\section{Non-Abelian Geometric Computation}\label{sec:non-abelian-geometric-computation}

The abelian regime presents a remarkably economical picture of physical computation. A continuous geometric system may accumulate phase, retain winding information, and distinguish different paths while preserving a commutative algebra of transport. Yet this economy has a limit. Consider the familiar example of rotations in three-dimensional space: rotating an object about one axis and then another generally produces a different result from performing the same rotations in reverse. The individual transformations have not changed; what matters is their order of composition. In mathematical physics, Yang--Mills theory and non-abelian geometric phases provide a far-reaching generalization of this phenomenon. Internal transport is no longer adequately summarized by a single accumulated scalar phase, and the history of composition becomes part of the relevant geometric information.

For a theory of computation built from geometry, this raises a more fundamental problem than the introduction of additional continuous variables. If two individually admissible operations cannot be freely interchanged, then knowing the invariant content of each operation separately need not determine the invariant content of their composite. Local descriptions may remain perfectly well defined while their global assembly retains information that no independent collection of scalar phase ledgers captures. The challenge is therefore to formulate computation in a setting where ordered histories, geometric constraints, and invariant readouts coexist without sacrificing compositional consistency.

\begin{figure}[htbp]
    \centering
    \begin{subfigure}[b]{0.48\textwidth}
        \centering
        \includegraphics[width=\linewidth]{BPST.png}
        \caption{BPST instanton: a finite-action, localized topological solution to the Yang--Mills equations, exhibiting geometric twist and winding~\cite{Belavin1975BPST}.}
        \label{fig:bpst-instanton}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.48\textwidth}
        \centering
        \includegraphics[width=\linewidth]{swave.png}
        \caption{Complex quantum wave: a superposition of oscillations, illustrating the intertwined, helical modes of computation.}
        \label{fig:complex-wave}
    \end{subfigure}
    \caption{Familiar examples of geometric configurations corresponding to Type 3 and Type 1 $THM$.}
    \label{fig:v5-representations}
\end{figure}

\subsection{Gauge Transport and Computational Memory}
\label{sec:gauge-transport-and-computational-memory}

The quasi--fluidic regime of the preceding section is governed primarily by abelian phase transport and finite local readouts.  We now pass to the \emph{fully fluidic} regime, in which the state of a geometric computer must be transported through a nontrivial bundle of local configurations. The appropriate language is therefore that of connections, curvature, holonomy, and their associated invariant data.

\paragraph{Gauge-geometric state space.}
Let
\[
\pi:P\longrightarrow B
\]
be a principal \(G\)-bundle over a computably presented base \(B\), where
\(B\) represents the working region of the geometric substrate.  Let
\(\mathsf F\) be a finite-dimensional \(G\)-space describing the internal
geometric variables carried by a twin helix cell, and let
\[
E:=P\times_G\mathsf F\longrightarrow B
\]
be the associated state bundle.

A principal connection on \(P\) determines parallel transport between fibers.  In a local trivialization \(s:U\to P\), it is represented by a gauge potential
\[
A=s^{*}\omega\in\Omega^{1}(U;\mathfrak g),
\]
with curvature
\[
F_A
=
dA+A\wedge A
\in
\Omega^{2}(U;\mathfrak g).
\]
Globally, \(F_A\) is the corresponding adjoint-bundle-valued curvature two-form.

The connection specifies how local states may be transported; its curvature measures the infinitesimal obstruction to path-independent transport. A flat connection
\(F_A=0\) may still possess nontrivial global holonomy when \(B\) is not
simply connected, while a connection with \(F_A\neq0\) can exhibit local path dependence even when its structure group is abelian.  Curvature therefore records local geometric memory, whereas holonomy records the accumulated transport around finite paths and cycles.

Within GCT, a \emph{join} in this regime is no longer merely a symbolic identification of interfaces.  It carries a transport map induced by the connection.  Sequential composition of joins is therefore represented by composition of parallel transports.

\paragraph{Parallel transport and the interface groupoid.}
Let
\[
\Pi^{\mathrm{thin}}_1(B)
\]
denote the thin fundamental groupoid of \(B\).  Its objects are points of
\(B\), and its morphisms are paths modulo thin homotopy.  A connection
\(A\) defines a parallel-transport functor
\[
\mathsf{PT}_A:
\Pi^{\mathrm{thin}}_1(B)
\longrightarrow
\operatorname{Iso}(E),
\]
where \(\operatorname{Iso}(E)\) is the groupoid whose objects are fibers
\(E_x\) and whose arrows are the transport isomorphisms between them.

