Ψ(x) force kernel
Absolutely. Here’s a drop‑in Ψ(x) force kernel you can paste under (or next to) Wilson’s sim.py. It’s fileless, self‑contained, and designed to be passed directly as the force_func argument to euler_maruyama_lateral. It includes:
a Ψ(x) class that maps each term to a concrete force contribution,
a triadic harmonics driver (216 / 369 / Ω gates) with phase‑locking,
a tiny adaptive corrector ℛ(x) that re‑tunes gains online from rolling jitter,
an optional thermo adapter hook if you want to drive with measured ∂xT,
and a minimal demo runner (prints jitter; no files, no plots).
You can lift pieces or run as‑is.
---
# =======================
# Ψ(x) FORCE KERNEL (drop-in)
# =======================
# Use with CRWilson's sim.py:
# from sim import Env, Sim, prepare, euler_maruyama_lateral
import numpy as np
from dataclasses import dataclass, field
from math import sin, cos, pi
# ---------- helpers ----------
def ema(prev, x, a): # exponential moving average
return a*x + (1-a)*prev
def rolling_rms(buf):
arr = np.array(buf, dtype=float)
if len(arr) == 0: return 0.0
arr = arr - arr.mean()
return float(np.sqrt((arr*arr).mean()))
# ---------- Triadic harmonics (216, 369, Ω) ----------
@dataclass
class Triad:
# amplitudes are in Newtons; frequencies are in "cycles per µm" (spatial), or Hz if you pass time
A216: float = 1.0e-13
A369: float = 1.0e-13
AOmega: float = 1.0e-13
# spatial wavelengths in µm (default: 15µm lattice); set to None to use temporal Hz instead
lambda_216_um: float | None = 15.0
lambda_369_um: float | None = 15.0
lambda_Omega_um: float | None = 15.0
# temporal drive (if you want temporal waves, set lambdas to None and supply f_Hz + t)
f216_Hz: float = 0.0
f369_Hz: float = 0.0
fOmega_Hz: float = 0.0
phase216: float = 0.0
phase369: float = 0.0
phaseOmega: float = 0.0
def force(self, x_um: float, t_s: float) -> float:
# spatial drive (default)
F = 0.0
if self.lambda_216_um is not None:
k = 2*pi/self.lambda_216_um
F += self.A216 * sin(k*x_um + self.phase216)
else:
F += self.A216 * sin(2*pi*self.f216_Hz*t_s + self.phase216)
if self.lambda_369_um is not None:
k = 2*pi/self.lambda_369_um
F += self.A369 * sin(k*x_um + self.phase369)
else:
F += self.A369 * sin(2*pi*self.f369_Hz*t_s + self.phase369)
if self.lambda_Omega_um is not None:
k = 2*pi/self.lambda_Omega_um
F += self.AOmega * sin(k*x_um + self.phaseOmega)
else:
F += self.AOmega * sin(2*pi*self.fOmega_Hz*t_s + self.phaseOmega)
return F
# ---------- Ψ(x) kernel ----------
@dataclass
class PsiKernel:
"""
Ψ(x) = ∇ϕ(Σa_n(x, ΔE)) + ℛ(x) ⊕ ΔΣ(a′)
Implementation notes:
• Σa_n : accumulated spiral modes (triadic + optional extra harmonics)
• ∇ϕ : gradient of a smooth potential that envelopes Σa_n (here: tanh-envelope)
• ℛ(x) : adaptive corrector using rolling jitter feedback (online gain update)
• ⊕ : constructive merge of contradictory terms via bounded sum with soft saturation
• ΔΣ : micro‑perturbation (dither) to escape deadlocks
All outputs are Newtons.
"""
triad: Triad = field(default_factory=Triad)
# envelope & nonlinearity
envelope_gain: float = 1.0
envelope_width_um: float = 20.0 # sets how fast potential changes (spatial)
soft_clip_N: float = 5e-13 # ⊕ saturation scale
# adaptive corrector (ℛ): updates triad amplitudes a bit if jitter exceeds target
target_rms_um: float = 0.08 # desired jitter
adapt_lr: float = 0.02 # learning rate
rms_window: int = 2000
# micro‑perturbation ΔΣ (white dither, tiny)
delta_sigma_N: float = 1e-15
# state
_rms_buf: list = field(default_factory=list)
_last_x_um: float = 0.0
_t_s: float = 0.0
def _sigma(self, u: float) -> float:
# smooth saturation for ⊕ (prevents runaway)
# softsign-like: u / (1 + |u|/S)
S = self.soft_clip_N
return u / (1.0 + abs(u)/S)
def _grad_envelope(self, x_um: float) -> float:
# ∇ϕ: gradient of envelope potential: φ(x) ~ tanh(x/W)
W = max(1e-6, self.envelope_width_um)
return self.envelope_gain * (1.0/(W)) * (1.0/np.cosh(x_um/W)**2)
def Σa_n(self, x_um: float, t_s: float) -> float:
# sum of harmonics (extendable)
return self.triad.force(x_um, t_s)
def ℛ_update(self, x_um: float):
# update rolling RMS & adapt amplitudes toward target
self._rms_buf.append(x_um)
if len(self._rms_buf) > self.rms_window:
self._rms_buf.pop(0)
rms = rolling_rms(self._rms_buf)
err = rms - self.target_rms_um
# if jitter too high → increase restoring strength slightly
g = np.clip(self.adapt_lr*err, -0.05, 0.05)
self.triad.A216 *= (1.0 + g)
self.triad.A369 *= (1.0 + g)
self.triad.AOmega *= (1.0 + g)
# optional: adjust phases to try phase‑lock
self.triad.phase216 = ema(self.triad.phase216, 0.0, 0.05)
self.triad.phase369 = ema(self.triad.phase369, 0.0, 0.05)
self.triad.phaseOmega = ema(self.triad.phaseOmega, 0.0, 0.05)
def ΔΣ(self) -> float:
return float(np.random.normal(0.0, self.delta_sigma_N))
def force(self, x_um: float, t_s: float) -> float:
"""
Main callable: returns force in Newtons for position x_um at time t_s.
