This article develops a unifying framework for relating formal theories across logic and computation by combining categorical semantics with order-theoretic structure. We organize proof systems and models through polarity (Galois connections/adjunctions) and residuation, and study the role of lax/oplax morphisms in transporting theorems between theories. On the operational side, we analyze rewriting—highlighting local confluence, Church–Rosser properties, and critical-pair criteria—while connecting these to algebraic/quantitative semantics via residuated structures. The paper also explores completeness and soundness across ω-algebraic (omega-algebraic) domains and other dcpo-based models, and sketches a path-style semantics for reachability/derivation with thresholded values. For first-time readers, the included “Order & Polarity Card” acts as a compact guide that maps common orders, polarities, and their proof-theoretic rules (intro/elim), helping navigate between categorical viewpoints (adjunctions, monoidal closure, enrichment) and concrete rewriting or proof search. Overall, the work aims to make “relativity of theories” precise: when and how results, constructions, and normalisation arguments in one setting can be transferred to another through structured translations that respect order, polarity, and residuation. Keywords category theory, categorical semantics, adjunction, Galois connection, polarity, residuation, residuated lattice, quantale, enrichment, monoidal closed category, order theory, partial order, domain theory, dcpo, ω-algebraic, omega-algebraic, lattice theory, rewriting systems, term rewriting, local confluence, Church–Rosser, critical pairs, proof theory, natural deduction, sequent calculus, normalization, equality reasoning, lax morphism, oplax morphism, functorial translation, completeness, soundness, model theory, program semantics, abstract interpretation, reachability, weighted/path semantics, structural rules, introduction/elimination rules, logical frameworks, mathematical logic