AI 中文总结
该研究将 BCJ 分子构造化为自适应逆问题，通过固定规范编译相关内容，经运动学和图对称性约化后应用于单圈五胶子纯杨 - 米尔斯理论，确定特解与核，验证结果，还给出搜索规则等可对提议进行评估。
AI 中文摘要
逆问题由矩阵方程及其未知空间、物理数据、等价关系和验证测试组成。我们将 Bern--Carrasco--Johansson（BCJ）分子构造公式化为一个精确的自适应逆问题。固定的科学规范确定理论、图约定、系数域、局域性和幂次计数、切割数据、可观测量等价性以及独立检验。每个有限工作规范编译为\(\mathcal{P}_\sigma=(A,b;\mathcal{S};\mathcal{T})\)，其中\(Ax = b\)重构分子系数，\(\mathcal{S}\)对解纤维进行分类，\(\mathcal{T}\)在留出信息上对其进行测试。左零障碍确定一致性所需的候选分子基方向，而候选测量在右核上的作用确定有信息的新切割方程。我们在四个点手动说明这些步骤，并将其应用于单圈五胶子纯杨 - 米尔斯理论。经过运动学和图对称性约化后，候选分子基包含 1127 个独立坐标。组合的最大、盒、三重和双切割的秩为 920，给出 207 维仿射解纤维。精确重构确定一个特解和完整的有序核。指定的\(R_{12345}\)色环读出\(\mathcal{S}\)消除每个核方向，因此整个纤维代表一个可观测量类。一个已发表的前向极限分子位于此纤维中，新的切割、独立的积分约化和螺旋度振幅基准验证了结果。显式搜索规则和代理接口可以提出修订。确定性编译和精确评估评估每个提议。
英文摘要
An inverse problem comprises a matrix equation together with its unknown space, physical data, equivalence relation, and validation tests. We formulate Bern--Carrasco--Johansson (BCJ) numerator construction as an exact adaptive inverse problem. A fixed scientific specification determines the theory, graph conventions, coefficient field, locality and power counting, cut data, observable equivalence, and independent checks. Each finite working specification compiles to $\mathcal{P}_σ=(A,b;\mathcal{S};\mathcal{T})$, where $Ax=b$ reconstructs numerator coefficients, $\mathcal{S}$ classifies the solution fiber, and $\mathcal{T}$ tests it on held-out information. Left-null obstructions identify candidate numerator-basis directions needed for consistency, while the action of candidate measurements on the right kernel identifies informative new cut equations. We illustrate these steps by hand at four points and apply them to one-loop five-gluon pure Yang--Mills theory. After kinematic and graph-symmetry reduction, the candidate numerator basis contains 1127 independent coordinates. The combined maximal, box, triple, and double cuts have rank 920, giving a 207-dimensional affine solution fiber. Exact reconstruction determines a particular solution and the complete ordered kernel. The specified $R_{12345}$ color-ring readout $\mathcal{S}$ annihilates every kernel direction, so the full fiber represents one observable class. A published forward-limit numerator lies in this fiber, and fresh cuts, independent integral reductions, and helicity-amplitude benchmarks validate the result. Explicit search rules and agent interfaces can propose revisions. Deterministic compilation and exact evaluation assess each proposal.
Comments71 pages, 11 figures. Companion software and compact
