XiZeroSet
XiZeroSet extracts the zero locus of the xi function on a completed-ξ surface. Number theorists studying pairing invariants for zeros under reflection and conjugation in Recognition Science cite this set when building deviation and defect collections. The definition proceeds by direct set comprehension that selects points where the xi component vanishes.
claimFor a completed-ξ surface equipped with a function ξ : ℂ → ℂ that satisfies ξ(1 − s) = ξ(s) and ξ(conj s) = conj(ξ(s)), the associated zero set is { s ∈ ℂ | ξ(s) = 0 }.
background
The module records the minimal functional-equation symmetry surface needed for Vector C. It encodes only the completed zeta symmetries that supply pairing data on zeros, yielding reflection and conjugation invariants for zeroDeviation and zeroDefect without yet imposing a d'Alembert-style composition law. The CompletedXiSurface structure consists of a map xi : ℂ → ℂ together with the two axioms xi(1 − s) = xi(s) and xi(conj s) = conj(xi s). Upstream structures supply the J-cost convexity, spectral emergence of gauge content, and ledger factorization that frame the broader Recognition setting in which this surface appears.
proof idea
The definition is a direct set comprehension that selects those complex numbers s at which the xi component of the surface evaluates to zero.
why it matters in Recognition Science
This definition supplies the zero set used to construct deviation and defect sets in the same module, supporting the derivation of pairing invariants under the functional equation. It fills the role of extracting the kernel for symmetry analysis in the completed-ξ surface for Vector C. In the Recognition framework it underpins the minimal symmetry data required before adding stronger zero-location constraints from the phi-ladder or eight-tick dynamics.
scope and limits
- Does not impose any zero-location constraints beyond the two symmetry axioms.
- Does not compute explicit zero locations or densities.
- Does not connect the zero set to physical constants or the phi-ladder.
- Does not derive a composition law for the xi function.
formal statement (Lean)
34def XiZeroSet (Ξ : CompletedXiSurface) : Set ℂ :=
proof body
Definition body.
35 { s : ℂ | Ξ.xi s = 0 }
36
37/-- The set of zero deviations realized by zeros of a completed-ξ surface. -/