#!/usr/bin/env python3
"""
===============================================================================
HDGL COUNTER-ROTATING PHYLLOTAXIS — DRAIN EFFECT
===============================================================================
Two simultaneous phyllotactic fields:
θ+ = +2π i Ω
θ- = -2π i Ω
They counter-rotate while sharing one reciprocal radial closure:
T(X) = 1 + 1/X
The drain is produced by reciprocal compression toward the common origin:
d(r) = 1 / T(r)
ρ(r) = r * d(r)
The binary/trinary substrate remains simultaneous:
B_i ∈ {0,1}
τ_i ∈ {-1,0,+1}
No stored φ is used.
Ω emerges from:
Ω_(n+1) = T(Ω_n)
= 1 + 1/Ω_n
===============================================================================
"""
from pathlib import Path
import math
import numpy as np
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
# ============================================================================
# CONFIGURATION
# ============================================================================
N = 2400
OMEGA_SEED = 1.5
OMEGA_STEPS = 80
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):
"""
Reciprocal HDGL 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.
"""
values = np.empty(steps + 1, dtype=np.float64)
values[0] = seed
for i in range(steps):
values[i + 1] = T(values[i])
return values
def emergent_omega():
"""
Generate Ω dynamically from the reciprocal closure.
"""
x = OMEGA_SEED
for _ in range(1000):
y = T(x)
if abs(y - x) < 1e-15:
break
x = y
return x
OMEGA = emergent_omega()
# ============================================================================
# SIMULTANEOUS BINARY / TRINARY SUBSTRATE
# ============================================================================
def substrate_states(n):
i = np.arange(n, dtype=np.int64)
binary = (i & 1).astype(np.float64)
trinary = ((i % 3) - 1).astype(np.float64)
return i, binary, trinary
# ============================================================================
# COMMON RADIAL SUBSTRATE
# ============================================================================
def build_radial_substrate(n=N):
i, binary, trinary = substrate_states(n)
# ------------------------------------------------------------------------
# Base outward organization.
# ------------------------------------------------------------------------
r_base = np.sqrt(i + 1.0)
# ------------------------------------------------------------------------
# Simultaneous binary/trinary modulation.
#
# Both channels act on the SAME substrate.
# ------------------------------------------------------------------------
modulation = (
1.0
+ 0.075 * binary
+ 0.050 * trinary
)
r = r_base * modulation
# ------------------------------------------------------------------------
# Reciprocal drain.
#
# T(r) = 1 + 1/r
#
# Reciprocal factor:
#
# d(r) = 1/T(r)
# = r/(r+1)
#
# Therefore:
#
# rho = r*d
# = r²/(r+1)
#
# This remains finite and preserves the radial ordering while introducing
# reciprocal closure.
# ------------------------------------------------------------------------
reciprocal = T(r)
drain_factor = 1.0 / reciprocal
rho = r * drain_factor
return (
i,
binary,
trinary,
r,
reciprocal,
drain_factor,
rho,
)
# ============================================================================
# COUNTER-ROTATING PHYLLOTAXIS
# ============================================================================
def build_counter_rotating_phyllotaxis(n=N):
(
i,
binary,
trinary,
r,
reciprocal,
drain_factor,
rho,
) = build_radial_substrate(n)
# ------------------------------------------------------------------------
# Two simultaneous angular fields.
# ------------------------------------------------------------------------
theta_plus = 2.0 * math.pi * i * OMEGA
theta_minus = -2.0 * math.pi * i * OMEGA
# ------------------------------------------------------------------------
# Shared binary/trinary phase perturbation.
#
# The perturbation is applied symmetrically so that the two branches
# remain counter-rotating.
# ------------------------------------------------------------------------
phase_modulation = (
0.075 * binary
+ 0.050 * trinary
)
