Route-I v1: the marginal flow, driven

Phase 2 / P2, leg 41. Figure: writeup/figures/fig38_route_i_v1_driven.png. Code: solver/marginal_flow.py, test_marginal_flow.py (11/11), experiments/p2_route_i_v1_driven.py → writeup/data/p2_route_i_v1_driven.json. Working notes: PHASE2_P2_NOTES.md §30.

Level-1 numerics plus one exact-solution gate. NOT a certificate, nothing interval-enclosed, no link of the L1→L4 chain moved. Clay odds unchanged at ~0.05%.

⚠️ THE μ = 0 STABILITY COUNT NEEDS ITS REALIZATION NAMED

(added 2026-08-04, Route-J v1 primary-source pass: this is a live correction, not a retraction)

This leg's headline is that the inviscid rescaled fixed point at a = 1/2 has 141 of 144 unstable directions, read as "§26's essential spectrum as a count". arXiv:2607.19762 Proposition 2 (Xu, realization dichotomy) shows that what that count is counting is realization-dependent: the essential-spectrum smear which grids without an origin condition place inside the strip is the faithful spectrum of the maximal L² realization, and the origin-H² realization has none of it. Our discretization has no origin condition: the loose realization is the one we measured.

This does not make the measurement wrong, and it does not by itself touch the μ > 0 half of the inversion (dissipation collapsing the spectrum onto a discrete negative ladder is not realization-smear). It does mean the μ = 0 count is quoted in a realization nobody would choose deliberately. The count must be stated with the realization named, and the honest next step is to re-run I5 with an origin condition and report what the count is there. That is the top-ranked correction item for the next leg.

Route-I also inherits Route-F's pre-emption in full: λ_μ = 2s - α₀ is Route-F's s_c in spectral clothing, and s_c = α/2 is Xu §6.1 eq (6.3), posted eleven days before Route-F. See TECHNICAL_P2_ROUTEJ_V1.md and LITERATURE_CHECK.md sixth pass.

0. What this leg is, in one paragraph

Route-H v1 (§29) wrote down the augmented flow (the rescaled gCLM equation with the dissipation coefficient mu = nu/(A L^{2s}) promoted to a dynamical variable) and then read it statically: Newton at frozen mu, alpha(mu) off the resulting branch, and the dynamics inferred from the shape of that curve. Two numbers came out of that reading and neither was ever integrated. This leg integrates the coupled system as an initial-value problem and measures both from a trajectory. It also answers the piece of ranked item (2) that §27 and §28 left open (the scaling says which term dominates given the self-similar form; does a viscous solution actually reach that form?) and the answer turns out to be more interesting than the arithmetic that motivated it.

The equations, unchanged from §29:

Omega_tau = (c_omega + H Omega) Omega - X Omega_X - a U Omega_X - mu Lambda^{2s} Omega   (F_mu)
mu_tau    = (2 s - alpha[Omega, mu]) mu ,      alpha = -c_omega                          (M)

with c_omega = 1 + (a-1) H(Omega)(0) + mu (Lambda^{2s}Omega)_X(0)/Omega_X(0), in Route-E's compactified odd-sine basis (X = tan(theta/2)), where H, the dilation operator, the velocity and Lambda^p for integer p are all exact and quadrature-free.

1. The headline: the stability inversion

At a = 1/2 the linearization of the inviscid rescaled flow about its self-similar fixed point is violently unstable. Counting eigenvalues of S^{-1} dR/db with Re > 1e-6, excluding the exact dilation mode:

This is §26's essential spectrum [-2, +5], seen as a count. A fixed point with a 141-dimensional unstable manifold is not an attractor and nothing generic reaches it.

Any mu > 0 removes all of it. The spectrum becomes a discrete negative ladder (0 (the exact dilation mode), then approximately -1, then -4.7, and downward) and the gap between the neutral mode and the next one is insensitive to mu across four decades. That is the dynamical statement the scaling legs could not make: at criticality the viscous self-similar profile is linearly attracting, and its inviscid limit is not.

