Route-NB v1, TECHNICAL: the compactified-basis coefficient decay of HL_S2_nonsymmetric

Leg 55. Exploration route. Runner experiments/p2_route_nb_v1_targetnorm.py; module solver/target_norm.py; gates test_target_norm.py (23/23); curated data writeup/data/p2_route_nb_v1_targetnorm.json; figure writeup/figures/fig50_route_nb_v1_targetnorm.png; novelty log writeup/novelty/leg_55.md. Every number in this document is a field of the JSON, with one declared exception: the n = 3201 convergence confirmation in §4.1 (and the 6.3e-14 residual it reports) is a separate run, not part of the curated payload, and is labelled as such at both mentions.

0. The gate, in its pre-committed wording

Gate. Do HL_S2_nonsymmetric's compactified-basis coefficients decay fast enough that ‖·‖_{ℓ¹_w} is finite for at least one admissible s < 0.394, with the exponent stable across the resolution ladder?

Answered: yes. p = 1.3937 at the headline domain (1.3963 at the largest domain), finite at s = 0, s = 0.3 and, marginally, s = 0.39; resolution drift 3.8e-04.

§7 states what the yes-branch does and does not license, because the gate's yes-branch and the ban clause's literal text are not about the same s.

1. The object and the basis

HL_S2_nonsymmetric is solver/bordered_hl.py's bordered steady system for the 1D Hou–Luo model (Chen–Huang–Li arXiv:2604.01868 §2.5/§4), solved by damped Newton on the origin-clustered grid X = c sinh ρ, c = 0.5. Every solve in the curated JSON converges to ≤ 2.3e-14 (the separate n = 3201 confirmation of §4.1, which is not a JSON field, reaches 6.3e-14). The unknowns are Ω, V = Θ_X and three gauge constants (c_l, c_ω, c_r).

solver/spectral_certificate.py compactifies with the tangent half-angle map X = tan(θ/2), in which the three operators of the a = 0 CLM problem are exact. Its own profiles are odd, so it uses an odd sine series. The target is non-symmetric, so Route-NB uses the full circle:

h(θ) := Ω(tan(θ/2)),   θ ∈ (−π, π),   h(θ) = Σ_{k∈ℤ} c_k e^{ikθ},  c_{−k} = conj(c_k)

with coefficient magnitude in the complex-exponential convention

|ĥ_k| := |c_k| + |c_{−k}| = 2|c_k|,   k ≥ 1.

Gate 2 of test_target_norm.py checks that theta_of_X is the same half-angle convention as spectral_certificate.moebius_power, to 1.3e-15, so the projection lands in the certificate's basis and not a neighbouring one.

Convention independence. The real-basis alternative |a_k| + |b_k| satisfies 2|c_k| ≤ |a_k| + |b_k| ≤ 2√2 |c_k|; gate 6 verifies the bracket. A change of convention is worth at most √2 in the norm and nothing in the exponent.

1.1 Why the exponent is a statement about one point of the circle

X → ±∞ is θ → ±π, a single point, near which X ≈ 2/(π−θ). A far field Ω ~ A_± |X|^{−α} becomes

h(θ) ~ A_± ((π−θ)/2)^α,

a branch point of order α. Elsewhere the profile is smooth and contributes super-algebraically. Hence the prediction p = 1 + α, already asserted (not measured) by spectral_certificate.coefficient_decay_exponent. Non-symmetry only makes A_+ ≠ A_−; it does not change p.

1.2 The norm

‖h‖_w = Σ_{k≥1} w_k |ĥ_k|,   w_k = (1+k)^s (algebraic) or 1 (flat)

using spectral_certificate.weight_vector. Gate 7 confirms the flat class is the algebraic class at s = 0, entry for entry. For |ĥ_k| ~ C k^{−p}:

‖h‖_w < ∞  ⟺  p − s > 1.

2. The instrument, and its two failures

2.1 Pipeline

compactify interpolates the grid-borne profile in ρ (the variable the data are uniform in; interpolating in X would put a 745-to-0.04 spacing ratio inside one stencil) onto a staggered uniform θ grid of size M, then FFTs. The stagger θ_j = −π + 2π(j+½)/M matters: θ = ±π is simultaneously the one point where X is infinite and the point the exponent comes from, so a grid landing on it cannot be evaluated. The half-cell shift is a pure phase and does not touch any |c_k|.

