The Golden Ratio in Human Alphabets: A Deterministic Rule on Letter Names Generates Exact Algebraic Constants

A deterministic substitution rule applied to the written names of letters produces exact algebraic constants as the Perron eigenvalues of the resulting incidence matrices.

Across 36 scripts, nine scripts produce three exact quadratic Perron constants:

φ² (2.618), 1+√2 (2.414), and 2+√2 (3.414).

These constants are not evenly distributed across language families.

The quadratic x²−3x+1, giving φ²,

appears in two to three independent script lineages at three different core sizes, against a random baseline of 5.8% at the 3×3 core size. 

The probability of a shared quadratic Perron factor across three lineages is 0.009. The finding is structural, not numerical: the constant is determined by the shape of the name graph, not the language family.

The complete incidence matrices are available as open data at https://doi.org/10.5281/zenodo.23440579, allowing independent verification of every result. The proprietary transliteration operator is withheld under a staged disclosure agreement.