Proposed coupled implicit-donor temporal work experiment

This new research route preserves quadratic source b2b85ee0226bf2f1ef61313b610a049bc587fb21 and evidence c6c88365c1f52ed9dc0644e2ed48e64b3e24d567, including all rejected cells. It changes the finite momentum discretization explicitly. It does not retroactively validate their exact physical momentum-flux integral, change any public step, raise a tolerance, or add a native/Lean claim.

Two candidate routes and smallest selected experiment

A genuine space-time variational route would test conservative momentum with constant component tests and the varying velocity test needed for work. The old quadratic cell with six accelerations and sixteen constant pressure coefficients supplies only 22 unknowns for 22 constant-test equations. An additional independent work condition therefore needs richer temporal trial/test spaces and a compatible pressure/divergence pairing; appending one scalar gate alone gives no solvability or consistency theorem. A discrete-gradient route likewise must account for changing mass in the kinetic-energy chain rule, material-cap constraints and pressure’s zero work on the relevant time test. A scalar energy correction does not automatically preserve component momentum or the actual material geometry. These are viable research routes, but neither has been derived or implemented for this fitted moving space.

The smaller experiment reuses the qualified changing-mass implicit donor and backward-Euler identities in the fitted-height formulation. Retain the same reviewed six-coordinate chart and quadratic material/divergence-compatible path, and integrate the physical face MASS rates on that path. Route those face masses with endpoint donor velocities as an explicitly first-order finite MOMENTUM discretization. Use endpoint full positive strain and its pressure adjoint. This is not the integral of the varying donor momentum along the path. The extra dissipation is separately named and measured, never called viscosity.

Proposed finite system before implementation

Use the frozen two-column fixture, constant density 3, viscosity 1/20 and thickness 1, fixed impermeable free-slip bottom and natural weak cap traction. All 22 xy coefficients are retained; the invariant third field is zero. With reviewed lift z=J(q) eta, initial (q0,eta0), six unknown accelerations alpha, and sixteen endpoint pressure coordinates pi, declare

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

Rebuild each physical PS point and its true derivative. At every quadrature or recorded point test all 24 D z=0 and D zdot+Ddot z=0 rows and cap material motion. For each unordered face, oriented i to j, integrate separately

Fplus_ij = integral max(f_ij(s),0) ds,
Fminus_ij = integral max(-f_ij(s),0) ds,
F_ij = Fplus_ij-Fminus_ij,
f_ij(s) = physical relative fluid/mesh face MASS rate.

No marginal fit is allowed. Require G_i=m1_i-m0_i+sum_j F_ij within the unchanged 128 eps (m0_i+m1_i) gate. At the endpoint write U1=R z1 and route

cBE_i = sum_j [Fplus_ij U1_i - Fminus_ij U1_j].

Solve all 22 reduced conservative momentum equations

rBE = M_r1 z1-M_r0 z0 + R^T cBE + h K_r1 z1 + h B1^T pi = 0,
D1 z1 = 0,    B1=-Q1^T W_A1 D1.

The first equation is solved through the path’s six alpha and sixteen pi. The full divergence and cap chart remain independent checks. Atmospheric-relative pressure is Q1 pi, with its consequential constant mode, no pressure pin.

Exact discrete work identity and fixed gates

Define lumped energy T(U,m)=sum_i m_i |U_i|²/2, and

D_BE  = sum_i m0_i |U1_i-U0_i|²/2,
D_mix = sum_{unordered ij} (Fplus_ij+Fminus_ij)|U1_i-U1_j|²/2,
D_mu  = h z1^T K_r1 z1,
W_p   = h z1^T B1^T pi,
W_G   = sum_i G_i |U1_i|²/2,
W_r   = z1^T rBE.

Pairwise antisymmetry and the changing-mass polarization identity give exactly

T1-T0 + D_BE + D_mix + D_mu + W_p + W_G - W_r = 0.

For nonnegative face transfers and positive old mass, D_BE,D_mix>=0. D_mu>=0 uses the full endpoint symmetric strain. Exact endpoint divergence makes W_p=0. The work residual and GCL work remain explicit. Gate both |W_r| and the ledger error at the preceding 128 eps energy scale including the old/new energies, BE increment, mixing, strain, pressure, momentum-residual and GCL work magnitudes. Gate every component of finite momentum and its rate-normalized norm at the existing 1e-11 scale; retain full strong constraints/material mismatch there. Do not hide pointwise continuous momentum or exact integrated physical momentum: measure and report both against their original gates as different equations. A discrete-work pass does not label those ODE residuals zero.

The loss uses Fplus+Fminus, not abs(Fplus-Fminus). Sign reversals can transport both ways; net-zero transport can still mix. Corruption tests must reject omitted BE/mixing losses, false claims about physical momentum, wrong pressure adjoint, independently edited masses and zero-balance face circulation.

Consistency and qualification plan

Conditional on a smooth nonsingular chart and smooth spatial operators, z1=z0+h zdot0+O(h²), m1=m0+h mdot0+O(h²), and Fplus/Fminus=h max(plus/minus f0,0)+O(h²) (Lipschitz also at sign changes). Endpoint donor momentum differs from its actual path integral by O(h²); endpoint strain and pressure forces have the same first-order quadrature error. The finite equations divided by h thus approach the previously qualified semidiscrete coupled momentum equation. The quadratic cap position is compatible with the same limiting material velocity. This is a conditional local consistency argument, not a nonlinear existence, stability or global accuracy proof. D_BE is an explicit temporal dissipation of order h² per smooth step; D_mix/h approaches the semidiscrete donor loss.

