A local, generally covariant field theory of modified inertia

Abstract

VERSION 3: the static nonlinearity of the delta sector -- v2's sharpest named gap -- is CLOSED, and it closes by a PARITY THEOREM. K IS ODD IN π TO ALL ORDERS. For a static khronon T = t + π(x) the unit normal is nμ = (1/w, −∂iπ/w) with w = √(1−(∂π)2). Under π → −π the normalisation w is INVARIANT -- it depends on (∂π)2 -- while ∂iπ flips, so K[−π] = −K[π] EXACTLY, verified on the full three-dimensional closed form with NO series truncation. In one dimension the divergence has the exact closed form K = −π'' (1 − π'2)−3/2 -- odd numerator, even bracket -- whose expansion has ZERO even orders through fifth, and the general 3-D cubic reduces in 1-D to exactly −(3/2)π''π'2, so the 3-D formula is VALIDATED against the exact result rather than asserted. THEREFORE K2 IS EVEN AND THE delta SECTOR HAS NO STATIC CUBIC. Its leading static self-interaction is QUARTIC, with quartic/quadratic = 3π'2 -- of ORDER (∂π)2 with an O(1) coefficient. (A draft wrote 'exactly (∂π)2'; the coefficient is 3, and the script records the correction.) AND ON THE ALIGNED STATIC FOLIATION IT VANISHES OUTRIGHT: a static, shift-free metric gives Kij = 0 identically, so both K.K and K2 vanish at EVERY order and only η aiai survives; and δ(K2)/δT = 2K δK vanishes with K, making that foliation a SOLUTION of the K sector rather than an imposed ansatz. THE NONLINEARITY IS PRICED AND IT IS TINY. |∂π| is the tilt of the foliation against the local frame, ~ v/c: (∂π)2 = 1.5e-6 for the solar system against the CMB frame, 5.4e-7 galactically, 1.1e-5 for clusters, 1.0e-2 even for a 0.1c probe. So NO VAINSHTEIN-TYPE SCREENING RADIUS EXISTS -- the nonlinearity never reaches O(1). It would need |∂π| → 1, a foliation boosted at near-light speed relative to the local frame, which happens near a black-hole horizon and nowhere else. COMBINING THE SECTORS: the η cubic dies by carrying a time derivative, the delta cubic dies by parity in π -- two different mechanisms -- so THE LEADING STATIC SELF-INTERACTION OF THE WHOLE KHRONON SECTOR IS QUARTIC, worth a factor of 811 over a surviving cubic. The result is not cosmetic. WHAT REPLACES IT, and it is narrower. (i) The full T field equation around a real source is NOT solved -- the aligned foliation is shown consistent with the K sector and the nonlinearity priced GIVEN |∂π| ~ v/c, but π(r) is not obtained, so a larger |∂π| than the kinematic estimate is not excluded. (ii) |∂π| → 1 near a black-hole horizon is exactly where this analysis fails; UNIVERSAL HORIZONS in Lorentz-violating gravity (Blas & Sibiryakov 2011 PRD 84:124043) are not addressed, and this is now the sharpest structural gap. (iii) The η sector's quartic is uncomputed -- only that its cubic vanishes. NOTHING ELSE CHANGES IN v3. Every equation, result and caveat of v1 and v2 stands, including that a0's VALUE is NOT derived, that κ = 1/2 remains FITTED, that the theory GAINED two free parameters rather than fewer, and that NO claim is made regarding particle physics, the Standard Model or unification. One further verification script is added (22/22). VERSION 2: the strong-coupling scale -- v1's sharpest named risk -- is COMPUTED, and it does NOT threaten the phenomenology. Restoring the khronon fluctuation T = t + π gives ln N = −πdot + πdot2/2 + (∂π)2/2, so ai = −∂iπdot and Kij = −∂i∂jπ. At λ = ξ = 1, η = 0 the khronon action VANISHES IDENTICALLY (K.K − K2 = k4π2 − k4π2 = 0), so π is pure gauge in general relativity, as it must be -- and a prespecified λ = 3/2 decoy does NOT cancel. Everything surviving is therefore proportional to the small parameters, S2 ~ MPl2[−δ(∂2π)2 + η(∂iπdot)2]. That is the entire origin of the worry, now stated precisely. AN INDEPENDENT CROSS-CHECK FALLS OUT. That action's dispersion ηω2k2 = δk4 gives cs2 = (λ−1)/η -- EXACTLY the PPN-corner limit the spin-0 section obtained from the UNITARY-GAUGE ζ with the ADM constraints eliminated, here from the STUECKELBERG π in flat space. Two gauges, two variables, one answer; a prespecified decoy δ/(2η) is rejected. AND THE STATIC NONLINEARITY OF THE η SECTOR VANISHES. The leading self-interaction is −2πdot(∂iπdot)2 − 2∂iπdot ∂jπ ∂i∂jπ, and every term carries a time derivative, so the whole cubic part vanishes for static configurations -- no static Vainshtein-type screening radius from that sector (a decoy cubic (∂xπ)3 does NOT vanish, so this is a property of the actual terms). Derivative counting then gives Λsc ~ √η MPl/cs ≈ 7.7e14 GeV at η = 1e-7: the PPN-preferred corner does lower the cutoff, but only as η1/2, and from the Planck mass. AND THE CONCLUSION IS ROBUST TO THE POWER. Scanning Λsc = ηpMPl over p = 1/2, 1, 2, 3, 4 -- deliberately allowing powers far worse than the derived one -- Λsc exceeds EVERY scale at which the theory is APPLIED in every case: the Milky Way's orbital frequency (5.7e-31 eV), its inverse size (7.8e-28 eV), 1/AU (1.3e-18 eV) and 1/metre (2.0e-7 eV). At the pessimistic p = 4 there are still 26 orders of galactic