Route-M2P v1 (leg 125), Chen's γ=2 dissipative gCLM candidate: full text, constants, first Y₀

Branch leg/125-m2p-v1. Module solver/dissipative_profile.py (new, sole owner). Tests test_dissipative_profile.py, 20/20. Runner experiments/p2_route_m2p_v1_promotion.py. Data writeup/data/p2_route_m2p_v1_promotion.json. Figure fig61. no_dynamics_run: true: every solve here is of a steady equation. Claims no stage. Everything is floating point; no number in this document is a certificate.

0. What this leg was sent to do

Leg 63's screen produced the only screen-passing candidate in 63+ legs: gCLM with full Laplacian dissipation (γ=2), a near 1/2, blow-up proved analytically by J. Chen, arXiv:1908.09385 (Nonlinearity 33 (2020) 2502). It did not read the full text, transcribe the constants, or measure Y₀. Three debts, discharged in that order.

The novelty pass ran and was committed first (writeup/novelty/leg_125.md, commit bb3f356), five verbatim queries, links not counts. Its two operative findings:

• No CAP or validated-numerics certificate of any dissipative self-similar gCLM profile exists in the searched literature. Every CAP hit returned was inviscid. So a measured Y₀ here is a novel data point regardless of its value.
• Chen is analytic: weighted L² and weighted H⁴ energy estimates (§2.3–2.5), no interval arithmetic anywhere. This confirms at full-text depth Papers/MANIFEST.md's 2026-08-04 index-depth entry: an exclusion, not a CAP precedent.

1. The full-text read

1.1 Theorem 1.1, verbatim (p.3)

Theorem 1.1 (Finite time blow-up for a close to ½). Consider (1.1) with Lω = −∂ₓₓω. There exists δ > 0 such that for a ∈ (½ − δ, ½ + δ), 0 ≤ ν ≤ 1, (2.1) develops a self-similar singularity in finite time for some C_c^∞ initial data.

1.2 The profile, verbatim (eq (2.2), p.4)

Chen studies the inviscid problem first. For a = ½ the steady self-similar equation

(c_l x + a U) Ω_x = (c_ω + U_x) Ω ,   U_x = HΩ ,  a = 1/2

has, in his words, "the following analytic self-similar solution":

All of these are machine-readable in chen_constants() with the provenance string attached to each, and test_dissipative_profile.py checks their internal consistency (b² = 3/8, 1/b² = 8/3, 2c_l + c_ω = −1/3).

1.3 The sentence that decides the leg (§2.6, p.12, verbatim)

"Since ν(t) converges to 0, such profile is the same as the inviscid profile associated with a."

with Remark 2.1 (p.11): "(2.43) implies that the diffusion term in (2.6) vanishes as t → +∞", and §1.5 (p.4): "In Section 2, we construct the self-similar profile for the inviscid gCLM (1.1) with a = ½."

There is no γ = 2 self-similar profile in the paper. The dissipation is genuinely present in the equation and genuinely handled by the proof, but it is handled as a vanishing perturbation: Chen states the plan explicitly on p.4, "if we add the diffusion term, such term is asymptotically small compared to the nonlinear term in the equation of the self-similar variables. In our later analysis, we will treat the diffusion term as a small perturbation."

1.4 The a-neighbourhood is unquantified, as leg 63 suspected

Eq (2.41), p.11 fixes δ and ν₀ only through

(1 + C₂ + C₃ + C₂C₃ + C₂²)(δ + ν₀) < 1/1000

with C₂ (from (2.39)) and C₃ (from (2.40)) unnamed universal constants. So the width of the a-window is not extractable from the paper. Recorded as None with the provenance string, never bounded, lesson 73. Related, and worth stating because it is easy to misread: Theorem 1.1's "0 ≤ ν ≤ 1" is not a large-viscosity claim. (2.41)–(2.42) reach it by choosing the initial length scale C_l(0) large so that ν(0) ≤ C_l(0)⁻² ≤ ν₀; the rescaled viscosity is driven small.