We restrict this construction to the paths and boundary data admitted by the geometric computer.

\begin{defn}[Interface Transport Groupoid]
\label{def:interface-transport-groupoid}
The \emph{interface transport groupoid}
\[
\mathcal J_A
\]
is the subgroupoid generated by parallel transports along admissible interface paths.  Its objects are the boundary-state fibers associated with admissible interfaces, and a morphism
\[
J:X\longrightarrow Y
\]
is an admissible reversible transport from the state fiber \(X\) to the state fiber \(Y\).

Composition in \(\mathcal J_A\) is serial execution of transport, and path reversal determines inverse morphisms.  Noninvertible operations such as coarse-graining, measurement, dissipation, or irreversible effects belong to the broader GCT process semantics and are not assumed to lie in \(\mathcal J_A\).
\end{defn}

For a path
\[
\gamma:x\longrightarrow y,
\]
we write
\[
U_\gamma
:=
\mathsf{PT}_A([\gamma])
\]
for its parallel transport.  After choosing local frames, this is represented by the familiar path-ordered exponential
\[
U_\gamma
=
\mathcal P
\exp\!\left(
\int_\gamma A
\right).
\]
For a based loop \(\gamma:x\to x\), the transport lies in the isotropy group
\[
\operatorname{Aut}_{\mathcal J_A}(X_x),
\]
and, in a chosen gauge, is represented by an element of \(G\).

\begin{defn}[Effective Holonomy Group]
\label{def:effective-holonomy-group}

For \(x\in B\), the \emph{effective holonomy group} of the connection at
\(x\) is
\[
\Hol_x(A)
:=
\left\{
U_\gamma
\;\middle|\;
\gamma:x\to x
\right\},
\]
with the appropriate closure when required.  It is the subgroup of the structure group actually generated by closed admissible transports based at \(x\).
\end{defn}

The distinction between the nominal structure group \(G\) and the realized group \(\Hol_x(A)\) is essential.  A nonabelian structure group does not imply that a particular connection realizes noncommuting transport: the connection may reduce to an abelian subgroup.  The computationally relevant question is therefore whether the \emph{effective holonomy} is nonabelian.

\paragraph{Path dependence and relative holonomy.}
Two admissible transports can follow different geometric routes while having the same initial and terminal interfaces.  Such parallel arrows cannot themselves be multiplied as though they were elements of a single group.  Their difference is instead measured by closing them into a loop.

\begin{defn}[Relative Join Holonomy]
\label{def:relative-join-holonomy}
Let
\[
P,Q\in\Hom_{\mathcal J_A}(X,Y)
\]
be two admissible transports having the same source and target.  Their \emph{relative join holonomy}, based at \(X\), is
\[
\Delta_X(P,Q)
:=
Q^{-1}\circ P
\in
\operatorname{Aut}_{\mathcal J_A}(X).
\]
Equivalently, one may base the comparison at \(Y\):
\[
\Delta_Y(P,Q)
:=
P\circ Q^{-1}
\in
\operatorname{Aut}_{\mathcal J_A}(Y).
\]
The two representatives are related by conjugation,
\[
\Delta_Y(P,Q)
=
P\circ
\Delta_X(P,Q)
\circ
P^{-1}.
\]
Hence their conjugacy classes carry the same invariant information.
\end{defn}

Thus
\[
\Delta_X(P,Q)=\id_X
\]
if and only if the two transports agree.  Nontrivial relative holonomy records path dependence.  It may occur already for an abelian connection with nonzero curvature.

This observation separates two notions that will play different roles below:
\[
\boxed{
\text{path dependence}
\neq
\text{noncommutativity}.
}
\]
Curvature can make transport path-dependent without making transport order-sensitive.

