Level 4: the counting dual — theta > 0, configuration-free

Disposition

Level 3 left two named gaps in promotion order: a bound on the adversary's value valid over all configurations (its worst case was measured over two families), and the multi-pair slack partition. This level closes the first with an elementary counting theorem whose one-sided evaluation delivers

secured theta* = 0.1   (grid; worst depth-cell margin +1e-5, which is
                          ~10% RELATIVE to that shallow cell's slack)

for the level-3 per-pair inequality with the adversary side now configuration-free, and reduces the second to a measured frame with the remaining step named. The gap between the secured 0.1 and level 3's measured 0.9 is the counting bound's documented looseness (dropped non-adjacent payments, cell sups), not a change in the mathematics.

The theorem (the "rational dual object" of the hierarchy row)

Partition the offset line into cells of width delta. For any finite configuration X:

• same-cell pairs are within delta, so each pays internal mass at least K_delta = min over [0, delta] of omega^2;
• adjacent-cell pairs are within 2 delta, so each pays at least K_{2delta};
• non-adjacent payments are dropped — one-sided in the adversary's favour.

With F_j >= sup over cell j of f+ (f = -2W is the damage) and c = 1 - theta, the adversary's value is at most

max over n_j >= 0 of  sum_j [ n_j F_j - c K_delta n_j(n_j - 1)
                              - c K_{2delta} n_j n_{j+1} ],

a tridiagonal chain program solved exactly by dynamic programming over the cell chain. The chain term is what secures: without it (equivalently, delta past the first zero of omega, where K_{2delta} = 0) the bound's free density is 1/delta and the securing scan fails by an order of magnitude (test_chain_charge_beats_the_plain_bound: 19.2 vs the plain bound at y = 0.45). At theta = 1 every charge vanishes and the cap is infinite — matching level 3's dense-regime failure exactly.

One-sided cells (the "one-sided continuum cell bounds")

Cell sups of the damage come from closed-form centre values of C = Re Phi2(g+iy), S = -Im Phi2(g+iy) inflated by elementary Cauchy–Schwarz Lipschitz constants:

|dS/dg| <= (L/2) aL sigma(y)     |dC/dg| <= (L/2) aL E(y)
|dS/dy| <= (L/2) aL E(y)         |dC/dy| <= (L/2) aL sigma(y)

so f+ <= 4 (Sbar^2 - Cund^2)+ / (aL)^2 holds on the whole (g, y) cell. The sup-evaluation grid (step 0.0125, inflation ~2–4% of |S|) is decoupled from the capacity partition delta — coupling them was the first implementation's fatal looseness. Beyond the fine region the far tail uses the depth-scaled majorant

|S| <= psi_S(y)/g^2,
psi_S = (c_rho/w) sinh(yL/2) + 4 y cosh(yL/2) + y^2 aL sinh(yL/2)

(||(phi^2 sinh(y.))''||_1 via the paper's (2.13) norms), which vanishes linearly in y — a depth-blind tail majorant would silently fail the shallow depth cells, where the slack vanishes too. The depth ladder is geometric with ratio 1.18 refined to 1.04 from y = 0.09, and the slack is always evaluated at the shallow lip of each depth cell while damage is inflated to the deep lip.

The soundness incident, recorded because it is instructive

The first implementation failed its own projection control: the measured level-3 lattice adversary exceeded the "bound" at two depths. The leak was a factor-2 error in the damage (f = -2W carries factor 4 against the C/S split, not 2) hidden inside otherwise-plausible margins. The control that caught it is the hierarchy's own level-4 row — projection to every lower level must reproduce its controls — and it is now a permanent test (test_bound_dominates_level3_adversaries). A bound that had only been compared against its own cells would have shipped wrong.

Measured record

• Domination (projection rule): measured level-3 profits 0.138 / 1.889 / 6.998 at y = 0.1 / 0.3 / 0.45 sit under caps 0.191 / 3.901 / 19.230. The level-2 +-g* cluster and the single-point LAW I reduction are also dominated (tests).
• Lesion: dropping the Lipschitz inflation produces genuine sup violations (0.115 / 0.347 / 0.725 at the three probe depths) — the inflation is load-bearing, not decorative.
• Tightness: at y = 0.1 the cap is within 39% of the measured adversary; at y = 0.45 within 2.7x. The looseness is concentrated at depth, where the dropped non-adjacent payments matter most.

Gap 2: the halved-slack partition frame

Charging each pair-pair interaction half to each participant against half of each pair's 8 sigma^2 + 8 sigma^4 slack:

Stacked pairs (coincident ordinates) are positive (+8.94, +25.58), so pair-density concentration is self-defeating, as on-line density was. The remaining step is the pair-pair analogue of this module's cell bound at combined depths y ± y' — the same machinery, one more layer — and it stays named, not claimed.

What is and is not delivered

Delivered: the counting theorem (exact, elementary); its one-sided evaluation securing theta = 0.1 configuration-free at the single-pair reduction; the chain mechanism identified as the load-bearing sharpening; the projection/lesion control suite; the partition frame with measured coverage.

Not delivered, named in promotion order: (i) the arb-enclosure pass over the same finitely many cells (evaluation is double precision with the inflation margins doing the one-sided work — the binding margins are ~10% relative, far above float noise, but the enclosure pass is what the graduation pipeline requires); (ii) the theta gap 0.1 → 0.9 (the moment problem over the psd kernel FT[omega^2] >= 0, which the counting bound deliberately avoids needing); (iii) the multi-pair closure. No proportion is computed; the Phase-6 gate stays closed.

Reproduction

.venv/bin/python hunts/frontier_math/counting_bound.py      # ~5.5 min
.venv/bin/python -m pytest -q -o addopts='' \
    hunts/frontier_math/test_counting_bound.py              # ~40 s

Deterministic throughout: fixed grids, exact DP, no random search.