Gate-5 property 6: the question is not well posed, and that is the finding

Status: settled, in the sense that matters, and explicitly not settled in one arm. Nothing here is a result in the repository's sense, and nothing here is evidence for or against RH (docs/08). A hunt is exploratory.

Parameters were preregistered in MISSION.md and committed at c29c876, before any winding number was computed. results.json was written after. The commit order is checkable in git log.

The claim under test

docs/09 gate #3, property 6:

in a box strictly off the critical line, the completed function has no zeros

Answer

Property 6 as stated has no truth value, because the box is a free parameter of the sentence and the verdict is a function of it. Holding the σ-band fixed at [0.7, 0.9] and changing only the height window flips the verdict between DISTINGUISHES and VACUOUS. Both poles are realised, in the same band, by the same code, at the same precision.

That is not a failure to settle the question. It is the answer to a different and better question, and it is the one that had to be answered first: you cannot ask whether property 6 distinguishes until you say which box, and the gate as written does not say.

Once a box is named, each named box does have a verdict, and those are reported below.

The table

Zeros counted by the argument principle (zeta.epstein.count_zeros_box), same routine for every function. - means not attempted; see What I could not settle.

All three rivals satisfy property 6 on B2 and on B4. On B2 that is three independent RH-violating functions passing a property zeta passes, in a box strictly off the critical line, which is exactly what "vacuous" means in gate #3's sense.

Full per-count records, including evaluation counts, wall-clock and the working precision each count ran at, are in results.json.

Reading the flip

B1 and B2 and B3 share a σ-band. Every one of them is "a box strictly off the critical line". In B1 the Davenport–Heilbronn function has a zero and zeta does not, so property 6 separates them. In B2 and B3 neither has a zero, so property 6 holds of the rival too and separates nothing. Same property, same band, same code, opposite verdicts.

The mechanism is not subtle and is worth stating plainly: an RH-violating function does not violate RH everywhere. Davenport–Heilbronn has a positive proportion of its zeros on the critical line, and its off-line zeros are sparse. A box that misses them sees a function that looks exactly like one satisfying RH. So a local zero-free statement cannot carry a global distinction, and property 6 is local by construction.

What I chose, and why

The choices are in MISSION.md with their reasons; the short version:

• σ ∈ [0.7, 0.9]: strictly off the line, strictly inside the strip, clear of the Λ_Q pole at s = 1, and containing 0.80852, the real part of the Davenport–Heilbronn off-line zero this repository pins. That containment is declared non-blind: B1 exists to realise the "a rival fails" pole, not to supply a verdict.
• B2 and B3 are blind: width-4 windows at low and middling height, chosen for short boundary, with no knowledge of where any rival's zeros are.
• σ ∈ [1.05, 1.55] for B4, the band where zeta is zero-free as a theorem rather than as a computation, testing whether property 6 is anything other than the Euler product that property 5 already tests.
• Total boundary length, not box area, is the cost. The prior attempt's box has boundary 20.6 at t ≈ 90, which is the most expensive place the Epstein evaluation can be asked to work. The four boxes here total 27, but three sit low.

The defect that cost this hunt its high-t Epstein arm

The first run of probe.py used dps = 20 everywhere, as preregistered, and did not return on the first Epstein count. The cause is a real and reproducible precision defect in zeta.epstein.epstein_completed, measured at σ = 0.8 on form (2,1,3) by comparing dps = 20, 40 and 80:

The cause is structural. epstein_completed returns d^{s/2} · (first + second/√d + 1/(√d(s−1)) − 1/s), whose last two terms are O(1/t), while the answer decays like |Γ(s)| ~ exp(−πt/2). So about πt / (2 ln 10) = 0.6822 · t digits cancel. At dps = 20 the phase of Λ_Q is noise above t ≈ 30, and count_zeros_box's adaptive bisection then recurses to its depth limit (45) on every segment. It does not fail; it does not return.

This is very likely what killed the previous attempt too. That run asked for exactly this: Epstein counts at working precision in a box reaching t = 90, and reported a timeout. The natural reading was "the contour is long". The measurement above says the contour length was not the binding constraint: at t = 90 the count was being driven by a phase that carried no correct digits, and no amount of waiting would have produced an answer. That reading is offered as a diagnosis consistent with the symptom, not as a verified account of a run this hunt did not observe.

The response was a documented deviation from MISSION.md: Λ_Q is evaluated at dps = 20 + ceil(0.6822 · t_max), a rule that is a function of the box alone and was written from the accuracy table above, before any Epstein winding number existed. ξ and f keep dps = 20; both were checked to be unaffected. The deviation is recorded here rather than folded into the preregistration.

What I could not settle

The Epstein arm on B1 and B3. The guarded precision the rule demands is dps ≈ 79 for B1 and dps ≈ 50 for B3, at roughly 9 s and 3.5 s per evaluation respectively, against 24 and 124 boundary evaluations per form before adaptive refinement. That is about 15 minutes per box for the two forms together, and it did not fit this run. It is a cost, not an obstacle: both are reachable in about half an hour of compute, and nothing about them is uncertain except the numbers.