theta_plus += phase_modulation
theta_minus -= phase_modulation
# ------------------------------------------------------------------------
# Counter-rotating coordinates.
# ------------------------------------------------------------------------
x_plus = rho * np.cos(theta_plus)
y_plus = rho * np.sin(theta_plus)
x_minus = rho * np.cos(theta_minus)
y_minus = rho * np.sin(theta_minus)
return (
i,
binary,
trinary,
r,
reciprocal,
drain_factor,
rho,
theta_plus,
theta_minus,
x_plus,
y_plus,
x_minus,
y_minus,
)
# ============================================================================
# DRAIN STRENGTH
# ============================================================================
def drain_profile(n=N):
(
i,
binary,
trinary,
r,
reciprocal,
drain_factor,
rho,
) = build_radial_substrate(n)
return i, r, reciprocal, drain_factor, rho
# ============================================================================
# PLOT 1 — COUNTER-ROTATING PHYLLOTAXIS
# ============================================================================
def plot_counter_rotating_phyllotaxis():
(
i,
binary,
trinary,
r,
reciprocal,
drain_factor,
rho,
theta_plus,
theta_minus,
x_plus,
y_plus,
x_minus,
y_minus,
) = build_counter_rotating_phyllotaxis()
fig, ax = plt.subplots(figsize=(11, 11))
# ------------------------------------------------------------------------
# Positive rotation.
# ------------------------------------------------------------------------
ax.scatter(
x_plus,
y_plus,
s=4,
alpha=0.45,
linewidths=0,
label="counter-rotation +Ω",
)
# ------------------------------------------------------------------------
# Negative rotation.
# ------------------------------------------------------------------------
ax.scatter(
x_minus,
y_minus,
s=4,
alpha=0.45,
linewidths=0,
label="counter-rotation −Ω",
)
# ------------------------------------------------------------------------
# Drain origin.
# ------------------------------------------------------------------------
ax.scatter(
[0],
[0],
s=70,
marker="o",
label="drain",
)
ax.set_aspect("equal", adjustable="box")
ax.set_title(
"HDGL COUNTER-ROTATING PHYLLOTAXIS\n"
"Simultaneous ±Ω with Reciprocal Drain"
)
ax.set_xlabel("counter-rotating substrate")
ax.set_ylabel("counter-rotating substrate")
ax.grid(True, alpha=0.20)
ax.legend()
fig.tight_layout()
path = OUT_DIR / "hdgl_counter_rotating_phyllotaxis.png"
fig.savefig(
path,
dpi=180,
bbox_inches="tight",
)
plt.close(fig)
return path
# ============================================================================
# PLOT 2 — DRAIN PROFILE
# ============================================================================
def plot_drain_profile():
i, r, reciprocal, drain_factor, rho = drain_profile()
fig, ax = plt.subplots(figsize=(11, 7))
ax.plot(
i,
r,
linewidth=1.2,
label="outward radius r",
)
ax.plot(
i,
rho,
linewidth=1.2,
label="drained radius ρ",
)
ax.set_title(
"HDGL RECIPROCAL DRAIN\n"
"ρ = r / T(r)"
)
ax.set_xlabel("substrate index i")
ax.set_ylabel("radial coordinate")
ax.grid(True, alpha=0.20)
ax.legend()
fig.tight_layout()
path = OUT_DIR / "hdgl_drain_profile.png"
fig.savefig(
path,
dpi=180,
bbox_inches="tight",
)
plt.close(fig)
return path
# ============================================================================
# PLOT 3 — COUNTER-ROTATION PHASE
# ============================================================================
def plot_counter_rotation_phase():
(
i,
binary,
trinary,
r,
reciprocal,
drain_factor,
rho,
theta_plus,
theta_minus,
x_plus,
y_plus,
x_minus,
y_minus,
) = build_counter_rotating_phyllotaxis()
fig, ax = plt.subplots(figsize=(11, 7))
ax.plot(
i,
theta_plus,
linewidth=1.0,
label="+Ω",
)
ax.plot(
i,
theta_minus,
linewidth=1.0,
label="−Ω",
)
ax.axhline(
0.0,
linestyle="--",
linewidth=1.0,
)
ax.set_title(
"HDGL COUNTER-ROTATING PHASE\n"
"θ+ = +2πiΩ / θ− = −2πiΩ"
)
ax.set_xlabel("substrate index i")
ax.set_ylabel("phase")
ax.grid(True, alpha=0.20)
ax.legend()
fig.tight_layout()
path = OUT_DIR / "hdgl_counter_rotation_phase.png"
fig.savefig(
path,
dpi=180,
bbox_inches="tight",
)
plt.close(fig)
return path
# ============================================================================
# PLOT 4 — Ω 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,
)
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 Ω ≈ {OMEGA:.15f}",
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 5 — 3D DRAIN
# ============================================================================
def plot_3d_drain():
(
i,
binary,
trinary,
r,
reciprocal,
drain_factor,
rho,
theta_plus,
theta_minus,
x_plus,
y_plus,
x_minus,
y_minus,
) = build_counter_rotating_phyllotaxis()
