THE VANISHING POINT ON DISCRETE TERMS

Abstract. Within the real numbers 0.999… = 1 is a theorem: the decimal denotes the limit of its partial sums

0.9, 0.99, …, and that limit is 1. The proof rests on two assumptions no finite procedure carries out: a completed

infinite tail of nines, and the completeness of the real numbers, under which every bounded increasing

sequence converges to its least upper bound. This paper examines the equality operationally, under the

program’s hypothesis that physical reality is discrete and harmonic, with no completed infinity. The suite verifies

four exact results about the partial sums. Every partial sum, read as the integer of its digits, has digital root 9, and

no operation performed on it reaches root 1; reaching 1 requires supplying the missing unit, which is the

rounding itself. The ratio of 1 to the n-th partial sum is exactly 10ⁿ/(10ⁿ − 1), superparticular for every n, the first

being 10/9, the just minor whole tone. At every finite n the identity behind the one-line proof is 10·xₙ

= 9 + xₙ₋₁, and the equation 10x = 9 + x misses by exactly 9·10⁻ⁿ; it holds only of the limit. Deep partial sums agree with 1

to every finite precision, yet each differs from it by exactly 10⁻ⁿ. What physical process could supply the missing

unit remains open. The paper is refuted by a partial sum whose digital root is not 9, a non-superparticular ratio, a

partial sum satisfying 10x = 9 + x, or a finite test separating the limit from every deep partial sum. Every exact

claim re-derives in a public suite at github.com/thefirsthorstmann/g-theory-verify.