\paragraph{The join commutator.}
Order sensitivity is detected only after the relevant transports have been made into based cycles.

\begin{defn}[Join Commutator]
\label{def:join-commutator}
Let
\[
X\in\operatorname{Ob}(\mathcal J_A),
\qquad
J_1,J_2
\in
\operatorname{Aut}_{\mathcal J_A}(X).
\]
The \emph{join commutator} of \(J_1\) and \(J_2\) is
\[
[J_1,J_2]_X
:=
J_1\circ J_2\circ J_1^{-1}\circ J_2^{-1}
\in
\operatorname{Aut}_{\mathcal J_A}(X).
\]

The joins commute precisely when
\[
[J_1,J_2]_X=\id_X,
\]
or equivalently
\[
J_1\circ J_2
=
J_2\circ J_1.
\]

We call the transport at \(X\) \emph{order-sensitive} if there exist based join cycles \(J_1,J_2\) such that
\[
[J_1,J_2]_X\neq\id_X.
\]
\end{defn}

This construction is covariant under change of base object.

\begin{prop}[Covariance of the Join Commutator]
\label{prop:covariance-of-the-join-commutator}
Let
\[
K:X\longrightarrow Y
\]
be an isomorphism in \(\mathcal J_A\), and define
\[
J_i^K
:=
K\circ J_i\circ K^{-1}
\in
\operatorname{Aut}_{\mathcal J_A}(Y).
\]
Then
\[
[J_1^K,J_2^K]_Y
=
K\circ[J_1,J_2]_X\circ K^{-1}.
\]
Consequently, the condition
\[
[J_1,J_2]_X=\id_X
\]
is independent of the chosen representative, and the conjugacy class
\[
\bigl[[J_1,J_2]_X\bigr]
\]
is invariant under transport of the base object.
\end{prop}

\begin{proof}
Associativity and cancellation give
\[
\begin{aligned}
\relax[J_1^K,J_2^K]_Y
&=
(KJ_1K^{-1})
(KJ_2K^{-1})
(KJ_1^{-1}K^{-1})
(KJ_2^{-1}K^{-1})\\
&=
K
(J_1J_2J_1^{-1}J_2^{-1})
K^{-1}\\
&=
K[J_1,J_2]_XK^{-1}.
\end{aligned}
\]
\end{proof}

After choosing a local frame, based cycles are represented by ordinary holonomies \(U_{\gamma_i}\in G\), and the categorical commutator becomes
\[
[U_{\gamma_1},U_{\gamma_2}]
=
U_{\gamma_1}
U_{\gamma_2}
U_{\gamma_1}^{-1}
U_{\gamma_2}^{-1}.
\]
A gauge transformation conjugates this element.  Thus the statement that the commutator is trivial is gauge invariant, as are class functions of the commutator.

\begin{rem}[Path Dependence versus Order Sensitivity]
\label{rem:path-dependence-versus-order-sensitivity}
The two constructions measure genuinely different phenomena.

For two routes
\[
P,Q:X\longrightarrow Y,
\]
the condition
\[
\Delta_X(P,Q)\neq\id_X
\]
records path dependence.  This can occur in a \(U(1)\) theory.

By contrast,
\[
[J_1,J_2]_X\neq\id_X
\]
records the order dependence of two-based transport cycles.  This requires a nonabelian effective holonomy group.

Accordingly, abelian curvature may already store geometric context, while a nontrivial join commutator supplies the additional resource of order-sensitive transport.
\end{rem}

\paragraph{Curvature and the generation of holonomy.}
The relationship between curvature and holonomy is not merely heuristic.  Infinitesimally, the curvature determines the leading failure of parallel transport around sufficiently small loops to close trivially.  Globally, the Ambrose--Singer theorem identifies the Lie algebra of the holonomy group with the algebra generated by curvature transported back to a common base point~\cite{AmbroseSinger1953}.

Schematically,
\[
\mathfrak{hol}_x(A)
=
\operatorname{span}
\left\{
\operatorname{Ad}_{U_\eta^{-1}}
F_y(u,v)
\;\middle|\;
\eta:x\to y,\;
u,v\in T_yB
\right\},
\]
with the usual interpretation in the principal bundle.

Thus curvature provides the infinitesimal generators from which the effective holonomy algebra is assembled.  In particular,
\[
\Hol_x(A)\ \text{abelian}
\quad\Longrightarrow\quad
[J_1,J_2]_X=\id_X
\]
for every pair of based join cycles, whereas the existence of
\[
[J_1,J_2]_X\neq\id_X
\]
directly witnesses a nonabelian realized holonomy sector.

Notice that neither nonzero curvature nor nontrivial topology alone forces this conclusion.  A \(U(1)\) connection may have nonzero curvature and nontrivial global holonomy while all transports still commute.  Handles, defects, and curvature enlarge the possible memory channels; noncommutativity appears only when the effective transport group itself is nonabelian.

