module module high

`IndisputableMonolith.RRF.Physics.LeptonGenerations.Necessity`

Module RRF.Physics.LeptonGenerations.Necessity establishes the exact value E passive equals 11 and supplies interval bounds on the electron-muon and muon-tau transition steps along the phi ladder. A physicist deriving charged lepton masses from cube geometry would cite these results to fix the discrete generation parameters required by the framework. The arguments reduce via direct substitution from the imported electron mass necessity and alpha derivation together with interval arithmetic on powers.

claim$E_ {rm passive}=11$ exactly, together with interval bounds on the muon and tau residues obtained from the phi-ladder mass formula and cubic vertex deficits.

background

The module operates inside the Recognition Science derivation of lepton sector constants from the cubic ledger geometry. Lepton masses sit on the phi ladder via the yardstick times phi to the power of rung minus eight plus gap of Z, with the passive energy level fixed at 11. It imports the time quantum tau zero equal to one tick, the identity phi squared equals phi plus one, and the structural derivation of four pi from Gauss-Bonnet on the cube Q three.

proof idea

Exact statements such as E passive exact and cube faces exact reduce by direct substitution from the imported cube geometry and phi support. Bound lemmas such as step e mu bounds and predicted residue mu bounds apply the interval power function to the relevant phi exponents, confirming that the computed residues lie inside the stated intervals. The module is a sequence of one-line algebraic reductions and interval inclusion checks.

why it matters in Recognition Science

This module supplies the necessity arguments that fix the lepton generation parameters and feeds the higher mass hierarchy derivations in the Recognition Science monolith. It extends the T9 electron mass chain to the full charged lepton spectrum with the passive level at 11 and phi-based steps, consistent with the eight-tick octave and three spatial dimensions. The results close the discrete quantization of the observed lepton masses on the phi ladder.

scope and limits

Does not compute absolute lepton masses in SI units.
Does not prove that exactly three generations must exist.
Does not address the neutrino sector or mixing angles.
Does not derive weak couplings beyond the alpha band.

depends on (8)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Constants.Alpha module_import
IndisputableMonolith.Constants.AlphaDerivation module_import
IndisputableMonolith.Numerics.Interval.Pow module_import
IndisputableMonolith.PhiSupport module_import
IndisputableMonolith.Physics.ElectronMass.Defs module_import
IndisputableMonolith.Physics.ElectronMass.Necessity module_import
IndisputableMonolith.Physics.LeptonGenerations.Defs module_import

declarations in this module (29)

lemma E_passive_exact L51
lemma cube_faces_exact L56
lemma W_exact L61
lemma pi_gt_d6_local L66
lemma pi_lt_d6_local L69
lemma inv_4pi_lower L72
lemma inv_4pi_upper L84
lemma inv_4pi_bounds L95
lemma step_e_mu_bounds L100
lemma step_mu_tau_bounds L109
lemma predicted_residue_mu_bounds L122
lemma predicted_residue_tau_bounds L132
theorem phi_pow_neg963_lower L151
theorem phi_pow_neg962_upper L157
theorem phi_pow_residue_mu_lower L162
theorem phi_pow_residue_mu_upper L175
lemma phi_pow_residue_mu_bounds L186
theorem phi_pow_neg377_lower L196
theorem phi_pow_neg375_upper L202
theorem phi_pow_residue_tau_lower L207
theorem phi_pow_residue_tau_upper L217
lemma phi_pow_residue_tau_bounds L228
theorem predicted_mass_mu_lower L239
theorem predicted_mass_mu_upper L246
theorem muon_mass_pred_bounds_proven L257
theorem predicted_mass_tau_lower L268
theorem predicted_mass_tau_upper L275
theorem tau_mass_pred_bounds_proven L286
theorem lepton_ladder_forced_from_T9 L307