Relative Symmetry Principle and Relative Conservation Laws: Groupoids and Functional Geometry Physics

We present a complete mathematical formalization of the relative symmetry principle within the framework of Functional Geometry. By replacing Lie groups with groupoids and conserved currents with sheaf-theoretic objects, we generalize Noether's theorem from ``global symmetry → absolute conservation'' to ``relative symmetry → groupoid relative conservation''. The theory is founded on a single axiom: the self-referential groupoid fixed point _ _^_, which rigidly determines all physical structures from the arithmetic of the Heegner point = (1+√-163)/2 (class number one). The Connation field ---the sole entity of the universe---unifies all geometric quantities through the layer-turbulence duality = ⊕, generating spacetime, quantum mechanics, and fundamental interactions without free parameters. We derive three meta-conservation laws that replace classical conservation laws: (i) logical distance conservation from Gödel oscillation, (ii) total spectral weight conservation from self-similar closure, and (iii) layer-turbulence total flow conservation from discrete-continuous duality. These laws are not numerical constants but geometric objects relative-covariant with the observational framework = (λ, , , ARF). Experimental predictions include arithmetic thresholds in quantum error correction (≈ 19.37%), layer-vortex braiding in topological quantum computing, and self-referential fixed-point detection in large language models. All constants are rigidly determined: α⁻¹ = 137.035999177, ΩDM = 0.2656890744, = 161ln 2/576 ≈ 0.1937.