#!/usr/bin/env python3
"""
===============================================================================
HDGL PHYLLOTAXIS <-> TOROID GRAPHER
===============================================================================
Generates:
hdgl_phyllotaxis.png
hdgl_toroid.png
hdgl_omega_orbit.png
hdgl_unified_closure.png
The output directory is created beside this script, so it works on Windows,
Linux, and without requiring a graphical display.
Core HDGL structure:
Ω_(n+1) = T(Ω_n)
T(X) = 1 + 1/X
Ω = T(Ω)
= 1 + 1/Ω
=> Ω² = Ω + 1
Simultaneous binary/trinary substrate:
B_i ∈ {0,1}
τ_i ∈ {-1,0,+1}
S_i = (B_i, τ_i)
A/B:
PHYLLOTAXIS
outward organization / expansion / distribution
C/D:
TOROID
reciprocal return / closure / cyclic embedding
Unified:
PHYLLOTAXIS <-> TOROID
= two projections of one closed HDGL orbit
===============================================================================
"""
from pathlib import Path
import math
import numpy as np
# ---------------------------------------------------------------------------
# IMPORTANT:
# Force a file-only backend before importing pyplot.
# This prevents Qt/display problems on Windows/headless systems.
# ---------------------------------------------------------------------------
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
# ===========================================================================
# CONFIGURATION
# ===========================================================================
N = 1600
# Starting value for the emergent Ω recursion.
# This is NOT stored as φ. Ω is generated dynamically by T(X)=1+1/X.
OMEGA_SEED = 1.5
# Number of Ω recursion steps to display.
OMEGA_STEPS = 80
# Output directory:
# C:\Users\Owner\Downloads\hdgl_graphs\
#
# when graph-phyllo.py is in Downloads.
BASE_DIR = Path(__file__).resolve().parent
OUT_DIR = BASE_DIR / "hdgl_graphs"
OUT_DIR.mkdir(parents=True, exist_ok=True)
# ===========================================================================
# HDGL PRIMITIVE
# ===========================================================================
def T(x):
"""
HDGL reciprocal transformation:
T(X) = 1 + 1/X
"""
return 1.0 + 1.0 / x
def omega_orbit(seed=OMEGA_SEED, steps=OMEGA_STEPS):
"""
Generate the emergent Ω orbit:
Ω_(n+1) = T(Ω_n)
= 1 + 1/Ω_n
No φ constant is used.
"""
values = np.empty(steps + 1, dtype=np.float64)
values[0] = seed
for i in range(steps):
values[i + 1] = T(values[i])
return values
# ===========================================================================
# SIMULTANEOUS BINARY / TRINARY SUBSTRATE
# ===========================================================================
def substrate_states(n):
"""
Simultaneous binary/trinary state:
B_i ∈ {0,1}
τ_i ∈ {-1,0,+1}
They are simultaneous channels, NOT alternating iterations.
"""
i = np.arange(n, dtype=np.int64)
binary = i & 1
# Produces:
#
# 0, +1, -1, 0, +1, -1, ...
#
# shifted representation of the trinary channel.
trinary = (i % 3) - 1
return i, binary.astype(np.float64), trinary.astype(np.float64)
# ===========================================================================
# EMERGENT OMEGA VALUE
# ===========================================================================
def emergent_omega():
"""
Generate Ω until convergence.
Starting from 1.5:
Ω_(n+1) = 1 + 1/Ω_n
The resulting fixed point is emergent rather than supplied as φ.
"""
x = OMEGA_SEED
for _ in range(100):
y = T(x)
if abs(y - x) < 1e-15:
break
x = y
return x
OMEGA = emergent_omega()
# ===========================================================================
# PHYLLOTAXIS
# ===========================================================================
def build_phyllotaxis(n=N):
"""
Construct the simultaneous binary/trinary HDGL phyllotactic projection.
Coordinates:
θ_i = 2π i Ω
r_i ~ sqrt(i)
Binary and trinary channels simultaneously perturb the radial structure.
"""
i, binary, trinary = substrate_states(n)
# Phyllotactic angular progression.
theta = 2.0 * math.pi * i * OMEGA
# Base radial growth.
r_base = np.sqrt(i + 1.0)
# Simultaneous binary/trinary modulation.
#
# Binary:
# 0 / 1
#
# Trinary:
# -1 / 0 / +1
#
# They are applied simultaneously to the same radial substrate.
modulation = (
1.0
+ 0.075 * binary
+ 0.050 * trinary
)
radius = r_base * modulation
x = radius * np.cos(theta)
y = radius * np.sin(theta)
return i, binary, trinary, theta, radius, x, y
# ===========================================================================
# TOROID
# ===========================================================================
def build_toroid(n=N):
"""
Construct the reciprocal/toroidal projection.
