RESULTS: gate-5 property 6 is VACUOUS

Verdict: VACUOUS. Two of the three preregistered boxes were decided for all four functions, and in both of them all three RH-violating rivals satisfy property 6 exactly as ζ does. Under gate #3's own rule a property that a rival shares distinguishes nothing, so property 6 cannot be the load-bearing step of an RH argument.

Grade: measured. One route, floating-point mpmath at stated precision with a two-precision replication guard. Nothing here is evidence for or against RH (docs/08); a hunt is a probe, not a result.

Parameters were fixed in MISSION.md and committed at 623e800 before any zero count was computed. The probe ran afterwards. The commit order is the evidence and it can be checked.

The table

Counts are zeros in the closed rectangle, by the argument principle, with the winding required to be within 1e-6 of an integer.

"Satisfies property 6" means a count of 0. In boxes A and C every rival satisfies it, so shared_with is all three and the battery's rule returns VACUOUS. Box B is reported not decided, honestly: its two Epstein cells were never run, for the reason in the cost section below.

Every decided cell passed the preregistered adequacy guard by a wide margin. The worst relative disagreement between dps = D and dps = D + 15 at a box corner, across all ten decided cells, was 3.5e-22 against a 1e-6 threshold.

Whole probe: 641.6 seconds. No contour collision occurred, so the §5 nudge rule never fired and no box was moved.

The instrument checks out where it can be checked

Box B's Davenport-Heilbronn cell returns 1, and the zero it is counting is the one already pinned in zeta/epstein.py,

rho = 0.80851718245663738555335196060684... + 85.69934848537759217192926770894...i

which sits inside box B with 0.09 of clearance in σ and 0.5 in t, and which matches Spira, Math. Comp. 1994. So the counting routine recovers a known off-line zero at exactly the place the literature puts it, in the one cell where an independent answer exists. That is the only positive control available here and it passed.

What was chosen, and why

The three boxes and the precision rule were picked on cost grounds and on one measured numerical fact, before any count existed.

The boxes. Box A is the a priori box: the tallest strictly-off-line box inside the critical strip that all four functions could be counted on at an adequate precision within budget. Its height was set by what the Epstein evaluation costs, not by where anybody's zeros are. Box B is the witness box, chosen to be as favourable to DISTINGUISHES as any box can be: it contains a rival's off-line zero that was known in advance, so one rival was guaranteed to fail there. Box C sits in the half-plane σ > 1, where ζ is zero-free by the Euler product and where the rivals are known to have zeros somewhere, because that is where a well posed version of the property would have to live.

The precision. A calibration run before MISSION.md was written measured only timings and magnitudes, never a count. It found that

epstein_completed(0.8 + 85.7i, (2,1,3))  =  1.2e-34    at dps = 20
                                         =  1.617e-58  at dps = 60 and at dps = 90

against the analytic magnitude |(√d/π)^s Γ(s)| = 2.64e-58. The dps = 20 answer is round-off noise, wrong by twenty-four orders of magnitude, because Λ_Q is exponentially small in t while the terms of its Mellin-split representation are O(10^-2). The digits lost are about π t / (2 ln 10) ≈ 0.6822 t, which is the preregistered precision rule: dps = 20 + ceil(0.6822 · t_max), giving 28 for boxes A and C and 79 for box B.

What could not be settled, and what it would take

Box B's two Epstein cells are NOT-DECIDED-BY-COST, from arithmetic recorded before either was attempted:

Epstein (2,1,3) at dps 79:  14.26 s/eval x 22 forced segments x 2 = 627 s
Epstein (1,1,6) at dps 79:  13.93 s/eval x 22 forced segments x 2 = 613 s

against a 480-second cell cap. Deciding them needs roughly 21 minutes of mpmath at dps = 79, or a faster evaluator: ball arithmetic through python-flint, or an Epstein routine that does not build an exponentially small number out of O(10^-2) pieces. Neither was in scope here.

Their absence does not change the verdict. Boxes A and C are each decided for all four functions, and either one alone returns VACUOUS.

