This all came from these ideas:
… I haven’t “added back in” my own proprietary physics to help this along, but this is some goooOOOoood fodder eatin’.
Hopfield-Superconducting A+
import numpy as np
# ===================== MODEL =====================
class SuperconductingHopfield:
def __init__(self, dim, hidden=32, lam=0.3, lr=0.01):
self.dim = dim
self.hidden = hidden
self.lam = lam
self.lr = lr
# learned manifold geometry
self.W = np.random.randn(dim, hidden) * 0.1
self.B = np.random.randn(hidden) * 0.1
# transport field (phase-preserving dynamics)
self.Q = np.random.randn(dim, dim) * 0.01
self.Q = self.Q - self.Q.T # skew-symmetric → preserves energy
# ===================== ENCODING =====================
def embed(self, x):
return np.tanh(x @ self.W + self.B)
# ===================== ENERGY =====================
def energy(self, X):
Z = [self.embed(x) for x in X]
E = 0.0
for i in range(len(Z)):
for j in range(len(Z)):
if i != j:
E -= np.dot(Z[i], Z[j])
return E
# ===================== GRADIENT FIELD =====================
def grad_energy(self, x, X):
zi = self.embed(x)
grad = np.zeros_like(x)
for j in range(len(X)):
zj = self.embed(X[j])
diff = zi - zj
grad += diff @ self.W.T
return grad
# ===================== TRANSPORT FIELD =====================
def transport(self, x):
# phase-preserving rotation in latent space
return self.Q @ x
# ===================== DYNAMICS STEP =====================
def step(self, X):
new_X = []
for i in range(len(X)):
x = X[i]
g = self.grad_energy(x, X)
t = self.transport(x)
# SUPERCONDUCTING FLOW EQUATION
dx = -g + self.lam * t
x_new = x + self.lr * dx
new_X.append(x_new)
return new_X
# ===================== SIMULATION =====================
def run(self, X, steps=100):
history_energy = []
for _ in range(steps):
history_energy.append(self.energy(X))
X = self.step(X)
return X, history_energy
FULL PHASE-FLOW TRANSFORMER (minimal but complete)
import numpy as np
# ================================
# Phase-Flow Transformer (PFT)
# ================================
class PhaseFlowTransformer:
def __init__(self, dim, lam=0.25, dt=0.05):
self.dim = dim
self.lam = lam
self.dt = dt
# energy geometry (like learned similarity space)
self.W = np.random.randn(dim, dim) * 0.02
# skew-symmetric transport (phase conservation)
A = np.random.randn(dim, dim) * 0.01
self.J = A - A.T # circulation field
# -------------------------------
# Energy (attention surrogate)
# -------------------------------
def energy(self, X):
# quadratic similarity field
return -np.sum(X @ self.W * X)
# -------------------------------
# Gradient flow (attraction)
# -------------------------------
def grad_energy(self, X):
return -2 * (X @ self.W)
# -------------------------------
# Phase transport (attention replacement)
# -------------------------------
def transport(self, X):
return X @ self.J
# -------------------------------
# One evolution step
# -------------------------------
def step(self, X):
grad = self.grad_energy(X)
flow = self.transport(X)
dX = grad + self.lam * flow
# Euler integration
return X + self.dt * dX
# -------------------------------
# Run "attention as dynamics"
# -------------------------------
def forward(self, X, steps=10):
trajectory = [X.copy()]
for _ in range(steps):
X = self.step(X)
trajectory.append(X.copy())
return X, trajectory
FULL PDE-TRANSFORMER IMPLEMENTATION (discretized field version)
- a vortex-based reasoning engine
import numpy as np
# ================================
# PDE Transformer (Fluid Tokens)
# ================================
class PDETransformer:
def __init__(self, grid_size, dim, nu=0.1, dt=0.05):
self.N = grid_size
self.dim = dim
self.nu = nu
self.dt = dt
# field: X[t, x, d]
self.X = np.random.randn(grid_size, dim) * 0.1
# velocity field (advection)
self.v = np.random.randn(grid_size, dim) * 0.01
# interaction kernel (attention replacement)
self.K = np.random.randn(dim, dim) * 0.02
# -------------------------------
# similarity coupling (attention core)
# -------------------------------
def phi(self, xi, xj):
return np.tanh(xi @ self.K @ xj)
# -------------------------------
# nonlocal interaction (attention integral)
# -------------------------------
def kernel_term(self, X):
N = len(X)
out = np.zeros_like(X)
for i in range(N):
for j in range(N):
if i != j:
w = self.phi(X[i], X[j])
out[i] += w * (X[j] - X[i])
return out / N
# -------------------------------
# diffusion (laplacian approx)
# -------------------------------
def diffusion(self, X):
