Advancing viscous liquid: bounded goal and current obstruction

The end goal is one genuinely advancing, narrowly supported viscous liquid step: accepted liquid geometry, all required momentum components, compatible pressure and strain, and one physical clock advance. It is not achieved by the static MAC materialization milestone. The constant-field repair and static MAC bridge source/evidence heads remain frozen on their review branches. This companion is a research checkpoint identifying missing contracts; it adds no production stepping mode and claims no advancing rendered simulation.

Minimal remaining sequence

1. Select one physical liquid domain and discrete velocity/pressure spaces for both the old and candidate geometry. Derive face masses, wet face flux areas, surface normals, active support and all six strain samples from that same domain. Restrict initial work to constant density/viscosity, bottom-attached columns, stationary free-slip solid walls, no adhesion/capillarity and a 2D slice extruded in the third direction. Keep height/thickness/slope and arithmetic rejection. A first bounded implementation can refuse support changes as well.
2. Advance volume and every momentum component that can become nonzero with the same routed mass transfers. Account for phase/momentum boundary exchange, wall forces, remap mixing and store rounding. Surface motion must determine changing normal-face support and masses, rather than admit unrelated cap donors. Conservative momentum does not follow from conservative phase alone.
3. Assemble compatible pressure and Newtonian symmetric strain on candidate geometry. Derive the free-surface stress condition and its discrete boundary terms. Solve/gate the composed momentum/pressure equations, not merely the old inviscid atmospheric projection followed by unqualified damping.
4. Verify candidate phase mass, all momentum components, measured work/energy, true linear residual, actual stored divergence and common physical time. Publish existing velocity, pressure, phase and both geometry/time identities only after one final cancellation barrier. Keep scratch capacities explicit; reuse existing candidate/geometry owners without accepted-state snapshots.
5. Qualify repeated advancing execution using an analytic or controlled physical flow, independent numerical checks, cancellation/failure from nonzero accepted states, and actual rendered replay. Include refinement and any accuracy failure. Until these pass, the advancing end goal remains pending.

This is one dependency chain toward an advancing step, not a sequence of isolated operators relabeled as completed liquid simulation. The concrete obstruction is currently at the first three links. The existing transaction machinery already provides a suitable publication pattern, but cannot supply missing mathematics.

A moving global-flat closed layer is not an invariant class

For an incompressible constant-density liquid under a common height H(t) and fixed footprint of total area A>0, its mass is M=ρAH. With sealed solid walls, no sources, and a material free surface, mass conservation gives

ρ A [H(t+Δt)−H(t)] = 0,   hence H(t+Δt)=H(t).

Equivalently A Hdot=0. There is no nontrivial common-height motion within that closed global-flat class. At a flat material surface the kinematic condition is Hdot=u_n(surface). Different surface-normal velocities produce different column heights; zero total surface flux only conserves the integral, not flatness. This does not preclude viscous velocity evolution with stationary geometry. Such a stationary-surface invariant flow is a distinct supported goal and cannot be substituted for a requested moving-surface step without making that scope explicit. Open inflow/outflow or volume sources could change a common height, but require their own boundary momentum and stress work; none is silently added.

The exact native accepted field retained at evidence/column-mac/demo/pulse-y-0.25-jacobi/case.json supplies a concrete witness. Its physical height is 0.5625 m on four 0.25 m² columns. The virtual pressure surface slots have accepted f32 normal velocities

(−0.27636364102363586, 0, 0, +0.27636364102363586) m/s.

Their area-weighted sum is zero. Interpreting them as material surface velocities would give a height spread of 0.0005527272820472718 m after an Euler interval of 0.001 s. This is a hypothetical kinematic interpretation of a frozen static field, not a production advance or an accuracy claim. The current static API correctly refuses to use it for transport. The witness establishes why globally flat mass/projection support is insufficient for generic subsequent motion. Its actual export also omits positive normal kinetic energy, so re-prescribing only the two tangential components is not conservative three-component evolution.

Geometry and normal momentum need a shared interpretation

The current phase reconstruction describes bottom-attached columns. Its old inviscid pressure crossing uses interpolated center heights. The conservative parcel remap interprets columns as stepped prisms with piecewise constant slab means. The static MAC bridge defines pressure fluxes on a flat collapsed-slab control volume and a virtual top face. These are declared restricted models; they do not yet define one varying-height moving domain and strain space.

For unequal heights, adjacent final collapsed slabs have different endpoints. A horizontal interface must be split into overlaps and exposed liquid/air segments. A row can intersect more than one neighboring slab. Assigning the old nearest-neighbor coefficient or a full-box strain weight does not derive that geometry. Reconstructing a smooth graph instead changes volumes and normals and requires explicit reconciliation with the authoritative phase fractions. A bounded overlap graph is possible under the existing slope restriction, but its face incidence, velocity basis and exposed-surface degrees of freedom must be derived before implementation.

Normal momentum is not an optional export detail in a moving flow. The virtual surface unknown needs a physical support and motion rule. For example, consider the last normal interval [(L−1)h,H], of length ω=H−(L−1)h. The static bridge’s top-face mass is

m_top^bridge = ρ A (ω−h/2).