Consequences for the table, stated exactly:

• B1's verdict is partial. DISTINGUISHES holds on the rivals actually run. An Epstein zero inside B1 would not change it; an Epstein absence would flip B1 to VACUOUS and would strengthen, not weaken, the well-posedness finding.
• B3's verdict is not partial in any way that matters. VACUOUS is already forced by Davenport–Heilbronn alone, because gate #3 asks the structure be ungrantable to every rival, so one rival sharing it settles the box. Running the Epstein arm on B3 cannot change the verdict; it would only add two numbers.

So the well-posedness conclusion does not depend on the missing arm: it needs B1 to be DISTINGUISHES-or-not-VACUOUS and B2/B3 to be VACUOUS, and Davenport–Heilbronn supplies both on its own.

What a well posed version would be

The gate is asking for a global zero-free statement and has written a local one. Three repairs, in increasing order of how much they actually say:

1. Name the box in the gate. Make property 6 read "no zeros in σ ∈ [0.7, 0.9], t ∈ [85.5, 85.9]". Then it has a truth value and it DISTINGUISHES (on the rivals run here). It is also worth almost nothing, because the box was chosen knowing where a rival's zero is. A gate whose discriminating power comes from having been shown the answer is measuring its own selection criterion, which is the trap hunts/README.md names under control role #1.
2. Quantify over all boxes in a band: "for every t, the completed function has no zeros with σ ∈ [1/2 + δ, 1]". This is RH with a margin. It distinguishes, and it is not checkable in a box, which is the honest content of the observation, not a defect of the repair.
3. Quantify over the band where zeta has a theorem: "no zeros with σ > 1". Zeta satisfies this by the Euler product, unconditionally. Davenport–Heilbronn does not: Davenport and Heilbronn proved in 1936 that it has zeros in σ > 1, and the same is classical for Epstein zetas of class number greater than one. This repair is checkable, it distinguishes, and it is the interesting one, but note what it has become: it is the Euler product again, which is property 5, which the battery already records as DISTINGUISHES.

That last point is the substantive content of this hunt beyond the well-posedness finding. Every repair of property 6 that both (a) has a truth value and (b) distinguishes, turns out to be either RH itself or the Euler product. Property 6 does not appear to be an independent sixth property. It is a local shadow of properties the battery already has, and its apparent independence comes entirely from the box it does not name.

B4 was designed to test exactly this, and it came back VACUOUS: in σ ∈ [1.05, 1.55], t ∈ [10, 14], all four functions have no zeros. That is the asymmetry B4 could not escape, and it is worth stating as a result rather than as a disappointment. Zeta's 0 in that band is a theorem and holds for every t; each rival's 0 is a measurement in a window of height 4, and is silent about the band as a whole. Davenport–Heilbronn's zeros in σ > 1 exist by the 1936 theorem; a width-4 window that misses them is not evidence that they do not. So even the good repair, restricted to a box, reports VACUOUS, the discriminating content lives in the quantifier over all t, which no box can carry.

Controls

• Rival (hunts/README.md #1): the rivals are the battery's own, and the trap named there applies to B1 and is declared, not evaded.
• Precision response (#4): the Epstein arm was re-run at raised precision precisely because the dps = 20 phase failed to respond as a real quantity should; the accuracy table is the response curve. ξ and f were checked to be insensitive at dps = 20 over these boxes.
• Self-consistency: zeta's count is 0 on all four boxes, so no box invalidated itself under the preregistered rule.
• Not run: no lesion or surrogate control. Neither is meaningful for a winding-number count whose oracle is integrality.

Loose threads

• The Epstein arm on B1 and B3. Two counts per box at dps 79 and 50. About half an hour of compute settles both. Nothing conceptual is missing.
• epstein_completed loses 0.68·t digits and says nothing about it. A caller asking for dps = 20 at t = 44 gets four correct digits and no warning. A height-aware guard, or a returned accuracy estimate, would have saved this hunt an hour and probably saved the previous attempt its fifty minutes. This is a defect in a core module and this hunt may not fix it; it is reported.
• count_zeros_box has no wall-clock or evaluation ceiling. When the phase is noise, the adaptive bisection recurses to depth 45 and the call simply never returns. A depth-exhaustion counter that raised ArithmeticError, the failure mode the routine already documents, would have converted a hang into a diagnosis in seconds.
• The battery counts Davenport–Heilbronn on f and the others on their completions. Harmless in σ > 0, where the completion factors are zero-free and pole-free, and this hunt's boxes are all in σ > 0. It would not be harmless for a box reaching σ ≤ 0, where the trivial zeros enter and the two conventions genuinely differ.
• Where are the Epstein σ > 1 zeros? Davenport–Heilbronn's theorem guarantees they exist for class number greater than one, but this hunt found no height for them and did not look. A hunt that located the lowest one would turn repair #3 above from a citation into a computation.
• Is any gate-5 property independent of properties 4 and 5? This hunt suggests property 6 is not. Whether 1–3 are shadows of each other in the same way is not asked anywhere and is cheap to check.