TECHNICAL, Route-DECR v1 (leg 318): "DOMINATED → ENCLOSED", and the criterion that decides it

Runner experiments/p2_route_decr_v1_scoping.py (3.9 s) · data writeup/data/p2_route_decr_v1.json · §0a pass writeup/novelty/leg_318.md + writeup/data/p2_route_decr_v1_lit.json · figure writeup/figures/fig84_route_decr_v1_knife_edge.png · blog sibling BLOG_P2_ROUTEDECR_V1.md · journal experiments/journal/leg_318.md.

Scoping leg. Nothing built. No solver module written or edited; solver/viscous_novelty.py is leg 197's and was imported read-only. requirements.txt untouched. plan_of_record.py untouched and byte-identical to origin/main. No ban lifted, re-posed or weakened.

0. The gate, in its pre-committed and immutable wording

Gate. Does the leg state a general structural criterion for viscous-term enclosure that is falsifiable on at least one banked model of this repository? : yes → Bank the criterion together with the named falsification test. This still does not become a lane. : no → Report VACUOUS and stop. The route dies; it is never redrafted and never becomes a lane.

Answer: yes. The criterion is stated in §2, its five named falsification tests in §4, and it does not become a lane, §6 says why that is not a formality but the criterion's own consequence. The gate was not edited and this leg does not think it is wrong.

1. What is being asked, and what "prior art" it inherits

Leg 240 measured a distinction and named its two sides. The viscous term of a self-similar blow-up certificate is either

• DOMINATED: the certified object is the ν = 0 system and viscosity is admitted as a decaying error under a scalar inequality on the scaling exponent. This is what arXiv:2208.09445, arXiv:2310.05325 and (independently, with the computer removed) arXiv:2501.15701 all do; or
• ENCLOSED: ν sits inside the certified problem, so the certified object itself depends on ν. This is what Dähne–Figueras do for complex Ginzburg–Landau (arXiv:2410.05480) and what arXiv:2404.04054 does for viscous Burgers and the nonlinear heat equation.

Leg 240 asked whether a particular author group had closed that gap and answered NO, at full text, twice over. It did not ask why: whether the gap is a contingency of effort or a structural fact. That is this leg's question, and the gate demands the answer take the form of a falsifiable criterion, not an explanation.

Inherited prior art, stated up front and not claimed (full pass in writeup/novelty/leg_318.md): the sub/critical/supercritical classification of a dissipative term against a scaling is decades-old folklore, Kiselev's survey arXiv:1009.0540, arXiv:1408.5499, arXiv:1809.04373, arXiv:2503.11095 and a dozen others populate that cell. The criterion below rests on that classification and claims none of it. What the §0a pass found unoccupied (nine ANDed queries, every zero audited component-wise against a verified-working AND operator) is the cell where the classification is turned into a decision rule for a certificate designer with a test that can kill it.

2. The criterion: ENCLOSURE IS CRITICALITY

You cannot enclose an object that does not exist. A ν-dependent self-similar profile exists only when the dissipative term is scaling-critical for the blow-up ansatz. DOMINATED is therefore not a weakness of anyone's method; it is the signature of a subcritical dissipative term.

Setting. An evolution equation ∂_t u = N(u) + ν D u, with N the inviscid part and D dissipative and homogeneous of order m under x ↦ λx. Fix a self-similar ansatz u(x,t) = (T−t)^{−a} U(x/(T−t)^b), s = −log(T−t), chosen so that N is autonomous in s; let A be the exponent set admissible for the inviscid profile problem. In the rescaled equation the dissipative term carries a prefactor e^{−δ_dis s}; call δ_dis the dissipation-criticality defect. For a scalar operator of order m with the ansatz's own b, δ_dis = 1 − m·b. For a system whose dissipation coefficient carries a field weight, BCG's ν/ρ ∼ S^{−1/α}, δ_dis is the paper's own.

