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 (torresᵣobalinoᵣadar9ᵥ3. py) runs all experiments from Sections 14–16 and reproduces every table in the paper. The dashboard (torresdashboardᵥ3. 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.