An Inverse-Square Law from a Future-Phase-Position Acceleration Map and Harmonic Closure in a Closed Two-Body AB Phase System (v4: Integrable First-Order Feedback with Conserved Quantity and Envelope Preservation)

Version 2 of the supplementary paper on the two-body AB acceleration experiment. v1 reanalyzed the published AB acceleration data, defined the acceleration map alpha = R omega^2 centered on the future phase position, and derived the inverse-square law with respect to phase-cell width via harmonic closure |omega_n| dtheta_n = Omega. v2 retains the full v1 content and adds: (i) the zero-order no-go theorem — in the v1 dynamics class the angular speed is frozen, so the readout acceleration accumulates into no velocity measure and cannot act on the closure quantity; (ii) the kinematic consistency condition d|v|/dtau = |alpha| as the acceptance criterion of a first-order readout; (iii) the velocity-feedback dynamics domega/dtau = kappa*alpha as a conservative working hypothesis, with the exact integral omega = omega_0 + kappa*dchi/dtau; and (iv) numerical experiments J, K, L: kinematic consistency holds at machine precision (max error 8.61e-16), the angular-speed variation is bounded and oscillatory, the harmonic acceleration readout survives modulation within 0.96%, the closure-residual envelope does not grow, the inverse-square distance exponent survives to first order (-1.999937891, integrated form), and the slow envelope decrease at strong feedback vanishes in the RK4 continuum limit, identifying it as a discretization artifact. Self-citation [1] is updated to Basic Axiom System v6, whose Axiom 17 lifts the v1 acceleration map to the axiom layer; the feedback rule of v2 is a working hypothesis outside that zero-order readout. Japanese and English texts, TeX/PDF, six new figures, experiment code, and reproduction data are included. Version v3 leaves the text unchanged and replaces the two embedded figures of the English edition (the future-phase-position mechanism diagram and the published-AB second-difference figure) with English-labeled versions; both language editions are recompiled with the v3 DOI. Version v4 corrects the kinematic consistency condition to dv/dtau = R*kappa*alpha, unifies the continuous model with the discrete map via g(omega) = 4 sin^2(omega/2), removes an invalid compactness argument, and adds the theorem that the continuous feedback system is integrable: it possesses the exact first integral omega - omega_0 = kappa * dchi/dtau and the conserved quantity I = chi^2/2 + (1/kappa^2) * intomega_0ᵒᵐᵉᵍᵃ (s-omega_0)/g(s) ds, with the closed-form antiderivative F(s) = ln sin(s/2) - ((s-omega_0)/2) cot(s/2). The corollary confines omega to (0, 2pi), yields |chi|_max = sqrt(2I) at every turning point (exact envelope preservation), and shows the orbits are periodic around the unique equilibrium. Experiment M verifies the conservation numerically (RK4 relative drift of I down to 1e-10 at dt = 0.01; first-integral drift at machine precision) and quantifies the discrete-map envelope decrease as non-conservation of I (8.7-43.7 percent), with |chi|_max = sqrt(2 I_0) confirmed to 1.4e-13. The strict-closure versus quasi-stationary-residual distinction and the tangential geometry of C_f are also clarified.