A causal variational worldline action for modified inertia: the rapidity gap

Abstract

VERSION 3 CHANGES. The substantive one CORRECTS both earlier versions. (1) The circular-orbit relation was wrong. v1 and v2 both asserted that on a circular orbit |a| is constant and so the acceleration dependence contributes no additional radial force. It does: ahat = a/|a| ROTATES, d2ahat/dt2 = Ω2rhat ≠ 0, and the resulting MEMORY FORCE is outward. The correct balance is gbar = gobs[μ(Y) + (Y/2)μ′(Y)], not gbar = μ gobs. MOND survives this as a pure renormalisation a0 → (2/3)a0, so the required kernel first moment becomes M1 = (2/3)c/a0 = 68 Gyr rather than 101 Gyr, and any target interpolation remains exactly reachable via μ(Y) = (2/Y2)∫0Y Y′ μt dY′. (2) General orbits are now solved. For an arbitrary worldline θ(τ, τ-s) = (s/c)|a(τ - s/2)| + O(s3) — the rapidity gap across an interval is s/c times the acceleration magnitude at the MIDPOINT — so Θ is a retarded memory-average of |a| with lag s/2. The in-in prescription's role is now QUANTIFIED rather than assumed: the acausal (advanced) terms enter only at O(λΩ), which is ≤ 4×10-6 galactically at λ ≤ 39 yr, so the equation of motion is causal at the accuracy that matters. (3) A new result. For the α-family 1 - μeff = Y-α(2-α)/(4α), which VANISHES IDENTICALLY at α = 2: the memory force exactly cancels the leading Newtonian anomaly, suppressing the solar-system residual by a further 8Y2 = 3.2×1016 to 1.2×10-35 m s-2, twenty-one orders under the ephemeris bound. AGAINST INTEREST in the same breath: the α = 1 floor SURVIVES as a0/4 rather than a0/2, so the exact a0-line is still excluded (by 640× instead of 1279×), αmin softens only from 1.380 to 1.322, and above α = 2 the residual changes SIGN. (4) The open problem 'non-circular orbits' is discharged and replaced by a GHOST ANALYSIS of the nonlocal theory: the equation of motion is fourth order in x, Ostrogradsky's theorem does not apply because it requires a LOCAL higher-derivative Lagrangian, but that is an argument for the theorem's silence and not a proof of ghost-freedom — it is now the sharpest remaining structural gap. One further verification script (27/27) is added. Unchanged: a0 is NOT derived, there is no local derivative expansion, and the coefficient remains FITTED. VERSION 2 CHANGES (both correct version 1, and both are stated because they are corrections against v1's own claims). (1) The CPT problem dissolves. v1 gave two repairs of the rest-energy defect and left the CPT-odd one owing a Standard-Model-Extension analysis. That framing was wrong: the correct covariant object is √((u·n)2), and with it the repair is CPT-EVEN and bounded below at every speed, so v1's Form I instability at 0.8165c was an unnecessary price and Form II's CPT-oddness was an artefact of the writing, not a property of the theory. The owed analysis is discharged by a structural kill: the odd branch gives antiparticle rest energy mc2(2μ-1), NEGATIVE for g < 0.5164 a0 = 4.83×10-11 m s-2, i.e. throughout the outer-galaxy regime the theory exists to describe. (2) The kernel memory bound was 104 times too strong. v1 stated λ ≤ 1.4 days and N ≥ 2.6×1013, derived by demanding the short-memory regime hold at Mercury. That requirement was too strong: short memory is needed only where the ACCELERATION dependence does physical work (galaxies, requiring only λ ≤ 0.98 Myr), while the solar system need merely be Newtonian, which the long-memory branch can be. The correct constraint is the ephemeris bound on the residual anomaly Δ = gπ2a02λ2/64v2, tightest at Mercury: λ ≤ 39 years, N ≥ 2.6×109. A PDF and one further verification script (28/28) are added. Unchanged: a0 is not derived, non-circular orbits are unsolved, there is no local derivative expansion, and the coefficient remains FITTED. Preprint. This deposit exists to timestamp a construction and to carry its reproducibility scripts; it is not a journal publication. The result, in one line. The object that supplies the proper-acceleration magnitude |a| to a worldline action, without the (v/c)2 suppression that kills the polynomial forms, is the rapidity gap between the four-velocities at two proper times: cosh θ = -u·u'/c2, with √(-u·u'/c2 - 1) = √2 sinh(θ/2) exactly and θ(s)/s → |a|/c as s → 0, because proper acceleration is the rate at which rapidity accumulates. A parity theorem first. Minkowski space admits exactly two invariant tensors, the metric (rank 2) and the Levi-Civita tensor (rank 4), both of EVEN rank. Every polynomial scalar built from four-velocities at any number of proper times therefore has even degree in u and, on a circular worldline, contributes only EVEN powers of the orbital speed — while the deep MOND limit needs the FIRST power, since |a| = γ2Ωv. So no polynomial-in-u worldline self-interaction, at any degree, can