bibkey: euhuangkao2026zerorun authors: Sen-Peng Eu, Zai-Ting Huang, and Louis Kao year: 2026 title: “Zero-Run Spectra of the (3,2) Raney numbers Modulo Primes” doi: null url: https://arxiv.org/html/2609.25742v1 claim: “Conjecture 7.1: for fixed positive k,r and a prime p not dividing kr, the tied left-to-right record values of the zero-run sequence of the Raney numbers modulo p belong to a finite union of families (a*p^m+b)/c, with a,b,c depending only on k,r,p.” strata_touched:

• D5/S1/Recurrence/Raney/ZeroRunFamilies license: citation-only triage: anchor

Zero-run spectra of Raney numbers

Verified locator

The primary source is arXiv:2609.25742v1, submitted on 2026-09-22. Its HTML version at https://arxiv.org/html/2609.25742v1 gives the title and authors above. No later arXiv version was listed when the source was refreshed on 2026-09-24.

Literal source claim

Conjecture 7.1 states:

Fix k, r and a prime p not dividing kr. The left-to-right record values of the zero-run sequence of R_(k,r)(n) mod p belong to a finite union of families of the form (a p^m + b)/c, where a,b,c depend only on k,r,p.

The paper defines left-to-right records with ties allowed. Its zero-run sequence is formed from the actual infinite residue sequence, rather than from terminal runs created by cutting off a finite prefix. The source does not assert that every term of every displayed family must occur.

The Raney number used in the repository is the literal natural quotient

R_(k,r)(n) = r * binomial(k*n+r,n) / (k*n+r).

The quotient is proved integral before reduction modulo p; it is not replaced by division in ZMod p, which would be invalid when the denominator vanishes modulo p.

Formal scope

D5/S1/Recurrence/Raney/ZeroRunFamilies.raney_zero_run_coefficient_families proves a stronger statement than record containment. For every positive k,r and prime p not dividing k*r, it supplies one finite set of integer triples (a,b,c), with positive c, before arbitrary endpoints are chosen, and proves that every actual maximal finite zero interval has length L satisfying

c*L = a*p^m + b

for some triple in that set and some natural m. It also proves unbounded nonzero support, so no infinite zero tail is silently treated as a finite run. Tied record lengths are therefore included. No converse realization statement for the coefficient families is made.

The construction represents the finite evaluation word by states Fin(max(k-1,r-1)+1) x Bool. The Boolean component separates ordinary binomial values from the guarded predecessor value at n=0. Lucas transitions handle carries, the zero digit handles the predecessor borrow, and the fixed word equation includes leading zeroes. The literal Raney residue is read as the ordinary state at r-1 minus (k-1) times the predecessor state. Eventually, at indices p^j, this residue equals k modulo p; the coprimality hypothesis makes it nonzero.

The paper’s (3,2) results and the Catalan (2,1) specialization are known cases and receive no separate repository credit in this delivery.

Bounded prior-resolution evidence

A bounded search of the repository, arXiv, Crossref, and a fresh global issue query for Raney Conjecture 7.1 found the source, known cases, and unrelated records, but no full settlement in the searched scope. An OpenAlex request returned HTTP 429 and supplies no absence evidence. These observations do not certify worldwide novelty or first-publication priority.