Route-E v1: the rescaled gCLM flow has no eigenvalue available for a Hopf bifurcation

Phase-2 P2, Route E (the DSS lane), leg 1. Figure: writeup/figures/fig34_p2_route_e_v1_spectrum.png. Data: writeup/data/p2_route_e_v1_spectrum.json. Module: solver/rescaled_spectrum.py (gates: test_rescaled_spectrum.py, 8/8).

Level-1 numerics plus two small exact computations. Not a certificate, not a proof, not Clay progress. Plain float64 throughout; nothing here is interval-enclosed.

⛔ NOVELTY RETRACTED, ONE OPEN QUESTION CLOSED, ONE RESULT RE-CLASSIFIED

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

1. The point-spectrum negative is pre-empted. arXiv:2607.19762 Theorem 2 (Xu) proves the full point spectrum of the CLM linearization on the odd origin-H² realization is exactly {0, 1} (the symmetry modes, no embedded eigenvalues) in our own normalization Ω = -y/(y² + 1/4). This leg is confirmed but not novel.

2. THE ESSENTIAL-SPECTRUM CONTINUUM IS RE-CLASSIFIED, and this is the important one. Xu's Proposition 2 (realization dichotomy): the essential-spectrum smear that grids without an origin condition place inside the strip is the faithful spectrum of the maximal L² realization; imposing the single second-derivative condition at the origin removes the entire non-symmetry family (explicitly, u_λ(y) = y^{1-λ}/(y+i/2)², L² but with u'' ∉ L² at the origin). Our discretization has no origin condition, so we were rendering the loose realization and did not know there was a choice. The DSS conclusion survives (in the tight realization there is no continuum and no complex pair, so there is still nothing to bifurcate) but "the non-symmetry spectrum is continuous" must be stated as a property of the realization, not of the operator.

3. α(1/2) = 3 is exact and known, AND THIS LEG'S OPEN QUESTION IS ANSWERED. This writeup recorded that "what stays unexplained is why α = 3 lands on a round rational while α = 5 does not". arXiv:2207.07548 §1 answers it: exact pole-dynamics solutions exist at a = 0 and a = 1/2 and, per Lushnikov et al., nowhere else. c_l(1/2) = 1/3 is exact; verified here to 7.7e-5 by integrating their a = 1/2 system cold. The α = 5 point at a = 0.5821792673 is a property of our instrument (Λ⁵ is a finite matrix there), not of the problem.

4. The α(a) branch and a_c are published. a_c (Lushnikov–Silantyev–Siegel) is 0.6890665; Xu's recompute is 0.6888 (0.04%); ours is 0.693493 (0.64%). We are the least accurate of the three and should quote theirs.

See TECHNICAL_P2_ROUTEJ_V1.md and LITERATURE_CHECK.md sixth pass.

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

Route D spent sixteen legs on link L1 of the chain: a certified self-similar blow-up profile for a 1D toy model. Leg v15 (the literature check) found that L1 is very probably occupied territory: computer-assisted interval/Newton–Kantorovich certification is routine for the groups working this family. That did not invalidate a single measurement, but it did re-price the lane, and it promoted a different item to "the swing":

Nečas–Růžička–Šverák (1996), extended by Tsai: exactly self-similar blow-up for 3D Navier–Stokes in the natural scaling class is ruled out. Therefore the entire "find a self-similar profile and certify it" template cannot be pointed at NS as posed. The candidate class that survives is discretely self-similar (DSS).

In dynamic-rescaling variables the translation is exact, and it is why this leg is cheap:

A global search for a periodic orbit is expensive and needs somewhere to start. But one mechanism would both produce a periodic orbit and tell you where it is: a Hopf bifurcation off the self-similar branch, a complex-conjugate pair of eigenvalues crossing the imaginary axis as a moves. If such a pair exists the DSS lane opens with a concrete starting point; if there is no eigenvalue capable of crossing, that route is closed and the lane has to be entered some other way.

That is a question about a spectrum, and a spectrum is one dense eigenvalue solve.