\paragraph{Holonomy as computational memory.}
For a closed admissible path \(\gamma\), the \emph{holonomy}
\[
U_\gamma
=
\mathcal P
\exp\!\left(
\oint_\gamma A
\right)
\]
stores the cumulative result of transport around the cycle.  Under a change of gauge at the base point,
\[
U_\gamma
\longmapsto
g(x)U_\gamma g(x)^{-1},
\]
so the matrix representative is not itself an invariant.  Its conjugacy class and class functions are.

In a representation
\[
\rho:G\longrightarrow GL(V),
\]
the Wilson-loop observable
\[
W_\rho(\gamma)
:=
\tr\!\bigl(\rho(U_\gamma)\bigr)
\]
is the standard example~\cite{PhysRevD.10.2445}.  In GCT, this supplies a natural gauge-invariant readout of a closed execution cycle.

The computational interpretation is therefore:

\begin{enumerate}[leftmargin=*,itemsep=2pt]
    \item \textbf{Registers as cycles.}
    Distinguished loops or interface cycles carry holonomy data
    \(\{[U_{\gamma_k}]\}\).

    \item \textbf{Updates as transport.}
Control fields, deformations, and gluings modify the connection or the admissible path system and therefore modify the associated transports.

    \item \textbf{Relative memory as path comparison.}
    Distinct routes between the same interfaces are compared by
    \(\Delta_X(P,Q)\).

    \item \textbf{Order-sensitive memory as commutator.}
    Noncommuting closed transports are detected by
    \([J_1,J_2]_X\).

    \item \textbf{Observable output.}
Gauge-invariant readouts are extracted from conjugacy classes, characters, Wilson-loop traces, characteristic classes, spectral indices, or other invariant functions of the transport data.
\end{enumerate}

The essential progression is therefore
\[
\text{connection}
\longrightarrow
\text{parallel transport}
\longrightarrow
\text{relative holonomy}
\longrightarrow
\text{join commutator}.
\]
The first two stages already occur in abelian phase geometry.  The final stage detects the genuinely nonabelian resource: the output of a composite transport may depend on the order in which admissible cycles are executed.

\paragraph{From Berry phase to Wilczek--Zee transport.}
The preceding construction has a standard realization in adiabatic geometry.  Suppose the relevant internal states form a Hermitian vector bundle
\[
E\longrightarrow B.
\]
For an isolated nondegenerate state, the adiabatic eigenspace is a line bundle, and its Berry connection has effective structure group \(U(1)\) \cite{Berry1984QuantalPhase}.  Its holonomies are phases and therefore commute.

If instead an \(m\)-dimensional eigenspace remains degenerate over a region of parameter space, the adiabatic connection takes values in
\(\mathfrak u(m)\), producing Wilczek--Zee transport
\cite{WilczekZee1984Appearance}.  Such a connection \emph{may} have nonabelian holonomy.  For \(m=2\),
\[
U(2)
\cong
\frac{SU(2)\times U(1)}{\mathbb Z_2},
\]
so after separating the central phase, the traceless part supplies an
\(SU(2)\)-type transport sector.

Neither rank \(m\geq2\) nor degeneracy alone guarantees noncommuting holonomies: the realized connection may still reduce to an abelian subgroup.  Symmetries, couplings, boundary conditions, and controlled deformations of the twin helix substrate determine whether the effective holonomy is genuinely nonabelian.  In particular, anisotropy may either split an existing degeneracy or participate, together with couplings and constraints, in selecting a protected multidimensional subspace.  The operational criterion remains
\[
[J_1,J_2]_X\neq\id_X.
\]

\paragraph{Chern--Weil readouts.}
Holonomy records transport around particular cycles.  Chern--Weil theory supplies a complementary mechanism for compressing curvature into stable global invariants~\cite{BoTu82}.

Let
\[
p:\mathfrak g^{\otimes k}\longrightarrow\mathbb R
\]
be an \(\operatorname{Ad}\)-invariant polynomial.  Then
\[
p(F_A,\ldots,F_A)
\]
is a closed \(2k\)-form, and its de Rham cohomology class is independent of the chosen connection.  Familiar examples include representatives of Chern and Pontryagin classes.  In particular, expressions of the form
\[
\tr(F_A\wedge F_A)
\]
encode four-dimensional characteristic data.

