Experimental Systems · Building Block

Trefoil Streams

Three (2,3) torus-knot data streams wound around one golden toroid at the cube-root-of-unity offsets. Their quaternion centroid is an emergent 4th phase that collapses to the torus core at equilibrium — and a Polycentria Dual-Helix governs it: an oracle proposes, an exact certifier decides.

Stream quadrance scaling sₖ

Stream α s₀ 1.00

Stream β s₁ 1.00

Stream γ s₂ 1.00

Phase drift on β,γ Δψ 0.00

Emergent phase Φ₄ — equilibrium meter

off-core residual ρ = minorR·|Σ sₖ e^iψₖ| 0.000

Ψ₄ = Σₖ₌₁³ (Sₖ + Sₖ⁻¹) → 1 + ω + ω² = 0

At balanced scalings and cube-root offsets the three minor-radius phasors cancel exactly — Φ₄ falls onto the torus core (the "zero point"). Imbalance re-grows ρ linearly: a δ bump on one stream gives ρ = minorR·δ.

Topological parity channel

ρ = ‖Z‖ flags that a stream drifted; the argument arg(Z) points at which — no training, exact. (Reference: 100% single-stream localization over 3000 trials.)

Polycentria Dual-Helix

Governance — oracle proposes, certifier decides

Drift the streams off balance and observe them through noise. Helix A (geometric oracle) and a blind-random baseline each spend the same budget of Helix B exact-residual evaluations; we measure the Oracle Bias ε = (random best ρ) − (oracle best ρ).

method	best ρ found
blind random	—
geometric oracle	—

Oracle Bias ε (±2σ over trials) —

Like the Router, ε > 0 because the objective is the geometry. The certifier's ρ is exact and oracle-independent — the chosen equilibrium is always correct, whatever the oracle proposed.

Reference

All numbers above are computed in-page and match the Python reference trefoil_streams.py (run it for the exact-arithmetic checks in ℚ[√3] and ℚ[√5], the linearity table, and the governance ε with its 2σ band). The 3D view adapts the triple-trefoil equilibrium algorithm: Frenet-frame quaternion rotation P' = q·P·q⁻¹, three twistor pairs, and a centroid core.