Gate-check, answered rather than re-pasted. (a) Which link does this move? None. It does not advance L1→L4. It is lane scoping. (b) Is another L1 leg better? No: v15 re-priced L1 downward and named this as the swing. (c) Is there a cheaper experiment that would tell us the route is dead? This was it, which is why it was worth doing before any DSS search machinery was built.

1. The flow, and the one modelling choice in it

gCLM on the line: omega_t + a u omega_x = omega u_x, u_x = H(omega). Dynamic rescaling omega(x,t) = A(t) Omega(X,tau), X = x/L(t), dtau/dt = A gives

Omega_tau = (c_omega + H Omega) Omega - c_l X Omega_X - a U Omega_X ,   U(X) = int_0^X H(Omega) dX'

The two gauge functions (c_omega, c_l) carry the two scaling freedoms. This leg fixes c_l = 1 (a choice, and it is stated as one) and determines c_omega from the normalization that freezes the origin slope. Requiring (Omega_tau)_X(0) = 0 for odd Omega gives, exactly,

c_omega[Omega] = 1 + (a - 1) H(Omega)(0) .                                        (N)

At a = 0 this is the value-based normalization the project has used since Spike 0 (solver/gclm_rescaled.py), so this is the same flow continued in a rather than a new convention. Worth saying out loud: the project's existing residual used c_omega = 1 − H(Omega)(0) at every a, and that version has no fixed point at all for a != 0, differentiating the residual at the origin gives R_X(0) = -a H(Omega)(0) Omega_X(0) != 0. (N) is the repair, and it is forced, not chosen.

With c_l fixed, dilation survives as a symmetry: Omega(X) -> Omega(X/mu) is again a fixed point, and H(Omega)(0) is dilation-invariant so (N) is untouched.

2. The discretization: the far field costs nothing here

Compactify with X = tan(theta/2) and expand the odd profile in sines. Three operators are then exact on the whole line, with no domain truncation and no quadrature:

H(sin k theta) = -cos k theta + (-1)^k
X d/dX         = sin(theta) d/d theta          <- the DILATION term is bounded and exact
d/dX           = (1 + cos theta) d/d theta

The (-1)^k is not decoration: H^2 = -1 only modulo constants on the line, and that constant is exactly what makes H(Omega) vanish at X = infinity.

The velocity is exact too. Writing N_k(t) := ((-1)^k - cos k t)/(1 + cos t), the identity 2 cos t cos kt = cos(k+1)t + cos(k-1)t gives

N_{k+1} = -2 N_k - N_{k-1} - 2 cos k t ,      N_0 = 0,  N_1 = -1,

so every N_k is a trig polynomial, the 1 + cos t in the denominator always cancels, and U = sum_k b_k int_0^theta N_k is elementary. There is no quadrature anywhere in the build. (Gate 2 checks this as a polynomial identity out to k = 30, not merely at the anchor.)

The a = 0 fixed point is a single mode. Omega_0 = -sin theta = -2X/(1+X^2), with H(Omega_0) = 1 + cos theta = 2/(1+X^2) and c_omega = -1. It nulls the residual to 1.1e-16 at every K tested. Sixteen Route-D legs worked in this same compactified variable; this is the first time the self-similar (dilation) anchor has been written in it, and it is one Fourier mode.

3. Two eigenvalues are exact, at every a, and they are symmetry

Before computing anything, write down what has to be there. Let L be the linearization of the flow at a fixed point. Then

L (X Omega_X) = 0                              (dilation)
L (Omega)     = -Omega + X Omega_X             (amplitude)

The first is the dilation symmetry: the orbit is a curve of fixed points, so its generator is in the kernel. The second follows by substituting the profile equation into L(Omega) and using (N). So span{Omega, X Omega_X} is invariant with matrix [[-1,0],[1,0]]:

lambda = 0 (dilation) and lambda = -1 (amplitude), for every a.

A Jordan-like pair, present at every parameter value, carrying no dynamical information. Any DSS-relevant eigenvalue has to be something else. structural_pair_defect measures both identities rather than trusting them: 3.0e-15 / 3.3e-16 at a = 0 and 1.3e-10 / 9.1e-15 at a = 1/2, with the 2x2 block itself exact to 6.1e-16.

