Quantum Kinematics from Renormalization Symmetry: A Timeless Reconstruction of Hilbert and Statistical Structure

Version 4: Major upgrade completing the Quantum Mechanics kinematics reconstruction. Convex structure and statistical states are now derived rather than assumed, and an entropy functional is introduced from RG sector decomposition. The role of the axioms has been clarified within a unified timeless framework, and technical derivations have been expanded in appendices. We present a reconstruction of Quantum Mechanics kinematics from renormalization symmetry in a timeless, scale-invariant framework, extending previous derivations to include statistical structure. Starting from a minimal state space with no assumed linear, convex, or probabilistic structure, and imposing resolution independence, refinement composability, and observable consistency, we derive the core mathematical framework of quantum theory. Renormalization group (RG) consistency induces idempotent coarse-graining maps, contractivity, and a natural decomposition into independent sectors. From composability of indistinguishable refinements, a linear state space emerges without postulating superposition. Projective invariance and normalization then give rise to convex structure, interpreted as coarse-grained equivalence classes of refinements, thereby deriving statistical mixtures without assuming probability axioms. An RG-invariant scalar functional, constrained by homogeneity, additivity over independent sectors, and refinement stability, is shown to be quadratic. This induces a norm satisfying the parallelogram law, yielding an inner product and Hilbert space completion. Continuous distinguishability of coherent combinations introduces a phase degree of freedom, selecting complex scalars within the stated assumptions. The convex and sectoral structure further leads to the emergence of statistical states and an entropy functional associated with coarse-grained decomposition, providing the minimal ingredients required for quantum statistical description. Normalization of the quadratic functional uniquely yields the Born rule. Thus, within the stated assumption class, linearity, complex structure, Hilbert geometry, convex state space, and probabilistic interpretation arise as consequences of renormalization symmetry and refinement consistency in a timeless setting. The resulting framework provides a unified foundation for quantum theory that naturally interfaces with renormalization-based approaches to quantum gravity.