IndisputableMonolith.Physics.MixingDerivation
MixingDerivation derives the leading CKM element |V_us| as the golden-ratio projection phi^{-3} minus a 3/2 alpha radiative term fixed by cubic-face counting, together with parallel expressions for |V_cb|, |V_ub| and the three PMNS sin^2 theta predictions. A physicist fitting quark or neutrino mixing data would cite the module for its parameter-free outputs that land inside PDG bands. The derivations are assembled from geometric overlap rules imported from CKMGeometry and MixingGeometry rather than from free parameters or numerical fits.
claim|V_{us}| = phi^{-3} - (3/2) alpha, with |V_{cb}| = 1/24 and |V_{ub}| = alpha/2 obtained from rung differences on the phi-ladder; sin^2 theta_{12}, sin^2 theta_{23}, sin^2 theta_{13} likewise fixed by torsion weights and eight-tick closure.
background
The module operates inside the Recognition Science setting in which mixing matrices are forced by cubic voxel topology and the phi-ladder. Core objects are the golden projection phi^{-3} (torsion overlap of the three-generation constraint on the cubic ledger) and the cabibbo_radiative_correction (1.5 alpha arising from the six faces of the cube). It imports the RS time quantum tau_0 = 1 tick from Constants and the hypothesis from CKMGeometry that CKM elements are not arbitrary parameters but outputs of ledger geometry. PMNSCorrections supplies the integer coefficients (6, 10, 3/2) that appear in the PMNS angle formulae, including the reactor prediction sin^2 theta_{13} = phi^{-8}.
proof idea
The module is a derivation module whose structure consists of a sequence of named definitions: vus_derived applies the golden projection minus the radiative term; cabibbo_correction_geometric isolates the 1.5 alpha factor; vcb_derived and vub_derived repeat the pattern for the heavier elements; pmns_weight, pmns_prob and the three sin2_theta*_pred definitions encode the neutrino-sector weights from eight-tick octave closure. No tactic scripts are shown; each line is a direct algebraic transcription of the geometric rules supplied by the imported modules.
why it matters in Recognition Science
The derivations supply the explicit expressions consumed by the CKM module for the Jarlskog invariant and by CKMElementScoreCard for the Phase-2 scorecard that compares V_us_pred = phi^{-3} - (3/2)alpha against the PDG value 0.22500. They also feed ParticleSummary and PMNSScoreCard for the full set of mixing predictions. The module therefore closes the T11 step of the CKM Matrix Geometry hypothesis and supplies the concrete phi-ladder and torsion-overlap content required by the eight-tick octave and D=3 spatial constraints of the Recognition Science chain.
scope and limits
- Does not derive CP phases or Jarlskog invariant; those appear only in the downstream CKM module.
- Does not treat four or more generations.
- Does not perform numerical fits or error propagation; outputs remain exact symbolic expressions.
- Does not address running of alpha or higher-order radiative corrections beyond the 1.5 alpha term.
used by (4)
depends on (4)
declarations in this module (25)
-
theorem
vus_derived -
theorem
cabibbo_correction_geometric -
theorem
vcb_derived -
theorem
vub_derived -
theorem
vcb_geometric_origin -
def
pmns_weight -
theorem
pmns_weight_eq_phi_pow -
def
pmns_prob -
def
sin2_theta12_pred -
def
sin2_theta23_pred -
def
sin2_theta13_pred -
theorem
pmns_theta23_match -
theorem
atmospheric_correction_geometric -
theorem
pmns_theta13_match -
theorem
pmns_theta12_match -
theorem
solar_correction_geometric -
structure
MixingCert -
theorem
mixing_verified -
theorem
pmns_theta12_born_forced -
theorem
pmns_theta23_born_forced -
theorem
pmns_theta13_born_forced -
def
ckm_cp_phase -
def
jarlskog_pred -
theorem
jarlskog_match -
theorem
jarlskog_pos