Sources consulted on 2026-10-16:

* Faber and Pandharipande, Hodge integrals, partition matrices, and the lambda_g conjecture, https://arxiv.org/pdf/math/9908052, section 0.5, Theorem 1 and equations (4) and (5). This fixes the stack normalization, the multinomial coefficient, sum(d)=2g-3+n, the sine generating series and the even Bernoulli formula. The publisher record https://annals.math.princeton.edu/2003/157-1/p03 identifies the publication as volume 157 (2003), pages 97–124; the arXiv manuscript's opening header gives an inconsistent older year/volume. The draft cites the publisher's metadata.
* Macaulay2 HodgeIntegrals, https://macaulay2.com/doc/Macaulay2/share/doc/Macaulay2/HodgeIntegrals/html/_integral.html. Three displayed outputs checked exactly: (g,n,d)=(1,1,(0,)) -> 1/24; (2,2,(1,2)) -> 7/1920; (3,3,(1,2,3)) -> 31/16128. This is independent published software output, not a rerun of Macaulay2 in this session.
* Faber and Pandharipande, Hodge integrals and Gromov-Witten theory, https://arxiv.org/pdf/math/9810173, introduction. Explicit b_0=1 convention and one-point series.
* The author-hosted notes https://people.math.ethz.ch/~rahul/gwhm.pdf, Appendix B, separately list the genus-one 1/24 and genus-two one-point 7/5760 values.
* Joachim Kock, Notes on psi classes, https://mat.uab.es/~kock/GW/notes/psi-notes.pdf, section 1.7, discusses lambda_1=delta_irr/12 and stack weights. These notes contain editorial cautions and were not used as the definitive numerical source. The boundary check uses the stack gluing map of degree two, giving one twenty-fourth of the genus-zero psi intersection with two extra zero exponents.

The live and local skills were read. Their differences concern continuous parameter selection; the stricter local guidance was retained. This table has exact rational values, so the campaign's 100-digit approximation target is inapplicable.

Family #198's generic psi-only dimension constraint is not the constraint for Hodge integrals. Inserting lambda_g subtracts g from the available psi degree. The family says 'proven' for these exact rational tables; the current skill uses the stronger and appropriate label 'exact'.