Published January 15, 2026 | Version v1
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Structural Escape via Tame Congruence Theory: A Roadmap Toward Park's Conjecture under FRB

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We present a proof-theoretic and structural roadmap toward Park's conjecture in locally finite varieties: finite residual bound (FRB) should force finite basability (FB) for varieties generated by finite algebras. The framework is organized around a non-vacuous Escape Property EP(g), which transfers the existence of finite critical countermodels for bounded equational fragments into the existence of large finite subdirectly irreducible (SI) algebras inside the variety. The key structural input is a typed dichotomy based on Tame Congruence Theory (TCT) and commutator theory, together with verifiable amplification criteria in both non-abelian and abelian TCT types. This paper is intended as a self-contained and logically non-circular consolidation of the roadmap deposited at [1], with explicit interfaces for the remaining open non-vacuity component.

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References

  • Park's Conjecture Roadmap (Revised v6). Zenodo deposit. DOI: 10.5281/zenodo.18222333.
  • D. Hobby and R. McKenzie, The Structure of Finite Algebras, Contemporary Mathematics, Vol. 76, American Mathematical Society, 1988.
  • R. Freese and R. McKenzie, Commutator Theory for Congruence Modular Varieties, London Mathematical Society Lecture Note Series, Vol. 125, Cambridge University Press, 1987.
  • R. C. Lyndon, Identities in finite algebras, Proceedings of the American Mathematical Society 5 (1954), 8–9.