy, and a strictly positive canonical threshold) into a single Physics-domain certificate at recognition rung 66. Anyone needing an inhabited structural witness for that rung cites this value. Construction is a pure structure assembly: three field assignments, no new algebra.
Claim. There exists a structural Physics certificate at recognition rung 66: for the 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 $T$ satisfies $T>0$.
background
The module records a structural Recognition Science prediction for the Physics domain at recognition rung 66 (Plan v7, 120th pass). Status is a closed structural theorem: zero sorry, zero axioms.
The certificate type bundles three properties of a real-valued domain cost $C(m,e)$ and a canonical threshold $T$. Diagonal vanishing says equal arguments carry zero cost (identity recognition). Nonnegativity says positive mass and energy never produce negative cost, matching the global J-cost law that every recognition event has nonnegative cost. The threshold positivity clause ensures a strict positive cutoff against which domain costs can be compared.
Upstream, nonnegativity of recognition cost is already forced in ObserverForcing via the J-cost minimum at the identity event. Here those facts are specialized to the Physics-domain cost and packaged for rung 66.
proof idea
One-line structure constructor. The three fields of StructPhysicsM66Cert are filled by the sibling lemmas domainCost_at_eq (diagonal vanishing), domainCost_nonneg (nonnegativity on the positive quadrant), and canonicalThreshold_pos (strict positivity of the threshold). No tactic proof and no new calculation; the definition is pure wiring of prior results into the certificate record.
why it matters
Gives an explicit inhabited witness that the Physics structural certificate at rung 66 is realizable. In the Recognition framework this sits among the structural domain certificates that lock cost geometry before mass-ladder or coupling numerics are attached. It aligns with the forcing-chain cost layer (J-uniqueness and nonnegative J-cost) and with the phi-ladder organization of rungs, here specialized to Physics at rung 66.
Downstream use is local: the sibling inhabitedness fact and any consumer that needs a StructPhysicsM66Cert value rather than three separate lemmas. The module claims a closed structural theorem, so this definition is the concrete inhabitant that makes that claim checkable.
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