This monograph unifies a three-phase research programme in the foundations of physics and the philosophy of science into a single, self-contained argument. The programme addresses three interconnected foundational questions: Why is mathematics applicable to the physical world? Why do the fundamental constants of physics take the values they do? What role does the cognitive architecture of the physicist play in determining the form of physical law? The framework developed here — Structural Necessity Theory — answers all three questions within a single formal apparatus, proving theorems from explicitly stated axioms rather than adopting philosophical stances on aesthetic or pragmatic grounds. The central claim is that the relationship between mathematical structure and physical reality is structurally necessary at three distinct levels, each corresponding to one part of the monograph. Part I: Why Mathematics Works — Structural Necessity of Isomorphism Between Formal Systems and Physical Reality. The Persistence Filter Thesis (PFT) is established: any system that persists through time necessarily carries invariant structure; invariant structure induces algebraic structure; algebraic structure is a model of a substantive formal theory. The applicability of mathematics to physics is therefore not a coincidence but a consequence of persistence. A Dual-Source Decomposition is introduced, separating any physical theory T_phys into a form constraint T_form (arising from the cognitive-representational architecture of the theorist) and a content constraint T_cont (arising from empirical invariants extracted from the physical world). Part II: Can Fundamental Constants Be Derived as Irreducible Residues of Structural Self-Consistency? Five axioms characterising any system capable of sustaining persistent structure — persistence, self-reference, closure, non-degeneracy, and distinguishability — are formulated. It is proved that any model satisfying these axioms necessarily contains free parameters that are simultaneously irreducible (not derivable from the axioms alone) and non-contingent (constrained to measure-zero subsets of parameter space by the requirement of self-consistency). These parameters are termed irreducible residues. Two classes are distinguished: scale residues (dimensional constants such as c, ℏ, G) and coupling residues (dimensionless constants such as α ≈ 1/137). The central existence theorem is proved by two independent methods — a Brouwer fixed-point argument and a topological degree argument — and is illustrated by five computational toy models. Connections to asymptotic safety, the conformal bootstrap, and the swampland programme are established. The anthropic principle is shown to be a special case of a more general persistence filter. Nine objections are addressed and six falsifiable predictions are identified. Part III: Epistemic Filtering and the Cognitive Projection of Structural Necessity — Toward a Dual-Layer Theory of Fundamental Constants. A formal cognitive encoding operator Π: W → R is introduced, mapping mind-independent physical structure into a representational space subject to the admissibility constraints of a cognitive architecture. Three main theorems are proved by multiple independent methods. The Epistemic Residue Existence Theorem, supported by the Separation Inheritance Lemma, establishes that any cognitive architecture capable of stable theory formation necessarily produces a bounded, nonempty set of epistemically stable invariants functioning as fundamental constants. The Epistemic Stability Theorem establishes robustness under architectural perturbation. The Architecture Independence Theorem, using a metrological sufficiency condition that avoids circularity, establishes that dimensionless coupling constants are invariant across all sufficient cognitive architectures while dimensional constants may vary, with a Predictive Equivalence guarantee that no physical prediction is affected by this variation. Three additional toy models illustrate the framework. Four falsifiable consequences are identified, including a formally proved dimensional reduction bound. The paper engages directly with the Duff–Okun–Veneziano debate on dimensional versus dimensionless constants, Kant's transcendental idealism, structural realism, Tegmark's Mathematical Universe Hypothesis, Wheeler's "it from bit" programme, and the representational measurement theory of Krantz, Luce, Suppes, and Tversky. The status of running coupling constants, discrete dimensionless parameters such as N_gen = 3, and the diachronic process of theory formation are addressed. A documented peer review and revision history is included. The central result of the monograph is the dissolution of the false dichotomy between "mathematics reflects the world" and "mathematics is a cognitive projection." The Dual-Layer Theory shows that fundamental constants decompose into an architecture-invariant ontological core (the coupling residues — dimensionless, mind-independent, determined by self-consistency) and an architecture-dependent formal scaffolding (the scale residues — dimensional, mind-involving, determined jointly by the world and representational conventions). The "unreasonable effectiveness" of mathematics is decomposed into three reasonable necessities: structural necessity (persistence implies algebraic structure), parametric necessity (self-consistency constrains parameters), and representational necessity (cognitive architecture filters admissible representations). The monograph includes a General Introduction providing the overarching narrative connecting the three parts, a unified notation table, a guide to reading (including a 40-page fast-track path), eight computational toy models with numerical simulations, ten falsifiable predictions across all three parts, engagement with over twenty bodies of existing philosophical and scientific literature, and open-process exposition with explicit motivation, self-critique, and documented peer review throughout. Keywords: structural necessity, fundamental constants, persistence filter, mathematical applicability, structural self-consistency, irreducible residues, epistemic filtering, cognitive architecture, encoding operator, dual-layer theory, architecture independence, coupling constants, scale constants, dimensionless constants, Duff–Okun–Veneziano, structural realism, Kantian a priori, Wigner's problem, anthropic principle, fixed-point theory, Brouwer theorem, topological degree, renormalisation group, conformal bootstrap, asymptotic safety, swampland programme, measurement theory, Tegmark, information-theoretic physics, philosophy of physics, philosophy of mathematics, foundations of science License: CC BY 4.0 Author: Boris Kriger — ORCID: 0009-0001-0034-2903 Affiliations: Information Physics Institute, Gosport, Hampshire, United Kingdom; Institute of Integrative and Interdisciplinary Research, Toronto, Canada