Chapter 93

Topological Field Theory from S²

Introduction

The preceding chapters derived quantum mechanics, quantum information, and quantum thermodynamics from the geometry of $S^2$. This chapter derives topological quantum field theory from the same geometry, exploiting a structural separation that becomes visible only in polar field coordinates: the measure $du\,d\phi$ is flat Lebesgue, while the topology is carried by the constant monopole field strength $F_{u\phi} = 1/2$ and the boundary conditions on the polar rectangle $\mathcal{R} = [-1,+1] \times [0,2\pi)$.

Topological field theories are characterized by partition functions and observables that depend only on global (topological) properties, not on local geometry. In the polar formulation of TMT, this property is manifest: the flat measure $du\,d\phi$ has no geometric content, and the constant field strength $F_{u\phi} = 1/2$ is independent of position on $\mathcal{R}$. Topological invariants therefore reduce to elementary integrals over flat rectangles.

The central results are:

The Chern-Simons invariant on $S^2$ is a constant-integrand rectangle integral.
Topological invariants on the $N$-particle product space $(S^2)^N$ factorize trivially because flat measure respects the product structure.
Multi-monopole configurations have exact partition functions from polynomial integration on $[-1,+1]$.
The connection between TQFT and the concentration-of-measure results of Chapter 91: topological quantities are insensitive to the curved metric that drives concentration, while dynamical quantities are sensitive to it.

Scaffolding Interpretation

Scaffolding note: The polar field variable $u = \cos\theta$ is a coordinate choice, not a new physical assumption. The topological results in this chapter are coordinate-independent; the polar form makes their computation trivially elementary.

Chern-Simons Theory on the Polar Rectangle

The Chern Number as Rectangle Area

Theorem 93.1 (Chern Number from Constant Field on Flat Rectangle)

The first Chern number of the monopole bundle on $S^2$ is:

$$ c_1 = \frac{1}{2\pi}\int_{\mathcal{R}} F_{u\phi}\,du\,d\phi = \frac{1}{2\pi} \cdot \frac{1}{2} \cdot 2 \cdot 2\pi = 1 $$ (93.1)

where $F_{u\phi} = 1/2$ is constant, $\int_{-1}^{+1} du = 2$ (THROUGH range), and $\int_0^{2\pi} d\phi = 2\pi$ (AROUND period).

Proof.

The monopole field strength in polar coordinates is $F_{u\phi} = n/2$ for monopole charge $n$ (Chapter 60a, Key Result #7). For $n = 1$:

$$\begin{aligned} c_1 &= \frac{1}{2\pi}\int_{S^2} F = \frac{1}{2\pi}\int_0^{2\pi} d\phi \int_{-1}^{+1} du \; F_{u\phi} \\ &= \frac{1}{2\pi} \cdot 2\pi \cdot 2 \cdot \frac{1}{2} = 1 \end{aligned}$$ (93.18)

This is the simplest possible topological computation: a constant times a rectangle area. In spherical coordinates, the same integral requires $F_{\theta\phi} = \frac{1}{2}\sin\theta$ and $\int \sin\theta\,d\theta$, obscuring the constant nature of the integrand.

For general monopole charge $n$:

$$ c_1(n) = \frac{1}{2\pi} \cdot \frac{n}{2} \cdot 2 \cdot 2\pi = n $$ (93.2)

The Chern number is the monopole charge, computed as (constant field) $\times$ (rectangle area) $/ (2\pi)$. □ □

Property	Spherical $(\theta, \phi)$	Polar $(u, \phi)$
Field strength	$F_{\theta\phi} = \frac{n}{2}\sin\theta$ (variable)	$F_{u\phi} = \frac{n}{2}$ (constant)
Measure	$\sin\theta\,d\theta\,d\phi$	$du\,d\phi$ (flat)
Integrand	$\frac{n}{2}\sin\theta \cdot \sin\theta\,d\theta\,d\phi$	$\frac{n}{2}\,du\,d\phi$ (constant $\times$ flat)
Chern number	$\frac{n}{2\pi}\int \sin^2\theta\,d\theta\,d\phi$	$\frac{n}{2\pi} \times 2 \times 2\pi = n$
Physical insight	Topology hidden in $\sin^2\theta$ integral	Topology = constant $\times$ flat area

Chern-Simons Functional on $\mathcal{R}$

The Chern-Simons 1-form on $S^2$ in polar coordinates is particularly simple. Using the north-patch connection $A^{(N)}_\phi = \frac{1}{2}(1-u)$ (Chapter 60a, Key Result #6):

$$ \text{CS}[A] = \int_\gamma A^{(N)}_\phi\,d\phi = \frac{1}{2}(1-u_0) \cdot 2\pi = \pi(1 - u_0) $$ (93.3)

for a horizontal loop at THROUGH position $u_0$. This is linear in $u_0$: the Chern-Simons value is the THROUGH coordinate itself (up to a constant and the AROUND period).