C1: criticality (necessary)

• δ_dis > 0 on all of A ⟹ the rescaled system is genuinely non-autonomous in s, so every steady state of it solves the ν = 0 system. There is no ν-dependent profile object to enclose. DOMINATED is the strongest treatment available, and it is maximal, not provisional. Proof, one line: a fixed point of the rescaled flow requires the explicit s-dependence to vanish; e^{−δ_dis s} with δ_dis > 0 vanishes only in the limit, so the ν-term is absent from every fixed point.
• δ_dis = 0 at some b ∈ A ⟹ the rescaled system is autonomous with ν as a genuine parameter; the enclosed object exists and enclosure is structurally possible.
• δ_dis < 0 ⟹ dissipation outscales the nonlinearity, the ansatz fails, and the expected outcome is no blow-up at all.

C2: order compatibility (necessary for continuation from ν = 0)

Even where C1 holds, an existing ν = 0 certificate can be continued into ν > 0 only if D does not raise the differential order of the profile problem. If ord D > ord N, the ν > 0 problem is a singular perturbation: its solution set is not a deformation of the ν = 0 one, and the enclosure must be built ab initio at fixed ν > 0.

C3, bridge type (necessary at PDE level), cited to leg 315, not claimed here

The validated ODE→PDE bridge available in the literature is Zgliczyński's self-consistent a-priori bounds (math/0005247), whose hypotheses are dissipative; BCG's rescaled system is quasilinear hyperbolic. Leg 315 established this; this leg re-located math/0005247 independently through all:"rigorous numerics" AND all:"dissipative PDE" (2 hits, both dissipative objects) and does no more than cite it.

The criterion is NECESSARY, NOT SUFFICIENT

This is load-bearing and it is tested (FT5). Incompressible Navier–Stokes satisfies C1 exactly and C3 too, and is still closed, by the NRS/Tsai non-existence theorems, which plan_of_record.py's stage P0 gate already carries as its screen. A criterion that opened a route there would be refuted on the spot by a published theorem. This one screens routes out; it never opens one.

3. The sharp corollary, and the realization named (lesson 91)

THE KNIFE-EDGE: the enclosure exponent is the domination window's own excluded endpoint.

For BCG-type compressible implosion, leg 240 banked the viscous inequality in three papers' notations, agreeing to 1.78e-15: δ_dis = (r−1)/α + r − 2, with α = (γ−1)/2. Setting it to zero,

2(r−1) + (r−2)(γ−1) = 0   ⟹   r(1+γ) = 2γ   ⟹   r = 2γ/(γ+1),

which is exactly BCG's own threshold r > 2γ/(γ+1), i.e. exactly the lower endpoint of the banked domination window, at every γ. Numerically, against leg 240's bank read from its JSON rather than re-typed:

So the two treatments are complementary in the scaling exponent and share exactly one point, which neither occupies:

• DOMINATED owns the open interval (2γ/(γ+1), r_*(γ)). On the γ = 7/5 window the entire domination margin is δ_dis ≤ 0.1458980337503153: reproduced here to < 1e-6 of leg 240's own banked maximum, as a control.
• ENCLOSED could only live at r = 2γ/(γ+1), the endpoint the window excludes, because that is precisely where the domination decay rate hits zero (δ_dis at the window's lower endpoint, computed here: 8.88e-16, i.e. zero to a rounding).
• And at that one point enclosure is blocked twice over: by C2, because the Euler profile problem is first order in the similarity variable and admitting ν Δ makes it second order, a singular perturbation, not a continuation; and by C3, the hyperbolic-vs-dissipative bridge mismatch.

This is the structural answer to leg 240's measurement. The enclosure-eligible set has measure zero in r, and at its single point the cheap route in (continuation from the inviscid certificate) is closed by an order jump. Nobody failed to enclose the viscous term for want of effort.

4. The five named falsification tests

Every one is asserted in the runner, every one could have come out against the criterion, and one of them did, until its instrument was fixed (§5).

FT2's table, in full (including every abstention

The abstentions are the point. A criterion that "explains" rows it cannot see is unfalsifiable; the runner counts them and a control fails the process if fewer than two rows abstain.