1.5 The γ tension leg 64's review surfaced: resolved, and it was never a contradiction

The two have disjoint ranges of a and never meet. Neither yields a γ = 2 profile, and Theorem 1.1 does not need one. There is a third piece that removes the last of the tension: the blow-up data of Theorem 1.1 is class 3 (ω₀ odd, ω₀ ≤ 0 for x > 0; Remark 1.4, p.3), which is precisely the class in which L¹ is not conserved, Lemma 3.1(b), eq (3.5), gives only ‖ω(t)‖_{L¹} ≲ exp((1+a)∫uₓ(s,0)ds)‖ω₀‖_{L¹}. So the L¹-based criticality of §1.2 does not classify the blow-up data at all.

2. The obstruction, derived

From eq (2.7), the effective viscosity in dynamic-rescaling variables satisfies ν̇ = (2c_l + c_ω) ν (Chen writes this out explicitly on p.12). A steady self-similar state therefore requires (2c_l + c_ω) ν = 0: either ν = 0, or

2 c_l + c_ω = 0 .

The steady equation has an exact time-normalisation symmetry (Ω, c_l, c_ω) → (κΩ, κc_l, κc_ω) for every κ > 0 (verified directly: both sides scale by κ²), under which 2c_l + c_ω is covariant, not invariant. The banked gauge discipline in solver/gclm_family.py ("c_ω, c_l are a NORMALIZATION gauge, not results") applies. The invariant form is

**Δ  :=  2 c_l / |c_ω|  −  1**

Δ = 0  ⟺  c_l/|c_ω| = 1/2, the heat scaling ⟺ a γ=2 profile is admissible
Δ < 0  ⟺  dissipation vanishes in self-similar variables
Δ > 0  ⟺  dissipation would dominate

At Chen's eq (2.2): Δ = 2(1/3)/1 − 1 = −1/3 exactly, matching his own (2.40) bound "above by −1/3" derived by an entirely different route. The mechanism in words: the structure collapses like (T−t)^{1/3} while the diffusive length shrinks like (T−t)^{1/2}; the structure is always the larger of the two, so diffusion never resolves it.

Lesson-90 controls, written before the number was quoted. diffusion_consistency returns 0.0 exactly on the heat pair (1/2, −1) and +1.0 exactly at the a=0 CLM anchor (1, −1). Three distinct values across three known profiles; the functional varies and can report the other answer. Both are permanent tests.

3. Construction

solver/dissipative_profile.py. Reused, not rebuilt (standing ban; capabilities.py grepped first): gclm_family.GCLMResidual (sinh grid, whole-line Hilbert matrix via line_hilbert, the a-family residual convention), gclm_rescaled.sinh_grid, nk_bounds.budget. Added, because nothing in the repository had it:

1. A 4th-order cumulative velocity operator. The banked trapezoid operator floors the residual of the exact profile far above the level any Y₀ would need: lesson 86 ("a bound dominated by its own evaluation error is a statement about the code"). At n = 601 the Adams–Moulton panel rule drops the velocity error from 1.4375e−04 to 7.467e−07 (192.5×) and the residual sup from 2.4124e−04 to 3.7426e−06 (64.5×). Tested against the banked operator (test_high_order_velocity_beats_the_banked_trapezoid_one).
2. The dissipative steady residual R = (c_ω + HΩ)Ω − c_l XΩ_X − a U Ω_X + ν Ω_XX and its exact Jacobian (R is exactly quadratic in Ω, so D²F is a constant bilinear map and there is no linearisation error).
3. diffusion_consistency / nu_decay_rate.
4. The Y₀ measurement and the exact Z₂.

Gauge fixing. Two exact symmetries must be quotiented or Newton reports the conditioning of a gauge: amplitude (Ω,c_l,c_ω) → (κΩ,κc_l,κc_ω), and spatial dilation Ω(X) → Ω(X/μ) (same c's, ν → μ²ν). c_ω = −1 kills the first; Ω_X(0) fixed kills the second. c_l is left free and HΩ(0) is never imposed, so both are genuine outputs. Solved in least-squares form because the residual of an odd profile is odd and the square system is rank-deficient by construction.

