MISSION `r_401bbf`: the k=2 table with `zone_trade`'s inner prune disabled

The question

hunts/frontier_math/k2_closure.py::zone_trade maximises, over integer zone multiplicities with sum m <= 10, the adversary's zone trade

V(m) = sum_i cap_i*m_i - sum_i q_ii*m_i*(m_i-1) - 2*sum_{j<i} q_ji*m_j*m_i

by depth-first search. One of its two cuts is the line

if v > val - 1e-18 or m == 0:

which refuses to descend into a multiplicity that does not immediately improve the running value. That cut restricts the adversary's search, and a cut on the adversary's side is the unsound direction: wherever it bites, the trade comes out too small, the cell's deficit is understated, and the published margin is optimistic.

Run 78b4ce8e (hunts/r_a97060/, the enclosure pass) recorded the assumption in its §4 as unstated and load-bearing, with a measured delta of 0.0, but on six binding cells only, not on the 6600 the table publishes.

This hunt settles it on every cell. It does not extend the result, does not touch k >= 3, and does not touch the unequal-depth quantifier.

What "disabled" was taken to mean, and why it was widened

The brief says: re-run with the inner prune disabled. Merely deleting that line would leave the other cut in place:

if val + rem * caps[idx] <= best: return

an ordinary branch-and-bound bound whose admissibility rests on the same hypothesis the inner prune needs (nonnegative pair charges, so that no completion can gain more than rem copies of the largest remaining cap). Since the surviving components in this table are small (at most eight zones with a nonzero cap in any connected component of the whole 6600-cell table, six in the signed mode), the honest instrument is to drop both cuts and enumerate every one of the C(Z+10, Z) <= 43758 admissible multiplicity vectors, taking the exact maximum. That is a strict superset of what the inner-prune-free search explores, so a zero delta against it settles the brief's question a fortiori, and it settles the outer bound at the same cost.

Both trades are evaluated on the same zone data, cell by cell, so each cell's delta is an exact difference and not a comparison of two independently scanned tables.

Scope

May write: hunts/r_401bbf/; and, by the operator addendum of 2026-08-17 which overrides the generic "do not touch hunts/frontier_math/" line for this run only, hunts/frontier_math/k2_closure.py, its tests, and K2-TWO-SPECIES.md; plus the case-log entry in hunts/README.md.

May not write: zeta/, ontology/, harness/, meta/, lean/, any root markdown file, hunts/frontier_math/zeta23ext/ (a concurrent run owns it).

Standing: nothing here is evidence for or against RH (docs/08). The reserved word of zeta/rigor.py and the Lean arm does not appear in this directory, disclaimers included. The lexical guard reads the bytes.

id: r_401bbf
question: Does the inner prune in k2_closure.zone_trade ever understate the adversary's zone trade on any of the 6600 published k=2 tau-cells?
frontier: measured delta 0.0 on six binding cells (run 78b4ce8e, hunts/r_a97060 section 4); unchecked on the other 6594
proposed_attack: solve every cell's zone trade twice on identical zone data, once with the published search and once by exhaustive enumeration of all multiplicity vectors with sum m <= 10, and report the per-cell delta
dead_routes:
  - sampling binding cells only, which is what left the assumption open
  - re-tuning zones or widening a cap to rescue a cell, forbidden by the brief
required_oracles:
  - exhaustive integer enumeration of the multiplicity lattice
  - the published table's own recorded numbers as a reproduction check
  - a planted instance with a negative pair charge, on which the prune must be seen to bite
kill_conditions:
  - the exhaustive solver ever returns less than the pruned search on a real component
  - the transcription of the published search disagrees with the published search
  - the planted prune-biting instance is not detected, leaving a zero delta meaningless
agents_may:
  - search
  - derive
  - code
  - attack
agents_may_not:
  - declare novelty
  - declare theorem status
  - promote their own claim
  - re-tune the table

id: r_401bbf-2026-08-17-2b9b08aa
hunt: r_401bbf
started: 2026-08-17T21:50Z
finished: 2026-08-17T22:45Z
ran:
  - python hunts/r_401bbf/probe.py --quick --charges
  - python hunts/r_401bbf/probe.py --charges
outcome: the inner prune never bites on any of the 13200 cell evaluations, and it is sound by argument because every pair charge in this table is a square over 200
artifacts:
  - hunts/r_401bbf/results.json
  - hunts/r_401bbf/RESULTS.md