Route-SDSS v1 (leg 313): does leg 260's substantive obstruction survive a SEEDED search?

Gate, in its pre-committed wording:

Does leg 260's substantive obstruction survive seeding, with the (a)/(b)/(c) triple answered either way?

NO. Leg 260's substantive obstruction (verbatim, "Entry B's defining adjective UNSEEDED is incompatible with its object's only function space") holds in exactly one named realization: a finite-box, finite-energy spectral trawl measured in the unweighted L²(dy) norm on R³. It does not hold in the space leg 260's own answer (a) names (weighted L²(ρ), ρ = (1+|y|)^{-s}, s > 1; crossing measured here at exactly s = 1), and it does not hold in the algebraically compactified variable X = |y|/(1+|y|) that the one demonstrated seeded method actually uses, where the Type-I profile is the exact linear zero (1-X) with finite norm √(1/3) = 0.577350. All four of leg 260 §3.4's reasons dissolve under seeding, including the two leg 260 itself named as "the finding".

A different obstruction (not leg 260's, and not a price either) stands in its place, and is measured here. The (a)/(b)/(c) triple is answered in §§1–3.

This is the escalation branch. This leg does not lift the ban, and did not touch plan_of_record.py. The branch is leg/313-sdss-v1, parked, not merged to main. The ruling is the user's. The 3D-solver amendment keys off this same answer and is also the user's to action, not this leg's.

Novelty pass: writeup/novelty/leg_313.md, committed at af3b7e0 before the runner existed. Runner: experiments/p2_route_sdss_v1_scoping.py. Ledger: writeup/data/p2_route_sdss_v1.json, 30/30 self-tests pass. Figure: fig76 (writeup/figures/fig76_route_sdss_v1.png, rebuilt from the JSON alone by experiments/p2_route_sdss_v1_evidence.py, registered in writeup/build_figures.py).

Scope: paper scoping only. No construction, no search run. No solver was built, run, read into, or edited. Everything below is quadrature, a Chebyshev transform of a closed form, and arithmetic: exactly the scope leg 260's own scoping runner operated in.

No link of the L1→L4 chain moved. Clay odds stay ~0.05%. Nothing here is movement toward Clay. Answering a scoping question is choosing what not to try.

0. The named mechanism

THE COMPACTIFICATION TRANSPOSITION: the log-periodic block's regularity defect is an invariant of the entrance, not a property of the domain's far end. You can move it from infinity to a boundary point; you cannot delete it.

Leg 260's obstruction is a statement about a norm on an unbounded domain. Seeding lets the search work in a compactified variable, where that norm is finite, so the obstruction goes. But the same compactification that makes the norm finite carries the far-field block

r^{-1+iκ}   ↦   (1-X)^{1-iκ},        X = |y|/(1+|y|)

and (1-X)^{1-iκ} is the identical functional form to clause (a)'s recorded gCLM difficulty X^{1-iy} (PHASE2_P2_NOTES §26 / Route-E §4.1), a fractional power at a boundary point, now sitting at X = 1 instead of X = 0. The difficulty is conserved under the map. It is not about the seed, which is why seeding does not touch it.

Measured consequence: the compactified block's Chebyshev coefficients decay algebraically at fitted rate p = 3.000, not geometrically. See §1.2 for the cost that buys.

1. (a) THE FUNCTION SPACE (answered

1.1 The same object, three verdicts

The object is the Type-I profile |U(y)| ≤ C/(1+|y|) on R³ (Chae–Wolf arXiv:1610.09464 Thm 1.1, carried from leg 253 via leg 260) not re-derived here). Shell integrals ∫_R^{2R} / ∫_{R/2}^{R} started at R = 1e4, deep in the asymptotic regime, which is leg 260's own hard-won lesson and is reused rather than rediscovered.

Every weighted row agrees with the closed form 2^{1-s} to < 1e-3 relative, and the residual is explained, not tolerated: it is the O((2+s)ln2 / R) correction from using the faithful envelope (1+r)^{-1} rather than the pure power r^{-1}.

Positive control against a banked number. On leg 260's own integrand (the pure power |U| = r^{-1}, whose closed form is 2^{3-p}) this leg's quadrature reproduces leg 260's banked divergent p = 2 row 2.000035 to < 1e-4 (measured 2.000000). The 1.2e-4 gap between that and this leg's 2.000277 is the envelope, not an error, and is separated on purpose so the control tests the instrument rather than flattering it.

