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.