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Chapoton and Han Proposition 5.2 printed formula refutation

Abstract

Refutes only printed Proposition 5.2 in arXiv:2001.01449v1 at n = 1, t = 1; literature-attested, new_counterexample_claimed = false. No claim that the authors’ intended proposition is false or its proof has a gap; P’_n = G_n is an interpretation, not the printed text. No verdict on Conjecture 5.4. No priority claimed for these values or for identifying the Q’ to P’ misprint.

Frédéric Chapoton and Guo-Niu Han, On the roots of the Poupard and Kreweras polynomials, arXiv:2001.01449v1, printed page 8, Proposition 5.2, says For every n >= 1, the polynomial Q’_n is the Kreweras polynomial G_n. Rendered source pages 2, 3, 7, 8 and 9 were checked on 2026-09-10. Equation 5.4 defines Q’; the following paragraph defines P’ as Q’ divided by t minus 1. Both the proposition and proof still print Q’. The formal claim retains the entire polynomial equality for every natural n >= 1. It has no exclusion at n = 1.

Theorem 1.1 (At n = 1 and t = 1 the two sides are zero and two).

Lean statement: D5/S0/Certificates/ChapotonPropositionFiveTwoRefutation.result

Proof. Machine-checked in Lean as D5/S0/Certificates/ChapotonPropositionFiveTwoRefutation.result (✓ std3). ∎

Source. Repository-derived.

Commentary.

The public rho definition is imported directly from the frozen sister module. Section 5.1 on printed page 7 takes the constant term after floor(d/2) iterations on the fixed index space V_d. Each quotient in equation 5.4 has index 2n minus 1, including zero outer coefficients. At n = 1 the three numerators are 1 minus x squared, zero, and x squared minus 1. Multiplication verifies their exact quotients are minus (1+x), zero, and 1+x. All have index 1, so there are zero operator steps, and their constant coefficients are minus 1, zero, and 1. Thus Q’_1(t) = t squared minus 1. The equation rho(c(1+x)) = c is a derived index-one consequence, not a separately printed premise. Evaluation at 1 gives zero; page 2 prints G_1 = 1+x, whose value at 1 is two. Independent exact checks use integer arrays and monic long division with equation 2.1, and signed geometric sums with the symmetric-basis products of equation 2.2. Both give Q’_1 coefficients (-1,0,1), Q’_2 coefficients (-2,-2,0,2,2), and Q’_3 coefficients (-12,-12,-8,0,8,12,12), in ascending order. Dividing Q’ by t minus 1 gives P’_1 through P’_4 equal to the four G polynomials printed on page 2, coefficient by coefficient. Their ascending coefficient lists are (1,1), (2,4,4,2), (12,24,32,32,24,12), and (136,272,384,448,448,384,272,136). Independent G computations use equation 1.3 and equation 2.2 at D = 2. Section 5.1’s example (1,1,1,1,1) to (3,4,3) to (4) also agrees, as do all 36 entries printed in equation 5.6. There are 1239 checked exact divisions, zero control mismatches and no floating point. Matrix entries are controls only. The 2020 published version, Moscow Journal of Combinatorics and Number Theory 9, pages 163-172, DOI 10.2140/moscow.2020.9.163, still prints Q’_n in Proposition 5.2 and Q’_1 is 1+x in its proof on page 171. Page 170 prints equation 5-4 and Q’_n(1)=0; page 164 prints G_1=1+x. The actual values are therefore literature-attested. No separate formal erratum was located in the documented search scope as of 2026-09-10; this is a bounded search statement. new_counterexample_claimed = false. Only the printed assertion at n=1,t=1 is refuted. No claim is made that the authors’ intended proposition is false or its proof has a gap. P’_n=G_n is the suggested interpretation, not the printed text and not a universal theorem proved here. No verdict is given on Conjecture 5.4. Neither first discovery of the values nor first identification of this Q’ to P’ misprint is claimed. The formal calculation uses exact rational polynomial arithmetic. All finite equalities are local to the proof.

References