Q_tilde_down
Q_tilde_down assigns the natural number 2 as the magnitude of the down-quark charge in the sector-dependent generation torsion construction. Researchers deriving forced sector orderings from partition and asymmetry constraints cite this constant to separate up and down spans. The definition is a direct numerical assignment justified by scaling the lepton step of 6 by the fractional charge 1/3.
claimThe magnitude of the down-sector charge is defined by $Q̃_↓ := 2$.
background
The SDGT Forcing module shows that sector-dependent generation torsion is forced by three constraints: the partition of N₃ = 55 into overlapping consecutive-pair sums, lepton uniqueness requiring the middle pair to be 11 and 6, and charge asymmetry. The four step values {13, 11, 6, 8} are assigned to up quarks {13, 11}, leptons {11, 6}, and down quarks {6, 8}. Q_tilde_down records the absolute charge for the down sector as |6 × 1/3|.
proof idea
Direct definition that sets the constant to 2, with an inline comment tracing the value to the lepton step size scaled by the down-quark charge fraction.
why it matters in Recognition Science
This supplies the down charge magnitude required by the downstream theorems charges_distinct and larger_charge_larger_span to establish |Q̃_up| ≠ |Q̃_down| and the larger charge receiving the larger span. It completes the charge asymmetry constraint inside the SDGT Forcing Theorem, which selects the unique ordering (13, 11, 6, 8). In the Recognition framework it anchors the mass formula on the phi-ladder by fixing the sector torsion steps, while the single-principle derivation of the step counts themselves remains open.
scope and limits
- Does not derive the step values {13, 11, 6, 8} from the J-function or forcing chain.
- Does not compute particle masses beyond the charge magnitudes.
- Does not invoke the Recognition Composition Law or alpha band constraints.
Lean usage
theorem larger_charge_larger_span : Q_tilde_up > Q_tilde_down ∧ (13 + 11 : ℕ) > (6 + 8) := by unfold Q_tilde_up Q_tilde_down; constructor <;> omegaformal statement (Lean)
122def Q_tilde_down : ℕ := 2 -- |6 × 1/3| = 2
proof body
Definition body.
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