outband_intake: the information is worth between 0.005 and 0.009, and no known certificate can spend it

Verdict: the kill condition did NOT fire, and the hunt found a gap rather than a wall. The unconditional out-of-band positivity of Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh (arXiv:2306.04799, Theorem 1) is worth between about +0.005 and +0.009 at the measure level, landing the configuration class somewhere in [0.679, 0.682]; the third digit is not determined by the data. Separately, and this is the point of the hunt, the certificate machinery that would turn that into a theorem provably cannot reach it. What separates them is one structural obstruction, stated below with a counterexample.

Grade: measured, and only that. Two LP ladders on matched grids with a calibrated extrapolation. Nothing is kernel-checked, no theorem is claimed, and nothing here is evidence for or against RH (docs/08).

1. The measurement

Five matched rungs, once with out-of-band nonnegativity enforced on (1, 3] and once with bandwidth-one data alone. The discretisation restricts the adversary, so both ladders descend toward their limits from above.

The X = 240 out-of-band rung took 10598 s on a laptop, against a pre-run estimate of 866 s; see RUNS.md. The in-band ladder was extended alone, being roughly 300 times cheaper: 0.6756339 at X = 320.

The difference is the signal, and the method is calibrated before it is believed. Extrapolating d(X) = v_out - v_in by a + b·X^(-p) gives a = +0.0065. The same fit applied to the in-band excess over the record, whose true limit is known to be zero because the in-band LP measures the Montgomery-Taylor dual, returns +0.0018. That is the method error. The gain exceeds it by about three and a half times, so zero is excluded, though not overwhelmingly.

What the data does not determine is the third digit, and an earlier draft of this file claimed it. The draft read the difference-route value 0.6793 as sitting on Chirre, Goncalves and de Laat's RH-conditional 0.6792. Fitting the out-of-band ladder directly, rather than through the difference, gives 0.6815, one digit from the bandwidth-one configuration ceiling 0.6818. Two fits of the same five numbers land on two different famous constants, so neither landing is evidence of anything. refit.py reads the artifacts and prints all of it:

The honest statement is a range: the class value lies in about [0.679, 0.682] and the gain over the record in about [0.005, 0.009]. A three-parameter power law through five points does not fix the limit better than that, and the in-band ladder, whose limit is known, says the same: its own free fit misses zero by 0.0016 on six rungs. The coincidence with CGdL is withdrawn; the fact that the LP reaches the neighbourhood of the RH-conditional value from unconditional inputs survives, and that is a weaker and more defensible sentence.

A ratio test was tried first and is withdrawn. Comparing excesses over the record gives 2.808, 2.714, 2.780, 2.869, and an earlier draft of this file read that flatness as proof of a common limit. It is not. Over this range the two ladders decay at nearly the same rate, so the ratio is flat under both hypotheses and carries no information; the slight rise across the range is in fact what a positive limit predicts. The difference test with a calibrated error is the right instrument. Recorded because the withdrawn argument was stated confidently and was wrong.

Solver robustness: the out-of-band solve at X = 80 agrees to seven digits across highs, highs-ds and highs-ipm. The in-band ladder independently reproduces the published values in frontier_math/RESULTS-frontier-math.md §1.

1b. The information is local: 91% of it sits in (1, 1.5]

Lane B holds the grid at X = 80, J = 320 and sweeps how far past the band the positivity constraint is enforced.

It saturates almost at once. Three fifths of the value arrives by alpha = 1.25 and nine tenths by alpha = 1.5; doubling the reach from 1.5 to 3.0 buys another 7.5e-4. The 5.0 and 8.0 rungs were not reached before the run was stopped, and on this curve they cannot matter.

This is the most useful thing lane B could have said, and it changes the shape of the construction problem in §2. A certificate does not need to read far past the boundary. It needs to read a narrow strip just outside it, roughly alpha in (1, 1.5]. Whatever object eventually consumes this information has to be signed only on that strip, which is a much smaller demand than a kernel signed on a half-line, and it is the form any attempt on the §2 obstruction should take.

2. Why no certificate can spend it

The inertia argument that would consume out-of-band information 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. The source paper's own functional carries int |Khat|, and v >= 0 is exactly what discharges that modulus (hunts/frontier_math/paper_pin.py).

The requirement cannot be weakened to something a signed kernel satisfies. Replacing the Frobenius inner product by a weighted tr(X^H S Y S) with indefinite S keeps the rank half of the lemma, which needs no definiteness, and breaks the inertia half, which needs S^(1/2). Minimal counterexample: 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, worst slack +0.001321, so the instrument would have seen a violation had one existed.

The first-order perturbation law says where the gain lives. Adding out-of-band mass -eps where F >= Lbar moves the bound by dJ/deps = (J - Lbar)/g(0), which is adverse for every Lbar below about 1.3275. BGSTB gives Lbar = 0, so the direct channel pays nothing. The entire gain is indirect: negative out-of-band mass relaxes the constraint and licenses an in-band profile that is not an autocorrelation. That is precisely the profile the inertia framework cannot represent.

3. So the result is a gap, not a wall

The information supports somewhere between 0.679 and 0.682. Every known certificate construction is confined to 0.6725007. The difference is not a missing computation and not a frozen constant. It is that the only proof technology available insists on kernels that are squares, and the value lives at a kernel that is not one.

