Route-L1 v2: the certificate rebuilt where the operators are exact, and the one term that is left

Phase 2, leg 51. Stage L1 step two: move the certificate off the truncated finite-difference grid and into Route-E's compactified basis, where H, the dilation and the velocity are exact on the whole line. Code: solver/spectral_certificate.py, test_spectral_certificate.py (14/14), experiments/p2_route_l1_v2_spectral.py → writeup/data/p2_route_l1_v2_spectral.json → fig46. Deterministic, 53 s.

THE REPRESENTATION CHANGE DID EXACTLY WHAT THE PLAN SAID IT WOULD, AND THEN THE SURVIVING TERM FAILED. The operator gap vanishes, Y₀ becomes exactly zero in rational arithmetic, the truncation gap collapses into a single spectral-tail term, and that term has no bound in any weight class tested. The class that standard radii-polynomial work uses is the worst of them: the tail inverse grows by a factor ν per neglected mode.

0. What is being claimed, exactly

Stated before the numbers, and not widened afterwards:

On the a = 0 CLM fixed point written in the odd-sine basis of X = tan(θ/2), three of the four terms of the radii polynomial are finite and rigorous, Y₀ = 0 exactly, Z₁ on the finite block bounded in interval arithmetic, Z₂ finite from the Banach algebra structure of the basis. The fourth, the tail term, is measured to diverge in every weight class tried: flat ℓ¹, algebraic (1+k)^s for s ∈ [0, 2], and geometric ν^k for ν ∈ {1.05, 1.2}.

What is not claimed: that no space exists. What is measured is that no space in these three families does, on the friendliest object available, and that the obstruction has a name: the tail operator's diagonal is exactly zero and its kernel is the |X|⁻¹ far field. And the leg's ceiling (clause S7, pre-committed): the a = 0 CLM profile is one mode, analytic, and in every class considered, so a wall measured there bounds the difficulty for HL_S2_nonsymmetric from below.

1. Why the basis was changed

Leg 50 closed the radii polynomial in interval arithmetic, but around a finite-dimensional system built from stored finite-difference operators on |X| ≤ 745. The two gaps it left were named in the plan of record as the same gap: the certificate lives on a truncated grid. Route E's compactification removes the grid:

2. The exactness audit is executable, not a docstring (S1)

rescaled_spectrum.py has claimed the three identities since Route E. This leg makes them runnable:

• Exact rational check. w = (1+iX)/(1−iX), and w^k computed as a Gaussian rational for rational X, agrees with cos kθ + i sin kθ to 3.5e−15 over k = 1,2,3,5,9, machine epsilon times the size of k. The identity for H then follows from where w^k's only pole is: (1 − iX)^k vanishes at X = −i, in the lower half plane, so w^k is analytic in the upper one and the Hilbert transform on the line carries its real part to its imaginary part.
• The velocity's closed form. The recursion's constant terms are c_k = (−1)^k k exactly (defect 0.0, k = 1..16), and that is not a decoration, see §6.
• Independent numeric cross-check. solver/line_hilbert.py, an unrelated implementation, reproduces H(sin kθ) = −cos kθ + (−1)^k with a defect that falls like n^−1.00 on refinement: a rate, not a single agreement number.

3. The anchor's residual is exactly zero (S2)

Ω₀ = −sin θ, c_ω = −1, c_l = 1. In coefficient space the residual of an 8-mode profile has 18 modes and every one is Fraction(0). Not 1e−16, zero. The negative control: perturb b₄ by 2e−3 and 5 modes become nonzero.

This is a thing the grid certificate structurally could not do. Y₀ is not a small number in this basis; it is absent.

4. The four terms (S3)

Finite block, K = 256, rigorous (interval arithmetic on exactly-representable matrix entries), norms ‖b‖_w = Σ w_k |b_k|:

And the tail term, ‖T_tail⁻¹‖_w on modes 65 … M:

The geometric rows are quoted per mode because a power-law exponent would be meaningless there: the growth factor is ν, to three digits, in both cases. Every neglected mode costs the full weight of the mode.

5. The structural fact, and why it is not a tuning problem

The tail operator's diagonal is exactly zero. max|diag(T_tail)| = 0.0, not 1e−16.

