Route-F v1: the critical dissipation exponent is half the far-field decay exponent

Phase-2 P2, Route F (the viscosity lane), leg 1. Figure: writeup/figures/fig35_p2_route_f_v1_viscosity.png. Data: writeup/data/p2_route_f_v1_viscosity.json. Module: solver/fractional_gclm.py (gates: test_fractional_gclm.py, 6/6).

Level-1 numerics plus one elementary scaling derivation. Not a certificate, not a proof, not Clay progress. Plain float64; nothing here is interval-enclosed.

⛔ RETRACTION OF NOVELTY (added 2026-08-04, Route-J v1 primary-source pass)

This leg's headline is in the literature. s_c = α/2 is recorded as s*(a) = 1/c_l(a) in arXiv:2607.19762 §6.1 eq (6.3) (Xu, The spectral picture of self-similar collapse in the CLM equation), from the same rescaling argument, and posted 22 Jul 2026, eleven days before this leg. Our F6 map agrees with their Table 1 row by row (worst 3.1e-3, mean 1.2e-3; exact at a = 0 and a = 1/2). Note the exponent dictionary: their c_l is our β, our α = 1/c_l, and their Λ^σ is our (-Δ)^s with σ = 2s, so s_c = s*/2.

The measurements below are unaffected and none of them was found to be wrong. What survives as a contribution is narrow: F3's cross-check (α from a steady compactified solve against dp/ds from time-dependent periodic simulation, no shared grid, basis or fitted constant) is an independent dynamical validation of a formula that is theirs. Xu state the only quantitative validation of their branch is the a_c endpoint, so the cross-check is a genuine service; it is not a new result.

Two further corrections from the same pass: (i) s* is not the sharp blow-up/regularity threshold, Xu say so explicitly, "which for this family remains unknown", and cite Sakajo that at a = 0 blow-up persists at small viscosity regardless of the dissipation order; (ii) what lies ABOVE s_c is now known and this leg never had it: the balance switches to dissipation-against-stretching, β = σ c_l, with ω_t subdominant. See TECHNICAL_P2_ROUTEJ_V1.md §5.

0. Why this leg, and what it is allowed to conclude

The standing list of "what would actually be worthwhile toward a Clay-relevant candidate" has five items. Route-E v1 shut the cheap entrance to item (1), the DSS lane. This leg takes item (2), which is the only item on the list that probes the actual obstruction between a toy-model certificate and Navier–Stokes rather than polishing the toy:

Take a blow-up that exists, add dissipation, and determine the scaling at which dissipation kills it.

That is the Clay question in miniature. Navier–Stokes is hard because viscosity and the nonlinearity are exactly balanced at the blow-up scale; every real proof has to say something about which one wins. In a 1D model with a dial, that balance can be measured.

Gate-check, answered. (a) Which link does this move? None. It does not produce a certificate at any link. It maps the obstruction that separates L3 from L4, in a toy model where the map is computable. (b) Is another leg better? Route-E v1 re-priced the DSS lane upward (its cheap entrance is shut), which is what promoted this item. (c) Cheaper experiment that kills it? The derivation in §1 is the cheap part; the leg's cost is in testing it rather than believing it.

1. The prediction, and where it comes from

gCLM with fractional dissipation on a 2 pi-periodic domain:

omega_t + a u omega_x = omega u_x - nu (-Delta)^s omega ,      u_x = H(omega)

s = 1 is the ordinary Laplacian; s is the dial.

A self-similar blow-up has omega ~ (T-t)^{-1} and a length L ~ (T-t)^beta. Route-E v1 already computed beta, without knowing it would be needed here. In the dynamically-rescaled variables of solver/rescaled_spectrum.py the two rescaling ODEs A'/A^2 = -c_omega and L'/(LA) = -c_l integrate to A ~ 1/(alpha (T-t)) and L ~ (T-t)^{c_l/alpha}, where alpha = -c_omega is the profile's far-field decay exponent Omega ~ X^{-alpha}. With c_l = 1:

beta = 1 / alpha .