The total Chern-Simons invariant, integrating over all horizontal loops:

$$ \int_{-1}^{+1} \text{CS}(u)\,du = \int_{-1}^{+1} \pi(1-u)\,du = \pi \left[u - \frac{u^2}{2}\right]_{-1}^{+1} = 2\pi $$ (93.4)

This is the total monopole flux (Key Result #5), computed as a polynomial integral on $[-1,+1]$.

Topological Invariants on $(S^2)^N$

Factorization from Flat Product Measure

Theorem 93.2 (Topological Factorization on Product Rectangles)

For $N$ independent monopoles on $(S^2)^N$, topological invariants factorize as products of single-rectangle invariants:

$$ \mathcal{Z}_{\text{top}}[(S^2)^N] = \prod_{i=1}^{N} \mathcal{Z}_{\text{top}}[S^2_i] = \left(\mathcal{Z}_{\text{top}}[S^2]\right)^N $$ (93.5)

Proof.

The topological partition function on $(S^2)^N$ is:

$$ \mathcal{Z}_{\text{top}} = \int_{(S^2)^N} \exp\!\left(ik \sum_{i=1}^{N} \text{CS}[A_i]\right) \prod_{i=1}^{N} \mathcal{D}A_i $$ (93.6)

In polar coordinates, the path integral measure $\mathcal{D}A_i$ for each $S^2_i$ factor is built from the flat Lebesgue measure $du_i\,d\phi_i$ on $\mathcal{R}_i$. The Chern-Simons functional $\text{CS}[A_i] = \pi(1 - u_i)$ depends only on the $i$-th rectangle's THROUGH coordinate. Therefore:

Step 1: The exponential factorizes: $\exp(ik\sum_i \text{CS}_i) = \prod_i \exp(ik\,\text{CS}_i)$.

Step 2: The flat product measure factorizes: $\prod_i du_i\,d\phi_i = d\mu_{\text{flat}}(\mathcal{R}^N)$.

Step 3: Each factor integrates independently over its own flat rectangle $\mathcal{R}_i$.

The factorization is exact because: (a) the flat measure has no cross-terms between rectangles, (b) the constant field strength $F_{u\phi} = 1/2$ per rectangle creates no inter-rectangle coupling, and (c) the Chern-Simons functional is additive. □ □

Why this matters. In spherical coordinates, the product measure $\prod_i \sin\theta_i\,d\theta_i\,d\phi_i$ carries angular weights that, while technically factorizable, obscure the independence of the $N$ factors. In polar coordinates, the independence is manifest: each rectangle contributes its flat area independently, and the topological partition function is literally the $N$-th power of a single-rectangle integral.

Multi-Monopole Topological Sectors

For a system of $N$ particles with monopole charges $\{n_1, \ldots, n_N\}$, the total topological charge is:

$$ c_1^{\text{total}} = \sum_{i=1}^{N} n_i $$ (93.7)

Each particle contributes $c_1(n_i) = n_i$ from its own rectangle (Eq. eq:ch60z-chern-general). The topological sector is labeled by the total Chern number, which is an integer sum — no geometric computation required beyond rectangle-area counting.

The topological partition function in sector $Q = \sum n_i$ is:

$$ \mathcal{Z}_Q = \sum_\{n_i\: \sum n_i = Q} \prod_{i=1}^{N} \mathcal{Z}_{n_i}[\mathcal{R}_i] $$ (93.8)

where $\mathcal{Z}_{n_i}[\mathcal{R}_i]$ is the single-rectangle partition function with monopole charge $n_i$, computed as a polynomial integral on $[-1,+1]$.

