Route-NG v1: the no-go, stated as a proposition, and the class it is a theorem on

Stage NG, leg 58. Gate answer: YES. Data: writeup/data/p2_route_ng_v1_nogo.json. Runner: experiments/p2_route_ng_v1_nogo.py. Figure: fig55_route_ng_v1_nogo.png, rebuilt from the curated JSON alone by experiments/p2_route_ng_v1_nogo_evidence.py. Novelty pass: writeup/novelty/leg_58.md, run and committed before any construction.

Gate, in its pre-committed wording. Does the no-go admit a proof for a named class of approximate inverses strictly larger than block-diagonal, with hypotheses that provably contain the a = 0 CLM linearization?, YES. The yes-branch fires: the repository has a Tier-3-shaped negative theorem; write it as a standalone claim with its sharpness control, and escalate publication scoping to the user. This branch is therefore parked, not merged.

The claim is narrow on purpose and the scope line is load-bearing. What is proved is a statement about approximate inverses with A₂₁ = 0. What is measured is everything else, and leg 54's battery is the whole of it. "No A we tried" is still not "no A", and this leg does not pretend otherwise.

0. What seven legs held, and what was missing

Legs 51–57 produced every part of a negative result and assembled none of them: a named mechanism (the unbounded part is off-diagonal, a shift, while the standard tail estimate needs a multiplier), an inequality that covers the block-diagonal case (MM-1), a battery over seven shapes of A bottoming at Z₁ = 8.9591 against a block-diagonal baseline of 10.4584, a positive control that reports the other answer, and a literature classification saying the case is unpublished.

What was missing was not more measurement. It was the write-up as a proposition, with the one open mathematical question named honestly, and, if possible, an extension of MM-1 past the block-diagonal case.

The extension exists, and it was sitting in the repository in two halves.

1. The proposition

Setting. L is the a = 0 CLM steady linearisation in the compactified odd-sine coefficient basis, bordered with the far-field amplitude as an extra unknown and its matching condition as an extra equation (leg 52's repair, leg 53's assembly). It is split at mode K as

L = [[G, B],
     [C, T]]

with G the finite block on modes 1..K plus the gauge and amplitude auxiliaries, and T the tail block on modes K+1..M. The norm is weighted ℓ¹, w_k = (1+k)^s.

Hypotheses.

• (H1) the split is the one the radii-polynomial method uses (a finite block carrying the auxiliaries, and a tail the certificate must handle by an explicit operator rather than a numerical inverse;
• (H2) the tail block T has a kernel in the space: T h = 0 with 0 < ‖h‖_w < ∞;
• (H3) A is a bounded approximate inverse with A₂₁ = 0) its tail rows do not couple back to the finite block. A₁₁, A₁₂ and A₂₂ are otherwise arbitrary.

Conclusion. For every such A,

Z₁ = ‖I − A L‖_w  ≥  1 + ‖A₁₁ B h‖_w / ‖h‖_w  ≥  1.

The radii polynomial requires Z₁ < 1, so no approximate inverse in the class closes the certificate: at any split K, in any weight class with s < 1.

Proof. Test the operator on x = (0; h):

(I − A L) x  =  ( −(A₁₁ B + A₁₂ T) h ;  h − A₂₁ B h − A₂₂ T h ).

T h = 0 kills the A₁₂ term and the A₂₂ term; A₂₁ = 0 kills the third. So (I − A L) x = (−A₁₁ B h ; h), whose ℓ¹_w norm is ‖A₁₁ B h‖_w + ‖h‖_w. Divide by ‖x‖_w = ‖h‖_w. ∎

A₁₂ and A₂₂ never appear. That is exactly why the class is strictly larger than block-diagonal, and exactly why the argument stops at A₂₁ ≠ 0.

1a. The class, and in what sense it is strictly larger

𝒜_upper  =  { A = [[A₁₁, A₁₂], [0, A₂₂]] : A₁₁, A₁₂, A₂₂ bounded }

Block-diagonal is the single point A₁₂ = 0, A₂₂ = A_tail inside it. 𝒜_upper also contains leg 54's gs_upper shape (one step of block Gauss–Seidel, tail swept first, A₁₂ = −Γ⁻¹ B A_tail), which leg 54 measured as a separate shape. Of leg 54's seven shapes, two are inside the class (block_diag, gs_upper) and five are outside (gs_lower, schur, ff_lift, and the two inadmissible ones): all of which have A₂₁ ≠ 0. Both facts are checked numerically in the runner rather than asserted from the construction: the in-class shapes measure ‖A₂₁‖ = 0 exactly, and ‖A₁₂‖ > 0 on gs_upper, which is what makes the containment strict.

