response_forced
plain-language theorem explainer
Every self-similar dressing equals the forced continuum weight on nonnegative loads: g(t)=φ^{-t}. Anyone closing the α pipeline without a unit or derivative calibration cites this. The proof pins the step to ρ via step forcing, then applies T9 continuum uniqueness (factorization + antitonicity).
Claim. Let $g$ be any self-similar dressing (factorizing, antitone on $[0,\infty)$, and satisfying the balance $g(1)=1/(1+g(1))$). Then for every real $t\ge 0$, $g(t)=\varphi^{-t}$, i.e. $g$ coincides with the forced continuum weight.
background
Module M5 (Calibration Forcing) eliminates the unit-linear-response normalization that M1 still carried. A self-similar dressing is a response $g:\mathbb{R}\to\mathbb{R}$ with three structural premises only: factorization over independent loads (ledger additivity), antitonicity on nonnegative loads (more load never helps), and the single-step balance $g(1)=1/(1+g(1))$ — the same balance that forces the T9 measure step.
Upstream, step_forced already derives $g(1)=\varphi^{-1}$ from that balance plus positivity (itself from factorization: $g(1)=g(1/2)^2>0$). The reciprocal of the step solves the T6 fixed-point equation $r^2=r+1$. Separately, MeasureForcing's continuum layer states: any factorizing antitone weight with calibrated step $f(1)=\rho$ equals $\rho^t$ for all $t\ge 0$ (multiplicative Cauchy pinned by monotonicity; no power-law class restriction). Here $\rho=\varphi^{-1}$ and the continuum weight is that map $t\mapsto\rho^t$.
proof idea
Introduce $t\ge 0$. First identify the step: rewrite $D.g,1$ by step_forced to get $D.g,1=\rho$. Then apply continuum_weight_forced to $D.g$, feeding the structure's factorization and antitonicity hypotheses together with that step equality. The conclusion is exactly $D.g,t=\mathrm{contWeight},t$ on $t\ge 0$. No derivative or unit convention appears.
why it matters
This is the M5 closure step: the dressing of the α seed carries zero calibration input. Its form, rate, and step are forced by the same two structural facts (factorization, self-similar balance) that force the recognition measure (T9 continuum layer; step linked to T6 via $\varphi$).
Downstream, alphaInvGenesis_from_selfSimilar rewrites the forward α object as channel budget times $D.g$ of the spectral load, using this equality. natural_display identifies every M1 calibrated response (in log-$\varphi$ units) with every self-similar dressing. Both facts are bundled into CalibrationForcingCert and feed the top-level AlphaGenesisCert. The residual normalization worry for α genesis is discharged: no CODATA, no derivative condition, no unit choice.
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