Proposed coupled temporal prototype on the same fitted two-column domain

The pressure/path counterexample source eb7b354b7c20521357ca258728420c56ef96296a and evidence 00c502f1fe5be7f9c527ebb14ba5b78e450cb031 remain frozen. This separate research prototype targets their missing space-time coupling, with the same two-column periodic graph, density 3, viscosity 1/20, thickness 1, fixed impermeable free-slip bottom, weak homogeneous cap traction, initial (y,0,0) and zero external load. No public stepping API is changed.

Proposed system and acceptance gates, before prototype implementation

Let x contain four independent cap coordinates, R the unchanged admissible trace embedding, z all 22 potentially nonzero xy velocity coefficients, D(x) the full 24 microtriangle divergence rows, Q(x) its 16-dimensional image basis, B(x)=-Q(x)ᵀ W_A(x) D(x), and M_r=Rᵀ diag(m_i I_3) R. Physical PS points and their derivatives are rebuilt from this cap and its material motion, including the corrected periodic neighbor-center intersections. The current fixture has zero third momentum and a decoupled third strain block; this is a full xy experiment, not omission of a component that becomes nonzero.

The coupled semidiscrete equations are

xdot = C R z,
M_r(x) zdot + B(x)ᵀ p = −g(x,z),
D(x) z = 0,

g = Rᵀ diag(mdot_i I_3) R z + Rᵀ c(Rz,f) + K_r(x) z.

C extracts the independent cap xy velocities. f is the actual physical dual face flux of fluid minus differentiated PS mesh motion; c is its shared full vector donor convection. K_r is the existing full symmetric strain. Neither masses nor fluxes are fitted after the solve. Differentiating the constraint gives

D zdot = −Ddot(x;xdot) z.

At each evaluation solve the pressure saddle system

[ M_r  Bᵀ ] [ zdot ] = [ −g                    ],
[ B     0 ] [ p    ]   [ Qᵀ W_A Ddot(x;xdot) z ].

Check all 24 strong differentiated constraint rows as well as the 16 projected rows. This uses solved pressure, not pressure supplied from a separate force mesh. Pressure coefficients are coordinates in the changing image basis; the physical reconstructed pressure and its adjoint force must be checked.

To avoid drift from merely integrating the differentiated constraint, select a locally nonsingular 16 by 16 block of D on the initial graph. With the six remaining coefficients eta, define a geometry-dependent kernel chart H(x):

z=H(x) eta,  H_free=I,  H_pivot=−D_pivot⁻¹ D_free.

The ten integrated variables are (x,eta). At every evaluated or interpolated point reconstruct z from this chart and gate the full D z residual. Use the same material xdot=C R z in Ddot, mass derivatives and physical fluxes. The saddle solve determines zdot; its free rows determine etadot. The chart chain rule is checked against zdot, rather than silently projecting momentum onto a new velocity. Every point remains conditional on chart/rank/geometry checks; no mesh-family inf-sup theorem is imported.

The first finite trajectory is a tightly controlled ODE research reference, with a fixed upper bound on steps/evaluations. An adaptive Runge–Kutta dense curve is an approximation, not an exactly constrained material trajectory by itself. Its derivative must be evaluated analytically and tested separately. Use the actual derivative of this stored curve, rather than replacing it by the RHS, when rebuilding mesh motion and integrating the relative physical flux. Reconstruct velocity through H even on dense points. Check its actual chain-rule derivative and the pressure-dependent momentum residual there.

Acceptance gates are fixed before any result:

• full strong divergence, differentiated divergence, true xy momentum residual and material cap velocity mismatch at or below the existing 1e−11 scale;
• positive geometry/masses and the full 16-mode pressure image at every sampled stage/dense/quadrature point; measured conditioning reported, no uniform bound;
• actual positive/negative face integrals from the stored curved trajectory, with 16/32-point refinement measured per accepted dense interval and local GCL checked at the preceding native 128 eps (m_old,i+m_new,i) scale;
• horizontal momentum and the complete integrated energy ledger, using shared vector donor loss, full strain, pressure work, measured actual-curve momentum residual and the measured GCL-defect work; no unexplained energy drop;
• two independently refined coupled trajectories, complete output replay and meaningful corruption controls, preserving any failed trial without changing these acceptance tolerances.

