A Falsifiable Informational Hypothesis for the Origin of Rotational Symmetry
This paper introduces the Orientation-Free Relational Metric Hypothesis (ORMH), a falsifiable informational framework addressing the physical origin of continuous spatial rotational symmetry ($\mathrm{SO}(3)$). While modern physics relies on Noether’s theorem to link rotational invariance with angular momentum conservation, the fundamental reason why physical actions contain no absolute background direction remains an open question. Formulated under the scientific governance of the Unified Informational Physics Ontology (UIPO) V2.0, ORMH models rotational symmetry as an emergent property of an underlying relational state space rather than an axiomatic primitive.
At the fundamental level, physical states are specified by relational pairwise distance matrices $D_{ij}^2 = \Vert{}x_i – x_j\Vert{}^2$ and internal chiral pseudoscalars $Q$, rendering global rigid embedding rotations mathematically redundant. The framework extends this relational primitive by modeling emergent spatial geometry through the second moment of locally sampled relational directions, $Q_{ab}^{(N)} = \frac{1}{N} \sum_{k=1}^N u_k^a u_k^b$. For an orientation-neutral, mixing relational ensemble, coarse-graining over $N$ independent samples yields an exact statistical scaling law: root-mean-square (RMS) metric anisotropy falls as $N^{-1/2}$, or $V^{-1/2}$ ($\ell^{-3/2}$) across coarse-graining volumes in three spatial dimensions. Before temporal averaging, this yields a directional light-speed propagation anisotropy RMS of $\sigma_{\delta c/c} = \frac{1}{\sqrt{5N}}$, providing a quantitative finite-scale signature that distinguishes ORMH from exact-continuum Lorentz symmetry and regular-lattice discretizations.
ORMH explicitly separates the origin of rotational symmetry from local rotational motion and angular-momentum acquisition. Using relational configuration-space holonomy ($R_\gamma = \mathcal{P}\exp(-\oint_\gamma A)$), internal structural cycles produce net orientation changes without requiring an absolute external frame or violating global conservation laws. When recurrent relational updates coexist with spatial scale growth, state trajectories trace logarithmic spirals provided the ratio of fractional scale growth to angular progression ($\chi = \frac{H_I}{\Omega_I}$) remains stationary. Crucially, the hypothesis requires common-metric coupling: any physical metric perturbation tensor $B_{ab}$ must influence all colocated propagating fields universally once sector-specific response functions are accounted for.
The manuscript establishes a multi-stage, preregistrable experimental program to test the hypothesis against conventional comparators:
- Multi-volume rotating resonator tests evaluating $V^{-1/2}$ anisotropy RMS scaling.
- Cross-sector correlation analyses across colocated clock and cavity systems.
- Spatial decorrelation and temporal averaging models ($\ell_I, \tau_I$).
- Cosmological background constraints evaluating large-scale anisotropy and vorticity limits.
ORMH provides explicit scope-of-rejection rules: detection of a robust, non-dynamical vacuum preferred axis, incompatible cross-sector anisotropy tensors, or empirical scaling exponents excluding $-1/2$ reject the core stochastic extensions. The paper includes explicit non-claims, clarifying that ORMH does not replace established astrophysical torque mechanisms for macroscopic angular-momentum acquisition, nor does it claim to derive intrinsic quantum spin without formal recovery of $SU(2)$ representations.