Route-V v0, the novelty gate: certification under dissipation was done in 2024, and here it is re-derived

Phase 2 / P2, leg 48. Code: solver/viscous_novelty.py + test_viscous_novelty.py (8/8), experiments/p2_route_v_v0_novelty.py → writeup/data/p2_route_v_v0_novelty.json → fig43. Working notes: PHASE2_P2_NOTES.md §37. Deterministic, 84 s.

Stage V ("switch dissipation on and ask whether the radii polynomial still closes") is pre-empted. Dahne–Figueras (arXiv:2410.05480) verify whole branches of self-similar singular solutions of the complex Ginzburg–Landau equation, continued in the dissipation parameter from the conservative NLS limit, in interval arithmetic. This leg re-derives their published zeros to 1.8e−07, reproduces their branch to 3.0e−06 in the dissipation parameter across its whole length, and locates their fold at ε* = 0.0606365 against their own figure's 0.0606361. The plan's pre-committed fallback applies: stage V is closed, stage C-PILOT is next. No chain link moved; Clay unchanged at ~0.05%.

1. Why the gate exists, and why it is a ban rather than a habit

plan_of_record.py carries stage V behind a gate with both branches written down in advance:

FIRST: has anyone already done certification-under-dissipation for a self-similar blow-up profile? yes → report it, fall back to stage C-PILOT, and do NOT spend the leg. no → proceed to the deliverable.

and a standing ban: building stage V's measurement before its novelty check has reported. The reason is leg 42, which read the primary sources for the first time and deleted seven of twelve standing novelty claims. Stage V was proposed as a speculation of exactly that kind, and was flagged as one when it was written.

The gate is asked here the way Route-J (§31) asks literature questions: not with a paragraph asserting that a check happened, but with code that re-derives the published result from the published equations. A literature claim that cannot be re-run decays at the rate of memory (banked lesson 68).

2. The answer: YES, and the paper is Dahne–Figueras

arXiv:2410.05480, Dahne & Figueras, Self-Similar Singular Solutions to the Nonlinear Schrödinger and the Complex Ginzburg–Landau Equations (Oct 2024). Their equation is

i u_t + (1 − iε) Δu + (1 + iδ) |u|^{2σ} u = 0

and ε is a dissipation dial: ε = δ = 0 is the focusing NLS, conservative; ε > 0 is Ginzburg–Landau, in which the Laplacian acquires a dissipative real part. The Zakharov ansatz reduces blow-up to a singular ODE for the profile Q(ξ), and their Theorem 4.1 proves the existence of eight continuous branches of self-similar singular solutions in Case I (d = 1, σ = 2.3), each born at an NLS solution and followed as ε grows, verified along the whole branch by a rigorous shooting method in interval arithmetic. Theorem 4.4 does the same in Case II (d = 3, σ = 1) and verifies only parts of the branches.

That is stage V's question (does the certificate still close as dissipation turns on, and where does it stop closing) asked and answered two years ago, with rigour this project does not have. A second, weaker precedent (arXiv:2404.04054) certifies self-similar profiles of parabolic PDEs including a viscous Burgers equation by Newton–Kantorovich in a weighted Sobolev space, without following a dial.

The one hole, stated precisely because it is tempting. Twelve arXiv queries are logged in SEARCH_LOG; the four asking for the fluid version (a viscous Boussinesq/Euler/CLM blow-up certified under a viscosity dial) return nothing. That hole is real. It is not what stage V asked for. Narrowing "has anyone done certification under dissipation" to "has anyone done it for our model" after seeing the answer is the move leg 42 deleted seven claims for.

3. The re-derivation (V0-2), which is what makes the gate a check

Their profile ODE is integrated from the origin by an independent RK4 with a phase-resolving step, and matched at ξ₁ to a three-term far-field expansion derived in this module, not transcribed: with p = −1/σ − iω/κ and Q = γ ξ^p (1 + a₁ξ^{−2} + a₂ξ^{−4}),

a₁ = [(1 − iε) p(p+d−2) + |γ|^{2σ}] / (2iκ)
a₂ = {(1 − iε) a₁ [p(p+d−2) − 2(2p+d−1) + 6] + |γ|^{2σ}[(σ+1)a₁ + σ ā₁]} / (4iκ)

The value equation defines γ by a fixed point, leaving the derivative equation as two real equations in the two real unknowns (μ, κ). Newton on that:

The gate was pre-committed at 1e−06 on the j = 1 rows: 1.8e−07, VERIFIED. The j=4 row is the worst at 1.1e−05 and the reason is stated rather than hidden: they match at ξ₁ = 25, we clip to 20, and §4's ladder shows that is exactly what far-field truncation costs.

