Route-H v1: the marginal case, where every scaling argument returns zero information

Phase-2 P2, Route-H v1. Code: solver/critical_dissipation.py + test_critical_dissipation.py (10/10); driver experiments/p2_route_h_v1_critical.py → writeup/data/p2_route_h_v1_critical.json → fig37. Deterministic, not a logged Tier run.

Rigor level: 0–1: plain float64 throughout. Nothing is interval-enclosed, nothing here is rigorous, and none of it is a statement about Navier–Stokes. It is a statement about a one-dimensional toy whose only relevance is that it can be put at the point where NS sits.

⛔ THE CLOSED FORM (E) IS PRE-EMPTED: settled, not suspected

(added 2026-08-04, Route-J v1 primary-source pass)

This leg recorded (E) as "at high risk of being known" and declined to claim it. That was the right call, and it is now confirmed. (E) is arXiv:2207.07548 §5.3 equations (57)-(58) (Ambrose–Lushnikov–Siegel–Silantyev), under the parameter map ω₋₁(0) = -(1+μ₀)κ, v_c(0) = κT with κ = ν/μ₀. Verified: worst relative difference 6.5e-15 over three parameter sets × four times; their blow-up-time formula (59) returns our T with absolute error 0.0; and their evolution law dv_c/dt = ω₋₁(0)+ν reduces to -κ, which is (E)'s own. What survives is (E)'s use as a known-answer gate: which is what it was built for, and which is strictly better now that it carries a citation.

α₁ = 0 at a = 0 is likewise confirmed-and-known: ALS's self-similar form (61) carries ν inside the profile with the exponents fixed at c_l = β = 1, i.e. a one-parameter family of viscous self-similar blow-ups. We measured, numerically, the existence of an exactly-known family.

α₁ = +0.133683 at a = 1/2 is NOT pre-empted by Tier 1 (criticality at a = 1/2 is σ = 3, which is in neither ALS nor Xu) but it is also not searched beyond Tier 1, and the honest label is unsearched, not novel.

A correction to this leg's framing, and it goes the useful way. §29 says at criticality "the scaling argument returns zero information" and stops there. ALS §5.1 supplies what lies above criticality, which this leg never had: the balance switches to dissipation-against-stretching, β = σ c_l with ω_t subdominant, and the mechanism is a double pole whose residue B = -12iν is proportional to ν and is therefore absent inviscidly. See TECHNICAL_P2_ROUTEJ_V1.md §5.

Also settled: ALS §5.1 correct Schochet (CPAM 1986)'s constant to K± = 24(3±√6) from the printed 12(6±√6). We checked from our own side: residuals 5.24e-16 vs 2.40e-2, 13.66 decades apart. ALS are right.

0. The wall both previous legs stopped at

Route-F v1 (§27) measured the critical dissipation exponent for gCLM and Route-G v1 (§28) ported it to 2D Boussinesq, arriving at the invariant law

s_c = 1/(2β) ,      L ~ (T−t)^β .

Both then said the same thing about the point s = s_c itself: at criticality the dissipative and nonlinear terms balance identically, so the scaling comparison returns zero information.

That is not a footnote about a corner of the map. β = 1/2 for Navier–Stokes by dimensional analysis, so s_c = 1: the ordinary Laplacian, exactly. NS is the marginal member, and "every scaling argument about NS comes back empty" is a restatement of that fact rather than a separate difficulty.

So the marginal case is the case. This leg asks what actually happens there, in a model where the marginal point is reachable because α is a dial.

1. Promote μ to a dynamical variable

Take gCLM with fractional dissipation on the line,

ω_t + a u ω_x = ω u_x − ν Λ^{2s} ω ,      u_x = H(ω) ,

and Route-E's dynamic rescaling ω = A(t) Ω(X, τ), X = x/L, dτ = A dt, c_l = 1. The dissipative term neither vanishes nor survives with a fixed coefficient: it returns multiplied by

μ(τ) := ν / (A L^{2s}) ,

and the rescaling ODEs A'/A² = −c_ω, L'/(LA) = −1 turn that into an ODE for μ. With α := −c_ω (Route-E's output, and the profile's far-field decay exponent),

Ω_τ = (c_ω + HΩ)Ω − X Ω_X − a U Ω_X − μ Λ^{2s} Ω ,          (F_μ)
μ_τ = (2s − α[Ω, μ]) μ .                                     (M)

The pair (Ω, μ) is autonomous, and reading it is the whole leg.

