Governing-equation limit and pressure interpretation

Research source/evidence ef9c3dc3/598e9a06 and native source/evidence 3f0271d6/a9428641 are frozen. This additive audit changes no native code, acceptance allowance, Jacobi fixture or earlier failed counterexample. It checks which governing equations the selected time discretization approaches. A temporal self-convergence ratio alone cannot identify the physical equations being solved.

The target is the declared semidiscrete ALE donor system

Let z=J(q) eta, U=Rz, qdot=S eta, with S selecting the three cap speeds. The physical median-dual masses, relative fluid/PS-mesh face rates, pressure image and symmetric-gradient strain are those in the fitted-height formulation. Put M=R^T diag(m_i I) R, c=R^T route(f,U), and B=-Q^T W_area D. The tangent system previously derived and independently checked is

M a + Mdot z + c + K z + B^T p = 0,
D a + Ddot z = 0,     D z = 0,
a = J alpha + Jdot eta,     qdot=S eta.

Here p is an algebraic multiplier, and physical atmospheric-relative pressure is Q(q)p. It is determined from geometry/velocity and the differentiated constraint. There is no independent pressure evolution law or equation of state. The consequential constant pressure mode is retained; no gauge is pinned.

For the six-coordinate chart, define the square limiting matrix

A=[M J, B^T],
A [alpha0;p0] = -[M Jdot eta0 + Mdot z0+c0+K0 z0].

Under positive masses, a full-rank six-coordinate chart and a full-rank sixteen-mode pressure image, A is nonsingular: BJ=0; multiplying a homogeneous system by J^T gives the positive definite J^T M J, hence alpha is zero, then full rank of B^T makes p zero. The independent rational assembly solves this 22-row system and the full 38-row moving-constraint system consistently, on both fields. It also differentiates the full 38-row system and independently checks the chart form of its derivative and all 24 strong rows.

This target already contains lumped inertia and dissipative nodal donor transport. It is a declared spatial discretization, not the exact continuum Navier–Stokes PDE. Temporal consistency with it does not establish continuum spatial accuracy, pointwise free-cap traction, mesh-family stability or material calibration.

Small-step limit of the finite method

For unknowns (alpha_h,Pi_h) the method uses

eta(s)=eta0+s alpha_h,
q(s)=q0+s S eta0+s^2 S alpha_h/2,
z(s)=J(q(s))eta(s),
Fplus/Fminus = integral max(plus/minus f(s),0) ds,
E_BE = (M z)_1-(M z)_0 + C_BE + h(K1 z1+B1^T Pi_h)=0.

C_BE routes actual integrated face masses with endpoint donor velocities. For a bounded real solution branch with smooth nonsingular geometry/chart,

E_BE/h = M0(J0 alpha_h+Jdot0 eta0)
         + Mdot0 z0+c0+K0 z0+B0^T Pi_h + O(h).

Thus the unique bounded limit is the previously derived tangent solution: alpha_h -> alpha0, Pi_h -> p0. This conclusion compares explicit operators and a nonsingular limiting system; it is not inferred from self-convergence. The donor absolute value is Lipschitz. At zero initial face flux the leading one-sided sign is used; higher-order smooth expansions are conditional on that one-sided branch. Numerical sign discovery is not a sign-isolation theorem. A real-equation limit does not guarantee that arbitrarily tiny IEEE steps pass the unchanged Newton/arithmetic gates.

Independent leading coefficients and the expected order

Write u0=(alpha0,p0), and let udot0=(alphadot0,pdot0) be the derivative of that algebraic tangent solution along its actual DAE trajectory. This derivative is an observable; it is not a separately evolved pressure equation. On the quadratic path with fixed alpha0,p0, let

P2 = coefficient of s^2 in M(q(s))z(s),
cdot0 = right derivative of actual instantaneous donor momentum,
Fdot0 = derivative of K(q(s))z(s)+B(q(s))^T p0,
L = R^T route(f0, Udot0)/2.

Exact rational Taylor jets rebuild PS intersections, gradients, masses, full strain, pressure transpose and shared fluxes independently. They give

A udot0 = -(2 P2+cdot0+Fdot0),
A u1 = -(P2+cdot0/2+L+Fdot0),
(alpha_h,Pi_h)=u0+h u1+O(h^2),
u1-udot0/2 = -A^-1(L+Fdot0/2).

The independent differentiated full tangent system gives exactly the same udot0. These coefficients predict local velocity error O(h^2) and local cap position error O(h^3); accumulated velocity/geometry errors are observed first order. They do not constitute a global stability/convergence theorem.

The actual varying-donor momentum integral along the candidate curve with its constant pressure coefficient has defect

E_path = -h^2(L+Fdot0/2)+O(h^3),
E_path/h = O(h).

