Analytical proof of the Bulatov–Zhuk CSP dichotomy

Give an analytical proof of the dichotomy theorem stating that for all k ∈ N, finite alphabets E, and collections P of k-ary predicates over E, the problem P-CSP either admits a polynomial-time algorithm or else is NP-hard.

Background

The Bulatov–Zhuk dichotomy is a landmark result for general CSPs, with existing proofs relying on algebraic and universal-algebra techniques. An analytical proof could offer different insights and potentially suggest simpler algorithms or new structural tools aligned with the correlation-based perspective developed in this work.

Such a proof may bridge CSP complexity with harmonic analysis and inverse theorems on product spaces.

References

We finish this article by mentioning a few open problems for future research. PROBLEM 6.5. Give an analytical proof for the dichotomy theorem: for all k E N, finite alphabet E and a collection of k-ary predicates P over E, the problem P-CSP either admits a polynomial time algorithm or else is NP-hard.

— The Lens of Abelian Embeddings  (2602.22183 - Minzer, 25 Feb 2026) in Problem 6.5, Section 6 (Open Problems)