And the same two identities double as a free error bar. Because lambda = 0 is exact, its computed deviation measures the error of the whole spectrum at that parameter. At a = 0.2 the dilation defect is 8.0e-2 and the filter duly reports the dilation mode at -0.352; at a = 1/2 the defect is 1.3e-10 and it reports +8.9e-5. That single number is what licenses (and forbids) every quantitative claim below.

4. At a = 0 the rest of the spectrum is known in closed form

Set w = e^{-i theta} and Z = H Omega + i Omega, so the anchor is Z_0 = 1 + w. Writing the a = 0 linearization in s = delta Z:

L s = w s - ((w^2 - 1)/2) s_w - s(w=1) (1 + w) .

For s(1) = 0 the homogeneous problem L s = lambda s is a first-order ODE and integrates:

s_lambda(w) = (w - 1)^{1 - lambda} (w + 1)^{1 + lambda} .

The verification is one line (((w^2-1)/2) s_w = s (w - lambda), hence L s = lambda s) and admissibility does the rest: s -> 0 at w = -1 (decay at X = infinity) needs Re lambda > -1; s bounded and vanishing at w = 1 (i.e. at X = 0) needs Re lambda < 1.

So the a = 0 linearization carries a continuum of eigenvalues filling the strip -1 < Re lambda < 1, whose eigenfunctions carry a fractional power (w-1)^{1-lambda} at the origin. Demanding analyticity there forces 1 - lambda to be a non-negative integer, and with Re lambda > -1 that leaves exactly lambda = 0 and lambda = -1: the two structural modes and nothing else.