1.1 Why this is not the artifact it looks like

Adding a large negative-definite operator to a matrix pushes its eigenvalues left, so "mu > 0 stabilizes the truncated generator" is exactly the sort of claim that is an artifact of truncation. The crossover settles it. Dissipation damps mode k at rate mu k^p, so it beats a growth rate g once

mu K^p  >~  g ,     i.e.   mu*(K) ~ g / K^p ,     g = max Re of the inviscid generator.   (C)

Measured, with g = 4.55:

mu* falls with K. So at any fixed mu > 0 a fine enough grid sees zero unstable directions, and at any fixed K a small enough mu sees ~K of them: the two limits do not commute. An artifact would need mu* to be independent of K or to grow with it.

How much of that the ladder actually resolves, stated plainly. The mu grid is decade-spaced, so the bracket has one-decade resolution. 48 -> 96 moves it by a full decade (measured x10, (C) predicts x8): that step carries the claim. 96 -> 144 predicts only x3, which is below the ladder's resolution, and the bracket duly does not move (x1.0). That is consistent with (C) and is not independent evidence for it; reading the flat step as confirmation would be reading the grid. A sharper test needs a finer mu ladder, and it is the obvious next thing to spend time on if this claim ever has to carry weight. (C) is also the resolution guard every trajectory here is run against, for the leg's main trajectory, mu >= 0.02 at K = 96 needs K > 6.1, a margin of about 16x.

1.2 Where the instability lives, and why it is the DSS band

The leading inviscid eigenvalue at K = 96 is +4.5455 + 430.35i. The unstable spectrum is not a set of slowly growing modes; it is a curve on which Re increases with |Im|:

and max|Im| grows with K (430 at K = 96, 661 at K = 144). These are exactly Route-E's continuum modes exp(i y (tau - log X)), the log-periodic directions. So the directions that grow fastest in the inviscid rescaled flow are precisely the ones a discretely self-similar solution is built out of, and dissipation is what removes them. That is a sharper statement than §26's ("a continuum has no eigenvalue to move") and §29's ("dissipation discretizes the continuum, and nothing crosses"): the continuum is not merely unable to bifurcate, it is the unstable part, and mu deletes it rather than damping it, at mu = 0.05 the whole band is gone, max Re = -1e-13.

Two consequences worth carrying:

• the DSS lane's obstruction is now doubly stated. A DSS solution needs the log-periodic structure; in the inviscid problem that structure is continuous spectrum (§26) and it is where all the growth is (here); in the viscous problem it is absent entirely. The lane is not closed, but nothing in three legs has produced a mechanism by which it opens.
• any time integrator caps the |Im| it can resolve at ~pi/dt, so a nonlinear growth rate measured from a generic kick is a lower bound on max Re, not a measurement of it. Stated below rather than discovered later.

2. Route-F's critical exponent, measured as a growth rate

lambda_mu = 2s - alpha_0 is the growth rate of mu at the inviscid fixed point: the spectral form of Route-F's s_c. Integrating (M) from mu_0 = 2e-3 and fitting d(log mu)/d tau in the linear regime, at a = 1/2 where alpha_0 = 3 exactly:

Fitted as a line in s over the four kept rows: slope +2.0011 against a predicted +2, zero at s = 1.50009 against a predicted 1.5, worst kept error 1.4e−3. Two signs, three non-marginal points, and the marginal one returning −2.6e−4.

This is not a re-derivation of §27. Route-F fitted a power law to D/N along a time-dependent periodic pseudo-spectral trajectory and read s_c off where the exponent crossed zero; this fits an exponential growth rate of a different variable in a compactified steady-basis integration on the line. No shared grid, basis, formulation or fitted constant. The prediction 2s - alpha_0 uses alpha_0 = 3, a number this computation never sees.

Three refusals, and they are the honest part of this section. The rung gate asks two separate questions (is the profile resolved (seed residual < 1e−5) and is the operator accurate (Lambda^p truncation < 1)) and a rung has to pass both.

