Domain Number Mathematics II : Finite Increment Calculus, Differential and Summation Domains, and Resolution-Preserving Gluing
Year: 2026
Type: preprint
Abstract: Domain number mathematics treats requested mathematical resolution as part of the problem data and permits relation-valued or region-valued targets to serve as exact mathematical objects. The first paper developed this viewpoint for arithmetic, algebraic relations, dependency-preserving finite relation networks, and elementary dynamics. The present paper extends the framework to differential, difference, integral, and discrete-summation relations and develops their interaction with finite gluing. The central construction is a finite increment relation $z_{i+1}-z_i=a_i+r_i,\qquad |r_i|\leq \varepsilon_i,$ which provides a common finite relational semantics for four familiar operations. Difference relations correspond to finite increments, discrete sums to their accumulation, differential relations arise when $a_i=h_iq_i$, and integral relations arise from the accumulation of cell contributions. Exact chaining, hiding, block composition, dependency-preserving coarse-graining, and later region compression are separated explicitly. Several structural results follow. Local increment domains compose exactly over finite blocks, whereas universal region compression yields additive residual bounds that can be sharpened when dependencies are retained. Difference and discrete summation satisfy an exact finite fundamental theorem, and the differential-integral pair appears as its variable-mesh analogue. Finite-resolution rational differential and summation problems may remain representable by rational finite relation networks even when zero-resolution limits involve transcendental functions or constants. We prove a class-relative zero-resolution certification barrier for transcendental singleton targets and for the exponential solution graph, while explicit rational refinement ladders converge to $\pi/4$ and $\log 2$. We then study finite gluing. Fixed-weight gluing commutes with differentiation, integration, difference, and summation. Variable-weight solution gluing generally does not: its differential defect is controlled by weight variation multiplied by local-model disagreement, with a parallel discrete formula. By contrast, direct gluing of differential or difference relations introduces no derivative-of-weight term. Positive relation-level gluing also propagates normalized residual certificates without increasing the worst local normalized resolution. These results provide quantitative design laws connecting model disagreement, transition width, and target resolution, and give a structural reason to glue relations before solving them when the target permits. The resulting framework makes precise one central claim of domain number mathematics: changing mathematical resolution can alter not merely numerical accuracy but the representation and certification class in which a problem can be treated.
Author: Ueoka, Yoshiki
Language: English
Subfield: Modeling and Simulation
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208824643
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