The compactified row is the operative one, and it is exact. Under X = |y|/(1+|y|) the Type-I envelope (1+r)^{-1} becomes exactly (1-X): a linear zero at the boundary, representable by any polynomial basis, with L²(dX) norm √(1/3), matched to < 1e-6. This is not a convenience of this leg's choosing: it is the discretisation the one demonstrated seeded method actually uses (Hou arXiv:2405.10916 computes "in a transformed domain on a uniform mesh, which maps back to a highly adaptive physical mesh."

So leg 260 §3.4 reason 2) "a finite-box spectral method represents finite-energy states; a trawl cannot recur near an object outside its own space" (is a true statement about a finite-energy box, and false about the space the seeded entrance runs in. Its realization is named, per lesson 91: the unweighted L²(dy) finite-box spectral trawl.

1.2 Clause (a)'s recorded difficulty, carried) and it survives, transposed

The ban's lift clause (machine-read from plan_of_record.BANNED at run time, never transcribed, self-tests S1.10/S1.12) requires carrying §26/§4.1's difficulty: building blocks of limited regularity at the origin (X^{1-iy}), and the viscous gCLM band absent entirely (max Re = -1e-13 at mu = 0.05).

Both are carried, and they part company:

• The viscous band's absence does not transfer. It is a gCLM statement, and clause (b) of the ban's own text says so ("Nothing about NS. gCLM's scaling structure is not NS's"). Leg 260 §1.1 already explained the mechanism with its crossover formula: the gCLM band sits where dissipation wins, the NS band where the rescaling drift wins by r². This leg transcribes that and re-derives none of it.
• The limited-regularity difficulty DOES transfer: by transposition, and it is the finding of §0. Under the same compactification, r^{-1+iκ} ↦ (1-X)^{1-iκ}.

Measured cost of that transposition, Chebyshev coefficients on [0,1], N = 2^20:

Cost law across the measured rows: n(κ) ≈ 791 · κ^0.954, max relative residual 3.9%, i.e. essentially linear in the log-periodic frequency. Reported as a shape, not an endpoint (lesson 72). At the cheapest κ ≠ 0 row the separation from the smooth control is 82.3× at the same tolerance on the same instrument.

Controls, both directions. * The κ = 0 row is a control that can come out differently: at κ = 0 the exponent 1-iκ is 1, so (1-X)^1 is a polynomial and the defect must vanish, and it does, max|a_n| = 2.1e-17 for n ≥ 3. Had it not terminated, the measurement below would have been an artifact of the transform rather than a property of the object. * The smooth positive control is entire and asymmetric. This matters, and the failure is kept in the artifact: the first draft used exp(-8(X-1/2)²), which is even about the midpoint, so all its odd Chebyshev coefficients vanish identically (max odd |a_n| = 4.2e-17) and the "first coefficient below tolerance" test reported n = 1 for a symmetry reason having nothing to do with smoothness, lesson 90 exactly, a control that could not come out differently. Replaced, and the criterion strengthened to "last index at or above tolerance", which no symmetry can fake.

A resolution correction, kept in the artifact. The first draft ran at N = 2^14. Under an 8× refinement the κ = 20 cost moved 29.8% (10841 → 14073) and the 1e-8 column saturated against the array (n = 16332 of N = 16384). Those numbers were measuring the array, not the object. The runner now works at N = 2^20 and carries a three-level study: the 1e-6 rows are stable to ≤ 0.5% against the 8×-coarser level. The 1e-8 column is withdrawn and is not quoted anywhere.

Extrapolation, explicitly not an NS number. At gCLM's leading |Im| = 430.35 (Route-I) the cost law gives 257,466 modes. Clause (b) forbids importing gCLM's frequency into NS, and this row is reported only to price an import that is not licensed. The NS log-periodic frequency is unknown: it is an output of the very search being scoped.

2. (b) THE OBJECT, answered, and the new obstruction is here

All three ban reasons hold only in gCLM while Phase 0's target is NS. Machine-confirmed from the ban's own lift text (S1.11): the clause contrasts gCLM with NS in as many words. That half of (b) is exactly as leg 254 and leg 260 recorded it, and it is unchanged.

The object under seeding is leg 251's NS3D-DSS-NONAXI-LAMBDA-LARGE: 3D incompressible NS, non-axisymmetric backward DSS, λ significantly larger than 1, profile outside L^∞_t L³, as a search target, a periodic orbit of the dynamically rescaled flow with period T = 2 log λ. The novelty pass re-located that equivalence from an independent published source: Chae arXiv:1306.0305 defines the asymptotically-DSS object as a solenoidal V̄(y,s) with V̄(y,s) = V̄(y,s+S₀), S₀ ≠ 0. DSS is time-periodicity of the rescaled field, published, not this repository's coinage.

