res_nu3
Definition of the third neutrino rung: integer base of -54 plus fixed quarter-rung phase offset of -1/4. Neutrino mass modelers in the Recognition Science framework cite this rung to obtain the atmospheric mass scale m3 approximately 0.056 eV. It is realized as a direct arithmetic combination of the base integer rung and the phase offset.
claim$r_3 := -54 - 1/4$, where the base integer is the deep-ladder rung for the atmospheric neutrino scale and the offset is the fixed quarter-octave phase shift.
background
The NeutrinoSector module formalizes the deep-ladder hypothesis for neutrino masses. Neutrinos occupy even integers in the -50 range, with the atmospheric scale at rung -54 yielding m3 approximately 0.056 eV and the solar scale at -58 yielding m2 approximately 0.0082 eV. The residue difference of 4 rungs suggests quarter-ladder spacing. Rung type supports fractional values; rung_nu3 supplies the integer base -54 while nu_phase_offset supplies the fixed -1/4 shift, equivalent to -2/8 of an octave.
proof idea
One-line definition that converts the integer rung_nu3 to Rung type and adds the nu_phase_offset value.
why it matters in Recognition Science
This rung enters NeutrinoMassScaleScoreCardCert and the row_nu3_frac bound, which checks the predicted mass fraction against 0.04985 to 0.04993. It also supports dm2_31_frac_pred and the squared-mass ratio theorem that recovers phi^7 from the 7/2 rung spacing. The construction fills the T14 neutrino sector step in the Recognition Science chain, where the deep ladder and phi-ladder mass formula produce the observed atmospheric and solar scales after display calibration.
scope and limits
- Does not derive absolute eV scales without the external MeV-to-eV calibration seam.
- Does not fix the neutrino mass hierarchy sign.
- Does not incorporate oscillation phase or mixing angles.
- Does not replace the full RS-native calibration pathway.
formal statement (Lean)
152def res_nu3 : Rung := ofInt rung_nu3 + nu_phase_offset
proof body
Definition body.
153
154/-- Upgraded solar rung: enforce \(r_3-r_2 = 7/2\) (→ φ⁷ in squared-mass ratio). -/