rho_lt_one

Module T9 forces the weighting on recognition states after the T0–T8 chain has fixed the shape of the law (unique cost $J$, scale $\varphi$, eight-tick period, $D=3$). On the lattice layer, any admissible weight rule must factorize over independent composition and obey the per-step self-similar balance $\rho=1/(1+\rho)$. That fixed point is uniquely $\rho=\varphi^{-1}$.

The constant $\rho$ is defined as $1/\varphi$. Upstream, $\varphi>1$ is already proved in Constants (re-exported through PhiSupport). The present comparison is the elementary half of the open unit-interval membership $\rho\in(0,1)$ needed for geometric-series convergence and normalized probability masses $P(n)=(1-\rho)\rho^n$.

Short term-mode comparison. Unfold $\rho=1/\varphi$, rewrite the goal by the criterion that $a/b<1$ when $b>0$ and $a<b$ (here $a=1$, $b=\varphi$), and discharge with the lemma $1<\varphi$. No further measure-theoretic content is used.

This bound is the contractivity hinge for the whole forced measure. Inside the module it feeds the geometric-series evaluation of the partition function $Z=\varphi^2$, positivity and normalization of probability masses, the exact mean rung $\langle n\rangle=\varphi$, the weak comparison $\rho\le 1$, and the continuum uniqueness theorem: any factorizing antitone weight with calibrated step $f(1)=\rho$ equals $\rho^t$ for all $t\ge 0$.

Downstream in AlphaGenesis residual targeting, $\rho<1$ yields $\log\rho\neq 0$ and strict decrease of the dressed $\alpha^{-1}$ in the load, so exactly one closing $\delta_2$ matches CODATA. Framework-wise it converts the T6-forced scale into the geometric $\varphi$-measure of T9 (weight $\varphi^{-1}$ per recognition step).