4.1 The member worth looking at is lambda = i y

Near the origin w - 1 ~ -2iX, so

s ~ X^{1 - i y}          against the time factor   e^{i y tau}
  =>  exp( i y ( tau - log X ) )
```: a wave travelling **outward in `log X` at unit speed**, exactly `tau`-periodic with period
`2 pi / y`. **That is the log-periodic structure a DSS solution is made of, and it is present
in this operator, exactly.** It is *continuous* spectrum, not a bound state: it is the dilation
transport carrying a scale-invariant wave to infinity. Nothing there can cross an axis. That
is the mechanism behind this leg's negative rather than a restatement of it, and gate (5b)
checks the identity with a numerical derivative and confirms the period.

---

## 5. The branch: `alpha(a)` is an output, and it runs away

Following the fixed point from the exact anchor by continuation, the far-field decay exponent
is a result, not a parameter: `c_omega Omega = (X + a U_inf) Omega_X` gives

Omega ~ X^{-alpha} , alpha(a) = -c_omega(a) .

Richardson-extrapolated over `K = 64/128/256`:

| `a` | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 |
| --- | --- | --- | --- | --- | --- | --- |
| `alpha` | 1.000000 | 1.141397 | 1.334497 | 1.617244 | 2.079464 | **3.000000** |

`alpha` increases with `a`; following it finely (`K = 192`, `da = 0.005`) the branch is lost at
`a = 0.65` with `alpha = 11.5` at the last good point, and `1/alpha` extrapolates linearly to
zero at **`a_c ~ 0.694`**, the tail becomes infinitely steep and the branch, as posed on the
whole line, ends. This is the self-similar analogue of what Route-D v12/v14 found for the
*traveling-wave* object (which ends at a finite radius `X_c`); different object, so the
agreement is a check rather than a repetition.

**One consequence governs everything numerical here.** A branch point of order `alpha` at
`X = infinity` means the sine coefficients decay *algebraically*: measured residual `K^-2`ish
at `a = 0.3` (`7.8e-2 -> 6.7e-4` over `K = 16..256`). There is exactly one exception on the
branch besides the anchor:

* `a = 0`, `alpha = 1` (exact, one mode;
* **`a = 1/2`, `alpha = 3`) `c_omega = -3.000000000000`, residual `1.4e-14` at `K = 192`**
  (`2.7e-1 -> 4.4e-2 -> 7.0e-5 -> 1.4e-11 -> 1.4e-14` over `K = 16..192`; geometric coefficient
  decay, i.e. analytic).

The `a = 1/2` point was *found*, not assumed: a scan of the fixed-point residual in `a` at fixed
`K` shows a single dip, ten orders deep, exactly there.

### 5.1 The odd-`alpha` rule: hypothesised from two points, tested at a third, and
### nearly discarded on two rungs of a ladder

`Omega ~ (pi - theta)^alpha` near `X = infinity`, so "`alpha` an odd integer makes the profile
smooth there" is the natural reading, and it fits both special points (`alpha = 1` and `3`).
That is a rule inferred from a two-point set with one degree of freedom, so E8b/E8c went and
found the third point: `alpha = 5` occurs at **`a = 0.5821792673`** (secant-solved to `5e-11`).

The `K`-ladder there is the interesting part, because **its first two rungs say the opposite of
its last four**:

| `K` | 96 | 128 | 192 | 256 | 320 | 384 |
| --- | --- | --- | --- | --- | --- | --- |
| `sup|R|` | 3.19e-2 | 1.12e-2 | 7.92e-4 | 2.87e-5 | 9.24e-7 | 2.55e-8 |
| implied order | (| 3.6 | 6.5 | 11.5 | 15.4 | 19.7 |

An order that **rises monotonically** is not an order at all) it is exponential convergence
seen before it has settled. **`alpha = 5` is an analytic resonance too, and the odd-`alpha` rule
holds at all three points.** The `alpha = 5` profile is simply steeper (`Omega ~ X^-5` against
`X^-3`), so it enters its asymptotic regime later; at the `K = 256` of the E8b scan its dip is
only `13x` deep, and by `K = 384` it is four orders.

*This paragraph replaced an earlier version of itself.* On the first two rungs (`3.19e-2 ->
1.12e-2`, order `3.6`) this note said the rule was **false** and that `a = 1/2` was special for
an unidentified reason. Two rungs of a ladder are not a rate, banked lesson (22) applied one
level down, and the correction is marked rather than quietly edited (banked lesson (35)).

What survives as genuinely unexplained is narrower and more specific: **`alpha = 3` lands at
exactly `a = 1/2`**, while `alpha = 5` lands at `0.5821792673`, which is not an evidently
special number. So the rule explains *why* those `a` are analytic; it does not explain why one
of them is a round rational.

**Novelty unchecked**: an exact exponent at `a = 1/2` in this family is precisely what may be
folklore to people who work on gCLM/De Gregorio, and the literature check is still blocked on
PDF access (v15's finding).
A two-parameter rational ansatz `Omega = -c X/(X^2+gamma)^2` reproduces the profile to
**7e-5** relative (the right *shape*, right down to the location and depth of the minimum) but the profile itself is computed to `2e-14`, so the ansatz is a near miss and **the closed
form was not identified.**

Because of all this, **every quantitative spectral statement below is quoted at `a = 0` or
`a = 1/2`**: the only two points on the branch where the fixed point is analytic. At generic `a` the instrument cannot resolve even the eigenvalues it is known to
have (§3), and no conclusion is drawn there.

---

## 6. The spectrum

### 6.1 What the filter said, and why the first reading of it was wrong

Refining `K = 96 -> 144` and keeping the eigenvalues that move less than `1e-2`:

| `a` | kept |
| --- | --- |
| 0 | `0` (dist 8e-16), `-1` (dist 6e-15) |
| 1/2 | `+8.9e-5` (dist 9e-5), `-1.000005` (dist 5e-6), **`-2.0073` (dist 5.6e-3)** |
| 0.1, 0.2 | two values, but the *dilation* one is reported at `-0.120` / `-0.352`, see §3 |
| 0.3, 0.4, 0.55 | nothing (the fixed point is not resolved well enough) |

The third entry at `a = 1/2` is not symmetry, and the honest first reaction was that the
fixed point has a genuine non-symmetry mode. A `K`-ladder makes it look even better: `-2.007293 / -2.001677 / -2.000467 / -1.999999` over `K = 96..256`, converging on `-2` exactly.

**It is not a mode. It is the left edge of the essential spectrum.** The two singular endpoints
of the operator fix that spectrum's extent: near `X = infinity` the local operator is
`c_omega + xi d/d xi` with `xi = pi - theta`, and near `X = 0` it is
`(c_omega + H Omega(0)) - theta d/d theta`, so

    c_omega + s_min  <=  Re lambda  <=  c_omega + H(Omega)(0) ,

