The modified-gravity arm: a Bekenstein-Milgrom field theory for a_0 = kappa c sqrt(G rho_Lambda)

Abstract

This paper switches the framework's operative arm from modified INERTIA to modified GRAVITY, and says why in its first section rather than its last. WHY THE SWITCH. In modified inertia the metric is unmodified and baryon-sourced -- not incidentally but by requirement, since the whole content is a modified RESPONSE to a Newtonian field -- and photons have no rest mass to modify. So modified inertia predicts M_lens = M_bar, hence M_dyn/M_lens = 1/f_bar ~ 6.4 in clusters against an observed 1.0-1.3: excluded at ~21 sigma, and still 4.2 sigma on a fivefold-inflated systematic. All three rescue routes were computed and closed: the enhancement cannot sit in BOTH the metric and the inertia (doing both gives a = nu^2 g_bar, wrong by one factor of nu); the worldline action DOES have Randers-Finsler form when mu depends on position, but the resulting null cone DEGENERATES at mu = 1/2 and the signature turns EUCLIDEAN below it, unavoidably, since every interpolation reaching the deep-MOND limit must cross mu = 1/2 -- on this framework's kernel at g_bar = 0.48 a0, the outer disc of a spiral galaxy; and Bekenstein-Milgrom's own point-particle limit is MEMORYLESS, so the modified-inertia action is not that limit either. 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. WHAT SURVIVES IS THE PART THAT MATTERED: a0 = kappa c sqrt(G rhoLambda) = cHLambda/Z = 9.36e-11 m/s^2 is a statement about the COEFFICIENT and is ARM-INDEPENDENT. Lambda-CDM has no reason to predict a relation between the dark-energy scale and the dark-matter-phenomenology scale; this framework has no freedom to avoid one. That asymmetry is the principal asset and it is NOT evidence the framework is correct. THE THEORY. S = -Int d^3x [(a02/8 pi G) F(X) + rho phi] with X = |grad phi|2/a02 and field equation div[mu(x) grad phi] = 4 pi G rho. The free function is NOT postulated. For the framework's in-force kernel nu = 1/(1 - e-sqrt y) -- adopted because BOTH power-law kernels fail solar-system ranging (alpha = 1 by 1279x the Earth/Mars bound; alpha = 2 by 8.5-12.4x the Mars budget, its 1/g tail binding at the SUN via the Jupiter reflex, ~2233 a0) -- the substitution u = sqrt(y) gives the whole theory in one parametric pair, mu = 1 - e-u, x = u2/mu, a proved BIJECTION whose inversion round-trips. The derivative is exact at every acceleration, F'(X) = mu = 1 - e-sqrt y, with limits F -> (2/3)X3/2 deep and F -> X - C0 + 4e-s(s3+3s2+6s+6), s = X1/4, Newtonian -- where C0 is a pure Lagrangian constant and therefore UNOBSERVABLE, since only F' enters the field equation. A trap is flagged because the source script caught itself in it: X = x2 and x = y/mu ~ u2 while u = sqrt(y), so the Newtonian residual is exponential in X1/4 = sqrt(y), NOT in sqrt(X). The field-theoretic point of the construction is that the ephemeris relief is a property of the LAGRANGIAN, not of an algebraic force law bolted on afterwards. SEVEN PROPERTIES PROVED RATHER THAN ASSUMED: convexity of F -- from which existence, uniqueness, Newton's third law, standard centre-of-mass motion and the EXACT baryonic Tully-Fisher relation all follow -- plus ellipticity (both eigenvalues), ghost-freedom, subluminality (the scalar sound speed), positive phantom density, the BTFR coefficient obtained FROM THE ACTION, and a Newtonian residual exponentially small in the action, evaluated at the Sun. The health is a CONSEQUENCE of one proved property rather than a list of separately-checked conditions, which is the structural reason this arm is on firmer ground than the one it replaces. OBSERVATIONAL STANDING, both ways. Galaxies: the RAR is reproduced at 0.108 dex (Upsilon = 0.70), better than regularised MOND's 0.122 -- but the penalty for moving a0 by 20% is under HALF A PER CENT, so the RAR is compatible and NON-DIAGNOSTIC of the coefficient, and two of the author's own claims about it were withdrawn for that reason. The BTFR intercept is mass-to-light degenerate; the Upsilon-free gas-dominated box [0.84, 1.36]e-10 contains both 9.36e-11 and 1.2e-10 and does not discriminate. Lensing: works, and that is the entire reason for the paper -- though the AMPLITUDE is a prediction of the relativistic completion, not of the non-relativistic theory written here. Clusters: UNCHANGED by the switch, and still the sharpest problem. eta = 2.334 (0.368 dex) at 2.0-4.1 sigma; a full measurement audit finds the observations largely hold up -- there IS a real 14 +/- 2% Chandra-vs-XMM mass error and it HELPS, but 14% against a needed 133% -- and the better-established systematic points the OTHER way. The wall is the COSMIC BARYON BUDGET: f_gas(r500) = 0.163 +/- 0.032 against Omega_b/Omega_m = 0.167 +/- 0.006, so headroom is at most 1.22x against a required 2.33x, and every knob turned favourably at once leaves a residual factor 1.68. If you are looking for the observation most likely to kill this framework, it is clusters. Wide binaries: the author's frozen pre-registration carries the MODIFIED-INERTIA target, and this arm predicts a different one -- gamma_v = sqrt(nu(y_ext)) = 1.2139 canonical / 1.2592 alt against the MI 1.1582 (range 1.1311-1.1964), DISJOINT ranges; moving the registered target is amendment-grade and is NOT done in this paper. It cuts both ways: the 2.68 sigma separation means DR4 can now DISTINGUISH the arms. Directional external-field test: the strongest thing the switch buys, and it required no new observation -- pure modified inertia predicted EXACTLY ZERO aligned asymmetry while AQUAL-class theories predict 1-4% with a definite sign, and a first firing gave Ahat = +2.95 at p = 0.029 WITH THE AQUAL-CLASS SIGN. A measurement that was evidence AGAINST the previous arm is evidence FOR this one. At p = 0.029 it is a lever, not a confirmation; N ~ 1157 galaxies would settle it. WHAT THE SWITCH COST, stated first-class because two of these were the headline results of the arm being abandoned: a0 = (2/3)c m2/g is GONE (that reading was the modified-inertia kernel's first moment; a BM theory has no kernel); and with it the zeta-pole no-go, a theorem about that moment, whose conclusion that a0 is a renormalisation condition may still hold but must be RE-DERIVED and is not inherited; the g-2 Lorentz-violation prediction is GONE in pure Bekenstein-Milgrom, which has no preferred frame at all -- it survives ONLY if the completion is AeST-type, so which completion is adopted DECIDES whether the framework keeps it; and the Cassini quadrupole tension is INHERITED (3-15 sigma RAR-versus-Q2), which modified inertia did not carry -- the item a referee will find first. The parity theorem remains TRUE but is no longer load-bearing. WHAT IS OWED: a relativistic completion (the theory as written is NON-RELATIVISTIC, so 'it lenses' is a statement about the sign and mechanism, not the AMPLITUDE); the wide-binary amendment; clusters; kappa = 1/2 itself (sixteen attempts to force it have failed, and the residue after tracing sqrt(6) to Friedmann and sqrt(pi) to the 3-d momentum measure is a single bare factor of two); and whether the g-2 prediction survives, which follows from the completion. WHAT IS NOT CLAIMED. Bekenstein-Milgrom is NOT new -- it is theirs, 1984 -- and the interpolating function is Milgrom's, 1999. What is offered is the COEFFICIENT, the kernel forced by ranging, and the closed-form free function with its health proved for that kernel. kappa = 1/2 is FITTED, not derived. And no claim is made regarding particle physics, the Standard Model or unification -- the author publicly withdrew earlier statements of that kind in June 2026 and does not restate them; the obstruction is specific, in that the framework's characteristic constant carries sqrt(pi) while all flavour observables are algebraic, so the structure is BLIND to the gauge sector. This is modified gravity, and it does not derive a0. 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. Claims found to be incorrect were withdrawn rather than quietly re-scoped; the modified-inertia arm's demotion is the largest of them and is recorded in the paper's first and sixth sections. Reproducibility. Seven scripts are included (11/11, 6/6, 18/18, 24/24, 23/23, 27/27 and 26/26 checks), each exiting non-zero on any internal failure and each carrying negative controls that must trip. Both a0 footings are carried on every dimensionful number.