A local, generally covariant field theory of modified inertia

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 n^μ = (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 K_ij = 0 identically, so both K.K and K² vanish at EVERY order and only η a_iaⁱ 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 a_i = −∂_iπdot and K_ij = −∂_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 ~ M_Pl²[−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 c_s² = (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 Λ_sc ~ √η M_Pl/c_s ≈ 7.7e14 GeV at η = 1e-7: the PPN-preferred corner does lower the cutoff, but only as η^1/2, and from the Planck mass.

AND THE CONCLUSION IS ROBUST TO THE POWER. Scanning Λ_sc = η^pM_Pl over p = 1/2, 1, 2, 3, 4 -- deliberately allowing powers far worse than the derived one -- Λ_sc 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 a_iaⁱ 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 = S_EH[g] + S_kh[g,T] + S_χ[g,u,χ] + S_m[g,T,χ,x], with the khronon sector N√h[K_ijK^ij − λK² + ξR⁽³⁾ + η a_iaⁱ], a_i = ∂_i ln N, and n_μ = −∂_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 ~ s^ke^−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, ∂_[μn_ν] = A_[μn_ν] 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 spin-1 sector is removed entirely; the four aether couplings collapse to THREE, (c₁+c₃)K.K + (c₄−c₁)a.a + c₂(tr K)², Jacobian rank 3. Nothing is lost: in the gauge |∂T| = 1 with T = t the coupling √((u.n)²) equals γ EXACTLY, so the earlier action is the UNITARY GAUGE of this one. Still CPT-even.

5. AND THE KHRONON'S OWN ACCELERATION IS THE NEWTONIAN FIELD: a^μ[n] = ∂^μΦ at linear order in a static weak field, computed from the Christoffels and with three prespecified decoys rejected. Before covariantisation the only acceleration in the theory was the particle's own |a|; MOND relates g_obs to g_bar, and g_bar now exists as a GEOMETRIC OBJECT.

6. THE SPIN-0 MODE IS HEALTHY, in an explicit nonempty window. The quadratic scalar action is DERIVED from the ADM pieces. The GR check passes first: at η = 0 the α constraint degenerates to ζ = 0, so general relativity has no propagating scalar -- reproduced, not assumed. Then c_s² = ξ(2ξ−η)(1−λ)/[η(1−3λ)] = (2−η)(λ−1)/[η(3λ−1)] at ξ = 1, with the tensor sector giving graviton speed² = ξ and a POSITIVE kinetic term -- so GW170817 makes ξ = 1 a MEASUREMENT to ~1e-15, not a choice. NO GHOST ⇔ λ > 1 or λ < 1/3; NO GRADIENT INSTABILITY ⇔ 0 < η < 2 -- and the two conditions COINCIDE, both excluding exactly the band 1/3 < λ < 1. In the PPN-safe corner c_s² → (λ−1)/η EXACTLY, so only the RATIO survives and the vacuum Cherenkov bound reduces to the single inequality λ−1 ≥ η > 0; superluminal is SAFE because there is a preferred frame. PPN and Cherenkov make INDEPENDENT demands: no conflict.

7. AND MOND FALLS OUT OF THE JOINT FIELD EQUATIONS, non-circularly. Modified inertia PRESUPPOSES an unmodified gravitational field, and the same PPN bound that made the khronon healthy is what guarantees it: khronon corrections to Φ are O(η, λ−1) ≤ 1e-7 while MOND needs Φ_bar only to ~1% -- FIVE ORDERS of margin. With the Newtonian metric established independently, the worldline equation gives μ(g_obs/a₀) g_obs = g_bar, hence g_obs = √(a₀g_bar) and v⁴ = G M a₀ exactly -- flat rotation curves and the baryonic Tully-Fisher relation -- with NO new parameter. Propagating content 2 (graviton) + 1 (khronon) + 0 (χ) = 3.

8. THE MASS QUESTION, CONFRONTED. Rest energy mc² (μ-independent), kinetic mμv²/2, so m_grav = m and m_inert = mμ, AND THEY DIFFER -- not a defect to explain away, it IS the definitional content of modified inertia. The WEAK EQUIVALENCE PRINCIPLE SURVIVES: m cancels, giving μa = g_bar with no reference to mass or composition, so universality of free fall is exact; what is violated is the EQUALITY m_i = m_g. And a galaxy's GRAVITATING mass is its BARYONIC mass.