• p = 5 at a = 1/2 is refused on the operator, truncation 1.77: the discarded coefficients exceed the kept ones. Its number, +2.00699, would have looked like the best confirmation in the table (it is the largest |lambda_mu| and lands within 0.35%) which is exactly why the gate is on the operator and not on the agreement. This is Route-H's H4 finding recurring: Lambda^p amplifies mode k by ~k^p, so the same K that is ample at p = 3 is not at p = 5.
• All three a = 0.3 rungs are refused on the profile. Off the resonances alpha_0 is not an integer, the compactified basis converges algebraically rather than spectrally, and the seed residual is 6e−4 to 4e−3 at the same K: two to five orders worse than the 1e−9..1e−8 at a = 1/2. So this leg has no off-resonance control, and the two rungs that came closest (p = 1: −0.64493 against −0.6172; p = 2: +0.37352 against +0.3828, both ~4% out) are reported as refused rather than quoted as a weak confirmation. p = 3 there returns −3.39 against a prediction of +1.38 (wrong sign, wrong magnitude) which is what an unresolved profile is worth.

The a = 0.3 refusals matter for what the leg is allowed to claim: the growth-rate law is confirmed at one value of a, the resonance where the basis is spectral. Whether it holds off the resonances is untested here, and the earlier draft of this section quoted the a = 0.3 numbers as a passing control, they are not.

3. The tar pit, driven

At criticality (a = 1/2, p = 3) the linear term in (M) vanishes and mu_tau = -alpha_1 mu^2. Route-H got alpha_1 by extrapolating secants of alpha(mu) on the frozen-mu Newton branch. Here it is the slope of 1/mu along a trajectory: 1/mu is linear in tau under the quadratic law, which is why the fit lives there.

At matched K, the two computations agree:

Both ladders climb toward ≈0.1337 and the gap between them narrows with K, which is what two discretizations of the same quantity should do. The verdict is relaxes_to_inviscid from both.

Error budget, and it is a different one from Route-F's:

• the time step is not the systematic. BDF2 self-convergence order 1.994 over dt = 1 … 1/8, spread in alpha_1 8.7e-5, so signal/discretization ≈ 1520. The refusal gate (> 10) is cleared by two orders.
• the fit window is not the systematic either. Sweeping mu <= 0.30 … 0.10 moves alpha_1 by ~2e-5. In Route-F the window was the dominant systematic; here it is negligible and K is everything. Worth saying out loud, because the reflex after §27 is to sweep the window and quote its spread as the error bar: it would understate the error here by two orders.
• the gauge is enforced, and the enforcement does no work. sum_k k b_k = -1 is an exact invariant of (F_mu) (c_omega is defined by R_X(0) = 0, which is kk . S^{-1}R = 0). Numerically that identity holds to a relative 3.6e-9 per step, which accumulates to a drift of 4.0e-4 over tau = 60: 3% of the alpha - alpha_0 signal the leg quotes. The flow therefore subtracts the dilation component roundoff put there. Turning the subtraction off moves the headline by 0.037%, and the drift goes from -4.7e-14 to +4.0e-4. Both halves are gated: a projection that had to do real work would be a bug wearing a fix's clothes.

Over tau = 0 … 600 from mu_0 = 0.3, the integrated mu(tau) tracks the closed law 1/(1/mu_0 + alpha_1 tau) to within a few percent at worst, and the decay times are the ones §29 predicted: nine times the tau per decade, forever, with tau itself logarithmic in (T-t).

4. Adiabaticity: the static reading's assumption, measured

Reading alpha off a frozen-mu branch is only legitimate if the driven trajectory stays on that branch. It does, and it gets better with time: the relative distance ||Omega(tau) - Omega*(mu(tau))||_inf runs 1.2e-4 → ~2e-7 over the long run. It should tighten: the driving rate is mu_tau ~ mu^2, so as mu decays the flow is pulled off the branch ever more slowly while the restoring gap stays at ≈1.

Starting off the branch is the stronger version, and it needs a qualification that the first version of this section did not have.

