The five current and 22 recovered historical entries follow the retained edition.
Nishibuchi
Yuta Nishibuchi, \(p\)-adic Congruences of Generalized Euler Numbers and Relations to Even Zeta Values, arXiv:2605.10677v2, 1 June 2026. https://arxiv.org/abs/2605.10677v2. The comparison to Conjecture 3.9 uses the source review and fixed-version record preserved with this edition.
gessel
Ira M. Gessel, Applications of the classical umbral calculus, arXiv:math/0108121v3, 22 November 2001, Section 9, in particular the Frobenius Euler congruence and the derivation following Theorem 9.2. https://arxiv.org/abs/math/0108121. The normalization and quantifiers of the ordinary Euler input were checked against the primary PDF in the strengthening checkpoint.
Cited in this paper (1 passage)
DLMF24
NIST Digital Library of Mathematical Functions, §24.10(i), Eq. 24.10.1 (Bernoulli denominator theorem). https://dlmf.nist.gov/24.10.E1. The official primary entry was inspected for the positive even-index denominator specialization in this edition.
Cited in this paper (1 passage)
Mathlib
The Mathlib Community, Mathlib, source revision aeb72f1e9981b6d35b426b16dca7759be3e7ba06, used with Lean 4.34.0-rc2. In particular, the polynomial, strict inverse-function and analytic inverse-function developments cited in Section 11. The exact local source paths and dependency records accompany the release.
PaperIXArchive
Komorebi P., Projector-Resolved Seed Calculus IX: original manuscript, formal supplements and cumulative continuation sources, preserved source collection, consolidated 8 September 2026. The original 28-page status-corrected manuscript and all subsequent release records remain in the accompanying archive.
Recovered historical references. The bibliography underlying Appendix D has been recovered from the original source package. Its 22 citation keys are retained below, with bibliographic identities checked against publisher and author records. This repairs the source attributions; it does not enlarge the historical chapter's stated proof or formalization scope.
Amice1964
Yvette Amice, Interpolation \(p\)-adique, Bulletin de la Société Mathématique de France 92 (1964), 117–180. https://doi.org/10.24033/bsmf.1606.
Cited in this paper (1 passage)
BarberoCerrutiMurru2018
Stefano Barbero, Umberto Cerruti, and Nadir Murru, Some Combinatorial Properties of the Hurwitz Series Ring, Ricerche di Matematica 67 (2018), no. 2, 491–507. https://doi.org/10.1007/s11587-017-0336-x.
Cited in this paper (1 passage)
CarlitzLevine1960
Leonard Carlitz and Jack Levine, Some Problems Concerning Kummer's Congruences for the Euler Numbers and Polynomials, Transactions of the American Mathematical Society 96 (1960), no. 1, 23–37. https://doi.org/10.1090/s0002-9947-1960-0115971-x.
Cited in this paper (2 passages)
Chen2004
Kwang-Wu Chen, Congruences for Euler Numbers, The Fibonacci Quarterly 42 (2004), no. 2, 128–140. https://doi.org/10.1080/00150517.2004.12428429.
Cited in this paper (2 passages)
Comtet1974
Louis Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, D. Reidel Publishing Company, Dordrecht, 1974. https://doi.org/10.1007/978-94-010-2196-8.
Cited in this paper (2 passages)
EverestVanDerPoortenShparlinskiWard2003
Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, Mathematical Surveys and Monographs, vol. 104, American Mathematical Society, Providence, 2003. https://doi.org/10.1090/surv/104.
Cited in this paper (2 passages)
Gouvea1997
Fernando Q. Gouvêa, \(p\)-adic Numbers: An Introduction, second ed., Universitext, Springer, Berlin, 1997. https://doi.org/10.1007/978-3-642-59058-0.
Cited in this paper (2 passages)
GrahamKnuthPatashnik1994
Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, second ed., Addison-Wesley, Reading, Massachusetts, 1994.
Cited in this paper (1 passage)
IrelandRosen1990
Kenneth Ireland and Michael Rosen, A Classical Introduction to Modern Number Theory, second ed., Graduate Texts in Mathematics, vol. 84, Springer, New York, 1990. https://doi.org/10.1007/978-1-4757-2103-4.
Keigher1997
William F. Keigher, On the Ring of Hurwitz Series, Communications in Algebra 25 (1997), no. 6, 1845–1859. https://doi.org/10.1080/00927879708825957.
Cited in this paper (1 passage)
Koblitz1984
Neal Koblitz, \(p\)-adic Numbers, \(p\)-adic Analysis, and Zeta-Functions, second ed., Graduate Texts in Mathematics, vol. 58, Springer, New York, 1984. https://doi.org/10.1007/978-1-4612-1112-9.
Cited in this paper (1 passage)
LidlNiederreiter1997
Rudolf Lidl and Harald Niederreiter, Finite Fields, second ed., Encyclopedia of Mathematics and its Applications, vol. 20, Cambridge University Press, Cambridge, 1997. https://doi.org/10.1017/CBO9780511525926.
Cited in this paper (1 passage)
Lucas1878a
Édouard Lucas, Théorie des Fonctions Numériques Simplement Périodiques, American Journal of Mathematics 1 (1878), no. 2, 184–196. https://doi.org/10.2307/2369308.
Cited in this paper (2 passages)
Mahler1958
Kurt Mahler, An Interpolation Series for Continuous Functions of a \(p\)-adic Variable, Journal für die Reine und Angewandte Mathematik 199 (1958), 23–34. https://doi.org/10.1515/crll.1958.199.23.
Cited in this paper (2 passages)
Robert2000
Alain M. Robert, A Course in \(p\)-adic Analysis, Graduate Texts in Mathematics, vol. 198, Springer, New York, 2000. https://doi.org/10.1007/978-1-4757-3254-2.
Cited in this paper (3 passages)
Roman1984
Steven Roman, The Umbral Calculus, Pure and Applied Mathematics, vol. 111, Academic Press, New York, 1984.
Cited in this paper (1 passage)
Serre1979
Jean-Pierre Serre, Local Fields, Graduate Texts in Mathematics, vol. 67, Springer, New York, 1979. https://doi.org/10.1007/978-1-4757-5673-9.
Cited in this paper (2 passages)
Stanley2012
Richard P. Stanley, Enumerative Combinatorics. Volume 1, second ed., Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2012. https://doi.org/10.1017/CBO9781139058520.
Cited in this paper (2 passages)
Sun2008
Zhi-Hong Sun, Congruences Involving Bernoulli and Euler Numbers, Journal of Number Theory 128 (2008), no. 2, 280–312. https://doi.org/10.1016/j.jnt.2007.03.003.
Cited in this paper (2 passages)
Washington1997
Lawrence C. Washington, Introduction to Cyclotomic Fields, second ed., Graduate Texts in Mathematics, vol. 83, Springer, New York, 1997. https://doi.org/10.1007/978-1-4612-1934-7.
Cited in this paper (1 passage)
Young1999
Paul Thomas Young, Congruences for Bernoulli, Euler, and Stirling Numbers, Journal of Number Theory 78 (1999), no. 2, 204–227. https://doi.org/10.1006/jnth.1999.2401.
Cited in this paper (2 passages)
Young2001
Paul Thomas Young, Kummer Congruences for Values of Bernoulli and Euler Polynomials, Acta Arithmetica 99 (2001), no. 3, 277–288. https://doi.org/10.4064/aa99-3-5.