A Geometric Recursion Formula for Prime Numbers - Each Prime Is the Smallest Non-Trivial Orbit Minimum, with a Geometric Shortcut

This is revision v12 of the recursive paper. It extends the previous version (v8 on Zenodo) by a complete engineering section that tunes the orbit wheel sieve through four standard layers and benchmarks each layer head-to-head against Walisch's primesieve 12. 7 on the same hardware. The mathematical core is unchanged. Let pₖ denote the k-th prime, Mₖ the k-th primorial p₁ p₂. . . pₖ, and Bₖ the set of orbit minima of the unit group modulo Mₖ under the natural mirror action a -> Mₖ - a. Then for every k >= 3 the next prime is exactly the smallest non-trivial entry of Bₖ: p₊+₁ = min (Bₖ \ 1). The identity is established constructively from the structure of the mirror group action on (Z / Mₖ Z) *. No external sieve or primality test is invoked. The formula is verified by direct computation for every prime from p₄ = 7 through p₁₁ = 31. The paper also identifies a purely geometric shortcut. The address space of Bₖ embeds canonically as a higher-dimensional Cartesian cuboid T₁, T₂ x 1,. . . , 3 x 1,. . . , 5 x. . . x 1,. . . , (pₖ - 1) /2. The two-element factor T₁, T₂ corresponds to the Stella octangula decomposition of the eight coprime classes modulo 30 into squares and non-squares, i. e. to the unique index-2 subgroup of (Z / 30 Z) * = Z₄ x Z₂. The transition from stage k to stage k + 1 is governed by three explicit geometric operations: step, filter, mirror. New in v12: a tuned C implementation of the orbit wheel sieve at stage k = 8 and a four-layer hybrid bench. The orbit residues that drive the recursion are also the wheel of a wheel sieve. A naive C implementation at k = 8 delivers an asymptotically stable 3. 8x speedup over a textbook sieve of Eratosthenes while producing the complete ordered list of every prime up to N. Section 15 of the paper adds four engineering layers on top of the naive version: - Hybrid 1: segmented sieve with a 256 KiB byte-array segment. - Hybrid 2: bit-packed segment of 32 KiB with a tiled bit-mask of length M₈ + segment-size, allowing byte-aligned memcpy or a single shifted copy as the pre-sieve. - Hybrid 3: bucket sieving for ray primes above 2¹8, with lazy next-pointer maintenance per segment. - Hybrid 4: thread-parallel segments via OpenMP, lock-free, with per-segment next-pointer rebuild at block boundaries. Each layer is benchmarked head-to-head against primesieve 12. 7 on the same sandbox CPU (Intel Xeon, 2 vCPUs). For each value of N in 10⁹, 10¹0, 2 x 10¹0, 5 x 10¹0, 10¹1 the paper reports pi (N), the rightmost prime in the output list, the wall-clock time of every hybrid stage, and the wall-clock time of primesieve. The single-thread gap to primesieve narrows from a factor of about 40x (basic) to 15x (H1) to 11x (H2) to a stable 9x (H3). With two threads (H4) the gap narrows further to about 6x. The geometry of the orbit wheel never changes through these layers; what changes is only how the bits are laid out in memory and how the ray primes are scheduled. Twin primes for free. The orbit wheel produces the complete ordered prime list. A single additional bitmap pass over the output, scoring positions i where both i and i+2 are flagged prime, gives the count of twin primes in 2, N at essentially zero extra cost. Counts verify exactly against primesieve --count=2: pi₂ (10⁹) = 3, 424, 506, pi₂ (2 x 10⁹) = 6, 388, 041. The same trick extends to any locally checkable prime constellation (cousin primes, sexy primes, prime triplets). Updated hardware projection. With Hybrid 3 as the single-thread baseline (10. 80 s for N = 10¹0 on one core), the linear extrapolation gives a compute-limited per-day reach on El Capitan of N ~ 10²0 instead of 10¹9, while the memory bound is unchanged. The orbit wheel produces complete ordered prime lists for N in the 10¹7 to 10²0 range within one day of large-cluster wall time, depending on which bound binds first. Honest framing. The orbit wheel sieve is not the fastest published prime sieve. primesieve, Atkin-Bernstein, and other tuned implementations remain faster in absolute wall time. The value of the construction is structural: the wheel comes from the geometry of the orbit-minimum recursion, it produces the complete ordered list and not only a count, and it hybridises cleanly with every standard engineering layer. Reference implementations supplied with v12: - bahnₛieveᵥ7. c Full Hybrid 4 implementation in C with OpenMP. Builds with gcc -O3 -march=native -fopenmp. Reproduces every row of the H4 bench table. - PrimeGeometryRecursiveₒrbitₘethod. py Brute-force enumeration, unchanged from v8. - PrimeGeometryRecursiveₜickₘethod. py Step, filter, mirror with explicit self-test, unchanged from v8. Keywordsprime numbers, primorial, coprime classes, orbit minima, mirror symmetry, 16-cell, Stella octangula, recursion formula, deterministic prime generation, wheel sieve, segmented sieve, bucket sieving, OpenMP, primesieve benchmark LicenseCreative Commons Attribution 4. 0 International (CC BY 4. 0) Related identifiers Cites: https: //zenodo. org/records/20600998 Cites: https: //zenodo. org/records/20607940 Cites: https: //zenodo. org/records/20626566