3.1 M2 (known-answer gate: Chen's closed form nulls the residual, and the floor converges

3.2 M3) Newton reconstruction: Chen's constants come back as outputs

Started from a 5% perturbation of the exact profile with c_l₀ = 0.30. Newton converges quadratically (residual history 2.5e−02 → 9.7e−04 → 4.6e−06 → 3.2e−11 → 1.5e−15).

Second-order convergence in all three columns. This is an independent numerical verification of eq (2.2) and of the reading of the steady equation, including the fact that Chen's printed advection coefficient is a = 1/2 multiplying U (the layout stacks it as a fraction).

3.3 M4: Δ(a), the central measurement

Every point converged to machine precision (residual RMS 5.91e−16 … 4.64e−15), n = 401.

At Chen's a = ½ the measured Δ is −0.333317 against the exact −1/3 = −0.333333 (agreement 1.6e−05), and against the 0 a γ=2 profile would need. The gap to admissibility at Chen's own advection value is 1/3.

Δ crosses zero, between a = 0.35 and a = 0.40. Located two independent ways:

• linear interpolation of the sweep: a* = 0.386344
• direct solve with c_l = 1/2 imposed and a free (M7, ν = 0): a* = 0.386496
• agreement 1.5e−04

3.4 M5, the dissipative branch on Chen's own a, searched not assumed

Newton on the full dissipative steady equation at a = ½, Δ measured:

Monotone increasing: on Chen's branch, adding viscosity moves the profile away from γ=2 admissibility, by up to 3.5× over the tested range. No zero crossing.

3.5 M7 (imposing Δ = 0, and why the result is a lead and not a claim

M7 does the converse of M5: it imposes (c_l, c_ω) = (1/2, −1)) exactly the condition a γ=2 profile needs, and solves for (Ω, a) at each ν.

Two controls, and both matter.

• The ν = 1 row collapsed to Ω ≡ 0. The trivial null solves the equation exactly, so its absolute residual is zero and it would have been reported as a perfect solve. Only the scale-invariant residual ‖R‖ / ‖(c_ω + HΩ)Ω‖ exposes it (5.486, and solution scale 2.20e−16). This is lesson 90 in its sharpest form and it is why M7 reports a relative residual at all.
• Dilation covariance fails as a check. Ω(X) → Ω(X/μ) maps a solution at ν to one at μ²ν with a unchanged, so a genuine one-parameter branch must have a independent of ν. Measured a ranges over [0.283354, 0.386496], it is not. The direct diagnostic: dilating the ν = 1e−02 solution by μ = √10 and evaluating at ν = 1e−01 gives relative residual 5.59e−02, against an undilated control of 1.707 (a 30.5× improvement, so the covariance is partly there and the remainder is interpolation error), but not the ~0 a resolved branch would give.

Diagnosis. In the odd subspace the system has (n−1)/2 independent residual rows for (n−1)/2 + 1 unknowns: short by one. Newton (minimum-norm least squares) therefore lands on an arbitrary point of a one-parameter set that depends on the starting guess. This leg has therefore NOT established that a γ = 2 profile exists at a*. It is recorded as an open lead with the failed check attached (lesson 76), and explicitly not as a result.

4. Y₀ and the budget

4.1 Which Y₀ is the honest one (lesson 86, applied in advance)

After Newton the discrete defect sits at the Newton floor, O(1e−15). That is a statement about the code, not about the mathematics, and it is never the headline. The honest Y₀ is the consistency defect, the discrete solution's defect in the continuous equation, proxied at each resolution by the exact profile's residual (§3.1). Both are in the JSON, both are on fig61(d), and which is which is labelled at both.

4.2 Leg 53's μ = 2 positive control, re-read against this candidate

Leg 53's dissipative positive control reached Z₁ = 0.9156181325483919 (writeup/data/leg_54_verify_headline.json, Z1_total; independently re-derived by VER-A at leg 54). It is used here as a fixed input (the most favourable Z₁ this repository has ever measured) held constant while Y₀ is measured. It is not assumed to transfer: it was measured on the a=0 CLM linearisation with Λ¹ dissipation, a different operator from Chen's a=½ Λ² one, and it is quoted here only to make the budget as generous as the banked record allows. Any honest Z₁ for this operator would be larger and the budget smaller.