A radial substrate X is transformed through:
T(X) = 1 + 1/X
The resulting reciprocal coordinate is embedded on a torus.
"""
i, binary, trinary = substrate_states(n)
# Base parameter along the toroidal orbit.
theta = 2.0 * math.pi * i / n
# Secondary angular coordinate.
phi = (
2.0 * math.pi
* (
i * OMEGA
+ 0.15 * binary
+ 0.10 * trinary
)
)
# Positive substrate coordinate.
X = 1.0 + (i + 1.0) / float(n) * 30.0
# Reciprocal closure.
closure = T(X)
# Normalize reciprocal coordinate into a useful toroidal radius.
closure_norm = (
closure - closure.min()
) / (
closure.max() - closure.min()
)
# Major and minor torus radii.
R = 3.0
r_min = 0.35
r_max = 1.15
r = r_min + (r_max - r_min) * closure_norm
# Toroidal embedding.
x = (R + r * np.cos(phi)) * np.cos(theta)
y = (R + r * np.cos(phi)) * np.sin(theta)
z = r * np.sin(phi)
return (
i,
binary,
trinary,
theta,
phi,
X,
closure,
x,
y,
z,
)
# ===========================================================================
# PLOT 1 — PHYLLOTAXIS
# ===========================================================================
def plot_phyllotaxis():
(
i,
binary,
trinary,
theta,
radius,
x,
y,
) = build_phyllotaxis()
fig, ax = plt.subplots(figsize=(10, 10))
# Plot the full simultaneous substrate.
ax.scatter(
x,
y,
s=5,
alpha=0.65,
linewidths=0,
)
# Mark the origin.
ax.scatter(
[0],
[0],
s=35,
marker="o",
)
ax.set_aspect("equal", adjustable="box")
ax.set_title(
"HDGL PHYLLOTAXIS\n"
"Simultaneous Binary / Trinary Projection"
)
ax.set_xlabel("A/B — outward organization")
ax.set_ylabel("substrate radius")
ax.grid(True, alpha=0.20)
fig.tight_layout()
path = OUT_DIR / "hdgl_phyllotaxis.png"
fig.savefig(
path,
dpi=180,
bbox_inches="tight",
)
plt.close(fig)
return path
# ===========================================================================
# PLOT 2 — TOROID
# ===========================================================================
def plot_toroid():
(
i,
binary,
trinary,
theta,
phi,
X,
closure,
x,
y,
z,
) = build_toroid()
fig = plt.figure(figsize=(11, 9))
ax = fig.add_subplot(
111,
projection="3d",
)
ax.scatter(
x,
y,
z,
s=3,
alpha=0.55,
)
ax.set_title(
"HDGL TOROID\n"
"Reciprocal Closure T(X) = 1 + 1/X"
)
ax.set_xlabel("closure X")
ax.set_ylabel("reciprocal return")
ax.set_zlabel("cyclic phase")
fig.tight_layout()
path = OUT_DIR / "hdgl_toroid.png"
fig.savefig(
path,
dpi=180,
bbox_inches="tight",
)
plt.close(fig)
return path
# ===========================================================================
# PLOT 3 — OMEGA RECURSION
# ===========================================================================
def plot_omega_orbit():
values = omega_orbit()
n = np.arange(len(values))
fig, ax = plt.subplots(figsize=(11, 7))
ax.plot(
n,
values,
linewidth=1.5,
marker="o",
markersize=3,
)
