Deterministic black-box polynomial identity testing (PIT) for read-once oblivious algebraic branching programs (ROABPs) is a central open problem in algebraic complexity, particularly in the absence of variable ordering. Prior deterministic algorithms either rely on order information or incur significant overhead through combinatorial isolation techniques. In this paper, we introduce an algebraic rigidity framework for ROABPs based on the internal structure of their associated matrix word algebras. We show that nonzero width-$w$ ROABPs induce word algebras whose effective algebraic degrees of freedom collapse to dimension at most $w^2$, independent of the number of variables. This rigidity enables deterministic witness construction via intrinsic algebraic invariants, bypassing rank concentration, isolation lemmas, and probabilistic tools used in previous work.Thus, we obtain the first order-oblivious deterministic black-box PIT algorithm for ROABPs, running in quasi-polynomial time $n\cdot(wd)^{O(w^2)}$. This establishes that algebraic rigidity alone suffices to derandomize PIT in this model, without assuming ordering information. The framework further isolates a single remaining obstacle to full polynomial-time complexity. We formulate a Modular Stability Conjecture, asserting that width-$w$ ROABPs are stable under hashing into cyclic quotient rings $\mathbb{K}[λ]/< λ^r-1 >$ once the modulus exceeds a polynomial threshold in $w$ and the individual degree. This conjecture arises naturally from the low-dimensional coefficient structure revealed by rigidity and is supported by extensive empirical evidence. Assuming the conjecture, our methods yield a fully polynomial-time deterministic black-box PIT algorithm for ROABPs, matching the complexity of the best-known white-box algorithms and reducing the black-box problem to a concrete algebraic stability question.

翻译：确定性黑盒多项式恒等式测试（PIT）对于只读无记忆代数分支程序（ROABPs）是代数复杂性理论中的一个核心开放问题，尤其是在变量顺序未知的情况下。现有的确定性算法要么依赖顺序信息，要么通过组合隔离技术引入显著开销。本文基于ROABP相关矩阵词代数的内部结构，提出了一个代数刚性框架。我们证明非零宽度-$w$的ROABP所诱导的词代数，其有效代数自由度维度坍缩至最多$w^2$，与变量数量无关。这种刚性使得我们能够通过内在代数不变量构造确定性见证，绕过了先前工作中使用的秩集中性、隔离引理以及概率工具。因此，我们首次得到了针对ROABP的顺序无关确定性黑盒PIT算法，其运行时间为拟多项式$n\cdot(wd)^{O(w^2)}$。这表明仅凭代数刚性就足以在该模型中实现PIT的去随机化，而无需假设顺序信息。该框架进一步分离出实现完全多项式时间复杂度的唯一剩余障碍。我们提出了一个模稳定性猜想，断言宽度-$w$的ROABP在哈希到循环商环$\mathbb{K}[λ]/< λ^r-1 >$后是稳定的，只要模数超过$w$和个体次数的多项式阈值。该猜想自然源于刚性所揭示的低维系数结构，并得到了广泛实证证据的支持。假设该猜想成立，我们的方法将产生一个完全多项式时间的确定性黑盒PIT算法用于ROABP，其复杂度与最知名的白盒算法相匹配，并将黑盒问题简化为一个具体的代数稳定性问题。