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docs(diagrams): add 4 remaining SVG figures for §8 HMM + §10 geometry

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- fig_hmm_regime: 3-state Bull/Neutral/Bear HMM state machine + emission table
- fig_viterbi_trellis: Viterbi trellis K=3 T=4 with MAP path highlighted
- fig_std_vs_nat_gradient: standard vs natural gradient property comparison panels
- fig_lie_group_hierarchy: GL/SL/O/SO/Sp/H Lie group tree with finance annotations

All 27 SVGs regenerated; mathematical_foundations.md now 0 remaining ASCII blocks.
_fix_diagrams3.py added for reproducibility.
This commit is contained in:
ThotDjehuty
2026-03-07 13:39:43 +01:00
parent 5d06f0ab57
commit 49be6c7bf7
30 changed files with 8808 additions and 1086 deletions
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docs/source/_fix_diagrams3.py
+141
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@@ -0,0 +1,141 @@
"""Replace the 4 remaining ASCII diagram blocks in mathematical_foundations.md
with {figure} directives pointing to the new SVGs.
"""
import pathlib
MD = pathlib.Path(__file__).parent / "theory" / "mathematical_foundations.md"
text = MD.read_text(encoding="utf-8")
# ── 1. HMM regime state machine → fig_hmm_regime ──────────────────────────
old1 = '''\
```
HMM regime state machine (K = 3)
┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄
A₁₂ → A₂₃ →
┌─────────────┐ ┌─────────────┐ ┌─────────────┐
│ State 1 │──────▶│ State 2 │──────▶│ State 3 │
│ Bull │◀──────│ Neutral │◀──────│ Bear │
└─────────────┘ └─────────────┘ └─────────────┘
← A₂₁ ← A₃₂
Emission B_k(y) = 𝒩(μ_k, σ_k²):
┌────────┬────────┬────────┬──────────────────┐
│ State │ μ │ σ │ Character │
├────────┼────────┼────────┼──────────────────┤
│ Bull │ +0.05 │ 0.12 │ high return, low vol │
│ Neutral│ 0.00 │ 0.18 │ flat, medium vol │
│ Bear │ -0.08 │ 0.35 │ crash, high vol │
└────────┴────────┴────────┴──────────────────┘
(self-transition: A₁₁=0.97, A₂₂=0.97, A₃₃=0.90)
```'''
new1 = '''\
```{figure} ../_static/diagrams/fig_hmm_regime.svg
:align: center
:width: 90%
HMM $K=3$ state machine with Bull / Neutral / Bear regimes and Gaussian emission
parameters. Self-transitions $A_{11}=A_{22}=0.97$, $A_{33}=0.90$.
```'''
# ── 2. Viterbi trellis → fig_viterbi_trellis ───────────────────────────────
old2 = '''\
```
Viterbi trellis (K=3, T=4)
┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄
State t=1 t=2 t=3 t=4
1 ○─────────▶○─────────▶○─────────▶○
╲ ╳
2 ○─────────▶●─────────▶●─────────▶○ ● = MAP path
╲ ╲ ╲
3 ○─────────▶○─────────▶○─────────▶○
δ_t(k) = max_j [δ_{t−1}(j) · A_jk · B_k(y_t)]
ψ_t(k) = argmax_j ← backtrack pointer
Traceback: z_4★ ← z_3★ ← z_2★ ← z_1★ via ψ
```'''
new2 = '''\
```{figure} ../_static/diagrams/fig_viterbi_trellis.svg
:align: center
:width: 82%
Viterbi trellis ($K=3$, $T=4$). Filled nodes mark the MAP (most probable) state
sequence; arrows show transition candidates. Backtracking via $\psi_t(k)$ recovers
$z_1^\star \to z_4^\star$.
