log_phi_lt_048122
plain-language theorem explainer
An explicit decimal upper bound $\log\varphi<0.48122$ on the natural log of the golden ratio. Cited inside the quarantined Alpha Genesis measurement-verdict chain that compares the first-order $\alpha^{-1}$ prediction to CODATA. The proof rewrites the log inequality as a comparison of $\varphi$ against $\exp(0.48122)$, then chains a certified $\varphi$ ceiling through a Taylor lower bound on the exponential.
Claim. Let $\varphi=(1+\sqrt{5})/2$ be the golden ratio (the RS self-similar fixed point). Then $\log\varphi<0.48122$.
background
This lives in the Alpha Genesis M7 quarantine module. The module formalizes a measurement-side objection: the first-order genesis value of $\alpha^{-1}$ sits more than $0.0007$ above the CODATA anchor, hence tens of thousands of $\sigma$ high before any seam correction. Nothing in the forward Alpha Genesis construction imports the module; CODATA enters only here through ExternalAnchors.
The constant $\varphi$ is the unique positive fixed point forced by the Recognition Composition Law and the T5–T6 uniqueness of the $J$-cost $J(x)=(x+x^{-1})/2-1$. Interval infrastructure already supplies a tight decimal ceiling $\varphi<1.6180340$ (W8Bounds / PhiBounds). A sibling private lemma proves the matching exponential floor $\exp(0.48122)>1.61804$ by a degree-12 Taylor remainder bound.
Together those two decimal facts pin $\log\varphi$ from above at a granularity coarse enough for the later residual comparison yet tight enough that the exponential load $f_{\mathrm{gap}}/\alpha_{\mathrm{seed}}$ stays under $0.0086705$.
proof idea
One short tactic proof. Rewrite the goal with $\log x<a\Leftrightarrow x<\exp a$ (using positivity of $\varphi$). Instantiate the certified upper bound $\varphi<1.6180340$ from W8Bounds. The private lemma $\exp(0.48122)>1.61804$ then yields $1.6180340<\exp(0.48122)$ by a one-line linarith, and ordinary real transitivity closes $\varphi<\exp(0.48122)$.
why it matters
Direct input to exponentialLoad_lt_0086705, which bounds the natural exponential load $f_{\mathrm{gap}}/\alpha_{\mathrm{seed}}$ from above. That load bound is the arithmetic step that lets the module certify $\alpha^{-1}{\mathrm{genesis}}>\alpha^{-1}{\mathrm{CODATA}}+0.0007$, the decisive measurement verdict of M7.
In the broader RS picture the bound is a constants-side certificate, not a forcing-chain step: $\varphi$ itself is already forced at T6, and the eight-tick / $D=3$ landmarks are upstream. The point of this lemma is purely metrological honesty: the first-order Alpha Genesis residual is excluded by CODATA before any higher-order seam correction is derived. The module remains quarantined precisely so that this external comparison never contaminates the constructive genesis path.
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