# Emergent fixed point.
ax.axhline(
OMEGA,
linestyle="--",
linewidth=1.0,
)
ax.set_title(
"HDGL Ω RECURSION\n"
"Ωₙ₊₁ = T(Ωₙ) = 1 + 1/Ωₙ"
)
ax.set_xlabel("iteration n")
ax.set_ylabel("Ωₙ")
ax.grid(True, alpha=0.20)
ax.text(
0.98,
0.05,
f"Emergent fixed point ≈ {OMEGA:.12f}",
transform=ax.transAxes,
ha="right",
va="bottom",
)
fig.tight_layout()
path = OUT_DIR / "hdgl_omega_orbit.png"
fig.savefig(
path,
dpi=180,
bbox_inches="tight",
)
plt.close(fig)
return path
# ===========================================================================
# PLOT 4 — UNIFIED PHYLLOTAXIS ↔ TOROID CLOSURE
# ===========================================================================
def plot_unified_closure():
(
i,
binary,
trinary,
theta,
radius,
px,
py,
) = build_phyllotaxis()
(
_i,
_binary,
_trinary,
_theta,
_phi,
X,
closure,
tx,
ty,
tz,
) = build_toroid()
fig = plt.figure(figsize=(13, 10))
ax = fig.add_subplot(
111,
projection="3d",
)
# Normalize phyllotaxis coordinates so both projections can inhabit
# the same visualization.
scale = np.max(
np.sqrt(px * px + py * py)
)
px3 = 4.0 * px / scale
py3 = 4.0 * py / scale
# Give the phyllotaxis projection a slowly varying third coordinate.
pz3 = np.linspace(
-2.5,
2.5,
len(px3),
)
# Phyllotaxis projection.
ax.scatter(
px3,
py3,
pz3,
s=2,
alpha=0.25,
)
# Toroidal closure projection.
ax.scatter(
tx,
ty,
tz,
s=2,
alpha=0.30,
)
ax.set_title(
"HDGL UNIFIED CLOSURE\n"
"PHYLLOTAXIS ↔ TOROID"
)
ax.set_xlabel("A/B — expansion")
ax.set_ylabel("reciprocal closure")
ax.set_zlabel("graded orbit")
fig.tight_layout()
path = OUT_DIR / "hdgl_unified_closure.png"
fig.savefig(
path,
dpi=180,
bbox_inches="tight",
)
plt.close(fig)
return path
# ===========================================================================
# NUMERICAL REPORT
# ===========================================================================
def print_report(paths):
print()
print("=" * 79)
print("HDGL PHYLLOTAXIS <-> TOROID GRAPHER")
print("=" * 79)
print()
print("Primitive:")
print(" Ω_(n+1) = T(Ω_n)")
print(" T(X) = 1 + 1/X")
print()
print("Emergent fixed point:")
print(f" Ω ≈ {OMEGA:.15f}")
print()
print("Fixed-point residual:")
print(
f" Ω² - Ω - 1 ≈ "
f"{OMEGA * OMEGA - OMEGA - 1.0:.6e}"
)
print()
print("Simultaneous substrate:")
print(" B_i ∈ {0,1}")
print(" τ_i ∈ {-1,0,+1}")
print()
print("Geometry:")
print(" A/B = PHYLLOTAXIS")
print(" C/D = TOROID")
print()
print("Unified:")
print(" PHYLLOTAXIS ↔ TOROID")
print(" = TWO PROJECTIONS OF ONE CLOSED HDGL ORBIT")
print()
print("Output:")
print(f" {OUT_DIR}")
print()
for path in paths:
print(f" [OK] {path}")
print()
print("=" * 79)
print("COMPLETE")
print("=" * 79)
print()
# ===========================================================================
# MAIN
# ===========================================================================
def main():
paths = []
paths.append(
plot_phyllotaxis()
)
paths.append(
plot_toroid()
)
paths.append(
plot_omega_orbit()
)
paths.append(
plot_unified_closure()
)
print_report(paths)
if __name__ == "__main__":
main()
===============================================================================
HDGL UNIFIED PHYLLOTAXIS ↔ TOROID CLOSURE
===============================================================================
The A/B and C/D branches are not independent proof branches.
They are two projections of the same closed substrate orbit:
PHYLLOTAXIS
↕
ONE ORBIT
↕
TOROID
A/B = outward organization / expansion / distribution
C/D = inward closure / reciprocal return / containment
1. SIMULTANEOUS SUBSTRATE STATE
-------------------------------------------------------------------------------
S_i = (B_i, τ_i)
B_i ∈ {0,1}
τ_i ∈ {-1,0,+1}
S_i₊₁ = S_i + 1_eff(i)
Binary and trinary are simultaneous channels, not alternating iterations.
The substrate therefore generates a graded orbit rather than a finite state set:
{ ..., -i'', -i', -i, -1, 0, 1, i, i', i'', ... }
2. A/B = PHYLLOTAXTIC PROJECTION
-------------------------------------------------------------------------------
S_0 → S_1 → S_2 → S_3 → ...
