Hunt r_f7cd45: which structural properties of ζ actually discriminate

Status: settled for the five properties issue #21 queued; the sixth was already settled by earlier hunts in this tree, and what this run adds to it is the number that says why it is expensive.

Publishes the artifact issue #21 (https://github.com/teal-sea/zeta-lab/issues/21) recorded as queued. Gate #3 of docs/09-new-ontologies.md: whatever structure an ontology grants ζ must be ungrantable to functions that share ζ's analytic structure and violate RH. A property those rivals also satisfy distinguishes nothing and cannot be the load-bearing step of an RH proof.

The three rivals are zeta.epstein.battery's defaults: the Davenport–Heilbronn function, and the Epstein zetas of the two discriminant −23 forms (2,1,3) and (1,1,6). All three are linear combinations of legitimate Euler products, which is the point: linear combination preserves the functional equation and destroys the primitive multiplicative structure (docs/09 §5.1).

Nothing here is evidence for or against RH (docs/08-why-it-is-hard.md). Gate #3 is eliminative, never probative: prime-blind ⟹ not an explanation, but not prime-sensitive ⟹ proof.

Reproduce: .venv/bin/python hunts/r_f7cd45/probe.py

The five properties (stage 1)

dps = 25, all four functions, every cell computed. ✓ = the function has the property.

This reproduces issue #21's table exactly, cell for cell, from a probe that runs end to end. Three of the properties most often cited as structural insight into ζ are worth nothing as discrimination: every RH-violating look-alike has all three. Only the prime structure kills the rivals.

Cost, measured: 24.3 s, 23.9 s, 853.8 s, 0.0 s, 0.0 s. Property 3 dominates the run by a factor of thirty, because it counts sign changes of Z on t ∈ [10, 40] for two Epstein forms. A narrower window would have bought the same verdict; that width was not tuned and should have been.

What was chosen, and why

• Property 4's pair set excludes m = 1 on purpose. The Epstein coefficient is a representation count with r(1) ≠ 1, so a pair (1, n) would fail on normalisation rather than on multiplicative structure. Restricting to coprime m, n ≥ 2 gives every rival its best shot. They fail anyway: for the principal form (1,1,6), r(2) = r(3) = 0 but r(6) = 2; for Davenport–Heilbronn, a₆ = a₁ = 1 but a₂a₃ = −κ² = −0.0807….
• Property 2 carries a degeneracy guard. A Z identically zero would satisfy "real-valued for real t" vacuously, so the claim first requires Z to be somewhere non-zero on the sample set.
• Properties 1 and 4 reuse zeta.epstein.claim_functional_equation and claim_multiplicativity rather than restating them, so the published verdict is the packaged claim's verdict. Properties 2, 3 and 5 are new here.
• Positive control throughout: ζ satisfies all five. A probe in which ζ failed a property it demonstrably has would be reporting a defect in itself.

The sixth property (stage 2): already settled, and here is the price tag

Issue #21 lists a sixth property, "in a box strictly off the critical line the completed function has no zeros", as unfinished at a 50-minute timeout. That is stale. Three hunts in this tree settled it after the issue was written:

• Hunt #13 (gate5_p6_a/): VACUOUS on an adversarially chosen box.
• Hunt #14 (gate5_p6_b/): VACUOUS on two of three boxes, with the Davenport–Heilbronn zero recovered as a positive control on the third.
• Hunt #15 (gate5_p6_c/): the question as written has no truth value, the box is a free parameter of the sentence, and holding the σ-band fixed while moving the height window flips the verdict.

This run reproduces that independently on two preregistered boxes (MISSION.md, committed at 53c8cd1, before any winding number was computed), σ ∈ [0.6, 0.9]:

B1 is VACUOUS and the undecided Epstein cells cannot overturn it: gate #3 asks the structure be ungrantable to every rival, so one surviving rival settles the verdict on its own. B2's D–H cell returns exactly 1, recovering the off-line zero this repository pins at 0.8085171824… + 85.6993484853…i. That is the run's positive control, and it is why B2 is undecided rather than vacuous: with D–H failing the property, the verdict turns on the Epstein arm, which this run could not afford.

Same property, same σ-band, same code, opposite verdicts one height apart. That is hunt #15's finding, reproduced.

