Quantum Kinematics from Renormalization Symmetry: A Convexity-Free Reconstruction in a Timeless Framework (v3)

This version significantly strengthens the derivation presented in v2. In the previous version 2, the state space was assumed to have a linear (vector space) structure. In the present version, both linearity and convexity are removed from the foundational assumptions and are instead derived from refinement and renormalization consistency. This yields a fully non-circular reconstruction of quantum kinematics from RG symmetry and minimal structural principles. We present a conditional reconstruction of quantum kinematics from renormalization symmetry in a timeless, scale-invariant setting. Starting from a bare state set, observables as real-valued functions, and refinement composability, we add explicit consistency axioms (notably RG path consistency) and regularity hypotheses, then derive structural consequences. Under the RG path-consistency axiom, coarse-graining maps are idempotent and contractive and induce sector decompositions. Observable consistency of refinement composition yields a universal associative composition law on observable coordinates; with continuity, this law is additive up to strictly monotone reparameterization. In the corresponding additive gauge, and after additive completion plus scalar-extension regularity, a linear state envelope emerges without postulating superposition a priori. We then introduce an RG-invariant scalar functional. Homogeneity, additivity on independent sectors, and refinement-multiplicity stability force quadratic scaling (up to normalization) within the stated assumptions. Continuous distinguishability of coherent combinations introduces phase structure; under the explicit scalar-algebra class assumption (finite-dimensional associative real division algebra) and commutative phase composition, the scalar field is selectedas C (excluding R and H in that class). The quadratic functional induces a norm satisfying the parallelogram law, hence an inner product and Hilbert-space completion (unique up to isomorphism in the reconstruction class). Normalization then yields the Born assignment as the unique additive scale-invariant weighting on orthogonal decompositions. Thus, the framework gives a non-circular derivation within the stated assumption class: linearity, complex structure, Hilbert geometry, and Born probabilities arise conditionally from renormalization consistency plus declared regularity and phase/composition hypotheses.