Opened 2026-10-16 as Hunt #134; Hunt #139 on main (renumbered at landing on 2026-10-21, because #134 on main is gauss_positivity and #136 to #138 are taken). Nothing in this directory is a result until the case log in hunts/README.md says how it ended. Grades used here: cited (a line of the paper's LaTeX source, quoted), derived (algebra done here from cited formulas, checked in exact rational arithmetic by this hunt's test), inferred (a reading of the paper's structure that no computation here checks). The reserved enclosure word belongs to zeta/rigor.py and appears nowhere in this directory.
The question
OpenAI's preprint of 2026-09-30 (docs/38, docs/39) proves that no Dirichlet L-function, zeta included, has a zero with real part above 7/8. RH is the same statement at 1/2. The lab has rebuilt the Lean statement and built four results on it, but nobody here has read past sections 1 to 3 of the 199-page argument (docs/38, "What was not checked"), and nobody has located the step that stops the method at 7/8 rather than at a smaller number. Hunt #120 (qrh_rival_step/) read the places where the Euler product is used and recorded that the paper's second Euler region reads x_r >= 7/8; it did not ask what fixes that value.
Which inequality in the 7/8 argument binds, as a function of the parameters the paper chooses, and what is the smallest target the same assembly would allow if those choices were re-optimized with every cited input held fixed?
The paper's source is public: github.com/openai/math at commit adc7f1241b42e322a6451854ab7e4b4c146bf78a, preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/build/paper.tex (16,677 lines, Apache-2.0). Line numbers below refer to that file.
Scope
- In scope. Reading the final assembly (Part II, Section "The seven-eighths bound", lines 15495 to 16677) and the lemmas it cites; rewriting each of its numerical inequalities as a function of the free geometry (the prime-slot length, the scale offset, the frequency scale, the contour positions); reproducing every printed margin from those general formulas in exact arithmetic as a control; then varying the free parameters to see which inequality fails first.
- Out of scope. Checking the paper's lemmas themselves (the moment inductions, the sieve, the detector). Those are taken as cited inputs, at the exponents the paper proves them with. Any statement about RH: a smaller half-plane, if the re-optimization found one, would still be a candidate about this assembly, not a theorem, and nothing here says anything about real part 1/2.
- Budget and stop rule. One session. No heavy compute: the arithmetic is exact rational algebra on a laptop scale. Stop when the binding constraint is located and stated with its parameter range, or when a cited input turns out to depend on the target in a way the reading cannot resolve, and report that wall.
id: seven_eighths_pin
question: In OpenAI's 7/8 zero-free half-plane argument (preprint of 2026-09-30), which inequality of the final assembly binds as a function of the chosen geometry, and what is the smallest target the same assembly allows when that geometry is re-optimized with every cited lemma input held at the exponents the paper proves?
frontier: the theorem is kernel-checked here as a formal statement (docs/38 section 7) but its argument is unreviewed by any person; sections 1 to 3 were read (docs/38); hunt 120 located the Euler-product step (Lemma 7.1, Section 7.2) and recorded the second Euler region condition x_r >= 7/8; the Part II endpoint certificate prints a margin of 49/440640 and the low estimate matches the Mellin signal exactly at 7/8 (C_II(7/8) = 3/16 = l_x/2 + b/12, line 8642); no location of the binding step was found in this tree
proposed_attack: express the Mellin target, the low exponent and every high-side exponent of Section 15495 to 16677 as functions of the slot length ell, the offset b, the frequency scale h and the contour positions; reproduce each printed constant from the general formulas in exact rational arithmetic; then vary the free parameters with the row-count inputs held fixed and record which inequality fails first and at what target
dead_routes:
- reading the 7/8 half-plane as information about real part 1/2, which docs/39 rules out
- treating a zero of a linear combination of L-functions as a counterexample to a statement about single L-functions, which hunt 120 and docs/38 section 6 rule out
- lowering the target by substituting a smaller number for 7/8 in the final section alone, without rederiving the row counts and the bootstrap that the final section inherits
required_oracles:
- the paper's LaTeX source at the pinned commit, quoted by line number
- exact rational arithmetic in Python fractions reproducing each printed constant of the paper
kill_conditions:
- a general-parameter formula fails to reproduce a constant the paper prints, at the paper's own parameter values
- a parameter treated here as free is fixed by a cited lemma this hunt did not read, and that lemma changes the binding constraint
- the derived relation between the slot length and the target fails at either stage of the argument (11/12 with no slots, 7/8 with slot length 1/6)
agents_may:
- read and quote the paper
- derive
- code exact arithmetic
- attack the derivation
agents_may_not:
- declare novelty
- declare theorem status
- claim any improvement to the half-plane as established
- promote their own claim