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

We present Graded Projection Recursion (GPR), a framework for converting certain blocked recursive algorithms into model-honest (i.e., reflects full bit complexity) near-quadratic procedures under bounded intermediate budgets. Part I gives a proof-complete {\em integral specification} of a three-band packing identity and a two-round middle-band extractor, and shows how per-node centering and sqrt-free dyadic l2 normalization (recursive invariant amplification) gives a sufficient packing base that grows linearly with the scaling depth (i.e., logarithmic bit-growth) in exact arithmetic. On fixed-width hardware, exact evaluation of the packed recursion generally requires either an extended-precision path or digit-band (slice) staging} that emulates b(n) bits of mantissa precision using w-bit words, incurring only polylogarithmic overhead; This leads to a soft-quadratic bit cost O(n^2) when b(n)=Θ(\log n) in the basic 2x2 recursion). Part II introduces execution-format comparators (e.g., IEEE-754), a drift ledger, and a decision-invariance theorem that supports commensurate-accuracy claims in floating arithmetic (and that cleanly accounts for any staged/truncated auxiliary drift). Part III provides case-study reductions (LUP/solve/det/inv, LDL^T, blocked QR, SOI/SPD functions, GSEVP, dense LP/SDP IPM kernels, Gaussian process regression, and representative semiring problems) showing how to export the kernel advantage without reintroducing uncontrolled intermediate growth. Part IV abstracts admissible packings and extractors via a master condition and an easily checkable BWBM sufficient condition, and sketches extensions to multilinear/multigraded kernels and non-rounding extractors (e.g., CRT and semiring bucket projections).