Route-DTOR v1 (leg 390): does Fefferman statement (D) delete CLAY_OBLIGATIONS §4, or re-price it?

Gate, in the gate's own wording. Does the (D)-variant ledger re-derive with every §1–§5 obligation either named-as-vacated or priced-as-transferred, AND does the §2 enumeration close with each row's ℝ³-only/carries-to-T³ verdict tied to its landed record?, YES.

CEILING: TIER 2, in this branch and in the other one. §6's two no-method obligations stay OPEN. No link of the L1 → L4 chain moved. Clay stays ~0.05%.

THE RETARGET DECISION IS THE USER'S AND THIS LEG MAKES NONE. This leg prices an option; it does not exercise it. Nothing below is a recommendation and nothing below should be read as one. The magnitudes point in both directions and they are reported at the same strength in both.

• Runner: experiments/p2_route_dtor_v1.py
• Ledger: writeup/data/p2_route_dtor_v1.json
• Evidence script (rebuilds every claim from the JSON, re-runs nothing): experiments/p2_route_dtor_v1_evidence.py
• Figure: fig106 (writeup/figures/fig106_route_dtor_v1.png), registered in writeup/build_figures.py
• Novelty pass: writeup/novelty/leg_390.md, committed before the runner existed (68b027f)
• Journal, with the pre-committed immutable gate: experiments/journal/leg_390.md
• CLAY_OBLIGATIONS.md: read, never edited. sha256 at read time f47a73ac27903d79f63435ba8f4726245059e0c09f931e15cb9c0ed7a2923bde, banked in the JSON. Three corrections are routed to integration verbatim and applied nowhere by this leg.

1. What (D) actually says, machine-read

Leg 381 banked Fefferman's statements verbatim. This leg does not retype them: it parses them out of writeup/data/p2_route_cloc_v1.json, splitting each breakdown statement at the phrase "for which there exist no solutions" into the conditions the data must satisfy and the conditions a solution would have to satisfy.

Going (C) → (D), the solution side drops {6, 7} and adds {10, 11}. But (6) and (11) are the same text (machine-compared, p, u ∈ C^∞(ℝⁿ × [0,∞)) both) so the diff is exactly one condition out and one condition in:

(7) bounded energy OUT, (10) periodicity IN.

(7) is confirmed to be the bounded-energy condition from its own banked text (it contains the string "bounded energy"); (10) is confirmed to be the periodicity condition the same way. §4's premise is therefore ABSENT from (D)'s own text. That much of leg 381's note survives verification intact.

Red path. Planting the string (7), back into (D)'s solution list makes the parser report the bounded-energy premise present: the check fires, so its green is worth something.

The gap this leg cannot close, stated rather than hidden. (D)'s data conditions (8) and (9) are not in this repository at all: leg 381 banked (4),(5),(6),(7),(10),(11),(A),(C),(D) and did not bank (8),(9), and this leg has no outreach. They are treated as UNREAD-IN-REPOSITORY and nothing below is priced from them. Every magnitude here rests on (D)'s solution conditions and on the machine-checked absence of (7).

2. The rigidity that decides the shape of the question

Before any price can be quoted, one thing has to be settled: what would a torus target even be?

Measured statement. If u(·,t) is L-periodic for every t and u is exactly λ-DSS with λ > 1, then applying the DSS relation shows u(·,t) is also L/λ-periodic, hence L/λⁿ-periodic for every n, hence constant in x. A non-constant exactly-DSS field on T³ does not exist.

This is elementary (the novelty pass says so). What is measured here is how fast it bites, in the Fourier band |m|_∞ ≤ M = 64:

The rational case is the interesting one: λ = 17/10 leaves exactly the multiples of 17 alive after one step (7 per axis, 7³ − 1 = 342 nonzero modes) and kills them at the second. The margin for the irrational case decays with band width at a measured exponent −0.99057 against a predicted −1, so the incompatibility is exact for every λ > 1 but not uniform: a field that is DSS only to a tolerance can be periodic to that tolerance if its content above wavenumber ~1/tolerance is negligible. That is a statement about approximate objects and it does not soften the exact one.

