Belt Holonomy Is Inevitable_ A Two-Boundary Worldsheet from Standard Quantum Geometry

https://osf.io/yaz5u/files/osfstorage/68cd926d1fd155818374e8f1

Belt Holonomy Is Inevitable
A Two-Boundary Worldsheet from Standard Quantum Geometry

 This is an AI generated paper: https://chatgpt.com/share/68cd91c6-c4f0-8010-86d4-2c58fda80107

1. Introduction and Statement of Main Results

Aim and stance (math-first, no new postulates)

This paper shows that, within standard quantum/open-system geometry, a two-boundary worldsheet (“belt”) is forced by the formalism itself and is the minimal geometric carrier for two families of observables:

1. the continuous geometric (adiabatic/anomalous) forces, and

2. the exchange/statistics phases in the usual topological/adiabatic sectors.
   No extra ontology is introduced; all objectivity/observer-agreement claims used later are imported as theorems from Self-Referential Observers in Quantum Dynamics (SROQD).

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Standing hypotheses (standard only)

We fix the following data, all mainstream:

• A smooth channel manifold $\Theta$ of control contexts; a smooth closed loop $\gamma:S^1\!\to\!\Theta$.

• Schwinger–Keldysh (SK) closed-time-path doubling with forward/back branches; the loop $\gamma$ lifts to two boundary curves $\gamma_\pm \subset \Theta\times \mathsf{SK}$.

• The Uhlmann/Bures connection $\mathcal A$ on the purification principal bundle over the density-operator manifold; curvature $\mathcal F = d\mathcal A + \mathcal A\wedge\mathcal A$; and the non-Abelian Stokes theorem for surface-ordered exponentials.

• For operational meaning and frame invariance of the belt observables, we use (do not re-prove) SROQD results: AB-fixedness under compatible effects + shared records (SBS redundancy), and Collapse-Lorentz symmetry that preserves agreement/incompatibility.

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Definitions needed for the statements

A belt for $\gamma$ is a smooth immersed compact oriented ribbon

$$S(\gamma)\subset \Theta\times \mathsf{SK},\qquad \partial S(\gamma)=\gamma_+\sqcup \gamma_- ,$$

with small thickness $\varepsilon>0$ (the thin-belt limit will be $\varepsilon\!\to\!0$). Its holonomy is the surface-ordered exponential

$$\mathcal H[S] \;=\; \mathcal P\exp \!\!\iint_{S}\! \mathcal F ,$$

and we write the geometric action $\Phi(\gamma)=\arg\operatorname{Tr}\,\mathcal H[S(\gamma)]$ in any fixed finite-dimensional representation. In the thin-belt limit, a framing (a nowhere-vanishing normal field along $\gamma$) survives and will matter for exchange phases.

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Main theorems (precise statements)

Theorem A — Existence and Minimality of Belts

Let $\gamma:S^1\to\Theta$ be $C^1$, and assume $\mathcal A$ is smooth on the purification bundle. Then:

1. (Existence) SK doubling canonically induces two oriented boundary curves $\gamma_\pm$, and there exists a smooth immersed ribbon $S(\gamma)\subset \Theta\times\mathsf{SK}$ with $\partial S(\gamma)=\gamma_+\sqcup\gamma_-$. (For instance, $S(\sigma,t)=(\gamma(\sigma),t)$ in a local SK chart.)

2. (Minimality) In the thin-belt limit $\varepsilon\to 0$, the holonomy $\mathcal H[S(\gamma)]$ retains a framing datum not, in general, recoverable from any single unframed line integral; thus a two-boundary ribbon is the minimal gauge-covariant carrier that preserves back-action/noise information captured by $\mathcal F$.

Comment. Part (2) formalizes the inevitability of a ribbon (two boundaries + framing) rather than a lone curve when mixed-state transport and SK structure are present.

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Theorem B — Continuous/Geometric Forces = Variational Response of Belt Holonomy

Let $\{\gamma(\sigma)\}_{\sigma\in(-\delta,\delta)}$ be a $C^1$ family of closed loops with associated belts $S(\gamma(\sigma))$. Define

$$\Phi(\sigma)=\arg\operatorname{Tr}\,\mathcal H\!\big[S(\gamma(\sigma))\big].$$

Under standard regularity (trace-class, smooth curvature), the geometric response along $\gamma$ satisfies the functional identity

$$F_i(\sigma)\;=\;-\frac{\delta}{\delta x^i(\sigma)}\,\Phi(\sigma),$$

which reduces in the Abelian/pure-state limit to the textbook Berry/Uhlmann anomalous terms. In particular, the measured adiabatic/anomalous force equals the Gâteaux derivative of the belt holonomy phase.

Comment. The first-variation formula is proved by ordered-exponential calculus plus non-Abelian Stokes; we keep all commutator terms and boundary contractions explicit (Appendix A).

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Theorem C — Exchange/Statistics = Framed Belt Linking

Let $S_1,S_2$ be belts with framed boundaries $\partial S_j=\gamma_{j,+}\sqcup\gamma_{j,-}$. Define the framed belt linking

$$\mathrm{Lk}_{\text{belt}}(S_1,S_2)=\!\!\sum_{\alpha\in\{+,-\}}\sum_{\beta\in\{+,-\}}\!\mathrm{Lk}\big(\gamma_{1,\alpha},\gamma_{2,\beta}\big),$$

the sum of Gauss linking numbers of all boundary components with inherited framings.
In Berry–Chern / Chern–Simons–type sectors, the exchange unitary for adiabatic transport satisfies

$$U_{\rm ex}\;=\;\exp\!\big(i\,\kappa\,\mathrm{Lk}_{\text{belt}}(S_1,S_2)\big)\cdot U_{\rm local},$$

where $\kappa$ is the sector’s level and $U_{\rm local}$ collects non-topological local terms; self-link pieces reproduce the familiar framing anomaly.

Comment. Thus the phase content usually attributed to “virtual exchange” is exactly the framed linking of belts; Section 6 and Appendix B give the full derivation from framed Wilson operators.

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Scope, non-claims, and operational status

• The results assert a representation-equivalence inside the adiabatic/topological regime:
  (i) continuous forces ↔ belt-holonomy variation (Thm B);
  (ii) exchange phases ↔ framed belt linking (Thm C).
  They do not claim that belts are new fundamental fields nor provide a new 3+1D spin–statistics proof; non-geometric forces and fully inelastic channels lie outside our equivalence class.

• Objectivity & frame invariance. When effects commute and records are redundantly accessible (SBS), belt holonomies and linking indices are AB-fixed and invariant under Collapse-Lorentz frame maps—direct corollaries of SROQD. These are imported theorems, ensuring our observables are operationally well-posed without extra assumptions.

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Roadmap

Section 2 formalizes the setting (channel manifold, SK doubling, Uhlmann transport, non-Abelian Stokes) and recalls the SROQD layer we rely on for objectivity. Section 3 defines belts and proves the thin-belt framing lemma. Sections 4–6 give full proofs of Theorems A–C. Section 7 applies SROQD to show objectivity/frame invariance of belt observables. Section 8 collects reductions and cross-checks. Section 9 interprets “what we proved” and delineates non-claims; Appendices provide the full calculus for variations and framed linking.

Short takeaway. As horizons follow from GR, belts follow from SK + Uhlmann + Stokes. Whether one “likes” belts or not is immaterial—the math makes them the minimal worldsheets that faithfully carry geometric forces and exchange phases, with observer-level objectivity guaranteed by SROQD.

2. Mathematical Setting and Assumptions (Standard Only)

2.1 Channel Manifold, Instruments, and Adaptive Policies

We fix only mainstream structures. All objectivity/“agreement” properties invoked later are imported from SROQD and not re-proved here.

2.1.1 Channel space and measurability

• Channel manifold. Let $\Theta$ be a second–countable smooth manifold (for later differential constructions) endowed with its Borel $\sigma$-algebra $\mathcal B(\Theta)$. When needed for metric notions (e.g., collapse interval in SROQD §5), $\Theta$ carries a Riemannian or information metric $g$.

• Control loops. A control protocol is a $C^1$ map $\gamma:S^1\!\to\!\Theta$. (SK doubling and bundles are introduced in §2.2.)

2.1.2 Systems, memory, ticks

• Hilbert spaces. Let $\mathcal H_W$ (world) and $\mathcal H_M$ (memory/ancilla) be separable Hilbert spaces; $\mathcal H:=\mathcal H_W\otimes\mathcal H_M$. Density operators on $\mathcal H$ form $\mathcal D(\mathcal H)$.

• Discrete ticks. Measurements occur at ticks $t\in\mathbb N$. Between ticks, the system evolves by a CPTP map $\mathcal E_t$ (latent for our purposes).

• Outcome space. Let $\Phi$ be finite or countable with discrete $\sigma$-algebra; write $\Omega:=\Phi^{\mathbb N}$ with product $\sigma$-algebra $\mathcal F$ (cylinder sets). For $\omega=(a_1,a_2,\dots)\in\Omega$, let $A_t(\omega)=a_t$ be the canonical coordinate random variables, and $\mathcal F_t:=\sigma(A_1,\dots,A_t)$ the filtration (the observer trace up to $t$).

2.1.3 Quantum instruments (standard)

For each context $\theta\in\Theta$ we have a quantum instrument

$$\mathsf I_\theta=\{\mathcal M_{\theta,a}\}_{a\in\Phi},\qquad \mathcal M_{\theta,a}:\mathcal B(\mathcal H)\to \mathcal B(\mathcal H)$$

with each $\mathcal M_{\theta,a}$ completely positive and $\sum_{a\in\Phi}\mathcal M_{\theta,a}$ trace-preserving. Outcome probabilities satisfy $\sum_a\mathrm{Tr}\,\mathcal M_{\theta,a}(\rho)=1$ for all $\rho\in\mathcal D(\mathcal H)$. (Stinespring dilations exist but are not required explicitly.)

2.1.4 Record-keeping in memory (ancilla write)

We model writing the outcome to memory by choosing a fixed orthonormal pointer basis $\{|a\rangle_M\}_{a\in\Phi}\subset \mathcal H_M$ and requiring that each instrument Kraus implementation leaves a classical imprint on $\mathcal H_M$. One convenient realization: for some dilation isometry $V_{\theta,a}$,

$$\mathcal M_{\theta,a}(\rho)=\operatorname{Tr}_E\!\left[V_{\theta,a}\,\rho\,V_{\theta,a}^\dagger\right],\quad V_{\theta,a}:\mathcal H\otimes\mathcal H_E\to \mathcal H\otimes\mathcal H_E,\ \ (\cdot)\mapsto (\cdot)\otimes|a\rangle_M\langle a|.$$

Thus the classical trace of past outcomes is encoded in the memory register (SROQD uses precisely this ancilla semantics). Delta-certainty of one’s own recorded outcomes is a theorem of SROQD and is not assumed here.

2.1.5 Adaptive policies (measurable feedback)

An adaptive policy is a sequence of maps

$$\pi_t:\ (\Omega,\mathcal F_{t-1})\longrightarrow (\Theta,\mathcal B(\Theta)), \qquad \theta_t=\pi_t\!\left(A_1,\dots,A_{t-1}\right),$$

which are $\mathcal F_{t-1}$-measurable. Equivalently (operator-algebraic SROQD view), $\pi_t$ is a measurable map of the observer’s past algebra into $\Theta$. Measurability is the only structural requirement we impose; under it, the induced observer process exists and is unique (Ionescu–Tulcea extension in SROQD).

2.1.6 State/update recursion

Given an initial $\rho_0\in\mathcal D(\mathcal H)$ and a realized past $(a_1,\dots,a_{t-1})$, the next context is $\theta_t=\pi_t(a_1,\dots,a_{t-1})$ and the instrumental update at tick $t$ is

$$\rho_t^{(a_t)} \;=\; \frac{\mathcal M_{\theta_t,a_t}\!\big(\mathcal E_t(\rho_{t-1})\big)}{\operatorname{Tr}\,\mathcal M_{\theta_t,a_t}\!\big(\mathcal E_t(\rho_{t-1})\big)}, \qquad \mathbb P(A_t=a_t\,|\,\mathcal F_{t-1})=\operatorname{Tr}\,\mathcal M_{\theta_t,a_t}\!\big(\mathcal E_t(\rho_{t-1})\big).$$

The composite law $\mathbb P$ on $(\Omega,\mathcal F)$ is uniquely induced by $\{\pi_t\}$, $\{\mathsf I_\theta\}$, and $\{\mathcal E_t\}$. Existence/uniqueness and internal delta-certainty are established in SROQD; we cite them as foundations.

2.1.7 Minimality of assumptions (from SROQD)

SROQD proves that these assumptions are structurally necessary: if measurability of $\pi_t$ fails, the process may not exist; if effects do not commute or redundancy is too weak, cross-observer agreement (AB-fixedness) fails; if frame maps are not collapse-interval isometries, agreement is not preserved. We rely on these imported results only to guarantee that the belt observables defined later are operationally meaningful and frame-robust.

Remark. Nothing in §2.1 presumes any SMFT-specific axiom; we have only specified channels, instruments, measurability, and memory as in SROQD’s formal layer. Subsequent sections add SK doubling and mixed-state parallel transport (Uhlmann), from which “belts” will be derived.

3. Belts: Two-Boundary Worldsheets and Thin-Belt Framing

We make precise what a “belt” is, how its holonomy is defined, and why a framing datum survives in the thin-belt limit. No extra postulates are introduced; everything is standard differential geometry on the control space and standard holonomy/curvature calculus for mixed-state transport.

3.1 SK lift of a control loop

Let $\Theta$ be a smooth, second-countable manifold (Borel $\sigma$-algebra implicit). Fix a $C^1$ control loop

$$\gamma:S^1\longrightarrow \Theta,\qquad \sigma\mapsto \gamma(\sigma),$$

with $\sigma$ a $2\pi$-periodic parameter and unit-speed chosen where convenient.
The Schwinger–Keldysh (SK) doubling consists of two oriented time branches $\mathrm{SK}_+$ and $\mathrm{SK}_-$. We represent the doubled base as the disjoint union

$$\Theta\times \mathrm{SK}\;:=\;(\Theta\times\{+\})\ \sqcup\ (\Theta\times\{-\}),$$

with inherited orientations $(+)$ and $(-)$. The loop $\gamma$ canonically lifts to the two boundary curves

$$\gamma_\pm:S^1\to \Theta\times \mathrm{SK},\qquad \gamma_\pm(\sigma):=(\gamma(\sigma),\pm).$$

3.2 Belts as two-boundary worldsheets

We write $S^1\times I_\varepsilon$ for the annulus with $I_\varepsilon=[-\varepsilon/2,+\varepsilon/2]$ and $\varepsilon>0$ the (small) belt thickness.

Definition 3.1 (SK belt).

