Route-D v15: the literature check, and what it does to the lane

Phase-2 P2, Route-D leg 15. Data: writeup/data/p2_literature_scope.json. No figure, because there is no measurement (this leg produced no numbers of its own, and manufacturing a plot out of a reading list would misrepresent what it is.

Rigor level: 0) this is scoping, not mathematics. And it carries a limitation severe enough to belong in the first paragraph rather than a footnote.

0. The limitation, stated first

Not one of the papers below was read. This container's network policy blocks arxiv.org and every publisher domain reached (WebFetch and curl both return 403 at the proxy's CONNECT step) so the only instrument available was web search, i.e. titles, abstracts and result-summary paraphrase.

That makes everything here a lead, not a fact. Identifiers (titles, authors, arXiv numbers, DOIs) are reliable; technical statements are paraphrases and need verifying by someone who can open a PDF. The whole point of doing this leg anyway is that a labelled unverified lead is useful and an unlabelled one is a liability, so every entry in the data file carries an explicit confidence and a must_verify list.

Read the rest with that in front of you.

1. Why this leg happened now

The standing "what would actually be worthwhile" list has carried the literature search as item (5) for three legs with the same note attached: it currently BLOCKS every novelty claim the project might make. The previous leg (v14) added a new one to that pile, an exact first integral of the profile equation, and flagged its own novelty as unchecked. Doing the check before building anything else on it is the cheap, obvious move, and the continuation prompt had just promoted it to run in parallel with the next brick.

It came back with more than expected, and the useful part is not about v14.

2. What the search found

(a) The object this project has been building a certificate for appears to have a rigorous existence proof already. arXiv:2603.25104: Huang, Tong, Wang, March 2026, Self-similar finite-time blowups with singular profiles of the generalized Constantin–Lax–Majda model. Search-level summary: the two-scale blowup's inner profile is governed by a traveling wave on the smaller scale, and the existence of those traveling wave solutions is established rigorously via a fixed-point method. 55 pages, 19 figures.

That is this project's object. Route-D's entire premise has been that certifying the two-scale traveling wave would be a Level-2 result. If the reading is right, existence is already a theorem, and by an analytic fixed-point argument, not a computer-assisted one.

The same abstract carries a second item that may matter more: the two-scale scenario is described as the a ≤ 0 case, with a > 0 giving one-scale self-similar blowups with regular profiles. This project has been working the two-scale object at a > 0 throughout. That single sign is the highest-value thing on the verification list, because it bears on whether the object is the right one at all, and it is also the least well established of anything here: it comes from one sentence of a search summary.

(b) Compactly supported profiles in this family are known, and so is reducing the profile equation to a scalar fixed point. arXiv:2305.05895, same group, 2023. gCLM admits exact self-similar blowups with interiorly smooth profiles for all a ≤ 1, either smooth on the whole line or compactly supported and smooth in the interior of their closed supports; existence via the fixed point of an a-dependent nonlinear map R_a, where a fixed point f gives Ω(x) = −x f(x) and the scaling constants are explicit integrals of f. The a = 0 and a = 1/2 profiles are strictly negative on (0,∞); the a = 1 profile is compactly supported.

So: v12's "the profile ends" is inside known territory as a phenomenon, and v14's reduction is inside known territory as a technique. (Note the pattern differs: theirs is compactly supported at a = 1 in the self-similar problem; ours is compactly supported at every a > 0 in the two-scale traveling wave problem. Different objects, and the difference is itself worth verifying rather than explaining away.)

(c) gCLM traveling waves have been computed since 2014. Okamoto, Sakajo, Wunsch, Steady-states and traveling-wave solutions of the generalized Constantin–Lax–Majda equation, DCDS 34 (2014) 3155–3170: steady states and traveling waves computed, asymptotic behaviour described, existence for every a > 0 argued numerically.

(d) Compactly supported self-similar De Gregorio solutions on the line. arXiv:2209.08232: infinitely many, mutually distinct under rescaling, ω ∈ H¹ odd and supported on [−1,1].

(e) The single most decision-relevant item: computer-assisted proofs in this exact family are routine. Interval arithmetic (INTLAB) plus Newton–Kantorovich is established practice here, the Chen–Hou 2D Boussinesq / 3D Euler rigorous-numerics papers, and, per search summary, computer-assisted proofs of asymptotically self-similar one-scale blowups for the CLM model itself.

