Andrew John Paton

Structural realism maintains that relational structure survives theory change yet lacks a fully articulated mechanism explaining why structural continuity persists across revolutions. This paper introduces the Constraint-Compatibility Persistence Law (CCSL), a domain-neutral persistence criterion stating that identity persists only within admissible constraint regions. Structural survival is explained as compatibility-filtered continuation rather than accidental retention. A formal lemma, boundary analysis, and falsifiability condition are provided. The framework applies across scientific, psychological, biological, social, and formal domains, demonstrating that persistence depends on dynamic compatibility with operative constraint frames rather than definitional identity alone. The account is necessary but not sufficient and introduces no new physical laws, functioning instead as a structural explanatory principle.

This paper presents a minimal structural framework for determining whether a state belongs to a system and whether system evolution may proceed. The framework isolates two necessary and sufficient conditions: admissibility and reachability. Admissibility specifies whether a generated state satisfies governing constraints, while reachability specifies whether the state lies on at least one permissible path from an initial condition. Together they define system membership and continuation without reference to domain-specific laws. This establishes a domain-neutral criterion for valid states applicable across formal, physical, computational, and dynamical systems.

This note presents a domain-neutral condition governing system continuation. The Unified Admissibility Equation states that a system persists if and only if its generated state satisfies admissibility constraints. By formalizing continuation as a binary admissibility relation, the result unifies logical, geometric, and dynamical representations of persistence within a single structural criterion. The formulation operates prior to domain-specific laws and therefore provides a minimal universal validity condition for modeled evolution across physical, biological, computational, and formal systems.

Scientific and formal disciplines routinely distinguish between persistence and collapse but typically treat them as domain-specific phenomena governed by separate mechanisms. This paper presents a unified structural account showing that persistence and collapse are complementary consequences of constraint-conditioned admissibility. Two previously established laws—the Constraint-Compatibility Persistence Law (CCSL) and the Constraint-Incompatibility Collapse Law (CICL)—are shown to jointly imply a general theorem: the Paton Constraint Corridor Theorem (PCCT). The theorem states that identity persistence is equivalent to the existence of at least one admissible identity-preserving trajectory within a constraint corridor. This result reframes stability, survival, and failure as boundary properties of admissible state space rather than dynamical accidents. The framework is domain-neutral and applies uniformly across physical, biological, cognitive, and formal systems.

This paper introduces the Paton Constraint Corridor Theorem, a domain-neutral structural result stating that a system persists as the same system over an interval if and only if there exists at least one admissible trajectory consistent with its governing constraint frames. The theorem formalizes persistence as an existence condition rather than a dynamical property and provides necessary and sufficient conditions for identity continuity across transitions. Corollaries establish collapse inevitability when admissible regions vanish and a structural survival condition across regime change. The result applies across scientific, mathematical, and conceptual domains and operates at the level of admissibility rather than dynamics.

Structural realism explains persistence of relational structure across transitions but lacks a domain-neutral account of why systems cease to persist. This paper introduces the Constraint-Incompatibility Collapse Law (CICL), a structural principle stating that identity fails whenever a system exits its admissible constraint region without an admissible recovery path. Collapse is thus explained as incompatibility-conditioned termination rather than spontaneous failure. A formal lemma, boundary analysis, implications, objections, and falsifiability condition are provided. Together with the Constraint-Compatibility Persistence Law (CCSL), this yields a unified structural account of persistence and collapse across domains.

Structural realism maintains that relational structure survives theory change yet lacks a mechanism explaining why structural continuity persists. This paper introduces the Constraint-Compatibility Persistence Law (CCSL), a domain-neutral persistence criterion stating that identity persists only within admissible constraint regions. The account explains structural survival as constraint-filtered selection rather than accidental retention. A formal lemma, boundary analysis, implications, objections, and falsifiability condition are provided.

The Domain Regime Resolution Boundary (DRRB) is a minimal operational clarity tool for identifying admissible overlap between General Relativity (GR) and Quantum Field Theory (QFT) regimes. It introduces no new physical laws and makes no unification claim. DRRB functions as a pre-analysis diagnostic gate to prevent uncontrolled regime mixing. It evaluates curvature scale, energy scale, and backreaction magnitude relative to Planck-scale thresholds. The output is a regime classification: admissible, caution, or inadmissible. DRRB is intended strictly as a conceptual and computational hygiene tool. Keywords GR–QFT interface; regime classification; Planck scale; curvature threshold; backreaction; diagnostic boundary; admissibility gate; philosophy of physics; theory structure

Structural realism maintains that relational structure survives theory change, yet it lacks a fully articulated mechanism explaining why structural continuity persists across scientific revolutions. This paper proposes constraint compatibility as a domain-neutral selection principle governing persistence. All persistent systems survive only insofar as they remain dynamically compatible with their operative constraint frames. Scientific revolutions eliminate incompatible structures through empirical constraint failure, while psychological systems exhibit the same structural dependence under internally regulated constraint regimes. Engaging the pessimistic meta-induction (Stanford 2006; Psillos 1999), the paper reframes theory survival as constraint-filtered persistence rather than accidental retention. A strengthened formal selection lemma is introduced, boundary behavior is analyzed, and a historical case (Newton → Einstein) demonstrates explanatory gain. The thesis is defended as a necessary (but not sufficient) condition for persistence across scientific, psychological, biological, social, and formal systems.

