# AI provenance

Paper: *Dense Algebraic Loci of Hodge Classes*, Deep Bhattacharjee and Ushashi Bhattacharya.

## What was done with AI assistance

The constructions on (F2) and (F3'), the search for a closure of the Hodge
conjecture, the text of the paper, the figures, the Lean 4 file and the
Python, Julia, C, C++ and shell programs were carried out by Deep
Bhattacharjee with the help of AI assistance (Claude, Anthropic), in sessions
in the Claude Code environment. The mathematical content was checked as
follows.

* Every hand proof in the paper was read against the cited literature
  (Markman, van Geemen, Schütt, Varesco, Buskin, Bloch, Buchweitz, Flenner,
  Moonen, Zarhin, André, Milne) and the earlier paper of the authors with
  Priyabrata Mandal.
* Every finite quantity (closures of the 256 subsets, Hilbert symbols and
  Hasse invariants, the 165 forms over Q(sqrt 6), the Fricke discriminants
  and Picard numbers, the dimension counts 57, 104 and 26, the Hecke algebra
  dimensions) is recomputed by programs in `code/` in at least two languages
  and the combinatorial and arithmetic ones are proved in
  `lean/HodgeClosure.lean` (forty-seven theorems, forty-five of them free of
  every axiom and two depending on propext and Quot.sound only).
* Two numerical ranks (`product_object.py`, `convolution_checks.py`) are
  computed in floating point; no statement of the paper rests on them. The
  rank behind Theorem VI.6 is also computed exactly (`annihilator_exact.py`),
  the Riemann-Roch polynomial and the thirty-one supports of Proposition VI.7
  (`riemann_roch_pieces.py`), and the invariants of the trace form behind
  Theorem V.12 (`zeta9_invariants.py`), with the Ext groups and chains at the
  CM point of Appendix A (`cm_point_ext.py`, Lean Section 6). A fourth
  round of AI sessions added Lemma VI.9 and Proposition VI.11 (convolutions
  with three line bundles fail at E_0^6); its claim that no Massey product
  had been computed was kept as stated, and the configurations it leaves
  are listed as open in the paper.
* The theorem on real multiplication by Q(zeta_9)^+ was found in a second
  round of AI sessions and checked against van Geemen and Schütt (2025,
  Sections 4.8 and 5.6), van Geemen (2008, Remark 7.9), Reiner's Maximal
  orders (Theorem 33.15) and Platonov and Rapinchuk (Theorem 7.8). The
  extension to Q(zeta_7)^+ through the Dickson deformation of the family of
  Artebani, Sarti and Taki (2011, Example 6.1 and Table 3), and the
  proposition on where the dihedral mechanism stops, were found in a third
  round; the invariants of the trace form are recomputed in
  `zeta7_invariants.py` and the Dickson identities, the inertness of 2 and
  the sign pattern of the conjugates are proved in Lean.

## What the AI did not do

The AI did not prove (F2), (F3') or the Hodge conjecture; the paper says so
and states at which step each construction stops. No result of the paper is
reported as proved on the strength of a machine computation alone.

## Where the AI was wrong and corrected

During the work three slips were found by the machine checks and corrected
in the text before release: the Hasse invariant at 2 of the form V_2
(it is -1, not +1); the Picard number of the twist family at level 5 (the
base change adds four fibres of type III); and the dimension of the dihedral
Hecke algebra ((n+1)/2, not (n+3)/2).

## Approval

Both authors approved the final version. Funding: none. Competing
interests: none.