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Recovering ℏ/2from Hidden Phase Geometry: Information–Energy Balance in C6 Bora Akta¸s October 2025 Abstract The standard quantum uncertainty bound, ∆x∆p≥ℏ/2, can be reinterpreted geometrically within the dual-norm framework of the C6phase manifold. Here, the apparent indeterminacy originates from the information exchange between visible and hidden phase sectors rather than stochastic fluctuations. By analyzing the norm balance between π-phase (visible) and ζ(3)-phase (hidden) domains, the uncertainty relation can be reconstructed as an equality of information–energy transfer, where ℏ/2 emerges from the geometry of projection rather than an imposed quantization rule. 1 1. Dual-Norm Framework and Energy Balance In the C6metric, ∥z∥2=∥zvis∥2− ∥zhid∥2, the difference between visible and hidden components defines a geometric potential. The corresponding energy ratio is Ehid Evis =ζ(3) π. Hence, the hidden phase stores a fraction ζ(3)/π of the visible energy, representing information not accessible through direct measurement. 2 2. Derivation of the Uncertainty Balance Associating phase and momentum fluctuations with their respective energy fractions, we obtain ∆θvis ∆pvis =ℏ 2 Evis +Ehid Evis =ℏ 21 + ζ(3) π. The additional term (ζ(3)/π) expresses the analytic curvature correction due to the hidden sector. However, the observable equality condition follows from restoring the conserved information flow: π ζ(3)∆θvis ∆pvis −ℏ 2= 0. This identity implies that the Heisenberg bound is not violated but recovered internally through the compensation between visible and hidden energy channels. 1 3 3. Interpretation: Information–Energy Duality The uncertainty limit thus represents a stationary exchange point where visible and hidden phase energies are balanced: Evis :Ehid =π:ζ(3). The constant ℏ/2 emerges as the invariant scale of this exchange. Belief in intrinsic randomness is replaced by a geometric mechanism: uncertainty reflects the self-consistent curvature of phase information flow. The hidden ζ(3) domain provides the analytic reservoir through which ℏ/2 remains constant across all measurable transformations. 4 4. Summary Within the C6phase geometry, the Heisenberg uncertainty limit is derived from norm conservation rather than postulated. The visible–hidden decomposition converts stochastic indeterminacy into an analytic energy exchange, where ∆θvis∆pvis =ℏ 21 + ζ(3) π holds locally, while global conservation enforces ℏ/2 as an equilibrium constant. Thus, uncertainty is not a limitation of knowledge but the geometric signature of balanced information between the πand ζ(3) phase sectors. 2