The Dark State Criterion: A Structural Test For Dark Matter Versus Modified Gravity

Abstract

The dark-matter problem is usually posed as a choice between an unseen material component and a modification of gravity. That dichotomy is not representation-invariant. A field can be moved between the matter and geometric sides of an equation, a local field can be eliminated in favour of a nonlocal functional, and a conserved fluid can be represented as a functional of the metric. The prior question is therefore not where a term is written, but what additional physical structure any successful explanation must contain and what would make that structure one identifiable carrier across different physical regimes. This paper separates that problem into two logically independent questions. The first is a question of necessity. We define a reference theory consisting exclusively of local, four-dimensional, generally covariant, purely metric, second-order gravitational field equations with universal minimal matter coupling. We then define the corresponding closed catalogue of admissible cosmological solutions containing only the declared known sector, explicitly excluding any cold-dark-matter component and restricting all initial conditions, boundary conditions, primordial conditions, population assumptions, and free parameters to preregistered sets. The associated prediction family consists of all observable probability distributions generated by that closed catalogue. Within this framework, Lovelock closure applies only to the field-equation class itself, catalogue closure is an independent physical requirement imposed on the solution catalogue, and empirical exclusion is a third operation acting on the prediction family. Once these three levels are kept distinct, the conditional result is exact. If the observed distribution cannot be generated by the declared prediction family, then every exact explanation of the observations must either leave the declared gravitational field-equation class or invoke a successful solution lying outside the declared solution catalogue. Furthermore, at least one such departure must make a genuine empirical difference to the predicted observations; purely decorative extensions are therefore excluded. Operationally, the same conclusion follows only after a statistically calibrated rejection of the reference hypothesis at the stated level of error control. The published value of the cold-dark-matter density parameter within the standard Lambda-CDM fit is not itself such a test. Consequently, the empirical antecedent remains OPEN – CALCULATION REQUIRED until a constrained joint fit has been performed using the preregistered raw likelihoods. The second question concerns identity rather than necessity. Physics supplies the field equations, constraint equations, conservation laws, Cauchy data, and observations. La Profilée supplies the structural reconstruction. The Frame is the identity-defining structure preserved through change. The Module is the admissible variation carried by the candidate. The Coupling binds that variation to the Frame. Q1 asks whether the candidate exists. Typed Q2 asks whether it remains the same candidate throughout its evolution. Q2a concerns persistence-class continuity, whereas Q2b concerns the complete identity-relevant history reconstructed through an independently fixed physical continuation bridge. Neither a name, nor the side of an equation on which a quantity appears, nor its degree-of-freedom count, nor agreement between two endpoint solutions can replace such a bridge. The resulting hierarchy is strict. Additional mathematical structure is not yet an identified carrier. An identified carrier is not yet an autonomous physical state. An autonomous physical state is not yet a microphysical identification. A residual-relevant structure qualifies as a single dark carrier across the declared regimes only if it admits a role-pure reconstruction into Frame, Module, and Coupling together with a closed typed-Q2 continuation bridge. Whether that carrier possesses autonomous reduced initial data is tested independently and is not part of the structural identity criterion. For every nonzero, residual-relevant timelike conserved current defined on a regular spacetime domain, La Profilée identifies the conserved material-tube charge as the Frame, the evolving density and flow as the Module, and the continuity equation as the Coupling. As long as the flow remains regular and invertible, it provides the required persistence-class representation for Q2a. Once the characteristic flow ceases to be invertible, however, that physical representation no longer supplies an admissible class-preserving continuation across the resulting break. Q2b continuity is therefore no longer instantiated across that region, and no continuous cross-regime identity is established there. Whether the current requires independent reduced initial data is an entirely separate physical question concerning autonomy rather than an additional structural role. The criterion is then applied to the nonlocal relativistic model of Deffayet and Woodard, which was designed to reproduce the cosmological behaviour normally attributed to cold dark matter while simultaneously recovering Modified Newtonian Dynamics in gravitationally bound systems. The model contains an exactly conserved quantity obtained by combining the metric-induced density with the nonlocal contribution. This conserved quantity, rather than the metric-induced density alone, is therefore the only candidate capable of providing cross-regime identity. Because the candidate is exactly conserved along every smooth flow line, any positive initial dark charge must remain positive throughout regular evolution. It cannot disappear or change sign through finite smooth expansion. Consequently, the proposed bound-system limit cannot be interpreted as a purely algebraic cancellation between two terms. Instead, the conserved carrier itself must be driven locally toward zero. Physically, this requires either a substantial outward transport of the conserved charge, an extreme local depletion, or a breakdown of smooth continuation. Using the published interpolation function of the model, the maximum nonlocal contribution remains small compared with the cosmological normalization required to reproduce the cold-dark-matter background. Under the normalization adopted in the source paper, a region that initially retains the mean cosmological dark-charge density carries a normalized conserved charge of approximately forty-five. Even before any nonlinear structure formation is considered, the proposed bound-system limit therefore requires depletion of more than ninety-four percent across the full positive interpolation range and more than ninety-nine percent throughout the published deep-MOND regime. If the conserved current clusters in the same manner as the cold-dark-matter mode it was designed to reproduce, the required depletion becomes even more severe as the local overdensity increases. The model-specific analysis closes the cross-regime Q2b continuity question for the explicit Appendix construction presented in the source paper. Within the published test spacetime, the future-directed velocity field generated by the auxiliary scalar defines an irrotational geodesic congruence. The characteristics of the scalar equation therefore coincide with the initially resting radial geodesics analysed in the Appendix. Near the centre, the exact solution shows that neighbouring characteristics converge until they reach a common focal point after a finite proper time. Throughout the entire regular evolution preceding that focal event, exact current conservation does not reduce the conserved carrier. On the contrary, it amplifies the central carrier continuously while the geodesic congruence contracts. Instead of approaching the near-vanishing value required by the proposed pure-MOND limit, the conserved continuation carrier grows without bound as the focal event is approached. At the focal time, the characteristic map ceases to remain invertible. The velocity field is no longer single-valued, and the regular continuation that previously carried the identity of the conserved current terminates. The published realization therefore admits only two possible outcomes on this domain. Either the conserved carrier survives and is amplified throughout the regular evolution before the focus, or the regular carrier itself terminates at the caustic. It does not provide a regular, Q2b-continuous history connecting the cosmological regime to the proposed pure-MOND bound-state profile. Accordingly, the typed-identity verdict for the published cross-regime continuation is CLOSED – NO for the explicit interpolation analysed in the source paper. At the caustic itself, the regular carrier no longer exists, so the existence verdict becomes negative and no identity verdict can be assigned beyond that terminal point. A pressure term, vorticity, higher-derivative regulator, or ultraviolet completion could alter this conclusion, but every such modification changes the physical coupling structure of the theory itself. Each therefore defines a genuinely new physical theory that requires a new structural audit rather than extending the audited realization unchanged.