First qualify a single cell on the old initial shear and nonzero-pressure field, with physical mass 16/32-point quadrature refinement, full path checks, independent exact-rational/high-precision work assembly and meaningful corruption controls. Then qualify bounded repeated host execution and temporal refinement against an independently refined coupled DAE/ODE reference on the same spatial discretization. Research cancellation/failure must preserve the complete nonzero accepted state at every barrier, using one accepted state and one candidate, no accepted-state snapshot history. Bound nonlinear evaluations, steps, quadrature and recorded outputs. Host research arrays are not a production memory-accounting proof.

No public step follows until native bounded solver/workspace/owner publication, actual cancellation/failure, arithmetic guards and repeated accuracy are qualified. Retain all continuum, pointwise traction, constrained positivity, density, adhesion/capillarity and surface-reconstruction limitations.

Bounded numerical quadrature and source context

Fixed 16/32 quadrature over an entire repeated cell failed at a physical face sign reversal; that trial is retained. The candidate now discovers sign brackets on sixteen subintervals and performs exactly twenty bisections per crossing, with at most 64 roots and 4,096 face evaluations. Each resulting subinterval uses the same 16/32 rules. This is a bounded numerical subdivision strategy, not an exact sign-isolation theorem. Missed roots or inaccurate subdivisions still have to pass the unchanged 1e-15 measured refinement and local GCL gates. Both positive and negative mass transfers are always integrated; no small flux is clipped. The first relative-iterate root finder failed near zero time; its log/source is retained. A 52-bisection trial is also retained, with the same physical gates. Bisection depth is a resource/quad partition choice, not a physical residual allowance.

The initial SciPy nonlinear iterate-status criterion stalled despite tiny measured residuals on a repeated cell. A bounded Newton loop now terminates on the actual rate-normalized residual at 1e-13, stricter than the unchanged 1e-11 acceptance gate. Each correction uses a bounded 22-column finite difference Jacobian; at most 200 equation evaluations are allowed. No solver status replaces work, momentum, path or mass checks.

Hairer’s local-coordinate/index-two notes motivate constraint charts but do not prove this temporal discretization. The publisher identifies McLachlan, Quispel and Robidoux’s discrete-gradient paper; its full text was not accessible here, so no theorem from it is imported. The changing-mass work identity and method comparison above are derived locally. The reference follows SciPy’s documented DOP853 interface, using installed SciPy 1.17.0; web documentation currently describes 1.18.0. The two reference runs change both tolerances and maximum step and report their actual end-state difference. Their agreement is numerical evidence, not an ODE error certificate or continuum solution.

Observed outcome: finite work passes, finer repeated solve rejected

Six cells (two initial states, intervals .05,.025,.0125) pass the actual rate-normalized 22-row momentum, full moving constraint, physical finite GCL, endpoint pressure work and unchanged discrete-work gates. At .05 on the initial shear, momentum-rate norm is 4.31e-14, residual work 1.45e-15 and ledger error 2.18e-16, below allowance 4.5653e-14. BE increment, donor mixing and physical strain losses remain distinct. Independent rational reconstruction checks the same changing-mass identity, not the old integrated physical momentum claim. That old equation still fails here: its .05 residual norm is about 1.14149e-4. Its pointwise continuous momentum gate also remains false.

Tangent acceleration/pressure error ratios are 2.087,2.043 on initial shear and 2.080,2.040 on the nonzero-pressure field. Endpoint versus true donor momentum convection differences refine at 8.118,8.059 for initial shear (vanishing initial relative flux) and 4.629,4.335 on the pressure field. The latter coarse ratio fails the experiment’s originally assumed 15-percent band around the leading factor four. The failed assertion log/source and a false order-diagnostic flag are retained. This asymptotic-rate diagnostic is distinct from the physical momentum/work acceptance equations; no failed diagnostic is relabeled passing.

Repeated host cases to .1 complete at .05,.025,.0125, with 2,4,8 steps and lumped velocity errors 3.44682e-5,1.74329e-5,8.76669e-6 against the refined coupled ODE reference. Geometry errors are 1.58577e-6,7.68853e-7,3.77818e-7. These three cases pass every recorded work, mass and finite momentum gate and show first-order behavior on this same spatial model. The .00625 case fails the declared bounded Newton 1e-13 convergence check. The existing 1e-11 physical gate and 128 eps work/mass scales are not loosened. Its failed log, empty unfinished output and partial completed-case records are preserved. Four-case repeated accuracy qualification is therefore rejected/incomplete. The complete-trajectory replay helper is not executed without a completed payload.

A separate actual rollback run cancels and injects failure at each of four barriers after a nonzero accepted host step, checks every accepted geometry, velocity coordinate, pressure coordinate, clock and stamp unchanged, rejects four invalid intervals, and successfully continues that same owner. This is research transaction evidence; native owner/memory/rollback is still pending. The replay GIF depicts recorded single-cell candidate snapshots, not a new native advancing simulation. The plot includes the three completed cases and explicitly marks the fourth rejection.

Next research work, saved without implementation

Investigate the finer-cell Newton floor while retaining its convergence check: first compare cancellation in M1 z1-M0 z0 with the algebraically identical nodal form R^T[M1 R(z1-z0)+(M1-M0)R z0], and independently recheck true residual and work. That numerical repair is not implemented or qualified here. Then repeat the failed fourth case, preserve complete step output incrementally, and qualify complete repeated replay, additional references and real-path rank/sign bounds. Only after that consider bounded native workspaces and the existing production owner/publication protocol. Native arithmetic/scale guards, publication rollback, memory accounting and continuum/traction accuracy remain separate obligations. A genuine space-time/discrete-gradient alternative remains research, with no demonstrated fitted-space solvability or energy theorem.

The parent requested a clean pushed checkpoint and stop under the strict usage ceiling. This packet freezes the current finite-work success and repeated-solve rejection; it makes no public simulation change and starts no further experiment.