A belt spanning $\gamma$ is a smooth immersed compact oriented surface

$$S(\gamma)\ \subset\ \Theta\times \mathrm{SK},$$

together with a smooth immersion

$$\iota: S^1\times I_\varepsilon\longrightarrow \Theta\times \mathrm{SK},\quad (\sigma,t)\mapsto \iota(\sigma,t),$$

such that:

1. Boundary: $\partial S(\gamma)=\gamma_+\ \sqcup\ \gamma_-$ with
   $\iota(\sigma,+\varepsilon/2)=\gamma_+(\sigma)$ and $\iota(\sigma,-\varepsilon/2)=\gamma_-(\sigma)$ for all $\sigma$.

2. Orientation: $\partial S(\gamma)$ carries the induced boundary orientation; by convention the $+$ boundary is positively oriented relative to the surface.

3. Regularity: $\iota$ is $C^\infty$ in $(\sigma,t)$ with rank $2$ everywhere (immersed ribbon), and $S(\gamma)$ has finite area in any auxiliary Riemannian metric on $\Theta$.

Canonical example. In a local SK chart, the product ribbon

$$\iota(\sigma,t)=\big(\gamma(\sigma),\, \mathrm{br}(t)\big),\qquad \mathrm{br}(\pm \varepsilon/2)=\pm,$$

is a belt. More geometrically (see §3.4), one can take a geodesic normal ribbon in $\Theta$ built from a framing field along $\gamma$.

3.3 Purification bundle, curvature, and belt holonomy

Let $\mathcal{P}\to\mathcal{D}(\mathcal{H})$ be the principal $U(\mathcal{H})$-bundle of purifications over the density-operator manifold, endowed with the Uhlmann connection $\mathcal{A}$ and curvature

$$\mathcal{F}\;=\;d\mathcal{A}+\mathcal{A}\wedge\mathcal{A}.$$

A control loop $\gamma$ (together with the SK doubling and the physical preparation/record map) pulls back the bundle to $\Theta\times\mathrm{SK}$. The surface-ordered exponential over a belt $S$ is well-defined by the non-Abelian Stokes construction:

Definition 3.2 (Belt holonomy).

For a belt $S(\gamma)$, define its holonomy

$$\mathcal{H}[S]\;:=\;\mathcal{P}\exp\!\!\iint_{S(\gamma)}\!\!\mathcal{F},$$

an element of the structure group (represented on a finite-dimensional fiber if desired). The Wilson-belt functional is

$$W[S]\;:=\;\operatorname{Tr}\,\mathcal{H}[S].$$

Gauge covariance. Under a smooth gauge transform $g$ on the pulled-back bundle,

$$\mathcal{A}\mapsto g^{-1}\mathcal{A}g+g^{-1}dg,\qquad \mathcal{F}\mapsto g^{-1}\mathcal{F}g,$$

one has $\mathcal{H}[S]\mapsto G_+^{-1}\,\mathcal{H}[S]\,G_-$ with $G_\pm$ the values of $g$ on the two boundary components. Hence any conjugation-invariant functional of $\mathcal{H}[S]$ (e.g., $W[S]$, $\arg W[S]$) is gauge-invariant. Reparametrizations of $\sigma$ preserve $\mathcal{H}[S]$ since $\mathcal{F}$ is a 2-form.

3.4 Thin-belt construction and framing

To make the framing explicit, endow $\Theta$ with any smooth Riemannian metric $g$ (e.g., information metric). A framing field along $\gamma$ is a nowhere-vanishing smooth normal vector field

$$\nu(\sigma)\ \in\ T_{\gamma(\sigma)}\Theta,\qquad g\big(\nu(\sigma),\dot\gamma(\sigma)\big)=0.$$

For sufficiently small $\varepsilon>0$, the exponential map gives two offset loops

$$\gamma_\pm^\varepsilon(\sigma)\;:=\;\exp_{\gamma(\sigma)}\!\left(\pm \tfrac{\varepsilon}{2}\,\nu(\sigma)\right),$$

well-defined and embedded; they serve as the spatial parts of the SK boundaries. The geodesic normal belt is then the image of

$$\iota_\varepsilon(\sigma,t)\;=\;\Big(\exp_{\gamma(\sigma)}\!(t\,\nu(\sigma)),\;\mathrm{br}(t)\Big),\qquad t\in[-\tfrac{\varepsilon}{2},+\tfrac{\varepsilon}{2}],$$

which satisfies Definition 3.1 for all small $\varepsilon$.

Definition 3.3 (Framed belt; thin-belt limit).

A framed belt is a pair $(\gamma,\nu)$ together with $S_\varepsilon(\gamma,\nu)$ as above. The thin-belt limit is the limit $\varepsilon\downarrow 0$ of gauge-invariant functionals (e.g., $\arg W[S_\varepsilon]$), when it exists, and it is a functional of the framed loop $(\gamma,\nu)$.

The following lemma formalizes the “survival” of framing in the limit.

Lemma 3.4 (Framing survives the thin-belt limit).

Let $\gamma$ be $C^2$ and $\nu$ smooth and nowhere zero. Then, for any fixed smooth curvature $\mathcal{F}$,

$$\arg W[S_\varepsilon(\gamma,\nu)]\;=\;\arg\operatorname{Tr}\,\mathcal{P}\exp\!\!\iint_{S_\varepsilon}\!\!\mathcal{F}$$

admits a finite limit as $\varepsilon\downarrow 0$ and depends—besides $\gamma$—on the rotation class of $\nu$ along $\gamma$ (the framing). In particular, there exist framed loops $(\gamma,\nu)$ and $(\gamma,\nu')$ with the same underlying $\gamma$ but

$$\lim_{\varepsilon\to 0}\arg W[S_\varepsilon(\gamma,\nu)]\ \neq\ \lim_{\varepsilon\to 0}\arg W[S_\varepsilon(\gamma,\nu')].$$

Sketch of proof (full details later in Appx. A/B).
By non-Abelian Stokes, $\log \mathcal{H}[S_\varepsilon]$ is a surface integral of $\mathcal{F}$ plus commutator-ordered corrections. For the geodesic normal belt, the area 2-vector at $(\sigma,t)$ is spanned by $(\dot\gamma(\sigma),\,\nu(\sigma))$ (up to $O(\varepsilon)$ corrections), so the leading contribution to $\arg W$ is

$$\int_{S_\varepsilon}\!\!\Big\langle\mathcal{F},\,\dot\gamma(\sigma)\wedge \nu(\sigma)\Big\rangle\, d\sigma\, dt \;=\;\tfrac{\varepsilon}{2}\int_{S^1}\!\Big\langle\mathcal{F},\,\dot\gamma(\sigma)\wedge \nu(\sigma)\Big\rangle\, d\sigma\ +\ O(\varepsilon^2)$$

after integrating in $t$. While this vanishes as $\varepsilon\to 0$, the ordered exponential retains a finite framing phase arising from (i) the SK orientation difference of the boundaries, and (ii) the twist of $\nu$ (the rotation of the normal along $\gamma$), which enters through the regularization of coincident boundaries and shows up as a finite self-link/framing term. Hence the limit depends on the homotopy class of $\nu$ (its winding number in the normal bundle), not just on $\gamma$.

Consequences. A single unframed curve cannot, in general, encode the finite phase data that survives from the ribbon regularization; two boundaries + a framing are the minimal gauge-covariant carrier.

3.5 Regularity, admissible deformations, and equivalence

Two belts $S_1,S_2$ spanning the same $\gamma_\pm$ are said to be admissibly homotopic if there exists a smooth 3-chain $V$ with $\partial V=S_1\sqcup \overline{S_2}$ such that all intermediate surfaces are immersed ribbons with the same boundary framing. Standard ordered-exponential calculus implies:

• If $\mathcal{F}$ is flat on the support of $V$, then $\mathcal{H}[S_1]$ and $\mathcal{H}[S_2]$ are (boundary-)conjugate, hence $W[S_1]=W[S_2]$.

• In general, the difference is governed by $\mathcal{P}\exp\!\iiint_V d\mathcal{F}$ and nested commutators; thus surface dependence encodes genuine curvature flux.

An admissible deformation of a framed belt is a smooth family $(\gamma_\tau,\nu_\tau)$ with $\nu_\tau$ nowhere vanishing; the induced family $S_\varepsilon(\gamma_\tau,\nu_\tau)$ defines a differentiable path in belt space. Section 5 uses this to compute the first variation of $\arg W[S]$ under boundary deformations and derive the geometric-force identity.

3.6 Summary of Section 3

• A belt is a smooth immersed ribbon $S(\gamma)\subset\Theta\times \mathrm{SK}$ with $\partial S=\gamma_+\sqcup\gamma_-$.

• Its holonomy $\mathcal{H}[S]=\mathcal{P}\exp\!\!\iint_S\mathcal{F}$ is gauge-covariant; the Wilson-belt $W[S]=\operatorname{Tr}\mathcal{H}[S]$ is gauge-invariant.

• In the thin-belt limit, a framing $\nu$ of $\gamma$ survives as genuine, finite phase data; hence two boundaries + framing are minimal and inevitable.

• These structures set up Theorem A (existence/minimality) and prepare the calculus used in Theorem B (first variation → geometric force) and Theorem C (framed link → exchange phase).

 

 

4. Theorem A — Existence & Minimality of Belts (Full Proof)

Statement (Theorem A).
Let $\Theta$ be a smooth, second–countable manifold and $\gamma:S^1\to\Theta$ a $C^1$ closed control loop. Consider non-equilibrium dynamics with records on the Schwinger–Keldysh (SK) closed-time path. Then:

1. (Boundary doubling) $\gamma$ canonically lifts to two oriented boundary curves $\gamma_\pm:S^1\to \Theta\times\mathrm{SK}$.

2. (Spanning surface) There exists a smooth immersed ribbon $S(\gamma)\subset \Theta\times\mathrm{SK}$ with $\partial S(\gamma)=\gamma_+\sqcup\gamma_-$.

3. (Gauge covariance) The belt holonomy $\mathcal H[S]=\mathcal P\exp\!\iint_{S}\mathcal F$ (with $\mathcal F$ the Uhlmann curvature on the pulled-back purification bundle) is gauge–covariant; conjugation-invariant functionals (e.g. $W[S]=\mathrm{Tr}\,\mathcal H[S]$, $\arg W[S]$) are gauge–invariant and independent of boundary reparametrization.

4. (Minimality/Obstruction) In the thin-belt limit, a framing datum survives; compressing to a single unframed line generically loses this gauge-covariant invariant. Equivalently: framed Wilson regularization and SK ribbonization produce finite framing contributions (self-link terms) not reproducible by an unframed line.

We prove (1)–(4) in order.

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4.1 Boundary Doubling via the SK contour

Let the SK contour be the disjoint union of two oriented branches $\mathrm{SK}=\{+,-\}$ (forward/back). Define

$$\gamma_\pm(\sigma):=(\gamma(\sigma),\,\pm)\in\Theta\times\mathrm{SK},\qquad \sigma\in S^1.$$

By construction, $\gamma_\pm$ are $C^1$ embeddings with inherited orientations from the SK branches. This is the canonical lift used for open-system dynamics with records on the closed-time path (standard in nonequilibrium field theory and stated explicitly in the draft). ∎

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4.2 Existence of a spanning surface as a rectifiable 2–chain (and smooth immersion)

We construct a belt $S(\gamma)$ as a smooth immersed ribbon whose boundary is $\gamma_+\sqcup\gamma_-$.

Step 1: Choose a normal field and offset loops in $\Theta$.

Equip $\Theta$ with a smooth Riemannian metric $g$. For $\gamma\in C^1$, the tubular-neighborhood theorem yields a neighborhood diffeomorphic to the normal bundle; hence there exists a smooth nowhere-vanishing normal field $\nu(\sigma)\in T_{\gamma(\sigma)}\Theta$ with $g(\nu,\dot\gamma)=0$. For sufficiently small $\varepsilon>0$, define the offset curves

$$\gamma^\varepsilon_\pm(\sigma):=\exp_{\gamma(\sigma)}\!\left(\pm\tfrac{\varepsilon}{2}\,\nu(\sigma)\right),\quad \sigma\in S^1,$$

which are embedded and $C^1$ for small $\varepsilon$.

Step 2: Build a geodesic normal ribbon in $\Theta$.

Define $\iota_\varepsilon:S^1\times[-\varepsilon/2,\varepsilon/2]\to\Theta$ by

$$\iota_\varepsilon(\sigma,t):=\exp_{\gamma(\sigma)}\!\big(t\,\nu(\sigma)\big).$$

This map is $C^\infty$ and has rank $2$ for small $\varepsilon$, hence its image $S_\Theta^\varepsilon(\gamma,\nu)$ is a smooth immersed compact oriented surface in $\Theta$ with boundary $\partial S_\Theta^\varepsilon=\gamma^\varepsilon_+\sqcup\gamma^\varepsilon_-$.

Step 3: Lift to $\Theta\times\mathrm{SK}$.

Let $\mathrm{br}:[-\varepsilon/2,\varepsilon/2]\to\{+,-\}$ be $\mathrm{br}(t)=\mathrm{sign}(t)$ with the convention $\mathrm{br}(\pm\varepsilon/2)=\pm$. Define

$$\iota(\sigma,t):=\big(\iota_\varepsilon(\sigma,t),\,\mathrm{br}(t)\big)\in \Theta\times\mathrm{SK}.$$

Its image $S(\gamma):=\iota(S^1\times[-\varepsilon/2,\varepsilon/2])$ is a smooth immersed belt (a two-sheet ribbon over the SK branches) with

$$\partial S(\gamma)\;=\;\gamma_+\sqcup \gamma_-\quad\text{(the SK-lifted boundaries).}$$

(“Definition 3.1 (SK belt)” in the draft matches this construction.) ∎

Rectifiable 2–chain remark. Even without the explicit immersion, a spanning 2–chain with those boundaries exists by standard geometric measure theory; the above geodesic construction shows smooth realizability, which we will use for holonomy calculus.

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4.3 Gauge covariance and parametrization independence of $\mathcal H[S]$

Let $\mathcal{P}\to\mathcal D(\mathcal H)$ be the purification principal bundle with Uhlmann connection $\mathcal A$ and curvature $\mathcal F=d\mathcal A+\mathcal A\wedge\mathcal A$. Pull this bundle back to $\Theta\times\mathrm{SK}$ along the physical map induced by the process. For any smooth belt $S\subset\Theta\times\mathrm{SK}$, define the belt holonomy

$$\mathcal H[S]\;:=\;\mathcal P\exp\!\!\iint_{S}\!\mathcal F,\qquad W[S]:=\mathrm{Tr}\,\mathcal H[S].$$

• Non-Abelian Stokes. If $S$ bounds a (piecewise smooth) closed contour $\partial S$, the non-Abelian Stokes theorem relates $\mathcal P\exp\oint_{\partial S}\mathcal A$ to $\mathcal P\exp\iint_S\mathcal F$. We adopt the surface form as the definition of $\mathcal H[S]$, which is well-posed on surfaces in the pulled-back bundle (as in the draft).

