IndisputableMonolith.NumberTheory.ZeroDoublingLaw
The ZeroDoublingLaw module defines the doubled zero defect as the J-cost of the squared deviation factor generated by composing functional equation reflection with the recognition composition law. Researchers analyzing symmetry constraints on zeta zeros inside the Recognition Science framework cite it when testing whether reflection data alone can force the critical line. The module supplies the core definition plus direct algebraic identities for its properties.
claimLet zeroDeviation(ρ) = 2(Re ρ − 1/2). The doubled zero defect is J(exp(2 · zeroDeviation(ρ))), where J(x) = (x + x^{-1})/2 − 1.
background
The module belongs to the NumberTheory section and extends the dictionary between zero location and J-cost. DiscretenessForcing establishes that J(x) = ½(x + x^{-1}) − 1 has a unique minimum at x = 1, equivalently cosh(t) − 1 in logarithmic coordinates. XiJBridge connects the completed xi functional equation ξ(s) = ξ(1−s) to J(x) = J(1/x) via the map x = exp(2(Re(s) − 1/2)), sending the critical line to the J-minimum. ZeroLocationCost supplies zeroDeviation(ρ) = 2(Re ρ − 1/2) and zeroDefect(ρ) = defect(exp(zeroDeviation(ρ))).
proof idea
This is a definition module, no proofs. It introduces the doubledZeroDefect definition and derives its equivalence to a cosh expression, recurrence, non-negativity, and vanishing on the critical line by direct substitution into the J-functional and its algebraic identities.
why it matters in Recognition Science
The module supplies the doubled zero defect imported by VectorCSymmetryOnlyNoGo, which exhibits a toy completed-ξ surface whose zeros lie off the critical line despite reflection and conjugation symmetries. It realizes the virtual next-step observable suggested by FE reflection plus RCL self-composition. The construction engages J-uniqueness (T5) and the recognition composition law from the forcing chain.
scope and limits
- Does not prove that zeta zeros lie on the critical line.
- Does not evaluate the defect numerically for known zeros.
- Does not handle compositions beyond doubling.
- Does not connect to the phi-ladder or mass formula.