The RPRM Manifesto: A Relational Framework for Mathematical Unification

The Relational Pressure Retention Model (RPRM) is presented as a relational framework for mathematical unification. It describes problems through explicit carriers, typed roles, operations, observations and missing information, and studies the conditions under which a change of representation preserves a question or a continuation. The manuscript develops definitions and scoped preservation results, with complete worked examples, counterexamples and a record of unresolved questions. Applications include arithmetic, finite systems, mechanical motion and declared scientific models. Written proofs, selected machine-checked proofs, finite computations and proposed experiments are identified separately. Version 1.0.2 restores the intended independent research scope. Detailed Fermat arithmetic and its complete certificate are retained as a supplementary study in the companion archive; the main paper gives a short scope summary. The independent result covers min(a,b) ≤ 4000, with the stated all-height subcases and exact decisions for supplied gaps. No independent unrestricted Fermat proof has been obtained. The theorem-inheritance closure framing of version 1.0.1 is withdrawn. The companion repository supplies executable examples and checks, 20 selected Lean declarations, a 2D Mechanical Motion Atlas, an agent handbook, a layperson guide, nine executable experimental packs and three specifications for scientific comparisons that remain unexecuted. Physical and biological proposals retain their stated model assumptions and unexecuted experimental obligations. The project aims to connect existing mathematics while preserving its meaning. Original material is available for reuse under the licenses described in the record and accompanying notices.