Phase-2 P2 (Route D v6: the first genuine upper bounds, and the discrete-ball trap

Status: Level-1 tooling + the project's first upper bounds on Route-D's Newton–Kantorovich constants) NOT a certificate. Five legs built the space (v3 the decay grading, v4 the price, v5 the smoothness scale) and measured in it. Every one of those numbers was a family-restricted maximum, a lower bound, and Z₁ was never bounded at all. A budget assembled from lower bounds is not a quantity a certificate can use, so this leg attacks the two gaps the v5 note named: bound Z₁ from the closed-form far field, and turn the family-restricted operator norms into genuine upper bounds analytically.

The second half turned out to be half right and half a trap, and the trap is the leg's most useful output. Clay odds unchanged (~0.05%).

Rebuild the figure from committed data (no re-derivation): python writeup/4_p2_lottery/p2_route_d_v6_evidence.py → writeup/figures/fig24_p2_route_d_v6.png (reads writeup/data/p2_route_d_v6_bounds.json; regenerate (deterministic, ~10 min) with python experiments/p2_route_d_v6_bounds.py).

Code: solver/nk_bounds.py + test_nk_bounds.py (6/6). Suite now 13 files green.

1. The discrete-ball trap (fig24-A)

The obvious way to turn a family-restricted lower bound into an upper bound is duality. The domain norm is a max of linear functionals, so

||A|| = max over those functionals of ||functional ∘ A||_{Y*},

and ‖·‖_{Y*} is a supremum over the codomain unit ball, which, on a grid, one computes over the discrete unit ball. That is unsound, and not marginally.

A discrete Hölder seminorm only inspects pairs of grid nodes. A grid vector that alternates in sign at the grid scale therefore reports a modest seminorm, while the trigonometric interpolant it actually stands for oscillates violently between nodes. Duality, asked for the worst direction, picks exactly such a vector. Measured on the extremizer it selects, comparing both norms over the same θ-range:

~J^2.03. A smooth element of the same decay class is faithful to 3%. The "worst direction" is not in the true unit ball at all, and the unboundedness it reports is the instrument's, not the operator's.

This is banked lesson (9), build the adversary, meeting its mirror image. In v4, random sampling missed the adversary and reported a boundedness that was false. Here a discrete norm invents an adversary and reports an unboundedness that is false. A test family that is too small errs one way; a ball that is too big errs the other. Both are the instrument.

2. What survives: duality from continuum-valid inequalities (fig24-B)

The sound route is to use only inequalities the continuum norm implies. Both of

|g_m| ≤ ‖g‖_Y / v_m ,        |g_m − g_{m₀}| ≤ ‖g‖_Y / q_{m,m₀}

hold for the continuum norm, because it dominates the discrete one on nodal values. Applying them to a functional row c and minimising over a reference index gives

‖c‖_{Y*} ≤ min_{m₀} [ |Σ_m c_m| / v_{m₀} + Σ_m |c_m| / q_{m,m₀} ] ,

two_point_dual. Minimising over a subset of m₀ keeps it valid (a min over fewer terms is larger), so it stays cheap. On the two parts of the domain norm:

The sup part saturates: J^{+0.006} over a 13× range in J. That is the project's first uniform upper bound on any part of ‖A‖, and it brackets the v5 family-restricted lower bound (~2.2–2.9) by about a factor 2.

The seminorm part grows like J^{+0.496} ≈ J^γ. It is a valid bound, so this is slack, not a verdict, and §1 identifies the slack precisely: the direction it prices is the grid-scale sign pattern that is not in the continuum ball. The continuum expectation is that this part is comfortably finite, because the inverse gains a whole derivative (h_X = −(g + H(h) + X h)/c, so g ∈ C^{0,γ} puts h in C^{1,γ}). Three routes were tried (the direct pair dual, the two-point dual, and bounding the seminorm through ‖h_θ‖) and all three grow. Closing this is now the sharpest open question in Route D. (It is not the gauge: every gauge-row choice tested gives the same J^γ, with the innermost row merely the smallest constant, consistent with v4 W5.)

3. The far-field modelling error, in closed form (fig24-C)

v3's far-field model is L h = −c h_X − h/X, whose inverse norm between the decay-graded sup norms is exactly 2/|α−2|. The full linearization at the anchor Ω₂ = −1/(1+X²), H(Ω₂) = −X/(1+X²) is DF h = h H(Ω₂) + Ω₂ H(h) − c h_X, so the modelling error is an exact two-term identity:

(DF − L) h = h [H(Ω₂) + 1/X] + Ω₂ H(h)
           = h / (X (1 + X²))  −  H(h) / (1 + X²).                      (E)

Verified against the collocation operator to 1.5 × 10⁻¹⁶ relative.