4.3 The measurement, with Z₀ = 0 (maximally generous)

Z₂ is exact for this operator (F is exactly quadratic, so D²F is a constant bilinear map) and is assembled as ‖A‖·‖B‖ in the sup norm.

Y₀ is over budget at every tested resolution, and the gap WIDENS under refinement, 6.024× over the ladder, because Z₂ grows (measured ≈ n^6.59) far faster than Y₀ falls (measured ≈ n^-3.99, i.e. the expected 4th order).

What that Z₂ is and is not. ‖A‖ is the sup norm of the pseudo-inverse of a Jacobian that is rank-deficient by the two gauge directions, so Z₂ is substantially an artefact of a gauge-unbordered discretisation rather than a property of the operator. A real certificate would border those directions, which is exactly what legs 51–53's bordered certificate does, and exactly where leg 53 found the margin runs out (Z₁'s block coupling, entry K/2, 43.15). So the honest reading is: the candidate does not clear the budget, and the reason it does not is not the reason the ledger expected. The magnitude is reported; the mechanism is named as not fully attributed.

5. The gate, answered in its pre-committed wording

"With Chen's theorem located and constants transcribed from the FULL TEXT, and the profile constructed at two or more resolutions, does Y₀ come in under the radii-polynomial budget at any tested resolution?"

NO. At every tested resolution (n = 601, 801, 1201) Y₀ exceeds the budget, by 4.866e+10 at the finest, and the gap widens under refinement.

The no-branch's second clause fires independently and is the more important of the two:

"If the full text does not support the abstract as read (no explicit profile at γ = 2, or the a-neighbourhood excludes every usable case), that lands here too: quote the located text verbatim and the candidate leaves the ledger's top slot on literature grounds, which is itself the finding."

It does not support it. There is no explicit profile at γ = 2; the located text is quoted verbatim in §1.3 ("Since ν(t) converges to 0, such profile is the same as the inviscid profile associated with a.", §2.6, p.12). And the a-neighbourhood is unquantified (§1.4).

The candidate is set aside as "identified, measured, not under budget", and it leaves the ledger's top slot on literature grounds. No retry without new information.

Why even a passing Y₀ would not have helped. The certificate would have certified the inviscid a = ½ profile, which Chen gives in closed form (eq (2.2)) and verifies by hand on p.4, and which Papers/MANIFEST.md's exclusion list already records as covered analytically for the entire smooth gCLM branch a ≤ 1 (HQWW arXiv:2308.01528, arXiv:2305.05895). This is the same degeneracy as the banned leg-51 clause (Y₀ exactly zero because the a=0 CLM profile is one basis mode) arriving for a new reason, and it is flagged here so it cannot be re-read as progress later.

Escalation note (report only). The gate's yes-branch would have escalated a full certificate-attempt leg to the user. The gate answered no, so there is nothing to escalate on that ground, and no certificate was built under this leg's authority.

Tripwire, checked and not tripped. ν is floated in M5 and M7 against the profile equation's own residual, Chen's object, never against a certificate's margin, a Z-constant or r_min. The one banked certificate constant used (leg 53's Z₁) is held fixed while Y₀ varies, the opposite direction of dependence from the banned one.

6. Honest ceiling, pre-committed and unchanged

Not movement on L1→L4; not Clay. Clay odds stay ~0.05%. In 125 legs no link of the chain has moved and this leg does not move one. What is claimed is the sub-goal: no CAP of any dissipative self-similar profile exists in the searched literature, so the measured Y₀, the measured Δ(a) curve and its crossing at a* = 0.386496 are novel data points, and the candidate that topped the ledger for sixty legs is now measured rather than assumed.

7. What a follow-up would have to do (not proposed as a leg, just recorded)

The a* ≈ 0.3865 corner is unclaimed by Chen and by everything the novelty pass returned. To turn the lead into a result someone would have to (i) supply the missing equation (a second normalisation that pins the one-parameter set M7 slides along) and (ii) re-run the dilation covariance check as a pass/fail gate, with a required constant in ν to the solve tolerance. Until both hold, there is no branch, only solutions of an under-determined system.