Umbral calculus formalisation: finite operator calculus and Rota's classification

I wrote all of this in two days with Claude Code. The mathematical content is from the first half of my masters thesis (arXiv:0907.3950, Macdonald Polynomials and Symmetric Functions), specifically the finite operator calculus of Gian-Carlo Rota developed in sections 1.2.1–1.2.3.

The programme starts with the Hopf algebra structure on R[X] (#39410): the coproduct Δ(X) = X⊗1 + 1⊗X, counit as evaluation at zero, antipode S(x) = −x. This is the additive group scheme 𝔾ₐ, and it is the coalgebra that makes "binomial type" meaningful.

Then compositional inverses of formal power series (#39458), filling a genuine gap in mathlib — it had multiplicative inverses and substitution but not the ability to invert f(g(x)) = x. This is necessary because umbral operators are indexed by delta series, and computing images requires the compositional inverse.

Delta operators and Rota's classification (#39465) are the core: a delta operator is a shift-equivariant operator that is essentially a power series in d/dx, and Rota's theorem says its basic sequence is always of binomial type. Monomials and falling factorials are the two classical examples.

The culmination is #39498, which builds the full umbral operator U_Q from its generating function, proves the lowering property Q(p_{n+1}) = (n+1)p_n via the duality pairing between power series and polynomials, and closes the circle back to the Hopf algebra. Zero sorry's across the entire chain.

Then #39636 adds ascending Pochhammer (rising factorials) as a third binomial-type family via the backward difference operator, and computes the Lah numbers as the change-of-basis coefficients between rising and falling factorials.

The dependency order for reviewers is: #39410 → #39458, #39465 → #39498 → #39636, where #39458 and #39465 are independent of each other but both depend on #39410.

The intended next step is generalising from R[X] to the ring of symmetric functions Λ. The finite operator calculus on polynomials is classical Rota; lifting it to Λ connects to the Macdonald polynomial theory developed in section 1.3 of the thesis. This would be, as far as I know, unprecedented in any proof assistant. I plan to begin this after the current PRs have had time to be reviewed.

Start by reading issues #39410, #39458, #39465, #39498, and #39636 in their stated dependency order, then sections 1.2.1–1.2.3 and 1.3 of the cited thesis. Done would mean a formalisation of the finite operator calculus generalised from R[X] to the ring of symmetric functions Λ; no target files or tests are named.