Route-TN v1: the (H, D) consistency defect of L1 step one, enclosed

Leg 56. Exploration route (not critical path). Gate answered NO. Runner experiments/p2_route_tn_v1_consistency.py · data writeup/data/p2_route_tn_v1_consistency.json · figure writeup/figures/fig51_route_tn_v1_consistency.png (rebuilt by experiments/p2_route_tn_v1_consistency_evidence.py) · novelty log writeup/novelty/leg_56.md · journal experiments/journal/leg_56.md.

Every number quoted below is in the curated JSON.

1. Scope, stated before anything else

solver/interval_certificate.py proves a statement about a finite-dimensional polynomial system built from the stored float matrices H, D. Its docstring names the two gaps between that and the continuum profile: the consistency of (H, D), and the far field beyond X_max. This leg bounds the first, at fixed reach.

This is not "closing the truncation gap by extending the domain" (banned; leg 47 measured that trend at +0.47 decades per unit ρ, the wrong sign). X_max = 745.2394128947751 is identical in every run at every rung. n is the only thing that moves. The far-field term is computed only so it can be subtracted from the measurement and reported in a separate column (discipline 75: two defects in the same problem are not the same defect).

Nothing below is a claim about HL_S2_nonsymmetric being certified, about the far-field gap, about the coefficient-basis work of legs 51–53, or about any link of the L1→L4 chain.

2. Why the naive quantity has no referent (discipline 73)

There is no ‖H_disc − H‖. H_disc : R^n → R^n; H maps functions to functions. The difference is defined only relative to a named class, and its magnitude is a property of that class as much as of the operator. Reported accordingly as a curve (§7), never as a scalar impersonating an operator norm.

3. The structural fact the leg turns on: corrected after VER-C's review

The first version of this section was wrong, and the correction is kept visible rather than quietly patched. It claimed that H_disc and D_disc are the exact Hilbert transform and the exact derivative of the same natural-spline interpolant, so that both consistency defects were "one interpolation error seen through two operators". They are not the same interpolant. §8's own ablation already contained the evidence against that claim; this section previously failed to reconcile the two.

line_hilbert_matrix is not a quadrature rule. Per solver/line_hilbert.py (Huang–Tong– Wang arXiv:2603.25104 App. C.1), it expands the data in the C¹₀ Hermite basis {P_i, Q_i} with node slopes from the natural cubic spline, and applies the closed-form exact Hilbert transform of each basis element. That much stands.

But it assembles source columns for INTERIOR nodes only (Hp_full[:, 1:-1] = HP, and likewise Hq_full) so the two endpoint basis functions are dropped. slope_matrix, by contrast, really is the full natural-spline slope operator at every node. Writing Π_n for the natural-spline interpolant and Π⁰_n for the C¹ piecewise cubic that matches f and the natural-spline slopes at the interior nodes and is zero with zero slope at ±M:

H_disc f = H(Π⁰_n f)|_[-M,M]               D_disc f = (Π_n f)'|_nodes
                ^^ a DIFFERENT interpolant

Measured, at n = 201, node 1 (X = −687.943), a = 1/2 (validation.H_is_endpoint_zeroed_check):

So the gated defect splits into two terms of completely different character:

H_disc f − H_M f  =  [H(Π⁰_n f) − H(Π_n f)]   +   [H(Π_n f) − H_M f]
                      ENDPOINT ZEROING            TRUE INTERPOLATION
                      O(f(±M)); does NOT          converges, order ≈ 1.95
                      converge at fixed reach

H is unbounded on L^∞, so the interpolation part cannot be bounded from ‖e‖_sup; it must be evaluated against an exact reference, which is what the closed forms in §4 are for.

What this changes and what it does not. The gated quantity is unchanged: it is the defect of the operator as implemented, i.e. the left-hand side. What changes is the attribution, and it matters for anyone reading this code next: the Hilbert side is not a slightly worse version of the derivative side, it is a structurally different, and much worse-conditioned, discretisation. §7.1 gives the split.

4. The test class, and its closed forms

u = X − b, two families of identical interior smoothness:

(a = 1/2, b = 0 in the odd family is exactly the CLM pair test_line_hilbert.py already gates against.) Partial fractions of the truncated integral, with A = C, B per family and u₁ = −M−b, u₂ = M−b:

π H_M f = (A/2) ln((u₂²+a²)/(u₁²+a²)) + (B/a)(arctan(u₂/a) − arctan(u₁/a)) − C ln|(x−M)/(x+M)|

which → π B/a = π H f as M → ∞, as it must. The truncation is formed directly from the small residual terms rather than as a difference of two nearly equal numbers.

