Nonuniform xy pressure solve and the material-path gate

This checkpoint continues the fitted-domain plan after native source 23f93f9b72815766680e07a76ce2a6b23f8d7fce and evidence 6d57d1af5ca377f63cfeb90f8af4e71e11f6c280. Both remain frozen. The next question is whether a nonuniform xy pressure/strain solve also determines an incompressible material-cap trajectory and conservative finite transfers. A pressure solve on one geometry is insufficient to answer that question.

Proposed smallest experiment and acceptance test, before implementation

Use the smallest supported periodic fitted graph: two columns, period one, fixed bottom (0,0),(1/2,0),(1,0) and cap (0,1),(1/2,5/4),(1,1). Reuse the corrected periodic Powell–Sabin construction, physical lumped mass, trace restrictions and full symmetric strain of the existing formulation. Set density 3, viscosity 1/20, thickness 1, time interval 1/20, initial velocity (y,0,0) and external load zero. This initial shear is exactly divergence free and bottom impermeable. It need not satisfy the natural viscous traction pointwise at the initial instant; the variational force solve imposes the natural boundary condition weakly. There is no pointwise traction or continuum accuracy claim.

Let D be the actual microtriangle divergence composed with admissible velocity embedding R, Q a basis for its image, W_A the microtriangle areas, B=-Qᵀ W_A D, M=Rᵀ diag(m_i I_3) R, and K=Rᵀ Eᵀ W_mu E R. The first solve is the full xy saddle system

(M + dt K) z + dt Bᵀ p = M z_old,
B z = 0.

Pressure is solved, including the constant pressure mode admitted by the free cap. No pressure gauge is removed. The zero third component remains zero because its strain block decouples; all potentially nonzero xy components are included. This is a force test on a declared geometry, not a transported step.

The mathematical gate for an advancing successor is one declared trajectory X(s), U(s), p(s), 0<=s<=dt, satisfying all of:

• the actual cap-node material condition Xdot_cap=U_cap and fixed bottom;
• strong microtriangle divergence D(X(s)) z(s)=0, with U(s)=R z(s), along that same path;
• physical shared transfers F_ij=integral f_ij(X,U,Xdot) ds, with separate positive/negative integrals when direction changes, and m_i(new)-m_i(old)+sum_j F_ij=0;
• all component momentum equations on the declared temporal quadrature, the pressure adjoint, full strain and measured transport/work residuals;
• positive geometry/masses, the existing arithmetic and rank/residual gates, bounded owner/workspace memory, and common-state publication after a final cancellation barrier. Failure/cancellation must preserve both accepted states.

Acceptance of a native general step requires these integrated gates, repeated execution, temporal refinement, independent replay and rollback from nonzero accepted states. Passing an endpoint saddle solve, a volume sum, or a fitted flux marginal alone does not enable that step. No tolerance is relaxed.

The first deliberately falsifiable proposal is the endpoint force solve followed by straight cap motion with its solved nodal velocity. Rebuild the physical PS mesh at every path point and embed the same reduced coefficients there. The cap then is material by construction. Recompute strong divergence and actual dual relative flux using the differentiated rebuilt geometry. If these fail, preserve an exact rational counterexample and derive the missing nonlinear space-time equations; do not relabel the static force solve as advancement.

Result: the force solve passes, its straight material path is refused

The exact-rational solve has 22 xy unknowns, 16 pressure modes, 19 geometric nodes, 16 periodic masses and 24 microtriangles. Momentum and strong divergence residuals, pressure work and the complete symmetric-strain backward-Euler work identity are exactly zero. Reconstructed pressure reaches approximately 0.0101967270162; it is an unknown solved from zero load and the initial shear. Vertical velocities become nonzero. Removing pressure or vertical momentum fails the equations. Horizontal total momentum is exactly conserved. No conservation of vertical momentum against the bottom reaction is asserted.

Now set

X_cap(s)=X_cap(0)+s (R z)_cap,  X_bottom(s)=X_bottom(0),
U(s)=R z,  0<=s<=1/20,

and rebuild the corrected PS geometry, including its true split-node derivative, at each path point. The cap is exactly material, all checked areas/masses are positive and total liquid mass is exactly 27/8 at the five rational snapshots. Yet microtriangle 6, on geometric nodes (0,12,7), satisfies

D_6(X(0)) z = 0,
d/ds [D_6(X(s)) z] at s=0 ≈ −0.00203121758941077 ≠ 0.