Note the C2 column separating DF-CGL from viscous Burgers: both are ENCLOSED, but only DF-CGL has no order jump, its ε multiplies the same Laplacian as the dispersive term. That is exactly why DF can follow branches from ε = 0 (their Thm 4.1) while nothing in the bank continues a viscous-Burgers branch down to ν = 0; 2404.04054 certifies at fixed ν, ab initio, as C2 says it must. C2 was written before that column was computed.

FT4 in detail: the test most able to kill the criterion

C1 is extended to density-dependent viscosity μ(ρ) = ρ^θ. BCG's viscous term enters as (μ/ρ)Δu; for constant μ the ρ^{−1} contributes the (1/α)(1−r) piece of −δ_dis = 2 − r + (1/α)(1−r). Replacing ρ^{−1} by ρ^{θ−1} = (ρ^{−1})^{1−θ} scales that piece by (1−θ):

δ_dis^(θ) = (1−θ)(r−1)/α + r − 2 ,      θ_*(γ,r) = 1 − α(2−r)/(r−1).

This is DERIVED here from BCG's own exponent bookkeeping, not transcribed from either target paper, and it is flagged as such everywhere it is used. θ = 0 recovers the published formula to 0.0 (exact), which is the control on the derivation.

• Prediction 1. θ = 1 (viscosity linear in density, the shallow-water degeneracy) gives δ_dis^(1) = r − 2 < 0 for every r < 2, hence supercritical: the implosion mechanism cannot survive. Verified negative across the whole banked γ=7/5 window. arXiv:2512.18545 proves exactly that: globally regular spherically symmetric solutions that cannot cavitate or implode, all γ ∈ (1,∞) in 2D and γ ∈ (1,3) in 3D.
• Prediction 2. Below θ_*(γ,r) the dissipation is subcritical and implosion survives, as DOMINATED, i.e. there is a threshold in the viscosity power depending on the adiabatic exponent. arXiv:2603.10141's abstract, verbatim: "We identify a threshold value, depending on the adiabatic exponent, such that, for any power below this threshold, there exists a class of smooth initial data … which implode … the degenerate viscous terms are not sufficiently strong to suppress the convective mechanism." Predicted values at the window midpoints: θ_*(7/5) = 0.08158712005267243, θ_*(5/3) = 0.046205799573696305.

What is honestly not done. Only the abstract of arXiv:2603.10141 was read. The threshold value is therefore an open, named, one-leg falsification test: read that paper at full text and compare its threshold with θ_*(γ,r). If they disagree, C1's extension to degenerate viscosity is refuted. The runner records this as "NOT REFUTED (sign test passes; threshold-value test NAMED AND NOT YET RUN)" and the leg does not claim the match.

5. The instrument that had to be fixed, not the tolerance

FT1 first came out REFUTED, and it was wrong. In float64, max |δ_dis(γ, r_crit(γ))| over 45 γ is 5.329070518200751e-15, and the natural 1e-15 tolerance therefore reported the knife-edge identity as FALSE: the leg's central claim, killed by its own test.

~~Diagnosis before belief (leg 302's lesson, and this is the same failure mode verbatim, an IEEE-double cancellation masquerading as a transcription error): (r−1)/α and (r−2) are two quantities of size ≈ 0.9756 at the worst γ that cancel exactly, so the residual is 5.462e-15 relative, about 25 ulp of the cancelled operands. That is what catastrophic cancellation costs; it is not evidence about the identity.~~

