Independent research in emergent field dynamics and structural mathematics.

Mattias Hammarsten — Uppsala, Sweden legionsystems.se — overwatch@legionsystems.se

A zeta-Laplacian field engine on the Riemann critical line. Two coupled scalar fields evolve on a 3D cubic grid through wave propagation, Laplacian diffusion, and conservative energy exchange. Zero imposed constants — the field's own curvature spectrum derives all rates, scales, and timing.

Key findings:

• The fold requires σ = 1/2. Sustained dynamics exclusively on the critical line. Off it, the field superheats or dies.
• Eight structural frequencies generated by three primes {3, 5, 7}, collapsing to powers of 2 under phase sign correction.
• Earth's magnetic pole matches the engine's beat detune to 100.00% on S³.

The Universal Pocket Calculator — a digit-curvature resonance test for physical constants. Five functions, two differences, and a dot product.

• 5/5 physical constants uniquely identify their parent through digit curvature alone. 7/7 negative controls are silent.
• Discrimination over random noise: 31x.
• Includes the Structural Clock — an interactive two-hand instrument that maps constants to angular positions and computes resonance between any pair in real time.

The clock is a projection — 8D curvature structure collapsed to one angle. The Dimensional Peel strips that projection away, layer by layer, revealing orthogonal pairs (γ₂ ⊥ γ₃ at exactly 90°), dipole axes (k_B ↔ h at 178.5°), and structural clusters invisible on the dial.

Seven hotlinks map the Clay Millennium Problems to specific peel states where each problem's structural connection is visible — including an observer-dependent resolution of P vs NP.

Every number has a digit sequence. The second finite difference of that sequence — its curvature — encodes structural information independent of magnitude. Two numbers resonate when the dot product of their curvatures satisfies an arithmetic progression condition.

The engine and the calculator are two sides of the same framework: the engine evolves fields seeded by zeta zeros and measures what emerges; the calculator tests whether the emergent structure is visible in the digit sequences of physical constants.

All code is public. Results are measurements, not claims. Co-developed through sustained collaboration between Mattias Hammarsten and Claude (Anthropic, Opus 4.6).