Sturmian-Dirichlet Value

Abstract

The exact Sturmian-Dirichlet value is a fixed affine combination of the golden ratio and the twisted cotangent constant.

Theorem 1.1 (The Sturmian-Dirichlet value has its exact golden-ratio form).

$$T_0=\frac{27-13\sqrt{5}}{24}=\phi -\frac{7}{4}+C_{\phi }.$$

Proof. Machine-checked in Lean as D5/S3/Constants/SturmianDirichletValue.sturmian_dirichlet_value_eq (✓ std3). ∎

Source. Repository-derived.

Commentary.

Define T0 as (27 - 13 sqrt(5)) / 24 and C_phi as (57 - 25 sqrt(5)) / 24. Mathlib supplies the golden ratio phi = (1 + sqrt(5)) / 2. Substitution and normalization over the reals prove T0 = phi - 7/4 + C_phi exactly.

The decimal printed in the source table is an explanatory approximation, not a second exact claim. A checked negative control changes 57 to 58 and proves that the resulting equality fails.

References

• Truth anchor: D5/S3/Constants/SturmianDirichletValue.sturmian_dirichlet_value_eq