Level 5: the enclosure pass and the multi-pair energy

Disposition

The directive set two promotion gates: rigorous enclosure of the level-4 cells, and closure of the general multi-pair interaction. Gate 1 is delivered: the level-4 counting dual re-evaluated end to end in ball arithmetic holds at its original resolution — with an instructive scare recorded below. Gate 2 is half-delivered and half-quantified: the exact multi-pair algebra collapses onto the single-pair kernel (LAW L), every collective mode tested survives, per-pair charging works at unit pair density — but the combined per-pair budget (on-line cap + pair charge) overdraws at every depth by a measured 0.15–0.39 of slack. That is the directive's OUTCOME B, with the deficit and the two candidate reversing estimates named exactly.

theta_full > 0 is therefore not yet claimed. The decimal gate stays closed.

Gate 1 — the hardened scan, and what almost went wrong

enclosure_pass.py re-evaluates every kernel-touching quantity of the level-4 scan in acb/arb ball arithmetic (128-bit): the closed-form Phi2 with an explicit geometric tail bound on the moment series, exact integer ramp coefficients, exact rational aL, and directed endpoints everywhere — damage sups from upper endpoints, K_delta and slack from lower endpoints, the chain DP total inflated by 1 + 1e-9 (covering its double-precision arithmetic on directed inputs; a fully rational DP would be possible and pointless at these margins). Ball radii at representative cells: 1e-16 to 1e-20, versus binding margins of ~10% relative.

The scare. A first hardened run at a 2x-coarser sup grid (step 0.025, 4 depth slices — chosen to halve runtime) FAILED the scan by up to −3.0 at theta = 0. Probing the failing cells at the original level-4 resolution (step 0.0125, 6 slices) restored healthy positive margins:

The Lipschitz inflation scales with the sup-grid step, so coarsening the grid is not a free economy — it is a change in the theorem's constants. The failure was the economization, not the balls and not the mathematics; the module's defaults are now pinned at the original resolution and the coarse failure is kept as a documented negative control.

The gate-1 record. The full hardened depth-ladder scan at original resolution (~30 minutes):

theta = 0.00: worst hardened margin >= +0.000011   OK
theta = 0.05: worst hardened margin >= +0.000011   OK
theta = 0.10: worst hardened margin >= +0.000011   OK
hardened theta* (grid) = 0.1

The level-4 value survives enclosure exactly, with the same shallow binding cells and the same ~10%-relative margins, now carried by directed ball endpoints rather than double-precision goodwill.

Gate 2 — LAW L and the exact multi-pair energy (Phase 2)

For pairs at ordinates t_r and depths y_r, the off-line block's Frobenius energy is exactly

||Qhat||_F^2 = sum_r [4 + slack_r] + sum_{r != s} T(t_r - t_s, y_r, y_s),

with each self term pinned by LAW K, and the cross term given by the new exact identity

LAW L:   T(dt, y, y') = W(dt, y - y') + W(dt, y + y')

— the single-pair signed kernel at the difference and sum depths (both slots are instances of LAW D's complex Poisson identity; measured defect against the independent route: 8.9e-16). The only sign-indefinite terms in the total energy are the on/off cross (controlled by the level-4 counting dual) and these T terms. Three structural consequences, each measured:

• the difference layer is nonnegative: W(dt, 0) >= 0 (min +6e-10 over the probe range), so equal-depth pairs interact negatively only through the depth-scaled sum layer — shallow-shallow negativity vanishes as sigma^2(2y) (measured inside the (1+m0) sigma^2(2y) envelope);
• negativity is depth-split: sigma^2(y ± y') <= 2 cosh^2(y'L/2) sigma^2(y) + 2 cosh^2(yL/2) sigma^2(y'), so every negative T is chargeable to its participants in proportion to their own sigma^2 — a shallow pair is charged asymptotically nothing (split weight < 0.02 at depths (0.01, 0.45));
• the far tail is depth-scaled: only S^2 pushes W negative, so |T|_- <= 2[psi_S(|y-y'|)^2 + psi_S(y+y')^2]/(dt^4 (aL)^2).

Phases 3–4 — the partition, measured, and the collective battery

eta(nu_p) = worst per-pair charge ratio over adversarial ordinate lattices, with the proportional depth split:

Resolved by depth at nu_p = 1: eta runs 0.30 (deep) to 0.49 (shallow).

The Phase-4 collective battery — stacked columns, alternating-depth dipoles, staggered lattices, shallow/deep sandwiches, strip-edge clusters, periodic pair words — survives at both nu_p = 1 and nu_p = 2 (worst cross/slack = −0.876, all totals positive). So the nu_p = 2 failure is a failure of the pairwise charging scheme, not of the energy: exactly the situation level 4 met on the on-line layer, where per-cell reasoning needed the chain counting dual. The dense pair lattice at spacing ~0.6 is promoted to the level-6 kill control, and the level-6 task is the counting dual on the T kernel at mixed depths (2D cells).

Phase 6 — theta_full: OUTCOME B, with the deficit quantified

The per-pair linear assembly requires, at every depth,

(level-4 cap)/slack + eta(y) <= 1.

Measured, at nu_p = 1:

The budget overdraws everywhere: theta_full > 0 does not follow from stacking the current bounds, and saying otherwise would be the sham the referee exists to catch. The deficit is 15–39% of slack, and the two estimates that can reverse it are named with their measured headrooms:

1. the level-4 caps are loose by 1.4x (shallow) to 2.7x (deep) against measured adversaries — the dropped non-adjacent payments; recovering half of that looseness clears the deficit at every depth;
2. the pair charge is the crude pointwise split, loose by ~2x against the measured dipole totals (e.g. worst dipole −2.84 against a charged prediction of −5.7 at (0.3, 0.3)).

Either sharpening suffices; both together clear the budget with a margin comparable to the deficits. This is the precise sense in which the directive's OUTCOME B applies: multi-pair recovery survives structurally (collective battery, eta < 1 at unit density), and the loss is in the stacked bounds, not the energy.

Definition-of-done classification

• Gate 1: delivered (hardened at original resolution; the coarse-grid failure recorded as a negative control, not smoothed over).
• Gate 2: OUTCOME B — positive theta at the single-pair level survives hardening; the multi-pair assembly overdraws by a quantified 15–39% of slack; the two reversing estimates are named and their measured headroom exceeds the deficit.
• OUTCOME C content also produced: the dense pair lattice (nu_p = 2, spacing in the T-kernel's negative band) defeats pairwise charging while collective energy survives — pinned as the level-6 kill control.
• The decimal gate (Phase 7) is not entered; no proportion is computed.

Controls ledger

Reproduction

.venv/bin/python hunts/frontier_math/pair_energy.py          # ~30 s
.venv/bin/python hunts/frontier_math/enclosure_pass.py       # ~30 min
.venv/bin/python -m pytest -q -o addopts='' \
    hunts/frontier_math/test_level5.py                       # ~45 s