That gap is the research problem this hunt hands forward, and it is the one shape the operator asked for: not a ceiling, not a bridge over somebody's theorem, but a construction nobody has. It also says something about why the CGdL value is conditional. They reach 0.6792 with an isolation input that costs RH; the LP reaches the same neighbourhood, within the §1 range, with unconditional inputs, which suggests RH is doing work in their proof that it may not be doing in the truth. That is a suggestion and not a measurement, since §1 cannot tell 0.6792 from 0.6815.

4. One open question that bounds this entire file

Whether the LP's constraint set is unconditionally valid for zeta's zeros has not been checked here, and it is load-bearing. The out-of-band constraint is BGSTB and is unconditional. The tau >= -1 bound is rho >= 0, the nonnegativity of a pair density, true of any point process and costing no hypothesis. An intermediate report claimed that bound smuggles in the diagonal-isolation drop and cited frontier_math/CLEAN-KILL-REPORT.md; that document concerns a different object, the -2 off-line block interaction in the rank-trace inequality, and does not support the claim. The in-band pair-correlation data is the one not audited, and until someone traces it to an unconditional source rather than to Montgomery's RH-conditional theorem, the §1 range is a class value and not an unconditional one. That audit is the cheapest next step and it decides whether §3's gap is worth funding.

5. Gate #3 fires, and what it does and does not mean

This hunt's own MISSION.md names it a kill condition: any candidate argument that also passes the Davenport-Heilbronn control distinguishes nothing. The battery was run after the rest of this file was written, and it fires.

riemann_zeta:        True
davenport_heilbronn: True     (10 zeros in the box)
epstein_2_1_3:       True
epstein_1_1_6:       True
shifted_product:     VOID     (see below)

The diagonal-isolation drop holds for the Davenport-Heilbronn function, which satisfies a Riemann-type functional equation and has zeros off the critical line, and for both discriminant -23 Epstein forms. It is prime-blind.

It does not kill the measured result, and the reason is a distinction worth keeping straight. Gate #3 exists to catch a structural property claimed to explain RH. The isolation drop claims nothing of the kind: it is a counting input, and counting the simple zeros of a function is meaningful whether or not that function's zeros lie on a line. Davenport-Heilbronn has simple zeros to count. So the drop passing on a rival is the expected behaviour of a counting tool, not the signature of a vacuous one.

What it does establish is strategic and it is not comfortable. Both of CGdL's zero-side inputs are prime-blind. Every piece of arithmetic content in this whole construction therefore sits on the prime side, in the in-band theorem. A hunt working the zero side can re-optimise a certificate; it cannot add arithmetic. That is independent evidence for the reading in §2, and it bounds what §3's gap can ever be worth.

The shifted_product row is void, not a finding. The harness returned False where it should have returned "undefined": that function had zero sign changes in the test box, so the claim was never evaluated on it. Recorded because the void row would otherwise read as the drop failing on a uniformly displaced zero set, which is not something this run established.

One lead, flagged as weak. The drop also survived every planted off-line configuration tried: zeta plus a pair at 1/2 ± 0.05, 0.2, 0.5 + 20i gave slack +2.01, +2.08, +2.67, and the Cauchy weight's pole at 2i is reached exactly when a planted zero leaves the strip, which is the mechanism this tree already cites for why BGSTB holds. That hints the drop may survive unconditionally under a depth bound on off-line zeros, which would be a route through the obstruction rather than around it. It is weak evidence: four to thirteen zeros, one box, one planted pair at a time, and the bulk-displacement case went untested for the harness reason above.

5b. A disagreement left standing

The adversary held, twice, that the LP earns its 0.0068 through rho = 1 + tau >= 0 and that this constraint is the diagonal-isolation drop, which would make the class value RH-conditional rather than unconditional. This file does not accept that, for the reason in §4: rho >= 0 is the nonnegativity of a pair density, true of any point process, and the document cited in support concerns the -2 off-line block interaction in the rank-trace inequality, a different object. The disagreement is recorded rather than resolved because it is decidable by the §4 provenance audit and should be settled by that audit rather than by whoever writes the last file.

6. Cost

About five hours of laptop compute, three of them the single X = 240 out-of-band rung, no CI, no formalization. A negative would have been reached before anything expensive was funded; so was a positive.

The doors

Active constraints at the optimum. One binds: the certificate kernel must be an autocorrelation. Everything in §2 is that constraint seen from a different angle, and §1 measures what it costs, between 5e-3 and 9e-3.

Frozen-constant inventory.

| off-line depth, taken as unbounded | never varied | The §5 lead, and the only door that goes through the obstruction rather than around it. Every planted off-line configuration left the isolation drop holding with slack +2.01 to +2.67, and the Cauchy weight's pole at 2i is reached exactly when a planted zero leaves the strip. If the drop survives under a depth bound on off-line zeros, existing machinery consumes the information with no new kernel class. Weakly evidenced, cheap to sharpen. |

Information class. The reach and truncation doors stay inside the data the family already reads and buy only confidence. Three doors matter and none requires reading more information: the kernel-class door, the provenance audit, and the depth-bound lead. The information is already in hand and unspent. That is what makes this different from the sieve wall of frontier_math §2, where the missing input is Hardy-Littlewood grade and genuinely absent.

Ranked, and the follow-up goes through the third. The kernel-class door is the largest prize and the hardest, since §2 shows the requirement is an identity. The provenance audit is cheapest and gates whether any of this is unconditional. The depth-bound lead is the one that could deliver the prize using machinery that already exists, which is why the next hunt should take it: fix the harness guard that voided the shifted_product row, then re-run on a fully displaced zero set, which is the one configuration the inertia machinery was built to survive.