# Verification report

For each fixed finite connected loopless multigraph G and degree d, all positive regular subdivision parameters with divisorial gonality at most d form an effective finite union of finitely generated additive submonoids. Consequently the gonality sequence eventually depends only on gcd(k,M) for a computable M. For every DSW tricycle it is exactly 5 for even k and 6 for odd k, with a necessary midpoint form for degree-five reduced divisors.

# Originating-system adversarial audit

1. Definition and parameters: graph is connected, finite, loopless; parallel edges counted. Divisor equivalence uses div=-L and integer potentials. Our k is the number of replacement edges, not inserted vertices. Degree exactly d is obtained from degree<=d by adding chips. Edgeless singleton handled separately.
2. Uniform bound: for a generic superlevel cut of a firing potential, the sum of positive differences across the cut is D(U)-E(U)<=d. This bounds every slope. No unproved global bound on firing duration is used.
3. Finite templates: all d(n+1) chip labels are partitioned into coincidence blocks, placed on original vertices or ordered along edges. Actual coincidences must be grouped; splitting each label separately can require artificial intermediate slopes exceeding d. This rejected shortcut is not used.
4. Completeness: all slope changes at degree-two vertices occur in the union of the witness supports. Integer actual gaps and bounded slopes give one enumerated cone. Original vertex witnesses suffice by DSW Cor2.5, not an unsupported sample-of-vertices inference.
5. Soundness at zero lengths: sum divergence equations on merged nodes; internal contributions cancel. Equal edge sums k>0 forbid merging distinct original vertices. Cycle equations integrate slope-times-length; zeros descend. Integer slopes and lengths give integer unit-vertex potentials. Forced witness labels remain at their original target vertices. No spurious metric-only divisor enters the discrete conclusion.
6. Gordan/Apéry step: pointed rational cones have finite integer generating sets by simplicial triangulation and bounded remainders. Nonzero points have k>0. Projection is exactly an additive monoid. Normalize by each generator-set gcd and compute residue minima; the exact membership rule and eventual divisibility follow because min(B) is itself a generator. The union need not itself be one monoid. Taking degrees1..n supplies a global finite tail table, since sum(original vertices) has positive rank on every subdivision.
7. Tricycle input: the degree-five reduced normal form and lower bound are DSW Lemma4.5/Cor4.6, not a new lemma. Every original tricycle has spokes>=2; regular subdivision gives lengths>=2k. Outer cycle arc lengths can be arbitrary. The proof uses their two disjoint arcs, not equal arc lengths.
8. Legal wave: at time0 two coincident root chips pay two boundary edges; later four boundary chips each pay one. Transition distances are a<b<=k, and root chips cannot arrive before a. At a=0 no firing is required. The third transition chip is stationary.
9. Burn after unequal arrival: chosen target is the empty marked endpoint of the first cycle. Its one chip cannot stop two arcs. The vacated transition reaches a second cycle with at most one chip, hence burns it. Root burns from an empty spoke when a>0, or from four spokes against two chips when a=0. The last cycle is reached by its two empty spokes; if both marked endpoints hold chips, the two remaining transition interiors are empty and supply extra fire. Interior chips then burn from both sides. DSW root-reduced properties are not incorrectly reapplied after firing.
10. Midpoints: the cyclic equations p_i=k-p_(i-1) force k/2 on three paths. Even midpoint waves reach all six marked vertices; rank-one test uses the seven-vertex loopless coarse model. Six marked chips give the upper bound by a legal simultaneous inward wave until the shortest spoke reaches r.
11. Exact finite checks: minimal graph all placements1<=k<=12 and every unequal pair:22386 complete wave/burn certificates,4674876 checked conditions. Unequal arcs/spokes1<=k<=6:1932 certificates,367752 conditions. Separately implemented Dhar search1<=k<=7 matches midpoint necessity. Apéry/DP comparison:298 generator sets and294381 conditions. These are finite regression checks, not a formal proof or external independent review.

No gap found in the complete written arguments. Scope and novelty remain separate from mathematical acceptance. No new full-source closure is claimed.

# Bounded literature and novelty review,25October2026

Queries included gonality regular subdivisions eventually periodic; gonality semilinear; gonality periodic subdivisions; tricycle gonality odd/even/parity; gonality subdivisions Hilbert basis; gonality Presburger; and exact tricycle graphs gonality. Primary results included arXiv2106.12568, the publisher DOI record, its institutional MPG version, BWZ2021, Luo2009/2011 and HKN2013. A Delft dissertation search result reprints the DSW work; it is not a new parity theorem. Broad unrelated semilinear/incidence-geometry results were not used.

DSW already proves the unit-metric minimum, failure of subdivision invariance, and the original tricycle six-to-five drop. Its Section3 and Remark3.7 already use the same homogeneous cycle constraints and allow zero lengths. BWZ provides finite integer certificates for arbitrary subdivision minimization. The present contribution is to retain the integral uniform scaling parameter, apply finite generation to all types, derive an exact all-k monoid description and eventual gcd law, and prove midpoint necessity yielding the entire tricycle parity sequence. These are consequences/refinements of credited methods, not discovery of the tricycle or a new proof of Baker's already-refuted invariance claim.

No matching all-k theorem was located in the bounded search. Unindexed, unpublished and all database results were not exhaustively checked. Absolute priority is not certified. No claim is made that the separate odd-fold-drop question for other graphs has been answered.

Six final PDF pages were directly visually inspected, with no clipping, overflow, missing symbols or unresolved citations. Built-in compiler and Tectonic succeeded. Author name is prominent; Mercury Software GmbH/contact is a footnote; AI assistance is ordinary prose.