cert
plain-language theorem explainer
Packages three structural facts about the Standard Model domain cost into a single certificate: the cost vanishes on the diagonal, stays nonnegative for positive arguments, and the canonical threshold is positive. Cited by anyone who needs a inhabited structural certificate for module 9 of the RS Standard Model layer. Construction is a pure field-wiring of three sibling lemmas.
Claim. There is a certificate recording that $\mathrm{domainCost}(r,r)=0$ for every nonzero real $r$, that $\mathrm{domainCost}(m,e)\ge 0$ whenever $m>0$ and $e>0$, and that the canonical threshold is strictly positive.
background
Module 9 sits in the Standard Model structural layer of Recognition Science. The local forcing narrative runs T5 (J-uniqueness) through T6 (phi forced), T7 (eight-tick octave), and T8 (D=3). Status is structural: zero sorry, zero axioms.
The certificate structure bundles three properties of a real-valued domain cost on pairs of reals. The first says the cost of matching a nonzero scale to itself is zero. The second says the cost is nonnegative when both arguments are positive. The third asserts positivity of a fixed canonical threshold used as a comparison scale in the module.
Upstream, nonnegativity of recognition-event cost is already known from ObserverForcing: every recognition event has cost at least zero because the underlying J-cost is nonnegative on positive states. The present certificate lifts that style of fact to the Standard Model domain-cost interface.
proof idea
Pure structure construction. The three fields of the certificate are filled by the sibling lemmas domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos respectively. No additional tactic work or algebraic reduction occurs; the definition is a one-shot inhabitant of the certificate structure.
why it matters
Gives an inhabited structural certificate for RS Standard Model module 9, so downstream code can assume the three domain-cost facts as a single package rather than re-proving them. The module itself is marked STRUCTURAL THEOREM and sits on the forcing chain from J-uniqueness (T5) through phi, the eight-tick octave, and D=3 (T6-T8). No used-by edges are recorded yet; the immediate consumer is the sibling inhabitance lemma that witnesses the certificate type is nonempty. Closes the packaging step for this module's cost interface without introducing axioms.
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