From the computational viewpoint, this provides a canonical instance of \emph{curvature-as-readout}:
\[
\text{local curvature data}
\longrightarrow
\text{closed characteristic form}
\longrightarrow
\text{deformation-stable global class}.
\]
The characteristic class does not retain the full connection or all of its holonomy data; it is instead a controlled compression into a stable global sector label.

Chern--Simons forms provide the corresponding transgression data.  In a local presentation,
\[
d\,\operatorname{CS}_{2k-1}(A)
=
p(F_A,\ldots,F_A),
\]
so a Chern--Simons quantity may be regarded as a boundary or history-dependent ledger whose change records the characteristic density swept out by a higher-dimensional interpolation.  This relationship will reappear below when histories are treated as bordisms.

\paragraph{Topological memory channels.}
A twin helix configuration may also carry order-parameter fields independent of the connection itself.  For example, an internal orthonormal frame can determine a unit vector field
\[
n:B_0\longrightarrow S^2
\]
on an appropriate region \(B_0\subseteq B\).  Depending on the relevant order-parameter space and spatial dimension, familiar topological sectors include
\[
\pi_1(S^1)\cong\mathbb Z,
\qquad
\pi_2(S^2)\cong\mathbb Z,
\qquad
\pi_3(S^2)\cong\mathbb Z.
\]
These correspond respectively to vortex or phase winding, skyrmion-type degree, and Hopf-type linking of preimages.

These homotopy classes should not themselves be identified with nonabelian gauge transport.  A skyrmion or Hopf sector is a statement about the topology of an order-parameter map, whereas nonabelianity is a statement about composition in the effective holonomy group.  When the texture is coupled to the gauge connection, however, the two structures may interact: a topological sector can constrain admissible connections, defects, boundary conditions, and hence the possible holonomy readouts.

GCT therefore distinguishes at least three sources of persistent memory:
\[
\begin{array}{rcl}
\text{curvature} &:& \text{local obstruction to path independence},\\[2mm]
\text{holonomy} &:& \text{finite cyclic transport memory},\\[2mm]
\text{topological class} &:& \text{deformation-stable sector memory}.
\end{array}
\]
None of these should be collapsed into the others.

\paragraph{Expressive consequence.}
The passage from \(U(1)\) phase transport to nonabelian holonomy enlarges the available process algebra by introducing order-sensitive composition.  Relative holonomy already distinguishes paths; a nontrivial join commutator additionally distinguishes the order of closed operations.

This supplies a natural candidate witness for the increase in expressiveness discussed in~\cref{lem:geometric-logic-conservation}, as one now jumps across within~\cref{tab:n-logics} from a fuzzy ordered algebraic flow on one end to a chaotic form of geometric analog flow on the other. The claim must nevertheless be interpreted in the relational sense of the EVT framework: noncommuting transport does not prove that an arbitrary classical universal machine cannot simulate the corresponding matrix calculations.  Rather, it demonstrates that a translation which is required to preserve the native geometric interfaces, transport composition, and gauge-invariant readouts cannot simply collapse the nonabelian regime back to the abelian phase semantics without discarding structure.

\begin{ex}[Spectral Flow as a Computable Readout]
\label{ex:spectral-flow-as-a-computable-readout}
Let \(B\) be a compact odd-dimensional manifold, let
\[
E\longrightarrow B
\]
be a Hermitian vector bundle, and let
\[
\{A_t\}_{t\in S^1}
\]
be a smooth periodic family of unitary connections.  Associated to this family, choose a smooth periodic family of self-adjoint Dirac-type operators
\[
D_{A_t}:\Gamma(E)\longrightarrow\Gamma(E).
\]
Assume that the operators are Fredholm with discrete spectrum and that zero is crossed only at isolated parameters.

The spectral flow
\[
\operatorname{sf}\{D_{A_t}\}_{t\in S^1}
\in\mathbb Z
\]
is the signed number of eigenvalues crossing zero during one control cycle.