Wilson Loops on the Flat Rectangle

Wilson Loops as Rectangle Area

From Chapter 60d (Key Result #75), the Wilson loop on $S^2$ in polar coordinates is:

$$ W_\gamma = \exp\!\left(\frac{iqg_m}{2} \cdot \mathcal{A}_{\text{rect}}\right) $$ (93.9)

where $\mathcal{A}_{\text{rect}}$ is the enclosed area on the flat polar rectangle. Because $F_{u\phi} = 1/2$ is constant, the Wilson loop depends only on the area:

$$ \mathcal{A}_{\text{rect}} = \Delta u \cdot \Delta\phi $$ (93.10)

where $\Delta u$ is the THROUGH extent and $\Delta\phi$ is the AROUND extent of the loop.

Theorem 93.3 (Wilson Loop Topological Invariance from Constant Field)

Wilson loops on $S^2$ are topological invariants: they depend only on the enclosed rectangular area, not on the shape of the loop. This follows from the constant field strength $F_{u\phi} = 1/2$.

Proof.

For any loop $\gamma$ enclosing area $\mathcal{A}$ on $\mathcal{R}$:

$$ W_\gamma = \exp\!\left(\frac{iqg_m}{2} \int_{\text{enclosed}} F_{u\phi}\,du\,d\phi\right) = \exp\!\left(\frac{iqg_m}{2} \cdot \frac{1}{2} \cdot \mathcal{A}\right) $$ (93.11)

Since $F_{u\phi} = 1/2$ is constant, the integral depends only on the total enclosed area, not on how the area is distributed. Two loops with the same enclosed area give the same Wilson loop value, regardless of their shape. This is the defining property of a topological invariant.

The fundamental loop (enclosing the full rectangle, $\mathcal{A} = 4\pi$) gives:

$$ W_{\text{full}} = \exp(iqg_m \pi) = (-1)^{2qg_m} $$ (93.12)

recovering the spinor sign for $qg_m = 1/2$. □ □

$N$-Particle Wilson Loops

For $N$ particles, the product Wilson loop on $\mathcal{R}^N$ is:

$$ W^{(N)}_\{\gamma_i\} = \prod_{i=1}^{N} \exp\!\left(\frac{iq_i g_{m,i}}{2} \cdot \frac{1}{2} \cdot \mathcal{A}_i\right) = \exp\!\left(\frac{i}{4} \sum_i q_i g_{m,i} \mathcal{A}_i\right) $$ (93.13)

Each factor is a constant-field integral over one flat rectangle. The $N$-particle Wilson loop is the exponential of a sum of rectangle areas — no inter-rectangle coupling.

The Topological–Dynamical Separation

Why Topological Quantities Are Insensitive to Concentration

Chapter 91 established that concentration of measure on $(S^2)^N$ separates into flat measure (which tensorizes) and curved metric (which drives concentration through Lipschitz conditions). This separation has a profound consequence for topological field theory:

Theorem 93.4 (Topological–Dynamical Separation)

On the polar rectangle $\mathcal{R}$:

Topological quantities depend on the flat measure $du\,d\phi$ and the constant field $F_{u\phi} = n/2$. They are independent of the curved metric $h_{uu} = R^2/(1-u^2)$.
Dynamical quantities (masses, coupling constants, concentration rates) depend on the curved metric.

Therefore, topological invariants are immune to the concentration- of-measure phenomenon, while dynamical observables concentrate as $N \to \infty$.

Proof.

Step 1: Topological integrals. All topological invariants on $S^2$ are integrals of the form $\int f(F)\,du\,d\phi$ where $f$ depends on the field strength $F_{u\phi} = n/2$ (constant) and the flat measure $du\,d\phi$. The metric $h_{uu}$, $h_{\phi\phi}$ does not enter.

Step 2: Dynamical integrals. Dynamical quantities involve the Lipschitz constant $\|df\|_g = \sqrt{h^{uu}(\partial_u f)^2 + h^{\phi\phi}(\partial_\phi f)^2}$, which depends on the inverse metric $h^{uu} = (1-u^2)/R^2$ and $h^{\phi\phi} = 1/(R^2(1-u^2))$. These are the quantities that enter concentration inequalities via $\mathbb{P}(|f - \langle f\rangle| > t) \leq 2\exp(-cNt^2/L^2)$.

