TECHNICAL (Route-CLOC (leg 381): verifying CLAY_OBLIGATIONS.md §4

Gate answer, in the gate's own wording: YES) "the reviewer's arithmetic survives (i) with the DSS modulation handled, AND (ii) confirms the bounded-energy reading against the primary text", with one refuted side-clause, one labelling correction, and one repaired derivation step, all routed to integration verbatim in §6 below.

Ceiling: Tier 2. No link of the L1 → L4 chain moved. Clay stays ~0.05%. Verifying an obligation is not such a link. This leg builds no certificate, no enclosure, and attempts no localisation.

• Runner: experiments/p2_route_cloc_v1.py (self-tests: ALL PASS)
• Figure: fig104, writeup/figures/fig104_route_cloc_v1.png, rebuilt by experiments/p2_route_cloc_v1_evidence.py from the curated JSON alone (nothing recomputed), registered in writeup/build_figures.py. Added by a later DOCS-lane pass that closes this leg's missing-figure audit finding; it plots this leg's banked numbers and changes none of them. Caption: fig104, Route-CLOC (leg 381). Panel A, the cutoff bill at the banked Type-I exponent α = 1: the nonlinear residual (ρ^{−1.4993}), the viscous residual (ρ^{−1.4999}, identical scaling to the nonlinear one at exactly α = 1), the divergence defect (ρ^{−0.4996}) and the pressure perturbation at the origin (ρ^{−1.9997}) all shrink with the cutoff radius over ρ = 10 … 1000, while the critical L³ tail stays flat (13.1764 → 13.1773, exponent 1.28e−05). Panel B, the term that never gets cheap: the cube of the discarded L³ tail grows by a constant 326.875 per decade of window, constant to 7.4e−10 across four increments, log-divergent, so no cutoff radius makes it small. Panel C, the DSS exponent equals the SS exponent: 0.499999942 vs 0.500000000, ratio 0.9999998844 (no factor), while dropping the log-periodic modulation, which swings G by 2.99× within one period, biases the fit to 0.4876, 2.47 % off. This is a Tier-2-ceiling verification result: no L1 → L4 link moved, Clay stays ~0.05 %. Panel C shows the banked scalars: the per-sample energy series E(s) and the factor G(s) are not in the curated JSON, so no fit-through-data overlay is drawn.
• Curated data: writeup/data/p2_route_cloc_v1.json, every number below is in it
• Novelty pass: writeup/novelty/leg_381.md, committed before the runner existed
• Journal: experiments/journal/leg_381.md

1. What was verified, and what "handled rather than dropped" cost

1.1 The derivation, independent of the reviewer's

Similarity variables at blow-up time T*, standard normalisation:

y = x / sqrt(T*-t),   s = -log(T*-t),   u(x,t) = (T*-t)^{-1/2} U(y,s)

The NS scaling symmetry u ↦ Λ u(Λx, T* − Λ²(T*−t)) acts in (y,s) as the shift s ↦ s + 2 log Λ. A λ-DSS solution is invariant at the single value Λ = λ, hence

DSS  ⟺  U(y, s + 2 log λ) = U(y, s)          [leg 260's framing, reproduced not assumed]

with exact self-similarity the degenerate case ∂_s U = 0. Then

E(t) = ∫_{ℝ³}|u|²dx = (T*−t)^{−1}·(T*−t)^{3/2}·∫|U(y,s)|²dy = (T*−t)^{1/2} G(s),
G(s) := ∫|U(·,s)|²dy,   G(s + 2 log λ) = G(s).

Structural reason no exponent can shift. The DSS group λ^ℤ is a subgroup of the same one-parameter scaling group that fixes the SS exponents. Restricting ℝ_{>0} to λ^ℤ cannot change an exponent; it can only replace the constant ∫|U|²dy by a periodic function of s. So the (T*−t)^{1/2} is not an artefact of exact self-similarity, and the modulation enters as a bounded positive prefactor oscillating in [min G, max G], which means E(t)(T*−t)^{−1/2} has no limit as t → T* unless G is constant.