If instead normal velocity were P1 between the physical face at (L−1)h and the surface at H, the integrated row-sum top mass would be

m_top^P1 = ρ A ω/2,
m_top^bridge − m_top^P1 = ρ A (ω−h)/2.

The neighboring interior P1 mass would be ρ A(h+ω)/2, rather than ρAh. Both partitions have the same total mass but differ whenever ω≠h. For the native quarter-cap witness, the bridge/P1 top masses are 0.046875/0.0390625 kg per column. This is not a defect in the bridge’s declared pressure metric, nor a proof that no compatible strain operator exists. It demonstrates that the simplest proposed P1 basis cannot be assumed to integrate that metric exactly. A chosen mass quadrature or a different basis needs its own consistency and boundary derivation. The earlier tangential-only P1 shear basis does not answer this normal-component question.

The existing within-column remap accepts explicit cap donors. It is not a certificate of actual intercolumn mass routing. A moving successor must either use shared physical phase/momentum fluxes directly or derive how their integrated transfers determine donors for all three component supports. Changing height by an unrelated prescription and subsequently remapping profiles would reproduce another externally prescribed sequence, not the intended advancing step.

Pressure and stress cannot be inferred from energy damping

With no capillarity and atmospheric pressure p_a, the material surface condition is

(−p I + 2μ D(u)) n = −p_a n.

For an axis-aligned flat surface this requires

p−p_a = 2μ ∂_n u_n,
∂_n u_t + ∂_t u_n = 0  for each tangential component, when μ>0.

Setting p=p_a additionally requires zero normal strain. That holds for the previous periodic tangential-shear restriction, with u_n=0 and no lateral variation. It does not follow from divergence-free generic velocity. Independent component damping need not satisfy tangential traction either. A positive semidefinite matrix by itself proves discrete dissipation, not that the matrix represents symmetric strain or the right surface boundary.

Batty and Bridson 2008, Sections 4–5 derive coupled viscous surface terms and a variational split. Their later time-dependent Stokes paper, Sections 3–5 addresses pressure/stress coupling. These primary results motivate the research; neither validates Rheon’s collapsed-slab geometry. A split treatment would need an explicit normal-stress approximation and its measured error. The current goal should instead derive a compatible coupled treatment before claiming it complete.

One candidate finite formulation on a fixed candidate domain is

(M + Δt Eᵀ W E) v + Δt Bᵀ p = q_adv,
B v = 0.

Here q_adv is conservatively transported momentum, M is derived candidate liquid inertia, B is integrated negative divergence, E is actual symmetric-strain sampling and W contains nonnegative viscosity/geometry quadrature. Free boundary velocity variations must encode the complete natural stress condition. This is not already the existing nearest-neighbor pressure PCG: its Schur complement contains the inverse of M+ΔtK, not M alone. A bounded solver needs true composed residual checks and declared memory. Merely inserting an arbitrary E would still leave geometry, rigid-motion consistency and boundary traction unqualified.

For a backward-Euler unforced fixed-domain candidate satisfying these equations, pressure work cancels if Bv=0 and

E_before−E_after = ½‖v−u_adv‖²_M + Δt vᵀKv.

Moving mass, advection/remap, gravity or prescribed-boundary work add separate terms. They must be measured rather than attributed to viscosity. A sensible first-order ordering is conservative phase/momentum transport from Ω_n, assembly on Ω_(n+1), then coupled stress/pressure there; it still requires the missing shared moving flux geometry and a complete work ledger. More accurate geometry coupling is a later question, not assumed by advancing the clock alone.

Precise research exit criteria and current status

The next research result must specify a bounded domain/basis/flux construction for the 2D extruded varying-height slice; all face masses and surface strain weights; shared three-component transport and boundary work; a pressure/stress formulation; and the applicable stability/solver/moving-energy gates. Hand matrices must test symmetry, true work adjoints, nonzero affine shear, constrained boundary behavior and local admissible rigid-motion null modes. Then the pieces can be implemented in one advancing owner transaction and qualified with repeated physical time and genuine phase/momentum/energy evolution. A controlled gravity relaxation is a possible eventual replay, but is not yet executed or validated.

MovingLiquid.lean contains seven conditional exact-real obstruction identities: constant common height from supplied conserved mass; zero common rate from a supplied zero-flux balance; nonflat Euler heights from unequal rates; balanced two-column volume exchange; zero normal strain from supplied atmospheric normal traction; positive energy of omitted nonzero normal velocity; and the algebraic P1/bridge top-mass difference. These prove none of continuum conservation, kinematic discretization, basis integration, geometry/strain assembly, solver convergence, Rust/IEEE behavior, transaction atomicity or physical validation.

The advancing end goal is blocked by the missing shared moving geometry and full pressure/stress discretization, not completed by this research checkpoint. Production refusal and every existing tolerance remain unchanged. Old evidence, including the recorded f32 viscosity accuracy failure, remains immutable.