zeroDefectSet_reflection_invariant
plain-language theorem explainer
The functional equation of the completed xi surface preserves the set of realized zero defects. Number theorists working on zero-location constraints in the Recognition framework cite this invariance when assembling pairing data for Vector C. The proof is a direct term-mode construction that unpacks the existential witness for a defect, maps it to its reflected zero, and recovers the same defect value via the pairing and invariance lemmas.
Claim. Let $X$ be a completed $xi$-surface. If $d$ is a real number realized as the zero defect of some zero of $X$, then $d$ is again realized as the zero defect of some zero of $X$.
background
The CompletedXiSurface is the structure carrying a function $xi:mathbb{C}to mathbb{C}$ together with the reflection axiom $xi(1-s)=xi(s)$ and the conjugation axiom $xi(bar s)=bar{xi(s)}$. The zeroDefectSet of such a surface is the set of all reals $d$ for which there exists $s$ with $xi(s)=0$ and zeroDefect$(s)=d$. This module supplies only the minimal symmetry surface required for Vector C, recording pairing data from the completed functional equation without any stronger zero-location hypotheses.
proof idea
The term proof first rcases the membership hypothesis to obtain a witness zero $s$ with $xi(s)=0$ and zeroDefect$(s)=d$. It then constructs the reflected point via functionalReflection, applies zero_pairing_under_reflection to confirm the image is also a zero of $X$, and invokes zeroDefect_invariant_under_functional_reflection to show the defect value is unchanged. The resulting triple witnesses membership in the same set.
why it matters
The result records the reflection invariance of the realized zero-defect set, a basic symmetry step inside the CompletedXiSymmetry module that supports construction of pairing invariants for Vector C. It relies on the upstream zero_pairing_under_reflection and zeroDefect_invariant_under_functional_reflection lemmas and closes the minimal functional-equation symmetry surface described in the module documentation. No downstream uses appear yet.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.