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Published September 19, 2026 | Version v2
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Arbitrary-Lag Obstruction to Positive Strong Shift Equivalence over Z+[t] and the Polynomial Baker Problem

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This paper proves an arbitrary-lag obstruction to positive strong shift equivalence over the polynomial semiring Z+[t]. For a specific pair of 3 × 3 polynomial matrices C and E4, any hypothetical finite positive strong shift equivalence chain is reduced to endpoint intertwining and power identities. Complete parametrization of the endpoint intertwiners, an exhaustive kernel dichotomy, adjugate factorization, and classification of a resulting positive polynomial norm equation reduce the problem to a scalar obstruction polynomial. Exact coefficient formulas show that this polynomial necessarily contains a negative coefficient for every admissible parameter pair, contradicting coefficientwise positivity.

Explicit positive elementary strong shift equivalences are then constructed between the original polynomial Baker matrices and the endpoint matrices. By transitivity, the arbitrary-lag obstruction gives a negative answer for the displayed polynomial Baker pair. The proof is algebraic and does not depend on finite-lag, finite-state, or numerical enumeration.

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References

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