What this PR establishes

This PR adds one common 2D solver-in-the-loop neural-correction benchmark for
all six differentiable Navier–Stokes grid solvers:

• JAX-CFD
• INS.jl
• PhiFlow
• PICT
• Warp-NS
• XLB

It is stacked on the opt-in canonical recurrent-state contract in #121.
JAX-CFD, INS.jl, PhiFlow, and XLB carry their native checkpoint through
training, evaluation, plotting, finite-difference validation, and
differentiation. PICT and Warp-NS are velocity-complete. Both canonical
velocity and native checkpoint state are stopped in the paired stop-gradient
control.

The benchmark makes two deliberately separate measurements:

1. absolute correctability, measured against a declared reference;
2. temporal-credit value, measured within each solver by pairing full
   differentiation through the autoregressive solver–corrector recurrence with
   the identical stopped-state recurrence.

The second comparison is the primary differentiability result. The benchmark
is therefore best read as a recurrent credit-assignment audit: does
backpropagating through several corrected solver transitions improve a policy
over local one-step supervision? It is not a claim that a VJP alone makes the
corrected solver accurate, nor that one numerical reference is universally
neutral.

Reference design

The nonlinear task is repeated against two independently discretized targets:

Both audits start from the same Fourier-prolonged continuum initial condition
and compare seeds 0 and 100 at frames 1, 8, 24, and 36. Training is rejected if
the p95 128²-versus-256² discrepancy exceeds 0.5%.

The finite-volume method uses an FFT only to solve the periodic discrete
Poisson equation. Transport and diffusion use conservative grid stencils, so
this is not a second spectral time integrator.

Common paired protocol

Every cell uses the same 32² candidate grid, viscosity, physical time step,
16 training ICs, eight held-out ICs, three paired model seeds, 200 optimizer
updates, recurrent training to t=1.92, and evaluation to t=2.88.

The learned component is the same zero-initialized Equinox periodic residual
CNN (56,098 parameters). Only the differentiable solver transition and its VJP
change between cells.

Training uses truncated backpropagation through eight consecutive correction
intervals. Each interval advances four native solver steps (4 × 0.02 = 0.08),
applies the corrector, and feeds both the corrected velocity and native solver
checkpoint into the next interval. There is no teacher forcing inside the
sampled window. The loss averages all eight corrected states and adds a
0.1-weight terminal loss on the last provisional solver state.

Held-out evaluation is fully free-running for 36 solver–corrector intervals:
after the initial condition, no reference field is fed back into the rollout.

The paired modes are:

1. full temporal credit: gradients flow through eight solver–corrector
   transitions, including solver velocity and native state;
2. local control: the identical forward recurrence and supervision, but
   recurrent velocity and native state are stopped at every solver output.

The solver-VJP lift is the geometric paired
error_local-control / error_full-temporal-credit ratio. A correction gain
above one separately means that the full-credit corrector beats the
uncorrected solver.

Nonlinear result and reference sensitivity

Absolute error and correction gain remain reference-dependent:

The native final errors move by at most 2.7% when the reference changes, but
the trained corrected errors move by as much as 40.3% (PhiFlow). PICT is the
only corrector that beats its native solver under both references. This
separates target agreement from non-convex corrector-training sensitivity:
very similar converged targets do not guarantee the same learned endpoint.

The paired solver-VJP conclusion is more robust, but not universal:

JAX-CFD, INS.jl, PhiFlow, and XLB retain a resolved full-VJP benefit under
both references. Warp-NS remains inconclusive. PICT is the important exception:
its paired VJP effect changes from significantly harmful under the spectral
target to significantly beneficial under the finite-volume target, even though
its absolute correction gain remains above one in both tasks.

Recurrent-state admission

All six solvers pass the same nonlinear and Taylor–Green recurrence gate:

Analytic and solver-specific controls

Analytic Taylor–Green control

All six solvers pass and train against the analytic Taylor–Green solution.
JAX-CFD, INS.jl, PhiFlow, PICT, and Warp-NS improve absolute error; XLB is
close to neutral. The paired VJP effect is small near this error floor, as
expected.

Solver-specific refined-reference control

Five solvers also complete a method-local task against their own 64²,
half-step target. Those absolute errors are not ranked across solvers because
the targets differ. PICT improves absolute error by 1.119×. The paired
solver-VJP benefit is resolved for JAX-CFD (37.0%), INS.jl (29.8%), XLB
(13.3%), and PICT (4.60%); Warp-NS is inconclusive.

PhiFlow is rejected before training in this control only. Its native state
closes the common-reference task, but the chaotic refined-reference audit has
long-horizon spikes: maximum coarse/fine closure residuals are 0.0101/0.0252
and reach 22.8%/70.7% of the refinement signal.

Interpretation and limits

• The paired VJP comparison is within solver: it isolates temporal
  differentiation while holding the forward recurrence and supervision fixed.
• A positive temporal-credit lift says the eight-step training graph helps
  relative to local supervision; it does not by itself say that the learned
  corrector beats the uncorrected solver.
• Absolute solver and corrected errors remain properties of the chosen target.
  The two-reference sweep tests whether the conclusions survive one independent
  discretization change; it does not establish a universally fair reference.
• The reference agreement reported here applies to this smooth low-wave-number
  initial-condition distribution, viscosity, resolution, and time horizon.
• A corrector can benefit from a solver VJP relative to stop-gradient and still
  fail to beat the uncorrected solver. Both quantities are reported.

Implementation state

• Detect native recurrence only when state is differentiable in both the
  Tesseract input and output schemas.
• Thread solver state through corrector training, evaluation,
  finite-difference checks, long-closure checks, and self-reference generation.
• Stop both velocity and native state in the paired control.
• Add an independent conservative finite-volume reference, exact Fourier
  prolongation for resolution audits, and a gated 128²-versus-256²
  space-time-convergence check.
• Register the spectral/finite-volume sensitivity study as the canonical
  opt-in optimization/solver_in_loop_reference_sensitivity experiment.
• Keep the existing drag-optimization registrations unchanged.

Complete merged JSON, plot inputs, full-field/physics plots, GIFs, tables, and
job provenance are in the
artifact index.

Validation and offline execution

All experiments ran offline on Kander through the shared Slurm/Pyxis solver
tooling. No hosted benchmark was triggered; this draft carries
benchmark:none.

• reference convergence audit: 1697345;
• all-held-out reference-disagreement audit: 1697383;
• pseudo-spectral reference production: 1697346, 1697347, 1697348,
  1697349, 1697350, 1697351;
• finite-volume reference production: 1697352, 1697371, 1697372,
  1697373, 1697374, 1697375;
• reference-sensitivity merge/render: 1697376;
• existing nonlinear, Taylor–Green, and method-local controls: see artifact
  index;
• numerical-source validation at 21db5ff: Ruff and format passed; 522 tests
  passed and three were skipped (1697344);
• final plot-only layout change: Ruff/format and the offline pre-merge renderer
  passed; hosted Python checks are reported on this PR head.

This remains a draft PR based on #121 until the canonical recurrent-state
contract lands.