# AI provenance: Even-intersecting families of lattice paths and the Catalan numbers

Author: Deep Bhattacharjee (sole author). No AI system is an author or contributor.

## How the work was produced

Deep Bhattacharjee chose the problem, Conjecture 4.1 of Shankar (arXiv:2607.23117), and directed the work. Claude (Anthropic), working through Claude Code, did the following:

- searched the literature;
- found and checked the proofs;
- wrote the verification programs in C, C++, Bash, Python, Julia and Lean 4;
- drafted the paper.

## How the results are checked

The proofs are complete by hand in the paper. In addition:

- `verification/c/star_lemma.c` checks, for every word length k ≤ 13:
  - the cell identity (Lemma 4.2);
  - the star lemma (Lemma 4.1);
  - the formula of Lemma 5.2.
- `verification/cpp/skeletons.cpp` checks:
  - that the skeleton systems are admissible, extremal and distinct, with C(m, m/2) of them for m ≤ 10;
  - by exhaustive search over all admissible systems of length m ≤ 5, that the extremal systems are exactly the skeleton systems;
  - the C_n families F_M for n ≤ 6.
- `verification/python/verify_sat.py` (SAT, CaDiCaL) checks:
  - that all extremal systems of length m ≤ 6 are skeleton systems;
  - Shankar's conjecture for n ≤ 5 (1, 2, 5, 14, 42 families, none larger);
  - Proposition 5.1 for k ≤ 7.
- `verification/julia` checks the bijection between skeletons and nonnegative paths and the induction step of Section 3.
- `verification/shell` checks small cases in Bash integer arithmetic.
- `verification/lean/LatticePaths.lean` contains Lean 4 kernel checks of the key lemmas and of the conjecture for n ≤ 3.

## What is claimed

The paper proves Conjecture 4.1 of Shankar in full: for every n there are exactly C_n families of North–East lattice paths from (0,0) to (n,n) with pairwise even intersections and 2^n members, namely the families F_M of the noncrossing perfect matchings M of {1,...,2n}. It also determines all equality cases of Shankar's Lemma 2.3, of which there are C(m, floor(m/2)).

## Labels given on the Hexagon form

- Text of the contribution: a mix of human-written and AI-generated.
- Human understanding at submission: some parts, but not all, checked and understood by a human.

## Where everything is

Release v1.1.0: https://github.com/creelie/eConjecturePrimitiveTrinomial/releases/tag/v1.1.0