The other half of the gap: what "the operators are exact data" was hiding

Route-TN v1, leg 56. Gate answered NO. Figure: fig51_route_tn_v1_consistency.png. Data: writeup/data/p2_route_tn_v1_consistency.json.

A theorem that says exactly what it does not say

Six legs ago this project got a real result: solver/interval_certificate.py closes a radii polynomial on the target profile HL_S2_nonsymmetric, in interval arithmetic, at n = 201, 401 and 801. Not a float rehearsal, a rigorous enclosure.

And it came with an unusually honest docstring. Here is the theorem it claims, in its own words:

Let H, D be the stored float matrices ... with THOSE matrices as exact data. Then ... there is a TRUE ZERO of that polynomial system within r of the stored float iterate.

WHAT IT DOES NOT SAY. Nothing about the continuum profile. The gap between the discrete system and the PDE is two further terms (the consistency of (H, D) with the operators they discretise, and the far-field tail beyond X_max) and neither is bounded here.

Two named gaps. Legs 51–53 spent themselves on the second one, in a different (Fourier) basis, and it ended badly: the block coupling of the approximate inverse came out at 43 where it needed to be under 1.

The first gap had never been measured at all. H and D are discretisations: H of the Hilbert transform, D of ∂_X, on a sinh-graded grid reaching |X| ≈ 745. The certificate treats them as exact. This leg asks what that costs.

It is not the banned move. "Closing the truncation gap by extending the domain" is banned in this repository: leg 47 measured that trend and it runs the wrong way. The domain here is frozen at X_max = 745.24 in every single run. The quantity is a discretisation defect at fixed reach, and the far-field term is computed only so it can be subtracted off and reported in its own column.

First: the quantity does not exist until you say what it means

The obvious phrasing, "bound ‖H_disc − H‖", is not a thing. H_disc is an n × n matrix taking grid vectors to grid vectors. H takes functions to functions. Subtracting them is a type error.

The difference only becomes a number once you name a class of functions, and then the number depends on the class you named. A function wiggling faster than the local mesh has an enormous defect; a smooth one has a tiny defect. Same grid, same matrix. So this leg reports the defect as a curve over a named class (panel C), never as a single number wearing an operator norm's clothes.

The class is the rational pair whose Hilbert transform, truncated Hilbert transform and derivative are all closed-form, which happens to include the exact CLM profile the repo already tests against.

Then: the two operators turn out not to be the same operator

I got this backwards on the first pass, and the correction is the most useful thing in the leg. My first draft said H_disc and D_disc were the exact Hilbert transform and the exact derivative of the same spline interpolant, so both defects were "one interpolation error through two operators". That is false, review caught it, and the ablation further down had already contradicted it: I just hadn't reconciled the two.

Reading solver/line_hilbert.py closely does change the shape of the problem, but not the way I first wrote it down. line_hilbert_matrix is indeed not a quadrature rule: it represents the data by a C¹ cubic spline and applies the exact Hilbert transform of that spline.

But it builds source columns for interior nodes only. The two endpoint basis functions are dropped. So the spline it actually transforms is not the natural-spline interpolant Π f that slope_matrix differentiates; it is Π⁰ f, the one pinned to zero value and zero slope at ±M:

H_disc f  =  H(Π⁰ f)          D_disc f  =  (Π f)'
                  ^^ a different, endpoint-zeroed interpolant

Handed a function that is ≈ 1/X out there, H_disc silently sets it to zero over the two edge cells. At n = 201, node 1:

So the defect is really two defects stacked:

H_disc f − H_M f  =  [endpoint zeroing]  +  [genuine interpolation error]
                      4.70e−03, flat        4.42e−07, order 1.95

The artifact is the entire measured defect (99.99% of it), which is exactly why the total refuses to converge. And the genuine interpolation part is still why H cannot be bounded from ‖e‖_sup the way D's can: the Hilbert transform is unbounded on L^∞, so it has to be evaluated. Hence the closed forms, and hence the rigorous log and arctan this leg had to build (series with proved remainders: np.log carries no ULP guarantee this project is entitled to assume).

