Abstract This manuscript develops a unified algebraic, computational, and physical framework for rep-resenting, certifying, and exploiting structured partitions of high- dimensional spaces. Serving as the operational initialization for the categorical classification of non-associative gauge vacua, this work translates determinantal sheaf theory and functorial doubling into exact, machine-verifiable algorithms. The text begins with fully certified octonion (k = 3) and sedenion (k = 4) case studies,followed by applied treatments of determinantal geometry, canonicalization via Luna slices, spectralmass gap diagnostics, and exact topological transport. It concludes with the formal specification of a minimal reproducible artifact bundle, establishing the exact computational standards required for independent verification. "The Moduli Degeneracy: The Real Post-Quantum Cryptography .arr Standard": Table of ContentsIntroductionPart I: The Initialization StagesChapter 1: Octonions as Initialization Stage: Functorial Origin, Determinantal Sheaf, and Certified Structure * 1.1 Introduction and overview * 1.2 Functorial origin of O * 1.3 Left multiplication family and determinantal loci * 1.4 Determinantal sheaf W * 1.5 Orbit reduction and canonicalization via Luna slices * 1.6 Certification mandate: Positivstellensatz and Thom encodings * 1.7 Main structural theorem (octonion determinantal classification) * 1.8 Worked canonical example (sheaf language and full certificates) * 1.9 Stabilizer computation and gauge projection * 1.10 Spectral gap, mass gap certification, and Davenport-Mahler * 1.11 Concluding remarks Chapter 2: Sedenions: Doubling, Zero Divisors, Determinantal Pathologies, and Certified Examples * 2.1 Introduction and overview * 2.2 Functorial doubling: S = D_beta(O) * 2.3 Algebraic pathologies: zero divisors, alternator, and cohomology * 2.4 Left multiplication, determinantal loci, and the sheaf W * 2.5 Primary decomposition and embedded components * 2.6 Main structural statements for S * 2.7 Automorphisms, derivations, and the induced G_2 action * 2.8 Stabilizer computation under the induced action * 2.9 Certification mandate for sedenion examples * 2.10 Canonical seeds and worked sedenion examples * 2.11 Topological and spectral pathologies * 2.12 Algorithmic complexity and practical heuristics * 2.13 Figures, incidence tables, and data artifacts * 2.14 Concluding remarks Part II: Operational Geometry and CanonicalizationChapter 3: Degeneracy Planes and Determinantal Geometry (Applied) * 3.1 Overview and objectives * 3.2 Determinantal ideals and schemes * 3.3 The determinantal sheaf W * 3.4 Local normal forms near transversal points * 3.5 Degeneracy planes and hyperplane arrangements * 3.6 Algorithmic detection and certification * 3.7 Local normal forms: computational extraction of slice equations * 3.8 Examples revisited: octonion and sedenion loci * 3.9 Incidence data, Hasse diagrams, and repository manifest * 3.10 Concluding remarks Chapter 4: Canonicalization and Stabilizers (Operational) * 4.1 Overview and objectives * 4.2 Preliminaries and notation * 4.3 Stabilizer extraction and representation computation * 4.4 Deterministic canonicalization with field extension handling * 4.5 Certificate invariants and mandatory verification checklist * 4.6 Numeric witness to exact reconstruction refined * 4.7 Complexity, heuristics, and mitigations * 4.8 Worked operational sketch with representation data * 4.9 Concluding remarks Part III: Physical Translation and Stability**Chapter 5: Algebraic Charge and Spectral Readings (Applied Technical) * 5.1 Overview and objectives * 5.2 Algebraic charge and the physical vacuum dictionary * 5.3 Transverse spectral extraction: isolating the physical mass gap * 5.4 Signature diagnostics, Sturm counts, and multiplicity handling * 5.5 Topological responses: holonomy, Chern evaluations, and loop safety * 5.6 Compatibility checks: representation vs spectral projectors * 5.7 Exact certificate format and verification checklist (revised) * 5.8 Complexity, heuristics, and practical safeguards * 5.9 Concluding remarks Chapter 6: Perturbations, Topology, and Minimal Artifact Bundle * 6.1 Overview and objectives * 6.2 Notation and standing hypotheses * 6.3 Algebraic stability of determinantal fibers * 6.4 Transverse spectral continuity and certified mass gaps * 6.5 Certified projector transport and holonomy * 6.6 Lyapunov-Schmidt reduction adapted to the determinantal sheaf * 6.7 Minimal artifact bundle (final specification) * 6.8 Verification recipes and automation notes * 6.9 Concluding remarks Part IV: Appendices: The Open StandardAppendix D: Artifact Bundle Specification * D.1 Scope and intent * D.2 High level bundle layout * D.3 File format specifications * D.4 Cryptographic and integrity conventions * D.5 Verification scripts and deterministic checks * D.6 Numeric precision and interval arithmetic policy * D.7 Reproducibility checklist * D.8 Provenance and authorship metadata * D.9 Notes on archival and decentralized reproducibility * D.10 Reproducibility checklist (duplicated section in source) * D.11 Concluding remarks Appendix E: Computational Benchmarks, Reproducibility Notes, and Extended Proofs * E.1 Overview * E.2 Benchmark summary * E.3 Verification log format and canonical examples * E.4 Container recipe and reproducibility notes * E.5 Extended technical lemmas and proofs * E.6 Practical troubleshooting and FAQ * E.7 Concluding remarks

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