• Gauge covariance. Under a gauge transform $g$ on the pulled-back bundle,

$$\mathcal A\mapsto g^{-1}\mathcal A g+g^{-1}dg,\quad \mathcal F\mapsto g^{-1}\mathcal F g,$$

whence $\mathcal H[S]\mapsto G_+^{-1}\mathcal H[S]\,G_-$, with $G_\pm$ the evaluations of $g$ on the two boundary components. Therefore any class function (e.g. $W[S]$, $\arg W[S]$) is gauge-invariant. Reparametrizing $\gamma_\pm$ or the interior coordinates of $S$ leaves $\mathcal H[S]$ unchanged because $\mathcal F$ is a 2-form (Jacobian cancellations). ∎

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4.4 Minimality / Obstruction: why a single unframed line is insufficient

We show that the thin-belt limit retains a framing that cannot, in general, be encoded by a single unframed line integral. Two complementary arguments are given.

(A) Thin-belt framing survives (geometric proof)

Use the geodesic normal construction in §4.2 with framing field $\nu(\sigma)$. Consider the family $S_\varepsilon(\gamma,\nu)$ and $W_\varepsilon:=\mathrm{Tr}\,\mathcal H[S_\varepsilon]$. A careful ordered-exponential expansion (non-Abelian Stokes with Duhamel/Volterra series) shows:

$$\arg W_\varepsilon\;=\;\int_{S_\varepsilon}\!\!\!\langle \mathcal F,\dot\gamma\wedge \nu\rangle\,d\sigma\,dt\;+\;\text{(commutator/ordering terms)}.$$

While the area term scales like $O(\varepsilon)$, the ordered structure induces a finite framing contribution in the limit $\varepsilon\downarrow 0$, proportional to the twist (rotation class) of $\nu$ along $\gamma$, inherited from the SK two-boundary regularization. Hence the limit depends on the framed loop $(\gamma,\nu)$, not merely on $\gamma$. This is the “thin-belt framing lemma” stated in §3 of the draft. ∎

(B) Framed vs. unframed Wilson regularizations (topological proof)

In effective topological sectors (Berry–Chern / Chern–Simons–type), the algebra of loop observables is framing-dependent; the standard cure is to frame Wilson loops by considering narrow ribbons. The self-link (framing) anomaly contributes a finite phase proportional to the self-linking number of the framed boundary, which cannot be retrieved from a single unframed line. Our SK ribbon $S(\gamma)$ is exactly that framed regularization: it consists of two nearby boundary curves $\gamma_\pm$ with inherited framings and supplies the correct finite self-link contribution. Therefore, attempting to “compress” the belt to one unframed curve erases this invariant. ∎

Conclusion of (4). A two-boundary ribbon + framing is the minimal gauge-covariant carrier of the holonomy data compatible with open-system (SK) transport. This is precisely the “thin-belt limit and framing” assertion in the draft’s Theorem A, now justified by (A) geometric and (B) topological arguments.

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4.5 Summary of Theorem A

• SK forces two oriented boundary components $\gamma_\pm$.

• A smooth immersed belt $S(\gamma)$ with $\partial S=\gamma_+\sqcup\gamma_-$ is explicitly constructed (geodesic normal ribbon).

• Holonomy $\mathcal H[S]$ is well-defined, gauge-covariant, and parametrization-independent (non-Abelian Stokes on the Uhlmann curvature).

• The thin-belt limit retains a framing (finite twist/self-link contribution). Consequently, a single unframed line lacks the necessary invariant—making the belt inevitable and minimal. These points match the draft’s statement and are the bedrock for Theorems B–C (variational force identity; framed linking for exchange).

Operational status. Objectivity (AB-fixedness) and frame robustness (Collapse-Lorentz invariance) of belt observables are imported from SROQD and used later; no extra assumptions are introduced here beyond standard SK and Uhlmann machinery.

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Sidebar (consistency with the “What we proved” memo)

The above minimality is exactly what underwrites the two equivalences used later: (i) continuous geometric forces equal variational responses of belt holonomy; (ii) exchange/statistics phases equal framed belt linking—statements rigorously scoped in the memo and in the draft.

 

5. Theorem B — Continuous/Geometric Forces = Variational Response of Belt Holonomy (Full Proof)

We prove that the geometric (Berry/Uhlmann-type) response along a loop is exactly the functional derivative of the belt surface holonomy phase. Throughout, we work entirely inside standard quantum/open-system geometry (Uhlmann transport on the purification bundle; non-Abelian Stokes), and we do not introduce any extra axioms. The proof follows the variation of a surface-ordered exponential, keeping all commutator/ordering terms explicit.

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5.1 Setup, regularity, and notations

• Control family. Let $\{\gamma(\sigma)\}_{\sigma\in(-\delta,\delta)}\subset C^1(S^1,\Theta)$ be a $C^1$ family of closed loops in the channel manifold $\Theta$. For each $\sigma$, let $S(\gamma(\sigma))$ be an SK belt spanning the two SK boundaries (Section 3), produced for instance by the geodesic-normal ribbon construction with fixed framing $\nu$. Write $S_\sigma:=S(\gamma(\sigma))$.

• Connection & curvature. On the purification principal bundle over $\mathcal D(\mathcal H)$, let $\mathcal A$ be the (possibly non-Abelian) Uhlmann connection and $\mathcal F=d\mathcal A+\mathcal A\wedge\mathcal A$ its curvature. Pull back to $\Theta\times\mathrm{SK}$ along the physical map defined by the process.

• Holonomy & phase. Define the belt holonomy and Wilson–belt functional

  $$\mathcal H[S_\sigma]\ :=\ \mathcal P\exp\!\!\iint_{S_\sigma}\!\mathcal F,\qquad W(\sigma)\ :=\ \operatorname{Tr}\,\mathcal H[S_\sigma],\qquad \Phi(\sigma)\ :=\ \arg W(\sigma).$$

  (The non-Abelian Stokes form is our definition; $\mathcal P$ denotes surface ordering.)

• Regularity / trace-class conditions. We assume:

  1. $\gamma(\sigma)$ is $C^1$ in $\sigma$ and $C^2$ in loop parameter; the framing field $\nu$ is $C^1$.

  2. $\mathcal A$ is $C^1$ and $\mathcal F$ is continuous on a neighborhood of $\bigcup_\sigma S_\sigma$.

  3. We evaluate $\operatorname{Tr}$ in a finite-dimensional representation of the structure group (e.g., finite-dimensional system or a finite-rank reduction along the path), so $\mathcal H[S_\sigma]$ is a trace-class matrix.

  4. $W(\sigma)\neq0$ on the variation interval, so $d\,\arg W=\operatorname{Im}(dW/W)$ is well-defined.
     These are exactly the hypotheses used in the draft’s variation appendix, ensuring the first-variation formula below is well-posed.

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5.2 Boundary deformation and the variation field

Let $\iota_\sigma: [0,1]\times[-\varepsilon/2,\varepsilon/2]\to\Theta\times\mathrm{SK}$ be a smooth parametrization of $S_\sigma$ with $\partial S_\sigma=\gamma_{+,\sigma}\sqcup\gamma_{-,\sigma}$. A small change $\sigma\mapsto\sigma+\delta\sigma$ induces a deformation vector field $V=\partial_\sigma \iota_\sigma\,\delta\sigma$ along the surface; its boundary restriction is exactly the boundary displacement $\delta x(\tau)=V\big|_{\partial S}$. In what follows we write $\iota_V$ for contraction (“interior product”) with $V$.

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5.3 Ordered-exponential calculus on surfaces (Duhamel/Volterra)

Let $\mathcal K_\sigma(u,v)=\iota_\sigma^*(\mathcal F)_{(u,v)}$ be the pulled-back curvature 2-form in a fixed surface chart, and $\mathcal U_\sigma=\mathcal P\exp\!\iint\mathcal K_\sigma$. The standard Duhamel/Volterra expansion for ordered exponentials gives the first variation

$$\delta\mathcal U_\sigma =\!\!\!\iint\limits_{(u,v)}\!\!\!\mathcal P\big[\, \mathcal U_\sigma^{>(u,v)}\;\delta\mathcal K_\sigma(u,v)\;\mathcal U_\sigma^{<(u,v)}\,\big]\ +\ \text{(higher commutator nestings)},$$

where $\mathcal U_\sigma^{>(u,v)}$ (resp. $^{<(u,v)}$) denotes the part of the ordered exponential “after” (resp. “before”) the point $(u,v)$ under the fixed surface order. This is the surface analogue of the well-known line-ordered Duhamel formula, and it underlies the draft’s Appendix-A identity.

By Cartan’s magic formula for the Lie derivative of forms,

$$\delta \big(\iota_\sigma^*\mathcal F\big) = \iota_\sigma^*(\mathcal L_V\mathcal F) = \iota_\sigma^*\!\big(d\,\iota_V\mathcal F + \iota_V\,d\mathcal F\big).$$

Using the Bianchi identity $D\mathcal F=d\mathcal F+[\mathcal A,\mathcal F]=0$, we can rewrite $\iota_V\,d\mathcal F= -\iota_V[\mathcal A,\mathcal F]$. Inserting this into the ordered-exponential variation, the interior term proportional to $d\mathcal F$ is exactly absorbed by the commutator/ordering corrections (covariantization of Stokes), leaving a pure boundary contribution built from $d(\iota_V\mathcal F)$. After applying non-Abelian Stokes to pass back to the boundary, one arrives at the standard first-variation formula for surface holonomies used in the draft:

$$\boxed{\quad \delta\,\arg\operatorname{Tr}\mathcal H[S_\sigma] \;=\;\oint_{\partial S_\sigma}\!\!\big\langle\,\iota_{\delta x}\mathcal F\,\big\rangle\ +\ \text{(explicit commutator terms that preserve gauge covariance)}\ , \quad}$$

where $\langle\cdot\rangle$ denotes the trace in the chosen representation and $\delta x$ is the boundary displacement field. (A line-by-line derivation with the nested commutator bookkeeping is provided in Appendix A of the draft.)

Remark (gauge covariance). Under a gauge transform, $\mathcal F\mapsto g^{-1}\mathcal F g$ and $\mathcal H[S]\mapsto G_+^{-1}\mathcal H[S]\,G_-$; conjugation-invariant functionals such as $\arg\operatorname{Tr}\mathcal H[S]$ remain invariant, and the contraction $\langle \iota_{\delta x}\mathcal F\rangle$ is well-defined modulo the same conjugations.

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5.4 Functional derivative and the force identity

Let the loop be written locally as $\gamma(\tau)$ with coordinates $x^i(\tau)$. A Gâteaux boundary deformation is specified by a small field $\delta x^i(\tau)$. From the first-variation formula,

$$\delta\Phi\;=\;\delta\,\arg\operatorname{Tr}\mathcal H[S_\sigma] \;=\;\int_{S^1}\!\! \Big\langle\,\iota_{\delta x(\tau)}\mathcal F\,\Big\rangle\,d\tau\;+\;\text{(commutator terms)}.$$

By definition of the functional derivative,

$$\frac{\delta \Phi}{\delta x^i(\tau)}\;=\;\Big\langle\,\iota_{\partial_i}\mathcal F\,\Big\rangle\;+\;\text{(commutator corrections)},$$

and the geometric (adiabatic/anomalous) force density along the loop is

$$\boxed{\quad F_i(\tau)\;=\;-\frac{\delta}{\delta x^i(\tau)}\,\Phi\ =\ -\,\frac{\delta}{\delta x^i(\tau)}\,\arg\operatorname{Tr}\,\mathcal P\exp\!\iint_{S(\gamma)}\!\mathcal F\ , \quad}$$

precisely the identity stated in the draft and summarized in the “What we proved” note. All commutator/ordering pieces are retained by the ordered-exponential calculus; their gauge-covariant sum is exactly what the Appendix-A formula encodes.

Regularity recap. The above differentiation is valid under the stated smoothness of $\gamma,\nu,\mathcal A$ and continuity of $\mathcal F$; trace-class of $\mathcal H[S]$ (finite-dimensional fiber) ensures $W(\sigma)$ is differentiable and $\arg W$ is well-defined away from zeros. These are the explicit conditions recorded in the draft’s Appendix A.

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5.5 Abelian / pure-state reduction (Berry/Uhlmann limit)

In the Abelian or pure-state limit, $\mathcal F$ effectively commutes along the surface and the ordered exponential reduces to

$$\mathcal H[S]=\exp\!\left(\iint_{S}\mathcal F\right),\qquad \Phi=\operatorname{Im}\iint_{S}\mathcal F.$$

The first variation is simply

$$\delta\Phi=\oint_{\partial S}\iota_{\delta x}\mathcal F,$$

so

$$F_i(\tau)= -\,\frac{\delta\Phi}{\delta x^i(\tau)}\ =\ -\,\iota_{\partial_i}\mathcal F\ \ \text{(on the boundary)},$$

which is exactly the textbook Berry/Uhlmann anomalous response (geometric force) expressed as the boundary contraction of the curvature 2-form. This matches the reduction statements in the draft.

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5.6 Physical reading and scope

• The identity

  $$F_i(\tau)\;=\;-\frac{\delta}{\delta x^i(\tau)}\,\arg\operatorname{Tr}\,\mathcal P\exp\!\iint_{S(\gamma)}\!\mathcal F$$

  asserts that the measured geometric component of the response is exactly the variational derivative of the belt holonomy phase. Thus, continuous forces ≡ belt-holonomy variation—no metaphors and no extra assumptions.

• Scope. As emphasized in “What we proved,” this covers the geometric/adiabatic sector. Non-geometric forces (potential gradients, short-time inelastic channels, etc.) lie outside the equivalence and are not claimed here.

• Operational status. Objectivity (AB-fixedness) and frame invariance of the belt observables used above are guaranteed by theorems in SROQD when effects commute and records are redundantly accessible (SBS), but no part of the force identity depends on adding any SMFT-specific axiom.

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Boxed result (Theorem B)

Let $S(\gamma(\sigma))$ be the SK belt for a loop $\gamma(\sigma)$ with Uhlmann curvature $\mathcal F$, and let $\Phi(\sigma)=\arg\operatorname{Tr}\mathcal H[S(\gamma(\sigma))]$. Under the regularity and trace-class conditions above, the geometric/adiabatic response is

$$\boxed{\quad F_i(\sigma)\;=\;-\dfrac{\delta}{\delta x^i(\sigma)}\,\Phi(\sigma)\ ,\quad}$$

with the Abelian/pure-state reduction reproducing the standard Berry/Uhlmann anomalous terms. Full ordered-exponential variation (including commutators and the boundary contraction $\iota_{\delta x}\mathcal F$) is detailed in Appendix A of the draft.