The number the defect has to beat, and why it is so small

A consistency defect is an addition to the residual. The certificate can absorb

τ  =  budget / ‖A‖_w

and nothing more. At n = 801 the budget is 3.5547e−10 and ‖A‖_w is 1.5418e+04, so

τ = 2.31e−14.

That is a brutally small number, and it is the certificate's own, not one this leg chose. It is small because Z₂ ≈ 1.4e+09, and the budget goes like (1−Z₁)²/2Z₂. Note also that τ is the friendliest possible threshold: it throws away every amplification the real perturbation would carry. A defect that fails against τ fails against the truth by more.

(The budget and ‖A‖_w re-derive to all printed digits from leg 46/50's stored run. Y₀ itself does not: 9.97e−12 here against 7.35e−12 stored, a Newton iterate landing slightly differently. It does not enter this leg's comparison, but it is on the figure so nobody has to take that on trust.)

The answer

The gate answers NO, by seven and eleven orders of magnitude. And the two fail for completely different reasons (that is the actual content of the leg.

The derivative behaves perfectly and is simply far too big. It converges at order 4.01, 4.03) textbook natural-spline order, measured, not assumed. It is just starting from 4.27e−07 and needs 2.31e−14. At order 4 that is n ≈ 52,000, i.e. a dense N = 104,000 interval system. Not reachable, but honestly diagnosed: nothing is wrong with D.

The Hilbert defect does not converge at all. 4.7287e−03 → 4.7131e−03 → 4.7041e−03. A factor of 1.005 over a fourfold refinement. Refinement is not a lever here. No n closes this.

Why: ablated, not asserted

The defect sits at the last interior node, next to the cut. line_hilbert_matrix builds source columns for interior nodes only, so the function it actually transforms is the one that vanishes, with vanishing slope, at ±X_max. Handed a function that is ≈ 1/X out there, it silently sets it to zero over the two edge cells.

That is a story, and this repository has learned the hard way (lesson 90) not to trust a control that could not have come out differently. So: two test families with identical interior smoothness and identical resolution demands, differing only in their value at the cut by a factor M/a ≈ 1490. Nothing in the code path knows which one it has.

The same dial that moves H by three orders of magnitude moves D by eleven percent, and the collapse factor matches M/a to within 1%. The mechanism is measured.

And this table is what should have told me §3 was wrong. A dial that leaves interior smoothness completely 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 has to happen: the dial changes f(±M), which is precisely what the endpoint-zeroing term is proportional to and what the interpolation term barely notices. The evidence was sitting in my own results section, contradicting my own mechanism section, and I shipped both.

And the enclosures are enclosures: width/value is 6.34e−08 for D and 1.15e−13 for H. These bounds are not their own evaluation error (lesson 86): the risk that was flagged as this leg's central one, and it did not bite.

What this does and does not mean

The gate's no-branch was written before the run and it is honoured: the collocation realization cannot carry L1, and the coefficient basis is the only lane left for it. No grid-basis repairs are proposed here: not the boundary-basis fix the mechanism section obviously invites, not anything else. The H artifact is why the no is sharp, but the no does not depend on it, in two independent ways: even with the Hilbert defect deleted outright, the derivative alone still needs n ≈ 52,000; and even with the artifact deleted instead, leaving only the genuine interpolation error, the Hilbert side needs n ≈ 4.4 million, because order 1.95 is so much weaker than order 4. Correct attribution makes the answer more negative, not less.

What genuinely changed is the reading of that honest docstring. The two named gaps were listed as peers. They are not. One of them, the far field, has been the subject of three legs. The other one, measured here for the first time, is larger than the certificate's entire budget by eleven orders of magnitude, and the far-field term is larger still. The certificate closes with margin to spare in the discrete world, and that margin has no purchasing power at all against either bridge to the continuum.

Nothing here is a statement about HL_S2_nonsymmetric being certified, about the far-field gap, or about the method in the coefficient basis. No link of the L1→L4 chain moved. In 56 legs, none has. Clay stays at ~0.05%, behind Walls 1 and 2.

The novelty pass ran first and returned PROCEED_NARROW with nothing banked: rigorous error bounds for spline-based Hilbert transforms inside computer-assisted proofs are established practice in exactly this literature, Chen–Hou–Huang do it on these very models. The links are in writeup/novelty/leg_56.md.