Step 3: Independence. Since topological integrals use $\{F_{u\phi}, du\,d\phi\}$ and dynamical integrals use $\{h_{uu}, h_\phi\phi}$, and these are independent quantities on $\mathcal{R}$, topological invariants are structurally decoupled from concentration phenomena. □ □

Category	Depends on	Example
Topological	$F_{u\phi} = n/2$ (constant), $du\,d\phi$ (flat)	$c_1 = n$, Wilson loops, CS invariant
Dynamical	$h_{uu} = R^2/(1-u^2)$ (curved)	$g^2 = 4/(3\pi)$, mass spectrum
Mixed	Both	Berry phase $= qg_m/2 \times \mathcal{A}_{\text{rect}}$

The Berry phase is classified as “mixed” because its value is topological (depends on enclosed area via constant $F_{u\phi}$), but which loops are physically accessible depends on the dynamics (which is governed by the curved metric).

Exact Results from Flat Polynomial Integration

The topological–dynamical separation enables exact computation of topological partition functions:

$$ \mathcal{Z}_{\text{top}}^{(N)}(k) = \left[\int_{-1}^{+1} du \int_0^{2\pi} d\phi \; \exp\!\left(\frac{ik}{2} \cdot \frac{n}{2} \cdot u \cdot \phi\right) \right]^N $$ (93.14)

The inner integral is over one flat rectangle with constant field. For the monopole Chern-Simons theory at level $k$:

$$ \mathcal{Z}_{\text{top}}(k) = \int_0^{2\pi} d\phi \int_{-1}^{+1} du \; \exp(ikn\pi(1-u)/2) = 2\pi \cdot \frac{2\sin(kn\pi/2)}{kn\pi/2} $$ (93.15)

which is a sinc function in $kn$ — an exact, closed-form result from elementary integration on the flat rectangle.

Spin-Statistics from Rectangle Topology

The spin-statistics connection (Chapter 60a, Key Result #99) gains its clearest form as rectangle topology:

$$ \gamma_{\text{exchange}} = F_{u\phi} \times \frac{1}{2}\mathcal{A}_{\text{rect}} = \frac{1}{2} \times \frac{1}{2} \times 4\pi = \pi $$ (93.16)

The exchange phase is (constant field) $\times$ (half the rectangle area). This is the content of the spin-statistics theorem reduced to a single line of flat-rectangle arithmetic:

The “half” comes from exchange $=$ half the full rotation on the two-particle configuration space $\mathcal{R}^2$.
The $4\pi$ is the full rectangle area $2 \times 2\pi$ (THROUGH range $\times$ AROUND period).
The $1/2$ prefactor is the monopole field strength.
The result $\pi$ gives $e^{i\pi} = -1$: fermion statistics.

A $4\pi$ rotation (full rectangle traversal) gives $2\pi$ phase $= e^{i2\pi} = +1$: identity, confirming $\sigma^2 = 1$.

Connections to Established TQFT

Witten-Type and Schwarz-Type Theories

The topological field theory on $S^2$ in polar coordinates provides a concrete realization of both types of TQFT:

Schwarz-type: The Chern-Simons theory on $S^2$ with constant field $F_{u\phi} = n/2$ is a Schwarz-type TQFT. The action $S = (k/4\pi)\int A \wedge dA$ depends only on the connection, not the metric. In polar form, the action is:

$$ S_{\text{CS}} = \frac{k}{4\pi} \int_{\mathcal{R}} A_\phi \cdot F_{u\phi} \, du\,d\phi = \frac{k}{4\pi} \cdot \frac{1}{2} \int_{-1}^{+1}(1-u)\,du \cdot \frac{1}{2} \cdot 2\pi = \frac{k}{4} $$ (93.17)

a constant independent of $R$ and all metric parameters.

Witten-type: The monopole harmonic theory, where topological quantities (Chern numbers, winding numbers) are computed from the BRST cohomology of the monopole connection, is a Witten-type TQFT. The BRST charge $Q = \oint A_\phi\,d\phi = \pi(1-u_0)$ is linear in THROUGH position.

Relation to 3-Manifold Invariants

The $S^2$ Chern-Simons theory at level $k$ gives the Jones polynomial of links in $S^2 \times [0,1]$. In polar form, this reduces to evaluating polynomial integrals on the flat rectangle: link invariants become combinatorics of polynomial overlaps on $[-1,+1]$.

**Figure 93.1:** Topological field theory on the polar rectangle. *Left:* On $S^2$, the monopole field lines are curved and the Wilson loop $W_\gamma$ encloses a region with variable field strength $F_{\theta\phi} = \frac{1}{2}\sin\theta$. *Right:* On the flat rectangle $\mathcal{R}$, the field $F_{u\phi} = \frac{1}{2}$ is constant everywhere. Wilson loops depend only on the enclosed rectangular area $\mathcal{A} = \Delta u \cdot \Delta\phi$. *Bottom:* All topological invariants reduce to (constant field) $\times$ (flat rectangle area) — elementary arithmetic.