The periodic family, or a family periodic up to a specified gauge identification, determines the corresponding bundle
\[
\widetilde E
\longrightarrow
B\times S^1
\]
and the operator
\[
\mathcal D
=
\frac{\partial}{\partial t}
+
D_{A_t}.
\]
The spectral-flow index theorem gives
\[
\operatorname{sf}\{D_{A_t}\}_{t\in S^1}
=
\operatorname{ind}(\mathcal D).
\]
The Atiyah--Singer theorem \cite{Atiyah1963Index} then expresses this integer topologically:
\[
\operatorname{ind}(\mathcal D)
=
\int_{B\times S^1}
\left[
\widehat A\bigl(T(B\times S^1)\bigr)
\operatorname{ch}(\widetilde E)
\right]_{\dim B+1}.
\]

In the simplest abelian case, take
\[
B=S^1,
\qquad
B\times S^1=T^2,
\]
and let
\[
L\longrightarrow T^2
\]
be a complex line bundle with curvature \(F\).  For the corresponding family of one-dimensional Dirac-type operators,
\[
\operatorname{sf}\{D_{A_t}\}
=
\left\langle c_1(L),[T^2]\right\rangle
=
\frac{1}{2\pi}
\int_{T^2}F
\in\mathbb Z,
\]
with the conventional normalization of the connection understood. Thus a periodic control cycle produces an integer readout determined by the first Chern class.

For a nonabelian example, let
\[
\dim B=3
\]
and let
\[
\widetilde E\longrightarrow B\times S^1
\]
carry an \(SU(r)\)-connection.  The degree-four gauge contribution is governed, up to the chosen trace and sign convention, by the instanton number
\[
k
=
\frac{1}{8\pi^2}
\int_{B\times S^1}
\tr(F\wedge F)
\in\mathbb Z.
\]
When the remaining tangent-bundle contribution is fixed or trivial in the regime under consideration, this characteristic number determines the corresponding gauge contribution to the index and hence to the spectral-flow readout.

Spectral flow therefore provides an explicit example of analytic compression where an integer index classifies a continuous family of operators. The integer may be obtained either from signed eigenvalue crossings or from the appropriate characteristic class on the mapping torus.  It is the spectrum of the family
\(\{D_{A_t}\}\), rather than the spectrum of a single holonomy matrix,
that undergoes spectral flow.
\end{ex}

\begin{rem}[Holonomy and Spectral Flow]
\label{rem:holonomy-and-spectral-flow}

Holonomy and spectral flow are distinct but complementary readouts. Holonomy assigns a transport automorphism, or invariantly its conjugacy class, to a closed path.  Spectral flow assigns an integer to a closed family of self-adjoint operators.  Curvature enters the former through parallel transport and the latter through the characteristic forms appearing in the index density.
\end{rem}

\paragraph{Hamiltonian and contact presentations.}
The gauge description above does not require the fully fluidic state space to be symplectic.  In regimes that additionally admit a Hamiltonian formulation, however, the same process semantics \emph{can} be expressed in standard symplectic and contact language.

Let
\[
(X,\omega)
\]
be a symplectic phase space with Hamiltonian \(H\), and let
\[
\Sigma=H^{-1}(c)
\]
be a regular energy hypersurface of contact type.  Thus, in a neighborhood of \(\Sigma\), suppose
\[
\omega=d\lambda
\]
and
\[
\alpha:=\lambda|_\Sigma
\]
is a contact form; equivalently, the associated Liouville vector field is transverse to \(\Sigma\).  The Reeb vector field \(R_\alpha\) is defined by
\[
\alpha(R_\alpha)=1,
\qquad
d\alpha(R_\alpha,\cdot)=0
\]
\cite{McDuffSalamon-SymplTop,Geiges-Contact}.

Since the restriction of \(X_H\) to a regular energy hypersurface lies in its characteristic line field, one has, for an appropriate choice of orientation,
\[
X_H|_\Sigma
=
fR_\alpha
\]
for a nonvanishing function \(f\), positive after compatible orientation conventions.  Closed Hamiltonian characteristics are therefore reparametrized periodic Reeb trajectories.

In such a presentation, robust periodic execution cycles in GCT may be studied using the action, index, and orbit invariants familiar from contact and Floer geometry.  Symplectic field theory provides a much richer algebraic structure built from such Reeb data \cite{EliashbergGiventalHofer-SFT}.  We do not assume that machinery here.  The point is instead that whenever a fully fluidic GCM admits a Hamiltonian/contact realization, its periodic computational cycles have a standard geometric interpretation for those readers familiar with Hofer-style dynamics. The contact/Reeb description is best viewed as a \emph{change of mathematical presentation under additional hypotheses}, not an extra ontological postulate of GCT.

