q6 claim ledger

Status: local exact value matching the published record. No improvement of that record is asserted. The q5 equality $m(59)=15$ is not rewound.

The checked 15-set p61_upper.json and the completed size-at-most-14 exclusion together give

m(61)=15.

OEIS A398173 already publishes this value (published_oeis.json, retrieved 2026-09-09). It is new to the local exact table.

The upper half is

A=\{0,1,2,3,4,6,15,21,22,24,42,49,55,56,58\}.

Python ordered-pair enumeration and the independent Rust reflection check agree that no ordered multiplicity lies in $\{1,2\}$ .

The lower half is the exact predicate

\nexists A\subseteq\mathbb Z/61\mathbb Z:\quad 2\le |A|\le14,\quad A\text{ has no unique sum}.

No existence monotonicity in cardinality is assumed. The argument reuses the q5 engines (C reflection cover; Rust unordered-pair brancher) with the affine root $\{-1,0,1\}$ and the AP-length ladder:

AP5: C UNSAT after 810,493 nodes (p61_ap5_cover.json); Rust UNSAT after 1,476,025 nodes (p61_ap5_exact.json).
AP4 but not AP5: 28 named midpoint classes for center $-1=60$ on the AP4 root. C and Rust both return UNSAT on every representative (ap4_not_ap5_classes.json, dual_ap4_not_ap5.json).
AP4-free: 26 named midpoint classes for center $1$ on the AP3 root. C and Rust both return UNSAT on every representative (ap4_free_classes.json, dual_ap4_free.json).

bash compute/q6/run_all.sh from the problem folder replays the witness, small controls inherited from q5, stored class decisions, and the AP5 ceiling. Full named-class searches can be restarted with ../q5/midpoint_cases.py and ../q5/dual_classes.py.

Falsifiers: a unique unordered representation in the saved 15-set; any valid set of size at most 14; a SAT result on the AP5 family or on a named-class representative recorded as UNSAT; a missing midpoint pair or invalid affine identification in the partition.