natural_display
plain-language theorem explainer
On nonnegative loads, every calibrated dressing response R and every self-similar dressing D agree after a pure change of units: R.g(ln φ · t) = D.g(t). Anyone closing the α-genesis calibration story cites this. The proof is a two-rewrite term: R is forced to exp(−·) and D is forced to φ^{−t}, which match under the log-φ rescaling.
Claim. For every dressing response $R$ (factorizing survival with unit linear response) and every self-similar dressing $D$ (factorizing, antitone on $[0,\infty)$, with step balance $g(1)=1/(1+g(1))$), and every $t\ge 0$, one has $R.g((\ln\varphi)\, t)=D.g(t)$.
background
Module M5 (Calibration Forcing) eliminates the unit-linear-response calibration that M1's dressing response still carries. A self-similar dressing keeps only ledger factorization, antitonicity on nonnegative loads, and the fixed-point balance $g(1)=1/(1+g(1))$ — the same W2 equation that forces the T9 measure step. From those alone, $g(t)=\varphi^{-t}$ on $t\ge 0$, with step $g(1)=\varphi^{-1}$ derived rather than set.
By contrast, a dressing response from ResummationForcing is a factorizing map with unit linear response at zero load. Upstream resummation forcing gives $R.g(\varepsilon)=e^{-\varepsilon}$ with no free resummation. The continuous T9 weight is $\mathrm{contWeight}(t)=\varphi^{-t}$. The present identity says the calibrated exponential is exactly that weight read in natural log units of $\varphi$.
proof idea
Term proof by two rewrites. First apply response_is_forced_measure to $R$ at load $t$, which (via the M1 resummation theorem) replaces $R.g((\ln\varphi),t)$ by $\exp(-(\ln\varphi),t)$. Second apply the in-module response_forced for the self-similar dressing $D$ at the same $t\ge 0$, replacing $D.g(t)$ by the forced continuous weight $\varphi^{-t}$. The two sides are definitionally equal, so the goal closes.
why it matters
This is clause 3 of the M5 certificate CalibrationForcingCert: the M1 calibrated response is only the natural-units display of the calibration-free object. Together with step_forced, response_forced, and alphaInvGenesis_from_selfSimilar, it discharges the residual normalization worry for the forward $\alpha$ seed: form, rate, and step are forced by factorization plus self-similar balance — the same structural facts that force the recognition measure (T9 / measure forcing, linked to the T5–T8 chain and $\varphi$). No CODATA input enters. Downstream the certificate bundles this equality so later $\alpha^{-1}$ genesis identities can quote a single verified package rather than re-open unit conventions.
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