Red path. λ = 1, no dilation at all, leaves 2 146 688 nonzero modes alive and never annihilates the band. The check can go red.

Consequence. (D) cannot be targeted by a native torus DSS object. It can only be targeted by periodizing an ℝ³-anchored one, which is why the price below is a periodization price and not a relabelling. Two routes exist, and both are priced.

3. Route 1: wrap the uncut profile. THE BILL.

The cheapest conceivable move: u_per(x) = Σ_{k ∈ ℤ³} u(x + Lk), no cutoff, no modification. For a profile with |u(y)| ≍ (1+|y|)^{−α} the image at lattice site k contributes ~(L|k|)^{−α} at the cell centre, so the question is whether Σ_{k≠0}|Lk|^{−α} converges.

Measured on the actual ℤ³ lattice (spherical truncation, |k|₂ ≤ R, R ∈ {12, 24, 48, 96}), by the shell increment S(2R) − S(R), the partial sum itself is monotone whatever α is and so cannot detect convergence, while the increment's exponent changes sign exactly at the threshold:

Worst error 4.620e-03. Bisection on the measured exponent locates the convergence threshold at α = 2.996995 (against the exact 3; 0.10% low, the residual bias of the finite-R increment fit).

At α = 3 exactly the sum is logarithmically divergent: increments per doubling of R are 8.74999, 8.71536, 8.71356, spread 0.0364, the same signature leg 381 measured for the critical L³ tail (326.875 per decade, spread 7.4e-10), and now the third appearance of that signature in this ledger.

No torus size cures it. S(R, α, L) = L^{−α} S(R, α, 1): the measured L-exponent at α = 1 is −1.0000000000000002 against a predicted −1 (error 2.2e-16). Enlarging the torus buys a constant prefactor and does not touch the R-divergence.

The bill, in leg 381's own currency

Both obligations are quoted in the same unit, the profile's certified far-field decay exponent α, so they are directly comparable. §4's numbers are machine-read from p2_route_cloc_v1.json check_B3, not retyped.

REPURCHASE FACTOR: 3.99399× in deficit, 1.99800× in ratio.

Statement (D) deletes an obligation that needed α > 1.5 and, by route 1, reinstates one that needs α > 3, the same currency, four times the deficit.

Red path. Asserting the α = 1 exponent against a deliberately wrong prediction (2.25 instead of 2.0) fails at tolerance 0.02 while the correct prediction passes: error 0.250452 planted against 4.516e-04 true.

4. Route 2: cut off first, then wrap. What it inherits, and what it adds

Since §2 above shows route 1 has no exactly-DSS object to apply to, the realistic route is: cut off at radius ρ, then wrap on a torus of side L > 2ρ.

What it inherits: leg 381's entire cutoff bill, unchanged, every number (machine-read from p2_route_cloc_v1.json, not retyped): nonlinear residual ρ^{−1.4993}, viscous ρ^{−1.4999}, divergence defect ρ^{−0.4996}, pressure perturbation at the origin ρ^{−1.9997}, and the critical L³ tail that does not shrink, 326.875 per decade of window, constant to 7.4e-10.

This is the load-bearing observation of the whole leg:

What (D) deletes is the ACCEPTANCE TEST (7), not the cutoff analysis that §4 transferred to §5.

What it adds: measured, and smaller than expected. On a smooth compactly-supported exactly divergence-free test field (u = curl(ψ e_z), ψ = exp(−2/(1−|x|²/ρ²))·x·y; divergence measured on a refinement ladder n = 41 → 321, residual falling 3.09×, 3.75×, 3.92× per halving, i.e. second- order truncation error and not a non-solenoidal field):

• direct image contamination of the cell: exactly 0.0 at L/ρ = 3.0, 2.5, 2.0001, disjoint supports. It becomes 0.0389 (1.00× the cell field's own sup) at L/ρ = 1.5 and 0.1554 (4.00×) at L/ρ = 1.0, so the check has a demonstrated red path.
• image pressure interaction, leading multipole. Modelling the image pressure at the cell centre as Σ_{k≠0} T_ij(Lk) M_ij with T_ij = (3k_i k_j/|k|² − δ_ij)/(4π L³|k|³): the magnitude sum is log-divergent (increments per doubling 0.5785, 0.6329, 0.6624, 0.6776, converging to the predicted log 2 = 0.69315; relative error 0.022371 at the top rung, N = 32), but the signed sum cancels shell by shell on the cubic lattice to 2.82e-17 cumulative at N = 32. Breaking the cubic symmetry (half-lattice k_x > 0) leaves 0.3458: the cancellation check can go red.

