AI PROVENANCE STATEMENT

Manuscript: Convolution Smoothing in the Turan Hierarchy:
The Blind Zone Extends from Order 43 to Order 61, and Empirically
to Order 68
Author: Changxia Lin (xstr_1115@qq.com)
ORCID: 0009-0008-0668-7444
Date: 2026-10-19
Relation: companion to hexagon:2610.00293v2 (the r <= 43 paper)

TOOLS USED
Conversational AI assistance was used for literature retrieval,
notation cross-checking, LaTeX typesetting, and the independent
recomputation of every numerical constant recorded in Sections 5
to 7 of the manuscript.

HUMAN CONTRIBUTIONS (all mathematical content)
- The multiplicity analysis of the beta-alphabet (Lemma 2.1):
  the observation that evenness of Xi forces every zero to
  contribute to beta with multiplicity two, so that the relevant
  constituent is a four-element multiset and not a conjugate
  pair, is due to the author. This is the single structural point
  that separates this paper from the earlier r <= 43 paper.
- The convolution identity h_r(Y) = rho^r W_r with
  W_r = sum_i U_i U_{r-i}, its generating function
  1/(1 - 2 z cos theta + z^2)^2, and the Chebyshev derivative
  identity W_r = (1/2) U'_{r+1}(cos theta), were derived by the
  author.
- The pole-free sign criterion of Lemma 3.5 was derived by the
  author, together with the observation that the corresponding
  tangent criterion is illegitimate because cos(m theta) changes
  sign throughout the range of interest.
- Theorem 3.9 (D(1,r) > 0 for all r <= 61, unconditionally) and
  its proof were found and written by the author.
- The closed form for the shift-1 row, Theorem 3.8, and the
  interpretation of the three thresholds 44, 62, 69 as
  constituent / multiplicity / background effects, are due to the
  author.
- The design of the numerical experiments (truncation study,
  offset scan, position scan, multiple-pair scan, shift scan,
  cancellation diagnostic) was the author's.
- The distinction between what is proved (61) and what is
  numerical (68) is stated explicitly in Section 8 and was
  insisted on by the author.

AI CONTRIBUTIONS
- Typesetting, cross-checking standard identities (Jacobi-Trudi,
  Littlewood-Richardson, Gegenbauer generating functions), and
  independent recomputation of numerical constants.

VERIFICATION
Every numerical constant was computed at 60 significant digits
(80 for the certification of W_r) and independently reproduced
by two routes: the stable recursion for the complete symmetric
functions, and the closed form of Theorem 3.8 combined with the
Chebyshev identity. Thresholds were rechecked at 50, 60 and 80
digits and under variation of the truncation N = 20, 30, 45, 60,
80. No threshold changed. Imaginary parts of all computed Schur
values are smaller than the real parts by at least 40 orders of
magnitude, as required by conjugate symmetry.

STATUS OF THE CLAIMS
Proved: Lemma 2.1, Proposition 3.1, Theorem 3.3 (convolution and
Chebyshev derivative identity), Lemma 3.5 (sign criterion),
Proposition 3.6 (W_r > 0 for r <= 61, W_62 < 0; certified finite
computation with margin of order 10^2 at 80 digits), Theorem 3.8
(shift-1 closed form), and Theorem 3.9 (the main theorem).
Numerical, not proved: the threshold 69 and the bound 68, the
scaling law r* approximately 1.5 pi/theta, the robustness tables,
and the monotonicity used in Lemma 3.10. Section 8 of the
manuscript gives the complete ledger.

NOVELTY
No claim of a proof of the Riemann hypothesis is made. Level-2
positivity is the theorem of Csordas, Norfolk and Varga (1986)
and is cited as such; levels 2 and, where known, higher levels
are not claimed as new. What is claimed as new is the
multiplicity analysis, the convolution identity and the shift-1
analysis, and the numerical determination of the true threshold.

STATEMENT
I confirm that I understand all parts of this contribution and
accept responsibility for its content.

Changxia Lin
xstr_1115@qq.com
ORCID 0009-0008-0668-7444