1b. What this adds over MM-1

MM-1 bounds the same sub-block but through the coupling column, giving

Z₁  ≥  |1 − K/2| · (w_{K+1}/w_K) · ‖A_tail e_{K+1}‖_w / w_{K+1}

which is measured to be an exact equality (1.89e−15), but whose |1 − K/2| prefactor vanishes at K = 2, so its RHS only clears 1 from K ≥ 6 (flat) / K ≥ 4 (algebraic), and which fixes A₂₂ = A_tail. The proposition above has no prefactor, no K restriction, and no constraint on A₂₂. It therefore closes MM-1's small-K corner within this class and enlarges the class at the same time.

2. (H2), measured, and the reporting error it nearly caused

The proof needs the tail kernel to be in ℓ¹_w. That the kernel decays like m⁻², and that this puts it in the space exactly when s < 1, is leg 51's: fredholm_sides, and test_spectral_certificate.py's gate 7 has said "in ℓ¹ for s < 1, so the tail operator is not injective there" since that leg. It is not claimed here. What this leg does is check it on the vector actually used, over a ladder of truncations.

‖h‖_w partial sums at K = 8, over M = 256, 1024, 4096, 16384:

The discriminator is the increment ratio, not the shape of the curve: and this is where the leg nearly reported the wrong thing. At s = 0.7 the partial sum is still visibly rising at M = 16384 and a "has it settled?" criterion calls it divergent. It is not: the increments shrink geometrically by 0.659 per rung, so the series converges (to about 61.3 by geometric extrapolation), as Σ m^{−2} m^{0.7} = Σ m^{−1.3} must. A boolean would have recorded a false negative on a hypothesis this leg's whole result depends on. Report a magnitude, never a boolean: applied to a convergence test.

At s = 1 the increments are constant (r = 0.9988), logarithmic divergence, and at s = 1.5 they double (r = 1.9974), power divergence. So the threshold is exactly s = 1, which is leg 51's fredholm_sides crossing reached from the other side, and s = 1 is separately banned already (leg 52: the one exponent at which bordering cannot help, because the kernel leaves the space at the same moment the cokernel functional enters the dual).

Both classes legs 51–54 actually used, s = 0 and s = 0.3, are inside the hypothesis. That is what "hypotheses that provably contain the a = 0 CLM linearization" means here, and it is why the gate can answer YES.

2a. Two realizations, named

On the infinite tail the recursion makes T h = 0 exactly, and the bound is the unconditional Z₁ ≥ 1 + floor. At finite M the truncation leaves a nonzero edge term, so the honest finite-M statement is

Z₁  ≥  1 + floor − ρ_M · (‖A₁₂‖_w + ‖A₂₂‖_w),      ρ_M := ‖T h‖_w / ‖h‖_w.

ρ_M is a pure truncation artifact (T h is zero on every interior row to 3.55e−15 and nonzero only at the truncation edge, both of which are gated in test_spectral_certificate.py ("analytic right null vector is a kernel" and "the far field is annihilated except at the truncation edge, and that falls like 1/M")) and it vanishes on the ladder:

i.e. ρ_M ∼ M^{−(1−s)}, degenerating exactly at s = 1 again. The gap between the two realizations is therefore predicted, not noise, and §3 holds the measurement to it.

3. The bound against leg 54's battery

The runner rebuilds leg 54's assembled object unchanged and measures the ĥ-column of I − A L for every shape, at K = 2, 4, 8, 16, 32 in both admissible classes, against both forms of the bound.

Instrument check first (lesson 85). At the configuration leg 54 reported its headline on (algebraic s = 0.3, K = 2, M−K = 1024), this leg re-derives block_diag = 10.4584 and ff_lift = 8.9591, leg 54's baseline and best admissible, to a relative gap of 7.157e−06. (That gap is BLAS reduction order, not drift: this runner pins the thread count precisely so a verifier gets the same digits, and leg 54's run did not.) The battery being compared against is the same battery.

The three things that could have gone wrong, and did not:

1. The finite-M bound is never violated. Over every in-class measurement the slack column − (1 + floor − ρ_M(‖A₁₂‖+‖A₂₂‖)) is at least +5.336e−04.
2. The measured columns never fall below 1, which is what the proposition forbids. The in-class minimum over the whole sweep is 6.0424: the corresponding full Z₁ is 9.5660.
3. The deviation from the infinite-tail equality is the truncation. The largest |column − (1 + floor)| anywhere in the class, divided by the truncation budget ρ_M(‖A₁₂‖+‖A₂₂‖), is 0.5047: under 1, so the entire discrepancy is accounted for by the edge term §2a predicts, with nothing left over.

A sample of the columns against the infinite-tail bound 1 + ‖A₁₁ B ĥ‖_w/‖ĥ‖_w:

So the bound is not merely valid, it is attained on that column, and the containment is strict in the direction that matters: over these in-class measurements ‖A₂₁‖ = 0 exactly while ‖A₁₂‖ reaches 218.97, so A₁₂ is genuinely free and genuinely large, and the bound does not care.

4. The split-placement audit: the referee's objection, measured

"You chose a split whose tail block is singular. Put the far-field amplitude in the tail instead and the tail block is invertible."

True, and it does not help. The alternative split is the same operator under a permutation of one index (the amplitude column and its matching row move from the finite block to the tail) so it is built that way and measured rather than argued.

