The Barbelo-Tesla Prime Sieve is a tabular visualization of prime number distribution based on iterative digit-sum reduction (digital root). All natural numbers are classified into nine digit-sum classes (1–9). Classes 3, 6, and 9 are structurally excluded, as all their members are divisible by 3 — with the single exception of the prime 3 itself. This leaves six active classes (digital roots 1, 2, 4, 5, 7, 8) in which all primes greater than 3 must appear. When primes are entered into the table, a recurring structural pattern emerges: 6 candidate positions · 1 certain gap · 2 candidate positions · 1 certain gap — repeating indefinitely. Additional empty positions arise from multiples of primes (classical sieving). The visualization is implemented as an interactive HTML5 table with adjustable range and composite overlay. Part of the Barbelo prime visualization series, complementing the Barbelo Prime Tree I (DOI: 10.5281/zenodo.20475174) and Barbelo Prime Tree II (DOI: 10.5281/zenodo.20497826).
Barbelo et al. (Sun,) studied this question.
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