TECHNICAL, Route-NLH v1 (leg 331): Breden–Chu's machinery against a nonlocal operator

Arithmetic ceiling. float64 throughout, not interval arithmetic. Every number below is a magnitude, reproducing constants rather than enclosing them. Nothing here is a rigorous enclosure and nothing here may be cited as one. solver/interval_mp.py was available and was not needed: the obstruction is 100+ orders of magnitude wide, not a precision question.

Ban status. No ban is lifted. The repository's ell^1-Fourier/radii-polynomial ban covers the ell^1_w coefficient basis (leg 54), the collocation basis (leg 56) and origin-H^2 (legs 163/176). This leg works in Breden–Chu's H^2(mu) with the Hilbert norm ||u|| = ||Lu||_{L^2(mu)}, the half-Hermite basis, Sobolev-embedding taming and a Poincaré tail: the same footing leg 256 banked. It claims no fourth space and does not constitute a lift.

1. Object

Breden–Chu (arXiv:2404.04054v2) Theorem 42 concerns the generalised viscous Burgers self-similar profile, their eq. (54), on

H^2(mu),   mu = e^{|x|^2/4} / Z,   ||u||_{H^2(mu)} = ||L u||_{L^2(mu)},
L = -Delta - (x/2) . grad,   L psi_m = lam_m psi_m,   lam_m = 1/2 + m,
psi_m = L_m^{(-1/2)}(x^2/4) e^{-x^2/4} / Zeta_m = kappa_m H_{2m}(x/2) e^{-x^2/4},

in the even (Neumann) sector, d = 1. This leg changes one letter of eq. (54):

L u - u/4 + u^2 W_t u = 0,     W_t = (1 - t) d_x + t Lam^{2a},     Lam^{2a} = (-Delta)^a.

alpha = 1/2 gives Lam = H d_x, the Hilbert-transform realization (lesson 91: the realization is named, not implied). t in [0,1] is a homotopy in nonlocality; t = 0 is eq. (54) verbatim.

2. The primitive

psi_m is not a Fourier eigenbasis. The identity FT[H_n e^{-y^2}] ~ H_n is false here: Hermite functions carry e^{-y^2/2}, psi_m carries e^{-y^2}. From the generating function e^{2zt - t^2} one gets a monomial amplitude,

FT`H_n(x/2) e^{-x^2/4}` = 2 sqrt(pi) (-i)^n (2 xi)^n e^{-xi^2},

whence, exactly, with no special-function dependency beyond lgamma:

Lam^{2a} psi_m (x) = (2 / (sqrt(pi) m! Zeta_m)) Int_0^inf xi^{2m+2a} e^{-xi^2} cos(xi x) dxi.   (*)

(*) is evaluated in logs on a Gauss–Legendre panel rule in xi, with panel density set by cos(xi x_max) and xi_max = sqrt(n+1) + 9.

Known-answer gates on (*), pre-committed

The alpha = 0 gate is the one that earned its place. The first (false) amplitude gave m = 0 exact to 1.2e-15 and every m >= 1 at ~100% relative error, identical at 286 and 960 panels and at x_max 14 and 20, resolution-independent, hence structural. Lesson 84: the known-answer probe had a window, and m = 0 was inside it.

3. Transcription and its grading

Every Breden–Chu formula is used unchanged with d_x -> W_t in the operator columns. Their bounds were graded before any number was produced:

• DERIVATION-FREE (their algebra is operator-agnostic): Y, Zbar11, Zbar21.
• TRANSCRIBED (their derivations invoke locality): Zbar12, Zbar22, Z2, Z3.

A TRANSCRIBED number is a magnitude produced by their formula, not a bound their theorem licenses for this operator. The grading is carried in the JSON.

4. Controls

The t = 0 regression is what licenses t > 0: the leg's bound code is bit-identical to solver/bc_weighted_sobolev.py's on the same coefficients. It could have differed.