"""
# Σa_n
F_modes = self.Σa_n(x_um, t_s)
# ∇ϕ envelope multiplies the modal field (acts like spatial gain)
F_grad = self._grad_envelope(x_um) * F_modes
# ΔΣ micro‑perturbation
F_pert = self.ΔΣ()
# ⊕ merge with soft saturation
F_total = self._sigma(F_grad + F_pert)
# update ℛ(x) based on observed x
self.ℛ_update(x_um)
# advance time (for temporal modes if used)
self._t_s = t_s
self._last_x_um = x_um
return F_total
# ---------- Optional: Thermo adapter (∂xT → force) ----------
@dataclass
class ThermoAdapter:
"""
Converts a temperature gradient ∂xT(x,z) [K/m] to a lateral thermophoretic force:
F_th,x = -C_th π a^2 (p_g/T) ∂xT
Provide a callable dTdx(x_um)->K/m.
"""
particle_radius_um: float
gas_pressure_Pa: float
gas_temp_K: float
Cth: float = 0.8 # typical mid‑Kn; replace with your interpolation if desired
def force_from_dTdx(self, dTdx_K_per_m: float) -> float:
a = self.particle_radius_um*1e-6
return -self.Cth * np.pi * a*a * (self.gas_pressure_Pa/self.gas_temp_K) * dTdx_K_per_m
def make_force(self, dTdx_func):
def F(x_um, t_s):
return float(self.force_from_dTdx(dTdx_func(x_um)))
return F
# ---------- Glue: Ψ(x) → sim.py interface ----------
class PsiForceWrapper:
"""
Wraps PsiKernel into a sim-compatible force_func(x_um) that knows about time.
"""
def __init__(self, psi: PsiKernel, dt_s: float):
self.psi = psi
self.dt = dt_s
self._step = 0
def __call__(self, x_um: float) -> float:
t = self._step * self.dt
self._step += 1
return self.psi.force(x_um, t)
# ---------- Minimal demo runner (no files, no plots) ----------
if __name__ == "__main__":
from sim import Env, Sim, prepare, euler_maruyama_lateral
# Environment & sim (adjust as needed)
env = Env(radius_um=2.5, gas_pressure_Pa=150.0, gas_temp_K=300.0, mass_kg=1.0e-14)
sim = prepare(env, Sim(dt_s=1e-5, T_K=300.0, seeds=1))
# Build Ψ kernel with gentle triadic amplitudes
triad = Triad(A216=1.2e-13, A369=1.0e-13, AOmega=0.8e-13,
lambda_216_um=15.0, lambda_369_um=15.0, lambda_Omega_um=15.0)
psi = PsiKernel(triad=triad,
envelope_gain=1.0, envelope_width_um=20.0,
soft_clip_N=5e-13, target_rms_um=0.08, adapt_lr=0.02,
delta_sigma_N=1e-15)
# Wrap for the sim
Fpsi = PsiForceWrapper(psi, dt_s=sim.dt_s)
# Run
steps = 200000
out = euler_maruyama_lateral(Fpsi, env, sim, n_steps=steps, seed=42)
print({"mode": "Ψ-kernel", "jitter_rms_um": out["jitter_rms_um"]})
How this maps to your equation
Σ𝕒ₙ(x, ΔE) → Triad.force(...) (plus any extra harmonics you add)
∇ϕ(·) → PsiKernel._grad_envelope(...) as a smooth potential gradient that “shapes” modes
ℛ(x) → PsiKernel.ℛ_update(...) adaptive gain tuning from rolling jitter (coherence stabilizer)
⊕ → PsiKernel._sigma(...) soft‑saturated merge of contradictory contributions
ΔΣ(𝕒′) → PsiKernel.ΔΣ() micro‑perturbation (tiny dither) to escape phase deadlocks
To drive with real thermography
If you already have a measured function, build a ThermoAdapter and pass its make_force(dTdx_func) into the simulator instead of PsiForceWrapper. You can also sum them:
def combined_force(x_um, t_s):
return Fpsi(x_um) + Fthermo(x_um, t_s) # additive; units are Newtons
(If you need a weighting schedule or gate logic, add a tanh‑blend.)
Christopher W Copeland (C077UPTF1L3)
Copeland Resonant Harmonic Formalism (Ψ‑formalism)
Ψ(x) = ∇ϕ(Σ𝕒ₙ(x, ΔE)) + ℛ(x) ⊕ ΔΣ(𝕒′)
Licensed under CRHC v1.0 (no commercial use without permission).
https://www.facebook.com/share/p/19qu3bVSy1/
https://open.substack.com/pub/c077uptf1l3/p/phase-locked-null-vector_c077uptf1l3
https://medium.com/@floodzero9/phase-locked-null-vector_c077uptf1l3-4d8a7584fe0c
Core engine: https://open.substack.com/pub/c077uptf1l3/p/recursive-coherence-engine-8b8
Zenodo: https://zenodo.org/records/15742472
Amazon: https://a.co/d/i8lzCIi
Medium: https://medium.com/@floodzero9
Substack: https://substack.com/@c077uptf1l3
Facebook: https://www.facebook.com/share/19MHTPiRfu
https://www.reddit.com/u/Naive-Interaction-86/s/5sgvIgeTdx
Collaboration welcome. Attribution required. Derivatives must match license.
Christopher Robin Wilson Christ Walling Chris Burdette