MISSION, gate-5 property 6: vacuous or distinguishing?

Status when this file was committed: nothing has been computed. No zero count, for any function, in any box, had been evaluated when this file landed. The commit order is the evidence, and it is the only reason the verdict below is worth reading.

The question

docs/09 gate #3: a claimed structural property of ζ is worth nothing as evidence if a function that shares the structure and violates RH satisfies it too. zeta.epstein.battery runs a claim against ζ, the Davenport-Heilbronn function f, and the two discriminant −23 Epstein zetas (2,1,3) and (1,1,6).

Property 6 is the open one:

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

Does it DISTINGUISH (true of ζ, false of all three rivals) or is it VACUOUS (some rival satisfies it too)?

Scope of this hunt

Nothing here is evidence for or against RH (docs/08). This is only about what separates ζ from RH-violating look-alikes. Nothing here is a result; hunts/ is a probe area.

What is fixed before any truth value is computed

1. The instrument

zeta.epstein.count_zeros_box(s0, s1, dps=…, fn=…): an argument-principle winding count over a closed rectangle, which raises ArithmeticError when the winding is not within 1e-6 of an integer (a zero on the contour, or an undersampled edge). Errors are recorded, never swallowed; a box is nudged only by the rule in §5.

Declared deviation from battery(), with its reason. battery is not called directly. Its rival interfaces hard-cap the working precision at dps=min(dps, 20) (zeta/epstein.py, epstein_interface and dh_interface). A cost calibration run before this file was written, timings and magnitudes only, no zero counts, measured that epstein_completed(0.8 + 85.7i, (2,1,3)) returns 1.2e-34 at dps=20 but 1.617e-58 at both dps=60 and dps=90, against an analytic magnitude |(√d/π)^s Γ(s)| = 2.64e-58. The dps=20 value is round-off noise, wrong by twenty-four orders of magnitude. So the battery's own instrument cannot count Epstein zeros above roughly t ≈ 25, and this hunt therefore calls count_zeros_box directly with an explicit fn and an explicit dps, while reproducing battery's verdict rule exactly (§6). The capped-precision defect is reported as a finding, not routed around silently.

Each function is counted through the completion the battery's own interface uses:

The contour evaluator is memoized on the sample point. This changes no mathematics (the integrands are deterministic functions of s) and roughly halves the evaluation count, because _arg_variation re-evaluates its segment endpoints.

2. The boxes, and why these

All boxes are strictly off the critical line, Re s ≥ 0.70 > 1/2 throughout, and avoid t = 0 and the Epstein pole at s = 1.

Box B's σ range brackets Re ρ = 0.8085 with 0.09 of clearance on each side and its t range brackets Im ρ = 85.6993 with 0.5 above and 0.5 below, so no edge runs near the known zero.

3. The precision, and the adequacy guard

The completed Epstein function loses relative precision to the Γ(s) decay in its Mellin-split representation: Λ_Q is exponentially small while the terms that build it are O(10^-2), so the digits lost are about π t / (2 ln 10) ≈ 0.6822 · t. Preregistered working precision:

dps_epstein(t_max) = 20 + ceil(0.6822 * t_max)
dps_zeta = dps_dh   = 20

giving dps = 28 for boxes A and C and dps = 79 for box B. ζ's xi and the Davenport-Heilbronn dh_f are evaluated as products and Hurwitz-zeta sums with no comparable cancellation, so they keep dps = 20.

Adequacy guard, and it runs before any count is read. At each box's top-right corner, evaluate the function at dps = D and at dps = D + 15. If the two do not agree to a relative 1e-6, the cell is recorded NOT-DECIDED-PRECISION and no count from it is used. This is an empirical check that does not depend on the analytic estimate above being right.

4. The cost caps

• Per cell (one function, one box): 8 minutes wall clock. On overrun the cell is NOT-DECIDED-BY-COST.
• Predicted-infeasible cells are not run. If (measured seconds per evaluation at the required dps) × (the box's forced-segment count) × 2 exceeds the cell cap, the cell is recorded NOT-DECIDED-BY-COST from the arithmetic, and the arithmetic is reported. Burning the cap to confirm an arithmetic certainty buys nothing.
• Global probe cap: 20 minutes. Cells are executed in this fixed order, box A all four, box B ζ and DH, box B Epstein (2,1,3) then (1,1,6), box C all four. Any cell not reached is NOT-DECIDED-BY-BUDGET.

