Internal Review and Sources

Report: TA-TR-2026-22. Version: 1.0. DOI: 10.5281/zenodo.23103274.

The analytic proof was internally checked for strict prefix separation, ancestor comparison signs, deterministic multiplicity bounds, unconditioned independent spins, bounded differences, the small-tail dichotomy, all-sweep union bounds and the liminf constant. The deposited finite verifier covers 5,161,156 rank sequences, 36,127,300 one-spin comparisons, 7,898 prefix cuts, and 5,912 statistic-identity permutations. Coverage limits are given in the supplement and receipt.

The mathematical review receipt binds the original reviewed sources and verification script. Publication additionally requires a separate review of the exact DOI-bound PDFs and manifest. No independent external peer review or separate final human line-by-line verification is claimed.

The main manuscript references the directly relevant original works of Padrol and Philippe; Maclagan; Edelman, Gvozdeva and Slinko; Bassino et al.; Gessel and Zhuang; McDiarmid; Stanley; and Pinsky. Known tree representations, ancestor rules and run statistics are credited. The contribution is the proved quantitative combination under the standard-cube prefix condition and all-sweep requirement; exhaustive historical priority is not certified.

The constant-density conjecture and deterministic explicit constructions achieving the general lower bound remain open here. This version contains no Gray-family extremal claim or biological-data claim.