# Verification report

For ordinary degree-n real polynomials whose derivative zeros are all mutually distinct, the number a_n of realizable Rolle words is at most (D_n+1)(2D_n+1)^n, D_n=n(n^2-1)/2. Relative to the classical count b_n of all interlacing words, log(a_n/b_n)=-n^2 log(n)/2+((3/2)log(2)-1/4)n^2+O(n log n). This answers the auxiliary asymptotic counting question attributed to Arnold by Shapiro and Shapiro.

# Originating-system adversarial audit

1. Ordinary monic degree-n coefficients form R^n. Scaling a nonmonic polynomial does not change its word. Fixed nonzero derivative leading coefficients exclude degree-drop or roots-at-infinity issues locally.
2. Every derivative pair resultant is a polynomial of ordinary coefficient degree <=2n-i-j by its Sylvester determinant. Specializing separately for each pair to p=x^n+x^i gives p^(j) proportional to x^(n-j) and p^(i)(0)=i!, so that factor is nonzero. A common specialization for all factors is not required: the coefficient ring is an integral domain. The sum bound is n(n^2-1)/2, checked independently.
3. Q includes Res(p,p'), hence every polynomial in its complement is simple. Real root count is locally constant there, so connected components meeting the all-real open set cannot escape into another root signature. Monic degree remains fixed.
4. Within a hyperbolic component, each derivative is simple and all-real by Rolle. Sorted roots are continuous, and no cross-level root can cross another without Q=0. Word is locally constant and therefore constant on a connected component. We only require a surjection from hyperbolic components to words, not the false/unproved assumption that a word locus must be connected.
5. Q also detects complex collisions, but every derivative root of a hyperbolic polynomial is real, so no realizable word is accidentally excluded. Extra nonhyperbolic components only enlarge the bound.
6. Existence for every n follows because a nonzero real polynomial Q cannot vanish throughout a nonempty open hyperbolicity domain. This supplies 0<=log(a_n), necessary for the two-sided logarithmic asymptotic.
7. Complement-to-zero-set graph is exactly tQ-1=0. Projection and c->(c,1/Q(c)) are continuous inverses. Its degree is at most D+1, its ambient dimension is n+1, giving Milnor exponent n, not n-1. Milnor's affine theorem does not require compactness. Connected-component count is b_0 for semialgebraic sets.
8. The classical factorial formula is an attributed theorem, not a newly established enumeration. Three separate Stirling sums give log b_n=(1/2)n^2log n+(1/4-(3/2)log2)n^2+O(n log n). The error includes replacing 2j-1 by 2j and N=n(n+1)/2 by n^2/2. Subtracting log b_n from log a_n=O(n log n) proves the exact stated logarithmic asymptotic, not an asymptotic equivalent of a_n.
9. Known lower bound2^(n-2) uses KS Theorem2/Proposition3 for n>=3 plus openness/density; n=2 is immediate. This comparison is not needed for the main theorem.
10. Checks: exact interlacing-poset DP agrees with the factorial formula for n1..10;99 exact degree-sum checks;20 symbolic resultants and nonzero witnesses in degrees2..5;9 all-pairs-coprime rational-root families with disjoint exact root isolations in degrees2..10. These are regressions, not formal verification or independent review.
11. Full-word classification remains open in the record. The auxiliary question is specifically SS Section2 Problem1. No neighboring peer record, smooth-function problem, probability distribution on coefficients, or full frozen-source closure is claimed.

No gap found in the complete written proof. Mathematical acceptance and priority uncertainty are separate.

# Bounded novelty search,25October2026

Searches included Arnold Rolle ratio polynomials; symbolic sequences real-rooted asymptotic; root arrangements asymptotic; Rolle realizable limit; Arnold symbolic sequences polynomials; hyperbolic polynomials derivative root arrangements Arnold asymptotic solution; symbolic sequences polynomials bound; and Kostov Arnold Milnor. Primary references were Shapiro2015, Shapiro-Shapiro MonthlyFinal, Kostov-Shapiro2002, Kostov2006, and Milnor1964. The OEIS A131811 entry was a bibliographic lead only, not the proof source. It still listed counts1,1,2,10,116 and the Arnold question. Nonprimary aggregators, unrelated sign-pattern papers, search-result dates and irrelevant hits were not used as mathematical authority.

The primary literature already describes collision resultants and the lack of free parameters, counts all Rolle words, determines degrees<=5, realizes every two-level (p,p^(k)) interlacing, and bounds real algebraic Betti numbers. All of these are credited. No matching explicit a_n bound or two-term log-density statement was located. This is a bounded search, not a proof that no unindexed or unpublished argument exists. The contribution is a short combination of established tools that answers the explicitly sourced auxiliary counting question.

All four final PDF pages were visually inspected. No clipped text, overflow, missing symbols or unresolved citations were found. Built-in compiler and Tectonic succeeded. The author name is prominent; Mercury Software GmbH/contact is a footnote; AI assistance is ordinary prose.