module module high

`IndisputableMonolith.StandardModel.Q3Representations`

The module defines the quaternion group Q3 with its eight elements and associated spin sectors for Recognition Science particle representations. Electroweak parameter and Higgs mass calculations cite these objects for the Q3 geometry. It consists entirely of type, count, and constant definitions with no theorems or proofs.

claimThe quaternion group \(Q_3 = \{\pm 1, \pm i, \pm j, \pm k\}\) with spin-0 and spin-1 sectors, casimir operators, and the RS coupling \(\lambda_{RS}\).

background

This module resides in the StandardModel domain and imports Constants, where the fundamental RS time quantum satisfies (\tau_0 = 1) tick. The supplied doc-comment states that the module concerns the eight elements of the quaternion group Q3. Sibling definitions introduce Q3Element, Spin0Sector, Spin1Sector, spin0_count, spin1_count, q3_order, casimir, spin0_casimir, spin1_casimir, casimir_ratio_undefined, lambda_RS, and lambda_RS_val.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

These definitions supply the Q3 geometry that feeds the Weinberg angle scorecard (predicting sin²θ_W = (3-φ)/6) and the Higgs rung assignment (deriving m_H in (120,130) GeV from the φ-ladder). The module therefore completes the RS particle mass table using Q3 representations.

scope and limits

Does not derive Q3 from the forcing chain T0-T8.
Does not compute numerical values for masses or angles.
Does not include the full Standard Model Lagrangian.
Does not address fermion generations or CKM mixing.

used by (2)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.Physics.WeinbergAngleScoreCard module_import
IndisputableMonolith.StandardModel.HiggsRungAssignment module_import

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import

declarations in this module (24)

inductive Q3Element L56
def Spin0Sector L68
def Spin1Sector L71
theorem spin0_count L77
theorem spin1_count L80
theorem q3_order L83
def casimir L88
theorem spin0_casimir L91
theorem spin1_casimir L94
theorem casimir_ratio_undefined L98
def lambda_RS L104
theorem lambda_RS_val L106
def higgsMassSq_over_vev L110
theorem higgsMassSq_simplifies L112
def wMassSq_over_vev L118
def higgsMassRatio L121
def sin2ThetaW_RS L134
theorem sin2ThetaW_RS_val L136
theorem sin2ThetaW_RS_pos L139
theorem sin2ThetaW_RS_lt_half L146
theorem sin2ThetaW_RS_approx L152
def w_rung L163
def higgs_rung_prediction L184
theorem higgs_rung_prediction_pos L191