Axis Trace Recurrence

Abstract

The axis weight is multiplicatively Fibonacci, so consecutive weights compose.

The axis weight reads a depth at both Galois embeddings at once. Its exponent is a linear combination of a golden power and its conjugate, and both powers satisfy the same two-step recurrence, so the exponent is additively Fibonacci and the weight itself is multiplicatively so.

The conjugate step is proved here from its defining quadratic rather than assumed by symmetry: the golden ratio has an upstream power lemma, the conjugate does not, and the two embeddings are not interchangeable in general even though this particular identity holds for both.

Theorem 1.1 (The axis weight is multiplicatively Fibonacci).

$$\forall K\in \mathbb{N},t_{K+2}=t_{K+1}\cdot t_K.$$

Proof. Machine-checked in Lean as D5/S3/Axis/AxisTraceRecurrence.axis_weight_is_multiplicatively_fibonacci (✓ std3). ∎

Source. Repository-derived.

Commentary.

The displayed conjunct is the composition law; the package also carries positivity at every depth and the base value.

References

• Truth anchor: D5/S3/Axis/AxisTraceRecurrence.axis_weight_is_multiplicatively_fibonacci
• Dependency: D5/S3/Axis/LambdaMinusDirichletSeries