Level 6a: theta_full = 0.02 — the overdraw closed

Disposition

Level 5 ended with the combined per-pair budget overdrawing by 0.151–0.387 of slack at every depth and two named reversing estimates. Both landed:

theta_full = 0.02  at unit pair density,
ball-hardened caps at every probe depth, worst combined budget 0.9991,
shallow pair charge exactly 0.

The directive's milestone equation — rigorous single-pair theta + global multi-pair closure ⟹ theta_full > 0 — now holds with the labels below, which are the honest boundary of the claim:

• caps: ball-hardened pentadiagonal counting bound (configuration-free on the on-line layer), evaluated at seven probe depths spanning the strip; the full-ladder hardened penta scan is the named compute follow-up (the shallow cells close at 0.57 with charge 0, far from binding);
• eta: measured-exact on its adversary family (single-worst-depth ordinate lattices at density nu_p = 1, worst phase, optimal split) — the configuration-free pair layer is the named level-6b task, with the dense pair lattice as its kill control.

The Phase-7 gate (reconnect the gap floor, compute the proportion) remains closed until 6b; nothing here computes a proportion.

Lever 1 — the pentadiagonal counting dual

The tridiagonal chain undercharged mid-range pairs: an adjacent-cell pair at distance 0.2 paid min omega^2 over [0, 0.5] = 0.315 against a true 0.77 — the min sits at the far end of a too-wide range. The refinement keeps the counting theorem's shape and narrows the ranges. Cells of width delta with three charges:

same cell        (d < delta)          : K1 = min omega^2 [0, delta]
adjacent cells   (d < 2 delta)        : K2 = min omega^2 [0, 2 delta]
second neighbours (delta < d < 3delta): K3 = min omega^2 [delta, 3 delta]

with 3 delta <= 0.9 keeping every range inside omega's first positive stretch, and the value solved exactly by a pentadiagonal DP (state = counts in the last two cells, the third-range interaction folded in as an upper envelope). Controls: with K3 = 0 it reduces to the level-4 chain exactly (< 1e-6 at delta = 0.3125); more charges only lower the value; and the bound still dominates every measured adversary of levels 3–5.

Measured caps (theta = 0, as fractions of slack, hardened):

The mid-depth gains (x1.26 at y = 0.25 over the tridiagonal, on top of the level-5 numbers having been conservative assumptions) are what close the budget; y = 0.45 is the binding depth with 0.0009 to spare.

Lever 2 — the optimal charge split, and a bug the controls caught

The split of each negative pair-pair term between its participants is free in the accounting, and eta is a max over rows, so the right split is a min-max problem. Solved by iterative reweighting toward equalised row ratios:

Recorded because it is instructive: the first optimiser raised eta to 0.75–0.78 — it summed worst-case lattices over all neighbour depths simultaneously, a density-6nu_p adversary. The ordinate density budget is nu_p in total, and the row masses are linear in per-depth counts, so the worst mix is the single worst depth: the charge is a max, not a sum. With that fixed, the min-max equalises at ~0.30 across the strip and the shallow end pays exactly zero, which is the depth-graded design doing its job.

The verdict

Float scan (fine theta grid):

Ball-hardened at theta = 0.02, all probe depths:

y = 0.01: 0.571   y = 0.05: 0.861   y = 0.15: 0.889   y = 0.25: 0.896
y = 0.35: 0.971   y = 0.45: 0.9991  y = 0.49: 0.980         all closed

theta_full = 0.02 is deliberately not optimised further (the directive's operating rule); its value is that it is positive. The binding margin at y = 0.45 is 9e-4 of slack — thin, and stated rather than rounded away.

What stands between this and Phase 7

1. Level 6b: the configuration-free pair layer — the counting dual on the T kernel at mixed depths, against the dense-pair-lattice kill control. The eta side of today's verdict is measured on its adversary family, not yet dual-bounded.
2. The full-ladder hardened penta scan (compute, not mathematics): the probe grid spans the strip and the shallow cells are far from binding, but the graduation pipeline wants the ladder.
3. Then Phase 7: the gap floor, every truncation/taper/census term one-sided, and the first computation of whether the retained theta_full * Delta_gap clears the error budget — the question the whole hierarchy exists to earn.

Controls ledger

Reproduction

.venv/bin/python hunts/frontier_math/penta_bound.py          # ~1 min
.venv/bin/python -m pytest -q -o addopts='' \
    hunts/frontier_math/test_penta_bound.py                  # ~45 s

The hardened assembly is penta_bound.hardened_assembly(theta=0.02) (~75 s). Deterministic throughout.