T' is invertible at every finite M, and ‖T'⁻¹‖_w diverges with M:

The same exponent 1 − s, with the opposite sign, to twelve digits. Either the tail block has a kernel (amplitude in the finite block) or its inverse is unbounded (amplitude in the tail). The obstruction is carried by the operator, not by where the amplitude is filed.

Scope: this is a measured ladder over M−K = 128…2048 at K = 8 with a fitted exponent. It is not a proof that ‖T'⁻¹‖_w is unbounded, and the proposition is not stated for that split.

5. Where the proof stops, and it is exactly one unit

A₂₁ ≠ 0 buys back one unit and nothing else: the term h − A₂₁ B h that hypothesis (H3) excludes. Comparing the ĥ-column of block_diag against ff_lift (the rank-one far-field lift) across the whole sweep, the removed amount is 0.9451 … 0.9990.

On the infinite tail that credit is exactly 1. At finite M it falls short, and the shortfall is the truncation defect and not something else: the largest deficit anywhere is 0.0549, and across all ten configurations the deficit never exceeds ρ_M, tracking it at a ratio of 0.856 … 0.997 while ρ_M itself moves over the sweep by a factor of 66 (9.77e−04 at flat K = 2 to 6.42e−02 at algebraic K = 32). Had the shortfall not been proportional to ρ_M, the mechanism would have been wrong.

The residual ‖A₁₁ B ĥ‖_w/‖ĥ‖_w survives every admissible lift.

That is the precise place the proof stops. Beyond it this repository has:

• leg 54's battery over seven shapes × class × gauge × split, best admissible Z₁ = 8.9591 against baseline 10.4584, a 1.167× improvement where more than 8× was needed;
• leg 54's MM4c, which refuted a shape-independent floor by an explicit rank-one counter-construction (floor → ~1e−16), surviving only because the total Z₁ then hits 5.7e+05.

So the residual term is beatable in principle, at a catastrophic cost in the rest of the operator, and no proof covers A₂₁ ≠ 0. The general no-go is a measurement over a battery and is described that way everywhere it appears in this repository.

6. Sharpness: the hypothesis cannot be dropped

Lesson 90's test is what would have had to change in the code for this control to report the other answer? Here the control varies μ, which changes the tail operator itself, so the answer is "the operator, and it does".

With Λ¹ dissipation the tail acquires a diagonal −μk: it becomes a multiplier, (H2) fails outright, and the proposition has no content. Measured at K = 16, M−K = 1024, algebraic s = 0.3:

Both of these shapes are inside the proved class. At μ = 0 the theorem forbids Z₁ < 1 and the measurement agrees at every K and in both weight classes; the kernel is gone by μ = 0.1 (σ_min jumps by more than two orders of magnitude) and by μ = 2 the same class is comfortably under 1. The hypothesis is necessary, not decorative.

Cross-leg number hygiene. Leg 53 reports 0.9156 for the μ = 2 algebraic configuration; leg 54 reports 0.6663. These are the same configuration measured in two conventions: sum of the four sub-block norms versus the true column-max. The proposition is stated in the operator norm, so 0.6663 is the number that bears on it; both are re-measured here and both are emitted to the JSON, because quoting one while arguing in the other is precisely the cross-leg error this repository has been burned by twice.

7. Scope, and what is not claimed

PROVED. The class A₂₁ = 0 (every block-diagonal and every block-upper-triangular approximate inverse, with A₁₂ and A₂₂ free) at every K, for every s < 1.

MEASURED, NOT PROVED. Every A with A₂₁ ≠ 0. For those there is leg 54's battery and nothing else.

Not claimed, per the novelty pass (writeup/novelty/leg_58.md, PROCEED_NARROW):

• the observation that a tail estimate presumes a dominant diagonal, folklore in print. Cadiot arXiv:2505.03091 §§2–3 states it, arXiv:2411.18361 restates it with a compactness justification. Cadiot's own hypotheses were located in the full text (a Fourier multiplier with |l(ξ)| ≥ l_min > 0, |l| → ∞, tail an infinite diagonal matrix) and this operator fails all three, so Cadiot does not contain this no-go and carries no positive result contradicting it. Leg 62 reads the same paper at greater depth by assignment, and its reading caps this one.
• the m⁻² kernel decay and the s < 1 threshold, leg 51's.
• the shapes of A, textbook preconditioning.
• anything about HL_S2_nonsymmetric, and anything about any link of the L1→L4 chain.

The ceiling, pre-committed. The object is the a = 0 CLM linearisation, whose Y₀ is exactly zero because the anchor is one basis mode, so the radii polynomial's root r = 0 is available for a degenerate reason and certifies nothing. A wall measured here bounds the real target's difficulty from below, and no more. Float64 throughout, no interval arithmetic: the proof is exact linear algebra on an explicitly constructed kernel, but every number in this document is a float measurement.

No link of the L1→L4 chain moved. Clay odds remain ~0.05%.