1a. Route-F's s_c is an eigenvalue

Linearize (M) at the inviscid fixed point (μ = 0, α = α₀). The μ-direction has growth rate

λ_μ = 2s − α₀ ,

negative exactly when s < α₀/2 = s_c. Route-F obtained s_c by fitting a power law to D/N along a time-dependent trajectory; this is the same number, and it says what it is: the stability exponent of the inviscid self-similar profile against the one direction dissipation opens. A scaling statement has become a spectral one. (fig37 A)

1b. At criticality the eigenvalue is exactly zero, and one number decides

Set 2s = α₀. The linear term in (M) is gone, so the outcome is set by the next one. Expanding α(μ) = α₀ + α₁μ + O(μ²),

μ_τ = −α₁ μ² + O(μ³) ,

so the entire marginal case reduces to the sign of α₁ = dα/dμ:

One number, and it is cheap. That is the payoff for writing (M) down.

1c. The gauge had to be re-derived

Route-E's c_ω = 1 + (a−1)H(Ω)(0) comes from freezing the origin slope, (Ω_τ)_X(0) = 0. The dissipative term contributes to that derivative (for odd Ω, Λ^{2s}Ω is odd and its X-derivative at 0 is not zero) so

c_ω = 1 + (a−1)H(Ω)(0) + μ (Λ^{2s}Ω)_X(0) / Ω_X(0).          (N′)

Banked lesson 48 says re-derive a gauge when you generalize it rather than extending it; this is that lesson applied to the previous leg's own gauge. Gate (3) finite-differences the resulting Jacobian, because (N′) contributes a quotient and a quotient rule is exactly the kind of thing that is wrong by one term. It matches to 4.0e−11.

2. Why the marginal problem is exactly representable here

Route-E's compactified basis (X = tan(θ/2), odd sines) makes H exact, and therefore

Λ = H d/dX  maps sines to sines exactly,

as does every integer power of it. So Λ^{2s} is an exact, quadrature-free matrix precisely when 2s is a positive integer: when s is a half-integer.

And criticality asks for 2s = α. Route-E found α(a) passing through the odd integers at isolated a: α = 1 at a = 0, α = 3 at a = 1/2, α = 5 at a = 0.5821792673. The critical dissipative problem is exactly representable at precisely the points where criticality can be posed at all. That is a coincidence of two conditions, not a design choice, and it is what makes this leg cheap. Away from those a the equations are still correct; Λ^{2s} is simply no longer a finite matrix.

§6 shows this alignment is weaker than it looks at the third point.

3. H1, the known answer: a closed-form viscous blow-up

At a = 0, s = 1/2 the marginal problem is solvable in closed form. With z = H(ω) + iω (analytic in the lower half plane for these profiles) one has Λz = i z_x, so the dissipative CLM equation becomes a complex Burgers equation

z_t + iν z_x = z²/2

whose characteristics are the constant complex shift x → x − iνt. Substituting the CLM profile gives, for every μ₀ > 0 and every ν > 0 with κ := ν/μ₀,

ω(x, t) = −2(1 + μ₀) κ x / ( κ²(T−t)² + x² )                  (E)

as an exact self-similar finite-time blow-up of the dissipative equation, with ‖ω‖_∞ = (1 + μ₀)/(T − t) and L = κ(T−t), i.e. β = 1, α = 1, s_c = 1/2: critical, for every μ₀.