1.2 The numerical falsifier

An explicitly-constructed synthetic field that is exactly λ-DSS and exactly divergence-free, poloidal, non-axisymmetric (two orthogonal axes), with tunable algebraic decay α and a strong prescribed log-periodic modulation:

Pψ = curl curl (ψ(r) e) = (ψ'' − ψ'/r)(ŷ·e) ŷ − (ψ'' + ψ'/r) e
U(y,s)  = m₁(s) Pψ_α + m₂(s) Pψ_α,   c·d = 0,   ψ_α(r) = (1+r²)^{(2−α)/2}

This field is not the route-4 candidate and is not claimed to be. Its only job is to catch an algebra error in the derivation.

1.3 The exponent, and what dropping the modulation costs

THE FACTOR: DSS exponent / SS exponent = 0.9999998844, i.e. 1 to 1.2e-7. There is no factor. The DSS exponent is the SS exponent.

What the modulation actually does, measured on the same object:

• relative oscillation amplitude of G over one period: 2.99 (a 299 % swing) (so this is not a small perturbation of the SS case, and it still moves no exponent;
• best-fit period of the residual: 1.0574 in s, against 2 log λ = 1.0613) 0.36 %, i.e. the residual is the log-periodicity and nothing else;
• the naive fit's 2.47 % bias is the price of dropping it. It is a windowing artefact, not an exponent, which is exactly the error a careless reader of the reviewer's one-line arithmetic would make, and exactly why the gate asked for the modulation to be handled.

1.4 The one step of the reviewer's arithmetic that is NOT valid as written

"∫|u|²dx at time t scales as (T*−t)^{1/2}∫|U|²dy, so bounded energy requires U ∈ L²(ℝ³)."

In the case of interest (leg 260's banked object, where U ∉ L²) both sides are +∞, and an identity between infinities cannot carry a conditional. The step needs replacing, not deleting. The truncated law is finite for every α and every ρ, and reduces to the reviewer's statement whenever both sides are finite:

E_ρ(t) = ∫_{|x|≤ρ}|u|²dx = (T*−t)^{1/2} ∫_{|y| ≤ ρ/√(T*−t)}|U|²dy
       ≍ [K(s)/(3−2α)] · ρ^{3−2α} · (T*−t)^{α−1}          (α < 3/2)

verified against quadrature:

Same conclusion as the reviewer's, reached by a derivation that is valid in the case that matters. §4's conclusion stands; §4's proof needed this repair.

2. Where leg 260's banked object sits, in numbers

Leg 260 banked, from Chae–Wolf arXiv:1610.09464 Thm 1.1 read in full by leg 253: every λ-DSS solution is Type-I, |U(y)| ≤ C/(1+|y|), i.e. α = 1.

Two magnitudes worth carrying:

• The L² divergence is linear. Shell integral ratio ∫_{10³<|y|<10⁶} / ∫_{1<|y|<10³} = 1000.32, three decades of window give three decades of mass, reproducing leg 260's own "diverges linearly" independently and on a different (poloidal, non-axisymmetric) field.
• α = 1 is exactly the critical case for the local picture. The fixed-ball energy exponent is −0.00026 against a predicted 0. So the total energy's divergence is a pure far-field statement, not a concentration statement: the energy inside any fixed physical ball is bounded and essentially constant right up to T*.

3. Step (iii): what the admissible-cutoff analysis would consume, as magnitudes

This leg attempts no localisation. Cutting off changes the equation's solution, so the obligation is transferred to §5's persistence question, not discharged. What follows is the bill, not an attempt to pay it.

3.1 Input 1, the certified decay exponent

Built by leg 382 (slot C, Route-DEXC), not by this leg, and this leg does not depend on its landing: every magnitude below is a function of α. What the cutoff analysis needs is a two-sided enclosure [α_lo, α_hi], and every bound below depends only on α_lo. The thresholds that change the answer are α_lo > 1 (fixed-ball energy decays; critical L³ tail finite) and α_lo > 3/2 (global L²). Currently available: α = 1 a priori (Type-I), plus a fitted per-candidate exponent from solver/dssp_screen.py::fitted_far_field_decay_exponent, and CLAY_OBLIGATIONS §8 bullet 2 is right that fitted is not sufficient.

