Uniformity without Projective Consistency: An Exact Counterexample for a Nested Binary Term Grammar

Abstract

Let T_0={L} and T_{r+1}={L} union {N(a,b):a,b in T_r}. On the nonleaf terms E_r, require each event N(a,b) to occur after its nonleaf children, and let mu_r be the uniform measure on the linear extensions of this poset. We study the restriction rho_43 that deletes the new level-4 events while preserving the relative order of the level-3 events. We prove that the pushforward of mu_4 under rho_43 is not mu_3. Two explicit orders on the 25 level-3 events have different numbers of level-4 extensions. If b_i is the number of T_2 terms, including the leaf, seen in a prefix of length i, the number of newly released events is (i+1)^2-b_i^2. The release profile of a depth-priority order dominates that of a level order pointwise and is strictly larger for 3<=i<=15. An explicit injection between admissible interleavings therefore gives a strict analytic fiber inequality. Two algorithmically independent exact computations reproduce both 1557-digit fiber counts; their reduced ratio is 614690215260160000/479048686862260621, approximately 1.2831476885707443. The analogous restrictions through level 3 are consistent, so 4-to-3 is the first failure in this grammar. The result is specific to this grammar, uniform measures, and restriction map.

Disclosure

“author declares no conflict of interest. Author contribution. The sole author formulated the problem, proved the structural results, implemented and checked the exact enumerations, and approved the manuscript. AI-assisted tools. Generative-AI tools were used during drafting, language editing, code review, and computational cross-checking. The author independently checked the proofs, references, and executable artifacts and assumes full responsibility for the contents.”

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Classification

Drafting limited passages

Multiplier

5

Verified

2026-08-23T05:17:36.333056Z

Structural counts