occ_one_forced
plain-language theorem explainer
Single-rung aging-charge attenuation is forced to equal φ⁻¹. Anyone deriving the BIT redshift kernel or the rung dilution law cites this as the n=1 base. The proof is a one-line application of the unique positive fixed-point theorem for ρ = 1/(1+ρ), using positivity and one-rung self-similarity of the occupation map.
Claim. The single-rung attenuation equals the reciprocal golden ratio: $\mathrm{occ}(1) = \varphi^{-1}$, where $\varphi$ is the unique positive solution of $x = 1 + 1/x$.
background
This module forces the shape of the dark-energy deviation kernel $K(z)$ in $w(z) = -1 + \delta w_0 \cdot K(z)$ from two premises: rung factorization (attenuation across $m+n$ $\varphi$-rungs multiplies) and single-rung balance (one rung solves the reciprocal fixed-point equation $\rho = 1/(1+\rho)$).
The occupation map $\mathrm{occ}$ records aging-charge attenuation on the $\varphi$-ladder of cosmic scale. The upstream lemma states that any positive $\rho$ with $\rho = 1/(1+\rho)$ equals $\varphi^{-1}$: clearing denominators yields the quadratic $\rho^2 + \rho - 1 = 0$, whose unique positive root is $1/\varphi$.
Locally, $\mathrm{occ}(1)$ is assumed positive and one-rung self-similar, so it sits exactly at that fixed point. The golden ratio $\varphi$ is the RS self-similar scale (forcing chain T6).
proof idea
One-line term proof. Apply self_similar_attenuation_forced to two facts about the occupation map: positivity at rung 1, and the one-rung self-similarity identity $\mathrm{occ}(1) = 1/(1+\mathrm{occ}(1))$. The upstream lemma then returns equality with $1/\varphi$. No further algebra is done here.
why it matters
This is the $n=1$ pin of the rung dilution law. The parent theorem occ_forced inducts from it (with the zero case and composition) to obtain $\mathrm{occ}(n) = \varphi^{-n}$ for every natural $n$, i.e. $1/(1+z)$ on the lattice $1+z = \varphi^n$.
That forced rung law is the paper's central geometric input: in the scale-free class $K_s(z) = (1+z)^{-s}$ it pins $s=1$ exactly, excluding volume dilution ($s=3$) and spacetime dilution ($s=4$). Downstream consequences include the CPL identification on the thawing line $w_a = -(1+w_0)$, the sum rule $w_0 + w_a = -1$, the band $w_0 \in (-1,-0.88)$, and the no-phantom bound $w(z)\ge -1$.
Framework landmarks: T6 ($\varphi$ as self-similar fixed point) and the BIT single-channel ($d=1$) selection behind the rung condition. The today-amplitude $\delta w_0 \in (0,J(\varphi)]$ remains open.
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