The one-line arithmetic that decides how hard the last mile is

Leg 381, Route-CLOC. Technical companion: TECHNICAL_P2_ROUTECLOC_V1.md. Data: writeup/data/p2_route_cloc_v1.json.

Last week an external reviewer wrote this repository a document called CLAY_OBLIGATIONS.md, a list of what a numerically-confirmed blow-up candidate would still owe before anyone could call it a proof. The document says of itself, in its own header, that the clauses resting on the reviewer's reading are the ones most likely to be wrong. And then its closing note says: if one clause here is worth verifying first, it is §4.

§4 rests on exactly two things. A one-line piece of arithmetic, and a paraphrase of the official Clay problem statement. This leg checked both.

The arithmetic survives. The paraphrase has an error in it.

The arithmetic

The candidate object is a discretely self-similar blow-up: it repeats itself, but only after a fixed zoom factor λ, not continuously. In similarity variables that means the profile is periodic in s = −log(T*−t) with period 2 log λ: it breathes, log-periodically, all the way into the singularity.

The reviewer's line was: the energy at time t scales like (T*−t)^{1/2} times the profile's own L² mass. That is textbook for the continuously self-similar case. The question this leg was given was whether it survives when the breathing is handled rather than dropped, and if not, by what factor.

It survives. The exponent is 1/2 for both, and the ratio of the two exponents came out 0.9999998844: one, to seven digits. There is no factor. The reason is structural rather than computational: the discrete zooms λ^ℤ are a subgroup of the same scaling group that fixes the self-similar exponents, and restricting a group cannot move an exponent. All it can do is turn a constant into a periodic function.

But "handled rather than dropped" turned out to be the operative phrase. The breathing on this object is not a small correction: the profile's L² mass swings by a factor of about three within a single period. Fit the energy naively, ignoring the modulation, and you get an exponent of 0.4876 instead of 0.5: wrong by 2.5%, and wrong in a way that looks exactly like a real physical finding. Put the log-periodic factor into the fit and the exponent snaps back to 0.5 to eight digits. The 2.5% is the price of dropping the modulation, and it is the mistake the gate was written to catch.

One more thing came out of the re-derivation, which the reviewer could not have caught in one line: in the case that actually matters (where the profile has infinite L² mass, which is the whole point of §4) both sides of the reviewer's identity are infinite, and an equation between infinities cannot prove anything. The fix is to run the same argument on a finite ball, where every quantity is finite and the conclusion is identical. Same answer; now with a derivation that holds where it is needed.

What the object's numbers actually are

Every discretely self-similar solution decays at least like 1/|y| in the far field: that is a theorem (Chae–Wolf), banked here at leg 260. Call that exponent α = 1. Then:

• finite total energy needs α > 3/2. The available bound gives α = 1. Short by exactly one half: the certified exponent would have to be 1.5× the one the literature guarantees.
• the energy inside any fixed ball, right up to the singular time, scales like (T*−t)^{α−1}. At α = 1 that exponent is zero, measured at −0.00026. The energy near the singularity is bounded and essentially constant. The infinity is entirely a far-field statement, nothing is piling up at the origin.

That second point is the useful one, and it is why localisation is the right instinct: the problem is out at infinity, so cut it off out at infinity.

Why that instinct doesn't finish the job

Cutting off changes the solution. §4 says the obligation is therefore transferred to §5, persistence, rather than discharged. This leg's job was to price that transfer, not to attempt it, and the prices are all clean powers of the cutoff radius ρ, each measured to better than 0.1%:

• the error the cutoff introduces into the equation does shrink: ρ^{-3/2}, both the nonlinear and the viscous part (and at exactly α = 1 those two scale identically, so neither can be neglected;
• the pressure, which is global and feels the cutoff instantly everywhere, is perturbed at the singular point by only ρ^{-2}) relative to the profile's own pressure that ratio vanishes. The pressure non-locality is not the obstruction, which is worth knowing, because it looks like one;
• but the part you throw away is not small in any critical norm. Its L³ size grows by a fixed amount for every decade you push the cutoff further out: constant to nine digits across five window widths. You cannot make the discarded tail negligible by cutting further out. You can only make the equation error small.

So the transfer to §5 is real, and now it is quantified: §5 inherits a problem where the forcing error is ρ^{-3/2} small but the thing removed is never small.

The paraphrase, and the error in it

The reviewer's summary of the Clay problem said the target is initial data that is smooth, divergence-free, decaying faster than any polynomial, with f ≡ 0, such that no smooth solution with bounded energy exists.

This repository had never actually cited the official problem statement. So this leg fetched it and read it. Four clauses check out, one needed a labelling fix, and one is wrong:

• bounded energy is confirmed; it is numbered condition (7), verbatim, uniform in time. §4's premise stands.
• the decay condition is confirmed and is in fact stronger than stated: it binds every derivative, not just the field. That happens to be the easy one: a compactly supported cutoff satisfies it for free.
• f ≡ 0 is wrong. Fefferman's breakdown statement on ℝ³ permits "a smooth f(x,t) … satisfying (4),(5)". The words "take f(x,t) to be identically zero" appear only in the two existence statements. A blow-up candidate is allowed a rapidly-decaying forcing.

Before anyone gets excited: that relaxation is not a shortcut, and the technical companion explains precisely why the obvious exploit, define the forcing to be the cutoff error, fails on the statement's own terms. But it is a real change to what the programme owes, and it is exactly the kind of thing you only find by reading the primary source instead of the paraphrase.

What this is not

It is not progress on the Clay problem. Verifying an obligation is not discharging one, and nothing here moved any link of this repository's L1 → L4 chain. The odds stay at ~0.05%. The numerics here are float64 quadrature of a synthetic field built specifically so that any algebra error in the derivation would show up as a disagreement: it is a falsifier, not a measurement of the real object. The certified decay exponent that §4 actually needs is being built by a different leg, and this one deliberately expressed every magnitude as a function of that exponent so it can be dropped in when it exists.

What it is: the most valuable clause of the obligations document, checked, corrected in two places, and priced.