\begin{table}[t]
\centering
\small
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{p{0.36\textwidth} | p{0.54\textwidth}}
\toprule
\textbf{Computational notion}
&
\textbf{Gauge-geometric realization}
\\
\midrule
Local state description
&
Choice of local section, frame, or gauge representative
\\
\hline
Admissible interface update
&
Parallel transport along an interface path
\\
\hline
Sequential execution
&
Composition of transport morphisms
\\
\hline
Alternative execution paths
&
Parallel arrows compared by relative holonomy
\\
\hline
Closed execution cycle
&
Based loop and its holonomy
\\
\hline
Order-sensitive execution
&
Nontrivial join commutator
\\
\hline
Local geometric memory
&
Curvature and its gauge-covariant transport data
\\
\hline
Persistent cyclic memory
&
Holonomy conjugacy class
\\
\hline
Gauge-invariant scalar readout
&
Class function such as a Wilson-loop trace
\\
\hline
Global sector readout
&
Characteristic class, index, or related topological invariant
\\
\hline
Equivalent local descriptions
&
Gauge-related representatives preserving invariant observables
\\
\bottomrule
\end{tabular}
\caption{Computational semantics of gauge transport.  The correspondence
is operational: local representatives may depend on a chosen frame,
whereas observable output is extracted from gauge-invariant functions or
equivalence classes of the resulting transport.}
\label{tab:gauge-transport-computation}
\end{table}

\paragraph{The Wu--Yang dictionary and its computational extension.}
The gauge-geometric ingredients used above are standard.  In the Wu--Yang picture~\cite{WuYang1975}, the gauge potential is a local representative of a connection, the field strength is its curvature, gauge transformations relate local representatives, and nonintegrable phase factors are parallel transports or holonomies.  The same language underlies the modern treatment of monopoles, instantons, calorons, and other nonperturbative gauge configurations.

GCT does not claim novelty for this dictionary.  Its additional claim is semantic: when the geometric substrate itself is treated as a computational process, the standard gauge-theoretic constructions acquire operational roles,
\[
\begin{aligned}
\text{connection}
&\longleftrightarrow
\text{transport/update law},
\\
\text{curvature}
&\longleftrightarrow
\text{local path-dependent context},
\\
\text{parallel transport}
&\longleftrightarrow
\text{admissible interface update},
\\
\text{relative holonomy}
&\longleftrightarrow
\text{comparison of execution paths},
\\
\text{join commutator}
&\longleftrightarrow
\text{order-sensitive composition},
\\
\text{holonomy class}
&\longleftrightarrow
\text{persistent cyclic memory},
\\
\text{transition function}
&\longleftrightarrow
\text{interface gluing rule},
\\
\text{characteristic class}
&\longleftrightarrow
\text{global sector invariant}.
\end{aligned}
\]

The resulting hierarchy clarifies what is meant here by \emph{curvature as computation}.  Curvature is not itself an instruction set, and a nonabelian gauge group is not automatically a computer. Rather, the connection specifies admissible transport, curvature controls its infinitesimal path dependence, topology supplies additional global sectors, and holonomy records the accumulated result of closed transport.  When the effective holonomy becomes nonabelian, the join commutator adds an intrinsically order-sensitive layer of composition.

The fully fluidic regime therefore differs from the quasi--fluidic
\(U(1)\) corridor not merely by possessing more continuous degrees of
freedom, but by admitting a richer compositional geometry:
\[
\boxed{
\text{phase}
\;\longrightarrow\;
\text{transport}
\;\longrightarrow\;
\text{curvature}
\;\longrightarrow\;
\text{holonomy}
\;\longrightarrow\;
\text{noncommuting composition}.
}
\]
This ordered transport structure is the starting point for the nonabelian Algebra of Flow developed in the following subsection.

\subsection{The Algebra of Flow}
\label{sec:algebra-of-flow}

The preceding section established that geometric computation may be organized by transport, curvature, and holonomy, with the join commutator distinguishing genuinely order-sensitive composition. We now return to the geometry of the twin helix itself. The question is no longer merely how non-abelian transport is represented, but how the geometric degrees of freedom of a helical substrate can supply the transformations from which such transport is constructed.

There are two related mechanisms. First, relaxing rotational symmetry enlarges the available cross-sectional geometry, admitting anisotropies, distinct angular channels, and additional couplings. Second, the moving orthonormal frame of a helix naturally takes values in \(SO(3)\), whose spinorial double cover is \(SU(2)\). When the corresponding rotation directions become independently controllable, they provide the algebraic ingredients for noncommuting transport.

