无射影一致性的均匀性：嵌套二元项语法的精确反例
Uniformity without Projective Consistency: An Exact Counterexample for a Nested Binary Term Grammar
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中文总结 AI 辅助
针对嵌套二元项语法，构造了4级均匀测度经限制映射推前后不等于3级均匀测度的精确反例，证明4到3级是该语法中均匀性与射影一致性不成立的首次层级失败。
中文摘要 AI 辅助
设T₀={L}，Tᵣ₊₁={L} ∪ {N(a,b):a,b ∈ Tᵣ}。在非叶项Eᵣ上，要求每个事件N(a,b)在其非叶子项之后发生，令μᵣ为该偏序集线性延拓上的均匀测度。我们研究限制映射ρ₄₃，其删除新的4级事件同时保留3级事件的相对顺序。我们证明μ₄在ρ₄₃下的 pushforward（推前测度）不是μ₃。25个3级事件上的两个显式序具有不同数量的4级延拓。若bᵢ为长度i的前缀中T₂项（包括叶节点）的数量，则新释放的事件数为(i+1)²−bᵢ²。深度优先序的释放轮廓逐点支配层级序的释放轮廓，且在3≤i≤15时严格更大。因此，可容许交错之间的显式单射给出严格的解析纤维不等式。两个算法独立的精确计算重现了两个1557位的纤维计数；其约简比为614690215260160000/479048686862260621，近似值为1.2831476885707443。通过3级的类似限制是一致的，故4到3级是该语法中的首次失败。该结果特定于该语法、均匀测度和限制映射。
英文摘要
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.