Added after rebase (leg 382 landed while this leg ran; conclusions unchanged, pointer recorded). Leg 382's instrument certifies the exact power-law exponent set and therefore answers EMPTY on every real input, no real profile is an exact power law, so it carries a tolerance δ and returns a certified enclosure of width ≈ 0.8686·δ (half-width ≈ 0.434·δ). Composing that with the thresholds above gives the requirement this leg's magnitudes actually impose on slot C's instrument: the usable quantity is α_lo = α_centre − 0.434·δ, so δ must satisfy δ < (α_centre − 1)/0.434 for the fixed-ball-energy and critical-L³ thresholds, and δ < (α_centre − 3/2)/0.434 for global L², while δ must simultaneously exceed the profile's own departure from an exact power law on the window (leg 382 measured critical tolerances δ* of 0.0697, 0.3157, 3.3529 on its three controls). Whether those two demands on δ are simultaneously satisfiable for the real candidate is not settled by either leg, and is named here as the composed open question rather than assumed away.

3.2 Input 2, the perturbation size as a function of cutoff radius ρ

Cutoff χ_ρ: 1 on |x| ≤ ρ, 0 on |x| ≥ 2ρ, quintic transition. All exponents below are measured on the synthetic field and agree with the closed forms to ≤ 0.1 %:

Absolute sizes at α = 1 (ν = 1, C the synthetic field's own amplitude), so the decay is visible as numbers and not only as slopes:

Three readings, each a magnitude:

1. At exactly the Type-I exponent the nonlinear and viscous cutoff residuals scale identically, both ρ^{-3/2} (1/2 − 2α = −α − 1/2 ⟺ α = 1). For α > 1 the nonlinear one is the smaller. So at the banked exponent there is no regime in which one of the two can be neglected.
2. The pressure non-locality is not the obstruction. |δp(0)| ≍ C²ρ^{−2α}, to be compared with the profile's own pressure scale (T*−t)^{−1}: the ratio is ρ^{−2α}(T*−t) → 0. Cutting off far away does perturb the pressure everywhere, including at the singular point, but by a relatively vanishing amount.
3. What does not go away is the tail's size in critical norms. Shown, not asserted: the cube of the L³ tail grows by a constant 326.875 per decade of window (spread across five window widths: 7.4e-10 relative). At α = 1 the discarded far field is not small in L³ no matter how far out the cutoff is placed.

3.3 What this does not discharge

The residuals do go to zero with ρ. Smallness of a residual is not persistence of a blow-up. What must be shown is that the localised solution still loses smoothness at a finite time, over a time interval of length ~T*, with the cutoff sitting at similarity radius ρ/√(T*−t) → ∞. That is CLAY_OBLIGATIONS §5, classified there as NO KNOWN METHOD, AND THE HARDEST ITEM, and this leg does not attempt it. §4's own sentence ("cutting off changes the equation's solution, so this obligation is not discharged by the cutoff) it is transferred to §5": is verified as correct, and the magnitudes above are what §5 would be handed.

4. Step (ii): the official Clay statement, read directly

Source: Charles L. Fefferman, Existence and smoothness of the Navier–Stokes equation, the official Clay Mathematics Institute problem description, https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf (TeX creation date 2006-08-04, last modified 2013-02-05). Fetched by this leg and its text layer extracted; this repository had never previously cited the problem statement itself (novelty pass §1).

Verbatim, the conditions CLAY_OBLIGATIONS §4 and §5 depend on:

(4)  |∂ᵅ_x u°(x)| ≤ C_{αK}(1+|x|)^{-K}         on ℝⁿ, for any α and K
(5)  |∂ᵅ_x ∂ᵐ_t f(x,t)| ≤ C_{αmK}(1+|x|+t)^{-K} on ℝⁿ × [0,∞), for any α, m, K
(6)  p, u ∈ C^∞(ℝⁿ × [0,∞))
(7)  ∫_{ℝⁿ}|u(x,t)|² dx < C  for all t ≥ 0   (bounded energy)

(C) Breakdown of Navier–Stokes solutions on ℝ³.  Take ν > 0 and n = 3.  Then there exist a
    smooth, divergence-free vector field u°(x) on ℝ³ and a smooth f(x,t) on ℝ³ × [0,∞),
    satisfying (4), (5), for which there exist no solutions (p,u) of (1),(2),(3),(6),(7)
    on ℝ³ × [0,∞).