```'''
# ── 3. Standard vs natural gradient (text comparison) → fig_std_vs_nat_gradient
old3 = '''\
```
Standard vs natural gradient
┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄
Standard: θ_{k+1} = θ_k − η·∇ℒ Natural: θ_{k+1} = θ_k − η·ℐ(θ)^{−1}∇ℒ
────────────────────────────────────────────
┌────────────────────┐ ┌────────────────────┐
│ Flat ℝᵈ geometry │ │ Riemannian metric ℐ(θ) │
│ Ignores curvature │ │ Adapts to geometry │
│ Slow on ill-cond ℐ │ │ Reparam invariant │
│ O(κ(ℐ)) iters │ │ O(1) on exp families │
└────────────────────┘ └────────────────────┘
On Gaussian / exponential family: ℐ⁻¹∇ℒ = MLE step → 1 iteration!
```'''
new3 = '''\
```{figure} ../_static/diagrams/fig_std_vs_nat_gradient.svg
:align: center
:width: 88%
Standard versus natural gradient: geometric properties. On exponential families
the natural gradient equals the MLE Newton step, achieving convergence in one
iteration.
```'''
# ── 4. Matrix Lie group hierarchy → fig_lie_group_hierarchy ─────────────────
old4 = '''\
```
Matrix Lie group hierarchy
┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄
GL(n,ℝ) ─ all invertible n×n real matrices
│
├──▶ SL(n,ℝ) det = 1
│
├──▶ O(n) RᵀR = I (orthogonal)
│ └─▶ SO(n) det = +1 (pure rotations)
│ ↳ portfolio factor rotation, PCA constraints
│
└──▶ Sp(2n,ℝ) preserves symplectic form ω
↳ Hamiltonian mechanics, PMP §4.2 / §10.4
H(n) Heisenberg ─ upper triangular, 1s on diagonal
↳ path-signature feature maps
```'''
new4 = '''\
```{figure} ../_static/diagrams/fig_lie_group_hierarchy.svg
:align: center
:width: 90%
Matrix Lie group hierarchy: subgroup inclusions and their quantitative-finance
applications. $SO(n)$ underpins PCA factor rotation; $\mathrm{Sp}(2n,\mathbb{R})$
governs Hamiltonian mechanics (PMP §10.4); $H(n)$ drives path-signature features.
```'''
replacements = [(old1, new1), (old2, new2), (old3, new3), (old4, new4)]
for i, (old, new) in enumerate(replacements, 1):
if old in text:
text = text.replace(old, new, 1)
print(f" Block {i}: replaced OK")
else:
print(f" Block {i}: NOT FOUND — check encoding/whitespace")
MD.write_text(text, encoding="utf-8")
print("Done.")
docs/source/_gen_diagrams.py
+223
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@@ -663,6 +663,226 @@ def fig_picard():
save("fig_picard")
# ════════════════════════════════════════════════════════════════════════════
# §8 HMM REGIME STATE MACHINE (K = 3)
# ════════════════════════════════════════════════════════════════════════════
def fig_hmm_regime():
from matplotlib.patches import FancyBboxPatch, FancyArrowPatch
import matplotlib.patheffects as pe
fig, ax = plt.subplots(figsize=(9, 4.2))
ax.set_xlim(0, 9); ax.set_ylim(0, 4); ax.axis("off")
states = [
(1.5, 2.6, "State 1\nBull", C2),
(4.5, 2.6, "State 2\nNeutral", GRAY),
(7.5, 2.6, "State 3\nBear", C3),
]
box_w, box_h = 2.0, 1.1
for (cx, cy, label, col) in states:
fancy = FancyBboxPatch((cx - box_w/2, cy - box_h/2), box_w, box_h,
boxstyle="round,pad=0.08", linewidth=1.6,
edgecolor=col, facecolor=col + "22",
zorder=2)
ax.add_patch(fancy)
ax.text(cx, cy, label, ha="center", va="center", fontsize=10,
