Trace Invariance Under Conjugation

Abstract

The golden trace is invariant under Galois conjugation.

Theorem 1.1 (Trace invariance).

$$\mathrm{trace}\left(\mathrm{conj}\left(x\right)\right)=\mathrm{trace}\left(x\right)$$

Proof. Machine-checked in Lean as D5/S0/Carrier/TraceConjugation.trace_conj (✓ std3). ∎

Source. Repository-derived.

Commentary.

Conjugating a golden integer preserves its integral trace. In coordinates, conjugation sends (a,b) to (a+b,-b), and both traces simplify to 2a+b.

References

• Truth anchor: D5/S0/Carrier/TraceConjugation.trace_conj
• Dependency: D5/S0/Carrier/Conj