Interpolation is local barycentric Lagrange of order L on the uniform ρ grid (no scipy in this project by design). Gates 4–5 check exactness on polynomials and that the order knob is really wired: order 2 errs 9.3e-05 where order 12 errs 3.3e-16, a ratio of 2.8e+11. An ablation on a dead knob proves nothing.

2.2 fit_exponent was wrong twice, and both times it returned a plausible number

Both failures were caught by a control, not by a test. Both are now regression gates (14, 15).

Failure 1: log-averaging. The first version binned log|ĥ_k| logarithmically and fitted the bin means of the logs. Negative control 1, Ω = 1/(1+|X|), obeys the identity h(θ) + h(π−θ) = 1, which annihilates every even mode; the FFT returns ~1e-18 there. Taking logs of those gave p = −0.06 for a spectrum whose k²|ĥ_k| is flat to three digits to 2.3% over k = 9 … 129 (0.6222 → 0.6355 → 0.6365) and to three digits from k = 33 on. A > 0 filter does not catch 1e-18.

Failure 2: sparse bins. Switching to a linear mean within log bins fixed the arithmetic but not the failure: one equal-width log bin at the bottom of the band caught a single annihilated mode (k = 10, value 9.8e-19), whose log is −41 against a trend of −12, and it captured the whole least squares. Result p = −0.25.

Failure 3, avoided. Equal-count bins cure the population problem but have k-widths that grow along the band, so the convexity offset between a bin's linear mean and its value at the bin's geometric-mean k drifts: biasing the fitted exponent by up to +0.06 on the calibration family.

The shipped fitter uses log-spaced bins merged until each holds ≥ 8 modes, with a linear mean inside each bin. Log spacing makes every bin span the same ratio of k, so the convexity offset is identical in every bin and moves the prefactor rather than the slope; the merge makes an annihilated sub-sequence a factor-of-two effect on the prefactor and none at all on the slope. The linear mean is also the physically right object: the norm under study sums coefficients, so what decides finiteness is the mass in a bin, not its geometric mean.

2.3 Calibration: the instrument's error bar, at the transform size actually used

calibration_family(X, α) = (1+X²)^{−α/2} has far field exactly |X|^{−α} and equals |cos(θ/2)|^α exactly, so p = 1 + α for every α. Sweeping α and asking the fitter to recover an exponent it was never told, at M = M_PRIMARY = 16384, the transform size every target measurement in this leg uses:

Systematic at the target's own α: +0.0039. Worst over the sweep: 0.0075. The bias is positive, the fitter over-estimates p, so a bias-corrected exponent is lower: 1.3937 − 0.0039 = 1.3898. All exponents below are quoted against this.

2.3.1 And the systematic is a property of (M, band), not of the fitter alone

This was wrong in the first version of this document and was caught by VER-B's review. The calibration was originally run at M = 65536 while every target measurement runs at M = 16384, and the finer grid's systematic (+0.0022) was quoted against the coarser grid's answer. It is 1.8× too small:

The mistake was not arbitrary and that is what makes it worth recording: at M = 65536 the finest θ cell reaches |X| = 2M/π = 4.17e+04, past the headline domain's X_max, so that grid could not have been used on the target without extrapolating. The calibration family is analytic and has no such limit, which is exactly why calibrating it on a finer grid than the target was measured on flatters the instrument. The control battery (§3) is now run at M_PRIMARY throughout for the same reason.

3. The controls

Positive-control window, pre-registered (lesson 84): | |ĥ₁| − 1 | < 1e-10 and max_{k≥2} |ĥ_k| < 1e-08. Passed at n = 201, 401, 801 (4.4e-11 → 1.8e-13 → 3.4e-15). It can fail: a wrong half-angle convention, a sign error in theta_of_X, or an off-by-one in the FFT phase each smear a single mode across all k.