The reference and its bounded output arrays are research host allocations. It adds no owner, clock, accepted state or simulation memory ledger. Public stepping would still require a bounded native temporal solver, reusable workspaces, rollback/cancellation from nonzero accepted states, repeated execution and native qualification. A Python research pass cannot enable that API.

Independent review and selected first finite-cell experiment

The independent review reproduced exact tangent compatibility on both the initial shear and the preceding nonzero-pressure field. The local model here independently reproduces reconstructed pressures 0.0103045276861 and 0.0101166010162, respectively, with exact strong acceleration/momentum/work/GCL and horizontal momentum-rate residuals zero. Thus the frozen straight-path failure is a temporal obstruction, not a spatial impossibility.

Following the review’s more specific finite-cell construction, reduce cap geometry to q=(x0,x1,h0), with h1=9/4−h0 and periodic endpoint (x0+1,h0). Its area is exactly 9/8. Use divergence rows [0,2,3,4,5,6,8,10,11,12,14,16,17,18,20,22], cap velocities (ux0,ux1,uy0), and selectors (z[0],z[1],z[4]) to define a six-coordinate lift z=J(q) eta. Its first three coordinates are qdot; the other cap vertical velocity is −uy0. The remaining three coordinates describe interior motion.

The selected bounded candidate cell is

eta(s)=eta0+s alpha,
q(s)=q0+s eta0[0:3]+(s²/2) alpha[0:3],
z(s)=J(q(s)) eta(s),  0<=s<=h.

Cap motion is therefore quadratic, and cap velocity follows the actual path derivative by construction. Rebuild all PS split-node derivatives from that motion. Solve for six alpha and sixteen pressure coefficients pi using all 22 integrated conservative momentum equations,

M_r(q(h)) z(h)−M_r(q(0)) z(0)
  + integral_0^h [c_r(s)+K_r(s) z(s)+B(q(s))ᵀ pi] ds = 0.

Physical pressure is Q(q(s)) pi, with no constant-pressure gauge removal. Integrate the actual vector momentum flux at each quadrature point; endpoint velocity times aggregate mass transfer is not this equation. No extra cap projection equations are appended to the momentum system.

The original ODE research proposal above remains a possible accuracy reference; this finite cell is a different temporal discretization. Its integrated momentum residual uses the same fixed 1e−11 gate. Pointwise momentum residuals are measured and retain their original gate result; an integrated pass is not relabeled as a pointwise ODE pass. Most critically, even a zero integrated vector residual does not imply a zero work defect. With

r(s)=(M_r z)' + c_r + K_r z + Bᵀ pi,

measure integral z(s)ᵀ r(s) ds independently. Require its magnitude and the complete unexplained energy-ledger error at the preceding 128 eps energy scale. If that work gate fails, this candidate is rejected; no tolerance changes and no public advancement follow from its divergence or momentum successes. This separate work gate is part of the independent review’s construction, not an energy theorem supplied by the coordinate chart.

Local-coordinate DAE methods motivate constraint-preserving representations, but their smoothness/conditioning assumptions require separate checks here. Ascher and Petzold, Sections 2.1–2.3 derive a constrained underlying ODE and discuss projection to retain constraints. The finite chart and momentum/work equations above are our fixture derivation; that paper does not validate them. Likewise the space-time conservation approach in Busto, Dumbser and Río-Martín’s abstract uses a different staggered FV/FE method. Its numerical claims do not transfer to this fitted velocity/pressure space or donor operator. The adaptive reference, if used, follows SciPy’s DOP853 documentation; its algorithmic description supplies no liquid acceptance or energy proof.

Observed coupled-cell result and work rejection

The reviewed quadratic construction closes the moving constraints and the actual physical mass flux on the .05 cell. The nonlinear solve uses 62 evaluations; all 22 integrated momentum equations have maximum component residual about 1.51e−16. Strong velocity divergence, strong acceleration compatibility and material cap mismatch stay below 1e−11 at all recorded quadrature points. The finite local GCL defect is about 1.29e−16, within the preceding native 128 eps (m_old+m_new) scale. The 16/32-point flux/work difference is about 1.06e−17. These are numerical checks of the published curve, not a universal quadrature theorem or a native advancing solver.

The complete energy ledger passes only when its actual residual work is included:

T_new−T_old + integral(D_adv+D_strain+pressure_work) ds
  −integral zᵀr ds + integral [sum_i G_i |U_i|²/2] ds = error,
G_i=mdot_i+sum_j f_ij.