fontweight="bold", color=col, zorder=3)
# Forward arrows A₁₂, A₂₃
for x0, x1, label in [(2.5, 3.5, r"$A_{12}$"), (5.5, 6.5, r"$A_{23}$")]:
ax.annotate("", xy=(x1, 2.85), xytext=(x0, 2.85),
arrowprops=dict(arrowstyle="-|>", color=C0, lw=1.5))
ax.text((x0+x1)/2, 2.98, label, ha="center", fontsize=9, color=C0)
# Backward arrows A₂₁, A₃₂
for x0, x1, label in [(3.5, 2.5, r"$A_{21}$"), (6.5, 5.5, r"$A_{32}$")]:
ax.annotate("", xy=(x1, 2.35), xytext=(x0, 2.35),
arrowprops=dict(arrowstyle="-|>", color=C1, lw=1.5))
ax.text((x0+x1)/2, 2.22, label, ha="center", fontsize=9, color=C1)
# Emission table
col_labels = ["State", r"$\mu$", r"$\sigma$", "Character"]
rows = [
["Bull", "+0.05", "0.12", "high return, low vol"],
["Neutral", " 0.00", "0.18", "flat, medium vol"],
["Bear", "−0.08", "0.35", "crash, high vol"],
]
row_colors = [[C2+"33", C2+"33", C2+"33", C2+"33"],
[GRAY+"33", GRAY+"33", GRAY+"33", GRAY+"33"],
[C3+"33", C3+"33", C3+"33", C3+"33"]]
tbl = ax.table(cellText=rows, colLabels=col_labels, loc="bottom",
cellColours=row_colors, bbox=[0.05, 0.0, 0.90, 0.42])
tbl.auto_set_font_size(False); tbl.set_fontsize(9)
for (r, c), cell in tbl.get_celld().items():
cell.set_edgecolor("#cccccc")
if r == 0:
cell.set_facecolor(C0 + "33")
cell.set_text_props(fontweight="bold")
ax.set_title(r"HMM Regime State Machine ($K=3$) — Emission $B_k(y)=\mathcal{N}(\mu_k,\sigma_k^2)$",
fontsize=11, pad=6)
plt.tight_layout()
save("fig_hmm_regime")
# ════════════════════════════════════════════════════════════════════════════
# §8.2 VITERBI TRELLIS (K=3, T=4)
# ════════════════════════════════════════════════════════════════════════════
def fig_viterbi_trellis():
from matplotlib.patches import Circle, FancyArrowPatch
K, T = 3, 4
state_labels = ["1 (Bull)", "2 (Neutral)", "3 (Bear)"]
map_path = {(1, 1), (1, 2)} # state index 1 = "2 (Neutral)" at t=2,3 (0-indexed t)
fig, ax = plt.subplots(figsize=(8, 3.8))
ax.set_xlim(-0.5, T + 0.5); ax.set_ylim(-0.5, K - 0.3); ax.axis("off")
# x-positions: t=1..4 → 0.5, 1.5, 2.5, 3.5
xs = [0.6 * (t + 1) for t in range(T)]
ys = [K - 1 - k for k in range(K)] # top = state 1
# Draw crossing / passing arrows (selective to show crossing)
arrow_kw = dict(arrowstyle="-|>", connectionstyle="arc3,rad=0.0",
color=GRAY, lw=1.1, alpha=0.55)
cross_kw = dict(arrowstyle="-|>", connectionstyle="arc3,rad=0.18",
color=GRAY, lw=1.1, alpha=0.45)
for t in range(T - 1):
for k in range(K):
for k2 in range(K):
rad = 0.0 if k == k2 else (0.18 if k2 > k else -0.18)
col = C0 if (k == 1 and k2 == 1 and t >= 1) else GRAY
alpha = 0.9 if col == C0 else 0.3
ax.annotate("", xy=(xs[t+1], ys[k2]), xytext=(xs[t], ys[k]),
arrowprops=dict(arrowstyle="-|>",
connectionstyle=f"arc3,rad={rad}",
color=col, lw=1.2 if col == C0 else 0.8,
alpha=alpha))
# Draw nodes
r = 0.14
for k in range(K):
for t in range(T):
is_map = (k == 1 and 1 <= t <= 2)
fc = C0 if is_map else "white"
ec = C0 if is_map else GRAY
circ = Circle((xs[t], ys[k]), r, facecolor=fc, edgecolor=ec, lw=1.8, zorder=4)
ax.add_patch(circ)
# Labels on left
for k in range(K):
ax.text(-0.1, ys[k], state_labels[k], ha="right", va="center",
