module module high

`IndisputableMonolith.StandardModel.Q3Representations`

The module defines the quaternion group Q₃ with its eight elements and associated spin-0 and spin-1 sector representations for the Standard Model in Recognition Science. Researchers deriving electroweak parameters or Higgs masses from the φ-ladder would cite these objects. It supplies group elements, sector counts, Casimir operators, and the scalar λ_RS through definitions only.

claimThe quaternion group $Q_3 = {±1, ±i, ±j, ±k}$, partitioned into Spin0Sector and Spin1Sector, with associated counts, order, Casimir operators, and the derived constant $λ_{RS}$.

background

Recognition Science derives physics from the J-cost functional equation and the forcing chain T0-T8 that yields D=3. This module imports the base time quantum τ₀ = 1 tick from Constants and sits in the StandardModel domain. It defines Q3Element as the eight quaternion elements, Spin0Sector and Spin1Sector for representations, spin0_count, spin1_count, q3_order, casimir, sector-specific Casimirs, and λ_RS.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

Supplies the Q₃ geometry used in WeinbergAngleScoreCard to obtain sin²θ_W = (3-φ)/6 and in HiggsRungAssignment to derive the Higgs mass from the φ-ladder using Q₃ geometry, completing the RS particle mass table.

scope and limits

Does not derive Q₃ from the Recognition Composition Law.
Does not assign specific particles to individual Q₃ elements.
Does not evaluate numerical masses or angles.

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