Native third velocity on the reviewed extrusion

This is the implemented successor to the frozen domain and derivation. Its domain is x in R/Z, 0 < y < H(x,t), z in R/Z, with both periods one and every field independent of z. The full velocity is (u,v,w). The bottom is impermeable with natural zero tangential traction; the material cap has atmospheric-relative natural traction in the restricted weak space. There are no extrusion end walls. The nonconstant initial third field is admissible weak data, not a classical pointwise traction benchmark. This is a two-dimensional, three-component family.

API and one publication

CoupledDiscreteFlow::new_extruded and step_extruded use the existing accepted frame, candidate frame, full nodal velocity, pressure and scalar workspaces. The state returned by state() contains the jointly accepted geometry, conservative nodal masses, all three velocity components, pressure, time and stamp. CoupledExtrudedReport reports the unchanged planar solve, the third solve, both third shears and the total energy ledger.

The cross-section is still the two-column Powell–Sabin family with bottom abscissae [0,.5,1], width one, density three, dynamic viscosity .05 and cap-height sum 2.25. The third field has twelve independent coefficients with the existing scalar trace embedding. It changes neither the cap’s material equation nor the planar pressure solve because partial_z=0.

On the qualified native target the conservative declared budget is 470992 bytes, including fixed caller/callee scratch reservation. The legacy zero-third budget remains 450320 bytes. The constructor checks the opt-in budget, and each step reuses existing scalar buffers and fixed dense factors. There is no heap allocation during a step, second accepted owner or rollback history. These are declared bounded reservations, not measurements of peak process memory.

All candidate geometry, velocity, pressure and reports pass their checks before the common BeforePublish barrier. Publication uses the existing accepted-frame swap and infallible pressure assignment. The old step refuses nonzero accepted third velocity; it cannot silently erase it. The opt-in zero-third step agrees bit for bit with the legacy published state on the tested sequence.

Equations and unchanged checks

The planar nonlinear path and actual integrated relative face fluxes are unchanged. Using their Fplus, Fminus and endpoint masses, the added solve is

Aw xi1 = Rw^T diag(m0) w0,
Aw = Rw^T diag(m1) Rw
   + sum_ij (Ri-Rj)^T (Fplus_ij Ri-Fminus_ij Rj) + h Kw1.

Kw1 is the third block of the existing full symmetric-strain assembly. Its physical loss is h mu sum_T |T| [(w_x)^2+(w_y)^2]; neither shear has a factor two. Pressure has no third force. Paired periodic z-face fluxes cancel locally. Changing-mass inertia includes m' w, and the independent semidiscrete reference integrates the same conservative equation, not a scalar equation on fixed mass.

The exact finite identity, with actual residual rw and local GCL defect g, is

Tw1-Tw0 + D_BE,w+D_mix,w+D_mu,w + sum_i g_i w1_i^2/2 - xi1^T rw = 0.

The third and full-vector ledgers are checked separately. Both stable and direct 34-row momentum rates must meet 1e-11. The old planar Newton gate remains 1e-13, at most seven iterations and 200 equation calls; root/quadrature bounds remain unchanged. Local GCL and work retain the original 128-epsilon factors. Arithmetic continues to reject nonzero subnormal intermediates and checked division failures. Failed candidates preserve the accepted state.

Pressure is still a freshly solved algebraic endpoint multiplier, not an evolved pressure state or a literal mean pressure. Its weighted force approximates the interval impulse; the multiplier itself is not that impulse. The frozen equation-consistency limitations, including the failed pressure-rate interpretation and the original .00078125 bounded-Newton refusal, remain valid.

Actual tests, replay and rendering

The eight new Rust contracts inspect bitwise snapshots of all published velocity, physical geometry, mass, pressure, time and stamp. They exercise all six old and three added cancellation stages after an accepted nonzero-third step, repeated quadrature/assembly callbacks, bitwise-identical retries, constant-third motion with changing mass, invalid intervals, wrong-API refusal, iteration failure, an actual checked-arithmetic third-solve failure, budget and trace refusal, both physical shears, total work, and zero-third legacy parity. These rollback tests use the original initial planar fixture. The pressure-state fixture is advanced and independently replayed; separate pressure-fixture rollback coverage is not claimed. The prior review’s shared original rollback finding remains shared.

examples/extruded_third.rs exports actual accepted owner states: twenty constructors and 248 steps from the two planar fixtures, nonconstant/constant third fields, and five intervals .05 through .003125 to common time .1. Every physical field and the actual owner stamp are mandatory in replay.py. The reader reconstructs 34 momentum equations, actual integrated GCL, both shears, separate and total work, physical pressure and geometry from those publications. It regenerates eight bounded DOP853 references at two declared qualities and recomputes all errors and refinement ratios. The original bands remain 1.8–2.2 for full/third velocity and 1.7–2.3 for geometry. Thirteen actual malformed-input controls must be rejected, including renderer-unsafe float indices that happen to compare equal to integers.

The nonconstant third-velocity lumped L2 error decreases from approximately .00365951 to .000250116 in the initial fixture and .00369077 to .000252321 in the pressure-state fixture. Its successive refinement ratios approach two (approximately 1.909, 1.952, 1.975, 1.987). These are conditional temporal results on this fixed spatial model, not spatial or continuum convergence.

render_native.py makes PNG/PDF figures, a 33-publication GIF and self-contained HTML from actual exported geometry, periodic identification and velocity. The HTML selects recorded fixture/interval/state, draws actual microtriangles and computes total energy and third momentum from the selected publication. It runs no second simulator. The Node VM checker verifies all 268 states, twenty cases, canvas coordinates/colors, selected default interval and play/wrap/pause controls. Browser rasterization requires the installed Chromium sandbox to work; a failed sandbox attempt is recorded explicitly, without disabling sandboxing.

No new Lean theorem is claimed. Existing conditional algebra is not a proof of the advancing IEEE implementation, root isolation or arbitrary meshes. There is no validated general 3D motion, continuum cap traction, variable material, adhesion, density variation, capillarity, surface reconstruction or full liquid simulator claim. Old Jacobi fixtures, failed readers and numerical refusals remain frozen and separately identifiable.