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.