So the pressure non-locality is not the obstruction on T³ either: the same verdict leg 381 reached on ℝ³, by a different mechanism. That is a credit on (D)'s side and it is reported as one.

5. §2's four screen rows, machine-read

Pre-registered rule (journal §0, rule 2, fixed before any number was computed): a row is ℝ³-ONLY if the hypothesis text in its landed record invokes whole-space structure T³ does not carry, or if the object class it constrains is defined by the dilation action §2 above shows does not act on T³. Both sub-verdicts are reported separately and never netted into one word.

Tally: 4 of 4 rows ℝ³-ONLY. 0 clearances carry to T³.

The two rows whose marker verdict is CARRIES-TO-T3 are instructive and are why the sub-verdicts are reported separately: NRS/Tsai's and Chae–Tsai's decisive content is not about the ambient space at all (Chae–Tsai's decisive clause is that its theorems hypothesise the rescaled Euler system, and that silence about NS is domain-independent). They still come out ℝ³-ONLY because the object class they speak about (exactly-self-similar, or DSS) is defined by a dilation that does not act on T³.

CLAY_OBLIGATIONS.md §2 calls the screen "LARGELY DISCHARGED, and this is the programme's strongest position." That is true of the ℝ³ target and, on this count, of no torus target: §2's rigidity screen re-opens in full, and a torus screen would have to be rebuilt against a periodic rigidity literature this repository has never searched (novelty pass §1). This is the known cost side and it is reported as a count, not as a word.

Red path. A planted control row with no whole-space marker and no ansatz marker ("a smooth solution of the heat equation with bounded gradient") returns CARRIES-TO-T3; a planted row with both returns ℝ³-ONLY. The rule is not one that returns ℝ³-ONLY for everything.

6. The §1–§5 disposition table

Vocabulary closed and pre-registered: VACATED, UNCHANGED, TRANSFERRED; a TRANSFERRED without a magnitude is a NO on the gate's first clause.

§6 is unchanged by this leg, as the spec requires of a scoping answer: item 1 (certified far-field decay and the admissible cutoff) OPEN, item 2 (persistence/stability under localisation) OPEN. If anything item 1 becomes more load-bearing under (D), because the certified far-field exponent α is the currency of the periodization bill as well as of the cutoff bill.

7. Corrections routed to integration, verbatim, applied nowhere by this leg

1. CLAY_OBLIGATIONS.md's (D) paragraph says (D) "carries no decay condition and no bounded-energy condition". Machine-read against (D)'s own banked text, the supportable statement is narrower: (D)'s solution conditions are (1),(2),(3),(10),(11) and contain no (7); its data conditions are (8),(9), whose verbatim text is not in this repository, so no claim about what they do or do not require is supportable here.
2. writeup/data/p2_route_cloc_v1.json check_D_clay_primary_text.verbatim_conditions banks (4),(5),(6),(7),(10),(11),(A),(C),(D) but not (8),(9). A leg with outreach should close that gap.
3. CLAY_OBLIGATIONS.md §2's "the programme's strongest position" is correct as written for the ℝ³ target and would need a scope word if (D) were ever adopted.

8. What this leg is not

It is not a retarget, not build authority, and not a recommendation. It priced an option in both directions: (D) genuinely deletes a Clay acceptance condition and genuinely sits on the favourable side of §1's domain census; and (D) genuinely re-opens §2 to zero clearances, genuinely leaves the cutoff analysis in place, and, if the uncut profile is wrapped instead, charges the deleted obligation back at four times the deficit. The choice between those is the user's, and this leg makes none of it.