Relation-Primary Arithmetic: Moving Problems Across Representations — A 133-Component Architecture Toward the Riemann Hypothesis

Relation-Primary Arithmetic A 133-Component Architecture for Moving Mathematical Problems Across Representations Relation-Primary Arithmetic (RPA) is a mathematical research architecture for moving a problem into the representation where its structure becomes easiest to see and use. It treats representation choice itself as part of mathematical problem-solving: transport the structure, determine what survives, identify what is lost, reconstruct what is needed, and cross-constrain independent views. Its central research cycle is: transport → compare → measure loss → lift → cross-constrain → exclude The governing idea is simple: A mathematical problem does not have to stay where you found it. RPA develops this principle across 133 linked mathematical components, moving among arithmetic coefficients, transfer operators, Dirichlet/Mellin symbols, prime-coordinate systems, ordered histories, finite and profinite state spaces, phase representations, Hardy-space geometry, Weil positivity, and spectral defect structures. The Core Architecture RPA repeatedly uses five mathematical moves: Transport Move the same structure into another representation where a difficult operation becomes simpler or native. Information Loss Determine exactly what a projection, scalarization, quotient, or coordinate change preserves, compresses, or destroys. Lift and Reconstruction When a representation loses necessary information, introduce the minimum additional structure needed to recover it. Rigidity and No-Go Results Use failed representations and incompatibility theorems to determine what any successful representation must contain. Cross-Constraint Represent the same obstruction in multiple controlled ways and ask what can satisfy all of them simultaneously. The result is not a single preferred representation of arithmetic, but a navigation system for representation space. Three Anchor Results AP-001 — Structure-Preserving Coefficient–Operator–Mellin Transport Establishes an explicit correspondence among arithmetic coefficients under Dirichlet convolution, scale-transfer operators under composition, and Dirichlet/Mellin symbols under multiplication. Prime-Generator Infinitesimal Rigidity Shows that under stated analytic and functional-equation constraints, an apparent first-order deformation of the primitive logarithmic generators collapses to rigidity. AP-109 — Exact Identification with Geometric Loss Demonstrates that a representation can remain exactly injective while still destroying geometric information needed for subsequent mathematics. Together they exhibit the central RPA pattern: exact transport → structural constraint → controlled information loss The Riemann-Hypothesis Frontier The late sequence AP-121–AP-133 carries the architecture into an RH-directed spectral program. A hypothetical off-critical zeta zero is transported through several linked analytic realizations: as a positive source, boundary current, Hardy-space defect, frequency leakage source, and obstruction to Möbius/Nyman approximation. The current frontier is explicit: Obtain critical-line norm control strong enough to force the off-line defect space to vanish. This release does not claim a proof of the Riemann Hypothesis. It provides a multi-representation defect architecture and isolates the remaining analytic obstruction. The Four-Document Edition 00 — Overview and Research Program The shortest complete introduction to the architecture, major results, and open frontier. 01 — Complete Mathematical Corpus The full 133-component architecture in one continuous mathematical document. 02 — Theorem and Proof Corpus AP-001–AP-133 The canonical statements and proofs in permanent AP order for direct adjudication. 03 — Discovery Morphology Case Study A longitudinal account of the representation shifts, failed carriers, repairs, and research transformations through which RPA developed. Each principal document is supplied in PDF for human reading and Markdown for search, comparison, AI-assisted reading, and reuse. Recommended order: Overview → Complete Corpus → Theorem and Proof Corpus. Research Archive RELATION-PRIMARY-ARITHMETIC-RESEARCH-ARCHIVE-V1_0.zip contains the reproducibility and development layer: TeX sources, experiments, validation tools, theorem-status metadata, audits, supporting research materials, and the complete AP-001–AP-133 Individual Proof Corpus in matched Markdown, TeX, and PDF editions. As a Research Instrument RPA is designed to help researchers ask: Where does this problem become easiest to see? What does this representation preserve? What information does it destroy? Can that information be reconstructed? Is the obstruction intrinsic, or created by the coordinates? What additional structure restores what was lost? Can another representation independently constrain the answer? What does a failed representation tell us about any successful one? Its intended contribution is therefore not merely a collection of results, but a reusable mathematical procedure for navigating representation choice itself. Summary Relation-Primary Arithmetic is a 133-component mathematical architecture for transporting problems across representations, tracking information loss, reconstructing missing structure, and cross-constraining independent mathematical views. Its current endpoint is an explicit RH-directed spectral frontier, while its broader research claim is that representation space itself can be treated as an active mathematical search space. Citation: Woffinden, A. (2026). Relation-Primary Arithmetic: Moving Problems Across Representations — A 133-Component Architecture Toward the Riemann Hypothesis, v1.0. Zenodo. DOI: 10.5281/zenodo.22731871Related theorem: Structure-Preserving Coefficient–Operator–Mellin Transport Theorem. DOI: 10.5281/zenodo.22731873License: CC BY-NC 4.0