Checked in closed form (no quadrature, no differencing) against ω_t − ω H(ω) + ν Λω = 0 over ν ∈ {0.05, 0.5, 2}, μ₀ ∈ {0.1, 1, 3}, t ∈ {0, 0.5, 0.9, 0.99}: worst relative residual 6.3e−16. The amplitude law is measured, not asserted: 4.0/4.0, 20.0/20.0, 200.0/200.0, 1998.5/2000.0.

Novelty: not claimed, and (E) is at high risk of being known. Explicit solutions of the viscous CLM equation by complexification go back to Schochet (CPAM 1986). See §9 and LITERATURE_CHECK.md. What this leg is for is (M) and α₁, not (E).

4. H2: a = 0 carries a line of viscous self-similar blow-ups

In rescaled variables (E) is the one-parameter family Ω_μ = −(1 + μ₀) sin θ with α ≡ 1 identically. So α₁ = 0 at a = 0: the marginal direction is neutral to all orders, and the viscous self-similar blow-ups form a line rather than a point.

Newton rediscovers it from a cold start, the "build the same object twice" gate:

α₁ = −3.9e−17. Verdict: neutral_line. (fig37 B)

Note what makes this a real gate rather than a tautology: the numerics do not know (E). They solve (F_μ) with the re-derived gauge (N′) at μ = 4, where the dissipative term is four times the size of everything else, and land on the closed form to 1e−15.

5. H3, a = 1/2, s = 3/2: the degeneracy does not survive, and α₁ > 0

At the second resonance Λ³ is exact and α₀ = 3.0000000000. α(μ) is now genuinely curved, and that curvature is a trap: a straight fit of α against μ over a finite window returns the chord, not the derivative at the origin. Over μ ≤ 0.2 the chord is 0.1264 while the extrapolated slope is 0.1337: a 5.5% error in the one number the verdict is quoted from. So the secants (α(μ) − α₀)/μ are fitted linearly in μ and extrapolated to μ = 0 (banked lesson 52's "add rungs until the exponent stops moving", applied to a derivative instead of a rate):

K-spread 9.1e−4, and the Λ³ truncation falls by 8× across the ladder while α₁ stops moving at the fourth digit. α₁ > 0, so the verdict is relaxes_to_inviscid. (fig37 C)

The truncation number is worth reading rather than hiding: it is not small in coefficient terms (8% at K = 96). What has to be true is that it falls with K and that α₁ stops moving anyway. Gate (8) asserts both, not just the second.

6. H4: the third point is NOT REACHED, and the refusal predicate is the finding

At a = 0.5821792673, α = 5, s = 5/2 the leg does not converge. Reported as a failure rather than dropped, on a two-rung ladder:

Both rungs refuse. Neither reaches a signal-to-residual ratio of 1, let alone the factor of 10 the gate asks for.

6a. The rung that would have shipped a sign flip

The K = 288 rung looks respectable. Its seed matches Route-E's own ladder for this point (α = 5 is steeper and reaches its asymptotic regime late), and a branch residual of 4e−4 is not obviously disqualifying. Fed to alpha_slope it returns

α₁ = −0.0017    →    verdict: dissipation_runs_away

which is the opposite sign to a = 1/2, and would have been a headline: "the marginal verdict flips at the third resonance."

It is not a measurement. The entire excursion of α across the μ-window is 2.0e−5, a factor of 19 below the residual of the solve it was extracted from. The derivative being fitted is an order of magnitude smaller than the error bar of the quantity it is fitted from.

Refusing on the residual alone would not have caught this: 4e−4 is a perfectly ordinary residual, and the guard worst < 1e−5 would have been the only thing standing between this leg and a wrong sign, on a threshold chosen before the number was seen. The predicate is now the stricter and more honest one:

drift  >  10 × worst residual                        (is there signal at all?)

which is a different question from "did it converge", and both numbers are recorded at every rung so the refusal can be audited rather than trusted. Gate (10) tests the predicate against these exact numbers, including that the sign it would have quoted is a real sign: the trap is live, not hypothetical.