The number this run adds: the precision floor, measured

The Epstein cells are marked inadmissible, not slow. Before spending any budget on a winding number, the probe ran each completed function up a dps ladder at 0.75 + 85.5i and asked where the value stops moving. 15 was the lowest rung tested, so a floor recorded as 15 means "≤ 15", not "exactly 15":

At dps = 15 the Epstein completed function returns ≈ 5e−30 where the true value is ≈ 2e−58: noise larger than signal by 28 orders of magnitude. At the dps = 20 that battery actually uses it returns ≈ 2e−33, still noise by 25 orders of magnitude. This independently confirms the cancellation defect hunts #14 and #15 reported (≈ 0.6822·t digits lost; 0.6822 × 85.5 ≈ 58, which is exactly the gap measured here) and puts a floor on it: dps ≈ 60 at t ≈ 85.5.

Two consequences worth stating plainly:

1. count_zeros_box's integrality check does not catch this. Noise winds to an integer as happily as signal does. A box count run below the floor returns a plausible small integer and is not a zero count. My own preregistered dps = 15 would have produced one; the probe skips those cells rather than publish them.
2. battery cannot clear its own floor. Both rival interfaces hardcode dps=min(dps, 20) into count_zeros_box (zeta/epstein.py:1091, :1141). No value the caller passes reaches the routine above 20, so the packaged property-6 route is below the convergence floor for every t ≳ 30 and cannot be fixed from the outside. This is a zeta/ defect, outside this hunt's scope, reported and not patched.

At dps = 60 one epstein_completed evaluation costs 8.8 s (measured), and a unit-height box boundary needs of order 10²–10³ of them. That is the price tag on the Epstein arm, and it is why three hunts in a row have declared it out of budget. It is not a hard problem; it is an expensive one, and the expense is a fixable implementation defect rather than a fact about the mathematics.

What this run could not settle

• The Epstein arm of property 6, on any box above t ≈ 30. Third independent time. Affordable only after the cancellation defect is fixed; the estimate above is 10²–10³ × 8.8 s per cell as the code stands.
• Anything about novelty. No literature search was run. These are original results of this laboratory in the provenance sense; whether the observation is new to the world is unsearched and is not claimed.
• Whether properties 4 and 5 are the only discriminators. Five properties were tested because five were queued. The battery admits any claim; nothing here bounds the space of properties that might also distinguish.

Grade: measured, float and mpmath, one route per cell, with ζ as a positive control on every property and the pinned Davenport–Heilbronn zero as a positive control on box B2. Not hardened: no enclosure carries these steps.

Loose threads

• battery's rival interfaces cap count_zeros_box at dps = 20. Why it might matter: it silently caps the packaged gate-#3 property-6 route below its own convergence floor for every t ≳ 30, and the routine's integrality check cannot detect the resulting noise, so the failure mode is a plausible wrong integer rather than an error. Hunts #14 and #15 reported the cancellation; this adds that the cap makes it unreachable from the caller. First step: change min(dps, 20) at zeta/epstein.py:1091 and :1141 to a floor derived from the box height, hunt #14's measured ≈ 0.6822·t digit loss plus guard digits, which at t = 85.5 predicts ~58 and brackets the 60 measured here, and pin it with a test that the count at t ≈ 85.5 is stable between dps = 60 and dps = 100. Landed: the clamp is gone from every interface (zeta/epstein.py, commit c2e97df); tests/test_interface_dps_is_honoured.py discovers each *_interface and checks the caller's dps reaches count_zeros_box. The caller still chooses the digits; hunts/gate5_p6_b/ gives the rule dps = 20 + ceil(0.6822 t_max).
• Property 3 cost 853.8 s of a 900 s stage for a verdict visible in seconds. Why it might matter: zeros_on_line is quantified over a window that nothing chose, and the same VACUOUS verdict follows from any window containing one zero of each function. First step: narrow LINE_WINDOW to (10, 16) in probe.py and confirm the five verdicts are unchanged; if so the whole stage runs in under a minute.
• Property 6 is still carried by the battery as a property. Why it might matter: three hunts have now concluded it is a (function, box) pair rather than a property, and hunt #13 already proposed retiring it. It keeps attracting budget, this run included. First step: the disposition is a zeta/ decision, not a hunt's; someone with that scope should either retire it or replace it with the box-free version hunt #13 drafted.
• Issue #21's "not yet settled" section is stale. Why it might matter: it is what routed this run at the sixth property, and it will route the next one the same way. First step: comment on issue #21 pointing at hunts #13, #14, #15 and at this artifact, and edit that section.