IndisputableMonolith.Verification.PhiNeZeroCert
Verification certificate that the golden ratio φ is nonzero as a real number. Downstream algebra that divides by φ or forms φ-powers cites this to discharge the nonzero side condition. The module is a thin wrapper around Mathlib and the PhiSupport fixed-point lemmas.
claimThe golden ratio $\varphi = (1+\sqrt{5})/2$ satisfies $\varphi \neq 0$ in $\mathbb{R}$. The module exports a named certificate of this fact for use in Recognition Science verification chains.
background
Recognition Science forces $\varphi$ as the unique positive self-similar fixed point of the cost functional (forcing step T6): $\varphi = 1 + 1/\varphi$, equivalently $\varphi^2 = \varphi + 1$. All rung masses, the eight-tick clock, and the constants $c=1$, $\hbar=\varphi^{-5}$, $G=\varphi^5/\pi$ are expressed as powers or multiples of $\varphi$.
Any such expression that divides by $\varphi$ or invokes field inversion needs a formal $\varphi \neq 0$ fact. The support module PhiSupport.Lemmas already records the elementary identities $\varphi^2 = \varphi + 1$, the fixed-point form, and uniqueness of the positive root; those rest only on real algebra and Mathlib's Real.goldenRatio API.
This verification module sits in the certificate layer: it packages the nonzero side-condition so later proofs never re-derive it.
proof idea
Thin certificate module. It imports Mathlib and the golden-ratio support lemmas, then records that $\varphi > 0$ (hence $\varphi \neq 0$) from the standard positive-root characterization. No deep argument: one-line appeal to positivity of the golden ratio already present in Mathlib / PhiSupport.
why it matters in Recognition Science
Keeps every later $\varphi$-ladder or constant derivation free of ad-hoc nonzero obligations. In the RS forcing chain, T6 installs $\varphi$ as the self-similar scale; mass formulae use $\varphi^{\mathrm{rung}-8+\mathrm{gap}(Z)}$ and the constants use negative powers such as $\varphi^{-5}$. Without a reusable certificate, each of those steps would re-prove $\varphi \neq 0$. The module has no further downstream edges listed yet; it is infrastructure for the verification domain rather than a leaf theorem.
scope and limits
- Does not prove uniqueness or the fixed-point equation for φ; those live in PhiSupport.
- Does not derive any physical constant or mass formula.
- Does not address φ in other rings or characteristic-p settings.
- Does not supply numerical bounds beyond the algebraic nonzero fact.