APEIRON MATHEMATICS Continuous Mathematics Through Infinite-Process Representation, Stability, and Invariant Preservation

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Original abstract

Apeiron Mathematics is a proposed framework for studying mathematical objects as finite observable values together with indefinitely continuing generative structures. The framework originated from a simple question: ordinary counting, [1,2,3,4,\ldots,] may continue indefinitely, but every finite written list terminates. Can continuation instead be carried within a numerical representation? The elementary identities [\frac19=0.\overline1,\qquad\frac29=0.\overline2,\qquad\ldots,\qquad\frac99=0.\overline9=1] provide the motivating example. A finite numerical value may be represented through an infinite sequence of finite stages. Apeiron formalizes this observation in two compatible layers. The representation layer uses the classical Banach algebra (c) of convergent real sequences and the limit projection (\pi:c\to\mathbb R). The generative layer represents a process by an initial state and evolution operator, [\mathfrak A=(x_0,F),\qquad x_{n+1}=F(x_n).] When the generated trajectory converges, its infinite evolution is mapped by the projection to its finite invariant value. Contraction theory supplies rigorous sufficient conditions for existence, uniqueness, and exponential convergence of that invariant. Summable perturbations and common-contracting switched systems are also treated. Apeiron does not redefine classical real arithmetic or claim that finite numbers are literally infinite. Its proposed contribution is a framework in which value, continuation, evolution, and preservation are retained simultaneously rather than collapsing an infinite process immediately to its limiting scalar. JJ and Cosmic Witter (ChatGPT) was here

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