The first term is pointwise. The second needs |H(h)| in the far field for an arbitrary h in the unit ball: exactly where v4's adversary lives, and exactly what the Hölder grading was introduced to pay for. Splitting the principal value at half-scale on the even kernel K(X,y) = 2X/(X²−y²):

|H(h)(X)| ≤ (1/π) [ I_near(X) + I_out(X) ] ,
I_near ≤ (4/3) S 2^γ (X/2)^γ / γ + (8/15) (1+(X/2)²)^{−α/2} ,
I_out  = ∫_{y≥0, |y−X|>X/2} (1+y²)^{−α/2} |K(X,y)| dy ,

with S = 2^γ (1+(X/2)²)^{−(α+γ)/2} the X-side Hölder envelope (v5 fixed the seminorm weight at α−γ in θ; converting with |dθ| ≤ 2|dX|/(1+X_min²) makes the X-side weight α+γ, not α−γ).

Using the even kernel is not cosmetic. A two-sided 1/(X−y) split diverges logarithmically as X → 0 (where the true value is 0, since H of an even function is odd) and loses a factor 2 in the far field. The even form is finite at the origin and recovers the sharp constant: X · bound → 1.681 against M_α/π = 1.669.

Feeding this into (E) and taking the supremum over X ≥ X₀ gives the first bounded piece of Z₁ in six legs. Validated against the collocation operator on nodes the grid resolves in X (|X| dθ ≤ 1, the project's standing convention) over an adversarial test set:

and it decays at the predicted rate X₀^{α−2} across the whole grading range (measured −0.874, −0.815, −0.740, −0.653, −0.556, −0.453, −0.347, −0.240 for α = 1.1 … 1.8, against predicted −0.9 … −0.2).

The same |H(h)| bound gives the first upper bound on the quadratic constant: C_Q ≤ sup_X (1+X²)^{1/2} · bound(X), finite because the bound decays like M_α/(πX). At α = 1.5, γ = 0.5 it is 3.13, against v5's family-restricted lower bound of 0.86–1.14.

4. Pricing Z₁ moves the optimum (fig24-D)

Z₁ has never before entered the optimization, and it changes the answer.

Two things stand out.

v5's joint optimum is dead. At α = 1.8, Z₁ is 2.3–4.3, not near the 1 it must beat, at every X₀ tested. The reason is structural: the modelling error decays like X₀^{α−2}, so at α = 1.8 the far field must be pushed out 10⁵× further to buy what α = 1.2 gets for free, while ‖A‖ = 2/(2−α) is simultaneously blowing up toward the α = 2 resonance. Both of v5's knobs were tuned against constants that did not include Z₁.

The optimum moves to α ≈ 1.2, and the conditional budget there is 1.18 × 10⁻². That number invites a comparison that must be made carefully: the GA residual floor at the a ≈ 0.5 two-scale boundary (measured on a > 0) is also ~10⁻². The two being the same order is not a statement that the boundary profile could be certified. The budget above is conditional and optimistic: it prices only the far-field part of Z₁, uses the far-field ‖A‖ (v4 confirmed that predicts the full gauged ‖A‖ to 6% on α ∈ [1.4, 1.7], which is not the range the optimum now sits in), and omits three constants entirely. The honest reading is that the target is no longer obviously out of reach by orders of magnitude, which is a change from v5, and nothing more.

5. The ledger

Three of eight moved from measured to bounded. Three remain open, and they are now named precisely enough to be attacked one at a time rather than scoped.

6. Honest ceiling

Everything here is plain float64: analytic bounds with numerically evaluated constants, gated against measurements. Nothing is interval-enclosed and nothing is rigorous: solver/interval.py has existed since v1 and still has not been pointed at any of this, correctly, because nothing has closed in float. This leg does not climb the rigor ladder; it converts three of the eight constants from things we had measured into things we can bound, kills one candidate optimum, and disqualifies one method. Even the success it is scouting would be a computer-assisted toy-model certification (Chen–Hou / Gómez-Serrano genre), not a Clay solve. 1D HL remains a toy model of the boundary behaviour of Hou–Luo / 3D axisymmetric Euler. Overall Clay odds ~0.05%.