(f) One extraordinary claim, recorded so a later session does not rediscover it and get excited. arXiv:2604.09949, April 2026, single author, claims stable finite-time singularity formation for 3D Navier–Stokes via a computer-assisted Newton–Kantorovich validation of a "5D-lifted" profile. Do not use, do not repeat as a result, do not re-plan around it. A solo preprint claiming a Clay Millennium Problem is overwhelmingly likely to contain an error, and nothing here is in a position to check it.

3. What this does to L1, said plainly

L1 in the standing chain is: a certified (Level-2, computer-assisted) self-similar blow-up profile for the 1D gCLM/HL toy model at some a > 0. Fifteen legs of Route-D have been aimed at it.

On the evidence above, L1 is occupied territory. The technique is routine for the groups working in this area, and the specific object has (apparently) had its existence proved by other means five months ago. Finishing the certificate would therefore be:

• a capability demonstration, real value, and exactly what standing brick (4) always said it was ("worth doing for the capability, not the result"); and
• a reproduction, not a novel contribution.

That is not a reason to be gloomy and it is not a reason to stop; it is a reason to stop calling it the lottery ticket. The honest version: the marginal value of the next Route-D leg just dropped, and it dropped for a reason external to the work rather than because the work went badly. v14's kill switch passing was real. It was also, plausibly, a rediscovery.

What this does NOT do: it does not touch the two structural walls (a search programme can only argue for blow-up; interval numerics only reaches toy models), it does not change the ~0.05% Clay odds, and it does not invalidate a single measurement in v3–v14. Being second is not being wrong.

4. v14's novelty claim, retracted to the right level

v14's writeups said the first integral's novelty was unchecked and that it was "exactly the sort of thing that is folklore to people who work on gCLM/De Gregorio". That hedge was correct and is now stronger than a hedge:

Presume the first integral is known. A group has published a fixed-point existence proof for this exact traveling wave, and the natural way to build such a fixed point for Ω H(Ω) = E Ω_X is precisely this reduction. Until arXiv:2603.25104 and arXiv:2305.05895 are read, the correct posture is that v14 rederived a known structure.

The mathematics of v14 is unaffected: the identity is verified three ways, the kill switch result stands, the reduced solver works. What changes is the framing, and it has been corrected in place in both v14 writeups with the change marked rather than silently edited (banked lesson 35).

5. Gate-check against the Clay chain

(a) Which link does this move? None: it re-prices L1, which is arguably more valuable than moving it. This is the first leg in fifteen to change the estimate of what finishing L1 would be worth, using information from outside the project.

(b) Is another L1 leg the best use of the next chunk? No, not as a novelty play. The recommendation changes: 1. Verify the two things that are load-bearing: the a-sign question in arXiv:2603.25104, and whether its fixed point is this first integral. Both need a PDF, i.e. a session with network access to arXiv, or a human. Until then, treat the two-scale a > 0 object as possibly the wrong object. 2. The DSS lane (standing item (1)) is now clearly the best swing. It was already ranked first for a theoretical reason (Nečas–Růžička–Šverák/Tsai rule out exactly self-similar NS blowup, so a Clay-relevant candidate must be discretely self-similar or otherwise non-generic) and the search did not turn up anyone doing a global profile search for one. That last inference is the weakest thing in this document: absence of search hits is not absence of literature, and it must not harden into "nobody has done it". 3. Finish the certificate anyway, but as a capability, cheaply, and stop calling it the result. It is still the pipeline every other lane needs.

(c) Is there a cheaper experiment that kills the whole route? Reading one paper. That is now the cheapest decisive act available to this project, and it is blocked by a network policy rather than by difficulty, which is worth saying out loud, because it means the binding constraint on the next decision is access, not compute and not cleverness.

6. What is NOT claimed

• No paper here was read. Every technical statement is a search-summary paraphrase.
• "L1 is occupied" is an inference from (e) plus (a), both at medium confidence. It should change what the project prioritises; it should not be written into a public claim.
• The DSS lane being uncrowded is the weakest inference in this document and is recorded as "not contradicted", never as established.
• Nothing here is mathematics. No new object, no new estimate, no new gate. The project's rigor ladder is exactly where v14 left it: Level-1 tooling plus one exact identity, no certificate, nothing interval-enclosed.