Hunt: gate-5 property 6, does "no zeros in a box off the critical line" distinguish?

Status when this file was committed: nothing has been computed. The parameters below were fixed before a single truth value was produced. The commit order is the evidence; see git log for this file versus results.json.

Scope

This hunt is exploratory. Nothing here is a result, and nothing here is evidence for or against RH (docs/08). The question is only which structural properties separate zeta from RH-violating look-alikes.

May write: hunts/gate5_p6_c/, and one case-log entry in hunts/README.md. May not write: zeta/, harness/, lean/, meta/, any other hunts/ directory, any root markdown file.

The question

docs/09 gate #3 property 6:

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

Does this DISTINGUISH (holds of zeta, fails on all three rivals in zeta.epstein.battery) or is it VACUOUS (some rival satisfies it too)?

The four functions are the battery's: zeta (via ξ), Davenport–Heilbronn (via f, equivalently F), and the Epstein zetas of discriminant −23, forms (2,1,3) and (1,1,6) (via Λ_Q).

The prior attempt, and why its parameters are not reused

A previous run computed count_zeros_box(0.6+80i, 0.9+90i) across four functions at working precision and timed out at 50 minutes. That box has a boundary of length 2·(0.3 + 10) = 20.6 and, at the module's forced subdivision step for t ≈ 90, roughly 176 forced segments per function, each costing at least two evaluations before adaptive refinement. The binding cost is the Epstein evaluation. Timed here, blind to any zero location, before choosing anything:

So the Epstein arm alone is three orders of magnitude more expensive than the other two, and the previous box asks it for something on the order of a thousand evaluations at its most expensive height. That is the whole reason that run did not finish. The design constraint is therefore: total boundary length, not box area.

(These are timings of function evaluation. They are measurements of cost, not of the property, and they were taken before any box was chosen. No winding number had been computed at the time this file was committed.)

What I am actually going to test, and why

Property 6 is box-dependent by construction: it quantifies over a box that the statement does not name. So the first thing worth deciding is not "does it distinguish" but "is the question well posed". The design is built to answer that directly rather than to gamble one box.

The instrument: hold the σ-band fixed and vary only the height window. If the verdict flips between two windows of the same band, then the property as stated has no truth value, the box is a free parameter carrying the answer.

The σ-bands

Band A: σ ∈ [0.7, 0.9]. Strictly off the critical line (0.2 clear of σ = 1/2), strictly inside the critical strip (0.1 clear of σ = 1, which is where Λ_Q has its pole and where the meaning of the claim changes). It contains 0.80852, the real part of the Davenport–Heilbronn off-critical-line zero this repository pins in OFFLINE_ZERO_RE. That containment is deliberate and declared: band A is chosen so that one preregistered window is known in advance to fail for one rival, which is what makes the flip-test meaningful. It is not a blind choice and is not reported as one.

Band B: σ ∈ [1.05, 1.55]. Strictly right of σ = 1, clear of the Λ_Q pole at s = 1. Zeta is zero-free here as a theorem (Euler product), with no appeal to RH and no numerics needed; the rivals have no such theorem. This band tests whether property 6 is anything other than the Euler product already tested by property 5.

The height windows

• W1 = [85.5, 85.9]: the literature-anchored window. Declared non-blind: Spira (Math. Comp. 1994) and this repository both place a Davenport–Heilbronn zero at 0.808517 + 85.699348·i inside it. Its job is to realise the "a rival fails" pole of the well-posedness question, not to supply a verdict.
• W2 = [10.0, 14.0]: blind. Chosen as a width-4 window at low height, where the boundary is short and the module's forced step is longest, hence cheapest. I do not know where any rival's zeros are in it.
• W3 = [40.0, 44.0]: blind, a second width-4 window at middling height, to check whether whatever W2 reports is typical rather than an accident of one window.

The boxes, fixed now

Total boundary length 27, against the prior attempt's 20.6, but spread over four boxes, three of which sit at heights where the Epstein evaluation is roughly half the cost it has at t = 86, and only one of which sits high.

Precision and route

• dps = 20 for every count. This is count_zeros_box's documented working precision and the value zeta.epstein's own interfaces clamp to (dps=min(_d, 20)); the winding number is an integer and the routine raises if the residual exceeds 1e-6, so more digits buy nothing here and cost linearly.
• Route: the argument principle, zeta.epstein.count_zeros_box, driven through the battery interfaces so that each function is counted by the same code. The default forced step is used unchanged.
• Truth rule, fixed now: property 6 holds for a function on a box iff count_zeros_box returns exactly 0 on that box. An ArithmeticError (zero on the contour, or undersampled edge) is recorded as undetermined and is not silently read as either verdict; the box is not nudged after the fact.
• Verdict rule, fixed now: per box, DISTINGUISHES iff zeta's count is 0 and every rival's count is non-zero; VACUOUS iff zeta's count is 0 and at least one rival's count is 0. If zeta's own count is non-zero the box is reported as invalidating itself, not as a result about zeta.

One thing to keep straight about "completed"

The claim names the completed function. In every box above, σ > 0, and in σ > 0 the completion factors are zero-free and pole-free: ξ vs ζ differ by s(s−1)π^{−s/2}Γ(s/2)/2, F vs f by (π/5)^{−(s+1)/2}Γ((s+1)/2), Λ_Q vs ζ_Q by (√d/π)^s Γ(s). Γ never vanishes and has no poles in σ > 0. So the counts are the same whichever of the pair is used, and the battery's mixed convention (zeta and Epstein counted on the completion, Davenport–Heilbronn counted on f itself) does not affect any number below. That mixed convention is noted as an observation about zeta/epstein.py, not as a defect this hunt fixes.

Kill conditions, declared

• If zeta's count is non-zero on any box, that box is discarded and said to be discarded, not repaired into agreement.
• If the flip-test does not flip, every box agreeing on one verdict, I report that single verdict and say the well-posedness question came back negative, rather than adding boxes until it flips.
• If the Epstein arm does not finish inside budget, the hunt reports not-settled for the Epstein rivals and says what it would cost. It does not substitute a cheaper rival or a cheaper property.

id: gate5_p6_c
question: Does gate-5 property 6 (no zeros of the completed function in a box strictly off the critical line) distinguish zeta from the three battery rivals, or is it vacuous?
frontier: properties 1-3 of the gate-5 battery came back VACUOUS, properties 4-5 DISTINGUISHES, property 6 open; the prior attempt on box 0.6+80i to 0.9+90i did not finish in 50 minutes
proposed_attack: hold the sigma-band fixed and vary only the height window, so that any change of verdict is attributable to the box alone and settles well-posedness before it settles the verdict
dead_routes:
  - the box 0.6+80i to 0.9+90i at working precision across all four functions, cost-bound on the Epstein evaluation at t near 90
  - raising dps to sharpen an argument-principle count whose output is an integer with a 1e-6 residual guard
required_oracles:
  - argument-principle winding number over a closed rectangle with an integrality residual guard
  - the pinned Davenport-Heilbronn off-critical-line zero, independently reported by Spira, Math. Comp. 1994
  - the classical theorem that zeta has no zeros in sigma greater than 1
kill_conditions:
  - zeta itself is counted with a non-zero winding in a box strictly off the critical line
  - the winding residual exceeds the integrality guard and the count is therefore undetermined
  - every box agrees on one verdict, so the box-dependence claim has no support
  - the Epstein arm exceeds budget and the verdict for those rivals is unavailable
agents_may:
  - search
  - derive
  - code
  - attack
agents_may_not:
  - declare novelty
  - declare theorem status
  - promote their own claim
  - nudge a box after seeing its count