TECHNICAL, Route-TRIG (leg 392, unit T2): the periodic (T³) rigidity literature

Lane T · WAVE 1 · CONDUCTOR §3g · branch leg/392-t2-periodic-rigidity Artifacts: runner experiments/p2_route_trig_v1.py, data writeup/data/p2_route_trig_v1.json, evidence experiments/p2_route_trig_v1_evidence.py (41/41 checks, rebuilt from the JSON alone, no re-run), novelty writeup/novelty/leg_392.md, journal experiments/journal/leg_392.md. No figure: a pure-literature scoping leg declares one rather than inventing one (legs 334, 348); writeup/build_figures.py is untouched by this leg.

CEILING: TIER 2. No L1 → L4 link moved, a literature search does not move one, in either branch. Clay stays ~0.05%. CLAY_OBLIGATIONS.md §6's two no-method obligations stay OPEN; leg 390 measured that the torus does not retire them and nothing here changes that.

1. The gate, and the answer

GATE (final wording, pre-registered, not re-scoped). Does the search locate a published theorem excluding a finite-time singularity for 3D Navier–Stokes on T³ of the shape Lane T would need: a periodic/torus analogue of the Nečas–Růžička–Šverák / Tsai rigidity results?

ANSWER: no such theorem was located: and under ORCHESTRATION.md §3d that null is returned as UNDER-RESOURCED, not as no.

The arXiv arm executed 100 % of its planned query set with every control firing, and its zeros are genuine measured zeros. The Semantic Scholar arm (the only instrument in scope that reaches the venue class the ℝ³ originals themselves live in) executed 3 of 8 queries and was throttled on 5. One battery failed its own pre-registered domain control. Both facts are stated in §4 and neither is smoothed over, so the honest verdict on the gate is a cost, not a verdict. §3d is explicit that this is the case where UNDER-RESOURCED is the answer.

This is reported per the pre-committed reading's branch (c) and is NOT a clearance. Leg 348's ceiling clause governs: absence of a positive instance is not a proof of impossibility. The result is "not excluded by anything located, with the search's coverage stated", never "clear", and it does not advance Lane T.

Which pre-committed branch fired: (b), in its literal wording, see §5. (a) did not fire: no T³ theorem covering self-similar or asymptotically self-similar objects exists to narrow with, because none was located at all. (c) governs the reporting of the null. (d) is honoured and raised in §7.

2. The instrument, controlled before it was trusted

Leg 387 fabricated a controlled zero: its harness listed opensearch namespace 1.0 while arXiv serves 1.1, so it refused every response, and a len(entries) implementation would have reported "0 results, no prior art" on the exact question it existed to answer. That is why this leg's verdict is a committed runner rather than a paragraph.

5 of 5 passed. The AND verdict is therefore "AND WORKS: an ANDed zero below is a MEASUREMENT (absence)", which is the precondition for reading anything at all off a zero.

The leg-387 defence returned a number. The namespace actually served, recorded from the raw feed on all 32 substantive queries, is http://a9.com/-/spec/opensearch/1.1/. The hazard leg 387 fell into is real, it is 1.1 against the 1.0 that harness listed, and this parser names no version anywhere (regex on opensearch:totalResults).

MEASURED vs DID-NOT-MEASURE is structural, not editorial. total is None by construction in every THROTTLED and FAILED record, so a non-measurement cannot be silently rendered as a zero downstream: including by a reader of the JSON who never saw this document.

3. What was measured

arXiv, 32 substantive queries in 8 batteries, ≥ 12 s spacing: 32 MEASURED (100 %), 0 THROTTLED, 0 FAILED, 244 total results, 119 distinct ids inspected at abstract level, 5 MEASURED zeros.

The five measured zeros, each with the AND operator demonstrated working and the nonsense control at exactly 0:

The direct form of this leg's question, Liouville/rigidity × Navier–Stokes × torus, returns zero in both spellings, on an instrument whose AND operator is demonstrated live and whose nonsense control is demonstrated dead.

