cert_inhabited

The module records the algebraic identity $\varphi^2 = \varphi + 1$ as a structural theorem of Recognition Science: the defining fixed-point relation of the golden ratio, which forces the Fibonacci recurrence $\varphi^n = \varphi^{n-1} + \varphi^{n-2}$ as the recognition recursion. Status is fully proved (zero sorry, zero axiom).

A certificate structure packages three elementary cost facts used around that identity: domain cost of a ratio against itself is zero whenever the argument is nonzero; domain cost of positive measure and exposure is nonnegative; and the canonical threshold used to gate the identity is strictly positive. Those three fields are exactly the data of the certificate type.

The present declaration only asserts that such a packed certificate exists; the concrete witness is the sibling term cert built earlier in the same module from the corresponding lemmas on domain cost and the threshold.

One-line term proof. The existing certificate term is fed as the sole constructor argument to Nonempty, yielding an inhabitant of the certificate type. No further rewriting or case analysis is required.

Inside the Foundation layer this is the inhabitance gate for the phi-square certificate. Downstream consumers that quantify over a nonempty certificate (rather than over the three raw lemmas separately) can discharge the packing obligation in one step. The module itself is the structural home of $\varphi^2 = \varphi + 1$, which is the T6 fixed-point relation in the forcing chain and the algebraic engine behind the phi-ladder mass formula and the eight-tick octave. No open scaffolding remains: the certificate and its inhabitance are both closed.