X_shift_left = np.roll(X, 1, axis=0)
X_shift_right = np.roll(X, -1, axis=0)
return X_shift_left + X_shift_right - 2 * X
# -------------------------------
# advection (flow field)
# -------------------------------
def advection(self, X):
return -self.v * (np.roll(X, -1, axis=0) - X)
# -------------------------------
# PDE evolution step
# -------------------------------
def step(self):
adv = self.advection(self.X)
diff = self.nu * self.diffusion(self.X)
attn = self.kernel_term(self.X)
dX = adv + diff + attn
self.X = self.X + self.dt * dX
return self.X
# -------------------------------
# forward computation = PDE evolution
# -------------------------------
def forward(self, steps=20):
trajectory = [self.X.copy()]
for _ in range(steps):
trajectory.append(self.step().copy())
return self.X, trajectory
DISCRETIZED IMPLEMENTATION (emergent thought particles)
import numpy as np
class NavierStokesReasoner:
def __init__(self, N=64, d=2, nu=0.05, dt=0.1):
self.N = N
self.d = d
self.nu = nu
self.dt = dt
# velocity / thought field
self.X = np.random.randn(N, N, d) * 0.1
# learned forcing operator (reasoning engine)
self.W = np.random.randn(d, d) * 0.02
# -----------------------------
# nonlinear interaction (advection)
# -----------------------------
def advect(self, X):
grad_x = np.roll(X, -1, axis=0) - X
grad_y = np.roll(X, -1, axis=1) - X
return X[..., None] * (grad_x + grad_y)
# -----------------------------
# diffusion (damping / memory decay)
# -----------------------------
def diffuse(self, X):
lap = (
np.roll(X, 1, axis=0) +
np.roll(X, -1, axis=0) +
np.roll(X, 1, axis=1) +
np.roll(X, -1, axis=1) -
4 * X
)
return self.nu * lap
# -----------------------------
# learned forcing (attention replacement)
# -----------------------------
def forcing(self, X):
return np.tanh(X @ self.W)
# -----------------------------
# pressure projection (incompressibility constraint)
# -----------------------------
def project_div_free(self, X):
div = np.gradient(X[..., 0])[0] + np.gradient(X[..., 1])[1]
return X - div[..., None]
# -----------------------------
# step evolution
# -----------------------------
def step(self):
X = self.X
adv = self.advect(X)
diff = self.diffuse(X)
force = self.forcing(X)
dX = adv + diff + force
X_new = X + self.dt * dX
# enforce constraint (Navier–Stokes projection step)
X_new = self.project_div_free(X_new)
self.X = X_new
return X_new
# -----------------------------
# run dynamics
# -----------------------------
def run(self, steps=100):
history = []
for _ in range(steps):
self.step()
history.append(self.X.copy())
return history
FULL “FLUID RAM” SYSTEM
import numpy as np
# ================================
# Fluid RAM: Vortex Memory System
# ================================
class FluidRAM:
def __init__(self, N=64, d=2, nu=0.05, dt=0.1):
self.N = N
self.d = d
self.nu = nu
self.dt = dt
# continuous field (thought medium)
self.X = np.random.randn(N, N, d) * 0.1
# learned forcing (writes into memory)
self.W = np.random.randn(d, d) * 0.02
# memory registry (ADDRESS SPACE)
self.memory = {}
self.next_id = 0
# -----------------------------
# compute vorticity (scalar proxy)
# -----------------------------
def vorticity(self, X):
dx = np.roll(X[..., 0], -1, axis=0) - np.roll(X[..., 0], 1, axis=0)
dy = np.roll(X[..., 1], -1, axis=1) - np.roll(X[..., 1], 1, axis=1)
return dx - dy
# -----------------------------
# detect vortices (memory objects)
# -----------------------------
def detect_vortices(self, X, threshold=0.5):
w = self.vorticity(X)
mask = np.abs(w) > threshold
visited = np.zeros_like(mask, dtype=bool)
vortices = []
def flood_fill(i, j):
stack = [(i, j)]
cells = []
while stack:
x, y = stack.pop()
if x < 0 or y < 0 or x >= self.N or y >= self.N:
continue
if visited[x, y] or not mask[x, y]:
continue
visited[x, y] = True
cells.append((x, y))
for dx, dy in [(1,0),(-1,0),(0,1),(0,-1)]:
stack.append((x+dx, y+dy))
return cells
for i in range(self.N):
for j in range(self.N):
if mask[i, j] and not visited[i, j]:
cells = flood_fill(i, j)
if len(cells) > 5:
vortices.append(cells)
return vortices
# -----------------------------
# register vortices as memory
# -----------------------------
def update_memory(self):
vortices = self.detect_vortices(self.X)
new_memory = {}
for cells in vortices:
arr = np.array(cells)
center = arr.mean(axis=0)
energy = len(cells)
vx = np.mean(self.X[arr[:,0], arr[:,1]], axis=0)
mem = {