Scientific models are commonly evaluated in terms of prediction, explanatory power, and empirical adequacy. Yet such evaluation presupposes a prior structural condition: that the candidate state remains admissible under the constraints defining its representation. This paper argues that structural permission precedes dynamical interpretation. A state that violates its defining constraints does not reveal exotic behaviour; it dissolves the referential frame required for behavioural analysis. The thesis introduces no ontology and asserts no new physical law. It isolates a minimal condition of intelligibility operative wherever modelling occurs.

This paper introduces a centroid-referenced admissibility framework for constrained dynamical systems. By expressing system evolution relative to an intrinsic centroid datum rather than an external coordinate origin, admissibility can be evaluated as deviation from internal equilibrium. Constraints are represented as measurable fields with tolerance bounds, yielding a scalar admissibility margin that determines whether continuation is permitted. A local directional admissibility set is defined, characterizing allowable infinitesimal motions. It is shown that tightening constraints causes admissible direction sets to shrink monotonically and potentially collapse to the empty set, producing a well-defined no-go condition. The formulation is consistent with classical stability and feasibility methods while providing a unified geometric representation.

Wormholes are frequently represented as tunnels that permit traversal or effective distance reduction through spacetime. Such interpretations persist despite repeated violations of admissibility, including instability under perturbation, causal inconsistency, and unrepayable constraint budgets. This paper introduces a scale-invariant admissibility principle that reframes wormhole-class phenomena as boundary surfaces rather than transport pathways. The principle asserts that any admissible wormhole-class structure must preserve the full causal, energetic, and distance ledger at every scale, while permitting only boundary-level mirroring—visibility, correlation, or informational alignment—without enabling traversal. Apparent shortcuts arise from matched, oppositely oriented constraints that refract description across a boundary; such refraction alters representation without permitting physical motion. This framework is viability-first and non-interfering: it does not modify governing equations, introduce new forces, or propose transport mechanisms. Instead, it functions as an interpretive admissibility filter that distinguishes physically meaningful structures from representational artifacts. By insisting on real, touchable interaction at the observer’s scale while preserving global constraint accounting, the principle explains why wormhole-class phenomena appear observable and structurally real yet remain non-traversable. No new physics is proposed.

Scientific frameworks routinely encounter boundaries that resist direct description, including singularities, horizons, undecidable regimes, and scale limits. In many cases, these boundaries are treated either as placeholders for future ontology or as regions requiring speculative explanation. This paper argues for a different approach: boundary regions may require precise definition without descriptive content. Within the Paton System, Tier-8 represents such a boundary. It is not a domain of entities or mechanisms, but a limiting condition inferred through breakdown behaviour, curvature budgets, and constraint saturation. This paper formalises Tier-8 as boundary knowledge rather than ontological territory, clarifying why definition at this level is necessary for responsible use while description is neither possible nor admissible. The result is a viability-first account of boundary reasoning that preserves continuity, prevents misuse, and establishes a principled stopping rule.

Atomic theory is often presented as a linear progression toward increasingly accurate representations of physical reality. This paper offers an alternative analysis, examining atomic models in terms of model admissibility rather than cumulative truth. Scientific frameworks are assessed according to the empirical and conceptual conditions under which they generate stable, predictive explanations, and the points at which they break down when applied beyond those limits. Tracing the progression from early atomism through classical, quantum, and modern field-based descriptions, the paper argues that model failure typically reflects boundary crossing rather than falsity. Later models do not invalidate earlier ones absolutely, but clarify the constraints that define their domains of applicability. The analysis is methodologically conservative, makes no new physical or ontological claims, and is fully compatible with established scientific theories.

Scientific explanations are typically assessed in terms of truth, correctness, or predictive adequacy. However, many failures in scientific practice arise not from falsehood, but from the continued use of otherwise correct models outside the conditions under which their application remains admissible. This paper distinguishes truth from admissibility, arguing that these concepts play logically distinct roles in scientific reasoning. Truth concerns whether a claim accurately describes a stabilised structure, whereas admissibility concerns whether further explanation, prediction, or application is permitted under prevailing constraints. By clarifying this distinction within a viability-first framework, the paper explains how breakdown can occur without error and highlights the importance of admissibility as a precondition for responsible scientific continuation.

Atomic theory is often presented as a linear progression toward increasingly accurate representations of physical reality. This paper offers an alternative analysis, examining atomic models in terms of model admissibility rather than cumulative truth. Scientific frameworks are assessed according to the empirical and conceptual conditions under which they generate stable, predictive explanations, and the points at which they break down when applied beyond those limits. Tracing the progression from early atomism through classical, quantum, and modern field-based descriptions, the paper argues that model failure typically reflects boundary crossing rather than falsity. Later models do not invalidate earlier ones absolutely, but clarify the constraints that define their domains of applicability. The analysis is methodologically conservative, makes no new physical or ontological claims, and is fully compatible with established scientific theories.

Scientific explanations often break down not because the underlying laws are wrong, but because they are applied outside the conditions where further reasoning remains admissible. This paper introduces the Paton System, a viability-first framework that treats admissibility as a pre-theoretical condition for scientific explanation. Rather than proposing new physical laws or mechanisms, the Paton System formalises a continuation gate that determines whether explanation, prediction, or optimisation may proceed at all. The framework distinguishes correctness, applicability, and viability, arguing that viability is logically prior to dynamical modelling. By making explicit a constraint that is typically applied implicitly across science and engineering, the Paton System clarifies scope limits, breakdown conditions, and stability while remaining fully compatible with existing theories across domains.