Ordinal Functions & Series Convergence

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Abstractions about limits, continuity, and series in real and ordinal analysis, including ordinal-indexed continuity such as continuous and normal functions, series and sum approximation like Riemann sums and Helly's selection theorem, and generalized calculus operators such as the differintegral, transseries and Z-transform.

9 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Continuous function (ordinal theory) — An ordinal-indexed sequence whose value at every limit index equals the supremum of its earlier values, usually considered together with monotonicity in transfinite constructions.
• Differintegral — A fractional-calculus operator D^q that unifies differentiation for positive order and integration for negative order, with integer cases recovered under a specified convention.
• Equidistributed sequence — Require the limiting frequency of sequence terms in every subinterval to equal that subinterval’s normalized length.
• Function series — An infinite series whose terms are functions, with its sum and inherited properties determined by a specified mode and domain of convergence.
• Helly's selection theorem — Every uniformly bounded sequence of monotone real functions on a compact interval has a subsequence converging pointwise, with bounded-variation generalizations providing compactness for measure and weak-convergence arguments.
• Normal function — An ordinal-valued function that is strictly increasing and continuous at limit ordinals, so its value at a limit is the supremum of all earlier values.
• Riemann sum — Approximate a definite integral by partitioning an interval, sampling the function once in each subinterval, multiplying each sampled value by subinterval width, and summing, with mesh refinement controlling convergence.
• Transseries — Formal generalized series built from nested powers, exponentials and logarithms and ordered by asymptotic dominance, supporting algebra, differentiation and solution of differential equations beyond ordinary power series.
• Z-transform — Represent a discrete-time sequence by a Laurent series in a complex variable together with its region of convergence, enabling shifts, convolution, recurrences, spectra, and system behavior to be analyzed algebraically.