IndisputableMonolith.Verification.OneLtPhiCert
Verification module packaging a certificate that the golden ratio strictly exceeds one. Downstream certificate consumers cite it whenever a proof needs $\varphi>1$ as a named, reusable fact rather than an ad-hoc inequality. The argument is a thin wrapper around the elementary golden-ratio algebra already proved in PhiSupport.
claimA named certificate asserting $1 < \varphi$, where $\varphi=(1+\sqrt{5})/2$ is the unique positive root of $x^2=x+1$ (equivalently $\varphi=1+1/\varphi$).
background
Recognition Science forces $\varphi$ as the unique positive self-similar fixed point of the cost calculus (forcing step T6). Algebraically this is the golden ratio: the unique $x>0$ with $x^2=x+1$, or equivalently $x=1+1/x$. Mathlib already records $\varphi^2=\varphi+1$; the local PhiSupport layer re-exports that identity, the fixed-point form, and uniqueness of the positive root, all from elementary real algebra.
Verification modules in this tree turn such background facts into named certificates that other certificate-driven proofs can import without re-deriving inequalities. The present module sits in that layer: it does not redefine $\varphi$, it only certifies the strict comparison $1<\varphi$ needed by rung, mass, and cost arguments that divide by $\varphi-1$ or take positive powers of $\varphi$.
proof idea
Module-level structure, not a multi-step development. It imports Mathlib and PhiSupport.Lemmas, then exposes a single certificate object (OneLtPhiCert) whose proof is a short appeal to the golden-ratio identities already in PhiSupport: from $\varphi=1+1/\varphi$ and $\varphi>0$ one obtains $\varphi-1=1/\varphi>0$, hence $1<\varphi$. No new analysis is introduced.
why it matters in Recognition Science
Strict positivity of $\varphi-1$ is a standing hypothesis in the phi-ladder mass formula, in eight-tick counting, and in any estimate that uses $\varphi^n$ growth or $J$-cost self-similarity. Packaging $1<\varphi$ as a verification certificate keeps those downstream arguments free of local inequality noise and ties them to the same PhiSupport base used by the forcing chain (T5--T6). The module currently has no recorded used-by edges in the mirror graph, so it functions as a leaf certificate ready for import by later verification or constants modules rather than as an intermediate lemma in a long chain.
scope and limits
- Does not prove uniqueness or existence of $\varphi$; those live in PhiSupport.
- Does not derive the numerical value of $\varphi$ or any decimal bounds.
- Does not address $\varphi$ as a forced RS constant beyond the algebraic golden-ratio facts.
- Does not supply ladder, mass, or $J$-cost theorems that consume the inequality.