Now compare the two terms at the blow-up scale, the nonlinearity is ~ omega^2, the dissipation ~ nu omega / L^{2s}:

      dissipation / nonlinearity  ~  nu (T - t)^{1 - 2 s / alpha}
  =>  the blow-up beats dissipation  <=>  s < s_c(a) = alpha(a) / 2 .          (SC)

Three things make (SC) worth an experiment rather than a paragraph.

• It is a statement about s, not about nu. For s < s_c the blow-up survives every nu > 0; for s > s_c dissipation eventually dominates for every nu > 0. So the transition must be nu-independent, which is a control the experiment can run on itself (F4).
• It is falsifiable as a slope, not as a threshold. The relation predicts the whole function p(s) = 1 - 2s/alpha, not just its zero. Measuring a line is a far stronger test than locating a transition by eye, and it is the difference between this leg and a blow-up/no-blow-up sweep.
• It says exactly where the Navier–Stokes difficulty lives. NS's natural scaling is L ~ (T-t)^{1/2}, i.e. beta = 1/2, i.e. alpha = 2, for which (SC) puts the ordinary Laplacian s = 1 exactly at s_c. NS is critical, and that is the whole problem. In gCLM alpha is a measured function of a and is free to move.

2. What is measured, and why not "did it blow up"

A binary blow-up test near a critical exponent is exactly the kind of measurement this project has learned to distrust: near s_c the blow-up is only asymptotically dissipation-free, so at finite compute the apparent threshold is biased and resolution-dependent, and the bias points the way the experimenter expects.

So the primary diagnostic is quantitative. Track D/N, the ratio of the dissipative to the nonlinear term at the peak, and fit

D/N ~ (T - t)^p ,        prediction:  p = 1 - 2 s / alpha .

p crosses zero at s_c. Measuring p across s gives a line whose slope, intercept and zero are all separately checkable.

One thing had to be fixed before this worked, and it is worth recording. Dissipation delays the blow-up, so fitting against the inviscid T_0 biases the exponent: every point came out above its prediction, with a shallower slope and a zero crossing ~10% high. That pattern is the signature of a wrong singular time, not a wrong exponent. The repair is that the run supplies its own T: for omega ~ 1/(T-t), 1/amp is linear in t, so extrapolating it to zero gives T from the run itself (recovered to 5.3e-5 relative on the inviscid case, where T is known exactly).

3. Results

F1, the known answer

At a = 0, nu = 0 the model is exactly solvable on the circle as well as the line: with z = H(omega) + i omega, z_t = z^2/2, so z = z_0/(1 - t z_0/2). Checked against an independent fine RK4 integration to 1.8e-14, and the solver reproduces it to 1.9e-9 at t = 2.5 (amplitude 2.41). The blow-up time is T = 2/max{H(omega_0) : omega_0 = 0}.

And alpha(0) = 1 exactly (Route-E v1's anchor is one Fourier mode), so (SC) predicts s_c(0) = 1/2, which is also the classical critical exponent for dissipative CLM. So a = 0 is a genuine known-answer gate, not a self-consistency check.

F2: the relevance line at a = 0, with nothing fitted

alpha = 1 exactly here (Route-E v1's anchor is one Fourier mode), so the prediction p = 1 - 2s contains no fitted input at all. At n = 8192, nu = 1e-3, fit window 0.40–0.94:

Fitted slope −2.068 against the predicted −2.000, and p = 0 at s = 0.5033 against the predicted s_c = 0.5000. Per-point fit RMS 0.016–0.065.

F3: the cross-check, which is the headline

alpha(a) was measured by a steady, compactified spectral solve on the whole line in a different module, for a different reason. The slope dp/ds is measured by time-dependent pseudo-spectral simulation on a periodic domain with dissipation. The two computations share no grid, no basis, no equation as posed, and no fitted constant. (SC) says the second is -2/alpha of the first.

The ratio is 1.022 ± 0.014 while alpha itself doubles. That is a uniform bias of about 2%, not an a-dependent failure: the shape of the relation (that the slope is -2/alpha with alpha supplied by an unrelated computation) is confirmed to sub-percent, and the normalisation carries a systematic quantified in F7.

