Raney Conjecture 7.1: zero-run record families

Eu, Huang, and Kao, arXiv:2609.25742v1, Conjecture 7.1, state:

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 source quantifies over positive k,r and prime p with p not dividing k*r. Its Raney numbers are

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

The source's left-to-right records allow ties and come from maximal zero runs in the full infinite sequence. The formal target was preregistered in issue #9544 with these parameters, the literal natural quotient, tied records, and the source-to-formal implication. It does not ask for a converse saying that every term of every family occurs.

The problem asks for a uniform description of record zero-run lengths for the entire Raney family, extending beyond the paper's (3,2) focus and the known Catalan (2,1) case. The delivered theorem proves the stronger containment of all actual maximal finite zero-run lengths in one finite family selected before the endpoints, and separately rules out an infinite terminal zero run.

Before this delivery the repository contained neither the general maximal-block coefficient theorem nor a formal bridge from the literal integral Raney quotient to a uniform finite-state fixed word. The source paper records Conjecture 7.1 rather than a proof of its full k,r,p statement.

A bounded prior search inspected this repository, the current arXiv source and its known cases, Crossref results, and a fresh global issue search for the exact conjecture. It found no full settlement in that scope. A broad Crossref query returned unrelated items. OpenAlex returned HTTP 429; that is not absence evidence. No worldwide priority or exhaustive literature certification is claimed.

MaximalBlockEvolution formalizes literal morphism powers, support stabilization, actual maximal intervals including first=0, and boundary-safe uniform versions of BKS Lemmas 11 and 12. MaximalBlockDescent uses one stabilizing power throughout, builds actual predecessor/image steps with literal context words, descends strictly in the right endpoint, and reaches finite families of early roots, bounded late root words, and bounded contexts.

BoundaryPivotTransport follows the complementary boundary letters through two consecutive descent edges. Rightmost left pivots and leftmost right pivots make both signed endpoint corrections functions of a finite paired pivot state. Finite-state repetition yields eventual periodicity. FinitePathDisplacements normalizes periods by the factorial of the paired-state cardinality, combines finite roots with finite correction paths, and solves the resulting affine recurrence. It chooses one finite set of integer (a,b,c), with c>0, before an arbitrary actual maximal interval and proves c*length=a*P^m+b.

ZeroRunFamilies then works with the literal natural quotient. A guarded adjacent-binomial identity proves integrality and the exact difference formula before reduction modulo p. With A=max(k-1,r-1), the state space Fin(A+1) x Bool stores ordinary values choose(k*n+a,n) and guarded predecessor values choose(k*n+a,n-1), taking the latter as zero at n=0. For each base-p digit, the transition records the bounded carry (k*d+a)/p; the Boolean branch records the predecessor borrow through a zero digit. Lucas' theorem proves the transition weights, including leading-zero compatibility, so the complete evaluation vector is a pointwise fixed p-uniform word.

The source residue is read exactly as the ordinary component at r-1 minus (k-1) times its predecessor component. For all sufficiently large j, the same construction proves R_(k,r)(p^j)=k modulo p; since p does not divide k*r, this gives unbounded nonzero support. Applying the generic actual-block theorem to the zero letter set proves the displayed coefficient equation for every maximal finite zero interval. Hence every tied left-to-right record value belongs to the finite union required by Conjecture 7.1.

This route uses neither a finite-prefix approximation nor an assumed automaticity theorem. It makes no converse-attainment claim. The (3,2) and Catalan cases are not counted as separate results.

A positive k,r, a prime p not dividing k*r, and an actual maximal finite zero interval whose length satisfies no equation c*L=a*p^m+b from the one finite coefficient set would refute the formal endpoint. An index cutoff beyond which every Raney residue is zero would refute its unbounded-support conjunct. Either finding would also invalidate the route from the stronger theorem to the source's tied record claim.

• Preregistration: issue #9544.
• Source: Eu, Huang, and Kao, arXiv:2609.25742v1, Conjecture 7.1.
• Literature mechanism: Bugeaud, Krieger, and Shallit, arXiv:0808.2544v2, Lemmas 10-12 and Corollary 13.
• Formal owner chain: MaximalBlockEvolution -> MaximalBlockDescent -> BoundaryPivotTransport -> FinitePathDisplacements -> ZeroRunFamilies.
• Final theorem: D5/S1/Recurrence/Raney/ZeroRunFamilies.raney_zero_run_coefficient_families.
• Its first conjunct is unbounded nonzero support. Its second chooses the finite coefficient set before all interval endpoints and covers every IsMaximalDeltaInterval {0} in the literal Raney residue sequence.
• The accepted implementation retains only the standard axiom profile propext, Classical.choice, and Quot.sound; canonical report and Freeze identities are generated by repository tooling rather than asserted here.

theorem. The formal result proves a stronger all-actual-runs statement and unbounded support, from which the literal tied-record assertion follows. Its proof_shape is content and its admission_basis is escape-witness: the finite-state Raney construction, exact integral readout, nonzero-support proof, and actual-block coefficient assembly are live new proof content. The delivery does not rely on the bind-only open-problem admission exemption.

The arXiv title, authors, version, submission time, and literal Conjecture 7.1 were read from the primary v1 pages on 2026-09-24. The BKS title, authors, version history, and relevant primary-body lemmas were also read directly. These bibliographic facts, the source-to-formal interpretation, and issue #9544 are not kernel-checked.

The negative prior-art result is bounded to the repository, arXiv, Crossref, and the fresh global issue query described above. OpenAlex was unavailable with HTTP 429. Worldwide priority, exhaustive absence of another proof, and novelty beyond the stated repository strengthening remain unverified.