An exact identity between electromagnetic and scalar vacuum response on circular worldlines

Abstract

**v1.3 (9/21/2026]) — scope and prior-art correction. No computed value changes.** An external literature check and an independent derivation found two things. (1) The Letter's master identity,G_EM = −2 ∂²_s G_sc, is not specific to circular motion: in Feynman gauge it holds for every worldline in flatspacetime (three-line proof; verified to 10⁻³⁰ on a deliberately non-circular, non-stationary worldline). It is thefield-strength counterpart of the known E² factor for derivative-coupled detectors, stated in Biermann et al.,Phys. Rev. D 102, 085006 (2020), Sec. 5. (2) The Letter's integral I(v) is exactly the small-gap integral ofBiermann et al. Eq. (3.9): T_circ/T_lin (small gap) = 2v√(1−v²)·I(v), so C_EM(v) = π/I(v) restates theirsmall-gap temperature. The Letter did not cite this work; the omission is recorded. What remains the Letter's: the explicit small-velocity coefficients π/6, π/15, 43π/840; the exact low-velocityshape (1−E)³ with a hard zero above E = 1 (generalised in Paper 8); and the derivation of the program's measuredC_EM = 6 and its intercept offset. All v1.2 numbers were independently recomputed and reproduce to every printeddigit. Earlier files are unmodified; corrections are append-only. This computes the thermometer, not the engine; the inertia coupling remains unsolved. **v1.2 (2026-07-29) — corrections from a cross-paper consistency audit> against Paper 8. No computed value in this Letter changed.** The manuscript masthead carries the v1.1 version DOI (10.5281/zenodo.21609658); this version's DOI is assigned by the record.>> An adversarial audit of this Letter against Paper 8 of the same program> graded nineteen claims: eight consistent, nine needing a qualifier, two> contradicted as written. Every *v* → 0 result survived independent> recomputation from the printed equations, and the two papers cross-validate> in both directions on the mathematics. The corrections are all matters of> scope, procedure, and named systematics.>> The two contradicted claims. **Corollary 2's prose** asserted field> blindness for "the electromagnetic sector"; the theorem holds for the> field-strength contraction that Eq. (2) defines, which freezes no> spatial-frame transport and carries only integer harmonics. A> Fermi–Walker gyroscopic dipole carries γ-dependent harmonics encoding the> Thomas precession rate, and is a counterexample at the wider scope. The> claim is rescoped; the omission is booked at face value and no cure is> claimed. **The successor predictions** *C*EM(0.3) = 5.776487> and *C*EM(0.6) = 4.993094 **stand verbatim** and are confirmed> to every printed digit by three independent routes, but they are the> *E* → 0 intercept of the estimator, not the reading of the frozen record> procedure, which on its own grid *E* = {*v*/2, *v*, 2*v*} returns 7.531107> and 10.089893. The procedural clause is corrected and the grid condition> *E*min ≲ 2.6 × 10⁻⁴ **absolute** is now stated.>> Nine qualifiers are repaired, three of them the abstract's uses of> "exact". In their place the abstract now carries a genuinely exact law,> *C*EM(*v*) = π/*I*(*v*), valid at all velocities, of which> 6/[1 + (2/5)*v*² + *O*(*v*⁴)] is the small-*v* expansion. This version> also supplies that law in closed form: *I*(*v*) = Σ *J*m*v*2m> with every coefficient a rational multiple of π, which *derives* rather> than checks the Letter's two closed-form integrals π/6 and π/15 and the> exact coefficient 2/5.>> Three deviations that previously carried no named systematic now carry> one, and two of the three are favourable. The retrodicted intercept's> "agreement at 0.5 %" is shown to be consumed by the frozen record's own> pointwise digit precision, which bounds the comparison at 0.51 %. The V1> commutator miss is re-diagnosed: both missing residuals are scalar, the> electromagnetic sector sits at machine epsilon at every window, and the> cause is contour-window truncation against the scalar kernel's 1/*s*²> tail — curable by window size alone. The booked misses are not repaired;> only their diagnoses are corrected.