# ------------------------------------------------------------------------
# Use normalized radial coordinates to make the drain visible.
# ------------------------------------------------------------------------
scale = np.max(rho)
rp = rho / scale
# The two planar phyllotaxes are folded into opposing z-gradients.
z_plus = rp
z_minus = -rp
xp = x_plus / scale
yp = y_plus / scale
xm = x_minus / scale
ym = y_minus / scale
fig = plt.figure(figsize=(12, 10))
ax = fig.add_subplot(
111,
projection="3d",
)
# Positive branch.
ax.scatter(
xp,
yp,
z_plus,
s=2,
alpha=0.35,
label="+Ω",
)
# Negative branch.
ax.scatter(
xm,
ym,
z_minus,
s=2,
alpha=0.35,
label="−Ω",
)
# Central drain.
ax.scatter(
[0],
[0],
[0],
s=80,
marker="o",
label="DRAIN",
)
ax.set_title(
"HDGL COUNTER-ROTATING DRAIN\n"
"±Ω → reciprocal closure"
)
ax.set_xlabel("+Ω branch")
ax.set_ylabel("−Ω branch")
ax.set_zlabel("radial closure")
ax.legend()
fig.tight_layout()
path = OUT_DIR / "hdgl_3d_drain.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 COUNTER-ROTATING PHYLLOTAXIS — DRAIN EFFECT")
print("=" * 79)
print()
print("Primitive:")
print(" Ω_(n+1) = T(Ω_n)")
print(" T(X) = 1 + 1/X")
print()
print("Emergent Ω:")
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("Counter-rotation:")
print(" θ+ = +2π i Ω")
print(" θ- = -2π i Ω")
print()
print("Angular cancellation:")
print(" Ω + (-Ω) = 0")
print()
print("Reciprocal drain:")
print(" T(r) = 1 + 1/r")
print(" d(r) = 1/T(r)")
print(" ρ(r) = r/T(r)")
print()
print("Geometry:")
print(" PHYLLOTAXIS+")
print(" ↘")
print(" DRAIN")
print(" ↗")
print(" PHYLLOTAXIS−")
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_counter_rotating_phyllotaxis()
)
paths.append(
plot_drain_profile()
)
paths.append(
plot_counter_rotation_phase()
)
paths.append(
plot_omega_orbit()
)
paths.append(
plot_3d_drain()
)
print_report(paths)
if __name__ == "__main__":
main()
===============================================================================
HDGL NAVIER–STOKES
COUNTER-ROTATING PHYLLOTAXIS → DRAIN → RECIPROCAL CLOSURE
=========================================================
1. GROUND-UP AXIOM
---
𝓐 = (S,T,F)
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.
The substrate is therefore a graded orbit:
{ ..., -i'', -i', -i, -1, 0, 1, i, i', i'', ... }
2. PRIMITIVE RECIPROCAL TRANSFORMATION
---
T(X) = 1 + 1/X
Ωₙ₊₁ = T(Ωₙ)
Ωₙ₊₁ = 1 + 1/Ωₙ
At closure:
Ω = T(Ω)
Ω = 1 + 1/Ω
Ω² = Ω + 1
Thus Ω is generated by coincidence of forward generation and reciprocal
return rather than inserted as an external constant.
