A Preferred Frame Is Forced: Static Weak-Field MOND with One Metric and Two Propagating Modes, and the Lensing Lock That Discharges the Locality Hypothesis

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Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-17
DOI
https://doi.org/10.5281/zenodo.22815997
Primary Topic
Pulsars and Gravitational Waves Research
Type
preprint
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preprint

A Preferred Frame Is Forced: Static Weak-Field MOND with One Metric and Two Propagating Modes, and the Lensing Lock That Discharges the Locality Hypothesis

Carl P. Zimmerman
Zenodo (CERN European Organization for Nuclear Research)
Pulsars and Gravitational Waves Research
preprint

A Preferred Frame Is Forced: Static Weak-Field MOND with One Metric and Two Propagating Modes, and the Lensing Lock That Discharges the Locality Hypothesis

Carl P. Zimmerman
preprint en

Abstract

A theorem: if a relativistic theory of gravity (i) reduces to Milgrom's modified Newtonian dynamics in the static weak field, (ii) carries a single physical metric to which matter couples minimally, and (iii) propagates exactly two gravitational degrees of freedom, then it contains a distinguished timelike direction — a preferred frame. Adding the elliptic boundary-value posing that Milgrom's equation actually uses, Frobenius makes that direction hypersurface-orthogonal: a preferred foliation, non-dynamical, with an instantaneous constraint. The proof is three steps, each with its own controls. (E) MOND's variable is not a local scalar of the metric: a uniform field leaves every curvature invariant exactly unchanged while y = |∇Φ|/a0 changes by 1488× at 1 pc from a star, and a constant deepening of the potential is an exact isometry, so no functional of the metric — local or nonlocal, at any order — can measure a depth. The lemma is proved against an adversarial control invariant that is exact on Schwarzschild to 3.6 × 10−15 and then breaks by 87% on a binary, 9× on a disc and 11× in a Plummer core, and is 0/0 in a uniform field; physically, local curvature at the Sun's radius is nearest-star dominated over 99% of the volume and misses the Galaxy by 1487×. (Q) A symmetric form annihilating all timelike directions is identically zero (rank-10 solve); for one distinguished direction the null space is nine-dimensional and is exactly the spatial projector, so MOND's squared gradient is a projector contraction. (T) The principal symbol of the required extra structure is hyperbolic (it propagates, N ≥ 3), degenerate (it needs the distinguished direction), or absent (local, so E applies; or matter-built, which fails in the vacuum where rotation curves are flat and lensing is measured); there is no fourth case for finite-order field equations. That finite-order proviso is the locality hypothesis, and it is discharged against the known nonlocal class by a mechanism that never uses locality: with no preferred timelike vector there is exactly one transverse symmetric operator, so every covariant scalar linear in the metric perturbation is a function of the d'Alembertian acting on the linearised Ricci scalar — the inverse d'Alembertian included, so nonlocality buys nothing — and therefore, for any frame-free covariant addition, with any weight and any coupling, ∇2(Φ + Ψ) = 8πGρ exactly: the lensing potential is welded to the baryons, and the rotation-curve and lensing equations cannot be modified separately. Adjoining one unit timelike vector makes the count two and unlocks them. This reproduces Soussa and Woodard's frame-free nonlocal result ("far too little lensing") and an independent proof that null geodesics are conformally invariant. The escape was nonetheless real, and that is recorded: the mode count of retarded-nonlocal gravity is 2, not 4 — the localised counting rule overcounts even general relativity, whose conformal mode carries a wrong-sign kinetic term — so the nonlocal challenger satisfies (i), (ii) and (iii) at once. The theorem survives because the only nonlocal theory reproducing both MOND and lensing carries a unit timelike vector built as a normalised gradient, hence hypersurface-orthogonal, whose own authors describe its consequence as preferred-frame effects. A seventeen-theory control table (general relativity, RAQUAL, phase-coupling gravity, TeVeS, Einstein-aether MOND, AeST, khronometric, Hořava, BIMOND, MOG, Horndeski/DHOST, dRGT, modified