Every radii-polynomial certificate in the literature splits the approximate inverse as A = A_K ⊕ A_tail with A_tail diagonal, because the unbounded part of the operator being certified is a multiplier (a Laplacian, a dispersion relation, a Λ^s) whose tail Λ_k → ∞ has a diagonal inverse 1/Λ_k that is both explicit and small. Here the unbounded part is the dilation transport sin θ ∂_θ, whose matrix is bidiagonal with entries ±k/2 and nothing on the diagonal. There is no A_tail of the standard shape, for any weight.

And the operator is not merely awkward to invert; it is not injective in the classes where the object lives. The homogeneous recursion

v_{m+1} = v_{m−1} + 2 g_m ,        h_m = v_m / m

gives a measured decay h_m ~ m^{−2.007} (predicted m^{−2}). A coefficient decay k^{−1−α} is a far field |X|^{−α}, so the kernel of the tail operator is the |X|^{−1} profile, in ℓ¹_w for every s < 1. Leg 46's truncation gap and leg 47's wrong-sign reach trend were not two facts about a domain size. They were this one mode, seen through a grid.

6. The velocity does not escape it either

c_k = (−1)^k k means each mode's velocity carries a term linear in θ. θ is not a trigonometric polynomial: its odd Fourier series is the sawtooth, coefficients exactly 1/m. So for any model with advection, and HL_S2_nonsymmetric has it, the exact velocity operator injects an algebraic 1/m tail regardless of how analytic the profile is. The far field does not leave when the grid does.

7. The positive control (S4)

A negative result needs an instrument that can say "bounded". Add Λ¹ dissipation, the same code path, −μk on the diagonal, turning the shift into a multiplier and nothing else:

Two things worth keeping. The instrument reports bounded the moment the operator has a diagonal, so the inviscid divergence is a measurement. And the geometric class needs five times more dissipation than the other two before it saturates: it is the fragile class on both sides of the experiment.

8. Float against exact (S5)

Banked lesson 86 says a bound dominated by its own evaluation error is a statement about the code. The quantity that decides this leg, the weighted inverse norm, was recomputed in exact rational Gauss–Jordan (fractions.Fraction, no floating point) at K = 16, 32, 64, 96, 128, ν = 1.1. Maximum relative gap against float: 8.8e−15. At K = 96 the exact answer is 62.0005; at K = 128 it is 609.17. The divergence is mathematics.

9. The window, from both sides (S6)

The object side: a far field Ω ~ |X|^{−α} gives coefficients ~ k^{−1−α}, so ‖Ω‖_w < ∞ for w_k = (1+k)^s iff s < α. For HL_S2_nonsymmetric, α = 0.394.

The operator side, measured: the tail's divergence exponent as a function of s is a U-curve with its minimum at s = 1.00 (exponent +0.64), rising to +1.28 at s = 0 and +1.27 at s = 2. At the object's own boundary s = α = 0.394 the exponent is +1.00.

The window is empty by 0.606 in exponent units, and no point of the curve touches zero. The class where the operator is least bad is the class where the target has infinite norm, and vice versa.

10. What this leg answers, and what it does not

L1's pre-committed gate was: does the polynomial close in interval arithmetic, tail included? The answer is no, and the term that ran out is named: the weight class of the tail, not the interval widening (three terms are rigorous and small) and not the spectral truncation as such (Y₀ is exactly zero).

The plan of record says that if it is the weight class, that is the contribution and it should be written up rather than filed as a failure. Route J's literature pass found no hit for the one methodological claim this touches (geometric ℓ¹ weights on bounded domains versus algebraic decay on an unbounded one) and this leg sharpens what that claim should be: the obstruction is not the profile's decay but the unbounded part of the operator being a shift instead of a multiplier, which is a statement about self-similar transport and not about any particular profile. It also predicts the literature's shape: the certified self-similar blow-ups that used this machinery (Dahne–Figueras, CGL) are dissipative, and the certified inviscid ones (Chen–Hou) use weighted energy estimates instead. That prediction is checkable and it has not been checked here.

No link of the L1→L4 chain moved. The object still has no proof of any kind.