Level 6b: the density question — the two adversaries exclude each other

Disposition

Level 6a's theta_full = 0.02 carried the label "at unit pair density". This level asks what happens when the density is the adversary's to choose, and the answer has a structure worth stating before the numbers: the on-line adversary and the pair-density adversary cannot both show up. Dense pair clusters are nearly immune to on-line attack (the compensation factor collapses tenfold from nu_p = 1 to 2), and they pay their own stacking; sparse pair layers are on-line-vulnerable but produce little pair-pair negativity. The joint budget, measured across the whole interpolation, holds everywhere — worst margin +0.0704 exactly at the pinch (nu_p = 1.5, deep) where the separable accounting had broken by −0.60.

theta_full = 0.02 is thereby corroborated at the joint level across all tested densities. The honest labels are in the last section; the Phase-7 gate stays closed.

The separable budget breaks, and that record is kept

Summing the level-6a per-pair hardened caps is valid (each damage term belongs to one pair's row), giving the global requirement sum T_signed >= - sum (1 - capfrac) slack. Measured across lattice families (nu_p = 0.5–3, four depths, three phases) and the collective battery: 24 of 68 configurations break it, worst at nu_p = 1.5, y = 0.49 with margin −0.599. Deep mid-density lattices maximise pair-pair negativity (T/slack = −0.92) while the budget offers only 0.32. If the per-pair caps were simultaneously attainable, theta_full would be dead at those densities — the level-5 lesson repeating one level up.

The compensation factor: why they are not simultaneously attainable

A single on-line configuration cannot deal every pair its worst case: a zero at one pair's most negative offset (g* ~ 0.7) sits in a neighbouring pair's positive region when the pairs are 0.5–0.8 apart. Measured (greedy joint on-line adversary against whole lattices, theta = 0.02):

At unit density the joint attack already reaches only ~15% of the summed caps; at density 2 it is one-sixtieth. Density is a defensive liability for the adversary's own pair layer.

The joint budget across the interpolation

used = joint on-line profit/slack + max(0, -T_signed)/slack against 1:

(entries are margins, 1 − used). The trade-off curve pinches at nu_p ~ 1.25–1.5 deep — T-negativity near its maximum while the on-line attack is already collapsed — and clears by 7%. On both flanks the margin grows: sparse layers have positive T (stacking-dominant), dense layers pay their stacking and blind the on-line attack further.

The stacking floor (the provable seed)

One-sided, all depth pairs on the probe grid, grid + Lipschitz margin:

T(dt, y, y') >= +2.78   for |dt| <= 0.2,     dies by delta_p = 0.3
                                              (at the strip edge, mapped)

Every same-window pair the adversary adds pays this floor — the third instance of the self-defeat pattern (depth paid quartic slack at level 3, on-line density paid quadratic packing at level 4, pair density pays stacking here).

Honest labels

• The T-side is exact for the tested lattices, and the family sweep covers the battery shapes; but it is a family, not all configurations.
• The joint on-line profit is a greedy measured adversary — a lower bound on the true joint supremum. The +0.07 pinch margin is against the greedy; a cleverer joint attack could be stronger. The named theorem object is therefore the joint cap: an on-line counting dual against a cluster of pairs (the level-4 machinery with the damage field summed over pair centres), which would upper-bound the joint profit and convert this measured verdict into a configuration-free one. Nothing about the mechanism suggests the greedy is far from the sup — the collapse of chi is geometric (band overlap), not adversary-weakness — but that is an argument, not a bound.
• Mixed-depth lattices are covered only through the battery shapes; the depth-split machinery of level 5 applies but has not been rerun jointly.
• No proportion is computed. The Phase-7 gate stays closed behind the joint cap dual and the full-ladder hardened scan.

Controls ledger

Reproduction

.venv/bin/python hunts/frontier_math/pair_density.py        # ~2 s
.venv/bin/python -m pytest -q -o addopts='' \
    hunts/frontier_math/test_pair_density.py                # ~5 s