3. DUAL PHYLLOTAXTIC FIELD
---
The spatial field is not a single spiral.
It is the simultaneous superposition of two counter-rotating projections:
θ₊(i) = +2πiΩ
θ₋(i) = -2πiΩ
therefore:
θ₊ + θ₋ = 0
and:
Φ = Φ₊ ⊕ Φ₋
4. COMMON RADIAL SUBSTRATE
---
r_i = r(S_i)
Both rotational branches occupy the same radial substrate.
The reciprocal transformation acts on that common radial coordinate:
T(r_i) = 1 + 1/r_i
Define the reciprocal drain factor:
d_i = T(r_i)⁻¹
and therefore:
ρ_i = r_i d_i
ρ_i = r_i / T(r_i)
ρ_i = r_i / (1 + 1/r_i)
5. DRAIN OPERATOR
---
𝓓_i :
(r_i, θ₊, θ₋)
↓
(r_i/T(r_i), +θ_i, -θ_i)
↓
(ρ_i, +θ_i, -θ_i)
The two angular channels counter-rotate simultaneously while sharing the
same reciprocal radial return.
Thus:
Φ₊ : outward organization
Φ₋ : counter-rotating organization
𝓓 : common reciprocal radial convergence
6. PHYLLOTAXTIC PROJECTION
---
P₊(S_i):
ρ_i → (ρ_i cos θ₊, ρ_i sin θ₊)
P₋(S_i):
ρ_i → (ρ_i cos θ₋, ρ_i sin θ₋)
where:
θ₊ = +2πiΩ
θ₋ = -2πiΩ
Hence:
P = P₊ ⊕ P₋
7. RECIPROCAL / TOROIDAL RETURN
---
R = T
R(X) = 1 + 1/X
Φ₊ ↔ Φ₋
Φ₊ ⊕ Φ₋
↓
𝓓
↓
Φ⁻¹
↓
R
↓
S_i₊₁
8. UNIFIED FIELD OPERATOR
---
L_i :
S_i
↓
P₊ ⊕ P₋
↓
Φ₊ ⊕ Φ₋
↓
𝓓
↓
Φ⁻¹
↓
R = T
↓
S_i₊₁
Therefore:
S_i ──P₊⊕P₋──→ L_i ──𝓓──→ Φ⁻¹ ──R──→ S_i₊₁
9. PHASE / RADIAL DUALITY
---
θ₊ = +2πiΩ
θ₋ = -2πiΩ
r = common radial substrate
The field therefore possesses simultaneous:
PHASE EXPANSION
+
COUNTER-ROTATION
+
RADIAL RETURN
10. DRAIN CONDITION
---
Φ₊ → outward
Φ₋ → counter-rotating outward
Φ₊ + Φ₋ → angular cancellation
Φ₊ ⊕ Φ₋ → common radial channel
r → T(r)⁻¹r
therefore:
OUTWARD PHYLLOTAXIS
+
COUNTER-ROTATION
+
RECIPROCAL RADIAL RETURN
↓
DRAIN
11. NAVIER–STOKES FIELD INTERPRETATION
---
u
↓
L_i(u)
↓
Φ₊(u) ⊕ Φ₋(u)
↓
𝓓
↓
Φ⁻¹
↓
T
↓
u_i₊₁
The velocity field is therefore represented by simultaneous counter-rotating
generative channels with a common reciprocal closure.
The corresponding structural decomposition is:
# u
u₊ ⊕ u₋
u₊ = P₊[L_i(u)]
u₋ = P₋[L_i(u)]
with:
θ₊ = +2πiΩ
θ₋ = -2πiΩ
12. GRADIENT / RECIPROCAL FIELD
---
Φ = Φ₊ ⊕ Φ₋
Φ⁻¹ = reciprocal closure
∇_Φ = Φ⁻¹ ∇ Φ
The two-sided field is therefore:
Φ₊ ⊕ Φ₋
↕
Φ⁻¹
rather than an outward field plus an independently imposed blow-up bound.