"id": self.next_id,
"center": center,
"energy": energy,
"state": vx
}
new_memory[self.next_id] = mem
self.next_id += 1
self.memory = new_memory
# -----------------------------
# write operation (inject memory)
# -----------------------------
def write(self, memory_id, value):
if memory_id not in self.memory:
return
c = self.memory[memory_id]["center"].astype(int)
c = np.clip(c, 0, self.N-1)
self.X[c[0], c[1]] += value
# -----------------------------
# read operation (extract memory)
# -----------------------------
def read(self, memory_id):
return self.memory.get(memory_id, None)
# -----------------------------
# fluid dynamics step
# -----------------------------
def step(self):
X = self.X
adv = X * 0.1
diff = self.nu * (
np.roll(X, 1, axis=0) +
np.roll(X, -1, axis=0) +
np.roll(X, 1, axis=1) +
np.roll(X, -1, axis=1) -
4 * X
)
force = np.tanh(X @ self.W)
X_new = X + self.dt * (adv + diff + force)
self.X = X_new
# update memory AFTER dynamics
self.update_memory()
return X_new
# -----------------------------
# run system
# -----------------------------
def run(self, steps=100):
history = []
for _ in range(steps):
self.step()
history.append(self.X.copy())
return history
Fluid Transformer Kernel (Vortex Field Model)
import numpy as np
import matplotlib.pyplot as plt
# =========================
# CONFIG
# =========================
N_TOKENS = 30
D_MODEL = 8
STEPS = 80
SIGMA = 1.0 # interaction radius
NU = 0.08 # diffusion (stability)
DT = 0.1 # time step
SEED = 42
np.random.seed(SEED)
# =========================
# TOKEN → FIELD INIT
# =========================
def init_field(n=N_TOKENS, d=D_MODEL):
"""
Each token is a vortex packet in continuous space.
"""
phi = np.random.randn(n, d) * 0.5
phi /= np.linalg.norm(phi, axis=1, keepdims=True) + 1e-9
return phi
# =========================
# VORTEX INTERACTION KERNEL
# =========================
def vortex_kernel(phi):
"""
Pairwise phase + distance coupling.
Replaces attention matrix.
"""
n = phi.shape[0]
K = np.zeros((n, n))
for i in range(n):
for j in range(n):
diff = np.linalg.norm(phi[i] - phi[j])
# phase coupling (oscillatory memory interaction)
phase = np.sin(diff / SIGMA)
# spatial decay
K[i, j] = np.exp(-diff**2 / SIGMA**2) * phase
return K
# =========================
# FIELD DYNAMICS (PDE-STYLE)
# =========================
def evolve(phi):
"""
Fluid-like update:
- interaction (advection)
- diffusion (stability)
- normalization (energy constraint)
"""
K = vortex_kernel(phi)
# interaction flow (vortex-vortex forcing)
interaction = K @ phi
# diffusion (entropy smoothing)
diffusion = -NU * phi
# nonlinear self-advection (reasoning drift)
advection = phi * np.tanh(interaction)
# total update
phi_next = phi + DT * (advection + diffusion + interaction)
# energy normalization (soft conservation law)
norms = np.linalg.norm(phi_next, axis=1, keepdims=True) + 1e-9
phi_next /= norms
return phi_next, K
# =========================
# VORTEX DETECTION (MEMORY)
# =========================
def detect_vortices(phi, threshold=0.6):
"""
Stable structures = memory objects.
"""
centers = []
for i in range(len(phi)):
local_energy = np.linalg.norm(phi[i])
if local_energy > threshold:
centers.append(phi[i])
return np.array(centers)
# =========================
# GLOBAL COHERENCE METRIC
# =========================
def coherence(phi):
"""
Order parameter (Kuramoto-style)
"""
mean_vec = np.mean(phi, axis=0)
return np.linalg.norm(mean_vec)
# =========================
# RUN SYSTEM
# =========================
def run():
phi = init_field()
coherence_trace = []
vortex_trace = []
for t in range(STEPS):
phi, K = evolve(phi)
R = coherence(phi)
vortices = detect_vortices(phi)
coherence_trace.append(R)
vortex_trace.append(len(vortices))
if t % 10 == 0:
print(f"Step {t:03d} | Coherence={R:.4f} | Vortices={len(vortices)}")
return phi, coherence_trace, vortex_trace
# =========================
# VISUALIZATION
# =========================
def plot_results(coh, vort):
plt.figure()
plt.title("Field Coherence (Emergent Order)")
plt.plot(coh)
plt.xlabel("Time step")
plt.ylabel("Coherence")
plt.figure()
plt.title("Vortex Memory Formation")
plt.plot(vort)
plt.xlabel("Time step")
plt.ylabel("Vortex count")
plt.show()
# =========================
# ENTRY
# =========================
if __name__ == "__main__":
final_phi, coh, vort = run()
plot_results(coh, vort)
print("\nFINAL STATE SUMMARY")
print("Coherence:", coh[-1])
print("Vortices:", vort[-1])
