Route-D v8, the codomain seminorm part of C_Q, and the first complete Z₂

Status: Level-1 tooling + upper bounds. NOT a certificate, NOT rigorous, NOT a Clay result. Plain float64: analytic majorants with numerically evaluated constants, gated against an exact second build and against measurements. Nothing here is interval-enclosed.

Figure: fig26 (writeup/figures/fig26_route_d_v8_quadratic.png) Data: writeup/data/p2_route_d_v8_quadratic.json Code: solver/hilbert_holder.py (+ test_nk_hilbert_holder.py, 6/6), experiments/p2_route_d_v8_quadratic.py, writeup/4_p2_lottery/p2_route_d_v8_evidence.py

0. What this leg was for

v7 closed the domain seminorm part of ‖A‖ and produced the project's first (α, γ) map made entirely of upper bounds, with one term still missing, and it said so:

Caveat that must travel with it: Z₂ here still omits the codomain SEMINORM part of C_Q, which is unbounded, and that omission is worst exactly where γ is smallest. So the location (1.4, 0.15) is provisional.

This leg bounds that term. The headline is not the bound: it is that after three consecutive legs in which replacing a lower bound by the honest upper bound cost the conditional budget an order of magnitude, this one costs 7%, and v7's provisional optimum survives unmoved, for a reason v7 got backwards.

Notation is v5's: X = tan(θ/2), w_β = sec^β(θ/2) = (1+X²)^{β/2}, domain ‖h‖_{α,γ} = S + T with S = sup w_α|h| and T the w_{α−γ}-weighted θ-Hölder seminorm; codomain the same with α → α+1. All pairs live in θ ∈ (0, π), because h is even and that is where the project's norms are defined (solver/decay_collocation.grid).

1. What actually has to be bounded, and with which weight

With Q(h) = h·H(h) and ψ = H(h), expand the product's increment about the inner point of the pair (θ_i = the one with smaller |θ|, where min(w) sits):

w_{α+1−γ}(θ_i) |Q(θ₁) − Q(θ₂)| / d^γ
    ≤ S · { w_{1−γ}(θ_i) |Δψ| / d^γ }   +   T · w_1(θ_i) B(θ_o)

using |h| ≤ S/w_α and |Δh| ≤ T d^γ / w_{α−γ}. Two things fall out and both matter:

• The second term needs no new work. w increases in |θ|, so w_1(θ_i) ≤ w_1(θ_o) and w_1(θ_i) B(θ_o) ≤ sup_θ w_1 B, which is exactly v6's C_Q sup-part quantity.
• The first term needs the weighted Hölder seminorm of ψ with weight 1 − γ, not α − γ. H does not inherit h's decay: for even h with nonzero mass, H(h)(X) → (∫h)/(πX) however fast h decays, so ψ's decay grading is 1 and its seminorm weight is 1 − γ by the same rule that gave the domain α − γ. Asking for α − γ here would be asking for something false, and would have produced an infinite constant with no explanation.

So the object of the leg is

T_ψ = sup_{θ₁≠θ₂} min(w_{1−γ}) |ψ(θ₁) − ψ(θ₂)| / |Δθ|^γ  ≤  b_sup S + b_semi T .

2. The estimate

Work relative to θ₁: put φ = θ₁ + t, θ₂ = θ₁ + σ, d = |σ|. In t nothing wraps, which is the whole reason for the change of variable: in absolute θ the pair (θ near π, φ near −π) is a pair of neighbours on the circle, so a near region defined as an interval of the line would leave a kernel singularity sitting in the "far" region.

Using p.v.∫cot = 0 on the circle, ψ(θ) = (1/2π) p.v.∫[h(φ) − h(θ)] cot((θ−φ)/2) dφ, and with N = [min(0,σ) − p·d, max(0,σ) + p·d]:

ψ(θ₁) − ψ(θ₂) = (1/2π) [ E_N + E_F + G ]
E_N = ∫_N [h−h(θ₁)] cot(−t/2) dt − ∫_N [h−h(θ₂)] cot((σ−t)/2) dt
E_F = ∫_F [h−h(θ₁)] [cot(−t/2) − cot((σ−t)/2)] dt
G   = [h(θ₂) − h(θ₁)] ∫_F cot((σ−t)/2) dt

Each piece gets an explicit majorant:

• increments of h by whichever norm part is cheaper at that point, |h(φ) − h(ref)| ≤ min( T |Δ|^γ cos^{α−γ}(θ_near/2), S[cos^α(φ/2) + cos^α(ref/2)] ) with folded distances Δ = ||φ| − |ref|| (the norm lives on (0,π) and h is even, so a pair straddling θ = 0 has increment zero and the seminorm knows it). The choice is made by a rule depending only on (φ, ref, α, γ), never on S or T, which is what keeps the result a genuine linear bound in (S, T) rather than a concave envelope;
• the kernels exactly, with the far difference in the stable form cot(t/2) + cot((σ−t)/2) = sin(σ/2)/(sin(t/2) sin((σ−t)/2)), which preserves the O(d) cancellation that makes E_F small;
• G in closed form: ∫_F cot((σ−t)/2) dt = −p.v.∫_N = 2 log|sin((σ−n₁)/2) / sin((σ−n₂)/2)| → 2 log(3/2) for p = 2.

The padding is a parameter, not a constant, and that mattered. The decomposition only needs N to cover the circle at most once, i.e. (1+p)d ≤ π. A first draft restricted the estimate to d ≤ (π−θ_i)/6, which is what the scaling argument needs, and the sweep then returned a constant a factor 12 too large, entirely from pairs just outside that cutoff where the crude pointwise route had to take over. Shrinking p for wide pairs instead of abandoning the estimate fixed it. Do not let the regime of an argument become the regime of the code.

