The free lunch, priced (leg 390)

There is a version of the Clay Navier–Stokes problem that quietly deletes the hardest thing on our obligations list. Fefferman's official statement comes in four parts, and part (D) asks for breakdown on the torus, periodic boundary conditions, instead of on all of ℝ³. Part (C), the ℝ³ version, requires a hypothetical blow-up solution to have bounded energy. Part (D) does not.

Our CLAY_OBLIGATIONS.md §4 is entirely the bounded-energy obligation and the localisation work it generates. If (D) deletes the premise, §4 evaporates. That is what a free lunch looks like.

Free lunches in this repository have a history of turning up on a later bill. So this leg did one thing: it read the price tag. It authorises nothing, it changes no target, and it makes no recommendation. When you finish this page you will know the magnitudes and you will still be facing the choice, which is the point.

The deletion is real

We didn't take the earlier note's word for it. The runner parses Fefferman's own banked text and splits each breakdown statement into what the data must satisfy and what a solution would have to satisfy:

• (C) solution conditions: (1), (2), (3), (6), (7)
• (D) solution conditions: (1), (2), (3), (10), (11)

Conditions (6) and (11) turn out to be the same text. So going from (C) to (D) exactly one condition leaves and exactly one arrives: (7) bounded energy out, (10) periodicity in. §4's premise is genuinely absent from (D). (Plant (7) back into the list and the check reports it present, the green light can go red.)

One honest gap: (D)'s data conditions (8) and (9) are not in this repository, and this leg has no outreach. Nothing here is priced from them.

Then the rigidity bites

Here is the thing that shapes everything else. Our target object is discretely self-similar: it reproduces itself under a zoom by a fixed factor λ. Zoom is exactly the thing a torus does not have.

If a field is both periodic on a box of side L and exactly λ-DSS, then it is also periodic on a box of side L/λ, and L/λ², and so on forever, which means it is constant. A non-constant exactly-DSS field on the torus does not exist.

We measured how fast that bites, in a Fourier band of 64 modes per axis. For λ = 1.7, one zoom step leaves 342 modes alive and the second step leaves zero. For λ ≈ 1.6487 (an irrational ceiling from the literature) the first step already leaves zero. Control: λ = 1, no zoom at all, leaves 2.1 million modes alive forever, so the check works.

So (D) cannot be attacked with a torus-native object. You have to take the ℝ³ object and wrap it. That is where the bill is.

The bill

Wrapping means summing infinitely many copies of your profile, one per lattice site. If the profile decays like distance^(−α), the copies sum like Σ|Lk|^(−α) over the 3D lattice, and that converges only when α > 3.

We measured it rather than quoting it. Using shell increments (a running total of positive terms is increasing whatever α is, so its slope can't detect convergence; the increment's exponent flips sign exactly at the threshold), out to a truncation radius of 96 lattice spacings, the measured threshold is α = 2.996995: the exact 3, to a tenth of a percent. Worst error anywhere in the table: 0.00462. At α = 3 exactly it is logarithmically divergent, adding a near-constant 8.71 per doubling.

Now put both obligations in the same unit, the profile's certified far-field decay exponent α:

The deleted obligation comes back at 3.99× the deficit. Making the torus bigger does not help: the L-dependence is a pure prefactor, measured exponent −1.0000000000000002 against a predicted −1.

Unless you cut first, in which case §4's work never left

The realistic route is: cut the profile off at radius ρ, then wrap. Do that and the wrapping bill vanishes (with a box wider than the support, the copies don't overlap at all: contamination measured exactly 0.0, and 4× the cell's own field strength when we deliberately shrink the box, so the check can fail).

But cutting off is precisely the work leg 381 already priced. Every number carries over unchanged, including the one that hurt: the critical L³ tail does not shrink, 326.875 per decade of window, constant to 7 parts in 10¹⁰.

So the sharpest way to say what (D) does:

(D) deletes the acceptance test, not the work.

And it re-opens the thing we were proudest of

CLAY_OBLIGATIONS.md §2 (the rigidity screen, four theorems from the literature that could have killed our object and don't) is described there as "the programme's strongest position."

We machine-read all four against their landed records, by a rule fixed in writing before any number was computed:

Four of four. Zero clearances carry to the torus. Two of them are ℝ³-only not because of the ambient space but because the class of object they talk about is defined by the zoom that the torus doesn't have. A torus target would need that whole screen rebuilt against a periodic-rigidity literature we have never searched. (Planted control rows come out "carries", both directions, so the rule isn't just saying ℝ³-only to everything.)

The credits, at full strength

It would be dishonest to list only the debits.

• (7) really is gone. Measured, not asserted.
• §1's named obstruction points the other way. The reason our profile-existence route is stuck is that every periodic-orbit certification method we could locate closes its tail estimate against a compact domain: census of 6 compact/periodic instances, 1 unbounded (stationary, 1D, different machinery), 0 unbounded periodic-orbit at any weight. A torus is the domain those methods want. The catch, stated plainly: that credit is collectable only by an object that lives on the torus, and the rigidity above says no such exact object exists. We report both and net neither.
• Pressure non-locality is not the torus obstruction either. The image-pressure magnitude sum is log-divergent, but on a cubic lattice the signed sum cancels shell by shell to 2.8e-17. Break the symmetry and it jumps to 0.3458, so the cancellation check is real.

Where that leaves it

Statement (D) is neither the free lunch nor a dead end. It removes a genuine Clay acceptance condition and sits on the favourable side of our §1 domain census; it also empties the rigidity screen to zero clearances, leaves the cutoff analysis exactly where it was, and, if you wrap the uncut profile, charges the deleted obligation back at four times the deficit.

Ceiling: Tier 2. §6's two no-method obligations stay open in every branch. No link of the L1 → L4 chain moved. Clay stays ~0.05%.

No retarget recommendation is made here. The decision is the user's, and this leg makes none of it.

Numbers: writeup/data/p2_route_dtor_v1.json. Full derivation: writeup/4_p2_lottery/TECHNICAL_P2_ROUTEDTOR_V1.md. Figure: fig106.