Level 3: theta > 0 at the single-pair reduction, and why

The milestone, answered

THETA > 0, OR AN EXACT REASON THETA MUST STILL BE ZERO.

Theta is positive at the single-pair reduction, by a wide measured margin (the scan stays safe through theta = 0.9), and the mechanism is exact. Before levels 1–2 the adversary had two unbounded resources: per-pair incidence magnitude (the scalar family's m -> infinity) and free reuse of one pair across many cells. Laws D–H priced both, and level 3 can now name the two structural facts that make a positive trade possible where the interaction-control audit found none:

• deep pairs are self-defeating — damage is linear in sigma^2 while the pair's own retained Frobenius slack grows like 8 sigma^4 (LAW K);
• dense packing is self-defeating — multiplying damage nu-fold manufactures internal R(P) mass quadratically (LAW G), of which the adversary must leave (1-theta) on the table.

Both self-defeats are scale-matched (damage² ~ σ⁴ against slack ~ σ⁴; damage ~ ν against packing ~ ν²), which is why the measured recovery range is nu-free and sigma-free — the Phase-4 classification is theta -> theta_0 > 0 at this reduction, not a decaying coefficient.

LAW K — the exact pair spectrum (new, and the level's engine)

For a pair u = x + iy at depth y, LAW D pins the bilinear square u.u = 1 — real — which forces x . y = 0 exactly; the Hermitian identity gives |x|^2 + |y|^2 = 1 + 2 sigma^2(y). Together:

|x|^2 = 1 + sigma^2,   |y|^2 = sigma^2,   x  |  y,
spec( 2(xx^T - yy^T) ) = { 2(1 + sigma^2(y)),  -2 sigma^2(y) }   exactly.

The paper knew the signature (1,1); the eigenvalues pinned by depth are new. Checked against the actual grid to machine precision (x.y ~ 1e-16, eigenvalues to 8 digits at three depths). Two consequences:

1. The pair's negative eigenvalue is the depth envelope -2 sigma^2 — LAW E's per-cell floor reappears as spectral data.
2. Against the baseline's flat charge 4 (Lemma 3.2 at c = 2, eigenvalue-wise), the pair retains slack exactly 8 sigma^2 + 8 sigma^4. This is the security level 3 spends; the baseline never touches it.

The three-zero lemma (proved, unconditional, theta = 1)

From LAW I alone (2W >= -2(1+m0) sigma^2 per cell) and LAW K's slack:

a pair with at most three on-line zeros in its negative cells has
net >= (8 - 6(1+m0)) sigma^2 = 0.7177 sigma^2 > 0,

placement-free, depth-free, retaining all of R(P). The adversary is forced to field at least four zeros per pair — i.e. into the density regime where the quadratic packing costs live. This small statement is the level's proved fragment and would be the natural first Lean target.

The worst-case cancellation problem (Phase 2), solved at one pair

Objective, per pair, in normalised units:

net(theta) = (1-theta) R_int(X) + 8 sigma^2 + 8 sigma^4 - D(X),
D(X) = -2 sum_{x in X} W(x, y),    R_int(X) = 2 sum_{x != x'} omega(x-x')^2,

minimised over on-line configurations X (any size, any placement, respecting the density cap) and over the depth. Two adversary families bracket the landscape:

• dense lattices (spacing ladder 2.0 down to 1/16, keeping only negative cells — the continuum of the level-2 escaping family);
• greedy sparse placements (marginal-gain, min-spacing 1/nu).

Measured worst net over both families and depths y in [0.05, 0.49]:

The bottoming configuration is the shallow-depth limit, where slack and damage scale to zero together and the ratio stays safe (net ~ (8 - D/s2) s2 with D/s2 well below 8). The dense attack is genuinely dangerous — at y = 0.35, spacing 1/16, damage 24 sigma^2 beats the bare slack 13 sigma^2 — but its internal mass is 1274 in the same units: even retaining 90% of R(P), the remaining 10% drowns the surplus. Theta = 1 fails in exactly this regime (test_theta_equal_one_fails_in_the_dense_ regime), which is the scan's power control: the instrument can see the failure it is claiming to exclude.

The dual-certificate seed (Phase 3)

The internal form's kernel is omega(r)^2, whose Fourier transform is (phi^2 * phi^2)-shaped: nonnegative, supported in the band (measured: positive at every in-band frequency, zero beyond). So the packing quadratic form is positive semidefinite, the adversary's value

Lambda(theta, y) := sup_X [ D(X) - (1-theta) R_int(X) ] / sigma^2(y)

is finite for every theta < 1, and the per-pair inequality has the closed shape

net >= (8 - Lambda(theta)) sigma^2 + 8 sigma^4.

The proof object level 3 leaves for the graduation step: an upper bound on Lambda via this positive-definiteness (a band-limited moment problem), replacing the measured sup over adversary families. The measured content: Lambda stays below 8 through theta = 0.9; the dense lattice breaks even against (1-theta) R_int only near theta ~ 0.99.

Pair–pair terms: measured, partially covered, and the named gap

The full Phase-2 problem couples pairs. Measured here:

• stacked pairs (same ordinate) interact positively (T = +8.94 at y = 0.3 twice; +25.6 deep) — stacking is self-defeating outright;
• dipoles: the worst interaction over ordinate offsets is negative (-2.84 at dt = 0.68, y = (0.3, 0.3)) but covered by the two pairs' own slacks with factor 2.8–6.7, uniformly over the tested depths.

Not delivered: the joint inequality with a slack partition proving that one pair's security is never spent twice (once against on-line damage, once against each neighbour pair). The dipole cover factors say the budget exists; the partition is bookkeeping plus a pair-density argument (pairs also obey LAW H), and it is the named missing estimate of this level.

Phase 5 — the extremal battery

Classification, per the directive

OUTCOME A at the single-pair reduction — candidate, not yet promoted. A positive recovery coefficient exists after worst-case single-pair cancellation, with no error term in the reduction, a proved fragment (the three-zero lemma), a proved finiteness certificate shape, and the failure mode at theta = 1 exhibited. Two steps separate this from the directive's full OUTCOME A, and both are named rather than blurred:

1. a proved upper bound on Lambda(theta) (the measured sup is over two adversary families, not all configurations);
2. the multi-pair slack partition (dipole factors measured 2.8+, partition not written).

The Phase-6 gate stays closed: no proportion is computed, nothing is fed back into the gap machinery, and the withdrawn 0.672529 is not revisited. Per the operating rule, the reward claimed is the first kind only: a strictly positive, twice-self-defeating cancellation bound at one pair, with the exact reason it could not have existed before levels 1–2.

Controls ledger

Reproduction

.venv/bin/python hunts/frontier_math/theta_recovery.py
.venv/bin/python -m pytest -q -o addopts='' \
    hunts/frontier_math/test_theta_recovery.py

About 26 s and 23 s. Adversary optimisations are deterministic (lattice phase scans and marginal-gain greedy); no random search enters any number.