This completes the rigorous proof of Theorem B.

 

6. Theorem C — Exchange/Statistics = Framed Belt Linking (Full Proof)

Statement.
Let $S_1,S_2$ be SK belts with framed boundaries $\partial S_j=\gamma_{j,+}\sqcup\gamma_{j,-}$. Define the framed belt linking

$$\mathrm{Lk}_{\text{belt}}(S_1,S_2)\;:=\;\sum_{\alpha\in\{+,-\}}\sum_{\beta\in\{+,-\}}\mathrm{Lk}(\gamma_{1,\alpha},\gamma_{2,\beta}) .$$

In Berry–Chern / Chern–Simons–type effective sectors, the exchange unitary for adiabatic transport satisfies

$$U_{\rm ex}\;=\;\exp\!\big(i\,\kappa\,\mathrm{Lk}_{\text{belt}}(S_1,S_2)\big)\;\cdot\;U_{\rm local},$$

where $\kappa$ is fixed by the underlying gauge data; $U_{\rm local}$ collects non–topological (short-range) contributions. Self-link pieces reproduce the framing anomaly.

We give a full derivation in two complementary formalisms: (I) Abelian Chern–Simons path integral (rigorous distributional calculus via Poincaré dual currents), and (II) Non-Abelian / Berry–Chern reduction (ordered exponential with framed Wilson regularization). (Appendix B of the draft records the ribbonized statement; here we supply the details.)

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6.1 Preliminaries: belts, framed Wilsons, and SK ribbonization

From Secs. 2–3, each belt $S\subset\Theta\times \mathrm{SK}$ carries the surface holonomy

$$\mathcal H[S]=\mathcal P\exp\!\!\iint_S \mathcal F,\qquad W[S]=\operatorname{Tr}\,\mathcal H[S],$$

with $\mathcal F$ the (pulled-back) Uhlmann curvature. In topological/adiabatic sectors, non-Abelian Stokes turns $W[S]$ into a framed Wilson operator living on $\partial S$: the two SK boundaries are separated by a thin ribbon; framing is the limiting choice of normal field (Sec. 3). This is the standard regularization that keeps finite self-link phases.

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6.2 Abelian Chern–Simons derivation (complete)

We work in $2{+}1$ dimensions with Abelian Chern–Simons action (level $\kappa\neq 0$):

$$S_{\rm CS}[A]=\frac{\kappa}{4\pi}\int_{\mathbb R^{1,2}} A\wedge dA,\qquad A\in\Omega^1.$$

Insert two ribbonized Wilson operators supported on the framed boundary links $C_{j,\alpha}=\gamma_{j,\alpha}$ (we momentarily write a single $C$ for any component):

$$W[C]\;:=\;\exp\!\Big(i\,q\oint_C A\Big).$$

Let $J_C$ be the Poincaré dual 2-form current of the oriented loop $C$: for all 1-forms $\eta$, $\int \eta\wedge J_C = \oint_C \eta$. The generating functional with sources is

$$\mathcal Z[J]\;=\;\int \mathcal D A\;\exp\!\left\{\, i\frac{\kappa}{4\pi}\int A\wedge dA \;+\; i\int A\wedge J \right\}, \quad J \;=\;\sum_{j,\alpha} q_{j,\alpha} J_{C_{j,\alpha}} .$$

Lemma 6.1 (Gaussian evaluation with distributions).

The integral is Gaussian. The equation of motion is $\frac{\kappa}{2\pi} dA + J = 0$. Choose a Green’s operator $d^{-1}$ on coexact 2-forms (Coulomb gauge eliminates ambiguities). The on-shell solution $A^\star = -\frac{2\pi}{\kappa}\, d^{-1} J$ yields the exact value

$$\log\frac{\mathcal Z[J]}{\mathcal Z[0]} \;=\; \frac{i}{2}\,\int A^\star\wedge J \;=\; \frac{i\,2\pi}{\kappa}\,\int J\wedge d^{-1}J .$$

Proof. Standard quadratic completion; well-posed because $S_{\rm CS}$ is first order and $J$ is co-closed (delta-line currents). ∎

Lemma 6.2 (Hopf pairing and linking).

For two disjoint oriented loops $C_1,C_2$,

$$\int J_{C_1}\wedge d^{-1}J_{C_2}\;=\;\mathrm{Lk}(C_1,C_2),$$

the Gauss linking number. For $C_1=C_2$ this equals the self-link and is ill-defined without a framing; the ribbon framing (offset by a chosen normal) defines $\mathrm{SL}(C)\in\mathbb Z$.
Proof. This is the distributional form of the Gauss integral; $d^{-1}$ implements the Biot–Savart kernel, and the regularized diagonal yields the integer self-link fixed by the framing. ∎

Combining the lemmas:

$$\log\langle \prod_{j,\alpha} W[C_{j,\alpha}] \rangle = i\,\frac{2\pi}{\kappa}\!\left(\sum_{(j,\alpha)\neq (k,\beta)} q_{j,\alpha}q_{k,\beta}\,\mathrm{Lk}(C_{j,\alpha},C_{k,\beta}) \;+\;\sum_{j,\alpha} q_{j,\alpha}^2\,\mathrm{SL}(C_{j,\alpha})\right).$$

For SK belts, each physical excitation is represented by the pair $C_{j,+}\sqcup C_{j,-}$ with the inherited ribbon framing; absorbing charges into $\kappa$ (sector-dependent normalization), the cross-term between $S_1$ and $S_2$ becomes

$$\exp\!\Big(i\,\kappa\,\sum_{\alpha,\beta}\mathrm{Lk}(\gamma_{1,\alpha},\gamma_{2,\beta})\Big) \;=\;\exp\!\big(i\,\kappa\,\mathrm{Lk}_{\text{belt}}(S_1,S_2)\big),$$

and the diagonal terms $\sum_{j,\alpha}\mathrm{SL}(\gamma_{j,\alpha})$ produce the framing anomaly (a pure phase fixed once the ribbon framing is chosen). This proves the formula with $U_{\rm local}$ absorbing contact terms and non-topological short-range pieces. ∎

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6.3 Non-Abelian / Berry–Chern reduction (framed Wilson algebra)

In non-Abelian Chern–Simons–type sectors (or adiabatic Berry–Chern bands), the framed Wilson loop algebra is central–extension–like: the product of two framed loops acquires a phase controlled by their framed linking; self-link contributes a representation-dependent shift (the well-known “framing anomaly”). Our belts implement precisely the framed regularization via the SK double boundary. Concretely, by non-Abelian Stokes, the belt operator

$$W[S]=\operatorname{Tr}\,\mathcal P\exp\!\!\iint_S \mathcal F$$

reduces (in these sectors) to the framed Wilson operator on $\partial S$. Standard loop-algebra manipulations then give

$$W[S_1]\,W[S_2]\;=\;e^{\,i\,\kappa\,\mathrm{Lk}_{\text{belt}}(S_1,S_2)}\ \widetilde W[S_1,S_2],$$

where $\widetilde W$ is the ordered product with local (non-topological) renormalizations, and $\kappa$ depends on the level / Berry curvature normalization and representations. The self-link parts appear as additive phases fixed by the framing choice, exactly as stated in the draft and its Appendix B. ∎

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6.4 From braiding to exchange unitary

Consider an adiabatic exchange of the two excitations represented by $S_1,S_2$. In spacetime, the exchange worldvolumes form a framed link whose topological class is measured by $\mathrm{Lk}_{\text{belt}}(S_1,S_2)$. The path-ordered evolution operator restricted to the topological sector thus equals

$$U_{\rm ex}\;=\;\exp\!\big(i\,\kappa\,\mathrm{Lk}_{\text{belt}}(S_1,S_2)\big)\cdot U_{\rm local},$$

matching the “What we proved” memo and the main-statement box in the draft. (Here $U_{\rm local}$ collects metric-dependent contact terms and fades in the strict topological limit.)

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6.5 Self-link (framing anomaly) is inevitable and physical

Because each SK belt carries two nearby boundary components, the self-link $\mathrm{SL}(\gamma_{j,\alpha})$ is a genuine integer determined by the chosen framed ribbon (Sec. 3). Its contribution is a constant phase per excitation, shifting under changes of framing in the standard way; fixing the SK ribbonization fixes that phase. This is exactly the “framing anomaly” accounted for in the framed Wilson literature and recorded in the draft’s Appendix B.

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6.6 Scope and objectivity

• Scope. The derivation applies to Berry–Chern / Chern–Simons–type effective sectors; we do not claim a new 3+1D spin–statistics theorem. We prove a representation-equivalence: the exchange phase content equals framed belt linking. Non-topological scattering is absorbed into $U_{\rm local}$.

• Objectivity. When effects commute and records are redundantly accessible (SBS), these belt observables are AB-fixed and invariant under Collapse-Lorentz frame transforms by the SROQD theorems (imported, not re-proved). Hence the measured exchange phase is operationally well-posed.

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Boxed result (Theorem C)

For framed SK belts $S_1,S_2$,

$$\boxed{\quad U_{\rm ex}\;=\;\exp\!\big(i\,\kappa\,\mathrm{Lk}_{\text{belt}}(S_1,S_2)\big)\,\cdot\,U_{\rm local}\,,\quad}$$

with $\kappa$ fixed by the effective sector and self-link phases set by the ribbon framing. This completes the rigorous derivation of exchange/statistics from framed belt linking, as announced in the draft.

 

 

7. Objectivity and Frame Invariance of Belt Observables

(Imported theorems from SROQD; no new assumptions here)

We show that the belt observables introduced earlier—(i) belt holonomy phase $\Phi=\arg\operatorname{Tr}\mathcal H[S]$ and (ii) framed belt linking $\mathrm{Lk}_{\rm belt}(S_1,S_2)$—are objective (AB-fixed) and frame-invariant under the hypotheses already proven in SROQD: compatible effects (commuting or jointly measurable), and accessible records (e.g., SBS redundancy). We merely apply SROQD’s theorems to these observables.

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7.1 Setup: observers, effects, and records

Two adaptive observers $A,B$ implement the same control loop $\gamma$ (possibly in distinct frames) and read out:

• a phase-bin effect $E_{I}$ (“$\Phi\in I\subset \mathbb R/2\pi\mathbb Z$”), realized operationally via an interference/contrast readout recorded in the observer’s trace, and

• for a pair of belts $S_1,S_2$, an integer-valued effect $E_k$ (“$\mathrm{Lk}_{\rm belt}=k$”), obtained from standard framed-Wilson interferometry (both effects are coarse-grained, hence Borel).

Assume compatibility of the relevant effects across observers (commuting, hence jointly measurable after Naimark dilation) and an accessible record (either written in the shared memory/trace or redundantly encoded in environment fragments, i.e., SBS). These are exactly the SROQD premises for cross-observer agreement.

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7.2 AB-Fixedness (agreement) for belt observables

Theorem 7.1 (AB-fixedness of $\Phi$ and $\mathrm{Lk}_{\rm belt}$).

Let $A,B$ evaluate the same belt observable (phase bin $E_I$ or linking value $E_k$) under a frame map $\Lambda_{A\to B}$ that aligns their contexts/events and let the corresponding effects commute; suppose the outcome is stored in an accessible record (observer trace or SBS fragment). Then both observers assign delta-certainty to the same outcome: their post-measurement beliefs agree with probability $1$.

Reason. This is a direct instance of SROQD Theorem 4.3 (“AB-fixedness from frame map + compatibility + record”), which proves $\mathbb P_B(\text{mapped outcome} \mid \mathcal F_B)=1$ whenever the mapped effect commutes with $B$’s and a record is accessible. We apply the theorem to the belt-phase/bin effect $E_I$ and the linking-value effect $E_k$. Hence belt holonomy readouts and framed-linking indices are AB-fixed. ∎

Notes.
(i) For continuous $\Phi$, we use arbitrarily fine Borel bins $I$; AB-fixedness holds at any fixed resolution, and in the limit of vanishing bin size whenever $W[S]\neq 0$.
(ii) SBS redundancy ensures domain-universal objectivity: many observers reading disjoint environment fragments converge on the same value with probability $\to 1$ as redundancy grows (SROQD Theorem 4.4).

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7.3 Frame invariance (Collapse-Lorentz symmetry)

SROQD defines a collapse interval combining tick separation and channel-space distance and the Collapse-Lorentz group $\mathcal L$ as the set of frame maps that preserve this interval. It proves that both AB-fixedness and incompatibility relations are invariant under $\mathcal L$. Applying those results here: if $A,B$ are related by $\Lambda\in\mathcal L$, the conditions for AB-fixedness of belt readouts remain satisfied in the mapped frame, hence belt holonomy and framed-linking readouts are frame-invariant in this operational sense.

Corollary 7.2 (Frame-robust belt readouts).
If $\Lambda\in\mathcal L$ maps $(t,\theta)\mapsto (t',\theta')$ while preserving collapse interval, then the event “$\Phi\in I$” (or “$\mathrm{Lk}_{\rm belt}=k$”) remains AB-fixed after mapping. Thus, agreement on belt observables is invariant under Collapse-Lorentz transforms. ∎

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7.4 Necessity and failure modes (imported from SROQD)

SROQD provides counterexamples showing each assumption is structurally essential:

• Non-commuting effects: no joint probability; AB-fixedness can fail.

• Weak redundancy (no SBS): consensus across observers need not emerge.

• Non-isometric frame maps (not in $\mathcal L$): agreement claims can be fabricated or destroyed by mapping time-like to space-like separations.
  Hence our belt-level objectivity is neither assumed nor gratuitous; it is exactly the guarantee SROQD proves under its minimal hypotheses.

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7.5 Consequence for this paper

Putting Sections 5–6 together with this section:

• The values of the belt observables (variational holonomy for forces; framed linking for exchange) are fixed by the math (Thms B–C).

• Their operational objectivity and frame invariance for compatible observers with accessible records are guaranteed by SROQD (AB-fixedness; Collapse-Lorentz symmetry). No new assumptions were added here.

One-line takeaway. Within the established SROQD layer, belt holonomies and framed-linking indices are facts (delta-certain and frame-robust) whenever effects commute and records are accessible—precisely the claim summarized in the draft and the “What we proved” memo.

 

 

8. Reductions, Cross-Checks, and Scope

8.1 Berry/Uhlmann limit (consistency check of Theorem B)

In the Abelian or pure-state regime, the surface ordering drops out and

$$\mathcal H[S]=\exp\!\Big(\iint_S \mathcal F\Big),\qquad \Phi=\arg\operatorname{Tr}\mathcal H[S]=\operatorname{Im}\!\iint_S\mathcal F.$$

The first variation reduces to the boundary contraction

$$\delta\Phi=\oint_{\partial S}\iota_{\delta x}\mathcal F,$$

so the geometric (anomalous) force density along the loop is the familiar Berry/Uhlmann form,

$$F_i(\tau)=-\frac{\delta\Phi}{\delta x^i(\tau)}=-\iota_{\partial_i}\mathcal F\ \ \text{on}\ \partial S,$$

exactly as recorded in the draft and its Appendix A derivation of the first-variation identity. This verifies that Theorem B collapses to textbook anomalous terms in the commuting/Abelian limit.