margin and 6 against the laboratory; at the derived p = 1/2, fifty. So the strong-coupling scale bears on whether this is a UV-complete QUANTUM theory -- which it never claimed to be -- and NOT on the phenomenology. AGAINST INTEREST, AND CORRECTING A CLAIM OF THE AUTHOR'S OWN RATHER THAN SOFTENING IT. A first draft asserted the cutoff clears every scale INCLUDING the LHC (1.4e13 eV). That is FALSE: at p ≥ 3 the cutoff falls BELOW collider energies, so the khronon effective theory would not cover the LHC at those powers. Harmless only because the matter-khronon coupling there is |B| ~ a02/8g2 = 1.1e-23, and the DERIVED p = 1/2 clears the LHC by 10 orders. Two further caveats: only the SCALING is computed, not the coefficient (the comparison is built to be insensitive to it, but a factor of 100 in the prefactor is excluded by nothing); and the δ-SECTOR'S STATIC NONLINEARITY IS NOT ANALYSED -- the η sector's static cubic dies, but the (∂2π)2 sector's need not, so a Vainshtein radius from THAT sector is unknown. That is now the sharpest gap in the construction, and it replaces the strong-coupling item in the limitations list. Flat space throughout. A second self-correction: a first cubic-order counter returned ZERO terms, which made a check pass VACUOUSLY on an empty list; it was found and replaced with explicit order bookkeeping. A NOTE ON WHICH THEORY THIS IS. The notorious λ → 1 strong coupling belongs to PROJECTABLE Horava gravity. The non-projectable 'healthy extension' carrying the aiai term is the known repair -- and this construction landed on it BY THEOREM rather than by choice, since the vorticity of a gradient-built n vanishes identically. NOTHING ELSE CHANGES IN v2. Every equation, result and caveat of v1 stands, including that a0's VALUE is NOT derived, that κ = 1/2 remains FITTED, that the theory GAINED two free parameters rather than fewer, and that NO claim is made regarding particle physics, the Standard Model or unification. One further verification script is added (24/24). Preprint. This deposit exists to timestamp a construction and to carry its reproducibility scripts; it is not a journal publication. It completes the worldline action of DOI 10.5281/zenodo.21845411 into a LOCAL, GENERALLY COVARIANT field theory with three healthy propagating modes. It does NOT derive the MOND acceleration scale, and it makes NO claim about particle physics, the Standard Model or unification. The theory. S = SEH[g] + Skh[g,T] + Sχ[g,u,χ] + Sm[g,T,χ,x], with the khronon sector N√h[KijKij − λK2 + ξR(3) + η aiai], ai = ∂i ln N, and nμ = −∂μT/√(−(∂T)2). 1. THE MEMORY KERNEL BECOMES A LOCAL FIELD. The general-orbit form Θ = ∫ds K(s) s |a(τ−s/2)|/c carries both a factor s and a lag s/2; the substitution s = 2u collapses both into one kernel G(u) = 4u K(2u), and for the minimal causal K = (N/λ)e−s/λ this is G = g u e−mu with m = 2/λ, g = 4N/λ. That IS the retarded Green's function of (d/dτ + m)2, so χ'' + 2mχ' + m2χ = g|a|/c: a damped oscillator whose FRICTION term is the memory. Verified in the transform domain, by the jump conditions, and closed NUMERICALLY to 1e-23 against the nonlocal convolution. The damping ratio is EXACTLY 1 (critical; zero discriminant, double root), and the order is FORCED: K ~ ske−s/λ gives order k+2, so the exponential kernel is the MINIMAL localisation. The extra factor of u traces to the rapidity gap being LINEAR in s -- the same fact that, in the companion no-go (DOI 10.5281/zenodo.21854464), puts the required kernel moment exactly on the pole of ζ. 2. AND a0 BECOMES A COUPLING RATIO: a0 = (2/3) c m2/g -- the ratio of the auxiliary field's MASS SQUARED to its COUPLING, rebuilt to 1e-25 on BOTH a0 footings, with the implied kernel weight N = 2.11e6 matching the independently derived bound of the earlier paper. Combined with the companion no-go (M1 is a logarithmically divergent moment), g/m2 is a RENORMALISED COUPLING -- which is what every coupling in every local field theory is. 3. THE GHOST QUESTION BECOMES A COMPUTATION, AND RESOLVES FAVOURABLY. The multiplier obeys the adjoint (d/dτ − m)2 -- anti-damped, the Bateman mirror -- and the (Θ,π) kinetic form is off-diagonal with signature (+,−): a ghost by the naive criterion. It is NOT a new degree of freedom: π is a COSTATE carrying a FINAL-value condition and no Cauchy data, it decays backward, and the count closes exactly at 2+2 = 4 = the order of the fourth-order equation of motion already derived. The indefinite metric is the Keldysh (−) branch of the in-in formalism already in use. This REPLACES the earlier 'Ostrogradsky is silent because it needs a local Lagrangian' hedge with a positive statement. 4. THE PREFERRED FRAME BECOMES DYNAMICAL, and the vorticity vanishes IDENTICALLY -- a theorem. For any gradient-built n, ∂[μnν] = A[μnν] with A = −∂ ln N, verified termwise, and the spatial projector annihilates n; the Christoffels drop out, so ω = 0 in EVERY metric. Consequence: this is the HYPERSURFACE-ORTHOGONAL (Horava) sub-case of Einstein-aether and the s