IndisputableMonolith.Physics.LeptonGenerations.FractionalStepDerivation
This module derives fractional steps for lepton generations by partitioning the 4π steradian solid angle into active and passive edge counts on the cubic lattice. Physicists constructing the phi-ladder mass formula for the electron-muon-tau sequence would cite these equalities when fixing the generation gap term. The module is a chain of definitions and direct equality lemmas that reduce solid-angle fractions to an alpha seed and a derived step value.
claimThe module establishes the total solid angle $4π$ steradians in $D=3$, defines the fractional solid angle, passive and active edge counts on the lattice, the alpha seed, and the derived generation step for leptons.
background
Recognition Science fixes $D=3$ spatial dimensions and obtains the total solid angle of $4π$ steradians directly from the geometry of the cubic ledger $Q_3$. The module imports the RS time quantum $τ_0=1$ tick from Constants and the alpha derivation that extracts $4π$ via Gauss-Bonnet vertex deficits on $Q_3$ (AlphaDerivation doc-comment). Sibling declarations introduce totalSolidAngle as the sphere surface measure, fractionalSolidAngle as its normalized portion, passiveEdgeCount and activeEdgeCount as the lattice edge partition, alphaSeed as the initial fine-structure seed, and generationStepDerived as the computed step separating lepton families.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the geometric fractions that close the lepton-generation derivation inside the Recognition framework, feeding the mass formula yardstick * phi^(rung - 8 + gap(Z)) and the Berry creation threshold phi^-1. It extends the upstream alpha derivation (4π from Gauss-Bonnet on Q_3) by converting solid-angle fractions into the generation step that distinguishes the three charged leptons. No downstream theorems are listed, indicating this block is an intermediate scaffold for the full lepton mass ladder.
scope and limits
- Does not derive numerical lepton masses.
- Does not treat quark generations or mixing angles.
- Does not incorporate electromagnetic or weak interactions.
- Does not address higher-dimensional extensions beyond D=3.
depends on (2)
declarations in this module (15)
-
def
totalSolidAngle -
theorem
totalSolidAngle_is_sphere_surface -
def
fractionalSolidAngle -
theorem
fractionalSolidAngle_eq -
def
passiveEdgeCount -
theorem
passiveEdgeCount_eq -
def
activeEdgeCount -
theorem
activeEdgeCount_eq -
def
alphaSeed -
theorem
alphaSeed_eq -
def
generationStepDerived -
theorem
generationStepDerived_eq -
theorem
fractional_step_is_forced -
theorem
alpha_step_relationship -
theorem
same_ingredients