Route-L v1, the 2D preconditioner: the stall attributed, and removed

Phase 2 / P2, leg 44. Figure: writeup/figures/fig41_route_l_v1_precond.png. Code: solver/port_certification.py (Route-L additions), test_port_certification.py (10/10), experiments/p2_route_l_v1_precond.py → writeup/data/p2_route_l_v1_precond.json. Working notes: PHASE2_P2_NOTES.md §33.

Step (iii) of the certification chain, an approximate inverse A with a computable Z₁, was declared unreachable on this discretization by Route-K. It is now reachable: an O(N) exact sweep takes the Krylov stall from 0.6623 (flat) to 3.3e−6. That is the first time in 44 legs a blocked link has opened. Steps (i) and (ii) remain blocked, for a different reason, and the obvious explanation for that reason was tested and refuted. No link of the L1→L4 chain moved. Clay odds unchanged at ~0.05%.

0. What this leg is, in one paragraph

Route-K stopped the port at step (iii) with the obstruction half identified: the leading-order radial dilation split took the stall 0.6623 → 0.3596 and the curve stayed flat, so something else of comparable size remained. Route-K named two candidates: the nonlocal Biot–Savart velocity, and the wall. This leg tests them by ablation. Both are wrong. The real term was not on the list, and once found it names its own preconditioner, which works. Then Newton is tried again, still fails, and fails differently, and the obvious explanation for the new failure is tested and does not hold either.

1. L1 (the attribution battery

Six ablations of the residual map, each with its own Krylov ladder, ranked by how much the ladder bends rather than by where it ends (lesson 72) an ablation that removes the obstruction makes the curve bend, and a curve that ends lower while staying flat has removed nothing). Gate: the un-ablated variant reproduces solver.rhs to 0.0, so the battery is measuring ablations and not a reimplementation.

The two candidates Route-K named are both eliminated.

• The nonlocal velocity is not the obstruction: removing it makes things worse. Freezing the Biot–Savart feedback (velocity and modulation held at the base state, so δu no longer responds to δω) takes the stall from 0.6623 to 0.7582. The nonlocal term is mildly helping the Krylov solve. This is the only row in the battery that is worse than the full problem, and it is the one Route-K's writeup put first.
• The wall is not the obstruction either: §2 measures that directly.

The angular transport is. Switching off s_β ∂_β bends the ladder to 0.3712 (gain 2.12); switching off both transport pieces bends it to 0.1608 (gain 4.68). And the decisive combination is in Route-K's own preconditioner: angular transport off plus the radial preconditioner runs to machine zero (0.0401 → 0.0171 → 0.0006 → 0.0 → 0.0). So the transport operator carries the entire obstruction, in two pieces, and nothing else in the equation contributes.

2. L2: the wall hypothesis, retired by measurement

Route-K found the relaxation's surviving defect at the wall, and inferred the Krylov stall might live there too. It does not. Taking the stalled solve at m = 160 and computing where its residual energy sits, in the first three of forty-eight angular nodes (proportional share 0.0625):

ω and η are flat across the domain, the obstruction is distributed, which is what a continuum looks like and is not what a boundary-layer problem looks like. Only ξ shows a genuine wall concentration, and ξ is not what carries the stall. Two different defects were being conflated: the relaxation's, which is at the wall, and the linear solve's, which is not.

3. L3 (the preconditioner, and the negative that made it obvious

3.1 What does not work, and why it is worth reporting

The natural move once "radial plus angular" is the answer is to invert each direction exactly and compose them) classic ADI. It is catastrophically worse than doing nothing: 0.9960 against 0.6623. The operator does not split, and composing two exact 1D solves carries a splitting error larger than the thing being fixed. This is kept in the artifact and in the module because it is load-bearing: it is what forces the correct construction rather than merely permitting it.

3.2 What does work

The radial upwinding is outward everywhere on this profile: measured s_ρ ∈ [0.390, 5.732], strictly positive, and reported as a magnitude so a marginal case would be visible rather than binary. That single fact makes the coupled transport operator block lower bidiagonal in the radial index with tridiagonal diagonal blocks:

[(c − s_ρ/dρ) I − s_β ∂_β] f_i  =  rhs_i − (s_ρ/dρ) f_{i−1}

so one outward sweep of Thomas inverts it exactly, in O(N), with no splitting error at all, because there is no split. Gated (test_7): applying the full transport operator to its own sweep returns the right-hand side to 9.5e−16.

The ladder stops being flat. Route-K's stall verdict classifies the sweep's curve as bending (gain 9.72 across the ladder, against the full problem's 1.05), which is the same classifier, on the same data structure, returning the opposite reading. A is constructible. Step (iii) is unblocked.

4. L4/L5: Newton still fails, differently, and the obvious explanation is wrong

With the linear solve working, the question Route-K left downstream becomes askable.

The linear solves now succeed: GMRES relative residual 2.5e−3 in 200 iterations, against 1.00 before. And Newton still does not converge. ‖F‖₂ creeps 0.8069 → 0.7378 over eleven steps, a factor of 1.09, with the line search accepting only λ = 1/32 at the first step and 1/64 thereafter.

That is a different failure mode, and the difference is the finding. A full Newton step being rejected while the linear system is solved accurately is not a spectral problem; it is the signature of a near-null direction in DF: Newton computes a large step along a direction the residual barely sees, and the line search cuts it back.

The obvious candidate was the scaling gauge, and it is refuted. run(renorm=True) re-pins ω_x(0) and η_x(0) after every relaxation step; F carries no such constraint, so Newton is free to wander along the scaling symmetry. Applying the same projection inside Newton is a two-line test, and it makes things strictly worse: at the very first iteration the line search accepts no step at all (λ falls to 1/1024 and is still rejected), and ‖F‖₂ does not move from 0.8069.

So the near-null direction is not the scaling gauge. It is recorded as unidentified, which is the honest state, and the next leg's job is to find it: the direct route is to compute the smallest singular directions of the preconditioned Jacobian, which is now cheap because the preconditioner exists.

5. Where the chain stands

One link opened. It is the link Route-K called impossible on this grid, and it opened because the obstruction was attributed rather than guessed at: both guesses were wrong, and the battery is what found the right term.

What this is not. Not a certificate; nothing is interval-enclosed and Z₁ itself is still unmeasured (there is no profile to evaluate it at). Not a claim about the object: Chen–Hou certified this profile, and this leg is about our discretization. And not progress on the chain: (i) and (ii) are exactly where Route-K left them.

6. What must be built next, in order

1. Identify the near-null direction of the preconditioned Jacobian. Cheap now: a few inverse-iteration or Lanczos steps through M⁻¹DF. The gauge is ruled out; the candidates worth checking are the c_l/c_ω modulation's implicit dependence (which makes F an implicitly-defined map, not an explicit one) and translation along the profile branch.
2. A bordered system, once (1) names the direction: append the constraint that pins it, rather than projecting after the fact, which is what failed here.
3. Then the profile, the function space, Y₀, and Z₁.

7. Reproduce

.venv/bin/python -u experiments/p2_route_l_v1_precond.py     # ~13 min
.venv/bin/python writeup/4_p2_lottery/p2_route_l_v1_evidence.py
.venv/bin/python test_port_certification.py                  # 10/10