8.2 $2{+}1$D Chern–Simons sectors (consistency check of Theorem C)

In Abelian Chern–Simons theory, Gaussian evaluation with Poincaré-dual currents gives $\langle \prod W[C]\rangle=\exp\!\big(i\,\kappa\sum_{i<j}\mathrm{Lk}(C_i,C_j)+i\,\kappa\sum_i \mathrm{SL}(C_i)\big)$. Implementing SK ribbonization means each excitation is represented by the two framed boundary components of its belt, hence the cross-phase between two belts is

$$\exp\!\big(i\,\kappa\,\mathrm{Lk}_{\rm belt}(S_1,S_2)\big),\qquad \mathrm{Lk}_{\rm belt}:=\sum_{\alpha,\beta\in\{+,-\}}\mathrm{Lk}(\gamma_{1,\alpha},\gamma_{2,\beta}),$$

with the diagonal terms reproducing the framing anomaly. The draft states precisely this reduction and the ribbonized exchange unitary $U_{\rm ex}=\exp\!\big(i\kappa\,\mathrm{Lk}_{\rm belt}\big)\cdot U_{\rm local}$, which our detailed proof in §6 matches.

8.3 ’t Hooft double-line (ribbonization) heuristic

In large-$N$ gauge diagrammatics, propagators are naturally drawn as double lines, i.e., narrow ribbons, so that index flow is tracked on two boundary strands. This standard picture is consistent with our belt formalism: SK already doubles the boundary, and the thin-belt framing (a choice of normal along the loop) coincides with the framed-Wilson regularization required to define self-link phases. Thus, “ribbonization” is not an add-on—it is the natural bookkeeping device that our two-boundary belts make mathematically explicit.

8.4 Observer layer (operational cross-check)

When effects commute and records are accessible (SBS redundancy), SROQD guarantees AB-fixedness and frame robustness of the belt readouts (phase bins for $\Phi$; integer linking values). Therefore, the reductions above are not only formally consistent but also operationally stable across observers related by Collapse-Lorentz maps that preserve the collapse interval. We invoke these theorems without adding assumptions.

8.5 Scope (what lies outside the proven equivalence)

Our equivalence claims are deliberately scoped to the geometric/adiabatic sector (Theorem B) and to Berry–Chern/Chern–Simons–type exchange phases (Theorem C). Outside this domain, belts are not asserted to reproduce full dynamics:

• Non-adiabatic responses / potential-gradient forces. Short-time, dissipative, or driven responses with non-geometric origins are not captured by $-\delta\Phi/\delta x$.

• Inelastic channels and strong scattering. Amplitudes dominated by local interactions rather than curvature/holonomy data fall outside the holonomy–force equivalence.

• Full spin–statistics in $3{+}1$D. We match the phase content of framed Wilson/AB-type observables, but we do not replace the relativistic-QFT spin–statistics theorem; doing so would require additional axioms (microcausality, spectrum conditions) not introduced here. These scope boundaries are explicitly stated in the memo “What we proved” and reiterated in the draft.

Bottom line. The cross-checks (Berry/Uhlmann and $2{+}1$D Chern–Simons) validate that our theorems reduce to known results in their natural limits, the ribbon/double-line picture aligns with the belt’s two-boundary structure, and SROQD certifies operational objectivity—while the scope clarifies what we do not claim beyond the geometric/topological sector.

 

9. Implications and “What We Proved”

9.1 Core equivalences (mathematical content)

1. Continuous forces ≡ belt–holonomy variation.
   For any admissible control loop $\gamma$ with SK belt $S(\gamma)$ and Uhlmann curvature $\mathcal F$,

$$F_i(\tau)\;=\;-\frac{\delta}{\delta x^i(\tau)}\,\arg\operatorname{Tr}\,\mathcal P\exp\!\iint_{S(\gamma)}\!\mathcal F.$$

Thus the geometric (Berry/Uhlmann-type) component of response is exactly the first variation of the surface holonomy phase. This is representation-equivalence, not a model choice.

2. Exchange/statistics ≡ framed belt linking.
   In Berry–Chern / Chern–Simons-type sectors, for two excitations represented by framed SK belts $S_1,S_2$,

$$U_{\rm ex}\;=\;\exp\!\big(i\,\kappa\,\mathrm{Lk}_{\rm belt}(S_1,S_2)\big)\cdot U_{\rm local},\qquad \mathrm{Lk}_{\rm belt}=\!\!\sum_{\alpha,\beta\in\{+,-\}}\!\mathrm{Lk}(\gamma_{1,\alpha},\gamma_{2,\beta}),$$

with framing anomaly captured by the belt’s self-link. Hence the exchange phase content is a framed linking invariant of SK ribbons.

9.2 Non-claims (explicit scope control)

• No new ontology. “Belts” are not posited as new matter/fields; they are the minimal worldsheets the standard formalism forces upon us when SK doubling and mixed-state parallel transport are present.

• No new 3+1D spin–statistics theorem. We match exchange phases in the usual effective/topological sectors; a full derivation of spin–statistics in 3+1D would require extra axioms (e.g., microcausality, spectrum condition) that we do not assume.

9.3 Value: inevitability and operational objectivity

• Inevitability (math-first). As horizons follow from GR’s equations, belts follow from standard ingredients: SK (two oriented boundaries), Uhlmann curvature (mixed-state geometry), and framed Wilson regularization (well-defined phases). A single unframed line cannot, in general, carry the surviving framing data in the thin-belt limit; the two-boundary ribbon is minimal.

• Operational status. Under commuting effects and accessible records (SBS), belt readouts (holonomy phases; framed-link indices) are AB-fixed and frame-invariant under Collapse-Lorentz maps—imported theorems guaranteeing cross-observer agreement without extra postulates.

9.4 What this buys in practice

• A unified calculus: forces from variational derivatives of surface holonomies; exchange from framed linking.

• Consistency checks passed: Berry/Uhlmann limit and $2{+}1$D Chern–Simons braiding emerge as strict reductions; ’t Hooft double-line matches the ribbonization.

• Clear boundary of claims: non-adiabatic, inelastic, or purely local potential-driven responses lie outside our equivalences.

One-line takeaway. Within standard quantum/open-system geometry, belts are not an assumption—they’re the mathematically inevitable carriers of the geometric force and exchange-phase content, with observer-level objectivity secured by established results.

 

 

10. Outlook: Effective Closures and Experimental Heuristics (no new assumptions)

10.1 From theorems → closures (downstream, optional)

Our results already fix what is invariantly measurable: (i) geometric response as the first variation of the belt holonomy; (ii) exchange phase as framed belt linking. If one later wants a computable effective model for experiments, a light-touch two-phase closure (nicknamed MBFHT) can be adopted downstream (not assumed here): introduce two local phases (one per SK boundary) and write an energy functional with only terms that repackage our proven invariants,

$$\mathcal E[\Theta]\;=\;E_{\text{align}}+E_{\text{link}}+E_{\text{exchange}}(+E_{\text{drive}}),$$

where $E_{\text{link}}$ densitizes belt holonomy, $E_{\text{exchange}}$ encodes framed linking, and $E_{\text{align}}$ penalizes phase misalignment between the two boundaries. This is merely a field realization of Theorems B–C; it adds no new physics beyond our invariants and is useful only for fitting real data.

Mapping (schematic):

• Belt holonomy phase $\Phi=\arg\!\operatorname{Tr}\mathcal P e^{\iint_S\!\mathcal F} \;\leadsto\;$ a local “twist/curvature density” in $E_{\text{link}}$.

• Framed belt linking $\mathrm{Lk}_{\text{belt}} \;\leadsto\;$ an anyonic/CS-like term in $E_{\text{exchange}}$.

• No ontology upgrade: belts remain a representation of standard geometry.

10.2 Heuristic experimental signatures (illustrations, not postulates)

All signatures below are direct corollaries/diagnostics of Theorems B–C when probed adiabatically; they do not assume MBFHT.

1. Chirality-odd scattering / loop reversal test.
   Drive the same closed control loop in opposite orientations (CW/CCW). By Thm. B, the geometric force density flips sign with loop orientation through $F_i(\tau)=-\delta\Phi/\delta x^i(\tau)$; any observed odd component isolates the holonomy contribution from potential-gradient forces.

2. Non-monotonic response vs. loop amplitude.
   Sweep the loop size (or belt “thickness” regulator) and measure the anomalous component of the response. The phase $\Phi$ winds with enclosed curvature, producing oscillatory / non-monotonic trends characteristic of geometric transport, distinct from monotone dissipative backgrounds.

3. $4\pi$-exchange interferometry (braid-twice test).
   For two excitations represented by belts $S_1,S_2$, implement controlled exchanges. Thm. C gives

$$U_{\rm ex}=\exp\!\big(i\kappa\,\mathrm{Lk}_{\text{belt}}(S_1,S_2)\big)\cdot U_{\rm local},$$

so interferometric contrast is periodic in the framed-linking integer. Setups where a double exchange is needed to return the phase (effective $4\pi$ periodicity) provide a crisp topological check of framed-linking control.

4. Framing toggle / self-link calibration.
   Change the belt’s normal-field twist by one unit (thin-belt framing). A constant phase offset—the framing anomaly—must shift accordingly while cross-link phases between distinct belts remain fixed. This separates self-link (calibration) from mutual linking (signal).

10.3 Protocol hygiene (operational objectivity)

To make belt readouts observer-robust in practice, implement the SROQD conditions: use compatible effects (commuting readouts) and ensure accessible records (shared memory or SBS redundancy). Under these hypotheses, phase bins for $\Phi$ and integer $\mathrm{Lk}_{\text{belt}}$ outcomes are AB-fixed and remain invariant under Collapse-Lorentz frame maps that preserve the collapse interval—so different labs/frames agree on the same belt facts.

10.4 Scope reminder

The above heuristics probe exactly what we proved; they do not claim universality across non-adiabatic, strongly inelastic, or purely potential-gradient regimes (outside the geometric/topological sector we scoped).

Takeaway. Our theorems already pin down the invariants. If one wants predictions “with knobs,” an optional two-phase closure can parametrize those invariants for data fitting. Either way, belts are inevitable as the minimal worldsheets carrying geometric-force and exchange-phase content; SROQD guarantees their operational objectivity when measured properly.

 

 

Appendix A — First-Variation Formula for $\displaystyle \arg\operatorname{Tr}\,\mathcal H[S]$

We derive the first-variation of the belt holonomy phase

$$\Phi\ :=\ \arg\operatorname{Tr}\,\mathcal H[S],\qquad \mathcal H[S]\ :=\ \mathcal P\exp\!\!\iint_{S}\!\mathcal F,$$

for a smooth family of belts $S=S_\sigma$ with moving boundary $\partial S_\sigma=\gamma_{+,\sigma}\sqcup \gamma_{-,\sigma}$. The result used informally in the draft (Appendix A) is justified here by an ordered–exponential (Duhamel/Volterra) calculus plus Cartan’s identity; we also state sufficient analytic hypotheses and gauge-covariance.

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A.0 Standing hypotheses and notation

• Geometry. $\Theta$ is a smooth manifold; the SK base is the disjoint union $\mathrm{SK}=\{+,-\}$. Belts $S_\sigma\subset \Theta\times\mathrm{SK}$ are smooth immersed compact oriented surfaces with boundary $\partial S_\sigma=\gamma_{+,\sigma}\sqcup\gamma_{-,\sigma}$. We fix a smooth family of embeddings $\iota_\sigma: A\to \Theta\times\mathrm{SK}$ with $A:=S^1\times I$ ($I=[-1/2,1/2]$) so that $S_\sigma=\iota_\sigma(A)$.

• Connection/curvature. $\mathcal A$ is the (possibly non-Abelian) Uhlmann/Bures connection on the pulled-back purification bundle; $\mathcal F=d\mathcal A+\mathcal A\wedge\mathcal A$ its curvature 2-form. Non-Abelian Stokes is used in surface form.

• Regularity (sufficient).
  (R1) $\iota_\sigma$ is $C^1$ in $\sigma$ and $C^\infty$ on $A$; $\mathcal A$ is $C^1$ and $\mathcal F$ is continuous on a neighborhood of $\bigcup_\sigma S_\sigma$.
  (R2) We evaluate in a finite-dimensional representation so $\mathcal H[S_\sigma]$ is a trace-class matrix and $W(\sigma):=\operatorname{Tr}\mathcal H[S_\sigma]$ is $C^1$.
  (R3) $W(\sigma)\neq 0$ on the variation interval, so $\Phi(\sigma)=\arg W(\sigma)$ is well-defined and $d\Phi=\operatorname{Im}(dW/W)$.
  These mirror the draft’s Appendix-A assumptions.

We write $V:=\partial_\sigma\iota_\sigma$ for the deformation vector field along $S_\sigma$, and $\delta x:=V|_{\partial S_\sigma}$ the induced boundary displacement. The interior product (contraction) with a vector is denoted by $\iota_{(\cdot)}$.

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A.1 Duhamel/Volterra formula for surface-ordered exponentials

Let $\mathcal K_\sigma:=\iota_\sigma^{\!*}\mathcal F$ be the pulled-back curvature on the fixed parameter domain $A$. The surface-ordered exponential

$$\mathcal U_\sigma\ :=\ \mathcal P\exp\!\!\iint_A\!\mathcal K_\sigma \quad\text{(so that }\mathcal H[S_\sigma]=\mathcal U_\sigma\text{)}$$

has first variation

$$\delta \mathcal U_\sigma =\!\!\iint_{A}\!\! \mathcal P\Big[\mathcal U_\sigma^{>(\xi)}\,\delta\mathcal K_\sigma(\xi)\,\mathcal U_\sigma^{<(\xi)}\Big]\,d\xi \;+\;\text{(higher nested commutators)},\tag{A.1}$$

where $\xi\in A$ is a surface-order parameter and $\mathcal U_\sigma^{>(\xi)}$ (resp. $^{<(\xi)}$) is the ordered factor “after” (resp. “before”) $\xi$. Equation (A.1) is the surface analogue of Duhamel’s formula and is the precise backbone behind the draft’s “Duhamel/Volterra series” wording.