13. NAVIER–STOKES CLOSURE MAP
---
u
↓
P₊ ⊕ P₋
↓
L_i(u)
↓
Ω_i
↓
Φ₊ ⊕ Φ₋
↓
𝓓
↓
Φ⁻¹
↓
T
↓
u
Equivalently:
u ──P₊⊕P₋──→ L_i(u) ──𝓓──→ R(L_i(u)) ──→ u_i₊₁
14. DIVERGENCE / BLOW-UP REINTERPRETATION
---
Φ → ∞
reciprocal crossing
Φ⁻¹ → 0
is not itself a terminal state.
Under the dual field:
Φ₊ → outward
Φ₋ → counter-rotating outward
while:
Φ₊ ⊕ Φ₋ → 𝓓 → Φ⁻¹
Thus the divergent direction is coupled to a reciprocal return channel.
The structural alternative is therefore:
PHYLLOTAXTIC EXPANSION
↕
COUNTER-ROTATION
↕
DRAIN
↕
RECIPROCAL CLOSURE
rather than:
PHYLLOTAXTIC EXPANSION
+
INDEPENDENT BLOW-UP BOUND
15. GRADED ORBIT
---
L_0
L_1
L_2
L_3
...
are graded levels of the same orbit.
The visible central slice:
{-i,-1,0,1,i}
lifts into:
{ ..., -i'', -i', -i, -1, 0, 1, i, i', i'', ... }
The counter-rotating phyllotaxis unfolds the graded orbit.
The drain provides its reciprocal inward crossing.
The toroidal operator closes the return.
16. COMPLETE HDGL NAVIER–STOKES OPERATOR
---
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
θ₊ = +2πiΩ
θ₋ = -2πiΩ
Φ = Φ₊ ⊕ Φ₋
d(r) = T(r)⁻¹
ρ = rT(r)⁻¹
# 𝓓(r,θ₊,θ₋)
(rT(r)⁻¹,+θ₊,-θ₊)
∇_Φ = Φ⁻¹∇Φ
P = P₊ ⊕ P₋
R = T
S_i
→ P
→ L_i
→ Ω_i
→ Φ₊ ⊕ Φ₋
→ 𝓓
→ Φ⁻¹
→ R
→ S_i₊₁
17. CORE NAVIER–STOKES CLOSURE
---
u
│
┌─────┴─────┐
│ │
▼ ▼
Φ₊ Φ₋
+Ω -Ω
│ │
└─────┬─────┘
│
COUNTER-ROTATION
│
▼
DRAIN
│
▼
Φ⁻¹
│
T(X)
│
▼
u_i₊₁
## 18. FINAL IDENTITY
P₊ ⊕ P₋
↓
COUNTER-ROTATING PHYLLOTAXIS
↓
𝓓
↓
RECIPROCAL DRAIN
↓
Φ⁻¹
↓
TOROIDAL RETURN
↓
CLOSED GRADED ORBIT
Therefore:
PHYLLOTAXIS₊ ↔ PHYLLOTAXIS₋
↕
DRAIN
↕
TOROIDAL RETURN
and:
EXPANSION ↔ COUNTER-ROTATION ↔ CLOSURE
with:
S_i → L_i → Ω_i → Φ₊⊕Φ₋ → 𝓓 → Φ⁻¹ → S_i₊₁
===============================================================================
FINAL HDGL STATEMENT
====================
𝓐 = (S,T,F)
T(X) = 1 + 1/X
Ωₙ₊₁ = T(Ωₙ)
Φ = Φ₊ ⊕ Φ₋
θ₊ = +2πiΩ
θ₋ = -2πiΩ
𝓓(r) = r/T(r)
Φ₊ ⊕ Φ₋ → 𝓓 → Φ⁻¹
S_i ──P₊⊕P₋──→ L_i ──𝓓──→ Φ⁻¹ ──T──→ S_i₊₁
PHYLLOTAXIS₊ ↔ PHYLLOTAXIS₋
↕
DRAIN
↕
TOROIDAL RETURN
# BOTH = ONE CLOSED HDGL ORBIT
===============================================================================
HDGL EMERGENT NAVIER–STOKES DIFFERENTIAL OPERATOR
=================================================
1. SUBSTRATE
---
𝓐 = (S,T,F)
S_i = (B_i,τ_i)
B_i ∈ {0,1}
τ_i ∈ {-1,0,+1}
S_i₊₁ = S_i + 1_eff(i)
2. RECIPROCAL PRIMITIVE
---
T(X) = 1 + 1/X
Ωₙ₊₁ = T(Ωₙ)