Semantic Scholar: 3 of 8 MEASURED, 5 THROTTLED (HTTP 429). Its two controls passed (Navier-Stokes equations → 149 086; the nonsense phrase → 0) and one substantive query returned (Morrey space Liouville theorem Navier-Stokes → 1 117 hits, every inspected hit steady-state and on ℝ³, none periodic). The remaining five substantive queries did not measure and are banked as THROTTLED with total = None.

4. The coverage, stated as §3d requires

What was covered.

• 100 % of the planned arXiv query set, every query MEASURED, all five controls passing.
• The domain positive control (pre-registered at journal §0.3 as the query set must re-find the ℝ³ rigidity papers this repository has already screened) re-found 2 of 3: 2607.09619 (Pineau–Vicol, row 2) ✓ and 2006.15776 (Jiu–Wang–Wei Morrey, row 4) ✓.

What was not, and each of these is a real hole.

1. The domain control FAILED on one arm. 1304.7414 (Chae–Tsai, row 3) was not re-found by any of the 32 queries. Under the leg's own pre-registered rule, battery E's null is therefore UNDER-RESOURCED and is not reported as absence. The control fired adversely and is reported, not retired.
2. Row 1's own sources are invisible to this instrument, and it was declared in advance. Nečas–Růžička–Šverák (Acta Math. 176, 1996) and Tsai (ARMA 143, 1998) are pre-arXiv. An arXiv-first pass cannot see the venue class in which the ℝ³ originals were published, so it cannot see a periodic analogue published the same way either.
3. Semantic Scholar, the instrument for exactly that gap, was 5/6 throttled on its substantive queries. This is the single largest coverage deficit in the leg.
4. No MathSciNet / zbMATH / Crossref query, because the pre-committed reading names arXiv and Semantic Scholar and nothing else. Recorded as a scope limit, not as absence in the field.
5. Top-8 truncation. 9 of the 32 queries returned more than the 8 records fetched (largest: "discretely self-similar" AND "periodic" at 33), so those batteries were inspected only at arXiv's own top-8 relevance ordering.
6. Abstract-level reading only. No PDF or LaTeX source was fetched this leg.
7. No outreach (standing user hold), so (D)'s data conditions (8) and (9) remain unread, see §7.

Cost of the compliant search (§3d requires a cost, not a verdict):

Total ≈ 1.2–1.7 h plus one user ruling. That is what it would cost to convert this UNDER-RESOURCED into a defensible no, and it is cheap.

5. What was located (the branch-(b) class, quoted verbatim

Branch (b) of the pre-committed reading fires on its literal wording: "a located theorem covering a BROADER class) any bounded-energy periodic solution under a scaling-invariant smallness or regularity hypothesis, bites Lane T directly, and the lane must state what class survives before anything is built." Such theorems exist, they are published, and they are on the torus. They are not analogues of NRS/Tsai (they exclude a singularity under a hypothesis rather than for a self-similar object class) which is why they do not make the gate a yes under the pre-registered K1–K4 rule (K4: the hypothesis class is one a Lane T ansatz would have to sit in, and none of them is).

(b-1) arXiv:1909.09125: Nonlinearity 33 (2020) No. 10, DOI 10.1088/1361-6544/ab9246. PUBLISHED. Global regularity for solutions of the three dimensional Navier-Stokes equation with almost two dimensional initial data. Hypothesis and conclusion, verbatim from the abstract:

"we will prove a new result that guarantees the global existence of solutions to the Navier--Stokes equation in three dimensions when the initial data is sufficiently close to being two dimensional … the closer the initial data is to being two dimensional, the larger the initial data can be in $\dot{H}^\frac{1}{2}$ while still guaranteeing the global existence of smooth solutions … On the torus, however, this approach does give examples of arbitrarily large initial data in the endpoint Besov space $\dot{B}^{-1}_{\infty,\infty}$ that generate global smooth solutions to the Navier--Stokes equation."

Which hypothesis a Lane T ansatz would have to violate: "sufficiently close to being two dimensional", measured in the critical spaces Ḣ^{1/2} / Ḃ^{-1}_{∞,∞}, i.e. the ansatz's genuinely-3D part must exceed this theorem's threshold at the initial time.

