Collective Self-Gravity of Rotating Stellar Disks: A Mean-Field Kinetic Framework for Emergent Gravitational Acceleration in Galactic Rotation Curves

We ask a deliberately more fundamental question than is usually posed in the dark-matter-alternatives literature: can the collective, statistically correlated motion of the billions of stars in a rotating disk generate an emergent, non-additive gravitational effect that is not reducible to the sum of the individual stellar contributions? Flat galactic rotation curves are conventionally explained either by an unseen dark-matter halo or by a modification of dynamics at low accelerations (MOND); both treat the stellar disk as a passive tracer of an externally imposed potential. Here we instead build, from first principles of collisionless stellar dynamics, a Collective Self-Gravity (CSG) framework: starting from the coarse-grained stellar Vlasov–Poisson system, and using the well-established fact that a rotating, self-gravitating disk’s linear response to a perturbation is not bare but dressed by its own self-gravity — diverging in proportion to the disk’s proximity to marginal Toomre stability, as quantified by the swing-amplification theory of Goldreich & Lynden-Bell (1965) and Julian & Toomre (1966) and by the response-matrix (“dielectric”) formalism of Kalnajs (1971) — we construct a mean-field, coarse-grained description of the disk’s effective gravitational acceleration. We are explicit, at every step, about what is exactly derived, what is a standard approximation, what is a new physical assumption, and what is a free modeling choice among several defensible alternatives. The central, original hypothesis of this paper — clearly labeled as a hypothesis, not a theorem — is that this well-established dressing of the disk’s dynamical response, which is rigorously known to amplify transient non-axisymmetric perturbations, also leaves a residual, axisymmetric, time-averaged trace in the mean radial force budget once the disk’s local Toomre parameter Q(r) approaches or falls below unity. We show that the Toomre–Coherence Emergent Acceleration (TCEA) relation of earlier work, \( g_{\rm coh}(r) = \sqrt{g_N(r)\, a^\dagger}\bigl[1 - e^{-1/(\lambda Q(r))}\bigr] \), is recovered within this broader framework not as a postulated starting point but as the simplest member of a family of admissible saturating closures for a general collective coherence order parameter \( \mathcal{R}(Q) \in [0,1] \), singled out by (i) the disk’s known empirical self-regulation toward \( Q \sim 1 \)–2 (Sellwood 2014), which rules out the divergent, unsaturated closures naively suggested by linear swing-amplification theory, and (ii) the requirement that the \( Q \to 0 \) saturation limit reproduce the empirically established deep-acceleration asymptote of the radial acceleration relation. We construct a fully reproducible Python implementation (fixed seed, unit-tested, PEP8-compliant) and confront the resulting TCEA observable, together with three literature-standard alternatives (simple-interpolating-function MOND, the Navarro–Frenk–White halo, and the Burkert cored halo), with an illustrative representative rotation curve for NGC 3198 anchored to the qualitative and quantitative landmark features reported by Begeman (1989). On this single-galaxy benchmark, TCEA does not outperform the purely phenomenological alternatives (\( \chi^2_{\rm TCEA} = 198.77 \) vs. \( \chi^2_{\rm MOND} = 84.90 \), \( \chi^2_{\rm NFW} = 16.60 \), \( \chi^2_{\rm Burkert} = 11.28 \), all with \( k = 1 \)–2 free parameters over \( n = 20 \) radial bins), a result we report and analyze transparently rather than obscure. We show, through multi-parameter sensitivity and bootstrap uncertainty analysis, that this underperformance stems from a genuine, falsifiable structural prediction of the model — namely, that the coherence-correction term saturates once \( Q(r) \lesssim 1 \) across the accessible optical disk of a normal spiral, rendering \( \lambda \) observationally degenerate for single, dynamically “ordinary” galaxies and requiring a sample spanning a wide range of Toomre-Q regimes (dwarf irregulars, low-surface-brightness disks, bulge-dominated early types) to be discriminating. We present this outcome honestly as a motivating result for a specific, falsifiable, multi-galaxy observational program rather than as a completed empirical vindication, we identify explicitly which part of the CSG hypothesis chain is not yet rigorously derived (the passage from established, verified linear dressed-response theory to a genuine axisymmetric back-reaction) and what a rigorous derivation of it would require, and we lay out the roadmap, risk assessment, and falsification criteria required to elevate CSG/TCEA from a mathematically well-posed conceptual framework to an empirically and theoretically tested theory.