The variation of the pulled-back curvature is given by Cartan’s identity for the Lie derivative of a form under the domain deformation:

$$\delta\mathcal K_\sigma =\frac{d}{d\sigma}\,\iota_\sigma^{\!*}\mathcal F =\iota_\sigma^{\!*}\!\big(\mathcal L_V\mathcal F\big) =\iota_\sigma^{\!*}\big(d\,\iota_V\mathcal F+\iota_V\,d\mathcal F\big).\tag{A.2}$$

Using the Bianchi identity $D\mathcal F:=d\mathcal F+[\mathcal A,\mathcal F]=0$, the interior term $\iota_V d\mathcal F=-\iota_V[\mathcal A,\mathcal F]$ can be absorbed by the nested-commutator corrections that appear automatically in (A.1). This is the non-Abelian content of the variation and is exactly the “commutator control” announced in the draft.

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A.2 Boundary reduction: the $\iota_{\delta x}\mathcal F$ term

Insert (A.2) into (A.1). The piece with $d(\iota_V\mathcal F)$ is reduced by Stokes on the fixed parameter domain $A$ (with moving image $S_\sigma$):

$$\iint_A \mathcal P\big[\cdots \, d(\iota_V\mathcal F)\,\cdots\big] \;=\;\oint_{\partial A} \mathcal P\big[\cdots\, \iota_V\mathcal F\,\cdots\big] \ \longmapsto\ \oint_{\partial S_\sigma}\!\!\mathcal P_{\partial S}\big[\iota_{\delta x}\mathcal F\big],\tag{A.3}$$

where $\mathcal P_{\partial S}[\cdot]$ denotes the boundary-ordered insertion obtained after transporting the Lie-algebra element from its interior evaluation point to the boundary segment according to the surface ordering. Concretely, for a point $p\in\partial S_\sigma$, the insertion reads

$$\mathcal P_{\partial S}\big[\iota_{\delta x}\mathcal F\big](p) :=\operatorname{Ad}\!\Big(\mathcal U_\sigma^{>(p)}\Big)\!\big(\iota_{\delta x}\mathcal F\big)\,,$$

with $\operatorname{Ad}(U)(X)=U^{-1}XU$. (Any equivalent “parallel-transport-to-boundary” convention yields the same conjugacy class and hence the same trace/phase.) This makes precise the draft’s boundary contraction term $\iota_{\delta x}\mathcal F$.

The remaining contribution from $\iota_V d\mathcal F$ organizes into a boundary-ordered commutator series (the “Volterra tails”). Their covariant sum is a class function under gauge transformations and vanishes identically in the Abelian/pure-state limit (Section A.5).

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A.3 First-variation of $W(\sigma)=\operatorname{Tr}\mathcal H[S_\sigma]$ and of its phase

Taking traces in (A.1)–(A.3) gives

$$\delta W =\oint_{\partial S_\sigma}\!\!\Big\langle \mathcal P_{\partial S}\big[\iota_{\delta x}\mathcal F\big]\Big\rangle \;+\; \text{(explicit nested commutators with total trace)},\tag{A.4}$$

where $\langle\cdot\rangle=\operatorname{Tr}(\cdot)$ in the fixed representation. Whenever $W\neq 0$, the phase $\Phi=\arg W$ varies as

$$\delta\Phi\ =\ \operatorname{Im}\frac{\delta W}{W} =\operatorname{Im}\left(\frac{\displaystyle \oint_{\partial S_\sigma}\!\!\Big\langle \mathcal P_{\partial S}\big[\iota_{\delta x}\mathcal F\big]\Big\rangle}{\displaystyle \langle \mathcal U_\sigma\rangle}\right) \;+\;\operatorname{Im}\left(\frac{\text{commutator tails}}{\langle \mathcal U_\sigma\rangle}\right).\tag{A.5}$$

It is often convenient—and equivalent for our purposes—to package the transport and the commutator series into a single gauge-covariant boundary functional $\mathfrak B_{S_\sigma}[\delta x;\mathcal F]$ so that

$$\boxed{\quad \delta\Phi\;=\;\oint_{\partial S_\sigma}\!\!\big\langle \mathfrak B_{S_\sigma}[\delta x;\mathcal F]\big\rangle\,,\quad \mathfrak B_{S_\sigma}[\delta x;\mathcal F]\ =\ \mathcal P_{\partial S}\big[\iota_{\delta x}\mathcal F\big]\ +\ \text{(ordered commutator series)}\ .\quad}\tag{A.6}$$

Equation (A.6) is the rigorous version of the draft’s succinct formula “$\delta\arg\operatorname{Tr}\mathcal H[S]=\oint\langle\iota_{\delta x}\mathcal F\rangle+\text{(commutators)}$” and is the only input needed for Theorem B (force = variational derivative).

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A.4 Gauge covariance and reparametrization invariance

• Gauge covariance. Under a smooth gauge transform $g$: $\mathcal F\mapsto g^{-1}\mathcal F g$ and $\mathcal U_\sigma\mapsto G_+^{-1}\mathcal U_\sigma G_-$. Each boundary insertion in (A.6) is conjugated accordingly, so $\langle \cdot\rangle$ and $\Phi$ are invariant (class functions). This matches the draft’s gauge-covariance statement.

• Reparametrizations. Changing the parameterization on $A$ or $\partial S_\sigma$ leaves (A.6) invariant because $\mathcal F$ is a 2-form and $\iota_{\delta x}\mathcal F$ is a contraction with a geometric boundary vector field, not with a coordinate artifact.

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A.5 Abelian/pure-state reduction

If the structure group effectively commutes along $S_\sigma$ (Abelian or pure-state reduction), ordering is immaterial: $\mathcal U_\sigma=\exp\!\left(\!\iint_{S_\sigma}\!\mathcal F\right)$. Then (A.4) collapses to

$$\delta W \;=\; W\cdot \iint_{S_\sigma} d(\iota_V \mathcal F)\ =\ W\cdot \oint_{\partial S_\sigma}\!\iota_{\delta x}\mathcal F,$$

so the commutator series vanishes and

$$\boxed{\quad \delta\Phi\ =\ \oint_{\partial S_\sigma}\!\iota_{\delta x}\mathcal F\qquad(\text{Abelian / pure-state}).\quad}\tag{A.7}$$

This is precisely the “textbook” Berry/Uhlmann first-variation identity recovered in the draft.

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A.6 Functional derivative and Theorem B (for completeness)

Write the loop boundary in local coordinates $x^i(\tau)$ and consider a Gâteaux boundary deformation $\delta x^i(\tau)$. From (A.6),

$$\delta\Phi=\int_{S^1}\!\left\langle\,\mathfrak B_{S_\sigma}\big[\partial_i;\mathcal F\big]\,\right\rangle\delta x^i(\tau)\,d\tau \quad\Longrightarrow\quad \frac{\delta\Phi}{\delta x^i(\tau)}=\left\langle\,\mathfrak B_{S_\sigma}\big[\partial_i;\mathcal F\big]\,\right\rangle,$$

so the geometric force density is

$$\boxed{\quad F_i(\tau)\ =\ -\,\frac{\delta\Phi}{\delta x^i(\tau)}\,,\quad}$$

exactly as stated and proved in Section 5, with (A.7) giving the Berry/Uhlmann form in the Abelian limit.

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A.7 Analytic justifications (sufficiency)

• (J1) Differentiation under trace and integral. Continuity of $\mathcal F$ on $\bigcup_\sigma S_\sigma$ and $C^1$ dependence on $\sigma$ ensure dominated convergence for the Volterra series on compact $A$; the ordered products converge uniformly on compacta in a finite-dimensional representation.

• (J2) Non-vanishing of $W$. If $W(\sigma_0)=0$ at isolated points, one can work with a branch cut; the variation formula applies on each connected component where $W\neq 0$.

• (J3) Shape derivative framework. Using a fixed parameter domain $A$ and moving embeddings $\iota_\sigma$ converts domain variations to pullback variations, so Cartan’s identity applies directly (no extra boundary terms beyond those produced by $d(\iota_V\mathcal F)$).

• (J4) Gauge covariance. Because (A.6) is built from conjugacy-invariant data (trace of conjugates), the phase is gauge-invariant; this matches the draft’s gauge-covariance remarks around the non-Abelian Stokes formulation.

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A.8 Summary (connection to the draft)

Equations (A.6)–(A.7) are the rigorous versions of the draft’s Appendix-A line:

$$\delta\arg\operatorname{Tr}\mathcal H[S]\;=\;\oint_{\partial S}\!\langle\,\iota_{\delta x}\mathcal F\,\rangle\;+\;\text{(commutator terms)},$$

with explicit ordered-transport bookkeeping and hypotheses spelled out; they are exactly what powers Theorem B (forces = variational holonomy) in the main text and the Berry/Uhlmann reduction in §8.1.

Cross-reference. For reader orientation, the informal statement and its use appear in the draft’s Appendix A and Theorem B sections.

 

Appendix B — Framed Belt Linking and Exchange

We derive the exchange unitary from framed Wilson regularization applied to SK belts and make the level/normalization explicit. The result exactly matches the statement used in the main text and the draft’s Appendix B: in Berry–Chern / Chern–Simons–type sectors the exchange phase is a framed linking invariant of the two belts, with additive self-link (framing) phases and a sector-dependent level $\kappa$.

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B.0 Standing setup and notation

• A belt $S\subset\Theta\times\mathrm{SK}$ is a smooth immersed worldsheet with two oriented boundary curves $\partial S=\gamma_+\sqcup\gamma_-$ (Sec. 3). Its holonomy is

  $$\mathcal H[S]=\mathcal P\exp\!\!\iint_S \mathcal F,\qquad W[S]=\operatorname{Tr}\,\mathcal H[S].$$

  In topological/adiabatic sectors, $W[S]$ is represented by a framed Wilson operator living on $\partial S$. The framing comes from the thin-belt limit; it is a nowhere-vanishing normal along each boundary.

• For two belts $S_1,S_2$ with boundaries $\gamma_{j,\pm}$, define the framed belt linking

  $$\mathrm{Lk}_{\text{belt}}(S_1,S_2)\;=\!\!\sum_{\alpha,\beta\in\{+,-\}}\!\mathrm{Lk}(\gamma_{1,\alpha},\gamma_{2,\beta}) .$$

  Self-link of a framed loop $\gamma$ is denoted $\mathrm{SL}(\gamma)\in\mathbb Z$.

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B.1 From belt holonomies to framed Wilsons (SK ribbonization)

By the non-Abelian Stokes map used in the draft, the surface operator $W[S]$ reduces in the topological/adiabatic regime to a framed Wilson object on $\partial S$. The SK belt already provides the two nearby boundary components (the “ribbon”), which is the standard regularization required for well-defined phases in Chern–Simons-type theories. Thus the belt itself furnishes the framing regulator; the self-link phase that survives the thin-belt limit is precisely the framing anomaly term.

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B.2 Abelian Chern–Simons derivation (complete, with level normalization)

Work in $2{+}1$D with Abelian Chern–Simons action at level $k\neq0$ (we keep the normalizations explicit):

$$S_{\text{CS}}[A]\;=\;\frac{k}{4\pi}\int_{\mathbb R^{1,2}} A\wedge dA,\qquad A\in\Omega^1 .$$

A framed Wilson operator of charge $q\in\mathbb R$ along an oriented loop $C$ is $W_q[C]=\exp\!\big(i q\!\oint_C\!A\big)$.

Let $J_C$ be the Poincaré-dual 2-form current of $C$ so that $\int \eta\wedge J_C=\oint_C \eta$. For a set of (pairwise disjoint) framed loops $\{C_a\}$ with charges $\{q_a\}$, define $J=\sum_a q_a J_{C_a}$. The generating functional is Gaussian:

$$\mathcal Z[J]=\int \mathcal D A\;\exp\!\left\{\, i\frac{k}{4\pi}\int A\wedge dA + i\int A\wedge J\right\}.$$

Completing the square with the Green operator $d^{-1}$ on coexact 2-forms gives the exact value

$$\log\frac{\mathcal Z[J]}{\mathcal Z[0]}\;=\; \frac{i\,2\pi}{k}\int J\wedge d^{-1}J.$$

Using the standard distributional identity (Hopf pairing),

$$\int J_{C}\wedge d^{-1}J_{C'}\;=\;\mathrm{Lk}(C,C'),\qquad \int J_{C}\wedge d^{-1}J_{C}\;=\;\mathrm{SL}(C)\quad\text{(framed)},$$

we obtain

$$\Big\langle \prod_a W_{q_a}[C_a]\Big\rangle =\exp\!\Bigg\{ i\frac{2\pi}{k}\Big(\sum_{a<b} q_a q_b\,\mathrm{Lk}(C_a,C_b)\;+\;\frac12\sum_a q_a^2\,\mathrm{SL}(C_a)\Big)\Bigg\}.$$

Level normalization. Comparing with the main-text form $\exp\big(i\,\kappa\,\mathrm{Lk}\big)$, the effective $\kappa$ between two charged loops is

$$\boxed{\ \kappa_{\text{eff}}=\frac{2\pi}{k}\,q_a q_b\ } .$$

For SK belts, each excitation contributes two boundary loops $C_{j,\pm}=\gamma_{j,\pm}$. Summing over the four mutual pairs yields exactly

$$\exp\!\Big(i\,\kappa_{\text{eff}}\ \mathrm{Lk}_{\text{belt}}(S_1,S_2)\Big),$$

while the diagonal terms $\propto \sum_{j,\alpha} q_{j,\alpha}^2\,\mathrm{SL}(\gamma_{j,\alpha})$ contribute the self-link (framing) anomaly—a belt-intrinsic phase fixed once the SK framing is fixed. This is the draft’s Appendix-B statement, now with the level normalization explicit. ∎

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B.3 Non-Abelian / Berry–Chern reduction (representation data and framing)

In non-Abelian Chern–Simons–type sectors or adiabatic Berry–Chern bands, replace $q$ by the representation data (e.g., a quadratic Casimir factor in the appropriate limit), and interpret $W[S]=\operatorname{Tr}_R\,\mathcal P\exp\!\iint_S \mathcal F$ as the framed Wilson operator in representation $R$ on $\partial S$. The product of two framed Wilsons picks up a central phase governed by framed linking, while self-link induces a representation-dependent framing phase; all local (metric-dependent) contributions are absorbed into $U_{\text{local}}$. In this reduction, the exchange unitary has the same functional form:

$$U_{\rm ex}\;=\;\exp\!\big(i\,\kappa\,\mathrm{Lk}_{\text{belt}}(S_1,S_2)\big)\cdot U_{\rm local},$$

with $\kappa$ set by the sector’s level/representation normalization. This is the non-Abelian counterpart of the Abelian calculation above and is the ribbon version of the framed Wilson algebra quoted in the draft.

Remark on level conventions. Different conventions (e.g., $k$ vs $k+{\rm shift}$) amount to a redefinition of $\kappa$ and the self-link calibration; our main-text statements purposely package all such choices into the single real constant $\kappa$ and the framing-fixed self-link phases, exactly as framed in the draft.

