Route-D v10, a lower bound on ‖A‖ worth reading, and the first measured ceiling on sharpening

Status: Level-1 tooling + bounds. NOT a certificate, NOT rigorous, NOT a Clay result. Plain float64 throughout.

Figure: fig28 · Data: writeup/data/p2_route_d_v10_lower.json Code: solver/op_lower.py (+ test_op_lower.py, 6/6), experiments/p2_route_d_v10_lower.py, writeup/4_p2_lottery/p2_route_d_v10_evidence.py

0. Why

v9 ended with the sharpest statement this project has made about its own state: every bracket it quotes has a lower end that is a maximum over sign patterns, which is nearly meaningless, so "the bound is 50× too big" and "the operator really is that large" cannot be told apart, and they imply opposite decisions about the whole lane. v9 also paid for the confusion, turning the uninterpretable bracket into a "19× available gain" that was pure artifact.

This leg builds the adversary (banked lesson (9), applied to the operator rather than to the quadratic) and then reads the answer.

1. Why sign patterns fail here, and what replaces them

For a domain index i, the vector g = sign(A_i·)/v is the exact extremizer of the sup-to-sup problem. In this space it is a terrible direction, for precisely the reason v6's discrete-ball trap made famous in the other direction: a sign pattern's Hölder seminorm is enormous, so dividing by the full codomain norm throws away everything the numerator gained. At J = 400 these report 0.94 against an upper bound of 47, and, the tell nobody had looked at, they get worse as J grows (0.97 → 0.92 over J = 200..800). They were never converging to anything about the operator.

The Y-ball says what to look for instead: an element with finite codomain norm must decay at least like 1/v = cos^{α+1}(θ/2) and must not oscillate. So the family is 1/v times a slowly varying shape: powers, low-order cosines, Gaussian bumps over a wide sweep of centre and width, smoothed steps, and boxes. Any g gives a valid lower bound ‖A‖ ≥ ‖A g‖_X / ‖g‖_Y, so validity is free and the whole problem is construction.

A random ascent in a smooth cosine basis is run from the family's best, and adds essentially nothing (gain 1.000×). That is reported rather than dropped: a flat maximum is information about the problem's shape.

2. W1/W2, the numbers

At the reference (α, γ) = (1.5, 0.5):

At the operating point (1.4, 0.15), where the budget is actually evaluated, and has been for three legs:

2.74  ≤  ‖A‖  ≤  20.94          a factor of 7.7

The bracket that matters was never 50×. It is 7.7×. Quoting a bracket at the reference point was itself part of the confusion.

3. W3, what the extremizer is

A wide, far-field-supported, slowly varying shape: the winner at J = 400 is a bump centred at θ = 3.12 (i.e. X ≈ 93) with width 0.5, and the next four are its neighbours in centre and width. Nothing oscillatory comes close.

That is the same place every other Route-D finding has pointed at: v2's far-field degeneracy of the transport term, v3's resonance at α = 2, v6's matching radius X₀. The operator's worst direction is a broad far-field disturbance, not a local one, which is consistent with the physics of the problem and is a small independent check that the number means something.

4. W4: across the map

8–16× everywhere, and tightest at the optimum. The upper bound is worst exactly where the closure leans hardest on the interpolation inequality (large γ).

5. W5 (the verdict, which is the point of the leg

A perfect upper bound on ‖A‖) one reaching the adversary exactly, would multiply the conditional budget by the bracket and no more:

budget now                    2.45e-4
if ‖A‖ were exactly sharp     1.88e-3          (x 7.7)
GA residual floor             1e-2

This is the first measured ceiling on what sharpening can buy in ten legs. Read it in both directions and both matter:

• Better than it looked. After v7 the budget appeared to be ~40× below the residual floor with no idea how much was recoverable. The recoverable part of ‖A‖ is 7.7×, which lands ~5× short of the floor rather than 40×.
• Not enough on its own. ‖A‖ alone cannot close the gap even if bounded perfectly. Closing it needs C_Q's slack (~4×, v8 X2) as well, and those two together would only just reach the floor, with nothing left over for the three Z₁ items still open.

Caveats that travel with the number: the true norm is somewhere inside the bracket, not at its bottom, so 7.7× is itself an over-estimate of the achievable gain; the lower bound is still a finite family; and the budget remains CONDITIONAL (far-field Z₁ only).

6. Ledger and next

Coverage unchanged: seven of ten bounded, Z₂ complete. What changed is that the bracket on the dominant constant is now interpretable, so the next leg can be chosen on evidence: C_sup (elasticity ≈ 1 per v9, untouched since v6, and now with a measured ceiling on the payoff), then the Z₁ core↔far commutator, the change of ansatz, and the core discretization.

7. Reproduce

.venv/bin/python test_op_lower.py                                  # 6/6, ~8 min
.venv/bin/python experiments/p2_route_d_v10_lower.py               # ~20 min
.venv/bin/python writeup/4_p2_lottery/p2_route_d_v10_evidence.py   # fig28

8. Honest ceiling

Nothing here climbs the rigor ladder: float64, nothing interval-enclosed, no certificate, three ledger items open. The eventual success this line scouts remains a computer-assisted toy-model certification, not a Clay solve. Overall Clay odds ~0.05%.