For pairs too wide even for that (d > 2.6), and only those, the pointwise route is used: |Δψ| ≤ |ψ(θ₁)| + |ψ(θ₂)| with v7's split pointwise bound.

3. X1, the estimate, its convergences, and what it is worth

At the reference (α, γ) = (1.5, 0.5):

T_ψ  ≤  1.1936 · S  +  4.9410 · T

Flat to 4e-6 over a 4× refinement in each direction, and the per-pair quadrature is flat to 4e-4 over 150 → 2400 points. This refinement is a gate, not a nicety: the pair supremum is a grid supremum, which can only UNDER-report, the mirror image of v6's discrete-ball trap, where a set that was too big over-reported.

Gate: the decomposition, built twice. A majorant of a wrong decomposition is still an inequality about something, and no domination test would notice. So E_N + E_F + G is evaluated a second time with the true increments and compared against the exact conjugate (cos kθ → sin kθ): agreement to 5.6e-6 (quadrature-limited) over 7 pairs × 3 profiles. It caught two sign errors, one of them in a kernel identity that was also wrong in the module docstring.

Ablation: the estimate is the result. With only the pointwise route, all v6 and v7 had for this quantity, the same sweep returns b_sup + b_semi = 1452 instead of 6.13: 237× worse. And the per-pair rule matters: taking whichever route has the smaller coefficient sum inflates b_sup from 1.19 to 2.61, because at a near-tie it trades a large u_sup for a marginal gain in the sum. The route choice must be a single rule applied to both coefficients; mixing "the route that minimizes b_sup" with "the route that minimizes b_semi" is not itself a bound.

4. X2, the bracket

Measured against the family (anchor shape, cos kθ · f_α up to k = 256, square-wave partial sums, core and far-field bumps) at four (α, γ):

Valid everywhere, and ~4× lossy on the directions the family contains. Same order of slack as v7's closure. Per v7's own conclusion (the constants must be roughly sharp, not merely bounded) that slack is now the main quantity of interest, not the bound.

5. X3, the γ structure, and where v7's prediction went wrong

Both endpoint divergences are present and visible: b_semi = 18.6 at γ=0.05 (near region, ∫|t|^{γ−1} ~ 1/γ), falling to 4.94 at γ=0.5, rising again to 7.16 at γ=0.9 (far region, ∫d|t|^{γ−2} ~ 1/(1−γ)). So the complete C_Q bowls, with an interior minimum 3.330 at γ = 0.65.

But v7 predicted the omission would be worst where γ is smallest, and it is the opposite:

The reason is simple once seen: v6's sup-only C_Q already carried the same 1/γ near-region divergence, through the seminorm-paid half of its own |H(h)| bound. Nothing new blows up at small γ. The ratio is flat there and grows toward the Lipschitz end instead.

6. X4: the first complete Z₂ map

Z₂ = 2‖A‖C_Q at J = 800, with ‖A‖ = v6's sup-part dual + v7's closure and C_Q = v6's sup part (split by payer) + this leg's seminorm part. Every constant is an upper bound and nothing is omitted (the first Z₂ in this project of which that is true.

Optimum: (α, γ) = (1.4, 0.15), Z₂ ≤ 261.1) the same location v7's incomplete map reported (Z₂ ≤ 242.4), 7.7% higher. v7 called that location provisional because of this omission; the omission is now priced and the location holds.

7. X5, the budget, and the trend that does not continue

At the optimum, requiring Z₁ ≤ 0.5 from the far-field modelling error: X₀ = 3.2e3, implying J ~ 2.5e3 and a 6.2e6-entry dense core, inside what the existing collocation reaches.

(v7's own writeup quoted 2.0e-4, its value at the single row γ = 0.35; 2.58e-4 is v7's map priced over the same sweep as v8, which is the like-for-like number.)

Three order-of-magnitude losses, then one of 7%. The pattern v7 flagged as its most important negative (every time a lower bound is replaced by an upper bound, the budget loses an order) does not continue through this leg. Two reasons, both stated above: the old term already carried the new one's worst divergence, and v7's split-by-payer sharpening of |H(h)| recovers most of what the new term costs.

That is a real update on the lane question, in the good direction, and it is worth being precise about how much: it does not make the budget large (2.4e-4 is still ~40× below the GA residual floor), it removes one specific reason to expect the remaining three ledger items to be catastrophic.

8. X6, ledger after v8

Seven of ten bounded. The three that remain are all Z₁-side or representational; Z₂ is complete.

9. Reproduce

.venv/bin/python test_nk_hilbert_holder.py                        # 6/6, ~4 min
.venv/bin/python experiments/p2_route_d_v8_quadratic.py           # ~12 min -> JSON
.venv/bin/python writeup/4_p2_lottery/p2_route_d_v8_evidence.py   # fig26 from JSON

Deterministic: no GA, no seeds, no predicate lock (the logged-run discipline applies to stochastic Tier-1/2 runs; this is a tooling probe, entered in experiments/JOURNAL.md as a clearly-labelled non-logged entry).

10. Honest ceiling

v8 bounds the last unpriced constant in Z₂ and re-prices the budget with it. It does not climb the rigor ladder: everything is float64, nothing is interval-enclosed, and there is still no certificate, three ledger items remain open, all on the Z₁ side, and the bound itself is ~4× lossy. The eventual success this line scouts is a computer-assisted toy-model certification (Chen–Hou / Gómez-Serrano genre), not a Clay solve; 1D HL is a toy model of the boundary behaviour of Hou–Luo / 3D axisymmetric Euler. Overall Clay odds remain ~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.