# Verification report

Paper: Full Linear Homomesy Spaces for Rowmotion on Rooted Forests.
Author: Alper Ferudun, Mercury Software GmbH. Date: 29 September 2026.

The complete proof is a structural induction: root insertion changes one
marked cycle; a Cartesian-product deletion lemma computes the full and
deleted affine hulls; Gaussian elimination gives complete bases. A scalar
recursion determines their dimensions. A positive antichain homomesy
establishes its nonzero origin defect. Existing tree-orbit and Coxeter
mechanisms are explicitly credited.

The core standard-library checker independently enumerates lower ideals
by bit masks, computes rowmotion from its definition, and decomposes the
permutation into cycles. It compares full/deleted augmented-row RREFs,
marked averages and lengths, transitivity, kernels, and dimension formulas.
It covers all 3,047 unlabelled rooted forests of size at most 10, 275,986
ideals, 47,043 cycles and 2,646 toggle orders for forests of size at most 5.
There are 731,730 exact assertions. The extension checker adds 53,141,
testing positive antichain means and the closed rank formula on the same
forests, plus a 101-vertex star and a 100-vertex chain without enumeration.
Combined: 784,871 assertions per runtime, not 6,094 distinct forests.

Fresh isolated Python 3.9.6 and 3.13.5 runs pass and their reports agree
apart from runtime version. The general written proof does not depend on
finite exhaustion. The tests and audit are by the originating researcher;
they are not independent peer review or formal verification.

Scope: ordinary rowmotion on finite rooted forests; an attributed existing
theorem extends ideal-indicator conclusions to Coxeter ideal toggles.
No arbitrary-poset, antichain-promotion, piecewise-linear, birational or
sharp bit-complexity claim. The full source problem is not marked solved.
Novelty confidence remains limited; no absolute priority is certified.

Six-page PDF: native editor compilation, warning-free Tectonic export and
inspection of every final rendered page completed. Detailed hashes,
source reading extents and self-audit are retained in the source archive.