leptons_same_Z
plain-language theorem explainer
All three leptons receive the identical charge index Z derived from the minimal even polynomial on integerized charge. Researchers assigning masses via the phi-ladder in Recognition Science cite this when treating electron, muon, and tau with the same Z=1332. The proof is a direct reflexivity step on the shared ZOf definition for lepton flavors.
Claim. $Z_e = Z_μ = Z_τ$, where $Z$ is the lepton charge index obtained from the minimal even polynomial $Z(Q̃) = a Q̃² + b Q̃⁴$ with $a=b=1$ applied to the integerized charge $Q̃_e = -6$.
background
The Z-map derivation module computes the lepton charge index from charge integerization k=F(3)=6 (cube faces supplying independent 2D symmetry channels), yielding Q̃_e = 6·(-1) = -6. An even polynomial ansatz Z(Q̃) = a Q̃² + b Q̃⁴ is imposed for charge-conjugation invariance and non-negativity, with coefficient minimality selecting (a,b)=(1,1) to give Z_ℓ=1332. This is presented as a geometric structural input rather than a direct consequence of the T0-T8 forcing chain.
proof idea
The proof is a one-line term wrapper that applies reflexivity to the shared definition of ZOf across the three lepton constructors, confirming equality by construction without further computation.
why it matters
This theorem ensures uniform Z treatment for all leptons in the mass formula (yardstick · phi^(rung-8+gap(Z))), supporting consistent placement on the phi-ladder. It completes the Z-map derivation step that produces a single Z_ℓ value from polynomial minimality and feeds the alpha band and anchor definitions. No open scaffolding remains here; downstream mass assignments still require explicit rung and gap choices.
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