>> The MOND conclusion survives on a replaced premise: a field-specific> channel does exist, but γ is a function of *v* alone, so it introduces no> new dimensionful scale and nothing electromagnetic smuggles in *a*₀.>> The sealed v1.0 reproduction capsule> (SHA-256 `2b4518c6…126057e4`) and the v1.1 addendum are **unmodified**.> Corrections are append-only, continuing the v1.1 item numbering at 19.> The v1.2 addendum ships the corrected manuscript, an emitted grep-proof> of every edit old→new, an amended figure ledger, and a new executable> verification of the exact law.>> One open item is transmitted rather than closed: which spatial-frame> transport Eq. (2) implements. Until that is fixed, Paper 8's edge> amplitudes 4 and 4/3 and this Letter's 2 are amplitudes of different> objects.>> This computes the thermometer, not the engine; the inertia coupling> remains unsolved. Paper 3 of the Circular Vacuum Program. An exact identity between electromagnetic and scalar vacuum response on circular worldlines: C_EM = 6 derived, with the proved v->0 closed form 6*pi*k(E) = (1-E)^3 on 0 <= E <= 1 and exactly zero above the edge, and the finite-probe intercept offset retrodicted against the frozen record. Preregistered; misses booked at face value. FILES: manuscript PDF; Overleaf source zip; sealed reproducibility capsule with SHA-256 manifest; append-only addendum. THE PROGRAM This deposit is part of the Circular Vacuum Program, a preregistered computational series on detector response and vacuum entanglement along bound and circular worldlines. Each paper is deposited with its reproduction capsule, its frozen preregistration, and its deviation register. Scored values and verdicts are booked at face value, including misses, and corrections are appended rather than applied retroactively. Paper 1 — Detector response on circular worldlines in de Sitter spacetime: a sharp negative result for a single-temperature origin of the MOND acceleration scale. doi:10.5281/zenodo.21609173 Paper 2 — A kinematic pedestal above the thermal backbone: detector response on eccentric orbits in de Sitter spacetime. doi:10.5281/zenodo.21609579 Paper 3 — An exact identity between electromagnetic and scalar vacuum response on circular worldlines. doi:10.5281/zenodo.21609658 (v1.2 addendum: doi:10.5281/zenodo.21691002) Paper 4 — Vacuum dressing of bound worldlines in de Sitter spacetime: an infrared logarithm and exclusion bounds on vacuum-dressed modified inertia. doi:10.5281/zenodo.21609697 Paper 5 — Laboratory contact for a preregistered worldline-vacuum stack: retrodicting the Sokolov-Ternov polarization and its resonance structure. doi:10.5281/zenodo.21609721 Paper 6 — Entanglement harvesting on bound worldlines in de Sitter spacetime: the co-orbital map, a conformal split, and horizon degradation. doi:10.5281/zenodo.21609752 Paper 7 — Horizons against harvesting: the vacuum-entanglement competition on bound worldlines in de Sitter, BTZ, and Schwarzschild spacetimes. doi:10.5281/zenodo.21609782 Paper 8 — The edge law of circular vacuum response: the threshold exponent is the spatial dimension. doi:10.5281/zenodo.21694371 Paper 9 — Circular vacuum response at fixed velocity is an exact piecewise polynomial: the low-gap plateau splits at the Thomas frequency. doi:10.5281/zenodo.21747116 Paper 9 closes the arc's methodological line: at fixed finite velocity, with the Lorentz factor held exact in the phases, circular-worldline detector response is an exact piecewise polynomial whose breakpoints are the harmonics, which replaces the quadrature ladder the earlier papers computed against. Its qualitative result is that in (2+1) dimensions the Fermi-Walker object carries no integer harmonic at all. CEILING (binding, quoted exactly): this computes the thermometer, not the engine; the inertia coupling remains unsolved. METHODS NOTE: Produced in an AI-assisted workflow (Anthropic's Claude as computational and drafting assistant) under preregistered protocols, adversarial multi-round review, and independent recomputation; all results and interpretations remain entirely the responsibility of the author. No external funding; no competing interests. Binding scope sentence: this computes the thermometer, not the engine; the inertia coupling remains unsolved.