Run 37fb06a9-44a6-42d5-abc3-4c1342b1287b, 2026-08-18. Reads hunts/frontier_math/K2-TWO-SPECIES.md (section 5, the T1/T2 split), hunts/frontier_math/two_species.py, hunts/frontier_math/counting_lemma.py (defect #24's exact critical-lattice limit).
This is a probe, not a proof attempt. k >= 2 in full has been open for the life of the frontier hunt; k >= 3 is not claimed here and is not closed here. Nothing in this hunt is evidence for or against RH (docs/08). The reserved certification word is not used.
The question
K2-TWO-SPECIES.md section 5 splits blocker 2 into two obligations. This hunt attacks the first:
(T1), the centre gas.
2*sum_{p!=q} [Dam(2y, tau_pq) - Kpair(tau_pq)] <= 2k*Shq(y)*(1 - rho)for some atom reserverho > 0.
Settle it, or establish exactly where it resists and say so plainly.
What this hunt may and may not touch
Its own directory, hunts/frontier_math/ (per the run's scope override), and a case-log entry in hunts/README.md. Not hunts/frontier_math/zeta23ext/ (another run is live there), and not meta/, harness/, lean/, zeta/, ontology/, or any root markdown file.
id: r_b9552d
question: does the centre-gas obligation T1 of the two-species split admit a bound with a strictly positive atom reserve rho, over all centre configurations rather than only lattices?
frontier: k=2 equal depths closes under ball enclosures over 6600 cells in three cap modes (run 78b4ce8e, worst margins +0.0677 signed / +0.0146 unsigned); k>=3 open; T1 measured at 87.8% of the per-centre budget on the worst uniform lattice, with defect #24 giving the exact critical-lattice limit lim B/k = c2(0) - A^2 = 0.00517169408367867955 by Poisson summation at spacing 2*pi
dead_routes:
- per-pair domination, the cap is superadditive above the multiplicity threshold
- separation hypotheses, real zero gaps have no positive lower bound and a positive fraction of GUE gaps fails at any delta
- convex relaxation of the atom constraint, only integrality pays back the -nA^2, exact rational witness on file
- Cauchy-Schwarz on the positive-definite kernel, 18x to 111x too weak
- repulsion-free accounting, arithmetically dead from n=8
required_oracles:
- the identity residual against gram_form.slack_direct and kpair_identity.slack_k, which no accounting may contradict
- defect #24's Poisson-summation limit c2(0) - A^2, derived independently of any search here
- a planted-fault ladder on the configuration search, which must fire before a no-violation verdict is reported
- the numpy kernel checked pointwise against gram_form's scalar cmath path
kill_conditions:
- the reduction GAS = k*Shq - 2B + P fails to be an identity to double-precision rounding
- the configuration search reports no violation at any planted damage inflation, so it has no demonstrated power
- the run exceeds its time budget with no measured result
- a recorded dead route is being re-derived
agents_may:
- restate T1 exactly against the already-proved budget B and measure the residual
- search the configuration space for the gas extremum and report the search method and its power
- assess a named proof route and report it dead or open with the witness
- report that a recorded number did not reproduce, with every reading tried
agents_may_not:
- claim k >= 3, or claim T1 proved, from any measurement here
- use the reserved certification word
- assert that a recorded number is wrong when only a failure to reproduce it was observed
- describe a scan sup as an enclosureKill conditions actually reached
None of the kill conditions fired. The LP assessment of the certificate route (item E of probe_37fb06a9.py) did not settle at this cost, which is reported as such in RESULTS-37fb06a9.md rather than dressed as a bound.
Run 2: 113786a8-1f1c-4220-b772-15160a0274fa, 2026-08-20
Run 1's artifacts are preserved under their run-tagged names (RESULTS-37fb06a9.md, probe_37fb06a9.py, results_37fb06a9.json, HANDBACK-37fb06a9.json); the unsuffixed files are this run's.
Between the two runs, main withdrew the G4 counterexample (a735965) and closed gap B of hunts/frontier_math/LATTICE-EXTREMALITY-ROUTE.md (7efd506) with an explicit Fejér majorant. Run 1's threads 1 and 2 are therefore answered elsewhere, and the same question, T1's gas half, now sits entirely on that route's gap A, the sparse side rho < 1/(2*pi).
id: r_b9552d-run2
question: what must a Cohn-Elkies certificate be to close gap A of the lattice-extremality route (the sparse side rho < 1/(2*pi)), and can the Fejer family that closed gap B be it?
frontier: gap B closed on main 7efd506 with v = K_1(0)*(sin(x/2)/(x/2))^2, so J(T) <= LP_v(rho) for every configuration at rho >= 1/(2*pi), with LP_v(1/(2*pi)) = L = 0.11433003938654052 the uniform-lattice row; gap A open, the route document's own sparse illustration being +2.15 against a true -0.043 at rho*2*pi = 0.4
dead_routes:
- per-pair domination, the cap is superadditive above the multiplicity threshold
- separation hypotheses, real zero gaps have no positive lower bound
- convex relaxation of the atom constraint, exact rational witness on file
- Cauchy-Schwarz on the positive-definite kernel, 18x to 111x too weak
- repulsion-free accounting, arithmetically dead from n=8
- the G4 1,1,2,1,1,2,3 configuration, withdrawn on main a735965 as a two-sided number compared against a one-sided one
required_oracles:
- the numpy kernel k1_vec checked pointwise against gram_form.kernel's scalar cmath path
- the closed forms 2*c2(0), K_1(0) and cos^2(sqrt2/2)*(1+cosh 1) re-measured by independent scans rather than quoted
- two_species.centre_gas_row_closed as the lattice row L, computed on main and not here
- a planted-fault ladder on the sup f necessary condition, which must fire before a not-excluded verdict is reported
kill_conditions:
- sup f exceeds L/2, which would exclude every universal certificate and end the route
- the bound at c = 2*c2(0) fails to be constant in rho, which would mean the pinning argument is wrong
- the planted inflation of f fails to fire the necessary-condition test
- a recorded dead route is being re-derived
agents_may:
- derive the constants any density-independent certificate must hit, and test them
- report a family as unable to close a gap, with the frequency-domain witness
- state a bound weaker than the conjecture when it is the first of its kind on that side
agents_may_not:
- claim gap A closed, lattice extremality proved, T1 proved, or k >= 3
- use the reserved certification word
- describe a scan sup as an enclosure
- present the quoted Paley-Wiener factorisation and sampling steps as proved hereKill conditions actually reached, run 2
None fired. sup f = 0.018743 sits below L/2 = 0.057165 by a factor 3.05, so the route survives its necessary condition; the planted 3.05x inflation fires the test, so the test has power.