The intersection of two context-free languages is not generally context-free, but no geometric criterion has characterized when it remains so. The crossing gap (max(i'-i, j'-j) for two crossing push-pop arcs) is the natural candidate. We refute this: we exhibit CFLs whose intersection is CFL despite unbounded-gap crossings. The governing quantity is the inner segment measure: for crossing arcs inducing a decomposition w = P1 P2 P3 P4, it is max(|P2|,|P3|), the length of the longer inner segment between interleaved crossing endpoints. We prove a dichotomy for this measure: bounded inner segments imply context-freeness via a finite buffer construction; growing inner segments with pump-sensitive linkages imply non-context-freeness. The inner segment concept applies to all CFL intersections; the strictness of the resulting characterization depends on the language class. For block-counting CFLs (languages requiring equality among designated pairs of block lengths), the dichotomy is complete: the intersection is CFL if and only if the combined arcs are jointly well-nested. For general CFLs, the CFL direction is unconditional; the non-CFL direction requires pump-sensitive linkages whose necessity is the main open problem, reducing the general CFL intersection problem to a specific property of pump-sensitive decompositions.

翻译：两个上下文无关语言的交集通常不是上下文无关的，但尚未有几何准则刻画其何时保持上下文无关性。交叉间隙（即两条交叉推入-弹出弧的最大(i'-i, j'-j)值）是自然的候选度量。我们证伪了这一猜想：通过构造具有无界间隙交叉却仍保持上下文无关交集的CFL实例。核心度量是内段测度：对于诱导分解w = P1 P2 P3 P4的交叉弧，该测度定义为max(|P2|,|P3|)，即交错交叉端点间较长内段的长度。我们证明了关于该测度的二分定理：有界内段通过有限缓冲区构造保证上下文无关性；而具有泵敏感链接的渐增内段则蕴含非上下文无关性。内段概念适用于所有CFL交集问题，所得刻画严格性取决于语言类别。对于块计数CFL（要求指定块长对之间相等的语言），该二分法完备：交集为CFL当且仅当组合弧联合良嵌套。对于一般CFL，CFL方向无条件成立；非CFL方向需泵敏感链接条件，其必要性构成核心未解问题，这将一般CFL交集问题归约为泵敏感分解的特定性质研究。