Graded Projection Recursion (GPR): A Framework for Controlling Bit-Complexity of Algebraic Packing

Recursive algorithms that use algebraic packing may appear to achieve reduced computational complexity that does not reflect the dominating bit-complexity, i.e., the implemented performance would not exhibit the claimed scaling. This paper introduces Graded Projection Recursion (GPR), a formal framework designed to address this problem by defining a rigorous mechanism that provably controls bit growth if specific conditions are satisfied. We use GPR to develop a matrix multiplication algorithm that is model-honest and achieves a true near-quadratic bit-complexity. This framework also provides a general GPR Substitution Principle recipe for accelerating a significant class of related algorithms in a provably rigorous way.