Part I - CONTRADICTION: the geometry of structural conflict

Project website: https://www.integrodynamics.org/ This paper develops a formal framework for using contradiction to assess the geometry of structural conflict within human, institutional, and algorithmic decision systems. Structural conflict arises where a system's stated rationales, narrative justifications, and observable behaviours cannot be jointly sustained under symmetric examination. The framework models these elements through a rationale set R, narrative set N, behaviour set B, and a framed proposition f(P) designed to expose the structure, location, and magnitude of their incompatibility. Within this broader framework, the contradiction trap is a diagnostic construction in which every admissible response produces a minimal unsatisfiable closure and therefore incurs positive coherence cost. The resulting interaction is formalised as a one-move epistemic audit game in which contradiction is treated not as argumentative failure, but as an observable and information-bearing outcome. The framework operationalises structural conflict through rule-based, graph-informed, semantic-distance, and model-checking estimators of coherence cost. It distinguishes the existence of contradiction from its distribution across response branches, permitting structural inconsistency, asymmetric reasoning, and preferential commitment to be analysed separately. Secondary responses such as delay, reframing, avoidance, and misdirection are captured through a composite measure of meta-evasion. Branch-level coherence costs provide a basis for assessing whether otherwise symmetric alternatives impose materially different strains on the system's declared commitments. Where such asymmetry cannot be explained by legal, operational, informational, or stochastic necessity, it may increase the evidential weight of hypotheses involving selective or motivated reasoning. This inference is probabilistic rather than dispositive: contradiction establishes structural inconsistency, while any inference concerning intent requires an explicit necessity test and additional supporting evidence. By combining dialectical reasoning, inconsistency measurement, graph theory, information theory, Bayesian inference, and audit design, the framework provides a reproducible method for examining the geometry of structural conflict across legal analysis, organisational governance, algorithmic accountability, and other systems claiming neutrality. Ethical conditions concerning symmetry, transparency, proportionality, non-coercion, and appeal are treated as constitutive requirements of valid deployment. Keywords: contradiction analysis; geometry of structural conflict; structural inconsistency; contradiction trap; contradiction games; coherence cost; minimal unsatisfiable closure; epistemic auditing; motivated reasoning; algorithmic accountability; meta-evasion; reasoning integrity.