The action and the local domain

Section 3.6 starts with a smooth Lorentzian System Frame metric with negative timelike sign and its torsion-free, metric-compatible connection. It assumes a smooth real phase with units of action, fixed positive particle mass, positive finite source mass density and a single flow before flow lines cross. The velocity covector is the phase gradient divided by particle mass. Choose the phase orientation so that its raised velocity points to the future. [1]

Let ${\mathrm{𝑋}}_{\mathrm{MQ}}$ denote the dimensionless mass-shell invariant, defined as minus the inverse-metric contraction of the phase gradient with itself divided by squared particle mass and squared vacuum light speed. Let ${\mathrm{𝜌}}_{\mathrm{src}}$ be the independently varied mass-density multiplier, in kilograms per cubic metre, and $\mathrm{𝑐}$ vacuum light speed, in metres per second. Equation (1a) displays the timelike constraint Lagrangian density in joules per cubic metre. [1]

The action integrates this density over invariant four-volume. When all four coordinates have length units, divide the four-volume integral by light speed to obtain joule seconds. The phase and density are varied independently with fixed boundary data for those variations. Isolation assumes that other terms have vanishing phase and density variations on the branch. The contribution of other terms to metric variation must be assessed separately. [1]

Vary first, then apply the constraint

Density variation sets the coefficient of the density variation to zero. The resulting condition is Equation (2b). It fixes the squared velocity norm to minus light speed squared. The Lagrangian density vanishes when evaluated on this branch, but the unrestricted expression in Equation (1a) is needed for the variations. [1]

Table 1 identifies what is held independent in the proof. The phase enters through its gradient. Integrating the phase variation by parts gives zero covariant divergence of density times raised velocity, with a constant mass factor removed. In the metric variation the phase and density stay fixed. Differentiating the inverse-metric contraction gives the phase-gradient outer product with a minus sign. Multiplication by minus twice the derivative of the Lagrangian cancels the sign, the factor of two and the light-speed factors. Dividing the phase gradients by mass gives the velocity covectors. [1]

The variation of the volume element adds a term proportional to the metric times the Lagrangian density. This term vanishes on the constraint, leaving the velocity outer product multiplied by the mass density. In the local rest frame, the energy density is mass density times light speed squared and the spatial stresses vanish. Substituting the constraint before variation would instead erase the metric derivative that produces this result. [1]

Figure 1 separates variation from evaluation on the solution. The geodesic equation follows because the derivative of a gradient covector is symmetric for the connection used here. Contracting with the raised velocity converts the acceleration along the flow into half the gradient of the squared norm. The norm is constant, so the acceleration vanishes. Applying the product rule to the divergence of the dust stress gives a current-divergence term and an acceleration term. Both vanish. [1]

The reverse correspondence

Take smooth pressureless flow with positive density on the same metric, with the same mass-current normalization and locally exact velocity covector. On a contractible patch, irrotational geodesic flow provides such a covector. Integrate it and multiply by the fixed mass to construct the phase. Its gradient reproduces the covector, its norm gives the density equation and the dust continuity law gives the phase equation. The arbitrary integration constant leaves the flow invariant. [1]

The equivalence is local and classical. It is restricted to a smooth, single, irrotational flow with positive density and matching boundary data. Vortical flow, multistream regions, vacuum points, global phase topology and other matter species require separate treatment. The source’s comparison concerns the isolated matter solutions on a common metric. [1]

What remains for the metric and cosmology

To extend the comparison to the coupled equations, the gravitational action must have the corresponding Einstein–Hilbert metric variation, including the specified coupling and cosmological coefficient. The complete matter variation must supply the matching stress. The same independent variations and compatible initial and boundary data are needed, along with a lift of continuum solutions to admissible MQ configurations. The ability to represent a smooth metric perturbation does not establish its admissibility as a count variation. [1] [3]

For fixed coupling in the Einstein-form equation, the contracted Bianchi identity relates the gradient of the cosmological coefficient to the covariant divergence of the source stress. If the isolated dust is the only source, its stress is separately conserved, so the coefficient has zero gradient. An evolving coefficient in that setting needs compensating exchange with additional dynamics or a justified modification to the metric equation. The local proof supplies the conservation condition used in this argument. [1]

October 23, 2026 identifies the source version. This Brief was prepared November 3, 2026. The technical proof is in Section 3.6 of the manuscript available through the Institute. The linked public preprint provides related geometric context. [1] [2]