The General Bonelli Transform: Operational Calculus, Abelian Symmetries, and Supersymmetric Duality

We introduce a universal generalization of the Bonelli Transform, elevating it from a fixed-order geometric filter to an infinite-dimensional operational calculus over the space of integer partitions. By parameterizing the operator with an arbitrary arithmetic weight sequence $\Omega$, we establish that the set of all General Bonelli Transforms forms an Abelian group. We rigorously prove that the "Kaleidoscopic Filter" is a topological property of the General Transform under compact unit support. Furthermore, we expand the theory to include Fractal Prime Sieving, the Vacuum Eigenstate, Gauge Variations in the $\Omega$-field, the profound Bosonic-Fermionic Duality, Thermodynamic Condensation, Modular Connections, Homological Euler Characteristics, Ehrhart-Macdonald Polytope Duality, Hecke Algebra Isomorphisms, Hopf Algebra Endomorphisms, Tropical Polyhedral Retractions, Vertex Operator Truncations, Fourier Cyclotomic Nulls, Riemann Zeta Dirichlet Convolutions, Selberg Trace Trace-Class Operators, Fractional Order Differointegral Operators, Weyl Geometric Invariant Cohomology, Quantum Cluster Algebras, Non-Commutative Connes Integration, Categorical Functors, the Geometric Langlands Correspondence, Holographic Entanglement Entropy, Motivic Integration, Operadic Grafting, Chern-Simons Knot Surgery, the Bonelli-Weil Explicit Formula, GUE Spectral Rigidity, the Explicit Unitary Frequencies of the Riemann Zeros, Perturbed Spectral Zero-Shifting, Adelic Factorization, BPS Black Hole Microstate Extraction, Topological Quantum Error Correction, and Twistor Space Correspondence, bridging partition combinatorics with the absolute frontiers of algebraic geometry, harmonic analysis, higher algebra, and quantum gravity.