The full exact fraction is in research-qualification/normal-geometry.json. This is an exact nonzero derivative, not a fitted finite difference. It proves failure on sufficiently small positive intervals by continuity of the rational geometry formulas at the positive initial mesh. At s=1/20, maximum actual divergence is about 1.02675160113e−4 and maximum local continuity defect is 1.62767785196e−5. The global sum of that defect remains exactly zero. The physical Reynolds identity passes including the measured divergence term; setting that term to zero is what fails.

For this trace-constrained material cap, the exact nodal identity is

m'_i + sum_j f_ij = (rho b / 3) sum_{T containing i} A_T div_T U.

At s=0 its right side vanishes, but its exact derivative has nonzero entries. Consequently the finite local GCL defect has leading term

m_i(h)−m_i(0)+integral_0^h sum_j f_ij ds
  = h²/2 (rho b / 3) sum_{T containing i} A_T(0) Ddot_T(0) z
    + O(h³).

The exact nodal coefficients are retained in the geometry receipt. Independent 16/32-point quadrature of the actual physical relative faces measures maximum finite defects 4.06927623524e−7, 1.01733944226e−7 and 2.54337405270e−8 for intervals .05,.025,.0125. Refinement ratios near four are the quadratic failure of local GCL, not convergence of an accepted liquid simulation. Face quadrature differences are below 1e−15; they do not justify a universal quadrature bound. No flux is adjusted to force its marginals.

A native bounded diagnostic solves the same 38 by 38 saddle system from existing public FittedHeightWorkspace operators, then independently rebuilds/inspects these five candidate meshes. It publishes no advancing state. Its frame charges 23,584 heap bytes; assembly and solver use fixed stack arrays of 26,304 and 24,016 bytes. Input/output/render allocations are diagnostic host allocations, outside the simulation memory ledger. There are no accepted-state snapshots or new simulation authority. The production source and existing tests remain byte identical to the frozen ALE milestone. An initial unrefined native solve missed the 1e−11 pressure-coordinate comparison. The failed payload and traceback remain in trials; fused multiply-add elimination reduces the error below the same bound. A separately retained decimal/float path-bound failure was repaired by representing the requested decimal interval as an exact fraction, without extending the accepted interval.

Required next equations and concrete blocker

This counterexample rejects the stated force-then-straight-cap proposal. It does not prove that the fitted space cannot support general incompressible motion, or that every temporal integrator must use this proposal. Moving the force solve to the candidate endpoint alone still does not supply intermediate divergence or physical face integrals; it needs a coupled temporal derivation and its gates.

For a differentiable exact constrained trajectory with constant trace embedding R and changing geometry, differentiating D(X) z=0 gives

D(X) zdot = −Ddot(X; Xdot) z,
Xdot_cap=(R z)_cap,  Xdot_bottom=0.

The current static saddle solve enforces the first equation only with its geometry frozen; the counterexample leaves Ddot z nonzero. The conservative semidiscrete formulation instead requires

M_r(X) zdot + B_r(X)ᵀ p
  = −Rᵀ diag(mdot(X)) R z −Rᵀ c(R z,f(X,z)) −K_r(X) z,
B_r(X) zdot = Q(X)ᵀ W_A(X) Ddot(X; Xdot) z.

The second row is the pressure-image projection of the differentiated strong divergence equation, using B_r=−Qᵀ W_A D. It is sufficient only while the chosen image has the required rank and the full strong residual is independently checked. It introduces pressure and mesh dependence into the actual time integration, alongside the same physical momentum transport and changing masses. A finite successor may use another consistent temporal formulation, but must qualify its full cap/divergence/flux/work equations together. No such nonlinear finite integrator or native solver has been selected or qualified here. That space-time coupling is the concrete blocker to enabling the general public step.

The render and replay depict a rejected candidate path, not accepted motion. No new Lean theorem, native general pressure stepping mode, continuum spatial accuracy, pointwise free-surface traction, positivity, density variation, adhesion, capillarity or surface reconstruction claim is made. Original Jacobi fixtures, frozen arithmetic rejections and all previous numerical/proof failures remain intact. The restricted ALE API and the general public refusals are unchanged.