A local, generally covariant field theory of modified inertia

Abstract

VERSION 5: the paper now states its own FALSIFICATION TEST, with the power computed BEFORE the data -- and the honest headline is that the test is ASYMMETRIC. SAID FIRST, because it bounds everything that follows: the wide-binary prediction is NOT derived from this paper's steps 1-3. It follows from the external-field effect applied to the framework's interpolation kernel, both of which predate this construction. What sections 2-6 supply is the THEORY BEHIND the number -- a local, covariant, ghost-free action in which it is the prediction of something rather than of a rule. The number itself does not move. THE REGISTERED PREDICTION. γv = 1.1582, range 1.1311-1.1964 (radial) / 1.1339-1.2007 (magnitude), over both a0 footings and both external-field conventions, on the exponential kernel. This is the author's frozen, hash-stamped Gaia DR4 pre-registration as amended; nothing in this paper alters it. THE ONE DECISIVE STATEMENT, at the registered N = 30,000: a NEWTONIAN result across 2-30 kAU is evidence AGAINST at 4.74-7.10 sigma_tot. That figure is clean for a specific reason -- under a Newtonian truth the estimator's shape bias is identically zero -- so it survives every declared systematic. It is the number the theory should be judged on, and no re-hedging is permitted afterwards. THREE REGISTERED LIMITS ON THE TEST, all stated in advance. (1) The scoring bins collide with the prediction: 1.1582 falls in the interval pre-declared 'MG-side; MI disfavored', and the frozen modified-gravity target 1.137 lies INSIDE the MI range (0.77 sigma_tot from the point value), so a scorer executing the frozen table on a measurement AT the framework's own prediction would record the framework as DISFAVORED. The binding remedy -- report raw gamma_hat with sigma_fit and BOTH distances, never a single verdict word -- is implemented. (2) A nuisance parameter sits OUTSIDE its frozen window: 1.0575-1.0959 against [0.95, 1.05], whose pre-declared consequence is 'systematic-limited, no verdict -- reported, not repaired'. The exponential kernel fails this on both variance treatments where the retired power-law kernel passed both; it is a cost of the kernel change and is NOT repaired. (3) One corner is pre-declared UNSCOREABLE: on the magnitude convention the alternative-footing corner is 1.20069, i.e. 0.00069 ABOVE the >1.20 no-verdict edge, so a genuine detection there cannot be scored. Quoting only the radial convention would hide this at its convenient end. TWO FURTHER CHECKS, BOTH UNDERPOWERED, and that is stated rather than discovered afterwards. The anisotropy falsifier has its sign pre-declared -- perpendicular pairs must show the LARGER boost, and the opposite sense at ≥3σ falsifies the derived external-field effect INDEPENDENTLY of the aggregate -- and the 3-D eigenvalue spread is 0.176-0.199, strengthened by the kernel change. But only the sky-PROJECTED angle is observable, and averaging γ(ψ) = γ⊥ + (γ‖ − γ⊥)cos2ψ over isotropic 3-D orientations gives a projection dilution D = 0.2367 by Monte Carlo over the registered |b| > 15° sky, against the in-plane CLOSED FORM 4/(3π) = 0.4244. So the observable split is 0.042-0.047, reaching only 1.00-1.13σ at N = 30,000 and needing N ~ 2.1-2.7×105 for 3σ -- seven to nine times the registered sample. The gated branch, which must be scored alongside the ungated one, is only 0.03-0.06σ from Newton in the aggregate and would need N ~ 8.6×107 to separate: an aggregate-only scorer would report 'Newtonian' for a universe in which the gated branch is TRUE. Its only handle is an internal rise across the window, 0.001σ at 2 kAU to 1.56σ at 30 kAU -- falsifiable in SHAPE, but 1.56 is not 3. A watch item. What the two BRANCHES separate by is large: the ungated signal is 135x the gated amplitude. AND THE FORK OF SECTION 7 HAS ITS OWN TEST. Whether Θ is sourced by the particle's |a| (pure modified inertia) or by the khronon's |a[n]| = gbar is decided by a DIRECTIONAL external-field measurement, in which pure modified inertia predicts EXACTLY ZERO aligned asymmetry while an external-field-driven Θ does not. A separate front, not settled here. THE HONEST SUMMARY, which is in the paper and not only here: one number with teeth, three with context, one unscoreable corner, and one nuisance outside its window. That is a WEAKER test than the registration originally expected -- its section 1.5 anticipated a decidable outcome -- and the honest position is that DR4 can DISFAVOR this framework decisively and can SUPPORT it only weakly. Two verification scripts are added (31/31 and 20/20), bringing the total to eight. Sections 8-10 are renumbered 9-11. Nothing else changes: a0's VALUE is still NOT derived, κ = 1/2 remains FITTED, the theory still GAINED two free parameters rather than fewer, and NO claim is made regarding particle physics, the Standard Model or unification. VERSION 4: a FACTUAL CORRECTION, found by auditing this paper against the author's own frozen Gaia DR4 pre-registration, and made rather than left standing. Versions 1-3 stated that μ's shape is "the α = 2 interpolation that solar-system ephemerides force". THAT IS FALSE. α = 2 misses the Mars ranging budget by 8.5-12.4x, because its 1/g tail binds at the SUN via the Jupiter reflex (~2233 a0) rather than at a planet; α = 1 misses the Earth/Mars bound by 1279x. BOTH power-law kernels fail, and the framework's in-force kernel is the EXPONENTIAL one, ν = 1/(1 − e−√y), adopted in Amendment 8 of that registration on 2026-08-03 -- five days before v1 of this paper was deposited. Section 9 now says so. NOTHING STRUCTURAL MOVES, and the reason is worth stating precisely. The localisation of section 2 concerns the MEMORY kernel K(s), not the interpolation μ(Y), so a0 = (2/3) c m2/g is untouched and carries no reference to μ's shape. And section 6 used only the two limits -- ν → 1/√y deep and ν → 1 Newtonian -- both of which the exponential kernel satisfies, so v4 = G M a0 still follows. Section 6 now names the kernel used in the demonstration and records that the substitution is harmless. This was a WORDING-AND-CITATION defect carried in three published versions, not a structural one -- but it was a wrong statement about which kernel the framework holds, and it is corrected here rather than quietly re-scoped. Also recorded for the reader's orientation: the same exponential kernel has an INDEPENDENT Bekenstein-Milgrom field theory whose convexity, ellipticity, ghost-freedom, subluminality, positive phantom density and exact BTFR are proved (11/11 checks). That construction is modified GRAVITY; THIS paper's construction is modified INERTIA. They are different theories sharing one interpolation function, which is exactly the fork section 7 exposes -- and neither derives a0. No other change in v4. a0's VALUE is still NOT derived, κ = 1/2 remains FITTED, the theory still GAINED two free parameters rather than fewer, and NO claim is made regarding particle physics, the Standard Model or unification. No script changes; the six verification scripts are unchanged. 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 paramete