Domain Number Mathematics I : Resolution as Problem Data, Dependency-Preserving Finite Relations, and FRN-2

Abstract

Many mathematical problems are posed at a level of resolution stronger than what is required by a given task. This paper develops a framework in which requested mathematical resolution is treated as explicit problem data rather than only as a numerical-accuracy parameter. The contribution claimed here is not that intervals, semialgebraic regions, finite constraints, or quadratic computational graphs are individually new. The proposed contribution is their integration under three linked principles: (A) \emph{resolution is mathematical data}; (B) relation-valued and region-valued targets may themselves be exact mathematical objects; and (C) dependency-preserving finite relation networks provide a common operational semantics across arithmetic, algebra, and elementary dynamics. Point-valued, relation-valued, and region-valued representations are therefore separated from the independent question of exactness. As a first arithmetic model, positive integers are equipped with regions $I_n=[n-r_n,n+r_n]$. For the power law $r_n=\epsilon n^\alpha$, global covering, local reducibility, sequential factorization, and direct generation by a fixed finite multiplicative basis are distinct notions. Under the local relation $|n-ab|\le r_n$, the region-irreducible integers are exactly $2$ together with ordinary odd primes satisfying $r_p<1$; hence for every $\alpha>0$ only finitely many region primes remain. Nevertheless, for $0<\alpha<1$, no fixed finite multiplicative basis directly covers all sufficiently large integers at that resolution. This gives a first representative theorem of the framework: resolution can separate irreducibility from multiplicative generation. For algebraic equations, on the natural domain $Dom(D_P)=\left\{x:\sum_k|a_kx^k|>0\right\},$ $D_P(x)=\frac{|P(x)|}{\sum_k |a_kx^k|},$ a prescribed $\delta$-domain becomes an exact balance condition between positive and negative term groups. We then introduce FRN-2 (Finite Relation Network, degree 2). Its defining content is not the degree bound alone: FRN-2 integrates rational relations of degree at most two, domain guards, dependency semantics based on node identity and fresh copies, an observed/internal variable distinction, and a separation between exact relational operations and later region operations such as projection or certified compression. Every finite rational arithmetic computation can be represented in FRN-2, and real algebraic objects can be represented exactly over rational coefficients by extending the finite relation system rather than requiring the represented algebraic values to be adjoined as coefficient constants. The same language covers rational functions, finite and periodic continued fractions, iterated maps, fixed points, periodic points, convergence certificates, and elementary bifurcation certificates without expanding high-degree iterates. A resolution preorder is introduced for nested problem families: refinement shrinks admissible solution sets, and the original equality problem may appear as an intersection of progressively finer domainized problems. Finally, we describe a complementary role for the gluing method: domainization changes what level of information the problem asks for, while gluing assembles available local information to meet that specified target. The intended scope is not a replacement of point-valued mathematics, interval analysis, semialgebraic geometry, or numerical analysis, but an additional formulation layer connecting them through resolution-explicit and dependency-preserving semantics.