Earlier versions proposed Graded Projection Recursion (GPR) as a deterministic packed-recursion framework for model-honest near-quadratic dense matrix multiplication. This revised version withdraws the exact dense matrix multiplication theorem and the downstream consequences that depended on it with a conservative AMM framework. The local ingredients remain useful as local tools: the three-band packing identity, scaled middle-band extraction under certified gaps, centering and reconstruction identities, and row/column normalization bounds. The gap in the earlier argument is global: the proof relied on a bounded active-state realization that would remove first-mismatch terms through the recursion. For arbitrary dense inputs this would require an exact equality filter over the inner index. We formulate this obstruction as a target-slice/equality-filter problem and give a rank/capacity argument against the natural separable active-state realization. The positive replacement is a conservative approximate matrix multiplication framework. For chosen protected left and right query subspaces, the low/marginal part of AB is computed exactly and an unbiased AMM primitive is applied only to the high/high residual. The resulting estimator is unbiased, preserves protected queries exactly in every realization, localizes stochastic error to the residual subspace, and inherits residual output-norm or query-risk guarantees from the underlying estimator.

翻译：早期版本提出了分级投影递归(GPR)，作为一种面向模型诚实的近二次稠密矩阵乘法的确定性打包递归框架。本修订版撤回精确稠密矩阵乘法定理及其依赖的下游结论，代之以保守AMM框架。局部组件仍可作为局部工具使用：三带打包恒等式、认证间隙下的缩放中带提取、中心化与重建恒等式，以及行/列归一化界限。早期论证中的缺口是全局性的：该证明依赖于一个有界激活状态实现，通过递归消除首失配项。对于任意稠密输入，这需要在内部索引上使用精确等式滤波器。我们将此障碍表述为目标切片/等式滤波器问题，并针对自然可分离激活状态实现给出秩/容量论证。积极替代方案是保守近似矩阵乘法框架。对于选定的受保护左、右查询子空间，AB的低/边缘部分被精确计算，无偏AMM基元仅应用于高/高残差。所得估计量无偏，每次实现中精确保留受保护查询，将随机误差局限于残差子空间，并继承底层估计量的残差输出范数或查询风险保证。