The trap the brief named, and what it actually costs

Property 6 is box-dependent by construction, and the run shows the dependence rather than arguing about it: the Davenport-Heilbronn function satisfies property 6 on box A and on box C, and fails it on box B. The same rival, the same property, opposite truth values, decided by which rectangle you picked. So "property 6" as written is not one claim. It is a family of claims indexed by a box, and asking whether it is vacuous or distinguishing has no answer until the box is named.

That does not rescue it, because the box-indexed claims are vacuous almost everywhere and for a structural reason. For a fixed box B to distinguish, it would have to contain an off-line zero of the Davenport-Heilbronn function and an off-line zero of Epstein (2,1,3) and an off-line zero of Epstein (1,1,6), while containing no zero of ξ. Those three functions are unrelated in their zero locations, so a rectangle catching all three at once is a coincidence, not a construction. Box B is the demonstration: it was built around the one off-line zero this laboratory has pinned, and it still leaves two rivals with no reason to fail. Every box that misses any one rival's off-line zeros returns VACUOUS, and boxes A and C are two such boxes decided in full.

A well posed version, and where it lands. Quantify over boxes instead of fixing one:

for every box strictly off the critical line, the completed function has no zeros

This is well posed, and it does distinguish: it is false for all three rivals by Davenport and Heilbronn's 1936 theorem, quoted and not recomputed here, and it is expected true of ζ. But it is exactly the statement "the completed function has no zeros off the critical line", which for ζ is RH. A gate-#3 property that turns out to be the conclusion is not a step toward the conclusion, and a battery that returned DISTINGUISHES for it would be reporting circularity rather than progress.

The nearest version that is both well posed and not circular is the half-plane one:

the completed function has no zeros with Re s > 1

true of ζ by the Euler product, false of all three rivals by the same 1936 theorem. That is a real discriminator, and it is also not new information: it is the Euler product again, which is what property 4-5 already reported as DISTINGUISHES. Property 6 has no discriminating power of its own to add.

A defect in the battery, found on the way

zeta.epstein.battery cannot be used to answer any zero-counting question above roughly t = 25, and it will not say so. Both rival interfaces cap the working precision:

• epstein_interface(...)["count_zeros_box"] passes dps=min(dps, 20)
• dh_interface(...)["count_zeros_box"] passes dps=min(dps, 20)

For the completed Epstein function that cap is below the precision the function needs: at t = 85.7 a dps = 20 evaluation is noise twenty-four orders of magnitude too large, as measured above. An argument-principle count built on those values is the winding number of round-off, and it will usually still come out an integer, so count_zeros_box's integrality check does not catch it. A caller who asks battery a property-6 question at height gets a number with no warning attached.

This hunt therefore called count_zeros_box directly with an explicit fn and an explicit dps, reproducing battery's verdict rule unchanged, and declared the deviation in MISSION.md §1 before running. The defect is reported rather than patched: zeta/ is outside this hunt's write scope.

Loose threads

• Box B's Epstein cells. Decidable, at roughly 21 minutes of mpmath at dps = 79, or much less through a ball-arithmetic backend. They would not change the verdict, but they would close box B and make the box-dependence demonstration complete on all four functions rather than two.
• The capped precision in battery's rival interfaces. min(dps, 20) silently produces noise-driven counts above t ≈ 25. The fix is small: pass the caller's dps, or refuse a box whose height exceeds what the cap can support. Deserves an issue on this repository, and a test that a count at height disagrees with itself between two precisions.
• A precision-adequacy guard as a reusable instrument. The evaluate-at-D-and-D+15 check in probe.py is four lines and it is the only reason this run knows its numbers are real. Nothing in zeta/ offers it, and every hunt that evaluates a completed L-function at height needs it.
• The Epstein evaluator's conditioning. epstein_completed computes an exponentially small quantity as a sum of O(10^-2) terms, which is why it costs 0.6822 t digits. An evaluator that factors out the Γ(s) decay before summing would make Epstein zero-counting at height affordable, and would retire the cost barrier this hunt hit.
• Where the off-line zeros of the discriminant −23 forms actually are. Davenport and Heilbronn prove they exist. This laboratory has never located one, and box B shows why it matters: without a pinned Epstein off-line zero, no box can be built that puts all three rivals under the same test at once.
• The prepare-commit-msg hook is not installed in a fresh clone, so the Run-Id trailer this run's telemetry expects does not appear on commits unless it is written by hand. Noted where it was noticed.