bibkey: feldman2026missingzigzag authors: David V. Feldman year: 2026 title: “The Missing Zigzag: Cycles of Semitone Trichords and a Conservation Law in Equal Temperament” doi: null url: https://arxiv.org/html/2609.26114v1 claim: “Conjecture 9.3 gives the initial value and both parity recurrences for the balance-only count of labelled zigzag choices at n=3t, t>=2.” strata_touched:

• D5/S3/Combinatorics/Zigzag/Choices
• D5/S3/Combinatorics/Zigzag/FeldmanConjectureNineThree license: CC-BY-4.0 triage: anchor

Feldman’s balanced zigzag conjecture

Verified locator

The checked primary version is https://arxiv.org/html/2609.26114v1, Section 5, equation (2), Conjecture 9.3, and Open Problem 13.1. The checked arXiv abstract/API records submission on 21 August 2026; that date is taken from the record, not inferred from the identifier. No journal DOI or final publication date is asserted. The paper/docs license is CC BY 4.0.

Source semantics

For n=3t, t>=2, choose one directed form at each k=2,...,n-2 in ZMod n: I (1,k-1), II (k,1-k), III (-1,k), IV (k-1,-k), V (-k,1), or VI (1-k,-1). At k=2 only II, III, IV, and V are allowed. Form labels are distinct even if endpoint pairs coincide. A choice is balanced when outgoing minus incoming edge multiplicity is zero at every nonzero residue. The count a(t) imposes no step-sum closure, connectivity, Eulerian-circuit, or cycle-multiplicity condition.

Conjecture 9.3 asserts a(2)=4, a(2r)=4a(2r-1) for all r>=2, and (2r)a(2r+1)=6(2r-1)a(2r) for all r>=1. The repository’s stronger closed forms are a(2r)=4*6^(r-1)*choose(2r-2,r-1) and a(2r+1)=2*6^r*choose(2r-1,r) for every r>=1. They imply exactly those three clauses for the literal balanced count. They do not settle all of Open Problem 13.1, Conjecture 9.2, Hamiltonian existence, or the spectral claims.

Prior art and provenance

The author’s repository at immutable commit 5da74e8b5a1b19ff724a5c16b3b162c67bf172dd has Mzp.lean’s formPair for the six forms. Its BadPlan12 treats the stricter balanced-and-closed case at n=12; T6 is explicitly withdrawn; and its dynamic program includes even antipodes but omits the odd singleton. It supplies neither this all-n balance-only proof nor a reusable source-equivalent certificate. That code uses Lean 4.28 and Mathlib 8f9d9cff6bd728b17a24e163c9402775d9e6a365; this repository uses Lean 4.33 and Mathlib db584cd6d46c92f209a44c0f1c829460d327499d.

Choices.lean freshly transcribes the published Section 5 equation (2), compared against the author’s formPair; it imports no author code or certificates. The remaining modules construct a labelled finite path correspondence, a Laurent recurrence, and coefficient formulas in this repository. Any future actual source port would require preservation of its Apache-2.0 copyright, license, and NOTICE chain under A17.2; this delivery is not a port and requests no dependency upgrade.

A bounded publication screen covered the exact arXiv record/title/phrase, Crossref, the pinned author commit, and the linked Zenodo record. It found no later public proof or final journal article in that scope; the checked paper still lists the target as open. OpenAlex and Semantic Scholar were rate limited. Private, unindexed, and later work remains unverified. This is no worldwide absence or priority claim.