THE FINE STRUCTURE CONSTANT ON DISCRETE TERMS

Abstract. The reciprocal of the fine-structure constant is dimensionless and measured near 137.036, and

attempts to derive it have a long record of failure. The failures share a form: a formula matched to the measured

decimal, proposed after the measurement, and adjusted when the measurement moved. This paper separates

the constant into three strata and makes a different statement about each. The integer 137 is the unique prime

whose reciprocal has base-ten period eight and seventeen cyclic families, by an exhaustive check over the five

prime divisors of 10⁸ − 1. Its repeating block 00729927 obeys Midy’s theorem, and the four-nines complement

extends across the seventeen families. In the first decimal block the numerator 36 is over-determined by three

constructions that share no steps; its placement at the third decimal place follows through the nucleon masses

and remains open from the arithmetic alone. The rest value 137.036 stands 6.0 parts per billion above the

CODATA 2022 value, and every modern determination lies below it. Digits beyond the block are left to

measurement: named values are dense there, and the best determinations disagree. Twenty-eight times the rest

block is 1.008 exactly, and twenty-eight times the settled block 0.035999 is 1.007972, within 1.3 parts per million

of the mean of the proton and neutron masses. A measured α⁻¹ at or above 137.036 refutes the reading of the

block; an arithmetic error in any exhibit, or a census showing the three registers to be common, removes the

result it concerns. Every exhibit re-derives in a public suite at github.com/thefirsthorstmann/g-theory-verify,

where the exact arithmetic of Sections 2 to 7 is also formalized in Lean 4.