The seeded entrance has been operated: on a neighbour object. Hou arXiv:2405.10916 does all three things Route-SDSS's entrance needs: it seeds ("the initial condition for the dynamic rescaling formulation is obtained from a late-stage adaptive-mesh solution, rescaled via parabolic scaling invariance with a soft far-field cut-off"), it works on a transformed unbounded domain rather than a finite-energy box, and it continues in viscosity ν₀. It also names the obstruction seeding exists to defeat, scaling instability, which means a time-marching method "can only get close to the potential blowup without reaching arbitrarily near the blowup time."

And this is where the new obstruction is. That candidate is on the wrong side of the screen, three times over:

1. Axisymmetric. Leg 253's composition (Chae–Wolf Thm 1.1 with the axisymmetric Type-I exclusion, labelled there as that leg's own inference) kills axisymmetric DSS outright.
2. Generalized, not NS. Solution-dependent viscosity, effective dimension ≈ 3.188 (reported as apparently → 3 as background viscosity falls). Phase 0's target is NS.
3. Stationary, not time-periodic. It is a nearly self-similar profile, a fixed point of the rescaled flow, not a DSS orbit.

And the novelty pass's located gap closes the alternative: across every query and spelling variant, no source computes a genuinely discretely self-similar (time-periodic in s) blow-up profile for Navier–Stokes numerically. The genuinely time-periodic computations that exist are forward DSS (Tsai) or other equations (NLS log-log, Keller–Segel), where unstable modes are removed using symmetries and the spectral analysis of a compact linearised operator about an explicit ground state, "a luxury unavailable for NSE, since there is no explicit ground state and the linearized operator is not compact."

THE SEED SET FOR THE SCREENED OBJECT IS EMPTY. No published numerical DSS candidate for 3D NS survives the NRS/Tsai + axisymmetric screen, so a search "seeded from a known numerical DSS candidate" has, today, nothing to be seeded from.

This is a new obstruction, it is not leg 260's, and it is not a price: but it is an availability fact, not an impossibility. Two published recipes manufacture a seed rather than find one: Hou's own ν₀-continuation plan (drive c_l(ν₀) → 1/2, n(ν₀) → 3), and Chen arXiv:2605.15149 (14 May 2026), whose rigorous construction runs a fixed-point argument around a numerically constructed approximate profile. Manufacturing the seed is therefore the first item of the price, not a reason the route is closed, and saying which of those it is, is the user's ruling, not this leg's.

Wall 2, explicitly. The screened object is non-axisymmetric 3D with no symmetry reduction available. That is squarely on the far side of Wall 2, and this leg claims nothing otherwise.

3. (c) THE PRICE: answered, with a correction to leg 260

3.1 The build floor ROSE, and leg 260 told us to check

capabilities.py is read live by the runner, never hardcoded.

Leg 260 wrote that the floor "should be re-read, not cited" and predicted it would fall as this repository delivers modules. 53 legs later the module count is unchanged at 48, so the floor ROSE from 27 to ≈33. Leg 260's instruction was right and its prediction's direction was wrong; both are recorded. The five must-build modules are unchanged from leg 260 §3.1 (3D NS in similarity variables; 3D Leray/Biot–Savart; an unbounded-domain discretisation carrying an algebraic far field with log-periodic oscillation; a periodic-orbit search; a phase/gauge condition). The standing grep-first ban is honoured: 0 modules hold a periodic-orbit search, 0 hold a 3D velocity field, 0 hold a Leray projection.

Seeding does change one of the five in kind, though not in count. The periodic-orbit search stops being a global trawl and becomes a local Newton–Krylov / multiple-shooting continuation, 22-year-old published machinery at NS discretisation dimension O(10⁴) (Sánchez–Net–García-Archilla–Simó, J. Comput. Phys. 201 (2004) 13–33), whose own guidance is that good initial conditions matter most for exactly this task. It is still absent from this tree; it is no longer unprecedented.

3.2 The resolution demand, now measured rather than assumed

This is the contribution leg 260 could not make, because it had not measured the transposed block.

The deliverables remain incomparable in kind, exactly as leg 260 said: Phase 1's rung outputs a Grade-A certificate from a 1D ODE enclosure; this outputs a float candidate on the far side of Wall 2.

