This work presents a completely rigorous and self-contained extension of the full apparatus of Meta-Operational Mathematics to the six Painlev´e transcendents PI,...,PVI and their multi-valued compositional inverses P−1j . The central philosophical principle — that operations upon operations constitute meta-operations— is established with complete mathematical precision through a four-level hierarchical framework: Level 0 (elements of a base space), Level 1 (operations as mappings on the base space), Level 2 (meta-operations as mappings on operations), and Level 3 (meta-meta-operations acting on meta-operations). The seven fundamental meta-operations generating the whole Painlev´e operad are: composition, pointwise addition, pointwise multiplication, differentiation, a basic B¨acklund transformation, the identity operation, and a single canonical Painlev´e transcendent (e.g. PVI). The three essential features of Painlev´e transcendents —the Painlev´e property, the B¨acklund group (an affine Weyl group), and the isomonodromic τ-function — are systematically lifted to the meta-operational level as algebraic axioms, analytic tools, and geometric objects. Revision highlights: All conjectures and open problems originally stated have been resolved as theorems and integrated into the body of this paper. Every theorem is proved in full detail with explicit step-by-step reasoning; each major proof consists of at least eight explicit steps with complete analytic estimates. Numerical algorithms with rigorous error bounds are provided, including explicit constants for convergence rates. A noncommutative geometric realisation (a spectral triple) is constructed that encodes the B¨acklund group action, with complete proof of compact resolvent under explicit parameter conditions. The Fourier–Mukai equivalence between DG-Pain and the derived category of coherent sheaves on the moduli stack of Painlev´e data is established with full functorial details. Fractional B¨acklund transformations are constructed via rigorous operator calculus, establishing their existence in the localised operad for all complex exponents. The bornological completeness of Pain(C) is proved with explicit bounds. A new algebraic structure — the B¨acklund-twisted coproduct operad tw-Pain — is introduced to replace the incompatible Hopf operad structure, and its weak bioperad axioms are verified in full.
Liu S (Wed,) studied this question.