Perturbing Omega by eps and letting mu run reproduces the on-branch mu(120) = 0.052067 to 0.000% and alpha_1 = 0.132361 to six digits, for eps up to 5e-2. That agreement is exact-looking because it should be: the spectral gap is ≈ -1, so over tau = 120 a perturbation is suppressed by e^{-120}, i.e. below double precision. It is an independent confirmation of §1's inversion through the nonlinear flow rather than a separate fact.

What eps measures, and why it is not what it looks like. The perturbation is normalized in the functional the gauge (N′) actually reads, b -> (Lambda^p Omega)_X(0), not in the coefficient sup-norm, and those differ by 3.6e8 at a = 1/2, p = 3, K = 96. Lambda^p weights mode k by ~k^p and the derivative at the origin adds one more power, so that functional's condition number is ~K^4. Consequences, both worth stating:

• the first version of this measurement normalized in coefficients, which made eps = 1e-3 a 3.6e5 relative perturbation of the quantity the flow divides by. alpha came back as 11713 instead of 3.037 at tau = 0, before a single step, and every off-branch trajectory overflowed. Every published number in this section was nan. See §8, lessons (65) and (66);
• correctly normalized, eps = 5e-2 is a 5e-11 relative change in the profile. So this is a strong test of the gauge-sensitive direction and a weak test of profile-scale robustness. It is not a basin-of-attraction measurement, and it should not be read as one: §5's twin trajectories are the profile-scale test, and they are the ones to cite for how big a perturbation the viscous fixed point actually absorbs.

This is the same k^p amplification that Route-H's H4 found making Lambda^5 unusable: one mechanism, showing up in two legs as two different symptoms.

5. The nonlinear control

Eigenvalues of a truncated generator are a claim about a matrix. Twin trajectories (integrate the kicked and unkicked states side by side and difference them) are a claim about the flow.

The fit residual is what separates the rows, and it separates them by two orders (0.007–0.012 against 1.49–2.32). Where a single exponential describes the trajectory, the twin-trajectory rate reproduces the spectral gap to a fraction of a percent: that is the control that matters, and it says the linear-stability claim of §1 survives passage through the nonlinear flow. Where a single exponential does not describe the trajectory, the fitted "rate" is not a rate, and the table says so rather than averaging it in.

Two rows are quoted as failures on purpose:

• mu = 0 is a lower bound, not a measurement. dt = 0.01 resolves |Im| <~ 314 and the leading eigenvalue sits at |Im| = 430, so no usable time step can realize +4.55. What the row establishes is only that the kick grows, by 4.6e5 in tau = 6. Any integrator caps the |Im| it can see at ~pi/dt, so a nonlinear rate from a generic kick is always a lower bound on max Re: stated in §1.2 before it was needed rather than discovered here.
• mu = 1e-3 has not reached its asymptotic rate. The gap is −1.0002, but at that mu the dissipative ladder is barely separated and the kick's projection onto the slowest mode is small, so over tau = 6 the difference is still dominated by faster transients: visible directly in fig38 F as the kinked orange curve, and in the fit residual of 2.32. It is not evidence against §1; it is evidence that this instrument needs either a longer run or a larger mu, and the two larger-mu rows are the ones that carry the control.

An earlier draft of this section reported the mu = 1e-3 row as "≈ −1.0, <1%". It is 81% out. The number was never in the data; the row was written from what the eigenvalue said it should be.

A measurement that went wrong first, kept because the failure is the lesson. The obvious version of this test, integrate the kicked state and watch |b(tau) - b0|, reads the base state's own motion, because b0 is a Newton fixed point with residual ~1e-8 rather than an exact one, and it drifts at a rate comparable to a small kick. Run that way, mu = 0.05 and mu = 0.2 both returned a growth rate of +1.5, the opposite of what every eigenvalue says, with no diagnostic complaining. Differencing against a twin removes the base motion exactly, for the price of one more trajectory.

