A Coincidence Has No Single Price - The ladder underneath a numerical coincidence, and what it costs to climb

ABSTRACT

Somebody finds a formula: a few measured constants, raised to some powers, multiplying out to a number that matches an observation to a fraction of a percent. This note gives six checks that decide what such a claim is worth using arithmetic rather than instinct, and shows that all six are readings of a single object — the formula, written in logarithms, is a walk on a lattice generated by the constants it uses. The qualifying formulas are never alone. They lie on an evenly spaced ladder whose step is a continued-fraction convergent of one ratio computable from the constants before any data is examined, and whose length is set by how well the least-known input is measured. For the worked example the family is infinite today and becomes finite the moment the baryon-to-photon ratio is known to better than 1.08 percent, so the number of rival formulas is a fact about instruments rather than about nature. The ladder's single gap sits exactly where the formula stops depending on that input, which is the only region where such a claim is testable at all. The headline result is that a coincidence has no price until the searcher says what was fixed before looking: the same relation on the same data prices at one in twenty-six or one in three depending on the answer, and that gap is usually larger than the effect under discussion. The worked example is the author's own April 2026 dark-energy relation, used as a specimen rather than defended; the instrument applied to it was built afterwards and for unrelated work, which is what makes the exercise a test rather than a fit. No new theorems are claimed. Baker's theorem, continued fractions, the arithmetic metric and error propagation are all classical and cited. No credentialed prior-art pass has been run, so novelty language is restricted to "not located."

Keywords-numerical coincidence; base rate; reference class; continued fractions; convergents; rotation number; linear forms in logarithms; Baker's theorem;
exponent lattice; error propagation; temperament; dark energy; cosmological constant