This paper reframes relationship compatibility as a structural alignment problem rather than a psychological, emotional, or communication-based one. Most relational failures arise from a predictable mechanism: people use Emotional Reasoning Mode (ERM)—fast, associative, harmony‑seeking cognition—during decisions that require Logical Reasoning Mode (LRM)—slow, constraint‑aware, denotative cognition optimized for prediction and stability. When a relationship moves from low‑stakes social domains to high‑stakes structural domains (finances, cohabitation, legal entanglements, children, shared assets), ERM becomes maladaptive and systematically misaligned with the problem class. To correct this, the paper introduces a systems‑theoretic compatibility model built on three primitives: Cognitive Mode Selection (ERM vs LRM): identifying when modal mismatch, not personality or communication, is the driver of instability. [ Constraint Topology: mapping non‑negotiables as rigid or elastic constraints whose intersections determine whether a shared future is feasible. Only one partner may be rigid on any axis; dual rigidity collapses the topology. PLR Behavioral Gradients: a Parsimony–Leverage–Risk ternary model for describing resource posture, influence strategy, and risk tolerance, forming the gradient-based manifold where compatibility either stabilizes or diverges. Compatibility is defined as the intersection of constraint sets plus manageable PLR gradients under the correct cognitive mode. Misalignment emerges when rigid constraints collide, when modal switching fails during high‑stakes transitions, or when PLR divergence becomes unstable under load. The model explains why some relationships remain stable for decades while others fail suddenly when stakes increase—even if the early stages seemed harmonious. Beyond its human-facing implications, this paper presents a quantitative substrate for modeling, simulation, and prediction: ERM/LRM distinctions can be used for mode classification in dialogue systems. Constraint topology provides a formal framework for constraint‑set intersection, rigidity detection, and trajectory feasibility. PLR gradients offer a low‑dimensional embedding for behavioral prediction, stress testing, and multi‑agent alignment. The combined system is suitable for AI compatibility modeling, agent pairing, negotiation engines, co‑founder matching algorithms, and high‑stakes decision‑support tools. This is a foundational conceptual model, not a clinical protocol or prescriptive system. Its purpose is to provide clear, falsifiable structural primitives for predictive modeling, pre‑commitment analysis, applied assessment, and systems-level reasoning about relational stability. Positioning: AI‑first and non‑pop—this is a structural theory with operational hooks (ERM/LRM detection, constraint topology mapping, PLR embeddings) that can be implemented computationally or used to instrument high‑stakes human pairing. Related Works (Series Map). This paper is part of a six‑paper series developing a geometric framework for cognition, language, and applied inference: The Origin of Emergence — foundational mechanics of emergence https://zenodo.org/records/18807724). A Geometric Theory of Cognition — keystone theory https://zenodo.org/records/18807724). Language as Cognitive Geometry — structural form → patterns of thought (https://zenodo.org/records/18805427). Language as Constraint Geometry — tripartite attractor structure (https://zenodo.org/records/18807009). A Universal Decoder for Lost Oral Cultures — applied decoding under sparse priors (https://zenodo.org/records/18807013). Toward a Structural Model of Relationship Compatibility — applied compatibility modeling (DOI: 10.5281/zenodo.18050285). Series hub: GTC is supplemented by the two language pillars (LCG, LCnG) and two applications (UDLOC, SMRC). OoE provides the foundational emergence model that informs GTC.