fontsize=9, color=C0 if k == 1 else "black")
# x-axis ticks
for t in range(T):
ax.text(xs[t], -0.35, f"$t={t+1}$", ha="center", va="top", fontsize=9)
# Legend
ax.scatter([], [], color=C0, s=80, label="● MAP (Viterbi) path", zorder=5)
ax.scatter([], [], facecolor="white", edgecolors=GRAY, s=80, label="○ other nodes", zorder=5)
ax.legend(loc="upper right", fontsize=9, framealpha=0.9)
ax.set_title(r"Viterbi Trellis ($K=3$, $T=4$) — $\delta_t(k)=\max_j\,\delta_{t-1}(j)\,A_{jk}\,B_k(y_t)$",
fontsize=11)
plt.tight_layout()
save("fig_viterbi_trellis")
# ════════════════════════════════════════════════════════════════════════════
# §10.2 STANDARD VS NATURAL GRADIENT — PROPERTY COMPARISON
# ════════════════════════════════════════════════════════════════════════════
def fig_std_vs_nat_gradient():
from matplotlib.patches import FancyBboxPatch
fig, axes = plt.subplots(1, 2, figsize=(9, 3.0))
panels = [
("Standard Gradient\n" + r"$\theta_{k+1} = \theta_k - \eta\nabla\mathcal{L}$",
["Flat $\\mathbb{R}^d$ geometry",
"Ignores manifold curvature",
"Slow on ill-conditioned $\\mathcal{I}$",
"$O(\\kappa(\\mathcal{I}))$ iterations"],
C3, C3 + "18"),
("Natural Gradient\n" + r"$\theta_{k+1} = \theta_k - \eta\,\mathcal{I}(\theta)^{-1}\nabla\mathcal{L}$",
["Riemannian metric $\\mathcal{I}(\\theta)$",
"Adapts to manifold geometry",
"Reparametrisation-invariant",
"$O(1)$ on exp. families (MLE step)"],
C2, C2 + "18"),
]
for ax, (title, props, border, bg) in zip(axes, panels):
ax.set_xlim(0, 1); ax.set_ylim(0, 1); ax.axis("off")
fancy = FancyBboxPatch((0.03, 0.04), 0.94, 0.92,
boxstyle="round,pad=0.04", linewidth=2,
edgecolor=border, facecolor=bg)
ax.add_patch(fancy)
ax.text(0.5, 0.87, title, ha="center", va="top", fontsize=10,
fontweight="bold", color=border, transform=ax.transAxes,
multialignment="center")
y = 0.68
for prop in props:
ax.text(0.12, y, "• " + prop, ha="left", va="top", fontsize=9.5,
transform=ax.transAxes, color="#222222")
y -= 0.17
fig.suptitle("Standard vs Natural Gradient — geometric properties", fontsize=11, y=1.02)
plt.tight_layout()
save("fig_std_vs_nat_gradient")
# ════════════════════════════════════════════════════════════════════════════
# §10.3 MATRIX LIE GROUP HIERARCHY
# ════════════════════════════════════════════════════════════════════════════
def fig_lie_group_hierarchy():
from matplotlib.patches import FancyBboxPatch, FancyArrowPatch
fig, ax = plt.subplots(figsize=(9, 4.6))
ax.set_xlim(0, 9); ax.set_ylim(0, 4.6); ax.axis("off")
nodes = {
"GL": (4.5, 4.1, r"$\mathrm{GL}(n,\mathbb{R})$" + "\nall invertible $n\times n$", C0),
"SL": (1.8, 2.85, r"$\mathrm{SL}(n,\mathbb{R})$" + "\n$\det=1$", C2),
"On": (4.5, 2.85, r"$O(n)$" + "\n$R^\top R=I$", C1),
"Sp": (7.2, 2.85, r"$\mathrm{Sp}(2n,\mathbb{R})$" + "\npreserves $\omega$", C2),
"SO": (4.5, 1.55, r"$\mathrm{SO}(n)$" + "\n$\det=+1$ (rotations)", C2),
"Hn": (1.8, 1.55, r"$H(n)$ Heisenberg" + "\nupper triangular", C3),
}
notes = {
"SO": "portfolio factor\nrotation, PCA",
"Sp": "Hamiltonian\nmechanics, PMP",
"Hn": "path-signature\nfeature maps",
}
edges = [("GL","SL"), ("GL","On"), ("GL","Sp"), ("On","SO")]