Negative control 2 also has exact coefficient VALUES, |ĥ_k| = 2/(πk), matched to 1.0e-04 relative over k ≤ 512, and cross-checked against the repository's own spectral_certificate.sawtooth_coefficients (gate 10). It is simultaneously the instrument's hardest calibration: a jump is the worst non-smoothness on the circle, so it brackets the target's α-cusp from the bad side, while the positive control brackets it from the good side.

4. The two ladders

4.1 Resolution, flat

At the shipped domain ρ_max = 8 (X_max = 745.2), M = 16384, band k ∈ [32, 256]:

Drift 3.77e-04, an order of magnitude under the fitter's own systematic (+0.0039, §2.3). Separately confirmed at ρ_max = 12: p = 1.3937 → 1.3938 across n = 401 → 3201 (a separate run, not a JSON field). The measurement is not resolution-limited.

4.2 Domain, the ladder that moves, and then stops

n = 801, ρ_max = 8 → 14:

Three independent measurements (a Fourier fit on the circle, a log-log fit in physical space (bordered_hl.tail_exponent), and a ratio of two Newton unknowns) converge together. The k^{−1−α} law is confirmed to 1.5e-03, and the shortfall at the shipped domain is the profile not yet having reached its asymptotic tail at X = 745.

This is a diagnostic, not a repair. The live ban against closing the truncation gap by extending the domain concerns the certificate's truncation gap, which leg 47 measured to get worse with reach (+0.47 decades per unit ρ). Nothing here repairs that gap; the two quantities are different and both statements hold: the exponent improves with reach while the certificate's gap worsens.

Where the ladder stops, and why. Gate 1 measures the float round-trip X → θ → X: 4.0e-12 relative at X = 4.1e+04, 2.9e-11 at X = 3.0e+05. At X = 3.0e+05 the image θ sits 6.7e-06 from π and float64 tan/arctan lose about five digits. That, not the physics, is what bounds the ladder.

5. Ablations

Base for §5.1–§5.3: n = 801, ρ_max = 12, X_max = 4.07e+04, α = 0.39735.

5.1 The far-field closure: it DECIDES at the shipped domain

Spread where it fires: 0.190, on a quantity whose meaning is its third decimal. The finest θ cell of an M-point staggered grid sits π/M from the branch point, i.e. it reaches |X| = 2M/π = 1.043e+04 at M = 16384. A domain shorter than that must be extrapolated (X_max = 745: 14 sample points outside); a longer one never is.

At ρ_max = 12 all three rows are byte-identical, and that is not evidence the closure does not matter: it is evidence the closure is never consulted. Per lesson 90, identical numbers are reported with the reason they are identical (n_theta_points_outside_grid = 0) rather than quoted as a null result. The headline is measured where no closure is consulted. Gates 17–19 pin all of this, including that far_field="none" leaves NaN and the transform refuses it rather than inventing data.

5.2 Interpolation order (a real null, shown to be real

p is identical to five digits) again the lesson-90 tell. So the movement of the interpolant itself is reported beside it: order 2 shifts h by 8.7e-05 relative and shifts p by 3.5e-05. The knob is wired (gate 5) and the profile is resolved. That is a real null.

Spreads are taken within a domain: the two ρ_max values sit at genuinely different exponents, so a spread across both would report the domain ladder and mislabel it interpolation sensitivity.

5.3 Transform size and fit band

Spreads 0.0060 and 0.0068, comparable to the calibration systematic and an order of magnitude below the quantity being decided.

6. The norms

At n = 801, ρ_max = 12, p = 1.39374, C = 0.48560, α = 0.39735. Partial sums S_N = Σ_{k≤N} (1+k)^s |ĥ_k|; the tail is the integral bound 2^s C N^{−(p−s−1)} / (p−s−1).

At s = 1 the entry is not a large number. analytic_tail returns finite: false, bound: null, reason: "p − s ≤ 1: the weighted sum diverges; the tail has no value": when a quantity has no referent, say so instead of bounding it (lesson 73; gate 21). Where it is finite the bound is verified to dominate the true summed remainder (gate 22).

s = 0.39 is NOT RESOLVED, and the first version of this document got that wrong. Its margin is +0.00374 against a systematic of +0.00389 at the headline's own M: a ratio of 0.96×, i.e. the margin does not clear the instrument's own error bar. The bias is positive, so the bias-corrected margin is +0.00374 − 0.00389 = −0.00015: negative.

