Informational–Geometric Constraints as the Origin of Classical Mathematical Paradoxes A Relational Feasibility Framework Based on Stability Cost and Informational Openness

“The key to understanding the paradoxes of mathematics is not to eliminate them through stronger axioms, but to recognize them as the necessary signature of any system attempting to contain its own description under finite resources.” Abstract Classical “paradoxes” of mathematics—including G¨odel incompleteness, Russell’s para- dox, the Halting problem, Banach–Tarski, and Wigner’s “unreasonable effectiveness”—are typically treated as heterogeneous phenomena requiring separate foundational responses (axioms, meta-languages, or acceptance of non-constructive objects). This paper proposes a unifying diagnosis: paradoxes arise when mathematical objects are treated as absolute, cost-free entities rather than relational structures embedded in informational geometry with finite stabilization resources. We introduce a minimal formalism in which (i) mathematical representations are projections π of an underlying relational domain onto a formal system, (ii) representational and transformational acts carry an explicit stability cost functional J, and (iii) effective domains are informationally open, imposing intrinsic limits on verification. Within this framework, paradoxes are reinterpreted as signatures of four geometric mecha- nisms: (1) self-reference instability (metric degeneracy under π(π)), (2) boundary artifacts (treating limits as objects), (3) openness incompleteness (undecidability under inaccessible sectors), and (4) cost-neglect (non-constructive selections requiring divergent J). We prove general existence and instability results showing that any attempt to internalize its own projection operator forces either cost divergence or degeneracy of relational conditioning, yielding incompleteness or undefinedness without invoking ad hoc prohibitions. The paper derives corollaries for a canonical paradox family and formulates falsifiable cross-domain pre- dictions: constructions requiring non-measurable selection or arbitrary choice correspond to divergent stabilization cost and therefore cannot be physically realized in finite resources, while domains admitting stable low-cost relational embeddings yield robust effectiveness. The result is not a replacement of classical mathematics but a principled account of its lim- its and its effectiveness as consequences of informational–geometric feasibility. We provide a unified resolution pathway for eleven canonical paradox cases (foundational, computa- tional, geometric, semantic, and meta-theoretical) and connect feasibility claims to concrete physical realizability tests under finite resources