The Root-Gradient Principle for E10: Constructive Tensor Shadows and Finite-Rank Tests for Higher-Derivative M-Theory

This manuscript introduces a rigorous, finite-rank algebraic program to test the correspondence between the infinite-dimensional hyperbolic Kac-Moody algebra E10 and the local operator content of M-theory. It replaces the broad qualitative conjecture that E10 encodes M-theory with a sequence of falsifiable matrix-rank computations at each derivative order. The framework introduces the Kamboj Root-Gradient Filtration to assign derivative weight to positive roots, and a Constructive Tensor-Shadow map to convert root sectors into local curvature-flux monomials algorithmically. The mismatch between algebraic and physical quotients is isolated into an explicit obstruction complex. We prove a finite-rank reduction theorem, demonstrating that the correspondence holds at order d precisely when a canonically generated integer dictionary matrix has full quotient rank. The paper establishes structural theorems for Hamiltonian stability, supergravity admissibility, tensor-shadow exactness, and imaginary-root localization. Finally, it outlines a Five-Defect Completion Theorem, defining the exact numerical defects (operator, supersymmetry, dynamical, BFSS matrix limit, and graviton scattering) that must vanish to complete a non-perturbative formulation. An exact numerical protocol utilizing Peterson recurrence, Smith normal form, and modular rank verification is provided.