Route-D v9, the sharpness leg: a 32% better |H(h)| bound that buys nothing where it matters

Status: Level-1 tooling + upper bounds. NOT a certificate, NOT rigorous, NOT a Clay result. Plain float64 throughout; nothing is interval-enclosed.

Figure: fig27 (writeup/figures/fig27_route_d_v9_sharpen.png) Data: writeup/data/p2_route_d_v9_sharpen.json Code: solver/hilbert_pointwise.py (+ test_nk_hilbert_pointwise.py, 6/6), experiments/p2_route_d_v9_sharpen.py, writeup/4_p2_lottery/p2_route_d_v9_evidence.py

0. Why a sharpness leg

v8 left seven of ten constants bounded and Z₂ complete, which changed the question. The budget goes like 1/(‖A‖ C_Q); ‖A‖'s bracket was ~70× wide; bounding the last three constants can buy a constant factor each, while halving ‖A‖'s slack buys more than all three together. So: sharpen, don't cover.

The target picked itself. v7's closure reads T ≤ F(T) with the feedback carried by |H(h)|, and at the reference point the Hilbert term supplies 81 of the 93 units in the derivative bound: the closure is dominated by its own feedback, and the feedback is v6's |H(h)| bound, built from crude majorants (|cot| ≤ 2/|u|, a 4/3 here, a 16/15 there, the singular half charged wholesale to the seminorm).

The leg did sharpen it, by 32% at the reference point. The budget did not move. Sections 4–5 are why, and that is the result.

1. The bound

For even h, folding the conjugate-function integral onto (0, π) gives

psi(θ) = (1/2π) p.v. ∫_0^π h(φ) K_θ(φ) dφ ,
K_θ(φ) = 2 sin θ / (cos φ − cos θ) = −sin θ / ( sin((φ+θ)/2) sin((φ−θ)/2) )

Three properties, each of which removes something v6 needed:

• The decay is in the kernel. sin θ → 0 as θ → π. This is v6's even-kernel point in θ coordinates: H of an even function is odd, and a form that cannot see that cannot see the far-field decay either.
• p.v.∫_0^π K_θ dφ = 0 exactly: it is the conjugate of the constant function; the antiderivative −2 log|sin((φ−θ)/2)/sin((φ+θ)/2)| vanishes at both endpoints. So the principal value is handled by a global subtraction h(φ) − h(θ): no band, no matching scale, no remainder term. v6 needed all three and each cost a constant.
• The second form is the one to evaluate. Near θ = π both cosines approach −1 and their difference loses every significant digit; parameterising by the offset s = |φ − θ| makes sin((φ−θ)/2) = sin(±s/2) exact. The first draft produced NaNs at X ≳ 1e4 before this.

Result (Y1), against v6 across eight decades of X:

Gated by an exact second build (the same kernel, true integrand, against cos kθ → sin kθ): 1.5e-7. And the bound is nearly attained: on the anchor profile the measured |ψ| reaches 0.97 of it.

2. The payer rule: the part worth keeping

At each φ the increment can be charged to the seminorm or to the decay envelope, and any fixed rule gives a valid linear bound. v8 chose by comparing the two at S = T = 1. That is the wrong default, and generally:

the rule should be tuned to the ratio T/S of the answer, not to 1.

Introducing ρ (seminorm route iff ρ c_T ≤ c_S):

Two things fall out. There is an interior optimum, and the neutral rule ρ = 1 gives 74.7: worse than the 69.4 of the crude bound it replaces. In the closure T ~ 10 S, so charging work to T to minimise the S = T = 1 sum is charging the expensive account. And the two consumers want different ρ: C_Q maximises over the unit simplex where the ratio is O(1), so it prefers ρ ≈ 2. Both are valid bounds; each caller picks.

3. Y3, the new ‖A‖

At (α, γ) = (1.5, 0.5), ρ = 6:

Still flat in J (J^+0.0059, inherited entirely from C_sup). A 32% reduction, and the largest single improvement to ‖A‖ since the closure was built.

4. Y2b (the gain does not transfer

(1.4, 0.15) is the map's optimum) where the budget is actually evaluated, and has been for three legs. There the new bound is worth nothing at all.

The mechanism is the interpolation exponent. v7's closure is

T ≤ C(γ) (P/2)^γ (2S)^{1−γ}

and the |H(h)| bound enters only through P. At γ = 0.5 a 30% improvement in P moves T by ~15%; at γ = 0.15 it moves it by 4%. The operating point sits at small γ precisely because that is where ‖A‖ is cheap, and small γ is exactly where ‖A‖ stops caring about the estimate this leg improved.

5. Y6, the elasticity, and what it says to do next

Scaling each input of the closure by a factor and fitting the log-log slope:

‖A‖ is proportional to C_sup, v6's two-point dual on the sup part, and at the operating point barely sees the |H| bound at all. The last two legs (v8's codomain seminorm, v9's pointwise bound) both worked on inputs with elasticity ≤ 0.5, and both moved the budget by ≤ 7%. That is not a coincidence; it is the elasticity table read backwards.

What this does not license. It is tempting to compute how much is available by substituting a measured value for C_sup. An earlier draft of this leg did exactly that and reported a 19× available gain, which was an artifact: the "measured C_sup" was a family lower bound computed by dividing the sup part of an image by the full codomain norm of a sign pattern, whose Hölder seminorm is enormous. It is an order of magnitude below any plausible sharp value. Banked lesson (15), again, in a place we had already been warned about.

What is defensible: elasticity says C_sup is the input worth attacking; v6 B2's own bracket says C_sup is roughly 2× lossy; so ~2× is on the table there, not an order of magnitude. And the wider bracket (47× against a family lower bound of ~1.0) cannot be attributed at all, because a maximum over a handful of sign patterns says almost nothing about how lossy an upper bound is.

6. Y4/Y5 (map and budget

The complete Z₂ map (now choosing ρ per cell from {1, 3, 6, 12}) has its optimum at (1.4, 0.15), Z₂ ≤ 260.7) third leg running at the same place, essentially unchanged from v8's 261.1.

Three order-of-magnitude losses, then three legs of nothing. The losses stopped; so did the gains.

7. Ledger after v9

Unchanged in coverage (seven of ten bounded, Z₂ complete) with two status notes rewritten:

• ‖A‖ domain SUP part (C_sup), bounded (v6), and now identified as the dominant input: ‖A‖ is proportional to it, and it is the only remaining input with elasticity ≈ 1.
• C_Q sup part / ‖A‖ seminorm part: sharpened by this leg's |H| bound, materially at moderate γ and not at all at the operating point.

Open, unchanged: Z₁ core↔far coupling; Z₁ core discretization; the discrete↔continuum transfer.

8. Reproduce

.venv/bin/python test_nk_hilbert_pointwise.py                     # 6/6, ~6 min
.venv/bin/python experiments/p2_route_d_v9_sharpen.py             # ~35 min -> JSON
.venv/bin/python writeup/4_p2_lottery/p2_route_d_v9_evidence.py   # fig27 from JSON

Deterministic: no GA, no seeds, no predicate lock.

9. Honest ceiling

v9 improves one estimate by a third and moves the certification budget by 0.4%. It does not climb the rigor ladder: float64 throughout, nothing interval-enclosed, no certificate, three ledger items still open. The eventual success this line scouts remains a computer-assisted toy-model certification (Chen–Hou / Gómez-Serrano genre), not a Clay solve. Overall Clay odds ~0.05%. The honest best case for the whole Route-D leg is still "certifies the a = 0 traveling wave", which is already known in closed form.