Proposition: The retained signal is total diffusion

If the null admits an additional free-stage momentum-diffusion coefficient that can absorb $\lambda A_{r_{\mathrm{c}}}$, the four-time second-moment signature is matched exactly. In the quadratic gravity model, admitting this same shift throughout all required stages can also match the entire dynamics.

Therefore the certificate identifies gravity only relative to independently restricted competing sources. It is not source identification from heating alone.

Proof. The first assertion follows from equality of $D_{\mathrm{f}}+h\lambda A_{r_{\mathrm{c}}}$ in (eq:fourtime). For the second, matched quadratic coefficients give identical total generators and therefore identical complete instrument laws.