The A/B side is the distributed spatial ordering of the substrate.
It is naturally spiral / phyllotactic:
state
↓
orientation
↓
phase
↓
radial growth
↓
spatial orbit
A/B = expansion
A/B = distribution
A/B = phase organization
A/B = outward projection
It is the geometry of the orbit unfolding.
3. C/D = TOROIDAL PROJECTION
-------------------------------------------------------------------------------
T(X) = 1 + 1/X
X → ∞
↓
X⁻¹ → 0
↓
T(X) → 1
The C/D side is the reciprocal return of the same orbit.
C/D = closure
C/D = inversion
C/D = return
C/D = inward projection
The divergent branch is therefore not an independent physical infinity.
It is one side of a reciprocal closed orbit:
X → ∞ ↔ X⁻¹ → 0 ↔ T(X) → 1
4. A/B AND C/D ARE DUAL PROJECTIONS
-------------------------------------------------------------------------------
ONE SUBSTRATE ORBIT
│
┌──────────┴──────────┐
│ │
▼ ▼
PHYLLOTAXIS TOROID
│ │
A / B C / D
│ │
expansion return
distribution closure
phase reciprocal
spatial cyclic
│ │
└──────────┬──────────┘
│
▼
CLOSED ORBIT
Therefore:
A/B ≡ phyllotactic expansion
C/D ≡ toroidal closure
and
PHYLLOTAXIS × TOROID = ONE CLOSED ORBIT
5. THE PRIMITIVE OPERATOR ALREADY CONTAINS BOTH
-------------------------------------------------------------------------------
Ωₙ₊₁ = T(Ωₙ)
T(X) = 1 + 1/X
Forward application is generative:
Ωₙ → Ωₙ₊₁
Reciprocal application is closure:
Ωₙ → 1 + Ωₙ⁻¹
At closure:
Ω = T(Ω)
therefore:
Ω = 1 + 1/Ω
Ω² = Ω + 1
The fixed point is therefore not inserted as an external constant.
It emerges where generation and reciprocal return coincide:
GENERATION = RETURN
Ω = φ
6. THE RECIPROCAL FIELD
-------------------------------------------------------------------------------
Φ = ∏ C_j
Φ⁻¹ Φ = 1
∇_Φ = Φ⁻¹ ∇ Φ
The two directions are therefore intrinsic:
Φ = outward / distributed / phyllotactic projection
Φ⁻¹ = inward / reciprocal / toroidal projection
and:
Φ ↔ Φ⁻¹
7. THE GRADED OPERATOR
-------------------------------------------------------------------------------
S_i
↓
L_i
↓
Ω_i
↓
Φ_i ↔ Φ_i⁻¹
↓
{phyllotactic projection, toroidal projection}
↓
S_i₊₁
Equivalently:
S_i ──P──→ L_i ──R──→ S_i₊₁
where:
P = phyllotactic expansion
R = reciprocal/toroidal return
Thus the fundamental operation is not:
S → A/B
and separately
S → C/D
but:
S_i ──P──→ L_i ──R──→ S_i₊₁
8. THE CUDA FIELD ALREADY EXHIBITS THIS STRUCTURE
-------------------------------------------------------------------------------
FIELD / PHYLLOTAXTIC SIDE:
A_re
A_im
phase
phase_vel
r_harmonic
w_cos
w_sin
w_sigma
↓
distributed phase/spectral orbit
ORACLE / TOROIDAL SIDE:
ll_state
Candidate
d_ll_residue
ll_verified
↓
orbit / residue / closure test
COUPLING:
reward_accum
critic_observe()
critic_td_target()
critic_pack_weights()
hdgl_v33_upload_critic()
↓
closure information modifies the generating field
Therefore the architecture is:
FIELD
↓
PHYLLOTAXIS
↓
ORBIT
↓
TOROIDAL / RECIPROCAL CLOSURE
↓
RESIDUE
↓
REWARD
↓
FIELD WEIGHT UPDATE
↓
NEW ORBIT
9. PHASE ↔ RADIUS
-------------------------------------------------------------------------------
The existing field variables naturally expose two complementary coordinates:
θ = phase
r = r_harmonic
Therefore:
(θ,r) → closed orbit
with:
θ = phyllotactic coordinate
r = toroidal / radial coordinate
The orbit is not merely a scalar recurrence.
