AI 中文总结
本文精确刻画了CFG和MCFG学习中观察、碎片化、暴露等有限资源间的权衡，给出帕累托前沿、特征数据成本及调度阈值，证明这些资源在重构中本质不同。
AI 中文摘要
我们研究了在固定观察的CFG和MCFG学习中控制精确正重构的有限资源。对于一个显式的刚性CFG族，我们精确计算了安全观察大小、内部残余碎片化和特征数据成本。结果是一个两点帕累托前沿：在强制刚性见证固定后，精确重构归结为观察者纤维内部的连通性，且每个最小特征样本的可变部分都是完全二分纤维图的生成森林。对于有界扇出的MCFG重构，纯转移碎片化在子节点间倍增。因此，一个显式的扇出为二的族X_{k,r}在两种可比较的观察者下具有精确的特征数据成本2r[1+r(k-1)]和2rk^r。一个积分格不变量产生了仿射跨度下界和幺模性检验。在二元索引子族中，幺模性对于秩二和三的最小基数样本是充分的，但对于秩四则不然。随后出现两种不同的障碍：一种是与顺序无关的层状支撑冲突，另一种是与顺序相关的出现调度冲突。对于不相交的子需求，精确调度阈值是加倍约简槽颜色词中最大单色游程数；在双色情形下，这是一个交替阈值。Horn非锁定和笛卡尔锁定证书使这些约束显式化。分层复用可以用父根暴露换取局部扇出：对于混合秩四形状的自然关键模块库，分离、嵌套和交叉顺序的精确宽度-锚定前沿分别为{(2,1)}、{(2,2),(3,1)}和{(4,1)}。因此，观察、碎片化、暴露、算术跨度、层状兼容性、有序调度和分层复用是真正不同的有限资源。
英文摘要
We study finite resources governing exact positive reconstruction in fixed-observation CFG and MCFG learning. For an explicit rigid CFG family we compute safe observation size, internal residual fragmentation, and characteristic-data cost exactly. The result is a two-point Pareto frontier: after compulsory rigidity witnesses are fixed, exact reconstruction reduces to connectivity inside observer fibers, and the variable part of every minimum characteristic sample is a spanning forest of complete bipartite fiber graphs. For bounded-fan-out MCFG reconstruction, pure transition fragmentation multiplies across children. An explicit fan-out-two family X_{k,r} therefore has exact characteristic-data costs 2r[1+r(k-1)] and 2rk^r under two comparable observers. An integral lattice invariant yields an affine-span lower bound and a unimodularity test. In the binary-index subfamily, unimodularity suffices for minimum-cardinality samples through ranks two and three but not rank four. Two distinct obstructions then appear: an order-independent laminar support conflict, and an order-sensitive occurrence-scheduling conflict. For disjoint child requirements, the exact scheduling threshold is the largest monochromatic run count in the doubled reduced slot-colour word; in the two-colour case this is an alternation threshold. Horn nonlocking and Cartesian locking certificates make these constraints explicit. Hierarchical reuse can trade parent-root exposure for local fan-out: for the natural critical-module library of the mixed rank-four shapes, the exact width--anchor frontiers are {(2,1)}, {(2,2),(3,1)}, and {(4,1)} for separated, nested, and crossing orders. Thus observation, fragmentation, exposure, arithmetic span, laminar compatibility, ordered scheduling, and hierarchical reuse are genuinely distinct finite resources.
Comments72 pages. Includes an ancillary Python script reproducing the exact rank-four census and ordered laminar-trace audit