3.3 What leg 260 said "resists" no longer resists

Leg 260's §3.4 reason 4 ("a time-boxed negative would not be reportable", because a cold search over ~4.2e6 dimensions has no coverage metric) is a property of a cold search. A seeded Newton–Krylov continuation reports a residual, a condition number, and a continuation arclength before failure: its failure is a magnitude, in this repository's own required form. So the search step becomes costable in precisely the sense leg 260 found missing, which is the sense its gate's word "actionable" required.

4. All four of leg 260 §3.4's reasons, under seeding

Leg 260 named reasons 2 and 3 as "the finding". Both are the ones that dissolve most cleanly. That is the gate's answer.

5. A ban-scope observation, REPORTED, NOT ACTED ON

Machine-read from plan_of_record.BANNED at run time (S1.4–S1.6, S1.13):

• Entry A bans "any attempt to obtain a DSS orbit by BIFURCATION OFF A FIXED POINT of a rescaled flow (Hopf or otherwise), inviscid or viscous."
• Entry B's object clause is "a GLOBAL periodic-orbit search of a rescaled flow with no fixed point nearby to seed it."
• Neither entry names a search seeded from a known numerical DSS candidate.

A seeded search of that kind is not a bifurcation off a fixed point (Entry A's object) and is not a search with no seed (Entry B's object). Whether Entry B's substantive basis was intended to reach it is a question about the ban's wording, and a wording question is not a judgement an agent may make: precisely the shape of leg 304's escalation over Cadiot.

THE BAN STANDS IN FORCE, UNCHANGED, AND BINDS EVERY LEG. This leg has no authority over plan_of_record.py, did not touch it, and does not lift, re-pose, or weaken anything. The observation is handed to the user with the (a)/(b)/(c) triple attached, which is what the dispatched no-branch requires.

6. Honest ceiling

• A dissolved obstruction is not a discovery, and it is not permission. What this leg establishes is that leg 260's stated reason is scoped to a realization it did not name, and that a different obstruction, an empty seed set for the screened object, now occupies the position. Neither fact opens anything on its own.
• A THIRD obstruction, and the only one of the three that is a theorem. Located on this leg's MF3 re-audit of its own novelty pass (writeup/novelty/leg_313.md §8a): Chae–Tsai prove nonexistence for DSS solutions with time-periodic V under decay assumptions on the vorticity profile Ω = ∇×V, unique-continuation type, with S₀ > 0 the temporal period and conclusion V ≡ 0. That is precisely the object of §2, met in the nonexistence direction. The same rigidity line for Euler is Xue arXiv:1408.6619 and J. Nonlinear Sci. 10.1007/s00332-023-09975-1; arXiv:2602.17570 (2026) argues separately that producing NS singularities via Euler self-similar solutions must fail in the case it analyses. Reported at summary level only. Whether the decay hypothesis on Ω excludes the screened object or merely a decaying subclass is not known to this leg and is not guessed, and no claim in §§1–4 leans on it; a follow-up leg must read Chae–Tsai at full text. It does not alter the gate (it is not leg 260's argument) but any ruling on this route should be made with it in view, because it is the one item here that could close the route outright rather than merely price it.
• §0's transposition is arithmetic about a closed form, not a theorem about NS. That r^{-1+iκ} ↦ (1-X)^{1-iκ} is exact; that the NS far field genuinely carries a log-periodic block with a particular κ is hypothesis, and κ itself is unknown. The measured cost table is conditional on that structure, and every extrapolation beyond κ = 20 is labelled.
• The seed-set-is-empty finding rests on a literature search, not on a theorem. It is a located gap at abstract/summary level; no external PDF was read at full text this leg, and arXiv:2405.10916 is the one a follow-up leg should read fully. The (b) answer depends only on that paper's object being axisymmetric, which is in its title.
• The 1e-8 column was withdrawn, and the N = 2^14 numbers that appeared in this leg's own first draft (κ=20 → 10841) are superseded by the converged N = 2^20 ones (14149). Both are recorded rather than quietly replaced.
• Two of this leg's own controls were defective on the first pass, a symmetric smooth control and a tautological comparison, and both are described in §1.2 with their repairs. Lesson 90 caught them; they are kept in the artifact.
• Dependencies live on parked branches (legs 251, 253, 257, 260) and are carried with locators and parked status stated, never re-derived.
• No compute beyond quadrature, one Chebyshev transform, and arithmetic. No solver built, run, read into, or edited. No search run.
• Clay odds ~0.05%, zero links of L1→L4 moved, Walls 1 and 2 both stand.