Opened 2026-10-08. Hunt #121. Nothing in this directory is a result until the case log in hunts/README.md says how it ended, and even then nothing here is evidence about RH: every statement below is conditional on a hypothesis that nobody known to this laboratory has replayed. The strongest words used here are derived (an ordinary argument on cited inputs, written out), measured (one mpmath route at a stated precision) and cited (somebody else's statement, with its location). The reserved enclosure word belongs to zeta/rigor.py and appears nowhere in this directory.
The hypothesis
On 2026-10-06 OpenAI published github.com/openai/math. Its family 003 claims that every Dirichlet L-function, zeta included, is zero-free in Re s > 7/8, uniformly in the modulus, with a companion paper (2026-10-05) proving 11/12 by the same method. No replay by anyone known here, no peer review. The lab's reading of the release is 38-the-quasi-riemann-claim.md, the course's document 38, at commit 715d572 on branch claude/openai-math-release-2026-10-06; it is not on main as of 2026-10-08, which is why this directory names the file and never writes the usual bare reference to it.
Call the hypothesis QRH(theta): for every Dirichlet character chi mod q and every s with Re s > theta, L(s, chi) is nonzero except the principal pole, with theta = 7/8, and theta = 11/12 as a second instance. It is a hypothesis in this hunt, never a theorem. Every consequence drawn here is stated in the form "Under QRH(theta): ...", and if the claim is withdrawn or refuted by an independent replay, every such statement is void and this hunt says so.
The question
Which of this laboratory's priced walls and conditional results would move under QRH(7/8), by how much, and which would not move at all? Document 38 section 6 priced two consequences (the de Bruijn-Newman record does not move, the simple-zero proportions take no strip as input) and left one open: the q = 1 component of hunts/prime_pair_error/UPPER_BOUND.md (1), which SW_EFFECTIVE.md records as "a full power of N short of (31)". This hunt answers that one with every exponent explicit, extends the pricing to the q > 1 components, Theorems A and B, the signed-mean frontier, the ineffective constants, Li's criterion, and records the non-movers with citations.
id: qrh_conditional
question: Which of this laboratory's priced walls and conditional results move under the hypothesis QRH(theta) that every Dirichlet L-function is zero-free in Re s > theta, theta = 7/8 (and 11/12), by how much in the exponent, and which do not move at all?
frontier: hunts/prime_pair_error rank 1 at N^3 L^{-2H} (a full power short of (31)), rank 2 at N^{13/5} L^6, completed bound (1) at N^3 L^{-C}; Theorem A empty unconditionally; de Bruijn-Newman record 0.2 with 9/32 under the strip (document 38 section 6); Li coefficients nonnegative only as far as computed
proposed_attack: replace every prime-counting remainder the hunt reads by x^theta (log qx)^2, carry the exponent arithmetic exactly, and keep the hypothesis in every sentence
dead_routes:
- treating QRH(theta) as a theorem of this laboratory, it is an unreplayed claim and nothing here changes that
- re-deriving the de Bruijn-Newman consequence, document 38 section 6 already did that arithmetic and it is cited here
- bootstrapping the strip through the circle-method bound, the map theta to (2 + theta)/3 drifts toward 1 and has no fixed point below it
- reading a strip as a zero-density estimate, it bounds neither the number nor the heights of hypothetical off-line zeros
required_oracles:
- exact rational arithmetic on the exponents displayed in hunts/prime_pair_error/UPPER_BOUND.md and RESULTS.md
- mpmath at stated precision, dps 30, bisection on a one-variable sign change
- Davenport, Multiplicative Number Theory, chapters 16, 17, 19 and 20, as published
- Johnston and Yang, arXiv 2204.01980, Theorem 1.1, as read from the arXiv rendering on 2026-10-08
- the hunt's own recorded exponents in UPPER_BOUND.md, read and not re-proved
kill_conditions:
- QRH(theta) is withdrawn or refuted by an independent replay, in which case every conditional statement here is void and this hunt says so
- an exponent in RESULTS.md is shown not to follow from UPPER_BOUND.md's displayed inequalities plus the uniform remainder x^theta (log qx)^2, which retires that line and its row in the table
- the Johnston and Yang constants are found misquoted against the journal version, which voids the two crossover heights and nothing else
- a reader exhibits a consequence of the strip for rank 2 or for the Li coefficients at finite n that this hunt said does not follow, which reopens that row as unresolved rather than closed
agents_may:
- derive
- code
- measure
- cite
- price
agents_may_not:
- assign evidentiary status to any output of this hunt
- state any consequence of QRH(theta) unconditionally, or without the hypothesis in the sentence
- declare novelty
- declare theorem status
- promote their own claim
- edit anything outside hunts/qrh_conditional and the one entry in hunts/README.md1. What was read, and what was not
Read in full: hunts/prime_pair_error/SW_EFFECTIVE.md, UPPER_BOUND.md, RANK3_SCOPE.md, FRONTIER_2026_09_12.md, RESULTS.md sections 12 and 16 to 19 and its "The doors" section, the 2026-09-10 entry of HANDOFF.md, zeta/li.py's module docstring, document 38 sections 2 and 6, and pages 1 and 2 of the 11/12 preprint (/home/user/openai/math/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf), where Corollary 1.2 states that under its Theorem 1.1, for x >= 2, sup over 1 <= q <= x and (a, q) = 1 of |pi(x; q, a) - (1/phi(q)) Li(x)| << x^{11/12} log x with an absolute effective implied constant.
Read only at the header: SIEGEL_UNIFORMITY.md (its displayed bound (1) and the sentence saying when its exceptional terms are present), tb_bind_grh.py. Not read: the RANK3_BDH, RANK3_QUARTIC and RANK3_ROUTE documents, the remaining 180 pages of either preprint, and any Lean. Where RESULTS.md says a document was not read, the statement about it is limited to what its header says.
2. Rules binding this hunt
- A conditional result keeps its hypothesis in every sentence that uses it.
- Every number stated is pinned by
test_qrh_conditional.pyor hedged at the point of use. - Under
hunts/the reserved word is banned lexically; it is not typed here, not even to disclaim it. - No em dashes in prose written here; quoted material keeps its punctuation.
- This hunt does not assign evidentiary status to its own output. The grades it writes (derived, measured, cited) say what was done, not what it is worth.