Algebraic Structure of Primes in (Z/9Z)× — Geometric Invariance, Hierarchical Factorization, and the Bidirectional Helix

Description (English ) This paper investigates the multiplicative group (Z/9Z)× as a geometric framework for analyzing prime sequences. Rather than treating primes as independent statistical objects, we analyze the trajectory of primes as a dynamical system on the helix — a perspective that reveals structure invisible to classical equidistribution results. Exact results (no error bars): Theorem 7 (Coset Discriminator): Given N = p·q, N mod 9 determines deterministically whether p and q belong to the same or opposite cosets of (Z/9Z)×. Verified: 0 errors over 499,999 prime pairs. Proof follows directly from χ₂(ab) = χ₂(a)·χ₂(b). Proposition 8 (Optimal Level): The level k* where √^k*(N) ≈ p satisfies k* = ⌈log₂(log₂(N)/log₂(p))⌉ exactly. The search level is computed, not discovered by iteration. Empirically verified results (N = 500,000 primes, reproducible): η² = 12.2440% — canonical value, stable across protocols λ = −0.140879 — spectral gap of the transition matrix, within 95% CI Chebyshev bias: NR/QR gap ratio = 1.060, stable across scales Sequential memory: 3.7% → 6.7% for k = 1..5, monotonically increasing Proposition 9 (Multi-Scale Statistical Self-Similarity): The coset system under iterated square roots exhibits fast mixing (|λ₂| = 0.035, percentile 0.1% vs random baseline) with maximum-entropy stationary distribution H(π) = log₂(3) = 1.585 bits, until ergodicity collapses at level k* — exactly where factorization begins. Verified across 4 semiprime distributions and 3 discretization operators. RADAR-9 (Sections 14–16): A hierarchical factorization algorithm combining iterated square roots with the mod 9 algebraic filter. For unbalanced semiprimes N = p·q with p ≪ q, the algorithm locates p in O(log(log(N))) levels with ~33% candidate reduction per level via Theorem 7. Demonstrated on cases up to 134 bits in under 0.2ms. Epistemological note: This paper analyzes the sequence of primes as a trajectory — not individual primes nor arbitrary subsets. A critique from the equidistribution perspective (Dirichlet) is correct but answers a different question. The geometric pattern emerges on the trajectory of √^k(N) → p, not on primes as static objects. All numerical results are fully reproducible. The included Python code (torres_robalino_radar9_v3.py) runs all experiments from Sections 14–16 and reproduces every table in the paper. The dashboard (torres_dashboard_v3.html) provides interactive visualization of the helix geometry. Descripción (Español) Este paper analiza el grupo multiplicativo (Z/9Z)× como marco geométrico para la secuencia de primos. En lugar de tratar los primos como objetos estadísticos independientes, analizamos la trayectoria de la secuencia como sistema dinámico sobre la hélice — una perspectiva que revela estructura invisible a los resultados clásicos de equidistribución. Resultados exactos (sin barras de error): Teorema 7 (Discriminador de Cosets): Dado N = p·q, N mod 9 determina deterministamente si p y q pertenecen al mismo o distinto coset de (Z/9Z)×. Verificado: 0 errores sobre 499,999 pares de primos. Demostración directa desde χ₂(ab) = χ₂(a)·χ₂(b). Proposición 8 (Nivel Óptimo): El nivel k* donde √^k*(N) ≈ p satisface k* = ⌈log₂(log₂(N)/log₂(p))⌉ exactamente. El nivel de búsqueda se calcula, no se descubre por iteración. Resultados empíricos verificados (N = 500,000 primos, reproducibles): η² = 12.2440% — valor canónico, estable entre protocolos λ = −0.140879 — gap espectral de la matriz de transición, dentro del IC 95% Sesgo de Chebyshev: ratio NR/QR = 1.060, estable entre escalas Memoria secuencial: 3.7% → 6.7% para k = 1..5, monótonamente creciente Proposición 9 (Auto-similitud estadística multiescala): El sistema de cosets bajo raíces iteradas exhibe mezcla rápida (|λ₂| = 0.035, percentil 0.1% vs baseline aleatorio) con distribución estacionaria de entropía máxima H(π) = log₂(3) = 1.585 bits, hasta que la ergodicidad colapsa en k* — exactamente donde comienza la factorización. RADAR-9 (Secciones 14–16): Algoritmo de factorización jerárquica que combina raíces iteradas con el filtro algebraico mod 9. Para semiprimos desbalanceados N = p·q con p ≪ q, localiza p en O(log(log(N))) niveles con reducción de ~33% de candidatos por nivel. Demostrado hasta 134 bits en menos de 0.2ms. Todo es reproducible. El código incluido reproduce cada tabla del paper en 30 segundos.