A Constructive Field of Infinitesimals: Chunk and Permeate Approach

Naive infinitesimal reasoning, though intuitive, is inconsistent within classical logic, while rigorous nonstandard analysis relies on nonconstructive ultrafilters and the transfer principle.

We resolve this tension by building an explicit, totally ordered field R^{Z_<} that serves as a model for the (inconsistent) union of the real and hyperreal axioms, containing real numbers, infinities, and infinitesimals, using only real sequences and convolution.

The construction is underpinned by the "Chunk and Permeate" (C&P) strategy, a paraconsistent reasoning technique that isolates and manages contradictions locally without global collapse.

We equip R^{Z_<} with a two-tier topology, develop a calculus of microstable functions where derivatives and integrals permeate to their classical counterparts via a simple standard-part map, and introduce a fine-grained (k,n)-continuity hierarchy that captures infinitesimal smoothness invisible in classical or transfer-based models.

We further show that R^{Z_<} directly models Sergeyev's Grossone arithmetic, providing an explicit consistency proof, and we analyse the computability of field operations.

This work bridges paraconsistent logic, constructive mathematics, and nonstandard analysis, offering a transparent, computationally tractable framework for infinitesimal reasoning with potential applications in reverse mathematics and physics.