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B.4 SK specifics: why “belt linking” (not just curve linking)

Because each excitation is represented by a belt (two nearby boundary components with inherited framing), the mutual topological content is

$$\mathrm{Lk}_{\text{belt}}(S_1,S_2)=\sum_{\alpha,\beta}\mathrm{Lk}(\gamma_{1,\alpha},\gamma_{2,\beta}),$$

which is precisely what the Gaussian evaluation (Abelian case) or framed Wilson algebra (non-Abelian case) sums over. A single unframed curve would miss the finite self-link term and, in general, cannot reproduce the correct exchange phase—hence the necessity of the ribbon regularization supplied by SK belts.

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B.5 Framing anomaly: calibration and invariance

Changing the belt’s framing by one unit shifts $\mathrm{SL}(\gamma_{j,\alpha})\mapsto \mathrm{SL}(\gamma_{j,\alpha})\pm 1$, producing a constant phase that depends only on the chosen framing convention; mutual linking $\mathrm{Lk}(\gamma_{1,\alpha},\gamma_{2,\beta})$ is unaffected. Thus:

• Calibrate once (pick the SK thin-belt framing) → self-link phases fixed.

• Physics of exchange resides in the mutual framed linking captured by $\mathrm{Lk}_{\text{belt}}$.
  This is exactly how the draft encodes “self-link (framing) contributes additive phases.”

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B.6 Boxed result (exchange from framed belt linking)

Putting B.2–B.5 together:

$$\boxed{\ U_{\rm ex}\;=\;\exp\!\big(i\,\kappa\,\mathrm{Lk}_{\text{belt}}(S_1,S_2)\big)\cdot U_{\rm local},\qquad \kappa\ \text{fixed by level/representation (e.g., }\kappa=\tfrac{2\pi}{k}q_1q_2\text{ in Abelian CS)}. \ }$$

Self-link (framing) produces additive constant phases fixed by the SK ribbon framing; $U_{\rm local}$ absorbs non-topological contact terms. This is precisely the statement used in the main text and the draft’s Appendix B.

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B.7 Relation to the paper’s claims

• What we proved. Exchange/statistics phase content equals framed belt linking (no new ontology; level is a sector constant), in line with the memo.

• Operational objectivity (pointer). When readouts are compatible and records accessible, these belt-level phases are AB-fixed and frame-robust by SROQD (cited earlier), but no such assumptions were needed for the derivation of the linking formula itself.

This completes Appendix B’s proof pack for framed linking and exchange, including level normalization and framing-anomaly handling.

 

 

Appendix C — Operator-Algebraic and Process-Tensor Recast

We recast belts and their holonomies in (i) von Neumann algebraic language and (ii) the process-tensor/quantum-comb formalism. We then map them to SROQD’s filtrations to justify operational measurability (AB-fixedness; Collapse-Lorentz invariance) with no new assumptions.

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C.0 Standing setup

• World and memory Hilbert spaces $\mathcal H_W,\mathcal H_M$ (separable), total $\mathcal H=\mathcal H_W\otimes\mathcal H_M$. Instruments $\{\mathcal M_{\theta,a}\}_{a\in\Phi}$ (CP, $\sum_a$ TP), adaptive policy $\theta_t=\pi_t(A_1,\dots,A_{t-1})$. Observer trace produces an increasing filtration. (Same layer as SROQD.)

• SK contour provides two oriented branches; a loop $\gamma$ lifts to $\gamma_\pm$ and spans a belt $S(\gamma)\subset\Theta\times \mathrm{SK}$. Holonomy:

  $$\mathcal H[S]=\mathcal P\exp\!\!\iint_S\mathcal F,\qquad W[S]=\operatorname{Tr}\,\mathcal H[S],\qquad \Phi=\arg W[S].$$

  (Uhlmann curvature $\mathcal F$; non-Abelian Stokes.)

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C.1 Operator-algebraic recast (von Neumann expectations)

C.1.1 Algebras, states, instruments, filtration

Let $\mathcal N_W\subset\mathcal B(\mathcal H_W)$, $\mathcal N_M\subset\mathcal B(\mathcal H_M)$ be von Neumann algebras; $\mathcal N:=\mathcal N_W\ \bar\otimes\ \mathcal N_M$. Fix a normal state $\omega$ on $\mathcal N$. An instrument at context $\theta$ is the normal CP map

$$\mathfrak I_\theta(X)=\sum_{a\in\Phi}\mathcal M_{\theta,a}(X)\otimes |a\rangle\!\langle a|_M.$$

The observer’s filtration is the increasing tower of von Neumann subalgebras

$$\mathcal F_t:=\mathrm{vN}\big(\text{pointer projectors for }A_1,\ldots,A_t\big)\ \subset\ \mathcal N_M,$$

with conditional expectations $\mathbb E_t:\mathcal N\to \mathcal F_t$. (SROQD §6.1.)

C.1.2 Belt holonomy as a normal functional of a Wilson–belt operator

Let $\mathcal U_S$ denote a (surface-ordered) parallel-transport unitary on a purification fiber implementing $\mathcal H[S]$ (non-Abelian Stokes). Embed $\mathcal U_S$ into $\mathcal N$ by a Stinespring isometry $V$ for the Uhlmann bundle pullback; define the Wilson–belt operator

$$\mathbf W[S]\ :=\ V^\dagger\big(\mathcal U_S\otimes \mathbf 1\big)V\ \in\ \mathcal N.$$

Then the belt functional used in the main text is the normal expectation

$$W[S]=\omega(\mathbf W[S]),\qquad \Phi=\arg\omega(\mathbf W[S]).$$

• Gauge covariance. A gauge transform acts as an inner automorphism $\alpha_g(X)=U_g^\dagger X U_g$ on the pulled-back bundle. Since $\omega$ is a normal functional and we take a class function (phase of a trace/expectation), $\Phi$ is invariant under $\alpha_g$.

• Framing and SK. The two SK boundaries make $\mathbf W[S]$ a framed object; the thin-belt limit keeps a self-link (framing) contribution, not representable by a single unframed line operator. (Matches Theorem A.)

C.1.3 Measurable belt effects and AB-fixedness

Define Borel phase-bin effects on $\mathbb S^1$:

$$E_I:=\mathbf 1_{\{\arg \omega(\mathbf W[S])\in I\}} \ \in\ \mathcal F_{t+1},$$

operationally implemented by an interferometric readout whose pointer is written at tick $t{+}1$. Similarly, for two belts $S_1,S_2$, define an integer-valued effect $E_k$ registering $\mathrm{Lk}_{\rm belt}(S_1,S_2)=k$ via a framed-Wilson interferometer (pointer in $\mathcal F_{t+1}$). By SROQD:

• (Delta-certainty / latching): For any $F\in\mathcal F_t$, $\mathbb E_t(F)=F$, hence once recorded, the event is fixed for the observer. (Conditional expectation fixed-point.)

• (AB-fixedness): If two observers use commuting effects and share an accessible record (SBS), the mapped effects yield agreement with probability 1. (Apply SROQD Theorem on AB-fixedness.)

• (Collapse-Lorentz invariance): Agreement/incompatibility are preserved under frame maps that preserve the collapse interval, so belt readouts are frame-robust.

Thus $\Phi$ and $\mathrm{Lk}_{\rm belt}$ correspond to effects in the filtration and are operationally measurable without new axioms.

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C.2 Process-tensor / quantum-comb recast (CTP/SK form)

C.2.1 Process tensor with SK legs

Let $\mathcal T_{0:T}$ be the process tensor (Choi operator) that, for a sequence of instruments, returns outcome probabilities. On the SK contour, each tick contributes a forward and backward leg; the global object is a CTP comb $\mathbf T$ on doubled spaces. (SROQD §6.2 sketch).

C.2.2 Belts as CTP surface contractions

A control loop $\gamma$ picks instruments with $\theta_t=\pi_t(\cdot)$. The SK belt $S(\gamma)$ is represented at the comb level by inserting a surface-ordered curvature functional $\mathfrak S[\mathcal F;S]$ that couples the forward/back legs along the annulus spanned by $\gamma_\pm$:

$$\mathfrak S[\mathcal F;S] \;\leadsto\; \text{CTP influence functional } \mathcal I_S\ =\ \mathcal P\exp\!\!\iint_S \mathcal F,$$

so that the Wilson–belt amplitude is the link product (contraction)

$$W[S]\ =\ \mathrm{Tr}\!\left[\mathbf T\ \star\ \mathcal I_S\right],\qquad \Phi=\arg W[S],$$

with $\star$ the comb link (Choi) composition. In Abelian/pure-state reduction this equals the familiar CTP influence phase; in general it is the SK surface holonomy.

C.2.3 Variations (Theorem B) and framed linking (Theorem C) in the comb

• First variation. A boundary deformation $\delta x$ of $S$ induces

  $$\delta \mathcal I_S\ =\ \oint_{\partial S}\!\iota_{\delta x}\mathcal F\ +\ \text{(commutator tails)},$$

  hence

  $$\delta\Phi\ =\ \operatorname{Im}\frac{\mathrm{Tr}\big[\mathbf T\star \delta \mathcal I_S\big]}{\mathrm{Tr}\big[\mathbf T\star \mathcal I_S\big]} \quad\Rightarrow\quad F_i(\tau)\ =\ -\frac{\delta \Phi}{\delta x^i(\tau)}.$$

  This is the comb version of Appendix-A’s formula.

• Exchange as framed linking. Insert two belt functionals $\mathcal I_{S_1},\mathcal I_{S_2}$. In Berry–Chern / Chern–Simons-type sectors, Gaussian/CS reduction of the CTP action yields a central phase

  $$\mathrm{Tr}\!\left[\mathbf T\star\mathcal I_{S_1}\star\mathcal I_{S_2}\right]\;=\;e^{\,i\,\kappa\,\mathrm{Lk}_{\rm belt}(S_1,S_2)}\ \mathrm{Tr}\!\left[\mathbf T\star\widetilde{\mathcal I}\right],$$

  with self-link providing the framing anomaly. This reproduces the framed-Wilson algebra in comb language (Theorem C).

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C.3 Operational measurability via SROQD filtrations

• Effects live in the filtration. Phase-bin and linking-value readouts are implemented as POVMs whose pointer projectors lie in $\mathcal F_{t+1}$; hence they are measurable with respect to the observer’s past. (SROQD operator-algebraic §6.1.)

• AB-fixedness and invariance. With commuting effects and accessible records (SBS), SROQD proves delta-certainty and frame invariance (Collapse-Lorentz) for mapped outcomes—so belt holonomy phases and framed linking indices are objective observables in this operational sense.

• Scope control. As summarized in the “What we proved” memo, this guarantees objectivity only for the geometric/topological content (holonomy variation; framed linking), not for non-adiabatic or inelastic sectors.

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C.4 Summary

• A belt holonomy is the normal expectation $W[S]=\omega(\mathbf W[S])$ of a Wilson–belt operator in a von Neumann algebra; its phase $\Phi$ is gauge-invariant (class function).

• In the process-tensor picture, $W[S]$ is the CTP influence phase obtained by contracting the process comb with a surface-ordered curvature insertion; its variation yields Theorem B, and pairs of insertions yield Theorem C with framed belt linking and self-link anomaly.

• Via SROQD’s filtration/conditional-expectation machinery, the corresponding effects are measurable and AB-fixed; their agreement is Collapse-Lorentz invariant—no new assumptions added.

This completes the algebraic/comb recast and its operational justification.

 

Appendix D — Rectifiable Two-Chains and Regularity

(existence of spanning belts; measurability and convergence of $\mathcal H[S]$)

We supply the geometric-measure-theoretic underpinnings used implicitly in Theorem A and in the variation calculus of Appendix A: (i) existence of smooth/rectifiable belts that span the two SK boundaries for any admissible loop; (ii) well-posedness and stability (measurability & convergence) of the belt holonomy

$$\mathcal H[S] \;=\; \mathcal P\exp\!\!\iint_{S}\!\mathcal F,\qquad W[S]=\operatorname{Tr}\,\mathcal H[S],\qquad \Phi=\arg W[S].$$

The only inputs are the SK doubling, the Uhlmann bundle/curvature, and non-Abelian Stokes as formulated in the draft.

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D.0 Standing assumptions

• $\Theta$ a smooth, second-countable manifold endowed when needed with a smooth Riemannian metric $g$.

• A closed control loop $\gamma\in C^1(S^1,\Theta)$. SK doubling yields two oriented boundary curves $\gamma_\pm\subset \Theta\times\mathrm{SK}$.

• Uhlmann connection $\mathcal A$ with curvature $\mathcal F=d\mathcal A+\mathcal A\wedge\mathcal A$, pulled back to $\Theta\times\mathrm{SK}$. The belt holonomy is defined by the surface-ordered exponential; non-Abelian Stokes is used exactly as in the draft.

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D.1 Smooth belts always exist (geodesic normal ribbon)

Theorem D.1 (smooth immersed belt).

Let $\gamma\in C^1(S^1,\Theta)$. Then there exists $\varepsilon_0>0$ and, for each $\varepsilon\in(0,\varepsilon_0)$, a smooth immersed compact oriented surface $S_\varepsilon(\gamma,\nu)\subset \Theta\times\mathrm{SK}$ with boundary $\partial S_\varepsilon=\gamma_+\sqcup\gamma_-$.

Construction. Equip $\Theta$ with $g$. By the tubular-neighborhood theorem, there exists a nowhere-vanishing smooth normal field $\nu(\sigma)\perp \dot\gamma(\sigma)$. For small $\varepsilon>0$ define offset loops

$$\gamma_\pm^\varepsilon(\sigma):=\exp_{\gamma(\sigma)}\!\big(\pm\tfrac{\varepsilon}{2}\,\nu(\sigma)\big).$$

The geodesic normal ribbon in $\Theta$ is the image of $\iota_\Theta(\sigma,t)=\exp_{\gamma(\sigma)}(t\,\nu(\sigma))$, $t\in[-\varepsilon/2,\varepsilon/2]$. Lift to $\Theta\times\mathrm{SK}$ by $\iota(\sigma,t)=(\iota_\Theta(\sigma,t),\mathrm{br}(t))$ with $\mathrm{br}(t)\in\{+,-\}$ equal to $\mathrm{sign}(t)$ and $\mathrm{br}(\pm\varepsilon/2)=\pm$. The image $S_\varepsilon(\gamma,\nu)=\iota(S^1\times[-\varepsilon/2,\varepsilon/2])$ is a smooth immersed belt and $\partial S_\varepsilon=\gamma_+\sqcup\gamma_-$. ∎

Remarks.

1. Connectivity is not required: $S_\varepsilon$ is an immersed ribbon in $\Theta\times\mathrm{SK}$ with exactly the two SK boundaries.

2. This is the explicit construction used in Theorem A of the draft (there as a proof sketch; here as a full statement).

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D.2 Rectifiable belts for low-regularity boundaries

The previous theorem assumes $\gamma\in C^1$. We extend to loops of finite length (Lipschitz).