```

and in this space s_min = 1, because every sin k theta vanishes linearly at theta = pi. So the accessible strip is [c_omega + 1, c_omega + H(Omega)(0)], which is [0, 1] at a = 0 and [-2, +5] at a = 1/2. Both edges land where predicted: the measured spectrum spans [-2.007, +4.55] at a = 1/2, and c_omega + 1 = -2 is the value the "third eigenvalue" converges to.

The control that settles it costs nothing, because the project already had it: at a = 0 the same edge sits at 0, and 99% of the discretized spectrum sits on it. Nobody would call that an isolated eigenvalue. It is the same object at a = 1/2, moved to -2 because c_omega moved.

6.2 The verdict

The only grid-converged ISOLATED eigenvalues, at both points where the method can see, are 0 and -1: the two exact symmetry modes. There is no complex pair anywhere, nothing approaching the imaginary axis, and therefore no Hopf bifurcation off the gCLM self-similar branch.

Two things must be said alongside it, and neither weakens it:

• The flow is not spectrally stable. Its ESSENTIAL spectrum reaches c_omega + H(Omega)(0), which is +1 at a = 0 and +5 at a = 1/2, well into the right half plane, complex members included (the measured cloud runs to Im ~ ±150). Those directions are exactly the ones with a fractional power at X = 0: a corner at the origin grows relative to the profile. That is the familiar low-regularity essential instability of self-similar linearizations, it is norm-dependent, and (this is the point) a continuum has no eigenvalue to move, so it cannot Hopf-bifurcate.
• At a between the two resonances the instrument is too blunt to say anything, and §3's free error bar is how that is known rather than guessed.

6.3 The positive control: the part that makes the negative mean something

"Only two survived the filter" is evidence only if the filter would have reported more. The same filter, on the same operator with a smooth localized potential added:

The instrument can see an isolated unstable eigenvalue, twice over. There is not one.

6.4 What this does not say

• Not that gCLM has no DSS solution: only that one is not born from a Hopf bifurcation off the self-similar branch that continues from CLM. Periodic orbits can exist without a fixed point nearby that spawned them, and §4.1 shows the log-periodic directions themselves are present (in the continuum).
• Nothing about NS. gCLM's scaling structure is not NS's; the only reason DSS is interesting for NS is a theorem about NS.
• The essential spectrum's location is norm-dependent, and what is measured is the spectrum of a discretization, filtered for grid-independence. An eigenvalue embedded in the continuum can be missed by any such method. a = 0, where the closed form settles it independently, is the only place that risk is retired.
• One gauge only (c_l = 1 plus (N)). Changing the normalization moves the symmetry eigenvalues but not the transverse spectrum: standard, but an argument here, not a measurement.

7. What this cost, and what it bought

Cost: one module, one experiment, eight gates, and two follow-up measurements the first sweep forced. Bought:

1. The cheapest DSS mechanism is closed for the 1D family, with a positive control behind the negative and a mechanism (§4.1, §6.1) rather than an absence.
2. The self-similar branch of gCLM in the compactified variable: exact one-mode anchor, exact velocity operator, Newton continuation.
3. alpha(a), the far-field decay exponent map, as an output, with the branch's end.
4. The odd-alpha resonance family (alpha = 1 (exact), alpha = 3 at a = 1/2 (12 digits), alpha = 5 at a = 0.5821792673 (exponential ladder to 2.5e-8)) and the narrower open question it leaves: why alpha = 3 lands on a round rational. Novelty unchecked; closed form not identified.
5. Two exact statements about the a = 0 linearization: the closed-form continuum (w-1)^{1-lambda}(w+1)^{1+lambda} on the strip -1 < Re lambda < 1, and the log-periodic identification of its imaginary members.
6. The essential-spectrum edge formula [c_omega + 1, c_omega + H(Omega)(0)], confirmed at both exactly-resolved points, which is what turned a spurious "third mode" into a measurement.