Derivation Chain Summary

#	Statement	Justification	Reference
\endhead 1	P1: $ds_6^\,2} = 0$ on $\mathcal{M}^4 \times S^2$	Postulate	Part 1
2	Monopole connection $A_\phi = (1-u)/2$ linear	P1 $\to$ topology	Ch. 60a
3	Field strength $F_{u\phi} = 1/2$ constant on $\mathcal{R}$	$\partial_u A_\phi$	Ch. 60a
4	Measure $du\,d\phi$ flat Lebesgue	$\sqrt{\det h} = R^2$	Ch. 2
5	Chern number $c_1 = n$ from constant $\times$ flat area	Integral	This chapter
6	Wilson loops $= \exp(iqg_m\mathcal{A}/4)$ topological	Constant $F$	This chapter
7	$N$-particle invariants factorize on $\mathcal{R}^N$	Flat product measure	This chapter
8	Topological–dynamical separation: topology uses $\{F_{u\phi}, du\,d\phi$, dynamics uses $\{h_{uu}, h_\phi\phi}$	Independence on $\mathcal{R}$	This chapter
9	Spin-statistics $= \frac{1}{2} \times$ half-rectangle area $= \pi$	Constant field $\times$ flat area	This chapter
10	Polar: all topological invariants reduce to (constant) $\times$ (flat area) on $\mathcal{R}$	Polar dual verification	This chapter

Chapter Summary

Key Result

Topological Field Theory from $S^2$

Key results:

Chern number $c_1 = n$ from constant field $F_{u\phi} = n/2$ times flat rectangle area: $n/(2\pi) \times 2 \times 2\pi = n$.
Wilson loops are topological: $W_\gamma = \exp(iqg_m\mathcal{A}/4)$ depends only on enclosed rectangular area.
$N$-particle topological invariants factorize exactly on $\mathcal{R}^N$ from flat product measure.
Topological–dynamical separation: topological quantities use $\{F_{u\phi}, du\,d\phi\}$ (constant and flat); dynamical quantities use $\{h_{uu}, h_\phi\phi}$ (curved).
Chern-Simons partition function is a sinc function of $kn$: exact, closed-form from flat polynomial integration.
Spin-statistics theorem is one line: $\frac{1}{2} \times \frac{1}{2} \times 4\pi = \pi$; exchange phase from constant field times half-rectangle area.
Polar dual verification: Every topological invariant verified as (constant field) $\times$ (flat rectangle area), reproducing all spherical-coordinate results.

Fundamental insight: Topological field theory on $S^2$ is trivially elementary in polar coordinates because the two ingredients — constant field and flat measure — are the simplest possible. The apparent complexity of TQFT on $S^2$ in spherical coordinates is entirely a coordinate artifact.

Table 93.1: Chapter 60z results summary

Result	Value	Status	Reference
Chern number	$c_1 = n$ (constant $\times$ area)	PROVEN	Thm. thm:ch60z-chern-number
Wilson loop topological	$W = e^{iqg_m\mathcal{A}/4}$	PROVEN	Thm. thm:ch60z-wilson-topological
$N$-particle factorization	$\mathcal{Z}^{(N)} = \mathcal{Z}^N$	PROVEN	Thm. thm:ch60z-product-factorization
Topo–dynamical separation	$\{F, du\,d\phi\}$ vs $\{h_{ij}\}$	PROVEN	Thm. thm:ch60z-topo-dyn-separation
CS partition function	$2\pi \cdot 2\,\mathrm{sinc}(kn\pi/2)$	PROVEN	Eq. eq:ch60z-CS-partition
Spin-statistics	$\gamma = \pi$ (half-rectangle)	PROVEN	Eq. eq:ch60z-spin-stats

Verification Code

The mathematical derivations and proofs in this chapter can be independently verified using the formal and computational scripts below.

◆ Lean 4 Formal proof verification Coming Soon ● Python Numerical verification & plots Coming Soon ★ Mathematica Symbolic computation & CAS verification Coming Soon

All verification code is open source. See the complete verification index for all chapters.

