Opened 2026-10-21. Hunt #148. A lane of the campaign the owner directed on 2026-10-21 (route 6 of its second wave: a second probe in F_q[T]). The workflow that opened it proposed the number 141; that number and #138 to #144 are held for landings another workflow is renumbering, so this hunt takes #148. 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 a source's text, quoted with its file hash), derived (algebra done here and checked in exact arithmetic by this hunt's tests), inferred (a reading no computation here checks), measured (one route, float grade), hardened (independent routes agree exactly, or ball enclosures carry every step). Nothing here is kernel-checked. The reserved enclosure word belongs to zeta/rigor.py and the Lean arm and appears nowhere in this directory.
The question
In F_q[T], where the Fourier coefficients of theta functions on higher-order metaplectic covers can be computed (Hoffstein 1992, Patterson 2007 and 2014) and RH is a theorem (Weil), does a biquadratic-theta probe with quartic averaging have a scalar local Euler identity like Lemma 7.1 of OpenAI's 7/8 paper, that is, a principal row proportional to 1/L? What abscissa would the 7/8 template price for it?
Why F_q[T]
Hunt #136's probe lane (hunts/below_seven_eighths/PROBE.md, unlanded, in progress on hunt/below-seven-eighths) prices the 7/8 assembly as a function of the order k of the row characters and records two things by reading: lower-order twists of the cubic theta die (S1, S2), and S4 (biquadratic theta over Q(i), quartic twist, pole 1/4) is open because the number-field theta coefficients at squarefree indices are not determined for n >= 4. Over a rational function field those coefficients are finite sums of Gauss sums, so the identity can be tested exactly instead of by reading. The cubic case (n = 3 with sextic averaging) is the control: there the function-field analogue of the paper's construction should reproduce Lemma 7.1, and a failure would be a discrepancy to localize.
Starting point
- The template: OpenAI, The Quasi-Riemann Hypothesis, preprint of 2026-09-30, Section 7 (the Poisson representation and its Euler factors), lines 3723 to 4067 of
paper.texatgithub.com/openai/mathcommitadc7f1241b42e322a6451854ab7e4b4c146bf78a, SHA-25642a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3, read as data from<cache>/openai-math-adc7f124/paper.tex. No person has reviewed the paper's argument. - The function-field sources, as the workflow cited them from memory: Hoffstein, Invent. Math. 107 (1992) 61-86, and Patterson, Glasgow Math. J. 49 (2007) 243-255. Step one of this hunt is to verify or replace those citations (
SOURCES.md). - Hunt #136's order-k template: balanced probe 1 - 1/(2k), per-summand corner 1 - 3/(4k), perfect-row-count floor 1 - 4/(5k), so 11/12, 7/8, 13/15 at k = 6 and 7/8, 13/16, 4/5 at k = 4 (derived there, unreviewed).
Scope
- In scope. Exact computation over F_7[T] (n = 3, sextic averaging) and F_13[T] (n = 4, quartic averaging): power-residue symbols, Gauss sums in cyclotomic integers, Patterson's theta coefficients, the probe's original and Poisson representations, the local Euler factors, a Davenport-Heilbronn type rival, and a ledger of what the template charges against exact sizes.
- Out of scope. Any statement about number fields beyond an explicitly inferred analogy, any statement about zeta or RH over Q, and any edit to
zeta/,ontology/orharness/. A function-field verdict on S4 is evidence about the analogy, not a theorem about Q(i). - Compute. The swarm box (32 vCPU, 115 GiB, no swap), shared. Single-process Python, pytest with
-n0, at most 8 processes for any parallel numeric job, no job started while the 1-minute load average is above 64 orMemAvailableis below 30 GiB. A run longer than about twenty minutes is timed on one unit first and estimated inRUNS.md.
id: ff_probe
question: In F_q[T], where theta coefficients of higher-order metaplectic covers are computable and RH is a theorem, does a biquadratic-theta probe with quartic averaging have a scalar local Euler identity like Lemma 7.1 of OpenAI's 7/8 paper, a principal row proportional to 1/L, and what abscissa would the 7/8 template price for it?
frontier: hunt 136 records by reading that lower-order twists of the cubic theta die and that S4 (biquadratic theta over Q(i), quartic twist, pole 1/4) is open because number-field theta coefficients at squarefree indices are not determined for n >= 4; its order-k template prices k = 4 at 7/8 (balanced), 13/16 (corner) and 4/5 (floor); over F_q(T) the coefficients are finite Gauss-sum expressions (Hoffstein 1992, Patterson 2007 and 2014)
proposed_attack: verify the function-field theta coefficient formulas by brute force at q = 7, n = 3, build the F_7[T] analogue of the cubic-theta probe with sextic averaging and check its original and Poisson representations and a function-field Lemma 7.1 exactly, repeat with the biquadratic theta and quartic averaging over F_13[T], price the template against exact sizes, and run a Davenport-Heilbronn type rival through the local identity
dead_routes:
- lower-order twists of the cubic theta over Q(sqrt(-3)), killed by reading in hunt 136 (S1, S2)
- the classical theta over Q(i), whose coefficients vanish off squares (hunt 136, S3)
- reading a function-field identity or verdict as a theorem about number fields or about RH for zeta
- treating a zero of a linear combination of L-functions as a counterexample to a statement about single L-functions (hunt 120)
required_oracles:
- exact arithmetic in cyclotomic fields, with every equality checked as an equality of reduced polynomials modulo the cyclotomic polynomial
- brute-force enumeration of residue symbols and Gauss sums over F_q[T] at small degree
- PARI/GP through cypari2 as a second implementation sharing no arithmetic code with the first
- Arb ball enclosures from python-flint for the rival's zeros
- the cited source texts, quoted by line with their SHA-256 checked
kill_conditions:
- brute-force Gauss sums or residue symbols disagree with the cited function-field formulas at q = 7, n = 3, degree at most 3, and no corrected formula fits
- the original and Poisson representations of the function-field cubic probe disagree at any checked degree
- a planted wrong Gauss-sum sign or wrong residue symbol leaves the equality checks green
- the second implementation fails to reproduce an exact number reported here
- the rival's zeros all lie on |u| = q^(-1/2), or its local identity holds where the single L-function's does
agents_may:
- read and quote sources
- download sources into new cache directories as untrusted data
- derive
- code exact arithmetic
- attack derivations
agents_may_not:
- declare novelty
- declare theorem status for anything not kernel-checked
- transfer a function-field verdict to number fields as established
- promote their own claim
- push, publish or contact outside parties