A local, generally covariant field theory of modified inertia

Abstract

VERSION 6: the paper now carries its own most serious limitation, and it is a NEGATIVE result about the construction's STATUS. TAKEN AS A FUNDAMENTAL THEORY, THIS CONSTRUCTION PREDICTS THAT GRAVITATIONAL LENSING TRACES THE BARYONIC MASS ALONE -- AND THAT IS EXCLUDED AT ~21 SIGMA. The argument follows from the construction's OWN premise rather than from any observer's calibration: (i) the metric is unmodified and baryon-sourced, which is not an assumption imposed here but the premise section 6 VERIFIES to 1e-7, because modified inertia NEEDS grad^2 Phi = 4 pi G rho_bar; (ii) the worldline action is exactly proportional to m and vanishes identically at m = 0, so photons follow null geodesics of the baryonic metric; (iii) hence M_lens = M_bar while the anomalous dynamics read Newtonianly give M_dyn = M_bar/f_bar, so M_dyn/M_lens = 1/f_bar = 6.4 in clusters; (iv) observed: 1.0-1.3. On a deliberately generous 25% systematic that is excluded at 21 sigma, and still at 4.2 sigma if the systematic is inflated FIVEFOLD. Modified gravity predicts exactly 1, so the data do not merely disfavour this reading -- they land on the alternative. And it is a RATIO, so much of the mass calibration cancels and the lensing side assumes no equilibrium. AND THE ENHANCEMENT CANNOT SIT IN BOTH PLACES. Modified gravity alone gives a = ν gbar; modified inertia alone gives a = ν gbar; doing BOTH gives a = ν2 gbar, wrong by exactly one factor of ν. So the enhancement lives in the metric OR in the inertia, never both. Modified inertia and modified gravity are ALGEBRAICALLY IDENTICAL for massive test particles -- which is precisely why this framework could remain ambiguous about which it was -- and lensing is the observation that separates them. It selects the metric. THE CONSEQUENCE, STATED AS A DEMOTION. The lensing-viable arm of this framework is a modified-GRAVITY realisation of the same kernel (the AeST/TeVeS class), so sections 2-6 are at best an effective description of test-particle dynamics INSIDE a modified-gravity theory, not the fundamental theory. The abstract-level caption now says so, the limitations list is reordered to put this first, and 'what a referee should attack' now opens with it. It also exposes an inconsistency PREDATING this paper: using modified gravity for lensing and modified inertia for dynamics is not two descriptions of one theory -- by the table above they are two THEORIES, and the pair is inconsistent. A consistency problem INSIDE the framework, not a new observational tension. AND THREE PREDICTIONS ARE NOW IN NEED OF RE-DERIVATION, INCLUDING THE ONE THE PRE-REGISTERED TEST RESTS ON: the external-field anisotropy (pure modified inertia predicts EXACTLY ZERO), the g-2 Lorentz-violation scaling, and the wide-binary γv target of the frozen Gaia DR4 pre-registration. If modified inertia is only an effective limit, these are predictions OF THE LIMIT and must be re-derived in the modified-gravity theory rather than inherited. Section 9 should be read with that caveat attached. EVERY ESCAPE WAS COMPUTED, AND ONE CLOSES LESS CLEANLY THAN A FIRST DRAFT CLAIMED. The memory field is a costate and the μ-independent part of the matter stress-energy is still proportional to m, i.e. baryonic. A photon coupling cannot be bolted on -- the preferred-frame coupling also enters multiplied by m and vanishes at m = 0, so giving light its own effective metric IS the disformal/TeVeS construction. A non-baryonic component works but must be 84% of the cluster mass, at which point it lenses AND supplies the dynamics and MOND is not needed in clusters at all: a replacement, not a rescue. And the khronon is NOT intrinsically negligible -- that claim is corrected here. At η = 1 its effective density would be 159x the cluster gas, and η ~ 0.034 would exactly supply the missing mass, so that escape is closed ENTIRELY by the preferred-frame PPN bound (a factor 3.4e5) and by nothing else. If that bound moved, the escape reopens. WHAT MAY BE SALVAGEABLE, and it is NOT shown: in AeST/TeVeS-type theories the test-particle sector can carry an effective acceleration-dependent inertia, so the rapidity-gap construction may be recoverable as the POINT-PARTICLE LIMIT of the modified-gravity theory rather than as a rival to it. That would preserve the parity theorem, the localisation and a0 = (2/3)c m2/g while giving up the claim that inertia rather than gravity is what is modified. It is the next calculation. NOTHING IN SECTIONS 2-7 IS RETRACTED. Every equation and every check stands; what changes is the claimed STATUS of the construction. Two verification scripts are added (24/24 and 27/27), bringing the total to ten. a0's VALUE is still NOT derived and κ = 1/2 remains FITTED. 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 S