doubledZeroDefect
plain-language theorem explainer
doubledZeroDefect maps each complex ρ to the J-log cost of twice its deviation from the critical line. Researchers studying the Recognition Science derivation of the Riemann hypothesis cite this when building the Vector C data from functional equation symmetry. The definition is the direct composition of J_log with the doubled zero deviation.
Claim. Let ρ ∈ ℂ. The doubled zero defect is J_log(2 · δ(ρ)), where δ(ρ) := 2(Re(ρ) − 1/2) is the deviation from the critical line and J_log(t) := cosh(t) − 1.
background
The Zero Doubling Law module records the strongest concrete Vector C instantiation obtained from the functional-equation/J-symmetry bridge. The defect observable satisfies the doubling recurrence D(2t) = 2 D(t)^2 + 4 D(t). zeroDeviation(ρ) extracts the real-part deviation scaled by 2, as defined in ZeroLocationCost. J_log(t) is the defect in log coordinates given by J(e^t) = cosh(t) − 1, from DiscretenessForcing. This builds on the defect functional from LawOfExistence, which equals J for positive arguments.
proof idea
The definition is a one-line wrapper applying the J_log function from DiscretenessForcing to twice the zeroDeviation of ρ.
why it matters
This definition supplies the observable used in current_vectorC_attempt_data and the PureVectorCDoublingData structure. It realizes the doubling recurrence from the FE/RCL bridge, providing Phase 4 evidence for the interaction between functional equation symmetry and the RS defect. It advances the zero doubling law toward the full d'Alembert interface.
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