5. Results

5.1 The image leaves the space (M1)

Symbol |xi|^{2a} xi^{2m} e^{-xi^2} is non-smooth at the origin only through |xi|^{2a}, which is multiplied by a factor vanishing to order 2m; the forced tail is x^{-(1+2a+2m)}. Pre-registered, then measured at m = 0:

Against mu = e^{x^2/4} this is fatal. ||Op psi_0||^2_{L^2(mu), |x|<R}, alpha = 0.5:

The control saturates at 1; the nonlocal norm has no limit. Lam^{2a} does not map H^2(mu) into L^2(mu).

5.2 Finiteness is a horizon artefact (M1, rule horizon)

log10 of the truncated weighted norm² evaluated at each rule's own largest node, alpha = 0.5, m = 0 (logs because at n = 1500 it overflows float64):

n = 1500 is Breden–Chu's own published resolution for Theorem 42. Refinement makes this strictly worse: a larger n pushes the horizon further into the divergence. This is the sense in which the machinery "produces finite bounds" for a nonlocal operator: it does, and they are finite only because its quadrature stops before the divergence starts.

5.3 Quadrature exactness is lost (M3)

Their K-product Gauss–Laguerre rules (K = 6/4/2) are exact for polynomial×Gaussian only. Relative error against a refined Gauss–Legendre reference carrying the same t-mixture, n = 100, alpha = 0.5:

Y and Zbar21 compute tail terms as differences of rule-evaluated quantities, so an O(1) rule error enters those differences directly. (tail_sq < 0, the failure mode predicted in advance, did not occur on the tracked branch; recorded as a refuted prediction.)

5.4 The gate quantity: Z1 crosses 1 at finite t

Z1 < 1 is necessary: above it the radii polynomial has no positive root at all. Bisected crossing t*, bracket width 6.104e-06:

The machinery tolerates ~30–52% of one nonlocal operator. The crossing moves outward with n (0.4000 -> 0.4705 at alpha = 0.5), which taken alone would leave the endpoint open; 5.1 and 5.2 are n-independent and worsen with n, so the two ladders agree.

5.5 Which bound fails

n = 200, alpha = 0.5, along the tracked branch (t = 0 -> 0.5 -> 0.6):

The two that diverge are precisely the two graded TRANSCRIBED before the run.

5.6 What did not fail

sum_m sup|Lam^{2a}psi_m|^2 / lam_m^2 converges: 1.362 / 1.235 / 1.871 for alpha = 0.25/0.5/0.75 against 0.857 for the local control; fitted sup growth exponents 0.003 / 0.265 / 0.522. The sup/embedding route is not the obstruction. A subsequent attempt should not spend effort re-deriving Zbar22's sup-norm ingredients.

6. Lesson-90 handling

u == 0 solves the equation for every t, and its radii polynomial closes with Y ~ 1e-29, Z1 ~ 2.5e-3, closes = True. Newton falls into it once the nontrivial branch folds, which happens in all six (n, alpha) runs before t = 1. Every step carries collapsed_to_trivial_solution and verdict_is_meaningful; no closes = True in this data set is a certification. Fold located by bisection to width 2.441e-05.

7. Answer to the gate

No. Pointed at Lam^{2a} on its own weighted space, the machinery does not produce finite closing bounds of the same kind it produces for local operators. It produces apparently finite bounds whose finiteness is a quadrature-horizon artefact of a divergent integral (5.2), computed by a rule that has lost exactness by O(1) (5.3), and its contraction constant leaves the admissible region at 30–52% nonlocality (5.4), with the two failing bounds being exactly the two whose derivations used locality (5.5).

The obstruction is now measured rather than inferred, and it is not the one Remark 40 names. What breaks is the interaction between an algebraic tail and a Gaussian weight: the e^{|x|^2/4} weight, not nonlocality as an abstract property. Routing that distinction to the DM: it reframes the (iv_a) re-screen question as "which weight tolerates an algebraic tail", and a Gaussian one demonstrably does not.

8. Reproduction

.venv/bin/python experiments/p2_route_nlh_v1.py            # measure + fig85
.venv/bin/python experiments/p2_route_nlh_v1.py --figure   # fig85 from banked JSON

Data: writeup/data/p2_route_nlh_v1.json. Figure: writeup/figures/fig85_route_nlh_v1_nonlocal.png. Reads solver/bc_weighted_sobolev.py and solver/line_hilbert.py; edits neither.