bw, bh = 2.2, 0.76
for key, (cx, cy, label, col) in nodes.items():
fbp = FancyBboxPatch((cx-bw/2, cy-bh/2), bw, bh,
boxstyle="round,pad=0.07", lw=1.6,
edgecolor=col, facecolor=col+"22", zorder=2)
ax.add_patch(fbp)
ax.text(cx, cy, label, ha="center", va="center", fontsize=8.5,
multialignment="center", color=col, fontweight="bold", zorder=3)
if key in notes:
ax.text(cx + bw/2 + 0.15, cy, notes[key], va="center",
fontsize=7.5, color="#555555", fontstyle="italic")
for src, dst in edges:
sx, sy = nodes[src][0], nodes[src][1]
dx, dy = nodes[dst][0], nodes[dst][1]
ax.annotate("", xy=(dx, dy + bh/2 + 0.04), xytext=(sx, sy - bh/2 - 0.04),
arrowprops=dict(arrowstyle="-|>", color=GRAY, lw=1.4))
ax.set_title("Matrix Lie Group Hierarchy — subgroup inclusions and finance applications",
fontsize=11, pad=5)
plt.tight_layout()
save("fig_lie_group_hierarchy")
# ════════════════════════════════════════════════════════════════════════════
# RUN ALL
# ════════════════════════════════════════════════════════════════════════════
@@ -680,6 +900,9 @@ if __name__ == "__main__":
fig_mcmc_energy, fig_mcmc_trace,
fig_kl_asymmetry, fig_fisher_curvature,
fig_curvatures, fig_natural_gradient,
# new §8 & §10 diagrams
fig_hmm_regime, fig_viterbi_trellis,
fig_std_vs_nat_gradient, fig_lie_group_hierarchy,
]
for fn in funcs:
print(f" {fn.__name__} ... ", end="", flush=True)
docs/source/_static/diagrams/fig_bm_fan.svg
+35 -35
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@@ -6,7 +6,7 @@
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<dc:date>2026-03-07T11:02:30.202001</dc:date>
<dc:date>2026-03-07T13:33:12.904661</dc:date>
<dc:format>image/svg+xml</dc:format>
<dc:creator>
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L 68.455014 -153.98025
L 69.166619 -149.242201
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z
" style="stroke: #2e6be5; stroke-opacity: 0.13"/>
</defs>
<g clip-path="url(#pd553d71edc)">
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<defs>
<path id="m48c35e47ee" d="M 0 0
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@@ -1132,26 +1132,12 @@ $Y_t \mid Z_t=k \sim B_k(y)$.
**State machine diagram ($K=3$ regimes):**
```
HMM regime state machine (K = 3)
┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄
```{figure} ../_static/diagrams/fig_hmm_regime.svg
:align: center
:width: 90%
A₁₂ → A₂₃ →
┌─────────────┐ ┌─────────────┐ ┌─────────────┐
│ State 1 │──────▶│ State 2 │──────▶│ State 3 │
│ Bull │◀──────│ Neutral │◀──────│ Bear │
└─────────────┘ └─────────────┘ └─────────────┘
← A₂₁ ← A₃₂
Emission B_k(y) = 𝒩(μ_k, σ_k²):
┌────────┬────────┬────────┬──────────────────┐
│ State │ μ │ σ │ Character │
├────────┼────────┼────────┼──────────────────┤
│ Bull │ +0.05 │ 0.12 │ high return, low vol │
│ Neutral│ 0.00 │ 0.18 │ flat, medium vol │
│ Bear │ -0.08 │ 0.35 │ crash, high vol │
└────────┴────────┴────────┴──────────────────┘
(self-transition: A₁₁=0.97, A₂₂=0.97, A₃₃=0.90)
HMM $K=3$ state machine with Bull / Neutral / Bear regimes and Gaussian emission
parameters. Self-transitions $A_{11}=A_{22}=0.97$, $A_{33}=0.90$.
