Phase, Memory, and Floors: the corrected AC discriminator, a Preisach convergence theorem, and fluctuation limits for the Biefeld–Brown null test — Second companion note to the Project Montgolfier reconstruction

The first companion note ref built an equivariant-response framework for asymmetric-capacitor thrust and proposed an AC harmonic discriminator that isolates a hypothetical polarity-odd anomaly from all classical mechanisms. That discriminator, as stated, contained a hidden failure. Its "frequency-flatness" test was meant to reject a smooth memoryless anomaly's classical impostors on the grounds that classical mechanisms disperse; but a rate-independent dielectric memory-an electret, or any Preisach hysteron-is itself frequency-flat, and its steady minor loop is a non-analytic odd response that floods the high harmonics h₅, h₇, with a comb the flatness test cannot reject. Sub-onset operation kills the corona threshold but not the electret; sub-onset is not band-limited. We repair the discriminator by resolving each odd harmonic into its in-phase (cosine) and quadrature (sine) parts relative to the drive's turning instants. Two lemmas do the work: a memoryless odd response is pure in-phase (turning-point symmetry), and a rate-independent hysteresis loop emits its comb in quadrature, with the in-phase tail vanishing identically at Rayleigh (small-amplitude) order because the loop's branch-average backbone is linear. Beyond Rayleigh the backbone turns odd-nonlinear and leaks back into the in-phase channel, so channel purity is a second controlled small parameter, isolated by an amplitude-scaling-collapse test. The corrected discriminator projects onto the in-phase channel and treats the quadrature comb as a predicted classical signal rather than a contaminant. Three results become load-bearing and mutually reinforcing: (i) a Preisach convergence theorem bounding the degauss residual bias by w u₀² (1-r) / (1+r) plus a Maxwell-Wagner creep correction, which also bounds the in-phase leakage; (ii) a two-mechanism fluctuation-floor theorem-the measured loop area is the amplitude-dependent effective loss epsilon"ₑff= epsilon"ₗin+epsilon"ₕyst (U₀), so Callen-Welton applies to its small-signal (linear) part while the hysteretic remainder sets a drive-synchronous Barkhausen switching floor keyed to the elementary switching charge qₑff; the resulting limit is honestly an integration-time cost delta₉5 (T) sqrtSF/T, whose only time-independent parts are the systematic leakage terms and the 1/f knee, not the white-noise PSD; (iii) a corrected, closed-form AC error budget. We add exact composite theorems (a series-laminate 1/f enhancement cap, sharper than Clausius-Mossotti, plus Bergman-Milton bounds), a mean-field poling threshold, and a unified sloppy-model picture in which delta is structurally identifiable from DC data yet carries a variance inflated VIF-fold, so that only the orthogonal in-phase AC tail restores a usable Var () -though the restoration is subtler than a single amplitude admits: at fixed U₀ the anomaly's in-phase template is exactly degenerate with a free odd-polynomial backbone (the estimator seam), so identifiability comes only from the amplitude sweep, leaving a residual variance inflation set by its dynamic range and deflated toward O (1) by pinning the backbone with the quadrature comb. A measurement-chain section closes the instrument model: the phase reference lives in the mechanical admittance, so the in-phase/quadrature split needs per-harmonic phase calibration (with a resonant-detection variant that realises the matched filter in hardware at SNR gain Q), and high-voltage drive distortion is nulled by promoting the two-tone protocol to primary. All analytic claims are reproduced numerically in an accompanying self-checking pipeline; the pressure-law overlay and the full Bayesian run are stated as programme.