Mirny's Cosmodynamics: Notes. Note No. 4 Phase function g(ψ): three candidates and a narrow anapausta

In Note 1, the Mirny threshold was written as HPM(ψ)=φ2−110−ΔH g(ψ), where g(ψ) is the phase function of breathing and the slope at the saddle point D0=−7.77 is a measurable quantity. Here we show: (i) from the definition of D0D0 as local steepness follows a rigid relation ∣D0∣=ΔH⋅max⁡∣g′(ψ)∣, independent of the candidate's functional form; (ii) for ΔH=0.03–0.07, the required steepness is 155–259 per unit phase, and the 10→90% transition occupies 0.004–0.010 of the phase — the anapause is narrow for any reasonable candidate; (iii) the three canonical candidates (logistic, smoothed step, and narrow step) differ only in their shape inside the narrow transition; (iv) the test distinguishes not form but sharpness: a step between adjacent DESI bins confirms the narrow anapause, while a smooth slope across the whole zone is the way to die.