An Elementary Composite-to-Prime Ratio Sequence as an Independent Numerical Probe of Riemann Zero Statistics

Number-theoretic constructions that reach the statistics of theRiemann zeta zeros are almost always built from analyticmachinery. This note studies a construction with none of that: aninteger sequence y(x), defined only as the ratio of the cumulative sumof composite numbers to the cumulative sum of primes below x, frozenat primes and updated at composites. Examined at different scales,this minimal object exposes a hierarchy of number-theoretic depth,each layer reached by nothing more than elementary algebra and achange of normalization — from a one-line proof that every prime is alocal maximum, through a single closed-form Green's-function responsethat unifies every weighted variant of the construction tested (andadmits an exact twin theorem, with primes as local minima instead ofmaxima, for one such variant), to a direct, zero-free demonstrationthat Selberg/GUE-type suppression is visible in prime gap statisticsalone. An Italian translation of the full text accompanies this paperas an additional file in the same deposit. All numerical claims are tested throughout against surrogate andpermutation-based null models rather than taken at face value, andnegative results are reported alongside positive ones. No individualfact recovered in this note is new; the point is that all of them comefrom the same few lines of arithmetic, with depth accessed purely bywhere one chooses to look.