Endpoint nodes are excluded (2 of n; 799 interior at n = 801). At x = ±M the truncated transform is log-divergent, H_disc's boundary basis is one-sided and finite, and the two divergences cancel analytically but not in floating point.

5. Rigorous evaluation

New in solver/interval.py, because np.log / np.arctan carry no ULP guarantee this module may assume:

• ilog, exact reduction x = m·2^e (np.frexp), then log m = 2 atanh((m−1)/(m+1)) with |z| ≤ 1/3 and a proved geometric tail bound, plus a log 2 enclosure.
• iatan_small: alternating Taylor series for |t| ≤ 1/2; remainder bounded by the first omitted term. Domain guards raise rather than silently extrapolate, and test_interval.py gates that they fire.
• Both are checked against an independent 50-digit decimal reference. Max widths: ilog 3.55e−14, iatan_small 1.48e−14.

Discrete operators are applied by the compensated dot2_matvec on the midpoint, with the input enclosure's radius carried through by |Mat| @ rad. Both pieces round outward.

6. The comparison quantity (discipline 67)

A consistency defect enters as an addition to the residual: Y₀ → ‖A(F + δF)‖_w ≤ Y₀ + ‖A‖_w ‖δF‖_w. So the admissible defect is

τ = budget / ‖A‖_w

Re-derived here, not quoted (lesson 85). At n = 801: budget = 3.554656e−10, ‖A‖_w = 15417.7, both matching leg 46/50's stored values to all printed digits. Hence τ = 2.306e−14.

Y₀ itself re-derives as 9.97e−12 against 7.35e−12 stored (Newton landing at a slightly different converged iterate; both far under budget). Y₀ does not enter this leg's comparison, and both values are on panel E rather than being asserted away.

τ overstates what is admissible (clause TN-5): it discards every amplification the true δF carries, the profile's own norm, the S factor, the velocity operator. A defect that fails against τ fails against the real requirement a fortiori. Only that direction is claimed.

7. Results

X_max = 745.239 fixed; weighted sup norm with ν = (1+X²)^{p/2}, p = P_STAR = 0.39.

The ladder (odd, a = 1/2):

Against τ at n = 801: D is 1.854e+07 τ; H is 2.040e+11 τ; the far-field truncation is 1.139e+12 τ.

The two defects fail differently, and merging them would destroy the result (TN-7):

• D converges at order 4.01, the natural-spline order, measured. It is simply far too large. Extrapolating at that order, reaching τ needs n ≈ 52,163 (N = 104,329, dense).
• H does not converge: 1.0052× total over a 4× refinement. Refinement is not a lever.

7.1 Attributing the Hilbert defect (H_attribution)

The endpoint-zeroing term is the entire defect: its share at n = 801 is 1.00009. That is why the total does not converge: the artifact does not, and it dominates by four orders.

The genuine interpolation error converges, and it is still hopeless. At order 1.95, taking 4.4181e−07 down to τ = 2.3056e−14 needs n ≈ 4.43e+06: worse than the derivative side's n ≈ 5.22e+04, because order 1.95 is so much weaker than order 4. Correct attribution strengthens the NO rather than weakening it.

The scale curve (n = 801, odd), the answer to "the defect of what?": a = 0.125 → D 1.804e−03; 0.25 → 2.752e−05; 0.5 → 4.274e−07; ≥ 1 → floors at ≈ 7.13e−08. H is flat at 4.704e−03 across the whole range, because for large |X| the class member is ≈ −1/X regardless of a, and H's defect lives there. D's floor is the same far-field mesh effect (h ≈ 14.8 at |X| ≈ 745, n = 801) taking over from the origin.

8. Mechanism, ablated (lessons 85 and 90)

line_hilbert_matrix assembles source columns for interior nodes only (x[1:-1]); the endpoint P-columns are identically zero. The transformed object therefore has f(±X_max) = 0 imposed on it.

The ablation dials only the value at the cut, by M/a ≈ 1490, holding interior smoothness fixed. Nothing in the code path distinguishes the families.