6b. §2's "lucky alignment" was overstated, and is corrected in place

The worst Λ⁵ truncation is 1.7e4 at K = 192 and 1.5e3 at K = 288: the coefficients the fifth power throws off the end of the basis are three to four orders larger than the ones it keeps, because Λ^p weights mode k by k^p and so amplifies precisely the modes the single final truncation discards.

It does fall under refinement, roughly K^{−4}, and that is the honest statement rather than "refining does not help". But it starts so far above 1 that the observed rate puts the K needed to make Λ⁵ trustworthy at around 3000, which dense linear algebra does not reach.

So the alignment claim in §2 is real but weaker than it was written. Landing on an odd-integer α makes Λ^{2s} a finite matrix; it does not make the composite accurate. The alignment buys p = 1 and p = 3 and does not buy p = 5. The module docstring has been corrected in place.

6c. What this costs the leg

α₁ at the third resonance is unmeasured. The two-point trend (α₁ = 0 at a = 0, α₁ > 0 at a = 1/2) has no third point holding it up, and one leg has already been paid for reading a trend off two points (banked lesson 52). Read the sign at a = 1/2 as a measurement at a = 1/2, not as the start of a pattern.

7. H5, the DSS re-ask: dissipation does discretize the continuum

This is the leg's most interesting negative.

Route-E shut the DSS (discretely self-similar) lane's cheapest entrance with a mechanism, not just a null result: the only grid-converged isolated eigenvalues of the inviscid generator are 0 and −1, the two exact symmetry modes; everything else is continuous spectrum, and a continuum has no eigenvalue to move into Re > 0 as a complex conjugate pair. Critical dissipation is exactly the perturbation that could repair that, so the question is whether the continuum discretizes, and if it does, whether anything then moves or goes complex.

It discretizes. Route-E's convergence filter (coarse K = 96 against fine K = 144, tolerance 1e−3) applied to the dissipative generator at a = 0, s = 1/2:

(μ = 0.10 shows an empty-looking ladder because the −1 mode has already moved off the integer to −1.183216; it is the same two converged values as μ = 0.)

Turning μ on takes the converged count from 2 to 8 and grows a ladder of eigenvalues sitting on the negative integers. That is a real structural change: the mechanism Route-E used to shut the lane is gone.

And the lane stays shut anyway, for a better reason. Everything that condenses out of the continuum lands on the negative real axis. No converged eigenvalue reaches Re > 0 (worst +3.3e−13, i.e. zero), and none is complex: the two values flagged at μ = 2 are a pair at Re = −3 split by |Im| = 1.8e−05, a near-degenerate real pair resolved to noise, not a Hopf. (The driver now reports the largest |Im| rather than a boolean, because the boolean read "True" for that pair: the first version of this table would have overstated it.)

One mode does move, and it moves the wrong way for a bifurcation. The amplitude mode sits at

λ_amp(μ) = −√(1 + 4μ)                                          (measured)

to 2.1e−09 across the whole μ-ladder. At μ = 0 this is Route-E's exact symmetry eigenvalue −1; dissipation breaks the amplitude symmetry (rescaling ω changes the effective size of ν), which frees that eigenvalue to move, and it moves left. The amplitude mode becomes more stable. This formula is an empirical fit to six digits and is not derived; it is in the module because it is gated, not because it is understood.

The absence claim has a positive control (banked lesson 47): planting a localized potential in the same dissipative generator takes the filter from 6 converged eigenvalues to 9, one of them at Re = +1.58. So "nothing crosses" is a measurement and not a filter that cannot see. (fig37 D)

8. H7: the cross-check, through unrelated machinery

At s = 1/2 exactly the prediction from (M) is λ_μ = 0, i.e. the time-dependent D/N ~ (T−t)^p fit should return p = 0. That fit lives in a completely different instrument: periodic pseudo-spectral, RK4, integrating factor, fitted singular time. Nothing about the compactified steady solve enters it.

At n = 8192 (Route-F's resolution):

Mean p = +0.006, spread across ν 0.007, against a prediction of exactly 0, and the window spread within each ν (~0.04) is larger than the deviation from zero, so the honest error bar is the fit window, not the residual bias.

