doubledZeroDefect_eq_cosh_sub_one
plain-language theorem explainer
The doubled zero defect for complex ρ equals cosh of twice the zero deviation minus one. Researchers on the zero-doubling recurrence in Recognition Science cite this closed form to derive nonnegativity and vanishing conditions. The proof is a one-line simp that unfolds the doubled-defect definition and applies the J_log identity.
Claim. For ρ ∈ ℂ, let D₂(ρ) be the doubled zero defect and δ(ρ) the zero deviation. Then D₂(ρ) = cosh(2 δ(ρ)) − 1.
background
The Zero Doubling Law module records the concrete Phase 4 instantiation of the functional-equation/J-symmetry bridge, showing that the defect observable obeys the doubling recurrence D(2t) = 2 D(t)² + 4 D(t). The J-cost is introduced via J_log(t) = cosh(t) − 1, a convex bowl centered at t = 0. The defect functional equals J for positive arguments.
proof idea
One-line simp tactic that unfolds doubledZeroDefect and applies the J_log definition from DiscretenessForcing.
why it matters
This identity is the direct parent of doubledZeroDefect_nonneg, doubledZeroDefect_recurrence, and doubledZeroDefect_zero_iff_on_critical_line. It supplies the concrete self-composition law on doubled deviation that instantiates the FE/RCL bridge and links to T5 J-uniqueness (J(x) = cosh(log x) − 1) in the forcing chain.
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