one_sub_rho

Module T9 forces the weighting on recognition states after the T0–T8 shape chain (unique $J$, scale $\varphi$, eight-tick period, $D=3$). Any admissible lattice weight factorizes over independent steps and obeys the single-step self-similar balance $\rho=1/(1+\rho)$, which pins $\rho=\varphi^{-1}$.

Here $\rho$ is defined as $1/\varphi$. The golden equation $\varphi^2=\varphi+1$ (from phi_squared) is the only algebraic input beyond $\varphi\neq 0$. The identity $1-\rho=\varphi^{-2}$ is the normalization gap of the geometric series $\sum_n\rho^n$: the residual mass after the infinite ladder is exactly the inverse square of the forced scale.

Unfold $\rho=1/\varphi$. Record $\varphi\neq 0$ and the golden identity $\varphi^2=\varphi+1$. Clear denominators with field_simp, then finish by linear arithmetic on the golden equation: $1-1/\varphi=( \varphi-1)/\varphi$ and $\varphi-1=1/\varphi$ rearranges to $1/\varphi^2$.

This is the algebraic hinge of the forced measure. Downstream, partitionZ_eq_phi_sq rewrites the geometric sum $Z=1/(1-\rho)$ as $\varphi^2$; probMass_zero identifies the ground-state share $P(0)$ with $\varphi^{-2}$; and meanRung_eq_phi uses the same factor to prove the mean occupied rung equals $\varphi$ exactly.

In holography, one_sub_rho_eq_sq restates the identity as $1-\rho=\rho^2$, and probMass_eq_inv_pow obtains the closed form $P(n)=(\varphi^{n+2})^{-1}$. Framework-wise it converts T6 uniqueness of $\varphi$ into a concrete normalization constant for the Gibbs/geometric weight that T9 forces, closing the instance-selection gap left open by T0–T8.