Forced Coincidence Under Admissible Overlap (Euler's identity,Fundamental Theorem of Algebra, the Gauss–Bonnet theorem, and Noether's theorem)

This paper establishes a general structural theorem governing overlaps between admissible formal representational systems. It proves that when two such systems share a representable domain and satisfy minimal determinacy conditions—explicit formation rules, closure under transformation, fixed normalization, and absence of discretionary supplementation—at most one equivalence relation on the overlap domain is admissible. Any divergence would violate admissibility. The theorem is applied classificatorily to four results traditionally treated as distinct foundations: Euler’s identity / Euler characteristic, the Fundamental Theorem of Algebra, the Gauss–Bonnet theorem, and Noether’s theorem. In each case, the result is shown to arise as a forced coincidence at the interface of overlapping admissible representational regimes, rather than as a consequence of geometry, dynamics, ontology, or explanation. No derivations, reductions, or interpretive claims are advanced. The results are treated strictly as certificates of non-optional representational identity under admissible overlap. The paper is entirely structural, interpretation-independent, and does not introduce new primitives. Differences among the classical results are shown to reflect representational context only, not foundational distinction.