Clause-by-clause verdict on the reviewer's load-bearing paragraph (4 confirmed, 1 corrected, 1 refuted):

§4's premise, that the Clay statement requires bounded energy, is verified against the primary text, as numbered condition (7).

4.1 The refuted clause is a real relaxation, and it is NOT a shortcut

(C) permitting a forcing genuinely weakens the obligation: a breakdown candidate may carry an f, provided it is smooth and satisfies (5). It does not follow that the cutoff residual can be absorbed into f and §4 declared discharged. To do that one would define f as the residual of the truncated field; but (C) requires f to be smooth on ℝ³ × [0,∞) and to satisfy (5) while requiring that no smooth bounded-energy solution exists on [0,∞). So f must be specified past T*, where the candidate field does not exist, and cutting f off in time before T* removes exactly the forcing that was making the field a solution. The relaxation should be recorded; it discharges nothing.

4.2 One option named, with its cost, and no leg authorised by it

Statement (D), breakdown on ℝ³/ℤ³, carries no decay condition and no bounded-energy condition: its acceptance conditions are (10) periodicity and (11) smoothness only. A torus target would make CLAY_OBLIGATIONS §4 vacuous by construction. This is recorded as an option with its own cost, not a recommendation: the route-4 object is a DSS profile on ℝ³ and is not periodic, and re-targeting would re-open §2's rigidity screen from scratch, §2 being, per the obligations document, "the programme's strongest position". No leg is authorised by this note. It is integration's and the user's call whether it is worth a scoping question.

5. Ceiling, and what this is not

• Tier 2. Verifying an obligation is not a link of the L1 → L4 chain. Clay stays ~0.05 %.
• The numerics are float64 quadrature of a synthetic field. Nothing here is interval-certified, and nothing here measures the real candidate's α. A synthetic field can confirm an identity and can refute one; it cannot certify the object's decay.
• The α = 1 used throughout as "the banked value" is the a-priori Type-I bound (an upper bound on |U|, hence a lower bound on the decay rate), not a measurement of the candidate. If the real profile decays faster, every magnitude in §3 improves in the direction shown, and the thresholds in §2 say exactly how much faster it would have to be.

6. Routed to integration, verbatim

CLAY_OBLIGATIONS.md is integration's to amend; this leg edits no obligations file. The requested edits:

1. §4 is verified as a specification. The (T*−t)^{1/2} survives an independent derivation for genuine DSS with the modulation handled; the DSS exponent equals the SS exponent exactly (ratio 1 to 1.2e-7); the localisation problem is confirmed load-bearing; its inputs are named in §3 above. The DRAFT-UNVERIFIED header may be updated for §4 only: §1, §2, §3, §5, §6, §7 were not verified by this leg.
2. §4's derivation needs one repair. Replace "∫|u|²dx at time t scales as (T*−t)^{1/2}∫|U|²dy, so bounded energy requires U ∈ L²" with the truncated law E_ρ(t) ≍ ρ^{3−2α}(T*−t)^{α−1}, because in the case of interest both sides of the original are +∞. Conclusion unchanged; derivation now valid.
3. The target paragraph's "with f ≡ 0" is REFUTED and should be struck: statement (C) permits a smooth forcing satisfying (4),(5). Add the note in §4.1 above so the relaxation is not misread as a shortcut.
4. The labelling should read (C), not "direction (b)".
5. Add, in §4, that (4) binds every derivative of the data, and that a compactly-supported cutoff satisfies it trivially, so the decay-of-data clause is not the binding difficulty; bounded energy is.
6. §7 was not checked (the Millennium Prize rules: refereed journal, two-year wait, general acceptance). Its own text already says "check against the Clay Institute's own published rules before relying on it", and that instruction still stands: this leg read the problem statement, not the prize rules. Left to a successor.