We develop a graph path-sum framework for constructing determining equations for the Anderson localization-length exponent from a critical Green-function sector. The path sum is an all-depth representation; finite depth enters only as an explicit computational truncation. The critical Green function generates contact-irreducible interference shells; forest subtraction removes inherited lower-topology pieces; orbit reduction quotients finite path symmetries; and weighted Krylov--Lanczos compression maps the resulting hierarchy to a Jacobi operator. At finite depth the exponent is then defined by a finite Schur-complement equation, without assuming that the all-depth hierarchy has a simple closed form. For the three-dimensional orthogonal box-disorder problem we evaluate the hierarchy through depth four and obtain a working largest Jacobi eigenvalue $\lambda_4\simeq0.0345442$. Under an explicitly conditional critical-mode detuning-renewal closure and a Gaussian reference conversion $d_w^{(0)}=2$, the reference sequence $1.5000\to1.5227\to1.5398\to1.5528\to1.5537$ moves toward the independent transfer-matrix benchmark $\nu\simeq1.571$ at every computed depth. Because the nested Jacobi edges obey interlacing and the auxiliary renewal map is increasing, the monotonic direction is partly structural; the useful diagnostic is the size and stabilization of the finite-depth corrections. This auxiliary sequence is not a numerical root of the physical Schur equation, not a convergence theorem, and not an identification of the physical critical walk dimension with $2$. A separate transfer-Ward projection remains near $1.0225$, showing that Ward and physical disorder-detuning projections must be distinguished. The physical detuning tangent is generated internally by differentiating the fixed-disorder resolvent and exact box-cumulant functional, so the remaining microscopic task is a singly marked shell calculation rather than an additional empirical input. The graph-path formulation also exposes geometry that a one-parameter Euclidean dimensional expansion does not retain by itself. At minimal single-tail level, $\nu_0\sim\frac{2}{d_w(d_s-2)},$ so a walk/path dimension enters before higher-shell corrections. For the same normalized critical path kernel, spectral/return, walk, and volume-growth exponents obey $d_s d_w^*=2d_V$. Noninteger effective dimensions can therefore be carried by path statistics instead of by a formally noninteger number of Cartesian integration variables. The three-dimensional demonstration uses local vertex data, path scaling, and finite irreducible topology, but not finite-size flow data, a full finite-sample adjacency matrix, or a full lattice spectrum as fitting inputs. The same critical relation can serve as the middle, critical-region anchor in finite weak--critical--strong gluing, while the graph-path architecture provides a concrete extension route to other Anderson symmetry classes and to other kernel/resolvent theories after their theory-specific propagators and local vertices are rederived.
Yoshiki Ueoka· Zenodo (CERN European Organi...· 0 citations
We develop a graph path-sum framework for constructing determining equations for the Anderson localization-length exponent from a critical Green-function sector. The path sum is an all-depth representation; finite depth enters only as an explicit computational truncation. The critical Green function generates contact-irreducible interference shells; forest subtraction removes inherited lower-topology pieces; orbit reduction quotients finite path symmetries; and weighted Krylov--Lanczos compression maps the resulting hierarchy to a Jacobi operator. At finite depth the exponent is then defined by a finite Schur-complement equation, without assuming that the all-depth hierarchy has a simple closed form. For the three-dimensional orthogonal box-disorder problem we evaluate the hierarchy through depth four and obtain a working largest Jacobi eigenvalue $\lambda_4\simeq0.0345442$. Under an explicitly conditional critical-mode detuning-renewal closure and a Gaussian reference conversion $d_w^{(0)}=2$, the reference sequence $1.5000\to1.5227\to1.5398\to1.5528\to1.5537$ moves toward the independent transfer-matrix benchmark $\nu\simeq1.571$ at every computed depth. Because the nested Jacobi edges obey interlacing and the auxiliary renewal map is increasing, the monotonic direction is partly structural; the useful diagnostic is the size and stabilization of the finite-depth corrections. This auxiliary sequence is not a numerical root of the physical Schur equation, not a convergence theorem, and not an identification of the physical critical walk dimension with $2$. A separate transfer-Ward projection remains near $1.0225$, showing that Ward and physical disorder-detuning projections must be distinguished. The physical detuning tangent is generated internally by differentiating the fixed-disorder resolvent and exact box-cumulant functional, so the remaining microscopic task is a singly marked shell calculation rather than an additional empirical input. The graph-path formulation also exposes geometry that a one-parameter Euclidean dimensional expansion does not retain by itself. At minimal single-tail level, $\nu_0\sim\frac{2}{d_w(d_s-2)},$ so a walk/path dimension enters before higher-shell corrections. For the same normalized critical path kernel, spectral/return, walk, and volume-growth exponents obey $d_s d_w^*=2d_V$. Noninteger effective dimensions can therefore be carried by path statistics instead of by a formally noninteger number of Cartesian integration variables. The three-dimensional demonstration uses local vertex data, path scaling, and finite irreducible topology, but not finite-size flow data, a full finite-sample adjacency matrix, or a full lattice spectrum as fitting inputs. The same critical relation can serve as the middle, critical-region anchor in finite weak--critical--strong gluing, while the graph-path architecture provides a concrete extension route to other Anderson symmetry classes and to other kernel/resolvent theories after their theory-specific propagators and local vertices are rederived.
Yoshiki Ueoka· Zenodo (CERN European Organi...· 0 citations
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.
Yoshiki Ueoka· Zenodo (CERN European Organi...· 0 citations
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.
Yoshiki Ueoka· Zenodo (CERN European Organi...· 0 citations
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