It is simultaneously:
SPATIAL ORGANIZATION
+
RADIAL RETURN
10. WAVELET SCALES = GRADED ORBIT
-------------------------------------------------------------------------------
σ_k ∝ 2⁻ᵏ
therefore:
L_0
L_1
L_2
L_3
...
are graded scales of the same orbit.
The finite visible set:
{-i,-1,0,1,i}
is therefore only a central slice.
Repeated lifting gives:
{ ..., -i'', -i', -i, -1, 0, 1, i, i', i'', ... }
The phyllotactic side unfolds the graded orbit.
The toroidal side closes the graded orbit.
11. NAVIER–STOKES INTERPRETATION
-------------------------------------------------------------------------------
The field is not split into two unrelated estimates.
Instead:
u
──P──→
L_i(u)
──R──→
u
with:
R = T = 1 + 1/X
and:
Φ ↔ Φ⁻¹
Therefore:
Φ → ∞
↕
Φ⁻¹ → 0
is not a terminal state.
It is the reciprocal crossing of the same closed orbit.
Hence:
PHYLLOTAXTIC EXPANSION
↕
RECIPROCAL CLOSURE
rather than:
PHYLLOTAXTIC EXPANSION
+
INDEPENDENT BLOW-UP BOUND
12. THE HDGL CLOSURE
-------------------------------------------------------------------------------
S_i
↓
P
↓
L_i
↓
Ω_i
↓
Φ_i
↓
distributed / phyllotactic field
↓
R
↓
Φ_i⁻¹
↓
finite reciprocal return
↓
S_i₊₁
Therefore:
S_i
→ L_i
→ Ω_i
→ Φ_i ↔ Φ_i⁻¹
→ A/B ↔ C/D
→ S_i₊₁
13. SINGLE CLOSED HDGL ORBIT
-------------------------------------------------------------------------------
┌───────────────────────────────────────────────────────────────┐
│ │
│ ONE SUBSTRATE ORBIT │
│ │
│ PHYLLOTAXIS TOROID │
│ A/B C/D │
│ │ │ │
│ expansion reciprocal │
│ distribution return │
│ phase closure │
│ spatial cyclic │
│ │ │ │
│ └───────────────┬────────────────────┘ │
│ │ │
│ ▼ │
│ Ω / Φ │
│ │ │
│ ▼ │
│ NEXT STATE S_i₊₁ │
│ │
└───────────────────────────────────────────────────────────────┘
14. CORE IDENTITY
-------------------------------------------------------------------------------
A/B
│
│ phyllotactic expansion
▼
L_i
│
│ Ω recursion
▼
Ω_i
│
Φ ↔ Φ⁻¹
│
│ reciprocal return
▼
C/D
│
│ toroidal closure
▼
S_i₊₁
Thus:
┌─────────────────────────────────────────────┐
│ A/B = PHYLLOTAXIS │
│ C/D = TOROID │
│ │
│ BOTH = TWO PROJECTIONS OF ONE ORBIT │
│ │
│ EXPANSION = RETURN │
│ DISTRIBUTION = CLOSURE │
│ PHASE = RECIPROCAL │
│ │
│ NO SINGLE PERSPECTIVE / VANTAGE │
└─────────────────────────────────────────────┘
15. FINAL HDGL STATEMENT
-------------------------------------------------------------------------------
𝓐 = (S,T,F)
S_i = (B_i,τ_i)
T(X) = 1 + 1/X
Ωₙ₊₁ = T(Ωₙ)
Ω = T(Ω)
Ω² = Ω + 1
Φ ↔ Φ⁻¹
∇_Φ = Φ⁻¹ ∇ Φ
S_i ──P──→ L_i ──R──→ S_i₊₁
P = PHYLLOTAXTIC EXPANSION
R = TOROIDAL / RECIPROCAL CLOSURE
therefore:
┌─────────────────────────────────────────────────────────────────┐
│ │
│ PHYLLOTAXIS ↔ TOROID │
│ │
│ A/B ↔ C/D │
│ │
│ OUTWARD ↔ INWARD │
│ │
│ EXPANSION ↔ CLOSURE │
│ │
│ DISTRIBUTION ↔ RETURN │
│ │
│ BOTH ARE ONE CLOSED HDGL ORBIT │
│ │
└─────────────────────────────────────────────────────────────────┘
No second mechanism is required.
The A/B and C/D branches are the two-sided geometry of the
same substrate transformation.