For .05, energy decreases from approximately 0.803142280736 to 0.800306896842, yet the residual work is −2.86083482087e−8, versus the fixed allowance 4.56533530687e−14. The energy-ledger error is approximately −6.86e−16. Thus monotone energy and tiny unweighted momentum residual do not establish the required work balance. This candidate is rejected. Its pointwise momentum residual also fails the original 1e−11 gate, rather than being hidden by its integrated pass.

An independent rational reconstruction solves the 22-row chart at each point, rebuilds all physical fluxes and symmetric strains, and sums with 80-digit arithmetic. It gives residual work −2.86083482105e−8, independently agreeing with the energy/loss defect. Its integrated momentum maximum is about 7.15e−16; local GCL is about 3.78e−22. No NumPy momentum/strain/flux arrays or nonlinear-solver work metrics enter that reconstruction. Repeated cells at .025 and .0125 have work defects −3.54614759230e−9 and −4.41416889514e−10. Ratios near eight describe the cubic work failure of this rejected time approximation, not temporal convergence of an accepted liquid step. All three work gates fail with the same allowance.

The reason is visible directly in the equations. A zero integral r ds implies

integral zᵀr ds = integral (z−z_bar)ᵀ r ds,

for any constant z_bar, but does not make that weighted integral zero. If a small-cell residual has leading form r(s)=(s−h/2)b+O(h²) and z(s)=z0+s a+O(h²), then

integral_0^h zᵀr ds = h³ (aᵀb)/12 + O(h⁴).

This conditional expansion explains the observed failure order without assuming that this quadratic integrator is stable or that its pressure is a continuum solution. The missing condition is temporal momentum testing that controls its velocity-weighted work, while retaining conservative component momentum and physical cap/face coupling. It cannot be repaired by fitting face masses, relaxing residual tolerances or calling the extra energy loss viscosity.

Whole real-path chart and geometry certificate

Eight constant microtriangle row relations are verified symbolically for general (x0,x1,h0) with the declared volume constraint. Four are equal-divergence pairs on trace-constrained physical boundary splits; four are alternating relations at PS interior/periodic singular vertices. Their 176 entries against the full D(q) are identically zero. The selected 16 row selectors together with these eight relations form a nonsingular 24 by 24 row transformation. Hence solving the selected zero-divergence rows in the chart implies all 24 equations on this physical construction, rather than silently projecting an incompatible component. The symbolic area-weighted divergence identity also establishes the fourth cap velocity uy1=−uy0. Independent incompatible-component tests still retain a full strong-residual gate.

For the published .05 candidate, a 512-box exact rational interval certificate covers the entire actual binary-coefficient polynomial path. Every PS intersection and positive microarea is enclosed, and minimum shape quality stays above the unchanged 1e−6 gate. The chart and pressure-image matrices are nonsingular on each box: an approximate inverse is used only as an exact binary rational preconditioner, and ||I−C A(q)||_infinity<1 is proved by exact interval bounds. Maximum chart/pressure Neumann bounds are approximately .06172/.55261. Microarea and shape-quality lower bounds exceed .03814/.07506.

Interval endpoints are rounded outward to 200-bit dyadic rationals using integer floor/ceiling operations; the final Neumann midpoint/radius bound uses exact integer arithmetic. It does not trust floating inverse accuracy. The first 256-box resource budget was insufficient and its failed log is retained; the published host certificate has an explicit 1,024-box bound. Physical tolerances are unchanged. This is a certificate of this real rational path, not IEEE execution, an arbitrary-geometry or mesh-family bound, or a uniform inf-sup constant. The two shorter refinement cells have recorded numerical path checks, not this separate whole-interval certificate.

The production Rust/Cargo/test/example inputs are byte identical to the frozen pressure-path milestone; its actual Rust matrix and native replay evidence remain preserved. This packet runs new Python research, not a new Rust temporal build. No new Lean theorem or native public step is claimed. The remaining blocker is an energy-qualified temporal formulation, followed by bounded native solver, workspace/owner integration and rollback/repeated-accuracy qualification. Existing Jacobi fixtures, scale refusals, negative positivity evidence, pointwise traction, continuum accuracy, density/adhesion/capillarity/reconstruction limits remain intact.

Hairer, Sections III.2 and V.3 explains local coordinate partitioning and index-two constraint treatment. The specific chart, physical donor operator and failed integrated work above remain our independently checked fixture construction, not a theorem from those notes.