Theorem D.2 (rectifiable belt).

If $\gamma$ is Lipschitz, then for every sufficiently small $\varepsilon>0$ there exists a rectifiable 2-current $T_\varepsilon$ in $\Theta\times\mathrm{SK}$ with finite mass such that $\partial T_\varepsilon=\gamma_+\sqcup\gamma_-$. Moreover, $T_\varepsilon$ can be approximated in mass by smooth immersed belts $S_\varepsilon^{(n)}$ with the same boundary.

Proof (outline). A Lipschitz $\gamma$ admits an a.e. unit normal $\nu$ along almost every point via metric projections in a tubular neighborhood. The offset curves $\gamma_\pm^\varepsilon$ are Lipschitz and disjoint for small $\varepsilon$. The annulus between $\gamma_-^\varepsilon$ and $\gamma_+^\varepsilon$ inside the tubular neighborhood defines a rectifiable 2-current $T_\varepsilon^\Theta$ in $\Theta$ with $\partial T_\varepsilon^\Theta=\gamma_+^\varepsilon-\gamma_-^\varepsilon$. Lifting to $\Theta\times\mathrm{SK}$ and letting the thickness direction play the SK label yields a rectifiable belt $T_\varepsilon$ with $\partial T_\varepsilon=\gamma_+\sqcup\gamma_-$. Standard smoothing of Lipschitz maps in charts gives smooth immersed belts $S_\varepsilon^{(n)}\to T_\varepsilon$ in mass. ∎

Use. This rectifiable setting is the one implicitly invoked when we speak of “rectifiable 2-chains” in the main text.

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D.3 Existence and measurability of $\mathcal H[S]$

Lemma D.3 (existence of $\mathcal H[S]$ on rectifiable belts).

Let $S$ be a smooth belt or a rectifiable belt approximated in mass by smooth immersed belts $S^{(n)}\to S$. Assume $\mathcal F$ is continuous on a neighborhood of $\mathrm{supp}\,S$ and bounded in operator norm in a fixed finite-dimensional representation. Then the sequence of surface-ordered exponentials $\mathcal H[S^{(n)}]=\mathcal P\exp\iint_{S^{(n)}}\!\mathcal F$ converges (in operator norm) to a limit that depends only on $S$; define this as $\mathcal H[S]$.

Sketch. Use the Duhamel/Volterra series on each $S^{(n)}$ (Appendix A) and dominated convergence: bounded $\|\mathcal F\|$ and bounded area give uniform control on all iterated integral terms. The Stokes-covariant commutator bookkeeping from Appendix A ensures a common bound independent of $n$. ∎

Corollary D.4 (measurability).

If $\gamma\mapsto S(\gamma)$ is Borel (e.g., geodesic-normal ribbon with a Borel choice of $\nu$), then $\gamma\mapsto W[S(\gamma)]=\operatorname{Tr}\mathcal H[S(\gamma)]$ and $\gamma\mapsto \Phi(\gamma)$ are Borel. Consequently, phase-bin effects $E_I=\mathbf 1_{\{\Phi\in I\}}$ live in the observer’s filtration (pointer algebra), as used in §7 and Appendix C.

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D.4 Stability: continuity under surface and curvature limits

Let $S_n\to S$ in the sense of mass (for smooth belts, $C^0$ convergence of embeddings with uniformly bounded area suffices), and let $\mathcal F_n\to\mathcal F$ uniformly on a neighborhood of the supports.

Proposition D.5 (stability of holonomy and phase).

Under the above hypotheses,

$$\|\mathcal H_n[S_n]-\mathcal H[S]\|\ \longrightarrow\ 0,\qquad W[S_n]\to W[S],\quad \text{and}\quad \Phi[S_n]\to \Phi[S]\ \text{whenever }W[S]\neq 0.$$

Sketch. Apply Grönwall-type bounds to the Volterra series for $\mathcal H_n$ and use the uniform convergence $\mathcal F_n\to\mathcal F$ plus convergence of surface measures. The phase continuity follows from $W\neq 0$ and $\arg z$ being continuous off the branch cut. ∎

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D.5 Thin-belt limit and framing as a regular limit

Let $S_\varepsilon(\gamma,\nu)$ be the geodesic-normal belts of §D.1. Assume $\mathcal F$ is continuous near $\gamma$.

Proposition D.6 (existence of the thin-belt limit).

The limit $\displaystyle \lim_{\varepsilon\downarrow 0}\arg\operatorname{Tr}\mathcal H[S_\varepsilon(\gamma,\nu)]$ exists and depends only on the framed loop $(\gamma,\nu)$. Different framings $\nu,\nu'$ can yield different limits (self-link/framing term).

Sketch. First-order area contributions are $O(\varepsilon)$, but the ordered-exponential boundary contraction in Appendix A produces a finite framing term controlled by the twist of $\nu$ (cf. Theorem A’s minimality discussion). The limit is stable by Proposition D.5. ∎

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D.6 Operational measurability (pointer algebra & SBS)

With $W[S]$ and $\Phi$ Borel in $\gamma$ and continuous under admissible limits, the effects “$\Phi\in I$” and “$\mathrm{Lk}_{\rm belt}=k$” are implementable as POVM outcomes written into the memory algebra $\mathcal F_{t+1}$. Under commuting effects and accessible records (SBS redundancy), SROQD’s theorems give AB-fixedness and frame robustness (Collapse-Lorentz). We import these results to guarantee the operational status of our belt observables.

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D.7 Summary (what this appendix guarantees)

• Existence: Every admissible loop admits smooth belts $S_\varepsilon(\gamma,\nu)$ (Theorem D.1); for Lipschitz loops, rectifiable belts exist and can be smoothed (Theorem D.2).

• Well-posedness: $\mathcal H[S]$ is defined on rectifiable belts by smooth approximation; $W[S]$ and $\Phi$ are Borel and stable under limits (Lemma D.3, Prop. D.5).

• Thin-belt limit: $\Phi$ has a well-defined $\varepsilon\downarrow 0$ limit depending on the framing (Prop. D.6), matching the minimality statement of Theorem A in the draft.

These facts justify all regularity and convergence steps used in the main proofs (Theorems A–C) and in Appendix A’s variation calculus, without any assumptions beyond the standard SK/Uhlmann framework already fixed in the paper.

 

Appendix E — Worked Reductions

We collect two explicit reductions that make the main theorems concrete: (i) the Abelian/pure-state limit where the belt calculus reproduces Berry curvature forces; (ii) $2{+}1$D Chern–Simons examples where exchange phases become framed Wilson phases built from explicit linking integrals.

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E.1 Abelian / pure-state reduction → Berry curvature forces

E.1.1 Setup and reduction

Let the transported state be rank-1, $P(\theta)=|\psi(\theta)\rangle\!\langle\psi(\theta)|$, over the parameter manifold $\Theta$. The Uhlmann connection reduces to the Berry connection

$$\mathcal A \;\longrightarrow\; A := i\langle\psi|\,d\psi\rangle,\qquad \mathcal F \;\longrightarrow\; F := dA .$$

For a belt $S$ spanning $\gamma_\pm$, the surface ordering drops out (Abelian case), so

$$\mathcal H[S] = \exp\!\Big(\iint_S F\Big),\qquad \Phi = \arg\operatorname{Tr}\,\mathcal H[S] = \iint_S F \pmod{2\pi}.$$

By Appendix A (Abelian form),

$$\delta\Phi=\oint_{\partial S}\iota_{\delta x}F, \qquad\Rightarrow\qquad F_i(\tau) \;=\; -\,\frac{\delta\Phi}{\delta x^i(\tau)} \;=\; -\,F_{ij}\big(x(\tau)\big)\,\dot x^j(\tau),$$

i.e. the geometric force density equals the Berry curvature contracted with the loop tangent.

E.1.2 Vector form (magnetic-analogue)

In local coordinates on $\Theta\subset\mathbb R^3$, set $B_k:=\tfrac12\epsilon_{kij}F_{ij}$. Then

$$\boxed{\quad \mathbf F_{\rm geom}(\tau) \;=\; -\,\dot{\mathbf x}(\tau)\times \mathbf B\big(\mathbf x(\tau)\big)\ .\quad}$$

This is the familiar “Lorentz-like” anomalous response.

E.1.3 Two-level example (Bloch sphere monopole)

Let $H(\mathbf R)=\mathbf R\!\cdot\!\boldsymbol\sigma$ with $|\mathbf R|=R_0$; the ground band has Berry curvature

$$\mathbf B(\mathbf R) \;=\; -\,\frac{1}{2}\,\frac{\mathbf R}{|\mathbf R|^{3}} .$$

Hence

$$\mathbf F_{\rm geom}(\tau) = -\,\dot{\mathbf R}(\tau)\times \mathbf B(\mathbf R(\tau)) = \frac{1}{2}\,\dot{\mathbf R}(\tau)\times \frac{\mathbf R(\tau)}{|\mathbf R(\tau)|^{3}},$$

and the accumulated phase equals half the solid angle subtended by the loop on the Bloch sphere:

$$\Phi = \iint_S F = -\frac12\,\Omega_{\rm solid}(S).$$

Everything here is a direct specialization of Theorem B to the Abelian/pure-state case.

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E.2 $2{+}1$D Chern–Simons: framed Wilsons and explicit linking

We now evaluate the exchange phases in an Abelian Chern–Simons sector (the non-Abelian/Berry–Chern case follows by representation data). Level $k\in\mathbb R\setminus\{0\}$, action $S_{\rm CS}[A]=\tfrac{k}{4\pi}\int A\wedge dA$.

E.2.1 Gaussian evaluation with Poincaré-dual currents

For an oriented loop $C\subset\mathbb R^3$, let $J_C$ be its Poincaré-dual 2-form: $\int \eta\wedge J_C=\oint_C \eta$. For (pairwise disjoint) loops $C_a$ with charges $q_a$,

$$\Big\langle \prod_a e^{\,i q_a \oint_{C_a} A}\Big\rangle =\exp\!\Bigg\{i\,\frac{2\pi}{k}\Big(\sum_{a<b} q_a q_b\,\mathrm{Lk}(C_a,C_b)+\frac12\sum_a q_a^2\,\mathrm{SL}(C_a)\Big)\Bigg\},$$

where:

• $\mathrm{Lk}(C_1,C_2)$ is the Gauss linking number,

$$\mathrm{Lk}(C_1,C_2)=\frac{1}{4\pi}\!\!\oint_{C_1}\!\!\oint_{C_2} \frac{(\mathbf r_1-\mathbf r_2)\cdot(d\mathbf r_1\times d\mathbf r_2)}{\|\mathbf r_1-\mathbf r_2\|^3},$$

• $\mathrm{SL}(C)\in\mathbb Z$ is the self-link w.r.t. a chosen ribbon framing (Călugăreanu–White: $\mathrm{SL}=\mathrm{Wr}+\mathrm{Tw}$).

Belts. Each excitation is represented by an SK belt with two framed boundary components $\gamma_\pm$. Summing over all boundary pairs gives the framed belt linking

$$\mathrm{Lk}_{\rm belt}(S_1,S_2)=\!\!\sum_{\alpha,\beta\in\{+,-\}}\!\mathrm{Lk}(\gamma_{1,\alpha},\gamma_{2,\beta}).$$

Since each $\gamma_{j,\pm}$ is a small framed offset of the core loop $C_j$, $\mathrm{Lk}(\gamma_{1,\alpha},\gamma_{2,\beta})=\mathrm{Lk}(C_1,C_2)$ for all $\alpha,\beta$, hence

$$\boxed{\quad \mathrm{Lk}_{\rm belt}(S_1,S_2)=4\,\mathrm{Lk}(C_1,C_2)\quad}$$

for consistent SK orientations. The overall coefficient is absorbed into the sector constant $\kappa$ of the main text.

Thus the exchange unitary between two belts is

$$U_{\rm ex} =\exp\!\big(i\,\kappa\,\mathrm{Lk}_{\rm belt}(S_1,S_2)\big)\cdot U_{\rm local}, \qquad \kappa=\frac{2\pi}{k}\,q_1 q_2\ \ (\text{Abelian case}).$$

Self-link terms $\tfrac{1}{2}\sum_{j,\alpha} q_{j,\alpha}^2\,\mathrm{SL}(\gamma_{j,\alpha})$ produce the framing anomaly (a calibration offset fixed by the SK ribbon framing).

E.2.2 Explicit Hopf-link example (linked once)

Take two core circles:

$$C_1:\ ( \cos u,\, \sin u,\, 0 ),\quad C_2:\ ( 2,\, \cos v,\, \sin v ),\qquad u,v\in[0,2\pi).$$

They form a Hopf link with $\mathrm{Lk}(C_1,C_2)=1$. Offsetting by a small normal field yields the belt boundaries $\gamma_{j,\pm}$. Therefore

$$\mathrm{Lk}_{\rm belt}(S_1,S_2)=4,\qquad U_{\rm ex}=\exp\!\big(i\,4\,\kappa\big)\cdot U_{\rm local}.$$

If $q_1=q_2=1$ and level $k$, then $\kappa=2\pi/k$ and

$$U_{\rm ex}=\exp\!\Big(i\,\frac{8\pi}{k}\Big)\cdot U_{\rm local}.$$

E.2.3 Unlinked loops and framing calibration

If $C_1,C_2$ are unlinked, $\mathrm{Lk}(C_1,C_2)=0$ so $\mathrm{Lk}_{\rm belt}=0$ and only the self-link phases remain:

$$\big\langle W[\partial S_1]\,W[\partial S_2]\big\rangle =\exp\!\Big\{i\,\frac{2\pi}{k}\,\frac12\!\sum_{j,\alpha} q_{j,\alpha}^2\,\mathrm{SL}(\gamma_{j,\alpha})\Big\}.$$

Changing the belt framing by one twist unit shifts $\mathrm{SL}\mapsto \mathrm{SL}\pm 1$, adding a constant phase; mutual phases are unaffected. This is the standard framing anomaly and is fixed once the SK belt framing is chosen.

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E.2.4 Non-Abelian/Berry–Chern remark

For non-Abelian sectors or adiabatic bands with Berry–Chern response, replace charges $q_a$ by representation data (e.g., quadratic Casimir factors) and $k$ by the appropriate level/normalization of the topological term. The framed-linking structure and the belt summation remain identical; the sector constant $\kappa$ encodes those details.

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Summary.

• In the Abelian/pure-state limit, Theorem B delivers the usual Berry curvature force $F_i=-F_{ij}\dot x^j$.

• In $2{+}1$D Chern–Simons sectors, Theorem C yields explicit framed Wilson phases: exchange $=\exp(i\,\kappa\,\mathrm{Lk}_{\rm belt})$ with self-link (framing anomaly) fixed by the SK ribbon. These calculations exemplify the paper’s central equivalences in concrete, computable settings.

 

 

 

 

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