[LEG 337 CORRECTION: mechanism re-measured, FT1's verdict unchanged.] The paragraph above is struck: it named the wrong mechanism. Directly re-measuring which operand carries the rounding error (experiments/journal/leg_337.md has the full script and grid): at the worst γ = 1.075, α_of(γ) is computed with zero rounding error in float64, γ − 1.0 is exact by Sterbenz's lemma and the following halving is always exact, so α carries no ulp at all. All of the error lives in r_crit(γ) = 2γ/(γ+1), whose float64 value differs from the true rational 2γ/(γ+1) by −0.88 ulp(r) (≈ 1 ulp), from the addition-then-division inside r_crit. That single sub-ulp error in r is then amplified by δ_dis's own sensitivity to r at the root, d(δ_dis)/dr = 1/α + 1 = 27.667 at this γ (dominated by the 1/α = 26.667 term): substituting the exact rational r and α into δ_dis gives exactly 0; substituting the actual float64 r (with its −0.88 ulp error) and the actual float64 α (0 error) into δ_dis, evaluated in exact rational arithmetic, reproduces −5.424e-15, matching the observed float64 residual −5.329e-15 to within the small extra rounding of δ_dis's own float ops. So it is not cancellation between two ≈1-sized operands that costs precision; it is 1 ulp of pre-existing rounding error in one operand (r), amplified by the other operand's reciprocal (1/α). Confirmed across the full 45-γ grid: α_of never carries nonzero ulp error at any grid point; r_crit always does, and the residual scales with 1/α_of(γ) accordingly. "Leg 302's failure mode" as a category label is retracted with it: this is ill-conditioning under amplification near a root, not a subtraction-of-near-equal-terms precision loss. FT1's verdict is byte-unchanged: decided in exact rational arithmetic, residual identically 0, either way.

The fix is the right instrument, not a looser threshold. FT1's verdict is now decided in exact rational arithmetic (fractions.Fraction), where the residual over the same 45 γ is identically 0. The float number is kept in the JSON as the diagnosis, never as the measurement. Had the tolerance simply been relaxed to 1e-14, the test would have passed for the wrong reason and would have had no power to detect a genuine transcription error of size 1e-14.

6. Consequences: including the ones against this leg's own interest

It does not become a lane, and that is the criterion's own verdict, not an administrative restriction. The criterion's only positive statement about BCG's object is that enclosure is structurally possible on a set of measure zero in r, at a point where C2 forbids continuation and C3 forbids the bridge. There is nothing to build. The gate's yes-branch already said this; the mathematics agrees with it.

No link of L1→L4 moved. A statement about what could be enclosed is not a link moving. Clay odds stay ~0.05%, behind Walls 1 and 2.

It strengthens leg 240 rather than extending it. Leg 240's negative was about one author group's publication record; this leg says the negative would have held for any group, because the object they would have had to enclose does not exist at the exponents their theorem lives at. The two independent confirmations leg 240 already had, CGSS and the computer-free Shao–Wei–Wang–Zhang route, are exactly what C1 predicts: the domination is a property of the scaling, so removing the computer from the argument cannot change it.

What it says about this repository's own target is not encouraging, and is said anyway. For incompressible NS the viscous term is scaling-critical, C1 is satisfied, and the enclosure question was never the obstruction, NRS/Tsai non-existence is. Moving from the compressible object to the incompressible one trades a measure-zero enclosure window for a non-existence theorem. That is a worse position, not a better one, and the criterion is what makes the comparison sayable at all.

Adjacency declared, not entered. Leg 305 (Route-DWM) is live on whether the dominance-window deficit is sharp or slack via a per-constant ledger. This leg uses only the window's endpoint identity, touches none of 305's files, and re-derives none of its per-constant decomposition. Where the two meet: 305's certified width 0.0243163 against available 0.1666667 (shortfall exactly (7+3√5)/2 = 6.8541019662496845446, leg 300's corrected closed form, not the 6.855 this run corrected across seven surfaces) is cross-checked here to < 1e-9 purely as an internal consistency control on the banked width, and is not re-derived as a result of this leg.

7. Controls (lesson 90), asserted in code

All eight pass; the runner exits non-zero if any fails.

8. What this leg did not do

• Did not touch plan_of_record.py, requirements.txt, writeup/build_figures.py, solver/capabilities.py, or any other leg's territory. solver/viscous_novelty.py was imported, never edited.
• Did not build a solver module, a certificate, or any apparatus. This was a scoping leg.
• Did not read arXiv:2603.10141 or arXiv:2512.18545 at full text, abstracts only, and says so wherever their content is used.
• Did not register fig84 in writeup/INDEX.md or writeup/build_figures.py; those are shared and integration allocates centrally. The figure file is written directly, as leg 315 did with fig79.
• Did not widen into leg 305's per-constant ledger.
• Did not claim novelty for the criticality trichotomy, which is folklore.