rung_gap_is_seven_halves
plain-language theorem explainer
The effective rung difference between the third and second neutrino generations on the Recognition Science phi-ladder equals exactly seven halves. Neutrino mass modelers cite this to obtain the structural squared-mass ratio as exactly the golden ratio to the seventh without numeric bounds. The proof is a term-mode simplification that unfolds the second neutrino residue definition and reduces directly via the neutrino spacing equation.
Claim. Let $r_3$ and $r_2$ denote the effective rung positions of the third and second neutrino mass eigenstates on the phi-ladder. Then $r_3 - r_2 = 7/2$.
background
In the Recognition Science neutrino sector, neutrinos sit on the deep ladder with effective rungs near -54 for the atmospheric scale and -58 for the solar scale. The module treats these as fractional phi-powers after calibration against the electron structural mass (reused as MeV then converted by 10^6 to eV). The rung gap of 7/2 upgrades the integer residue difference of 4 into a precise fractional spacing that yields a pure power law for squared masses.
proof idea
The proof is a term-mode one-line wrapper. It unfolds the definition of the second neutrino residue then applies simplification against the neutrino spacing equation, which directly produces the difference 7/2.
why it matters
This theorem supplies the exact gap required by the downstream squared_mass_ratio_structural_phi7 result, which converts the difference into the structural squared-mass ratio phi^7. It completes the T14 neutrino sector step in the Recognition Science chain, where the phi-ladder mass formula and eight-tick octave produce the predicted splittings. The exact equality removes the need for interval bounds in the pure-law derivation.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.