rho
plain-language theorem explainer
The per-step recognition weight is fixed as ρ = φ⁻¹. Anyone working on the forced measure (Born weights, continuum dressing, α⁻¹ residual closure) cites this constant as the unique self-similar step factor. It is a one-line definition from the golden ratio constant.
Claim. Define the forced per-step weight $\rho := \varphi^{-1}$, where $\varphi$ is the unique self-similar scale of Recognition Science.
background
Module T9 closes the missing weighting rule after the T0–T8 forcing chain. That chain already forces the cost $J$, the scale $\varphi$, the eight-tick period, and $D=3$, but not how much weight sits on each recognition state. The lattice layer answers with two premises: factorization over independent composition, and per-step self-similar balance $\rho = 1/(1+\rho)$.
The second premise is the reciprocal fixed point of the ledger's own generator. By the BIT-kernel self-similar attenuation result, the only solution compatible with T6 uniqueness of $\varphi$ is $\rho = \varphi^{-1}$. Continuum weights then become the Gibbs rule with rate $\ln\varphi$, equivalently $w \propto \varphi^{-\mathrm{cost}}$.
Sibling facts record $\rho \in (0,1)$, positivity, and the lattice weight $w(n)=\rho^n$. Downstream α-genesis work treats $\rho$ as the base of the exponential dressing on spectral load.
proof idea
Pure definition: $\rho$ is introduced as the reciprocal of the already-forced constant $\varphi$ from Constants. No tactic proof; the mathematical content is the naming of $1/\varphi$ as the forced per-step weight. Downstream lemmas (rho_pos, rho_lt_one, step_self_similar forcing) justify why this value is the unique admissible choice.
why it matters
This is the primitive scalar of T9. Lattice weights become $\varphi^{-n}$; continuum weights become $\mathrm{contWeight}(t)=\varphi^{-t}$. Calibration forcing uses it as the forced single-step value of every self-similar dressing: response_forced proves $g(t)=\varphi^{-t}$ once $g(1)=\rho$.
Alpha-genesis residual targeting builds the unique closing load as $\log(\alpha^{-1}_{\mathrm{CODATA}}/\mathrm{budget})/\log\rho$ minus spectral load, and proves uniqueness because $\rho<1$ makes the dressed value strictly monotone. The same $\rho$ appears in continuum-weight forcing inside this module and in the canonical self-similar dressing used as the non-vacuous witness.
Framework landmarks: T6 uniqueness of $\varphi$ pins the ratio; the measure is the missing projection behind Born weights, chirality selection, and rung occupancy.
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