C_xi
Recognition Science sets the morphology coupling coefficient C_ξ to twice the golden ratio raised to the negative fourth power. Galaxy dynamics modelers cite this constant when fitting rotation curves under the RS gravity ansatz. The declaration is a direct real-arithmetic expression with no lemmas or hypotheses.
claim$C_ξ := 2 φ^{-4}$ where φ is the golden ratio.
background
The GravityParameters module derives seven galactic gravity parameters from Recognition Science. Three are marked DERIVED from φ, while C_ξ, p, and A carry an RS basis with explicit formulas. The module states that C_ξ follows the expression 2φ^{-4} and notes a 2% match to the fitted value 0.298 ± 0.015. Upstream imports supply the definition of φ together with positivity and structural predicates used elsewhere in the file.
proof idea
The declaration is a direct definition. It evaluates the real expression 2 multiplied by phi raised to the power of negative four.
why it matters in Recognition Science
C_ξ supplies the morphology coupling coefficient required by the GravityParameters table. It is referenced by the downstream positivity theorem C_xi_pos. The numerical value 2/φ^4 ≈ 0.292 aligns with the RS prediction derived from φ as the self-similar fixed point (T6) and supports the eight-tick octave structure in the forcing chain.
scope and limits
- Does not derive the exponent -4 from deeper axioms.
- Does not prove positivity of the coefficient.
- Does not link the parameter to specific observational datasets beyond the comment match.
- Does not address units or dimensional consistency.
Lean usage
theorem C_xi_pos : 0 < C_xi := by unfold C_xi; exact mul_pos (by norm_num) (Real.rpow_pos_of_pos phi_pos _)formal statement (Lean)
96def C_xi : ℝ := 2 * phi ^ (-(4 : ℝ))
proof body
Definition body.
97
98/-- C_ξ is positive.
99 Paper fitted value: 0.298 ± 0.015
100 RS prediction: 2/φ⁴ ≈ 0.292
101 Match: 2% -/