No substitution. A cell that cannot be decided is reported undecided. An easier box is never swapped in for a harder one, and no box is added after a count has been seen.

5. Contour collisions

If count_zeros_box raises ArithmeticError on a box, the box is nudged once, by the fixed rule σ range widened by +0.005 on the right and t range widened by +0.005 at the top, and the error, the nudge and the retry are all recorded. A second failure is NOT-DECIDED-CONTOUR.

6. The decision rule

For a fixed box B, the claim evaluated is

claim_B(F)  :=  (number of zeros of F in B) == 0

and battery's own verdict rule is reproduced without change:

• DISTINGUISHES on B iff claim_B is true for ζ and false for all three rivals.
• VACUOUS on B iff claim_B is true for ζ and true for at least one rival.
• If claim_B is false for ζ, the claim is not a property of ζ on that box and the box is reported as such.

And for the question as asked, which is not indexed by a box:

• If every decided box gives the same verdict, that verdict is reported for property 6, with the boxes named.
• If two decided boxes give different verdicts, property 6 is not well posed as stated, it is a box-indexed family of claims, not one claim, and a single VACUOUS/DISTINGUISHES answer to it does not exist. In that case the finding is the ill-posedness, the per-box verdicts are reported as the evidence for it, and RESULTS.md states what a well-posed version would be.
• If the rival cells needed to separate the two are undecided, the answer is not-settled with the cost arithmetic that makes it undecidable at this budget. That is a real answer and it will be given rather than dressed up.

7. What would make this hunt's own answer worthless

Recorded here so it can be checked against the commit history: choosing a box after seeing a count, widening a box until a rival's count changes, dropping a cell because its answer was inconvenient, or reading a count from a cell that failed the adequacy guard in §3.

The huntspec

id: gate5_p6_b
question: Does gate-5 property 6, the completed function has no zeros in a box strictly off the critical line, distinguish zeta from the three RH-violating rivals of zeta.epstein.battery, or is it vacuous?
frontier: properties 1-3 of the battery returned VACUOUS and 4-5 DISTINGUISHES; property 6 is undecided, and one prior attempt on the box 0.6+80i to 0.9+90i at working precision did not finish inside fifty minutes
proposed_attack: fix two boxes and a precision rule in advance, count zeros of all four functions in each by the argument principle, and read the battery's own verdict rule off the counts
dead_routes:
  - the box 0.6+80i to 0.9+90i at working precision, which did not finish in fifty minutes
  - epstein_completed at dps 20 above t roughly 25, where the Gamma decay in the Mellin split leaves only round-off, measured 1.2e-34 against a true 1.617e-58 at t = 85.7
  - epstein_functional_equation_defect as a check of anything, since it is zero by construction
required_oracles:
  - argument-principle winding count over a closed rectangle, rejected unless the winding is within 1e-6 of an integer
  - replication of every boundary evaluation at fifteen additional digits, agreeing to a relative 1e-6
  - the off-line Davenport-Heilbronn zero pinned in zeta/epstein.py and cross-checked against Spira, Math. Comp. 1994
  - the Davenport-Heilbronn 1936 theorem as published, quoted and not recomputed
kill_conditions:
  - a count is read from a cell whose two precisions disagree
  - a box is chosen, moved or widened after any count on it has been seen
  - the verdict changes between the preregistered boxes, in which case the single-verdict question is withdrawn as ill posed rather than answered
  - the winding fails to be an integer twice on the same box
agents_may:
  - evaluate the four functions and count zeros in the preregistered boxes
  - record a cell as undecided on cost, precision or contour grounds
  - report the capped-precision defect in the battery interfaces as a finding
agents_may_not:
  - substitute an easier box for a preregistered one
  - add a box after seeing a count
  - claim novelty for anything here, or use the reserved word
  - grade any of this above measured