This is a hunt. Nothing in hunts/ is a result, and nothing here is evidence for or against RH (docs/08).
A support run launched by a parent session working hunts/r_186989, to answer one bounded question independently and hand the answer back.
The bounded question
Let $S$ be $k$ primes and $A$ a set of distinct positive integers with every off-diagonal sum $a+b$ $S$-smooth; $g(k) = \max|A|$; the target is $\log g(k) = o(k)$.
Seek a clean size dichotomy: either $A$ is spread out enough that a descent/large-gap argument applies, or $A$ lies in a controlled interval where smooth-number counts apply. Optimise uniformly over $S$ and the scale of $A$. Return an explicit bound, or the exact normalisation/height barrier. Audit hunts/r_186989/RESULTS.md on the way.
Scope
May write: this directory and one case-log entry in hunts/README.md. May not write: any other hunt, zeta/, harness/, lean/, meta/, or any root markdown file.
What it measures
probe.py, stdlib only, ~20 s:
- re-verification from scratch of every witness in
r_186989/RESULTS.md§3; - exhaustive enumeration of primitive admissible 3-, 4- and 5-element sets through their sums (not their elements), across growing cutoffs, to see whether any height bound exists after normalisation;
- a rigorous two-sided sandwich (Rankin upper, simplex-volume lower) on $\log\Psi(N,S)/k$ as a function of $\log N/\log\mathrm{rad}(S)$, which is the counting horn's threshold.
id: support_95bb5cb7
question: Does a size dichotomy (large-gap descent vs smooth-number counting in a controlled interval) yield log g(k) = o(k) for Erdos 126, and if not, where exactly does it fail?
frontier: f(n) >> log n classical, unimproved since 1934; equivalently g(k) <= exp(O(k)). No upper bound on g(k) is established anywhere in this tree.
dead_routes:
- element-bounded clique search: reports one witness per k and licenses false claims about all optimal witnesses (Hunt #91, corrected here)
- a multiplicative composition law for g: it would refute the conjecture, not prove it (Hunt #91 section 4)
required_oracles:
- full trial division of every off-diagonal sum, recomputed from scratch, never reusing the search predicate
- Rankin's inequality for Psi(x,S), valid for every sigma > 0
- simplex lattice-point volume as an unconditional lower bound on Psi
kill_conditions:
- the counting horn's threshold turns out to depend on the choice of S or the scale, so no uniform statement exists
- primitive admissible sets of sub-extremal size turn out to have bounded height, which would revive the descent horn
- the two horns are found to overlap, which would settle the arm positive
agents_may:
- write hunts/support_95bb5cb7/ and one case-log entry in hunts/README.md
- report corrections to sibling hunts without editing them
agents_may_not:
- write any other hunts/ directory, zeta/, ontology/, harness/, lean/, meta/, or any root markdown file
- claim any upper bound on g(k), or any bearing on RH
- use the reserved word belonging to zeta/rigor.py and the Lean arm