Property	Spherical \((\theta, \phi)\)	Polar \((u, \phi)\)
Field strength	\(F_{\theta\phi} = \frac{n}{2}\sin\theta\) (variable)	\(F_{u\phi} = \frac{n}{2}\) (constant)
Measure	\(\sin\theta\,d\theta\,d\phi\)	\(du\,d\phi\) (flat)
Integrand	\(\frac{n}{2}\sin\theta \cdot \sin\theta\,d\theta\,d\phi\)	\(\frac{n}{2}\,du\,d\phi\) (constant \(\times\) flat)
Chern number	\(\frac{n}{2\pi}\int \sin^2\theta\,d\theta\,d\phi\)	\(\frac{n}{2\pi} \times 2 \times 2\pi = n\)
Physical insight	Topology hidden in \(\sin^2\theta\) integral	Topology = constant \(\times\) flat area

Topological Field Theory from S²

Introduction

Chern-Simons Theory on the Polar Rectangle

The Chern Number as Rectangle Area

Chern-Simons Functional on \(\mathcal{R}\)

Topological Invariants on \((S^2)^N\)

Factorization from Flat Product Measure

Multi-Monopole Topological Sectors

Wilson Loops on the Flat Rectangle

Wilson Loops as Rectangle Area

\(N\)-Particle Wilson Loops

The Topological–Dynamical Separation

Why Topological Quantities Are Insensitive to Concentration

Exact Results from Flat Polynomial Integration

Spin-Statistics from Rectangle Topology

Connections to Established TQFT

Witten-Type and Schwarz-Type Theories

Relation to 3-Manifold Invariants

Derivation Chain Summary

Chapter Summary

Verification Code

Category	Depends on	Example
Topological	\(F_{u\phi} = n/2\) (constant), \(du\,d\phi\) (flat)	\(c_1 = n\), Wilson loops, CS invariant
Dynamical	\(h_{uu} = R^2/(1-u^2)\) (curved)	\(g^2 = 4/(3\pi)\), mass spectrum
Mixed	Both	Berry phase \(= qg_m/2 \times \mathcal{A}_{\text{rect}}\)

#	Statement	Justification	Reference
\endhead 1	P1: \(ds_6^\,2} = 0\) on \(\mathcal{M}^4 \times S^2\)	Postulate	Part 1
2	Monopole connection \(A_\phi = (1-u)/2\) linear	P1 \(\to\) topology	Ch. 60a
3	Field strength \(F_{u\phi} = 1/2\) constant on \(\mathcal{R}\)	\(\partial_u A_\phi\)	Ch. 60a
4	Measure \(du\,d\phi\) flat Lebesgue	\(\sqrt{\det h} = R^2\)	Ch. 2
5	Chern number \(c_1 = n\) from constant \(\times\) flat area	Integral	This chapter
6	Wilson loops \(= \exp(iqg_m\mathcal{A}/4)\) topological	Constant \(F\)	This chapter
7	\(N\)-particle invariants factorize on \(\mathcal{R}^N\)	Flat product measure	This chapter
8	Topological–dynamical separation: topology uses \(\{F_{u\phi}, du\,d\phi\), dynamics uses \(\{h_{uu}, h_\phi\phi}\)	Independence on \(\mathcal{R}\)	This chapter
9	Spin-statistics \(= \frac{1}{2} \times\) half-rectangle area \(= \pi\)	Constant field \(\times\) flat area	This chapter
10	Polar: all topological invariants reduce to (constant) \(\times\) (flat area) on \(\mathcal{R}\)	Polar dual verification	This chapter

Result	Value	Status	Reference
Chern number	\(c_1 = n\) (constant \(\times\) area)	PROVEN	Thm. thm:ch60z-chern-number
Wilson loop topological	\(W = e^{iqg_m\mathcal{A}/4}\)	PROVEN	Thm. thm:ch60z-wilson-topological
\(N\)-particle factorization	\(\mathcal{Z}^{(N)} = \mathcal{Z}^N\)	PROVEN	Thm. thm:ch60z-product-factorization
Topo–dynamical separation	\(\{F, du\,d\phi\}\) vs \(\{h_{ij}\}\)	PROVEN	Thm. thm:ch60z-topo-dyn-separation
CS partition function	\(2\pi \cdot 2\,\mathrm{sinc}(kn\pi/2)\)	PROVEN	Eq. eq:ch60z-CS-partition
Spin-statistics	\(\gamma = \pi\) (half-rectangle)	PROVEN	Eq. eq:ch60z-spin-stats