```
### 8.2 Baum-Welch (EM)
@@ -1174,22 +1160,13 @@ iteration monotonically increases $\mathcal{L}(\theta)$ by Jensen's inequality.
**Viterbi trellis diagram ($K=3$, $T=4$):**
```
Viterbi trellis (K=3, T=4)
┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄
```{figure} ../_static/diagrams/fig_viterbi_trellis.svg
:align: center
:width: 82%
State t=1 t=2 t=3 t=4
1 ○─────────▶○─────────▶○─────────▶○
╲ ╳
2 ○─────────▶●─────────▶●─────────▶○ ● = MAP path
╲ ╲ ╲
3 ○─────────▶○─────────▶○─────────▶○
δ_t(k) = max_j [δ_{t−1}(j) · A_jk · B_k(y_t)]
ψ_t(k) = argmax_j ← backtrack pointer
Traceback: z_4★ ← z_3★ ← z_2★ ← z_1★ via ψ
Viterbi trellis ($K=3$, $T=4$). Filled nodes mark the MAP (most probable) state
sequence; arrows show transition candidates. Backtracking via $\psi_t(k)$ recovers
$z_1^\star o z_4^\star$.
```
**Viterbi (MAP path):** $\delta_t(k) = \max_j \delta_{t-1}(j)A_{jk} \cdot B_k(y_t)$, $O(TK^2)$.
@@ -1288,21 +1265,13 @@ The statistical manifold $\mathcal{M} = \{p(\cdot;\theta)\}$ carries the
**Standard vs natural gradient:**
```
Standard vs natural gradient
┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄
```{figure} ../_static/diagrams/fig_std_vs_nat_gradient.svg
:align: center
:width: 88%
Standard: θ_{k+1} = θ_k − η·∇ℒ Natural: θ_{k+1} = θ_k − η·ℐ(θ)^{−1}∇ℒ
────────────────────────────────────────────
┌────────────────────┐ ┌────────────────────┐
│ Flat ℝᵈ geometry │ │ Riemannian metric ℐ(θ) │
│ Ignores curvature │ │ Adapts to geometry │
│ Slow on ill-cond ℐ │ │ Reparam invariant │
│ O(κ(ℐ)) iters │ │ O(1) on exp families │
└────────────────────┘ └────────────────────┘
On Gaussian / exponential family: ℐ⁻¹∇ℒ = MLE step → 1 iteration!
Standard versus natural gradient: geometric properties. On exponential families
the natural gradient equals the MLE Newton step, achieving convergence in one
iteration.
```
**Natural gradient (Amari 1998):**
@@ -1344,23 +1313,13 @@ linearises the group at the identity.
**Matrix Lie group hierarchy:**
```
Matrix Lie group hierarchy
┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄
```{figure} ../_static/diagrams/fig_lie_group_hierarchy.svg
:align: center
:width: 90%
GL(n,ℝ) ─ all invertible n×n real matrices
│
├──▶ SL(n,ℝ) det = 1
│
├──▶ O(n) RᵀR = I (orthogonal)
│ └─▶ SO(n) det = +1 (pure rotations)
│ ↳ portfolio factor rotation, PCA constraints
│
└──▶ Sp(2n,ℝ) preserves symplectic form ω
↳ Hamiltonian mechanics, PMP §4.2 / §10.4
H(n) Heisenberg ─ upper triangular, 1s on diagonal
↳ path-signature feature maps
Matrix Lie group hierarchy: subgroup inclusions and their quantitative-finance
applications. $SO(n)$ underpins PCA factor rotation; $\mathrm{Sp}(2n,\mathbb{R})$
governs Hamiltonian mechanics (PMP §10.4); $H(n)$ drives path-signature features.
```
**Left-invariant control system on $G$:**