inertia, the Deser-Woodard nonlocal class, and this programme's own three constructions) contains no counterexample; the pattern is an exclusive OR — Lorentz invariance XOR two modes — reported as an empirical observation over an enumeration, not as a proof. Mode-counting controls return 2, 3, 3 and 5 for general relativity, general relativity + scalar, khronometric theory and Einstein-aether. A constructive companion result closes the complementary question on four independent counts: no foliation-independent scalar can replace the slice-built field. Because a preferred frame is not invisible, the theorem's constraint on the space of theories is turned into quantitative predictions with falsifiers, separated throughout into what is forced on any member of the class and what merely illustrates it in this programme's own action. The class is measured by the preferred-frame post-Newtonian sector — |α1| < 10−4, |α2| < 4 × 10−7, |α3| < 4 × 10−20 — and no member can set all three to zero by a symmetry it does not have. The corollary's instantaneous constraint is priced: where it has been computed, α3 = −1 as posed and −3 after the slip repair, i.e. 2.5–7.5 × 1019 over the pulsar bound, so the class faces a dichotomy — pay α3, or let the frame propagate — and this programme's own action takes the second horn with α3 = 0 and four modes. Read forwards, the lensing lock is a positive prediction with a size: every frame-free MOND theory lenses from the baryons alone, and the measured shortfall is 6.6× at cluster R500 and 46× (1.66 dex) at the deepest galaxy-lensing point, both ratios free of a0 and identical on the two footings. The extra structure has a measurable speed: |cT/c − 1| ≲ 10−15, satisfied structurally when the two aether couplings cancel, while gravitational Cherenkov bounds only slow modes (1 − cs ≤ 2 × 10−15 through a T00 vertex, 1.4 × 10−9 through a conformal one) and is switched off entirely by superluminality. And one fresh, sharply falsifiable prediction is stated with its price: the only live mechanism for the cluster shear shape — available only to a theory with a foliation, since a spatial smoothing needs a slicing — predicts a lensing-versus-dynamical mass slip of 0.02–2.3% for galaxies inside clusters against essentially none for the same galaxies in the field, decidable at 0.1% precision and measured nowhere near it today; its length is fitted, and at its ceiling it halves rather than cures the 9σ failure it was built for. Each prediction is stated as an observation and a threshold, and every dimensional number is carried on both a0 footings. What is NOT proved is stated explicitly. This is a statement about theories satisfying the three hypotheses and nothing more: it does not close MOND-like theories in general — three or more propagating modes, two metrics, or a non-minimal matter coupling are all untouched. The locality hypothesis is discharged against the known nonlocal class rather than in general, and the residual computation is named (a nonlocal curvature scalar, quadratic or higher, that separates the Newtonian from the lensing potential while remaining sensitive to the coherent coarse-grained field rather than to the nearest star). The strict conclusion is a frame; the upgrade to a foliation is an added hypothesis. Nothing here says whether nature is MONDian — hypothesis (i) is an assumption about a theory, not a claim about data, and no observation is used anywhere to test the theorem; the measured quantities in the predictions section run only in the other direction, saying how large an effect a member of the class would have to show and what would falsify it. Nothing here says whether a preferred-frame MOND theory is viable; the α3 price is re-derived for one chassis and the universal form of it remains a conjectured obstruction, not a theorem, while no post-Newtonian computation exists for the nonlocal class at all. Every number is produced by a committed script with checks that can fail (fable_independent_2026/L31_foliation_nogo.py, L39_nonlocal_modes.py, L27_foliation_scalar.py, and, for the predictions, L13_strong_coupling.py, L24_lensing_vs_dynamics.py, L19_cherenkov_applicability.py and L57_nonlocal_functional.py and their supporting lanes in the public repository). AI-assisted research draft; not peer reviewed.

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