# Verification report

## Result and scope

For a finite quandle with commuting right translations and orbit stabilizers H_i in L = Z^r, the ith orbit torsion group is

\[
T_i=\bigwedge^2L\big/\left(\sum_j e_j\wedge H_j+\bigwedge^2H_i\right).
\]

The manuscript proves the full decompositions of second integral rack and normalized quandle homology, the universal-coefficient formulas for second cohomology with any constant abelian coefficient group, and the sharp orbit-size annihilator. It gives a twenty-element example refuting the stronger proposed orbit-exponent annihilator.

The source's second-cohomology wording is retained. Integral second cohomology is torsion-free; the nontrivial integral torsion calculation concerns second homology and determines the additional cohomology factors for other constant coefficients. No twisted-coefficient, nonabelian-coefficient, arbitrary-medial-quandle, or higher-homology conclusion is claimed.

## Written argument

The proof credits the existing classification and class-two structure group of Darné and Lebed--Mortier, and the covering methods of Eisermann and Lebed--Mortier. It supplies the structure-group proof in the chosen right convention, computes the stabilizer commutator lattice exactly over the integers, identifies homology through a presentation covering, and checks that normalization removes one primitive free factor per orbit without removing torsion. Smith normal form of the stabilizer inclusion proves the annihilator.

The source and bounded literature audit are included. No absolute priority or exhaustive novelty certification is claimed.

## Exact finite checks

There are 52 distinct finite examples: 51 in `verify_lattice_formula.py` and one additional twenty-element example in `verify_annihilator.py`. The scripts perform 41,604 and 17,021 explicit conditions, respectively, for a total of 58,625 in each Python mode. Normal and optimized executions agree after removing timestamps and elapsed times.

The checks build operation tables, verify the quandle and commutation axioms, derive stabilizer lattices by Schreier generators, compute their Hermite and Smith forms, and compare predicted torsion and free ranks with the direct rack and normalized quandle boundary matrices, separately on each orbit.

The first orbit in the twenty-element example has orbit group (Z/2)^2 but torsion group Z/2 + Z/2 + Z/4. Thus its orbit-group exponent does not annihilate its torsion group. All three orbit components in that example are checked directly.

These finite calculations do not prove the general theorem. Both matrix routes use SymPy's Smith form implementation; they are not independent Smith algorithms. The scripts use explicit exceptions rather than assertions that disappear under optimized Python.

## Review status and reproducibility

The written proof is self-audited and AI-assisted. No independent expert review or proof-assistant verification has been completed. The acceptance record covers the stated theorem, not a claim that the wider dataset or the entire Oberwolfach report is solved.

The source archive contains the manuscript and executable checks, not third-party source PDFs. The package build record records portable compilation and replay results. The PDF is inspected page by page for layout, author formatting, formulas, and references before readiness is registered.