An Equivariant-Response Framework and a Quantitative Anomaly Bound for Asymmetric-Capacitor Thrust — Companion note to the Project Montgolfier reconstruction

This companion note develops the mathematics of the two-component (even/odd) model of asymmetric-capacitor thrust introduced in the reconstruction of Project Montgolfier. We promote the empirical "two-force" decomposition to a statement about a Z_2-equivariant nonlinear response under the polarity involution U->-U, and prove the central projection theorem in an exact, series-free form: the single-tone drive is antiperiodic, so any causal, time-invariant, fading-memory Z_2-equivariant operator-hysteretic ones included-sends the even part to the even harmonics and the odd part to the odd harmonics. The experimenters' half-sum/half-difference procedure is the quasi-static limit of this partition and, statistically, an exact orthogonalisation of the even coefficient from the odd ones. We extend the two-configuration symmetry to the Klein four-group V_4=Z_2^polxZ_2^lead of the four historical configurations, whose four character projections sort the geometric force, the polarised force, and the leading systematics into distinct irreps-so the anomaly channel is doubly protected. The frozen surface charge is a Z_2-odd order parameter whose history-dependence aliases an even-part asymmetry onto the anomalous odd channel, with an exact bias identity delta_spur=12(_+-_-) removed by a symmetric-conditioning protocol. Casting "is there an anomalous force?" as model selection, we show the anomalous term delta U|U| is strongly collinear with the classical U- U^3 over any bounded DC range (a design-independent variance-inflation floor set by V_/V_ alone), so DC data bound it only as 1/SNR. We correct a trap in the naive AC discriminator: above corona onset a classical threshold nonlinearity floods the high odd harmonics with a tail decaying as n^-2, slower than the anomaly's n^-3 (a general smoothness lemma), so bare "h_5 present" is not an anomaly signature. The repaired discriminator restricts the drive to U_0<U_on and identifies the anomaly by its signed template 1:15:-135:1105, by frequency-flatness (a memoryless anomaly does not disperse; every rate-dependent classical mechanism does), and-for the one rate-independent classical mechanism, the electret, which is itself frequency-flat and floods the same tail-by phase: the anomaly is purely in phase with the drive, the hysteresis comb in quadrature. A matched-filter Cramér-Rao analysis on the in-phase, classical-free tail harmonics removes the collinearity penalty (sqrtVIFapprox26) for a known hysteresis backbone; a free co-fit backbone is degenerate with the anomaly at a single amplitude, so identifiability comes from the amplitude sweep (developed in the companion analysis). We add ballistic-floor, bounded-enhancement, formulation-independence and quantitative Abraham-force (10^-11 N) results, a Navier-Stokes-free momentum-balance lemma for the thrust, and close with a revised falsifiable programme. No anomalous force is claimed; every statement is a classical bound or a falsifiable prediction, and all numbers are reproduced by an included script.