35. The unspent fact: what out-of-band positivity is worth, and why nobody can claim it

Hunt #110, hunts/outband_intake/. Read hunts/outband_intake/RESULTS.md for the measurements and the doors. This page is the front door: what the thread is, why it exists, and what a session picking it up needs to know before running anything.

Grade: measured, throughout. No theorem is claimed here and nothing on this page bears on RH (docs/08).

1. The situation

Every certificate in the public race for the proportion of simple on-line zeros of zeta reads pair-correlation data on [-1, 1] and nothing else. Call that the band. The configuration ceiling for that information class is 0.6818286874638, proved in the upstream development, and the entire public race sits between 0.6725007036794116 and 0.6734164909714992949. The race has therefore moved the record by 9.2e-4 in total.

In 2023 Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh proved (arXiv:2306.04799, Theorem 1) that the weighted form factor is nonnegative for every alpha, unconditionally. Conjugate-closure of the zero multiset writes it as an integral of a square, and the Cauchy weight's poles at ±2i cover every pair of strip zeros. This is information from outside the band, it costs no hypothesis, and this laboratory re-derived it before finding it (hunts/frontier_math §4 records that so nobody derives it a third time).

Nobody had ever measured what it is worth. That is what Hunt #110 did.

2. What it is worth

+0.0068, landing the configuration class near 0.6793.

The measurement is two linear-programming ladders on matched grids: one with out-of-band positivity enforced, one with band data alone. Both descend toward their limits from above, because the discretisation restricts the adversary. The in-band ladder is the control and the calibration, since its limit is known to be the record itself, so the same extrapolation applied to it returns the method error, 2.2e-3. The difference extrapolates to +0.0068, about three times that error.

For scale, that is roughly seven times the whole public race's progress. It also lands essentially on top of Chirre, Goncalves and de Laat's 0.6792, which is reached under RH. The same value falling out of unconditional inputs is the interesting part: it suggests RH is doing work in their proof that it may not be doing in the truth.

3. Why it is unspent

The only known technique for converting that information into a theorem runs an inertia count, and the count needs the evaluation form to be positive semidefinite. That forces the window's spectral density v = phi^2 to be nonnegative, and the pair weight in alpha-space is its self-convolution Khat = v * v, nonnegative everywhere by identity. The framework never has a free ghat. It has a free v >= 0, and v |-> v * v cannot produce a negative value.

The requirement does not weaken. Under an indefinite form the rank half of the lemma survives, needing no definiteness, and the inertia half fails, needing a square root. The witness is small enough to check by hand: one off-line pair with Q = t·[[0,1],[1,0]], S = diag(1,-1), c = 2 gives slack +2.0 at t = 1 and -0.5 at t = 1.5. Against 4000 random positive-semidefinite pairs the inequality was never violated.

So the value lives at a kernel that is not a square, and the machinery only builds squares.

4. Why this is a gap and not a wall, and why that distinction is the point

hunts/frontier_math §2 records a genuine wall: passing the band by widening it needs an unconditional upper bound on prime pair correlations, the loss is (C-1)·T^(lambda-1)·N/log T for any constant C > 1 + o(1), and the required input is Hardy-Littlewood grade. That is a famous open problem and no amount of cleverness in this laboratory opens it.

Hunt #110 is not that. The information is already in hand, already unconditional, and already measured to be worth something. What is missing is a construction. A gap of that shape is fundable in a way a wall is not, and telling the two apart is the main reason this page exists.

5. The one measurement that changed the shape of the problem

Sweeping how far past the band the positivity is enforced:

It saturates almost at once. Nine tenths of the value is inside (1, 1.5].

That shrinks the construction problem materially. The object nobody has does not need to be signed on a half-line. It needs to be signed on a narrow strip just outside the band. Any attempt on §3's obstruction should take that form, and a session that starts by trying to build a kernel negative everywhere is solving a harder problem than the one that pays.

6. What bounds the result, stated rather than buried

Three things, all in RESULTS.md at length.

The in-band data's provenance is unaudited. Until someone traces it to an unconditional source rather than to Montgomery's RH-conditional theorem, 0.6793 is a class value and not an unconditional one. This is the cheapest next step and it gates whether any of the rest is worth funding.

Gate #3 fires. The isolation input passes on the Davenport-Heilbronn function and on both discriminant -23 Epstein forms, so it is prime-blind. That does not kill a counting bound, since Davenport-Heilbronn has simple zeros to count and the gate exists to catch claims that explain RH. What it does establish is that all arithmetic content sits on the prime side: a hunt working the zero side can re-optimise a certificate but can never add arithmetic. That bounds what §4's gap can ever be worth.

An argument was withdrawn. A ratio diagnostic was stated confidently in the first draft and is wrong: over the measured range both ladders decay at similar rates, so the ratio is flat under either hypothesis and carries no information. The difference test with a calibrated error replaced it and reversed the verdict. Recorded because the withdrawn version nearly shipped as a negative result.

7. Where to pick it up

hunts/outband_intake/ carries MISSION.md with the kill conditions, RESULTS.md with the doors ranked, RUNS.md with the estimate written before launch, and the two sweep scripts with their checkpointed artifacts. The doors section ranks three:

1. The depth-bound lead, and the follow-up should take this one. Every planted off-line configuration left the isolation step holding with slack +2.01 to +2.67, and the failure mode arrives exactly when a planted zero leaves the strip. If the step survives under a bound on off-line depth, existing machinery consumes the information with no new kernel class. Weakly evidenced, cheap to sharpen, and it goes through the obstruction rather than around it.
2. The provenance audit, cheapest, and it gates everything.
3. The kernel-class construction, the largest prize and the hardest, now known to need signing only on (1, 1.5].