pmns_axes_symmetric
plain-language theorem explainer
The declaration asserts that the two primary axes in the F2Power 3 structure share identical flip counts of 2, implying equal Berry phases for the neutrino sector under the [4,2,2] Gray code. Neutrino physicists deriving PMNS mixing angles from Recognition Science would cite it to ground symmetry arguments for leading-order CP phase structure. The proof is a direct reflexivity step on the natural-number equality.
Claim. The flip counts of axis 1 and axis 2 in the three-dimensional binary vector space are both equal to 2: $flipCount(axis_1) = flipCount(axis_2) = 2$.
background
The PMNSMatrix module derives the neutrino mixing matrix from Recognition Science by quantizing angles via the golden ratio, targeting large mixing with maximal atmospheric angle near 45 degrees. Axis 1 and axis 2 are the first two weight-1 vectors in F2Power 3, defined as the standard basis elements (true, false, false) and (false, true, false). The local setting emphasizes that neutrino flavor mixing arises from φ-connections, with this axis symmetry enforcing equal Berry phases via the Gray code structure.
proof idea
The proof is a one-line wrapper that applies reflexivity to the equality of the natural number 2 with itself, confirming the shared flip count value for the two axes.
why it matters
This result supports the structural leading-order vanishing of δ_CP in the PMNS matrix, with the observed non-zero value arising only from torsion sub-corrections. It feeds the PMNSParameters and bestFitPMNS constructions in the same module and aligns with the Recognition Science mechanism for φ-quantized neutrino angles. The symmetry closes a step in the eight-tick octave framework by ensuring consistent Berry phases across axes.
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