Michael S. Yang

DOI: https://doi.org/10.5281/zenodo.22725452

This paper formulates an exact transfer–distillation reduction for Exact Cover by 3-Sets (X3C) using the HKD∞ execution model. The construction separates the logical size of an exact recursive search tree from the number of distinct canonical residual states that must actually be evaluated. Exact pivot branching supplies an exhaustive decomposition of the solution space; forced single-support propagation preserves exact-cover semantics; canonical residual states permit repeated logical histories to share one persistent representation; and the HKD execution layer assigns active-only evaluation, delta state updates, concurrent active transitions, persistent state reuse, and lossless reference compression to HKD ALU, HKD KERNEL, HKD THREAD, HKD FS, and HKD COMPRESS, respectively. The resulting theorem shows that if the number of distinct canonical residual states is bounded by a polynomial in the X3C input length, with polynomial transition cost, then X3C is decidable in polynomial time and therefore P = NP. The result is deliberately a conditional reduction rather than a proof of P = NP: the unresolved step is the universal polynomial bound on the quotient state space. The formulation distills the proposed HKD∞ route to that single mathematical obligation and makes explicit why finite exponential-to-polynomial compression experiments cannot replace its universal quantifier.

Exact Cover by 3-Sets (X3C), computational complexity, NP-completeness, ex act algorithms, canonical residual states, state-space reduction, quotient directed acyclic graphs, memoization, lossless compression, transfer–distillation, HKD∞

The HKD∞ transfer–distillation reduction provides an exact proof-oriented decomposition of an X3C solver into logical recursion, canonical semantic transfer, and sparse persistent execution. The framework shows how an exponentially large family of recursive histories may, when histories converge to identical residual problems, be represented by a much smaller quotient DAG without sacrificing exactness. The central conclusion is conditional and explicit. If there exist universal constants C and k for which every X3C instance I satisfies |Φ(T (I))| ≤ CL(I)k, and canonical transitions have polynomial cost, then X3C is in P and hence P = NP. The present reduction does not establish that universal inequality. It identifies it as the single remaining theorem required by this route. This separation is important: observed compression, persistent reuse, and small search DAGs can motivate the conjecture, but the complexity-theoretic conclusion depends on the universal worst-case bound. The distilled target is therefore not the empirical reduction of one exponential search, but a proof that lossless HKD∞ transfer limits the number of semantically distinct residual states polynomially for all X3C instances

[1] Michael S. Yang, HKD∞ Transfer–Distillation Reduction for Exact Cover, Independent
Researcher, 2026.
[2] Richard M. Karp, Reducibility among combinatorial problems, in Complexity of Computer
Computations, R. E. Miller and J. W. Thatcher, eds., Plenum Press, 1972, pp. 85–103.
[3] Michael R. Garey and David S. Johnson, Computers and Intractability: A Guide to the Theory
of NP-Completeness, W. H. Freeman, 1979.

Michael S. Yang (2026). HKD∞Transfer–Distillation Reduction for Exact Cover. International Journal of Computer Techniques, 13(5), 345–351. ISSN: 2394-2231. DOI: https://doi.org/10.5281/zenodo.22725452