9. NEW: A FORK THE COVARIANTISATION EXPOSES. Since a^μ[n] = ∂^μΦ, the theory possesses TWO acceleration scalars, and Θ may be sourced by either: Θ[|a|] is pure modified inertia, while Θ[|a[n]|] is a THIRD theory, external-field-driven, neither pure MI nor AQUAL, sharing the same interpolation function and the same a₀. The worldline formulation could not even express this fork. They are distinguishable by a directional external-field test, where pure modified inertia predicts EXACTLY ZERO aligned rotation-curve asymmetry. Exposed here, NOT resolved here.

AGAINST INTEREST, in the paper and in this record rather than buried.

• a₀'s VALUE is still not derived and κ = 1/2 remains FITTED. The companion no-go shows WHY a free-field correlator cannot supply it.
• The theory GAINED two free parameters (λ, η), not fewer. General covariance and ghost-freedom were bought with them, and nothing in modified inertia forces either -- so 'the scalar is healthy' means 'a viable region exists', much weaker than a prediction.
• The η → 0 corner that PPN prefers is the known STRONG-COUPLING corner, and that scale is NOT computed here. It is the sharpest remaining gap.
• Ostrogradsky is NOT evaded in the LOCALISED writing (the source is |a|, so the system is second-derivative in x); the exact action contains only u, which is why the two statements coexisted rather than contradicting.
• Exact localisation of the FULL bilocal cosh^-1(−u.u'/c²) is NOT available by this route -- the auxiliary-propagator trick needs linearity in the delayed field.
• The localisation is exact for the MIDPOINT form only; the relative error against the exact bilocal is O((λΩ)²) = 7.7e-7 galactically.
• BTFR normalisation BOTH WAYS: A_fw = 1/(Ga₀) = 80.5 vs McGaugh's fitted 47 M_⊙(km/s)^-4, a 0.234 dex mass offset; the ALT footing fits BETTER (1.42 vs 1.71); but A = 47 was set at Υ_3.6 = 0.5 so the framework needs 0.856, high yet inside the spread and near the 0.70 an independent RAR fit prefers, and the Υ-FREE gas-dominated a₀-line box [0.84, 1.36]e-10 CONTAINS 9.3619e-11. A mass-to-light question, not an a₀ question: neither a win nor a deficit.
• A first draft called the single-stream restriction the sharpest gap. That was wrong and is corrected in the paper rather than absorbed: Θ integrates over the particle's OWN past, so χ is a per-worldline INTERNAL variable and two stars crossing at a point simply carry different χ; and the metric never sees χ because m_grav = m is μ-independent, the first χ-dependent source being suppressed by (v/c)² = 5.4e-7. The restriction binds only a continuum rewriting undertaken for its own sake.
• Strong fields, black-hole universal horizons and nonlinear stability: not addressed. The spin-0 analysis is flat-space, quadratic order, scalar sector.
• Energy-momentum conservation for a time-nonlocal modified-inertia theory is CITED (Milgrom 1994), not proved here.
• NO claim regarding particle physics, the Standard Model, or unification. The author publicly withdrew earlier statements of that kind and does not restate them. A complete field theory of modified inertia is NOT a theory of everything.
• ν = √(1+1/y) is IDENTICAL to Eq. (9) of Milgrom, Phys. Lett. A 253, 273 (1999). The khronon sector is standard technology; what is new here is its use to complete THIS construction, the acceleration theorem, and the fork.

CREDIT. Khronon / hypersurface-orthogonal aether and the infrared limit of non-projectable Horava gravity: Jacobson 2010 PRD 81:101502; Blas, Pujolas & Sibiryakov 2010 PRL 104:181302 and 2011 JHEP 1104:018; Jacobson & Mattingly 2001 PRD 64:024028; Horava 2009 PRD 79:084008. Second-order relaxation: Israel & Stewart 1979 Ann.Phys. 118:341. Time-nonlocality of modified inertia and its conservation laws: Milgrom 1994 Ann.Phys. 229:384. Vacuum Cherenkov: Elliott, Moore & Stoica 2005 JHEP 0508:066. PPN: Will. Graviton speed: LIGO/Virgo GW170817. BTFR: McGaugh 2012 AJ 143:40. Auxiliary-field localisation of nonlocal actions, the Bateman dual and costate boundary conditions are classical.

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 -- among them a past-directed sign for n_μ, an assumed rather than derived reparametrisation invariance, an asserted BTFR band replaced by the computed value, and the overstated single-stream limitation.

Reproducibility. Every quantitative claim is produced by a committed script that exits non-zero if any internal consistency check fails, and each carries negative controls that must trip. The four scripts are included (27/27, 26/26, 30/30 and 32/32 checks). Both a₀ footings are carried on every dimensionful number.