zeroDefect_eq_J_log
plain-language theorem explainer
The zero-location defect attached to a complex number ρ equals the J_log function applied to its deviation from the critical line. Number theorists working inside the Recognition Science framework cite this equivalence when mapping zeta-zero positions onto the defect cost. The proof is a one-line wrapper that unfolds the definition of zeroDefect and applies the symmetry of J_log_eq_J_exp.
Claim. For any complex number ρ, the defect cost of ρ equals J_log of the deviation of ρ, where the deviation is 2(Re(ρ) − 1/2) and J_log(t) = cosh(t) − 1.
background
The Zero Location Cost module supplies the RS dictionary that translates zeta-zero location into defect cost: zeroDeviation ρ is defined as 2(Re ρ − 1/2) and zeroDefect ρ is defined as defect(exp(zeroDeviation ρ)). The critical line Re ρ = 1/2 is thereby identified with the zero-defect locus. Upstream, defect is the J functional on positive reals, while J_log(t) := cosh(t) − 1 is the same functional rewritten in logarithmic coordinates, with the relation J_log(t) = defect(exp t) already established.
proof idea
The proof is a one-line wrapper. It unfolds the definition of zeroDefect, then invokes the symmetry of J_log_eq_J_exp on the deviation of ρ.
why it matters
This identity is the bridge lemma that lets every subsequent property of zeroDefect (non-negativity, invariance under conjugation and reflection, the cosh closed form) reduce directly to the corresponding property of J_log. It is invoked by defectIterate_zero_eq_zeroDefect in the composition-law module and by the full suite of invariance and sign theorems inside ZeroLocationCost. In the framework it completes the dictionary step that connects zero locations to the J-uniqueness and eight-tick octave structure of the forcing chain.
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