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

Abstract

VERSION 5: the construction is now stated in ONE LINE as the paper's opening display. S = −mc2 ∫ [ μ(Θ) dτ + (1 − μ(Θ)) dt ] , Θ(τ) = ∫0∞ ds K(s) cosh−1( −u(τ)·u(τ−s)/c2 ) with dτ the proper time and dt the PREFERRED-FRAME (cosmological) time. This is not a new result — it is algebraically identical to eq. (20) of v4, certified in a new script (17/17) — but it was buried in §6, which badly undersold it. In this writing the two limits are the two most natural scalars a worldline possesses: μ → 1 gives S = −mc2∫dτ, the ordinary free relativistic particle; μ → 0 gives S = −mc2∫dt, a Lagrangian that is exactly velocity-INDEPENDENT, so the inertia vanishes identically while the rest energy stays mc2. So the action interpolates between proper time and cosmological time, with the interpolation variable the accumulated RAPIDITY GAP along the worldline's own past. Writing the second term with dt rather than γdτ IS the even root √((u·n)2), which is what makes the action CPT-even; the cosh−1 is forced rather than chosen, because no polynomial in u can produce MOND (the parity theorem) and the rapidity gap is the unique non-analytic escape; and the kernel's first moment M1 = (2/3)c/a0 is the only surviving parameter. A control in the new script confirms the assignment is load-bearing: exchanging which factor carries dτ and which carries dt preserves the rest energy but gives inertia m(1−μ), the wrong way round. NO PHYSICS CHANGES IN v5 — every equation, result, open problem and caveat of v4 stands verbatim, including that a0 is NOT derived and that μ's shape is the α = 2 interpolation forced by ephemerides. The line buys the STRUCTURE; it does not buy the coefficient, and κ = 1/2 remains FITTED. VERSION 4 CHANGES: perturbative stability is settled, and settling it required correcting a calculation of my own. A first attempt treated the transverse sector in isolation and reported a real runaway at p = √2 Ω, concluding the construction was dead. That was wrong. √2 Ω is EXACTLY the epicyclic frequency of a flat rotation curve, and recovering the standard STABLE oscillation frequency out of an instability calculation was the signature of the error: the Coriolis terms, the potential's η2 term (which cancels the rotating-frame +μΩ2η2/2 — omitting it alone makes even KEPLER look unstable), and the longitudinal-transverse mixing are all O(Ω), the same order as the effect, and all three had been dropped — as had the memory-force correction to the background balance. With all of them retained: the extra poles introduced by the memory force are PURELY IMAGINARY at |p| ≈ 2.45 Ω (oscillatory, Lee-Wick type), and the residual growing mode is Re(p) = 9.8×10-5 Ω in the deepest MOND regime, falling monotonically to zero by g = a0 — an e-folding time of ~1600 orbits, i.e. 363 Gyr for the Milky Way, about 26 times the age of the universe. That value is verified NOT to be root-finder noise: it is unchanged from 60 to 600 working digits. For g > a0 a large real root appears but tracks p = √(μ/B), the breakdown scale of the very truncation that produced it, and is spurious by the standard criterion for nonlocal theories — argued, not proved. So the theory is PERTURBATIVELY STABLE where MOND operates. It is NOT a proof of ghost-freedom: the 2.45 Ω oscillatory pole is a real prediction that has not been confronted with data, and a classically-stable Lee-Wick pole is still a ghost in the QUANTUM theory. The open problem list is updated accordingly — the quantum status of that pole is now the sharpest remaining structural gap. Scope of the stability result: quadratic order, non-relativistic, circular background, longitudinal term switched off, α = 2 kernel. One further verification script (14/14) is added, and the superseded one carries a withdrawal banner. Unchanged: a0 is NOT derived, there is no local derivative expansion, and the coefficient remains FITTED. 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 re