Scale-Lift Calculus for Fractional-Dimensional Objects and Differential-Algebra Extensions of Distributions A Two-Volume Research Manuscript Lifting, Reconstruction, and Coordinate Calculus

Let P : DP ⊆ R → Y be a partial observation and let ιΓ : X → DP be a selected section, so thatP ιΓ = idX . The manuscript studies operations on the selected representations and determineswhen their outputs are invariant under a change of representative.Volume I develops the underlying scale analysis. Discrete, continuous, and multiscaleparameter spaces lead to scale matrices, finite decompositions, and reconstruction in function,measure, and distribution topologies. Spatial derivatives, scale differences, continuous scalederivatives, and mixed differentials are constructed on their natural domains. Scalar and operator-valued normalizations describe nondegenerate asymptotic profiles. Hausdorff dimension, boxdimension, local measure dimension, graph dimension, and regularity exponents are then relatedto the Weierstrass, Cantor, and self-affine models. The final part equips continuous scale spacesand graph realizations with geometric structures, characterizes their behavior under tangentlimits and observation, and places these constructions in a universal projection geometry. Inlogarithmic depth, one fixed contraction law acts on every layered function; fractional regularityand dimension arise from the competition between that contraction and the growth of thefunction and its derivatives along the scale direction.Volume II applies compact-kernel regularizations to distributions. A moderate/negligiblequotient supports differentiation, nonlinear superposition on its natural domain, generalizedpoint values, and division by units. Sharp completion, relative differential generation, inverse andprimitive closure, scale recurrences, and formal differential solution adjunctions are developedas separate algebraic and topological constructions. The observation formalism keeps classicaloutputs, association, quotient equality, and internal nonlinear operations distinct