Bol operators, Cartier contraction, and local rigidity at ordinary partial weight one on Hilbert modular surfaces

This paper extends a recent article that appears in the 2026 Annals of Mathematics -- Lue Pan's second paper on locally analytic vectors in the completed cohomology of modular curves, whose program includes a classicality theorem for p-adic modular forms of weight one -- from modular curves to Hilbert modular surfaces. We work over a real quadratic field in which a prime p, at least 5, splits, and we study ordinary Hilbert modular forms of partial weight (1, k): weight one at one of the two embeddings and a regular weight k at the other. At such weights the usual weight-raising differential operator disappears in the weight-one direction, the two Hodge-Tate weights at the corresponding prime collide, and the standard routes to classicality and modularity break down. The paper separates what can be proved outright from what cannot yet be proved. Unconditionally, we show that the correct classical differential operator in the singular direction is the second-order Bol operator, and that no first-order operator exists on the flag line, even after factoring through an auxiliary algebraic vector bundle; any first-order singular operator is therefore forced to live on infinite-level period sheaves of the kind Pan uses, with the Bol operator as its classical shadow. We compute the relevant local analytic kernels in both first-order and second-order form; we construct the derived "old" quotient at Iwahori level and prove erasure criteria for its ordinary part, resting on a Cartier trace theorem that tolerates unit distortions, extra variables, and matrix coefficients, and on a mixed filtration-adic contraction theorem; we prove a diagonal compactness theorem showing that, in defect-one patching, the patched complex, the patched action, and the augmentation already follow from finite-level data alone; and we establish seven local criteria forcing a finite flat Hecke algebra to equal the weight algebra, together with component-support theorems for the integral question. The global conclusions are conditional on six precisely stated inputs, organized in three groups and each accompanied by evidence and by exact statements of what remains open. Assuming the branch-separation, patching, and transversality inputs, the deformation ring, the Hecke algebras, and the one-variable weight algebra at a fixed ordinary point all coincide, so the eigenvariety is smooth of dimension one there and the weight map is a local isomorphism. Assuming in addition the operator and comparison inputs, the classical forms of weight (1, k) are exactly the joint kernel of the two partial operators, in the spirit of Pan's theorem. A final input on integral component support upgrades this to a deformation-to-Hecke isomorphism on every component, and hence to modularity of every lift satisfying the stated conditions. The paper closes with an analysis of why each open input is a natural and fruitful target for further research. Computer checks of the key finite formulas are included as reproducible sanity evidence only; no proof depends on them.