self_similar_attenuation_forced
plain-language theorem explainer
Any positive real ρ satisfying the reciprocal balance ρ = 1/(1+ρ) equals φ⁻¹. Cosmologists deriving the BIT redshift kernel cite this as the single-rung fixed point. The proof reduces the balance to the quadratic x² + x − 1 = 0 and matches the unique positive root against the golden-ratio identity φ² = φ + 1.
Claim. If $\rho > 0$ and $\rho = 1/(1+\rho)$, then $\rho = \varphi^{-1}$, where $\varphi$ is the golden ratio (the unique positive root of $x^2 = x + 1$).
background
The 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 of aging-charge attenuation, and a single-rung balance. The second premise is the fixed-point equation $\rho = 1/(1+\rho)$ for the one-rung attenuation factor.
In Recognition Science, $\varphi$ is the self-similar fixed point forced at T6 of the unified chain, characterized by $\varphi^2 = \varphi + 1$. The reciprocal $1/\varphi$ is the natural single-step attenuation on the $\varphi$-ladder of cosmic scale (where scale steps are powers of $\varphi$). Upstream, phi_sq_eq records exactly $\varphi^2 = \varphi + 1$, and positivity/nonzero lemmas for $\varphi$ are standard.
This lemma isolates the algebraic content of premise 2 before it is installed into the abstract rung-dilution structure.
proof idea
From $\rho = 1/(1+\rho)$ and $\rho > 0$, clear the denominator to obtain the quadratic $\rho^2 + \rho - 1 = 0$. Invoke $\varphi^2 = \varphi + 1$ to show that $1/\varphi$ satisfies the same quadratic. Both roots are positive, so their difference squared is nonnegative; nlinarith closes uniqueness and forces $\rho = 1/\varphi$. The only named algebraic input is the golden-ratio identity; the rest is field simplification and linear arithmetic.
why it matters
This is the algebraic core of premise 2 in the Forced Redshift Kernel paper. Downstream, occ_one_forced applies it directly: the single-rung attenuation of any rung-dilution law equals $\varphi^{-1}$. That pins the full law $\mathrm{occ}, n = \varphi^{-n}$, which on the lattice $1+z = \varphi^n$ is $1/(1+z)$.
From there the module excludes volume ($s=3$) and spacetime ($s=4$) power kernels, forces the CPL thawing line $w_a = -(1+w_0)$ with sum rule $w_0 + w_a = -1$, and obtains $w(z) \ge -1$ (no phantom). The result sits at the T6 landmark: $\varphi$ as the unique self-similar fixed point, here realized as the unique positive reciprocal-balance root. Open items left untouched are the BIT aging mechanism itself, the $d=1$ channel selection, and the today-amplitude band $\delta w_0 \in (0, J(\varphi)]$.
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