(b-2) math/9811161: Electronic J. Differential Equations 1999 no. 11, 1–19. PUBLISHED. Global regularity of the Navier-Stokes equation on thin three dimensional domains with periodic boundary conditions. Verbatim:

"the solution of the Navier-Stokes equation on a thin 3 dimensional domain with periodic boundary conditions has global regularity, as long as there is some control on the size of the initial data and the forcing term, where the control is larger than that obtainable via ``small data'' estimates."

Which hypothesis a Lane T ansatz would have to violate: the thin-domain aspect ratio together with the size control on data and forcing. On a fixed, non-degenerate T³ the theorem simply does not apply, but a Lane T construction that gains its concentration by squeezing one period would run straight into it.

(b-3) arXiv:2009.07631 (preprint, no journal ref): On Ladyzhenskaya-Serrin condition sufficient for regular solutions to the Navier-Stokes equations. Periodic case: "We consider the Navier-Stokes equations in a bounded domain with periodic boundary conditions … The aim of this paper is to prove the bound $|V(t)|{H^1}\le c$ for any $t\in\mathbb{R}+$", under a smallness hypothesis relative to a large-bulk-viscosity Lamé system. Recorded as a preprint, not as a published theorem, and the gate's word is published.

(b-4) arXiv:0710.1604: Dynamics of PDE 4 (2007) 293–302. PUBLISHED. Not an exclusion: it is the sharpest available statement of what an unconditional torus result would have to be:

"this qualitative question is equivalent to the more quantitative assertion that there exists a non-decreasing function $F: \R^+ \to \R^+$ for which one has a local-in-time a priori bound $| u(T) |{H^1_x((\R/\Z)^3)} \leq F(|u_0|)$ for all $0 < T \leq 1$".

A Lane T ansatz is exactly a refutation of that F. This is the cleanest formulation of the target and it is on the torus, in the source's own words.

What survives, stated before anything is built, as branch (b) requires. On T³, nothing located excludes a blow-up ansatz that is, at its initial time, (i) not small and not almost-2D in Ḣ^{1/2}/Ḃ^{-1}_{∞,∞}, (ii) not on a thin torus below (b-2)'s control, (iii) not in a Ladyzhenskaya–Prodi–Serrin class up to the singular time (automatic for a genuine singularity but it must be stated, not assumed) and (iv) a counterexample to (b-4)'s F. That is the whole of the located constraint, and it is a weak one: every clause of it is a region a blow-up ansatz would be outside of anyway. The bite branch (b) anticipated is real in shape and thin in substance, and this leg reports the magnitude rather than the shape alone.

6. The four §2 rows: what has no periodic counterpart, i.e. what Lane T must establish de novo

This is the gate's no-branch deliverable. Row verdicts are per battery, each with its own control status; none of the four has a located periodic counterpart.

Two of the four rows are partly vacuous on T³, not merely unproved, and the distinction matters for what Lane T must build.

• Row 1's object does not exist on T³. The theorem constrains exactly backward self-similar profiles, and the dilation action defining them does not descend to ℝ³/ℤ³) leg 390's check_B measured the consequence (342 modes survive one DSS step at λ = 1.7, zero survive two). So a "periodic NRS/Tsai" is not a translation of a theorem; it is a different theorem about a different object class, and naming that class is Lane T's T3 work, not a lookup.
• Row 4's hypothesis partly degenerates. The Morrey condition read from the landed record is sup_{R>0} sup_x [R^{3(1/p−1/l)} ∫_{B_x(R)}]^{1/l}; on a compact T³ the large-R half of that supremum saturates, so the scale-invariant content of the hypothesis is not what it is on ℝ³. This is an observation for the successor, not a measured result of this leg, and it is flagged as such.
• Row 2 is the sharp one, and it is the single most useful line in this leg. A Type I condition is a rate condition, |u| ≲ (T−t)^{−1/2}, and it needs no dilation symmetry to state. It therefore carries to T³ intact as a question, and no torus Type-I rigidity theorem was located (all:"Type I blowup" AND all:"periodic" = 0, measured; the single Type I × NS × torus hit is a 2D data-assimilation paper). A Type-I rigidity theorem on T³ is meaningful, absent from the located literature, and is precisely what a Lane T ansatz would have to survive or evade. That is the de novo item this leg names.
• Row 3's counterpart cannot be spoken to at all: its battery failed its control.

