The wall has a name now, but only on part of the map

Route-NG, leg 58. Companion to TECHNICAL_P2_ROUTENG_V1.md. Data: writeup/data/p2_route_ng_v1_nogo.json. Figure: fig55_route_ng_v1_nogo.png.

For seven legs this project has been walking into the same wall and describing it slightly differently each time.

The setting: we are trying to build a computer-assisted certificate, the standard radii-polynomial machinery, for a self-similar profile of an inviscid transport equation. The machinery needs an approximate inverse A of the linearised operator, good enough that ‖I − A L‖ < 1. Everything else in the certificate had been made to work. That one number would not go under 1. It kept coming out around 10, and the best we ever managed, after spending every free choice in the construction, was 8.9591.

Leg 54 spent the last of those free choices, the shape of A, across seven different shapes, and reported the honest thing: a 1.167× improvement where more than 8× was needed. That is a battery of measurements. It is not a theorem. And there is a real difference between "every A we tried failed" and "every A fails", which is exactly the difference between a table and a result.

This leg's job was to find out which one we have.

The answer was in the repository, in two halves

Leg 51 had established that the tail operator has a kernel, a direction it sends to zero, and that this kernel, the far-field mode falling off like 1/|X|, actually lives inside the function space we work in. Leg 54 had noticed that on that one direction, one block of the error I − A L collapses to something no choice of A₁₂ can touch, and had even written down the dual remark about the other block.

Nobody had put the two block-rows in the same column and taken the norm.

Do that and the proof is three lines. Feed the operator the vector that is zero on the finite block and equal to the kernel h on the tail. Because T h = 0, two of the four terms vanish identically: they involve A₁₂ and A₂₂, and both get multiplied by T h. A third vanishes if A₂₁ = 0. What is left is the kernel itself, coming straight back out:

(I − A L)(0; h)  =  (−A₁₁ B h ;  h)

and therefore, dividing by the norm of the input,

For every approximate inverse whose tail rows do not couple back to the finite block, Z₁ ≥ 1. At every split. In every weight class with s < 1. Whatever A₁₂ and A₂₂ are.

A₁₂ and A₂₂ never enter the calculation. That is the whole trick, and it is also precisely why the argument stops where it does.

What this is worth, stated carefully

The class it covers, call it block-upper-triangular, is strictly larger than the block-diagonal A the method conventionally uses. It contains the block-diagonal shape as a single point, it contains a second shape leg 54 had measured separately, and it leaves two of the three off-diagonal blocks completely free. It also has no restriction on the split K, which the previous inequality did: that one carried a prefactor |1 − K/2| that vanishes at K = 2 and left a corner open. This closes it.

The class it does not cover is everything with A₂₁ ≠ 0, and that includes the shape that gave leg 54 its best number.

So the honest summary is a two-line scope, and it is on the front of the technical writeup:

• Proved: no block-diagonal or block-upper-triangular approximate inverse can close this certificate.
• Measured, not proved: everything else. Leg 54's battery, bottoming at 8.9591, is the entire evidence.

The part that is satisfying

You can measure exactly what breaking the hypothesis buys you. The one shape that couples the tail back to the finite block, a rank-one lift of the far field, removes one unit from that column and then stops. It buys back the kernel's own contribution, which is precisely the term the theorem's hypothesis excludes. The rest of the column, the part involving A₁₁, survives every admissible lift.

Measured across the sweep the credit is 0.9451 … 0.9990, not a flat 1. That is worth saying carefully, because the gap is the interesting part: on the infinite tail the credit is exactly 1, and at a finite truncation it falls short by an amount that should be proportional to the truncation error if the mechanism is what we think it is. It is. Across all ten configurations the shortfall stays below the truncation defect ρ_M and tracks it at a ratio between 0.856 and 0.997: over a sweep in which ρ_M itself moves by a factor of 66. The mechanism predicts the residue, on numbers nobody tuned.

Two ways this could have been fooling us, and both were checked

"You picked a bad split." The obvious objection is that we chose to put the far-field amplitude in the finite block, which is what leaves the tail block singular. Put it in the tail instead and the tail block is invertible, objection dissolved?

No. That alternative is the same operator with one index moved, so we built it that way and measured it. The tail block is indeed invertible at every truncation, and the norm of its inverse diverges as the truncation grows, at exactly the rate M^{1−s}. Which is the same exponent, with the opposite sign, at which the kernel defect vanishes in the original split. Either the block has a kernel, or its inverse is unbounded. The obstruction belongs to the operator, not to our bookkeeping.

"Your hypothesis is unfalsifiable." A no-go whose hypothesis always holds is not a theorem, it is a description. So the hypothesis is put on a dial: add dissipation, and the tail stops being a shift and becomes a multiplier with no kernel at all. Then the same class of approximate inverses (the one the theorem says can never get below 1) reaches 0.6663 and then 0.4026. The instrument can say yes. It says no here because the answer is no.

And a near-miss worth recording

The hypothesis needs the kernel to have finite norm in the working space. Checking that across weight exponents, the partial sums at s = 0.7 are still visibly climbing after 16384 modes, and a "has it converged yet?" test calls that a failure. It isn't one: the increments are shrinking by a factor of 0.66 each rung, so the series converges, just slowly. Meanwhile at s = 1 the increments are constant: a logarithmic divergence wearing the same costume.

Reading a boolean off those curves would have put a false negative on the one hypothesis the whole result rests on. Reading the increment ratio instead separates them cleanly: 0.25, 0.38, 0.66 converge; 0.9988 is log-divergent; 1.9974 is a power divergence. Report a magnitude, never a boolean: this time applied to a convergence test.

What it does not mean

It does not mean the underlying problem is settled, and it does not touch the Clay chain. The object measured here is the a = 0 CLM linearisation, a model whose certificate is degenerate anyway: its residual is exactly zero for a trivial reason. A wall measured on the easy object bounds the hard one's difficulty from below, and that is the only direction the inference runs. Nothing here says anything about the non-symmetric Hou–Luo profile, and no link of the chain moved. In 58 legs, none has.

What it does mean is that the repository now has one honest sentence where it used to have a table, and a precisely drawn line showing where that sentence stops being true.