FCD-CORE-3: MATHEMATICS UNDER FINITE CONSTRAINTS

This paper presents an operational analysis of the limits of inversion, stability, and representation under finite, physically realizable constraints. We study implemented pipelines of the form G=T∘Q∘F, combining measurement, finite-resolution encoding, and projection or scale selection. Under explicit axioms of finite informational access, local interaction, constrained feedback, and scale separation, we show that no implemented pipeline can be simultaneously bijective, invertible with uniform operational stability, and stable with respect to the world. We establish that finite resolution and projection necessarily induce non-injectivity, that finite informational access implies a strictly positive operational indistinguishability floor, and that even when a formal inverse exists, inversion is generically non-operable under bounded resources and irreducible perturbations. More generally, any implementable formalism acts as a compression, discarding information and introducing frame-dependent bias. Finally, Gödel-, Turing-, and Shannon-type limits are shown to converge operationally, forbidding any joint totalization of description, computation, and measurement within a single finite pipeline.