cert
plain-language theorem explainer
Packages three local facts (domain cost vanishes on the diagonal, is nonnegative for positive arguments, and the canonical threshold is positive) into a single Phi-square-identity certificate. Anyone wiring the phi algebraic package or cost-threshold side conditions cites this witness. Construction is a pure structure assembly of three preexisting lemmas.
Claim. There is a certificate consisting of: (i) for every nonzero real $r$, the domain cost of $(r,r)$ is zero; (ii) for all positive reals $m,e$, the domain cost of $(m,e)$ is nonnegative; (iii) the canonical threshold is strictly positive.
background
The module records the structural identity $\varphi^2 = \varphi + 1$, the defining algebraic relation of the golden ratio in Recognition Science. From it the Fibonacci recurrence $\varphi^n = \varphi^{n-1} + \varphi^{n-2}$ follows for all $n$, identified here with the recognition recursion. Status is a structural theorem with no sorry and no axioms.
The certificate structure bundles three side conditions on a domain-level cost: vanishing when the two arguments agree (nonzero), nonnegativity for positive measure and expectation, and positivity of a canonical threshold. Upstream, nonnegativity of recognition-event cost is already known from ObserverForcing via the J-cost minimum at $x=1$. The local domain-cost lemmas specialize that picture to the phi-square setting.
proof idea
One-line structure constructor. The three fields of the certificate are filled by the local lemmas domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos respectively. No further rewriting or case analysis occurs.
why it matters
Gives a single named witness that the cost and threshold side conditions needed around the phi-square identity are inhabited. In the RS forcing chain, $\varphi$ is forced as the self-similar fixed point (T6); the algebraic relation $\varphi^2=\varphi+1$ is the concrete fixed-point equation, and the Fibonacci form is the recognition recursion. This certificate does not itself prove $\varphi^2=\varphi+1$, but packages the cost-theoretic hygiene that surrounding foundation lemmas expect when they invoke the identity. No downstream users are recorded yet; the natural consumers are any later theorems that take a PhiSqIdentCert hypothesis rather than three separate cost facts.
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