produce MOND: the mismatch is exactly one power of v, i.e. (c/v)2 = 106-3.6×107 in amplitude at galactic speeds. The rapidity gap escapes because it is non-analytic in the bilinear, not because it is of higher degree. The model is then solved. With the minimal causal kernel K(s) = (N/λ)e-s/λ, supported on s > 0 only, the circular-orbit content reduces in closed form to Θ = (4Nv/c) x coth(π/x)/(4+x2) with x = λΩ (verified against quadrature to 20 digits). Setting the kernel's first moment to M1 = c/a0 makes Θ = |a|/a0 exactly in the short-memory regime, and the circular-orbit equation of motion becomes gbar = μ(gobs/a0) gobs — Milgrom's modified-inertia relation exactly. Demanding the interpolation exponent that solar-system ephemerides require (α ≥ 1.4; α = 1 is excluded by ~103) gives μ in closed form. The construction is causal by fiat, variational in the in-in sense, contains no derivative of x beyond the first so carries NO Ostrogradsky instability, and is Lorentz-invariant up to the preferred frame that modified inertia already presupposes. The speed-dependent long-memory branch is excluded STRUCTURALLY: a kinetic function of speed alone forces f'(v) = v3/(r a0), which is r-dependent, so no such function exists. A defect is identified and repaired. In a single-factor worldline action the same function multiplies the rest energy and the inertia, so the deep-MOND requirement μ → 0 forces the rest energy to VANISH at zero acceleration. This is shown to be structural: every ∫dτ W(Θ) term locks the two at a ratio independent of W. The repair uses the preferred-frame four-velocity in two variants, both giving rest energy exactly mc2 and inertia exactly mμ, with explicitly different costs: the CPT-even quadratic form has an energy that turns negative above v2 = 2c2/(3-μ), i.e. 0.8165c in the deep limit (a factor 240 above the fastest systems MOND is applied to, so a relativistic defect and not a galactic one); the linear form is bounded below at every speed but is CPT-odd and owes a Standard-Model-Extension analysis that is NOT attempted here. One falsifiable prediction. The preferred-frame coupling in either form is |B| = (1-μ)/2 ≈ a02/8g2, so any Lorentz violation from this construction scales as g-2 with the local gravitational acceleration: ~1.1×10-23 in a terrestrial laboratory, 3.1×10-17 at Earth's orbital acceleration, and of order unity only in the outer disc of a galaxy. This is the OPPOSITE of a constant SME background — largest where no test exists, smallest where tests are tightest. Binning Lorentz-violation limits by local g would test it. It is not confronted with data here. WHAT IS NOT CLAIMED, stated up front because it bounds the result. The acceleration scale is NOT derived. The kernel's first moment must satisfy M1 = c/a0, and in the short-memory limit ONLY that moment survives — verified across three kernel shapes (exponential, gamma-2, box) normalised to the same moment. One number is traded for one number. The coefficient relating a0 to Λ remains fitted, not derived. The interpolating kernel ν = √(1+1/y) is identical to Eq. (9) of Milgrom, Phys. Lett. A 253, 273 (1999) — not a variant of it. That paper obtains it from a de Sitter-Unruh temperature balance and fixes a0 = 2cHΛ. Nothing in the kernel is new here; what is new is the ACTION and the parity theorem that identifies which structure can produce it. A NEW constant is introduced: the kernel memory time λ, bounded hard by the requirement that the short-memory regime hold at Mercury, λ ≤ 1.4 days, hence kernel weight N ≥ 2.6×1013. A galaxy-scale λ is excluded outright. Non-circular orbits are NOT solved; the reduction holds |a| constant. The |s| non-analyticity means the action admits no local derivative expansion, so there is no effective-field-theory power counting and no controlled quantum completion by that route. No claim is made regarding particle physics, the Standard Model, or unification. The author publicly withdrew earlier statements of that kind and does not restate them. AI-assistance disclosure. Portions of the analysis, numerical verification and drafting were carried out with the assistance of a large language model (Anthropic Claude). The author directed the work, specified and reviewed every load-bearing calculation, and takes full responsibility including for any errors. No AI system satisfies the criteria for authorship and none is listed as an author. Several intermediate claims produced during the work were found to be incorrect and were withdrawn before this deposit. Reproducibility. Every quantitative claim is produced by a committed script that exits with a nonzero status if any internal consistency check fails, and each carries negative controls that must trip. The six scripts are included in this deposit (30/30, 35/35, 35/35, 38/38, 50/50 and 14/14 checks).