Papers by Andrew J Murphy
Preprint, 2025
Science has fundamentally transformed human understanding and technology, yet the philosophical f... more Science has fundamentally transformed human understanding and technology, yet the philosophical foundations explaining its enduring success remain contested. Prevailing philosophies-logical positivism, falsifiability, paradigm theory, and instrumentalism-each illuminate aspects of scientific practice but fail to capture why science produces robust, predictive, and progressively refined knowledge. This paper advances the Structural Isomorphism Thesis: science succeeds by developing conceptual frameworks whose internal relational structures recursively align with the persistent architecture of external reality. By articulating science as the recursive refinement of structural mirrors, this thesis offers a unifying and generative foundation for the philosophy of science, reconciling realism with pluralism and explaining scientific progress without metaphysical finality.
Preprint, 2025
We propose a reformulation of the Bekenstein bound in which entropy is not fundamentally constrai... more We propose a reformulation of the Bekenstein bound in which entropy is not fundamentally constrained by geometric surface area, but by the symmetry class of the boundary-encoded in its coset structure Σ ∼ = G/H. Using Haar measure on G/H, we show that both the volume and maximum entropy of a radially symmetric region arise from the same invariant object, independent of embedding geometry. This leads to a testable divergence: when symmetry and area disagree (e.g., in non-smooth or irregular boundaries), the entropy bound scales with symmetry, not surface extent. This reformulation suggests that information constraints in physics are fundamentally governed by symmetry, not geometry-offering a new group-theoretic foundation for gravitational thermodynamics.
Preprint, 2025
We present a fully formal, constructive framework for analyzing Fermat's Last Theorem (FLT) using... more We present a fully formal, constructive framework for analyzing Fermat's Last Theorem (FLT) using only integer arithmetic, symbolic polynomial manipulation, and classical descent. Central to the approach is a structural device-the Minimal Geometry of Descent Closure (MGDC)-which characterizes a recursive, norm-reducing descent over integer triples satisfying the Fermat equation. MGDC supports symbolic descent to contradiction without invoking elliptic curves, modular forms, or field-theoretic tools. All derivations are conducted within first-order arithmetic over Z, and the method is entirely finitary and syntactically verifiable. We show that any putative FLT counterexample must lie within the MGDC class, and that repeated application of the symbolic descent operator systematically eliminates such candidates. These results position symbolic descent, formalized through MGDC, as a candidate framework for a classical resolution of FLT and offer a new pathway for applying descent techniques to Diophantine problems.
Preprint, 2025
We present a unified framework for symbolic complexity, founded on a conserved decomposition law ... more We present a unified framework for symbolic complexity, founded on a conserved decomposition law and two derived constructs. This establishes the first invariant geometry for symbolic reasoning, enabling structure, randomness, and model coherence to be evaluated with the same formal precision long available to physics, mathematics, and information geometry. First, we define the Descriptive Invariant Duality (DID), an identity showing that every well-formed symbolic object admits a normalized partition into irreducible complexity and generative structure. This defines a canonical coordinate system over the 1-simplex and enables scale-invariant comparisons across symbolic domains. Second, we derive the Kolmogorov-Fisher-Curvature (KFC) bound, a geometric upper bound on symbolic complexity expressed as a curvature-weighted integral over Fisher information. This links algorithmic cost to representational strain and introduces a differential geometric approximation for symbolic complexity. Third, we introduce the Computable Triangulated Inefficiency Diagnostic (CTID), a falsifiability metric that quantifies the alignment between empirical compressors and symbolic model structure relative to the DID constraint. CTID is satisfied by physical systems obeying thermodynamic laws and violated by hallucinated or unjustified symbolic inference. We additionally define a scalar balance residual to detect proportionally symmetric but absolutely inaccurate decompositions. Together, these components define a framework for a new class of testable invariants and computable diagnostics for symbolic systems. It unifies entropy, curvature, and descriptive cost into a falsifiable symbolic mechanics, bridging mathematical formalism with broad applications across physics, logic, and the structure of inference itself.
Preprint, 2025
We examine structural limits on uniform algorithmic generation of NP witnesses, using a construct... more We examine structural limits on uniform algorithmic generation of NP witnesses, using a constructive framework grounded in Kolmogorov complexity and classical counting arguments. Within this setting, we isolate dense subsets of NP-complete instances whose valid witnesses are provably Kolmogorov-incompressible beyond a fixed threshold. We show that no deterministic polynomial-time algorithm can uniformly generate witnesses across these subsets without violating standard information-theoretic bounds. This suggests that witness generation for such instances lies fundamentally outside P, even though verification remains efficient. The approach avoids dependence on cryptographic assumptions, circuit lower bounds, or uncomputable constructs, and remains unaffected by relativization, natural proofs, or algebrization. By eliminating a broad class of candidate algorithmic strategies, the framework narrows the plausible space in which P = NP could still hold and offers a tractable new angle on the separation question.
Preprint, 2025
We derive closed-form expressions for the electromagnetic fine-structure constant α and the weak ... more We derive closed-form expressions for the electromagnetic fine-structure constant α and the weak isospin coupling constant α 2 , using only topological and symmetryconstrained properties of compact configuration spaces. Starting from the internal spinor-gauge structure of a minimal U(1) domain, we construct a five-dimensional norm-bounded configuration space with a four-sphere boundary. By evaluating a volume-to-surface amplitude and applying a canonical angular projection factor, we obtain:
Drafts by Andrew J Murphy
Zenodo, 2025
Microbial communities inherited within host populations may act as independently evolving reprodu... more Microbial communities inherited within host populations may act as independently evolving reproductive isolators. This short letter argues that microbiome divergence can form early barriers to reproduction, linking microbial ecology to speciation and conservation.
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Papers by Andrew J Murphy
Drafts by Andrew J Murphy