Where this sits relative to Clay (re-answered, not re-pasted)

Unchanged, and stated plainly: nothing here is a step whose success would resolve the Clay problem. This leg moves no link of the chain. What it does is stop the project spending several legs building a DSS search around a mechanism that does not exist in the family where its tooling lives. The two structural walls are untouched. Odds ~0.05%, unchanged.

The honest summary of the DSS lane after one leg: the lane is not closed, but the cheap entrance is. Entering it now costs a real build (a periodic-orbit search in the rescaled flow with no fixed point nearby to seed it) and §4.1 says where such an orbit would have to live (the log-periodic directions, which are continuous spectrum and of limited regularity at the origin). That is a much bigger commitment than this leg was, and it should be weighed against the alternatives (the Hou–Luo critical-viscosity map; the L1→L2 port to 2D Boussinesq) rather than taken by default.

Appendix: new lessons this leg paid for

(46) A symmetry audit is cheaper than an eigenvalue solve, and it predicts part of the answer. Two eigenvalues here were derivable in five lines from the flow's two symmetries. Doing that first meant the numerical spectrum arrived with its expected members already named, so "only two survived" read immediately as "nothing but symmetry" instead of looking like a result. Enumerate the symmetry modes before computing a spectrum; they are the null result's baseline.

(47) A null result needs a planted positive, not just a control. Banked (2) says ablate to attribute and (9) says build the adversary; this is the third member. When the finding is absence, show the instrument detecting a presence of the same kind. Planting a bound state and recovering it at +1.083 and +4.578 costs three lines, and is the difference between "there is no unstable eigenvalue" and "we did not find one".

(48) When you generalize a gauge, re-derive it: do not extend it. c_omega = 1 - H(Omega)(0) is correct at a = 0 and was being carried at every a. For a != 0 that flow admits no fixed point at all; one line of algebra at the origin catches it. This is banked (29) (a constant is attached to a point, not to a problem) wearing the gauge's clothes.

(49) The regularity of the object sets the convergence rate of everything built on it, and it can vary with the parameter. Here alpha(a) is an output and is an odd integer at exactly two points on the branch; there the method is spectral and everywhere else second-order, a ten-order accuracy swing driven by nothing but the parameter. Find where your object is smooth and quote your sharp numbers there, instead of quoting one accuracy for a whole sweep.

(50) An exactly known eigenvalue is a free error bar on every other one. The dilation mode is 0 by symmetry, so its computed value is the spectrum's error at that parameter: 0.35 at a = 0.2, 8.9e-5 at a = 1/2. That number decided which rows of the sweep were allowed to carry a conclusion, and it cost nothing to read. If a symmetry pins one eigenvalue, plot its deviation next to every claim you make about the others.

(52) Two points define a line through anything: and two RUNGS define a convergence rate through anything. "alpha odd integer => analytic" fitted the only two special points I had, so the third was located by secant and laddered. Its first two rungs implied order 3.6 and I wrote the rule off as false; four more rungs showed the order climbing 6.5 -> 11.5 -> 15.4 -> 19.7, i.e. exponential convergence that had not settled, and the rule was right after all. The mistake and its repair are the same lesson at two scales: before a fitted exponent becomes a claim, add rungs until the exponent stops moving. A steeper object reaches its asymptotic regime later, which is exactly when a short ladder is most misleading.

(51) A convergence filter can be fooled by the EDGE of a continuum, and the fix is a control point, not a tighter tolerance. The -2 at a = 1/2 converged to six digits under refinement and was not a mode: it was c_omega + 1, the accumulation edge of the essential spectrum in this space. Tightening the filter would have made it look better. What exposed it was evaluating the same quantity at a = 0, where the identical edge carries 99% of the discretized spectrum and is obviously not an eigenvalue. Before believing an isolated eigenvalue, ask where the continuum's edges are: and go and look at the same object somewhere you already understand it.