slug: mao-wang-2026-delannoy-square-negative-roots
bibkey: maowang2026narayana
doi: null
url: https://arxiv.org/abs/2607.01572v1
triage: theorem
motivation_gids:
  - D5/S1/Recurrence/Algebraic/DelannoySquareRoots.source_correspondence
  - D5/S1/Recurrence/Algebraic/DelannoySquareRoots.upper_circle_roots

Negative roots of the original Delannoy matrix square

Problem

Conjecture 4.1 of Mao and Wang, The Narayana transformation, asks whether the row polynomials of A² and D² have only real nonpositive roots. The target here is only the original path-count D² clause.

For every natural n and complex z, the exact target is

[ G_n(z)=0\quad\Longrightarrow\quad \operatorname{Im}z=0 \quad\hbox{and}\quad\operatorname{Re}z<0. ]

Here D(n,k) counts actual E/N/NE paths to (n-k,k) and G(n,k)=sum_(j=k)^n D(n,j)D(j,k). Row zero is the constant polynomial one. The target has no assumptions about root counts, interlacing, recurrence stability or real-rootedness of a surrogate family.

Motivation

The actual path-count square is a recent source question distinct from ordinary Delannoy root formulas and from the Eulerian clause.

Gap

The available formal prerequisite is source_correspondence: exact path enumeration, the original matrix-square generating function in every degree, nonvanishing of the ordinary row at 1-sqrt(2), and the universal quadratic divided difference. The theorem upper_circle_roots constructs at least N distinct actual squared-row roots in the central interval in every degree, where N counts ordinary-row zeros above sqrt(2)-1. The complex-root implication remains open in Lean.

Route

The remaining proof uses P_n(y)=(-1)^n G_n(-y) and the involution v=(1-u)/(1+u), with y=1/(uv). The formal divided difference uses u != 0, v != 0, u+v+uv=1 and yuv=1 and states

[ y(v-u)P_n(y)=\frac{T_n(-u)}{u^{n+1}}- \frac{T_n(-v)}{v^{n+1}}. ]

It is proved for every degree from the original squared rows. Applying it as a quotient additionally requires v != u. The standard Chebyshev-U description supplies the ordinary-row zeros. The threshold a=sqrt(2)-1 is excluded by the formal endpoint result.

The root construction has two complementary parts. On u=-1+sqrt(2) exp(i phi), 0<phi<pi, the formal argument crossing count in upper_circle_roots produces n-k distinct actual positive roots of P_n, where k counts ordinary-row zeros below a. Alternating real Chebyshev samples must produce k-1 further roots when k>=1, in a disjoint interval. Closed-arc argument continuity, endpoint phases, interior crossings, polar realness, actual-root correspondence, strict parameter injectivity and the central interval bounds are proved. The real-interval sample signs and their distinct roots remain obligations. Dividing out these roots leaves degree at most one; realness of the remaining factor and positivity of the original coefficients then recover the exact complex-root conclusion. When k=0, the circle count alone must supply all n roots.

Falsifier

Any natural row with a complex zero outside the strictly negative real axis would refute the target. Failure of an auxiliary count would refute that route without settling the original question.

Evidence

The formal definition negativeRoots is the exact original complex-root target. The theorem source_correspondence supplies the path, series, threshold and divided-difference prerequisites. It does not prove negativeRoots. The theorem upper_circle_roots proves the unconditional central-interval root count with its actual ordinary-row factors. It neither assumes the missing complementary count nor resolves the final target.

Triage

theorem: the target is a universal mathematical assertion. There is no formal resolution claim for Conjecture 4.1 or its D² clause.

ASSUMED-UNVERIFIED

Literature absence and publication priority are not established. The bounded comparison found no exact all-degree squared-row supplier; ordinary Delannoy formulas and generic Chebyshev, argument-continuity, IVT and polynomial division results do not by themselves discharge the complementary counts.