The caveat, stated rather than buried: every one of these runs ends under_resolved. The solver refuses to integrate past its spectral-tail criterion rather than return a number off an unresolved state (banked lesson 45). What softens it is the reached column, which is recorded for exactly this reason: each trajectory gets to 99.97% of the fitted singular time before refusing, so "stopped short" here means stopped short by three parts in ten thousand, not stopped early. It is still a cross-check with a stated limitation rather than a clean confirmation. (fig37 E)

9. Novelty and literature risk

Not claimed for (E). Explicit solutions of the viscous CLM equation by complexification go back to Schochet, CPAM 1986; the relevance exponent for dissipative gCLM is reported in arXiv:1908.09385 / arXiv:2207.07548. Route-F's s_c already sits at high risk of being pre-empted by the latter (see LITERATURE_CHECK.md §3 and the third pass), and Route-H's λ_μ = 2s − α₀ is the same statement in spectral clothing, so it inherits that risk in full.

What is plausibly new here is methodological, and remains unchecked because PDF access is still blocked:

1. μ as an autonomous variable of the rescaled flow, turning a scaling threshold into a stability eigenvalue and the marginal case into a quadratic normal form with one coefficient. Whether this is standard in the dynamic-rescaling literature is exactly the kind of thing that would be obvious to someone in the field.
2. α₁ = dα/dμ as the marginal invariant. The sign is the whole verdict.
3. The observation in §7: that critical dissipation discretizes the inviscid continuum onto the negative integers without producing anything that could cross.

Item 3 is the one worth a specialist's five minutes; items 1 and 2 should be presumed known until someone checks.

10. Where this sits relative to Clay

It moves no link of the L1→L4 chain. Three honest statements about what it does do:

• It converts Route-F/G's stopping point into an answerable question in a toy model, and answers it there: at criticality μ decays algebraically, μ ~ 1/(α₁τ), so a critical viscous solution relaxes onto the inviscid profile, but impossibly slowly. With α₁ = 0.1337, taking μ from 0.2 to 0.02 costs τ = 337 and to 0.002 costs τ = 3703, and τ is itself logarithmic in (T−t). Each decade costs nine times the last, because the linear term is gone. (fig37 F)
• It removes Route-E's mechanism for shutting the DSS lane (μ > 0 does discretize the continuum) while leaving the lane shut on the direct evidence. That is a strictly better epistemic position than before: the negative now rests on a measurement with a positive control rather than on "there is nothing there to measure".
• It is a toy. s = 3/2 is hyperviscosity, used because it is where criticality can be posed exactly, not because it resembles NS. (E) decays like 1/x and is therefore not finite energy: the honest analogy to Nečas–Růžička–Šverák, whose theorem excludes exactly self-similar NS blow-up in the finite-energy class, and whose escape route is precisely the slowly-decaying non-L² profiles Jia–Šverák constructed. (E) is the toy analogue of the escaping class, not a counterexample to NRS.

11. Reproduce

.venv/bin/python test_critical_dissipation.py                       # 10/10, ~4 min
.venv/bin/python -u experiments/p2_route_h_v1_critical.py           # ~8 min -> JSON
.venv/bin/python writeup/4_p2_lottery/p2_route_h_v1_evidence.py     # fig37 from JSON

Gates: Λ exact three ways (closed-form conjugate-Poisson identity to 1e−15, Λ² = −d²/dX² against a differently-built second derivative, and an independent line-grid Hilbert transform); μ = 0 reproduces Route-E bit for bit; the Jacobian including (N′)'s quotient term; (E) against the PDE in closed form; Newton rediscovering (E) from a cold start; the sign of the marginal verdict as a test in words, because "more dissipation ⇒ more dissipation" is the reflex reading and it is backwards; α = −c_ω still being the far-field exponent with dissipation on; the a = 1/2 K-ladder; and the positive control for the spectral filter.