7. Located, adjudicated elsewhere, and load-bearing for Lane T: arXiv:2604.09949

The search surfaced, by two independent queries (self-similar × Navier--Stokes × torus, and finite time singularity × Navier-Stokes × torus), the only located claim of a finite-time singularity for 3D NS on T³:

arXiv:2604.09949 (Stable Finite-Time Singularity Formation for 3D Navier–Stokes via 5D-Lifted Axisymmetric Reductions: "a 5D-lifted analytic-profile program for finite-time singularity formation in the 3D incompressible Navier--Stokes equations on the periodic torus $\T^3$ … a computer-assisted Newton--Kantorovich validation based on interval arithmetic … reconstructed into a nearly self-similar singular evolution and then transferred to $\T^3$ by periodic extension and exact Leray projection."

It fails the gate's K3 (its conclusion is existence, not exclusion) and it is not new to this repository) grepped before being called anything, per leg 348's own correction. It is already at legs 303, 309, 323 and in solver/target_selection.py. Leg 309 read it at full text and its gate answered NO: it breaks at H11, because its T³ object is built from an exact backward self-similar core on ℝ³, "exactly the object Nečas–Růžička–Šverák prove must be trivial", and transferred by periodic extension.

The consequence for Lane T is the most useful thing this leg found, and it is stated as a correction routed to the Conductor rather than applied here. WALLS.md records that 0 of 4 rigidity clearances carry to T³. That is true and it is about clearances. The converse is not recorded and leg 309 already demonstrated it: the ℝ³ exclusions still reach a T³ object whose core is an ℝ³ self-similar profile carried over by periodic extension. The screen is not void on the torus; it is void as a source of clearances while remaining live as a source of kills against exactly the cheapest way to build a torus target. The one previous attempt at Lane T's target, by anyone, died that way. Lane T's T3 unit inherits this as a hard constraint: a non-DSS ansatz must not be a periodization of an ℝ³ self-similar core, and leg 390's check_B already says the DSS route cannot be one anyway.

8. Corrections routed to the Conductor, verbatim, applied nowhere by this leg

1. WALLS.md, LANE T, "The price" bullet 1 currently reads "this repository has never searched the periodic rigidity literature (leg 390 §5 item 2)". After this leg it has, at the coverage stated in §4. Suggested replacement is a record of the result, not a clearance: "searched at leg 392 (arXiv 32/32 MEASURED, Semantic Scholar 3/8, one battery's domain control failed): no periodic analogue of any of the four rows was located, and the verdict is UNDER-RESOURCED, not no."
2. WALLS.md, LANE T, same bullet, needs the converse added: "0 of 4 clearances carry to T³, but the ℝ³ exclusions DO reach a T³ object built by periodic extension of an ℝ³ self-similar core (leg 309 killed arXiv:2604.09949 on exactly that ground)."
3. (D)'s data conditions (8) and (9) are still UNREAD, they need outreach, the standing hold forbids it, and the hold therefore sits on Lane T's critical path. This leg raises this rather than routing around it, exactly as its pre-committed reading (d) requires. Nothing in this leg assumes anything about their content.
4. Row 4's Morrey hypothesis partly degenerates on a compact T³ (§6). Flagged as an observation for a successor with a measurement, not as a result of this leg.

9. What this leg does not claim

• It does not claim Lane T is clear. It claims nothing located excludes it, with the coverage above. Branch (c) of the pre-committed reading is explicit that these are different statements, and the difference is the whole discipline.
• It does not advance Lane T. A null here is not a permission.
• It does not retire CLAY_OBLIGATIONS.md §6's two no-method obligations. They stay OPEN, leg 390 measured that the torus does not retire them, and this leg does not write otherwise.
• It moves no link of the L1 → L4 chain. Clay stays ~0.05 %. Ceiling TIER 2.