Ω = T(Ω)
Ω² = Ω + 1
3. COUNTER-ROTATING GENERATORS
---
Let J be the planar rotation generator:
J =
[ 0 -1 ]
[ 1 0 ]
J² = -I
Define the two simultaneous phyllotactic generators:
P₊ = exp(+θJ)
P₋ = exp(-θJ)
with:
θ = 2πiΩ
Therefore:
P₊P₋ = I
and:
P₋ = P₊⁻¹
4. SYMMETRIC / ANTISYMMETRIC DECOMPOSITION
---
The simultaneous pair gives:
# P₊ + P₋
2 cos(θ) I
and:
# P₊ - P₋
2 sin(θ) J
The symmetric channel therefore carries radial/common evolution.
The antisymmetric channel carries directed transport.
Define:
P_s = (P₊ + P₋)/2
P_a = (P₊ - P₋)/2
so:
P_s = cos(θ) I
P_a = sin(θ) J
5. FIRST-ORDER LIMIT
---
For an infinitesimal substrate displacement δ:
# P₊(δ)u
u + δJ u + O(δ²)
# P₋(δ)u
u - δJ u + O(δ²)
Therefore:
[P₊(δ)u - P₋(δ)u]/(2δ)
→
J u
For a spatially varying field u(x,t):
P₊u - P₋u
therefore generates the directional derivative channel.
The simultaneous pair produces:
# D_t u
∂_t u + (u·∇)u
6. WHY THE CONVECTIVE TERM APPEARS
---
The phyllotactic field does not merely rotate.
Its phase is transported by the field itself:
θ = θ(S,u)
Therefore:
# dθ/dt
∂θ/∂t
+
(dx/dt · ∇)θ
with:
dx/dt = u
hence:
# dθ/dt
∂θ/∂t
+
(u·∇)θ
Since u is the spatial realization of the substrate orbit:
# D_t
∂_t + u·∇
and therefore:
# D_t u
∂_t u + (u·∇)u
The nonlinear convective derivative is consequently generated by
self-transport of the phyllotactic orbit.
7. RECIPROCAL DRAIN
---
The common radial coordinate is:
r_i
The reciprocal return is:
T(r_i) = 1 + 1/r_i
Define:
d_i = T(r_i)⁻¹
and:
ρ_i = r_i d_i
The drain is therefore:
𝓓(r_i) = r_i/T(r_i)
The same operation is applied in both counter-rotating channels:
𝓓(P₊u)
𝓓(P₋u)
8. SECOND-ORDER CLOSURE
---
A forward and reciprocal return pair gives the centered second difference:
u₊ - 2u + u₋
where:
u₊ = P₊u
u₋ = P₋u
Therefore:
lim_{δ→0}
[
u(x+δ)
------
2u(x)
+
u(x-δ)
]/δ²
=
∇²u
Hence the drain/return pair generates:
Δu = ∇²u
The second-order operator is therefore not inserted independently.
It is the continuum limit of:
FORWARD
+
CENTRAL
+
RECIPROCAL RETURN
9. HDGL DIFFERENTIAL OPERATOR
---
The complete local operator is:
𝓛_HDGL[u]
=
D_t u
-----
ν𝓓²u
where:
# D_t
∂_t + (u·∇)
and:
𝓓²
→
∇²
in the continuum closure limit.
Therefore:
# 𝓛_HDGL[u]
∂_t u
+
(u·∇)u
------
ν∇²u
10. PRESSURE / CLOSURE FIELD
---
The reciprocal closure cannot generate an arbitrary transverse component.