F4: the nu-independence control

(SC) contains s and alpha and does not contain nu. A slope that moved with nu would mean the measurement is about the viscosity rather than the scaling. Over three decades: slope −1.866 / −2.051 / −2.110 at nu = 1e-2 / 1e-3 / 1e-4, a spread of 0.245.

Honest reading: the prediction −2 sits inside that range and the middle decade is within 2.6%, but the spread is not small, and the two ends are exactly the regimes where the asymptotic argument is stressed from opposite sides, nu = 1e-2 perturbs the blow-up itself, nu = 1e-4 makes D/N small enough that the fit is noise-limited. This control passes, but weakly, and it is the measurement most worth tightening if the leg is ever revisited.

F5, resolution

At s = 0.35 (prediction +0.300): p = +0.395 / +0.347 / +0.330 / +0.318 over n = 1024 / 2048 / 4096 / 8192. The whole ladder spans 7.6e-2; the two finest differ by 1.2e-2, and the sequence is monotone toward the prediction. So resolution contributes about 0.01: an order less than the window systematic below, which is why the leg is quoted with the latter.

F7: the fit window, swept rather than chosen

p = 1 - 2s/alpha is an asymptotic statement, so an early window has not reached it and a very late one is noise-dominated. Sweeping it is the honest error bar:

slope = −2.02 ± 0.09 (predicted −2) · s_c = 0.51 ± 0.05 (predicted 0.500)

Two things this makes visible. The single exponent at a given s moves by ±0.07 across windows and approaches the prediction monotonically from above: the signature of an asymptotic regime being entered, not of a wrong exponent. And the slope is far more robust than any single exponent, because every s shares the window and the bias cancels in the difference. That is why the leg's claims are about the slope and the zero crossing.

F6: the map, and the sentence to be careful with

With alpha(a) from Route-E v1, (SC) gives s_c(a) = alpha(a)/2. Since alpha rises through 2 on the branch, s_c rises through 1, the ordinary Laplacian.

The careful version of that sentence. For a above that crossing, the scaling says a self-similar blow-up of this family is not stopped by ordinary viscosity: the dissipative term is asymptotically negligible against the nonlinearity at the blow-up scale. That is a statement about gCLM's own scaling, and it is not a statement about Navier–Stokes. NS's alpha is pinned at 2 by dimensional analysis; it is not a free dial. What the map shows is what the NS difficulty is made of: NS sits exactly at the crossing this family walks through.

4. What this does not say

• It does not say a gCLM blow-up with s = 1 dissipation exists for a above the crossing. (SC) is a statement about which term dominates given the self-similar scaling; establishing that a solution actually reaches it is the whole difficulty, here as everywhere.
• It says nothing about NS beyond locating the difficulty. gCLM's nonlinearity, scaling and velocity relation are all different.
• The measured p carries a systematic of a few percent from the singular-time estimate and the fitting window, quantified in F5 and reported next to every number.
• Novelty unchecked, as for every leg since v15: s_c(0) = 1/2 for dissipative CLM is classical; the a-dependent map is what would need a literature check, and PDF access is still blocked (arXiv and publishers 403 at the proxy). Four legs now.

5. Where this sits relative to Clay (re-answered)

Nothing here is a step whose success would resolve the Clay problem, and this leg does not move L1, L2, L3 or L4. What it does is turn the sentence "any real proof must beat viscosity at small scales" into an equation with a measured right-hand side in a model where the arithmetic is checkable, and locate NS precisely at the marginal case. That is worth having as orientation, and it is worth nothing as evidence about NS itself. Odds unchanged, ~0.05%.

Appendix: new lessons

(53) When two legs measure the same constant through unrelated machinery, that cross-check is worth more than either leg's internal error bar. alpha from a steady compactified solve on the line and dp/ds from time-dependent periodic simulation share no grid, no basis and no fitted constant. Agreement there tests both computations at once, in a way no refinement of either could. Look for a second, structurally different route to a number you already have.

(54) A systematic that is uniform across a sweep is pointing at a shared input, not at the mechanism. Every measured p sat above its prediction with a shallower slope, which is what a wrong singular time does, and not what a wrong exponent does. The shape of a discrepancy names its cause faster than its size does. Read the pattern of the residuals before adjusting the model.