The Living Circuit — Harmonic Geometry Framework

John Davis Burlingame

Module 18 · v1.1 · Fluid Dynamics · NumPy Only

Harmonic
Geometric Fluid

A 2D fluid dynamics module that evolves through harmonic geometry rather than linear time. The coherence field tracks geometric order across the grid. Turbulence is failed harmony. Stability is restored phase alignment.

from harmonic_geometric_fluid import HarmonicGeometricFluid NumPy only · MIT licensed

The displacement approach

Standard Navier-Stokes evolves a fluid field through time — each step advances the system by dt. This module replaces that linear time axis with harmonic displacement. The fluid doesn't move through time. It moves through geometry.

The displacement operator D_φ combines three forces, each anchored to a universal geometric constant:

①

Convection

φ-scaled movement

The primary advection term, scaled by the golden ratio. φ sets the natural rate at which the field carries itself — not arbitrary, not tuned, but the ratio that appears wherever growth is stable.

conv = φ · (u₀·∂u₀/∂x₀ + u₁·∂u₀/∂x₁)

②

Rotation

θg golden-angle refraction

The rotational term, governed by the golden angle (2.39996 rad ≈ 137.508°). This is the angle that produces maximally uniform distribution in nature — sunflower seeds, phyllotaxis, optimal packing. Applied here as the rate at which the field refracts and turns.

rot = θg · (u₁·∂u₀/∂x₀ − u₀·∂u₁/∂x₁)

③

Damping

α fine-structure impedance

Strain damping scaled by the fine structure constant α ≈ 1/137. The same constant that governs electromagnetic coupling governs the drag coefficient here — the natural impedance of the field against its own deformation.

impedance = α · √(∂u₀/∂x₀² + ∂u₁/∂x₁² + ½(∂u₀/∂x₁ + ∂u₁/∂x₀)²)

Turbulence is failed harmony.
Stability is restored phase alignment.

The coherence field

Every step, the module tracks a coherence field C(x) across the grid — a measure of geometric order at each point. It evolves through four competing forces:

Diffusion

Coherence spreads

High-coherence regions pull neighboring cells toward alignment. Coherence is contagious — ordered regions stabilize those around them.

Strain growth

Energy input raises coherence

High-strain regions receive coherence — they're being worked, and that work lifts their geometric order. Rate: 0.8 × strain × (1 − C).

Saturation

Prevents lock at 1.0

Coherence can't reach perfect 1.0 — the saturation term resists it. A fully locked field is brittle. Rate: −1.2 × C × (1 − C)².

Harmonic reward

Phase-aligned regions are pulled up

Regions where the velocity gradient aligns with the golden angle θg receive a coherence boost. The field rewards its own geometry. Rate: 0.6 × cos(θg · ∂u₀/∂x₀) × (1 − C).

The floor

Coherence is hard-floored at PHI − 1 ≈ 0.618 — the golden conjugate. A field stabilized by φ cannot fall below 1/φ. This is not a clamp for numerical safety. It is the geometric minimum of the system's own basis.

Coherence field · φ = 1.618 · live simulation

Constants

Five universal geometric constants. Same constants used across all Living Circuit modules — the same relationships that appear in electromagnetism, biological growth, and signal propagation appear here in fluid geometry.

Symbol	Value	Role in HGF
φ	1.61803399	Primary convection scaling — the natural rate of self-similar movement
θg	2.39996 rad	Rotational refraction rate — golden angle, maximally uniform distribution
α	0.00729735	Strain impedance — fine structure constant as drag coefficient
π	3.14159265	Wave periodicity, angular normalization
e	2.71828183	Natural growth and decay in coherence evolution
φ−1	0.61803399	Coherence floor — golden conjugate, geometric minimum

API

Initialization

HarmonicGeometricFluid(grid_size=64, viscosity=0.001)

Initialize a grid_size × grid_size fluid field. Velocity initialized to zero. Coherence initialized at 0.85. C_floor set to PHI−1.

Seeding

seed_vortex(cx=None, cy=None, strength=0.1)

Inject a harmonic vortex into the velocity field. Center defaults to grid center. Strength controls peak velocity. Good starting condition for testing coherence recovery.

seed_uniform(ux=0.05, uy=0.0)

Seed a uniform laminar flow. Use to test displacement operator behavior in a clean field.

Stepping

step(dt_placeholder=0.1) → (u, C, total_energy)

One displacement step. dt_placeholder is a scaling parameter, not literal time — smaller values give more stable evolution. Recommended range: 0.01–0.1.

Returns: velocity field (2,N,N) · coherence field (N,N) · total energy scalar

Monitoring

is_stable(threshold=0.72) → bool

True if mean coherence exceeds threshold AND coherence std < 0.12. Both conditions required — high mean with high variance means the field is fighting itself.

get_summary() → dict

Returns step count, mean/min/max/std coherence, total energy, kinetic energy, stability flag, and C_floor. Use for logging and monitoring over long runs.

reset()

Reset velocity and coherence to initial state. Step count resets to zero.

Simulation output — v1.1

=== HarmonicGeometricFluid v1.1 ===

Zero velocity baseline:
  step              : 1
  mean_coherence    : 0.91705
  min_coherence     : 0.91705
  max_coherence     : 0.91705
  coherence_std     : 0.0
  total_energy      : 0.013761
  kinetic_energy    : 0.0
  is_stable         : True
  C_floor           : 0.618034

Seeded vortex — 20 steps:
  Coherence range   : 0.9170 – 1.0000
  C_floor held      : True
  Final stable      : True
  NaN detected      : False

v1.1 changes from v1.0 seed_vortex replaced nested Python loop with NumPy meshgrid — same output, ~100× faster on large grids. step() now computes independent convective terms for u[0] and u[1] — each velocity component advects correctly through its own gradients.

Pairs with

HGF operates on 2D grid fields — composable with any module that produces or consumes spatial signal arrays.

M07

Phase Reflection Agent

Monitor coherence drift over long runs. PRA sits above HGF and flags when the field is working too hard to stay stable.

M08

Impedance Variance Vectors

Pre-condition input vectors before seeding into HGF. Reduces initialization noise in high-resolution grids.

M10

Harmonic Coherence Filter

Score the output coherence field. Filter or gate downstream processing based on field quality.

M09

Impedance Network Simulator

Model signal propagation through the fluid field as an impedance network. Useful for photonic or acoustic applications.

NumPy only. Drop it in.

No framework dependencies. No external APIs. One file, one class, five constants.

Request the module → All modules HDRS — M17