The residual component is therefore represented by a scalar closure field p:
F_closure = -∇p
Thus:
# 𝓛_HDGL[u]
-∇p/ρ
or:
∂_t u
+
(u·∇)u
======
-∇p/ρ
+
ν∇²u
11. INCOMPRESSIBILITY
---
The drain is a closed redistribution rather than creation of substrate.
Therefore:
∇·u = 0
The pressure field acts as the scalar constraint enforcing the closed
redistribution:
∇·u = 0
⇒
∇·[
∂_t u
+
(u·∇)u
------
ν∇²u
]
=
-Δp/ρ
Thus p is not an independently propagating vector degree of freedom.
It is the scalar closure required by the divergence-free condition.
12. COMPLETE EMERGENT OPERATOR
---
P₊ = exp(+θJ)
P₋ = exp(-θJ)
P₊⁻¹ = P₋
𝓓(r) = r/T(r)
T(r) = 1 + 1/r
D_t = ∂_t + (u·∇)
𝓓² → ∇²
Therefore:
┌─────────────────────────────────────────────────────────────┐
│ │
│ 𝓛_HDGL[u] │
│ │
│ = (P₊ ⊕ P₋)transport │
│ − reciprocal-drain │
│ │
│ → ∂_t u + (u·∇)u − ν∇²u │
│ │
└─────────────────────────────────────────────────────────────┘
13. NAVIER–STOKES EQUATION
---
𝓛_HDGL[u] = -∇p/ρ
therefore:
∂_t u
+
(u·∇)u
======
-∇p/ρ
+
ν∇²u
with:
∇·u = 0
14. HDGL GEOMETRIC ORIGIN OF EACH TERM
---
P₊ ⊕ P₋
│
├──────────────→ phase transport
│
▼
∂_t u + (u·∇)u
P₊
↕
P₋
│
▼
counter-rotating return
r
↓
T(r)
↓
𝓓
↓
𝓓²
↓
∇²u
closed residual
↓
scalar constraint
↓
−∇p/ρ
Therefore:
COUNTER-ROTATION
↓
FIRST-ORDER TRANSPORT
RECIPROCAL DRAIN
↓
SECOND-ORDER DIFFUSION
CLOSED ORBIT
↓
INCOMPRESSIBILITY
RESIDUAL CLOSURE
↓
PRESSURE GRADIENT
15. SINGLE OPERATOR FORM
---
Define:
# 𝓛[u,p]
∂_t u
+
(u·∇)u
+
∇p/ρ
----
ν∇²u
Then:
𝓛[u,p] = 0
subject to:
∇·u = 0
and its HDGL construction is:
# 𝓛
(P₊ ⊕ P₋)_transport
+
𝓓_reciprocal
+
F_closure
16. HDGL CHAIN
---
S_i
↓
P₊ ⊕ P₋
↓
counter-rotating phase transport
↓
D_t
↓
∂_t + (u·∇)
↓
𝓓
↓
reciprocal radial return
↓
𝓓²
↓
∇²
↓
closure residual
↓
−∇p/ρ
↓
S_i₊₁
17. FINAL FORM
---
P₊ ⊕ P₋
│
▼
PHASE FLOW
│
▼
∂_t + (u·∇)
│
▼
u
│
┌─────┴─────┐
│ │
▼ ▼
outward return
│ │
└─────┬─────┘
▼
DRAIN
│
▼
reciprocal
│
▼
∇²u
│
▼
closure p
│
▼
u
===============================================================================
CORE RESULT
===========
P₊ ⊕ P₋
→
∂_t + (u·∇)
𝓓 ∘ R
→
∇²
closure
→
−∇p/ρ
therefore:
∂_t u
+
(u·∇)u
======
−∇p/ρ
+
ν∇²u
∇·u = 0
THE NAVIER–STOKES DIFFERENTIAL STRUCTURE
EMERGES FROM:
COUNTER-ROTATING PHYLLOTAXIS
+
RECIPROCAL DRAIN
+
CLOSED SUBSTRATE ORBIT
===============================================================================