The collapse matches M/a = 1490.5 to within 1%, and the same dial moves D by 11%. The control could have come out the other way (both moving, or neither): it did not.

The even family's H defect is also flat in n (3.33e−06 → 3.13e−06), so the non-convergence is a property of the cut and not of the particular family.

Reconciling this with §3, which the first version of this document failed to do. These numbers were already incompatible with "one interpolation error through two operators": a dial that leaves interior smoothness untouched cannot move a pure interpolation error by 1503× while moving the derivative's by 1.11×. Under the corrected reading it is exactly what must happen: the dial changes f(±M), which is precisely what the endpoint-zeroing term is proportional to and what the interpolation term is nearly blind to. §7.1 measures the same statement directly, by separating the two terms rather than inferring the split from a contrast.

9. The bound is not its own evaluation error (discipline 86)

This was pre-registered as the leg's central risk. At n = 801, enclosure width / value:

• D: 6.34e−08
• H: 1.15e−13

Both bounds dominate their own evaluation error by seven and thirteen orders. The risk did not materialise.

The closed-form reference is independently verified: test_interval_certificate.py (8) checks it against a direct principal-value quadrature sharing no code with it, over 12 (family, a, node) cases, worst absolute disagreement 4.44e−15. The comparison is mixed absolute/relative on purpose: the even family's truncated transform vanishes identically at X = 0 by symmetry, and a pure relative test there divides by a true zero.

10. Gates added

test_interval.py: ilog / iatan_small enclose an independent 50-digit reference; the domain guards fire.

test_interval_certificate.py (8)–(12): the reference matches independent quadrature; the ablation separates; D converges at order 4 ± 0.25; H is flat to within 5% over a 4× refinement, deliberately falsifiable, so that a future change making H converge fails the gate and forces a re-read; width/value < 1e−4 for both. All 12 gates pass (414 s).

11. Gate answer

Can the (H, D) consistency defect be enclosed by a rigorous bound that is smaller than leg 46's Y₀ budget at n = 801, with a convergence rate measured across n = 201/401/801?

NO.

Reporting the magnitude and the rate, as the no-branch requires: at n = 801 the defect is 4.704e−03 (Hilbert) and 4.274e−07 (derivative) in the certificate's own weighted sup norm, against an admissible τ = 2.306e−14, exceeding it by 2.04e+11× and 1.85e+07×. The rates across n = 201/401/801 are order 0.00 for the Hilbert defect (it does not converge) and order 4.01 for the derivative defect.

Per the pre-committed no-branch: the collocation realization cannot carry L1, and the coefficient basis is the only lane left for it: which makes L1's fate identical to MM's. No grid-basis repairs are proposed: not the boundary-basis change §8 obviously invites, nor any other.

That last clause is a consequence of the gate's own wording, not a claim about MM's outcome: leg 54 is live and unanswered as this is written, and nothing here predicts it.

The negative does not hinge on the H artifact: and correct attribution strengthens it. Two independent ways to see that:

1. With the Hilbert defect deleted outright, the derivative defect alone still requires n ≈ 52,163 at its measured order 4: a dense interval system of dimension 104,329.
2. With the endpoint-zeroing artifact deleted instead, so that only the genuine interpolation error remains (§7.1), the Hilbert side requires n ≈ 4.43e+06 at its measured order 1.95: an order of magnitude worse than the derivative side.

Repairing the endpoint basis would therefore move the binding constraint from 2.04e+11 τ to 4.43e+06 in n, and would not come close to closing anything. That is an argument for the robustness of the NO; it is not a proposed repair, which the no-branch forbids.

12. Ceiling

Pre-committed and honoured. Nothing above is a statement about HL_S2_nonsymmetric being certified, about the far-field gap, about the coefficient basis, or about the method's viability anywhere else. No link of the L1→L4 chain moved; in 56 legs none has. Clay remains ~0.05% behind Walls 1 and 2.

Novelty: PROCEED_NARROW, six queries, nothing banked. Rigorous error bounds for spline-based Hilbert transforms inside computer-assisted proofs are established practice in this exact literature (Chen–Hou–Huang, arXiv:2106.05422 and arXiv:2305.05660). Links, not counts, in writeup/novelty/leg_56.md. Leg 52's search-index flag on arXiv:2604.01868 stands; it surfaced inside a broad topical query here, which is recorded as a surfacing and explicitly not as a clearance.