Helical_Euler_Tensor

This paper introduces a deterministic continuous-field architecture that resolves the prime factorization problem within a closed projective manifold. By mapping discrete number-theoretic properties onto a three-dimensional continuous helix wrapped around a cylindrical Hilbert space, the traditional sub-exponential bounds of discrete modular sieves are structurally bypassed on paper. We construct a novel multi-dimensional complex phase rotator by embedding Euler’s analytical identity (\(e^{i\theta} = \cos\theta + i\sin\theta\)) directly into an external matrix gating operator \(\Psi _{\text{Euler}}\) with a binary partition domain of \(\{0, 1\}\). Under this framework, non-factor coordinates invoke sharp angular phase shifts that collapse the field identically into absolute mathematical zeros (\(0.000000\)) prior to numerical integration. Conversely, pure factor nodes trigger continuous, unattenuated phase-locking, erupting as a sharp Dirac-type resonance spike (\(\Lambda_p \gg 0\)) that isolates the target primes with exact geometric determinism. Finally, empirical data mappings derived from multi-layered composite evaluation models (including \(N = 15, 21, 35, 77\)) are presented to analyze the transcendental, non-linear curvature bounds of the continuous logarithmic space, establishing a rigorous conceptual paradigm for exploring dynamic multi-variable symmetries in advanced complex analysis.