Mathematical Foundations of Universal Admissibility

Abstract

Mathematical Foundations of Universal Admissibility Coding the Macroscopic Invariant Structure Driven by Dean A. Kulik September 2026 Abstract This report develops a registry-based framework for distinguishing local admissibility certificates from global realizability conditions across discrete computation, graph topology, feedback dynamics, delay systems, and quantum operations. In each domain, a local test—such as a continuation quotient, a declared 2-cell relation, a spectral loop-closure condition, or Gram-matrix positivity—defines a constrained boundary of allowable behavior. The analysis demonstrates that these local certificates do not generally determine the macroscopic object of interest: residual homology requires a completed relation complex; a local instability threshold does not determine the existence or basin of a stable periodic orbit; and a deterministic quantum map may become feasible only after the operational class is expanded to a heralded trace-nonincreasing process. The missing completion is formalized in each case, and an "admissibility registry" is proposed to record the state space, equivalence relation, local certificate, required global object, and resource coordinates. The framework is intended as a reproducible software specification: every registry entry is paired with declared assumptions, executable tests, and explicit falsifiers. 1. Introduction: The Local-to-Global Separation Principle The transition from localized computational thresholds to the global topological organization of physical and informational systems requires a precise mathematical framework devoid of ad hoc modeling. The underlying algorithmic structure of reality—whether analyzed through the discrete lens of finite state automata, the continuous manifolds of delay differential equations, the geometric bounds of quantum information, or the topological constraints of graph configuration spaces—can be understood not as a maximal collection of theoretical distinctions, but as a minimum stable distinction structure. This structure is what remains after all admissible mathematical identifications and boundary conditions are rigorously imposed. However, a fundamental category error persists in applied mathematics, theoretical computer science, and physics: treating a local mathematical derivative or threshold as identical to the global geometric object it permits. The central mathematical principle established in this report is formalized as follows: To code to the fundamental limits of computation and dynamics, one must evaluate an exact mathematical registry that classifies these boundaries. This requires stating the allowed class of transformations, calculating the local boundary, identifying what is strictly not represented by that local calculation, and systematically constructing the global object required for the actual physical or computational claim. This report formalizes the missing completion across four sharply bounded domains, providing a unified formal core and a strictly auditable registry. 2. Formal Core To align several distinct domains of "admissibility" under one design discipline, a common mathematical object is defined to model the state, transformation, and resource limits of any given system. We define the universal admissibility object as the tuple: where: : state or description space (the total space of available configurations). : admissible transformations (the allowed dynamic or logical operators). : equivalence or quotient relation (the topological or functional identifications). : retained resource coordinates (the required spatial dimensions, physical delays, or ancilla environmental parameters). : observables or output contract (the macroscopic structure that is mathematically guaranteed). Within this universal framework, the underlying principle dictates that: A local certificate evaluates a constrained boundary of allowable behavior. Such a certificate might take the form of a Jacobian crossing the imaginary axis, a positive-semidefinite matrix difference, a commuting logical square, or a localized quotient closure. Conversely, a global certificate is the complete mathematical manifestation that renders the phenomenon usable, such as an attracting invariant set in phase space, an explicitly generated homology class, a physically completed quantum instrument, or a finite minimal automaton quotient across all operational gaps. The transition from local certification to global realization defines the strict burden of proof required for establishing macroscopic existence. 3. Four Sharply Bounded Case Studies 3.1 Case Study 1: Graph Configuration Complex and Homological Defects When a discrete computational domain operates, it functions by systematically identifying equivalent states to prevent representational redundancy from masquerading as actual physical divergence. The formalism for establishing these fundamental identifications relies heavily on the homology of graph configuration spaces and their corresponding topological invariants. Configuration spaces of points are pivotal in modeling collision-free motion planning for autonomous agents, a problem domain deeply connected to geometric group theory and the topological complexity of systems. 3.1.1 Exact Graph Construction and Betti Notation In finite token-configuration complexes, a state graph possesses a 1-skeleton representing distinct local transitional moves. The relation layer of this space is formed by formally declaring 2-cell predicates. Consider the arithmetic group: with the generating set: For any index , the generator provides a directed edge: For the top generator , the operation: establishes an undirected, self-inverse edge, yielding a simple topological graph. To maintain strict notational consistency, the first Betti number is denoted as , reserving solely for contexts where it is explicitly redefined as the first Betti number in specialized topological configurations. For a connected finite graph, the exact algebraic calculation of the Betti number is: Under the specific edge-count convention where each generator edge is counted exactly once after identifying the self-inverse top-bit edges, the vertex and edge counts evaluate to: Therefore, the baseline Betti number calculates to: This algebra is rigorously reproducible solely if the graph is connected and the top-generator symmetry is explicitly mapped. The first Betti number fundamentally quantifies the number of independent cycles in the graph, forming a baseline for the higher-order relations and the broader study of graph braid groups. 3.1.2 State-Dependent Commuting Predicate To constrain the state space, 2-cell predicates are declared over commuting operations. When two operations commute, they are bounded by a 2-cell, effectively closing the loop. For a pair of generator indices (representing ordinary integer bit positions) and a local vertex , the square features vertices and . This square is declared "disjoint" exactly when the two operations modify distinct bits at vertex . The disjointness constraint is strictly state-dependent and can be verified via the boolean logic condition (the XOR/AND test), which functions as the exact bit-vector encoding of the mathematical support-set test: where the support set is defined as: Operationally, this means the addition in must not induce a carry chain that propagates into bit . This holds if and only if there exists a zero bit between and ; mathematically, some intermediate bit index must satisfy . To assure consistency across all configurations and evaluations, the following convention decisions are explicitly established: All additions: reduced modulo . All bit positions: represented by the canonical -bit word in . All support comparisons: use those canonical -bit representatives. Top-bit generator: undirected as a graph edge, but evaluated by the same canonical modular-addition support map. Wraparound is governed by reduction modulo ; bit positions remain indexed by . When , the intermediate set is merely . The condition violently collapses to the requirement that . If , adding flips bit and carries directly into , violating disjointness. 3.1.3 The Homological Defect and Discrete Morse Theory The residual topological dimension—representing the unclosed, macroscopic state distinctions—is measured by the algebraic quotient of these boundary cycles. Defining as the cycle space of the shared 1-skeleton, the boundary subspaces are formalized as follows: is the boundary span of the declared 2-cells. is the boundary span of all mathematically possible squares. The structure operates under the strict inclusions mapping the localized logical checks to the true geometrical boundaries. The homology of the declared and all-square complexes evaluates to: From these definitions, the homological defect is derived as the descent identity: This quotient is strictly a homological defect based on boundary subspaces. Determining this defect requires translating localized combinatorial conditions into a global topological evaluation. This is where discrete Morse theory, particularly the Farley-Sabalka discrete gradient field, becomes computationally indispensable. In this algorithmic approach, the cells of the complex are sorted into a Hasse diagram, and a discrete gradient field defines non-overlapping pairs of collapsible and redundant cells. Cells that cannot be paired in this manner are deemedcritical. According to discrete Morse theory, the homology of the underlying space is completely captured by a simplified Morse chain complex constructed strictly from these critical cells. By identifying exactly where diverges from local expectations—meaning an abundance of critical 1-cells that are not bounded by critical 2-cells—the defect geometrically isolates precisely what physical or logical structure is not repre