γ is deliberately NOT gated. They parameterise the solution manifold at infinity in their §7; our γ is the coefficient in our expansion. Two quantities with the same name and different definitions is how a transcription error hides, so they are kept apart (Route-J's rule: verified and transcribed are different words).

4. The guards, before the claim (V0-3)

• Step halving. Case II, 200 → 400 → 800 steps per oscillation: defect 4.994e−10 / 4.920e−10 / 4.915e−10. Halving moves it by 7.9e−12, two orders below the defect itself, so the residual floor is the far-field truncation, not the integrator.
• Far-field truncation. ξ₁ = 10 / 15 / 20 / 30 moves the returned κ by 1.2e−06 in total, and by 3.6e−08 across the last three. The ladder's shape is reported, not its tightest rung (discipline 72).
• The defect discriminates. At the published zero the defect is 8.2e−07 (relative to |Q'(ξ₁)| = 1.1e−02, i.e. 7.5e−05: the scale is named, not left implicit); perturbing κ by 1e−04 raises it 351×. A diagnostic that is small everywhere gates nothing (lesson 55).

5. The dissipation dial, and their fold (V0-4)

Continuing directly in ε cannot pass a turning point: below ε* there are two solutions, above it none. κ is monotone along the branch, so the continuation is run in κ, solving for (μ, ε), which makes the fold an ordinary interior point instead of a Newton failure.

129 converged records, dκ = 0.005, κ from 0.8531 down to 0.21:

fold          eps* = 0.06063648    kappa* = 0.554682
published     eps* = 0.06063610    kappa* = 0.554644
difference          +3.8e-07             +3.8e-05

The published curve is read as data, not eyeballed. Their Figs. 1 and 2 are pgf vector graphics, so the branch polylines and the axis tick marks are literally in the PDF content stream; read_df_figure calibrates on the ticks and returns (ε, κ) pairs. The calibration checks itself: the eight extracted curves' ε = 0 endpoints land on the eight κ of their Table 1, transcribed from a different page, to 1.0e−05, the width of a plotted line. Against that curve, our branch agrees to max 3.0e−06, rms 1.9e−06 over 13 samples spanning both sides of the fold.

6. The shape of their answer, which is the part stage V wanted (V0-5)

‖J⁻¹‖∞ for the shooting Jacobian is the float analogue of ‖A‖ in a radii polynomial, every certificate constant degrades with it, and it is called an analogue, not a certificate constant, because that is what it is.

eps = 0 (the NLS end)      ||J^-1|| = 7.15e-01     cond(J) = 8.4e+00
mid-branch (eps = 0.0486)  ||J^-1|| = 2.71e-02     cond(J) = 2.1e+01
nearest sample to the fold ||J^-1|| = 1.53e+00     cond(J) = 2.4e+03

Switching dissipation on does not degrade the margin: it improves it, by 26×. The degradation is entirely a fold phenomenon, and the fold's exponent is measured rather than announced: ‖J⁻¹‖ ∼ |κ − κ*|^s with

s = -1.061      (an ordinary quadratic fold predicts -1)

fitted on ten points at offsets 0.08 … 0.005 either side. Measuring an exponent instead of declaring a threshold is Route-F's rule; the exponent is checkable and a threshold near a critical point is biased in the direction you expect.

|Q'(ξ₁)|, the scale the proxy is implicitly divided by, moves only 1.09e−02 → 8.7e−03 over the top of the branch, so the 26× is not a units artefact (discipline 67: gate the quantity the measurement divides by).

7. What this is, and what it is not (V0-6)

Is: a novelty gate, answered YES, with the pre-empting result reproduced from its own equations well enough to be sure the pre-emption is real, and, as a by-product, an independent check on a published computer-assisted proof.

Is not: a certificate (nothing here is interval-enclosed; theirs is the rigorous result). Not about a fluid model: CGL is semilinear and radial; the Hou–Luo target is a transport model on the line. Not novel: reproducing somebody else's certified branch is not a result of ours. Not a chain link, and not evidence about Navier–Stokes. Clay stays at ~0.05%.

8. What it changes in the plan

Stage V closes as DONE (pre-empted), and the fallback the plan named before the check was run, stage C-PILOT, is next. The leg also hands C-PILOT something it did not have: branch_in_kappa is a known-answer object with a dissipation dial, whose answer is published and whose fold is now reproduced to 4e−07. C-PILOT's requirement is exactly an object with a known answer.

The banked lesson (82). A gate whose YES branch costs one leg and whose NO branch costs five is worth asking even when you are confident of the answer, and this one was asked because it was written down in advance, not because it felt necessary on the day. The result cost 84 seconds of compute and saved building interval arithmetic and a tail lemma for a question that was already answered.