A nonzero limit of E_path/h^2 is its ordinary first-order truncation coefficient, not a nonvanishing governing-equation defect. The correct momentum rate defect is E_path/h, which tends to zero. Pointwise momentum residual likewise is O(h) along the numerical path. Its old absolute gate is not relabeled zero.

Pressure: multiplier, endpoint display and weighted impulse

The constructor stores the instantaneous algebraic tangent pressure. A finite step solves Pi_h afresh; the previous stored pressure is not advanced or used as a dynamical initial condition. The numerical step publishes Q1 Pi_h on the accepted endpoint frame. This is a first-order algebraic pressure approximation associated with that finite step, not exact instantaneous tangent pressure. The force contribution is h B1^T Pi_h, an endpoint-weighted pressure impulse. Pi_h itself is not an impulse: omitting h or dividing the displayed pressure by h changes its dimensions and would produce a wrong or divergent limit.

Let p_alg,end be the tangent multiplier recomputed from the accepted endpoint, and p_avg the average of tangent coefficients on the true DAE trajectory. With coefficient coordinates kept explicit,

Pi_h-p_avg = h(u1_p-pdot0/2)+O(h^2),
Pi_h-p_alg,end = h(u1_p-pdot0)+O(h^2).

Both leading vectors are independently nonzero on both fixtures. Thus Pi_h is generally neither a literal coefficient average nor the exact endpoint multiplier, although both value errors vanish to first order. Physical pressure averages also contain the changing basis Q and cannot be identified with a mean coefficient vector. The actual pressure impulse has

h B1^T Pi_h - integral B(q(t))^T p_alg(t) dt
  = h^2[B0^T(u1_p-pdot0/2)+Bdot0^T p0/2]+O(h^3).

Its rate error is again O(h). The independent ODE/quadrature study checks this coefficient, not merely an error ratio.

A deliberately invalid pressure-as-evolved-state diagnostic is retained:

(Pi_h-p0)/h - pdot0 -> u1_p-pdot0 != 0.

For the first step after instantaneous initialization this derivative comparison fails even though the algebraic pressure value converges. A first-order pressure value is insufficient to validate its difference quotient. No pressure-rate accuracy claim is made. Repeated native pressure values are separately compared to the independent DAE at common physical time, using a declared labeled-mesh ALE pullback and reference-area RMS; that is temporal pressure evidence only.

Retained pressure-state coarse diagnostic and real counterexamples

The earlier pressure-state C_BE-C_path ratio 4.629 at .05/.025 still fails the original 15-percent band around four. Its exact leading expansion is h^2 L+O(h^3) with nonzero L. The normalized rate difference is h L+O(h^2) and vanishes; the nonzero delta/h^2 limit is an error coefficient. Initial shear has exactly zero f0 and L, hence donor difference O(h^3) and rate difference O(h^2). Endpoint-force truncation remains O(h^2) even there. Exact coefficients and finer diagnostics explain the coarse failure without retrospectively passing it or loosening any gate.

The independently retained wrong-equation variants genuinely fail as h tends to zero: omitting Mdot z leaves a pressure-state momentum rate defect with maximum component 0.00018051016017325; freezing the constraint to D a=0 leaves strong differentiated defect Ddot z, maximum 0.00203121758941077. Both are invisible on the initial exact-shear tangent, so that fixture alone cannot validate the moving equations. The selected method includes both terms. All prior failed pointwise/integrated gates, native zero-seed refusal and arithmetic/proof limitations remain unchanged. No new Lean theorem is asserted.

Next bounded physics proposal after this audit

Complete the existing formulation’s nonzero third velocity component on this same two-column, constant-density/viscosity, z-invariant extruded graph before introducing another material variable. The original full formulation retains all three components and both third-component strain entries; the present native coupled step qualifies their invariant zero slice only.

In this restricted extrusion, xy transport/geometry/pressure do not depend on w: partial_z=0, pressure has no z force, and full symmetric strain splits into xy and the two w shears. After the candidate xy solve, use the same physical face integrals/masses for a bounded 12-coefficient implicit w momentum/strain solve. The same accepted full nodal-velocity owner, candidate buffers, frame, pressure, clock/stamp and final barrier must publish all components together. Account the additional fixed matrix/scratch once; no parallel accepted phase or snapshots.

Qualify nonzero x/y/z transport with third momentum conservation, constant-w preservation, both w strain components, exact total/separate work, true residual, all cancellation/failure/retry states, budget exhaustion and actual refinement/ replay/render evidence. Retain constrained-embedding positivity limitations and all continuum/contact/reconstruction gaps. This is a proposal, not implemented physics in this checkpoint. Variable density, wetting/adhesion, capillarity and new surface reconstruction remain later separately derived features.