The first version quoted this row as finite on the strength of a 1.7× ratio against the M = 65536 systematic +0.0022 (§2.3.1): erring unsafe, in the one direction that matters. It is now reported as not resolved: the measurement cannot distinguish this class from the divergent side, and admissible_classes_with_finite_norm in the JSON contains only [0.0, 0.3], with 0.39 listed separately under admissible_classes_not_resolved.

Two things do point the right way and neither is sufficient: the domain ladder had not converged at this domain ((p−1) − α is still −0.0036), and against the largest-domain α = 0.39782 the margin would be +0.0078. The gate does not need this row: s = 0 and s = 0.3 carry it with margins 100× and 24× the systematic.

6.1 The second unknown

V decays roughly twice as fast, as the steady equation forces. Ω is binding and the gate is decided on Ω alone.

6.2 The window, both sides now measured

spectral_certificate.weight_window(α = 0.39735, s_operator = 1.0):

s_max_object = 0.39735     s_operator = 1.000     gap = +0.60265     empty = True

7. What the gate answer licenses, and what it does not

The gate answers yes on its own wording. The target's ℓ¹_w norm is finite for admissible s < 0.394, s = 0 (margin +0.394) and s = 0.3 (margin +0.094), with a resolution drift of 3.8e-04.

The gate's yes-branch says the ban clause is "factually wrong". That overstates it, and the overstatement should not be propagated. The clause reads:

does not have finite norm in the class where the operator is least bad**

The class where the operator is least bad is s = 1 (leg 51's divergence-curve minimum). At s = 1 the norm does diverge, margin −0.606. So the clause, read literally, is correct.

Leg 51 said so itself, and it is the strongest support for this reading. writeup/4_p2_lottery/TECHNICAL_P2_ROUTEL1_V2.md §9 already states, in the same breath as its window measurement:

"The class where the operator is least bad is the class where the target has infinite norm, and vice versa."

That is precisely the clause's content, written by the leg that raised it. The clause was never a claim that the target is outside every class: it is a claim about the coincidence of the two classes, and this leg's measurement confirms it rather than refuting it.

What this leg establishes is therefore narrower and more useful than "the ban is wrong":

1. The object side of the window is now a measurement (s_max = 0.397 ± 0.008) where it was previously a docstring derivation applied to an assumed α.
2. The k^{−1−α} correspondence is confirmed, to 1.5e-03, on the real target.
3. The window is empty by +0.603 in exponent units, with both sides measured, which confirms leg 51's published 0.606 (TECHNICAL_P2_ROUTEL1_V2.md §9: "The window is empty by 0.606 in exponent units") rather than adding a new quantity. The delta is that leg 51's object side was derived from an assumed α and this one is measured.
4. The target is comfortably inside the space at s = 0 and s = 0.3: the classes legs 52 and 53 actually used. So "the target was never in the space" is not an available explanation for those legs' failures. Leg 53's block coupling (Z₁ = 43.15 at its best split) stands as the operative reason, untouched by this leg.

Escalation. Whether the clause's text needs narrowing is the Decision Maker's call, not this leg's. It is flagged in the PR body. No ban was edited here.

Not claimed: that any certificate closes; anything about MM's gate; that k^{−1−α} or the value of α is novel (see writeup/novelty/leg_55.md Q1/Q4, the correspondence is classical, and arXiv:2308.01528 is the analytic prior art on far-field decay rates for Hou–Luo profiles). No link of the L1→L4 chain moved. Clay ~0.05%.

8. Reproduction

.venv/bin/python experiments/p2_route_nb_v1_targetnorm.py                       # ~4 min
.venv/bin/python experiments/p2_route_nb_v1_targetnorm_evidence.py             # fig50
.venv/bin/python test_target_norm.py                                            # 23 gates

solver/spectral_certificate.py and solver/bordered_hl.py are imported read-only; neither has a diff in this leg.