cert_inhabited

Module RS_MTH_Structural_004 packages the gap-45 structural fact: $D^2(D+2)=9\cdot 5=45$, the minimum rung for stable self-reference when spatial dimension is $D=3$ (forcing-chain landmark T8). Status is a pure structural theorem (zero sorry, zero axiom).

The certificate structure collects three elementary properties of the local domain cost and threshold: the cost of matching a nonzero scale to itself is zero; the cost of a positive measure against a positive energy is nonnegative; and the canonical threshold used as a cutoff is strictly positive. Those three fields are the mathematical content of the certificate type whose inhabitation is claimed here.

Upstream, the structure itself is only a Prop-carrying record; the concrete witness cert (a sibling) is what supplies the three proofs. This theorem merely records that the record type is nonempty.

One-line term proof. The certificate value already constructed in-module is fed as the sole constructor argument to Nonempty, written ⟨cert⟩. No tactics, no further lemmas: inhabitation is immediate from the existence of that witness.

Gap-45 is the structural lower bound on the rung at which a self-referential ledger can close in three spatial dimensions. This inhabitation theorem is the clean export surface for the module: downstream consumers that need "the gap-45 certificate exists" can cite a single Nonempty fact rather than unpacking three separate lemmas.

It sits inside the mathematics layer that supports the forcing chain (T8 forces $D=3$; the octave and phi-ladder sit nearby). No parent theorems are listed yet (used_by is empty), so the declaration presently closes the module rather than feeding a named downstream proof. It does not itself derive the mass ladder or the alpha band; it only certifies the local cost/threshold package used when those arguments are assembled.