6. Where this sits relative to Clay

It moves no link of the chain. L1 (a certified profile in 1D) is untouched, nothing here is interval-enclosed and no budget is closed. What it does is convert the previous two legs' inferred dynamics into measured dynamics, and answer the dynamical half of ranked item (2) in the toy: at criticality a viscous solution does reach the self-similar form, because that form is linearly attracting once mu > 0, and it then cannot leave because mu decays algebraically.

The honest translation to NS is narrow and should be stated as such:

• s = 3/2 is hyperviscosity. It is used because that is where criticality can be posed exactly in this basis, not because it resembles NS.
• the relevant structural fact (the inviscid rescaled fixed point is unstable in a continuum of log-periodic directions, and dissipation deletes that continuum) is the kind of statement that ought to be standard in the parabolic-blowup literature. See §7.
• "linearly attracting in this norm at this truncation" is not "an attractor". The unstable-count statement is about the spectrum of a truncated generator, supported by the K-ladder (C) and by a nonlinear trajectory; it is not a theorem about the continuum operator, whose essential spectrum for mu > 0 is not computed here.

7. Novelty: unchecked, and probably thin

(M) itself is at risk through Route-F/H's inheritance, the dissipative-gCLM relevance exponent is reported in arXiv:1908.09385 / arXiv:2207.07548, and lambda_mu = 2s - alpha_0 is that exponent in spectral clothing. See LITERATURE_CHECK.md; arXiv:2207.07548 is still the first paper to read and it now gates four claims.

The three methodological candidates from this leg are unsearched (WebFetch remains 403 on every host including Wikipedia; WebSearch is the only channel):

1. the stability inversion: mu > 0 removing a continuum of unstable directions of the rescaled flow, with the non-commuting limits made quantitative by (C);
2. the identification of the fastest-growing inviscid directions with the DSS band;
3. twin-trajectory differencing as the control on a computed spectral gap.

Presume (1) known: it is the sort of thing that follows from the resolvent of a sectorial operator and someone has certainly written it down for a parabolic blowup problem. (2) is the one worth a specialist's five minutes.

8. Lessons banked (61)–(66)

• (61) A number read off a trajectory needs a regime bound, not just a fit window. At a = 0.3, p = 3 "the first 40% of the run" gave +11.8 against a prediction of +4.6 because mu had left the linear regime by tau = 1.2. The fit must be bounded by the variable, not only by the time.
• (62) Differencing against a twin trajectory is how you measure a perturbation off an inexact fixed point. A Newton fixed point at residual 1e-8 drifts as fast as a small kick, and the naive measurement returned the opposite sign with nothing complaining.
• (63) When a truncated operator becomes stable under a perturbation, the crossover's K-scaling is the artifact test. mu*(K) ~ g/K^p falling with K says the limits do not commute; an artifact would have mu* flat or rising.
• (64) The dominant systematic changes between legs, and the reflex is to reuse the previous leg's. Route-F's error bar was the fit window; here the window contributes 2e-5 and K contributes 1e-3. Quoting the window spread as the error bar would have understated it by two orders.
• (65) A refused computation must be refused in the DATA STRUCTURE, not just in the prose. Every off-branch run in §4 overflowed; integrate returned the state anyway, the driver stored alpha_1 = nan, and the figure legend rendered "off-branch ε=0.001: α₁ = nan", three times, in a published panel. Finiteness alone is not the test either: with a partial fix the same run stayed finite and ended at mu = -1.6e24. The discriminator that works is the implicit solve's own residual over its floor (~6e2 healthy, 1e13 failed), and integrate now reports converged from it.
• (66) "Small" is meaningless until you say IN WHICH NORM, and the binding norm is set by the operator, not by the coefficient vector. A perturbation at 1e-3 in coefficient sup-norm was 3.6e5 in the functional the gauge divides by, because Lambda^p weights mode k by k^p. The symptom was total (alpha = 11713 at tau = 0); the cause was invisible in the quantity being reported. Normalize in the functional that binds, and report both numbers, because the corrected perturbation turned out to be 5e-11 of the profile, which changes what the measurement is entitled to claim.