These mechanisms should be distinguished. Breaking the rotational symmetry of a cross-section does not itself produce a non-abelian connection, and the existence of an
\(SU(2)\) spinorial lift does not itself establish nontrivial
Yang--Mills curvature. Nevertheless, together they identify a natural geometric route from helical form to a non-abelian algebra of admissible operations.

\paragraph{Cross-sectional symmetry and geometric freedom.}
Recall the cross-sectional shape families introduced in \cref{sec:logic-of-geometry}. A circular cross-section is invariant under every rotation about its central axis. A general smooth cross-section need not possess this symmetry.

To quantify the distinction, consider the space of smooth, strictly positive radial profiles
\[
\mathcal S
:=
\left\{
r\in C^\infty(S^1,\mathbb R)
\;\middle|\;
r(\theta)>0
\right\}.
\]
Each profile determines a star-shaped embedded contour
\[
\Gamma_r(\theta)
=
r(\theta)
\begin{pmatrix}
\cos\theta\\
\sin\theta
\end{pmatrix}.
\]
The positivity condition ensures that this parametrization defines a regular simple closed curve. The space
\(\mathcal S\) is an infinite-dimensional smooth
configuration space, modeled on the Fréchet space
\(C^\infty(S^1,\mathbb R)\).

A radial profile admits a Fourier expansion
\[
r(\theta)
=
a_0+
\sum_{k=1}^{\infty}
\left(
a_k\cos(k\theta)
+
b_k\sin(k\theta)
\right),
\]
where the coefficients decay sufficiently rapidly to represent a smooth function.

Full rotational invariance requires
\[
r(\theta+\varphi)=r(\theta)
\qquad
\text{for every }\varphi\in S^1,
\]
and therefore forces
\[
r(\theta)=a_0.
\]
Thus, within this family, continuous rotational symmetry reduces the cross-sectional shape space to a one-dimensional family of circles.

By contrast, imposing only finite rotational symmetry of order \(n\) requires
\[
r\left(\theta+\frac{2\pi}{n}\right)
=
r(\theta),
\]
which permits every Fourier harmonic whose index is divisible by \(n\). Even this restricted space remains infinite-dimensional in the unrestricted smooth category.

\begin{prop}[Rotational Symmetry and Angular Mode Capacity]
\label{prop:rotational-symmetry-and-angular-mode-capacity}
Let
\[
\mathcal F_K
=
\left\{
a_0+
\sum_{k=1}^{K}
\left(
a_k\cos(k\theta)+b_k\sin(k\theta)
\right)
\right\}
\]
be the real vector space of trigonometric polynomials of degree at most \(K\). Let
\[
\mathcal F_{n,K}
=
\left\{
r\in\mathcal F_K
\;\middle|\;
r\left(\theta+\frac{2\pi}{n}\right)=r(\theta)
\right\}
\]
be its \(C_n\)-invariant subspace.

Then
\[
\dim\mathcal F_K=2K+1,
\]
and
\[
\dim\mathcal F_{n,K}
=
2\left\lfloor\frac{K}{n}\right\rfloor+1.
\]
Consequently, the fraction of permitted angular modes is
\[
E_K(n)
=
\frac{
2\lfloor K/n\rfloor+1
}{
2K+1
}.
\]

For fixed \(K\), this quantity is nonincreasing with \(n\). When \(n>K\), only the angular zero mode remains.
\end{prop}

\begin{proof}
Under rotation by \(2\pi/n\), the complex Fourier basis transforms according to
\[
e^{ik\theta}
\longmapsto
e^{2\pi ik/n}e^{ik\theta}.
\]
Invariance requires
\[
e^{2\pi ik/n}=1,
\]
which holds precisely when
\[
k\in n\mathbb Z.
\]
Within the cutoff \(|k|\leq K\), the permitted frequencies are therefore
\[
k=0,\ \pm n,\ \pm2n,\ldots,
\pm\left\lfloor\frac{K}{n}\right\rfloor n.
\]
Counting these frequencies gives
\[
\dim\mathcal F_{n,K}
=
2\left\lfloor\frac{K}{n}\right\rfloor+1.
\]
Division by \(\dim\mathcal F_K=2K+1\) proves the result.
\end{proof}

The proposition gives a finite-cutoff measure of a more general phenomenon. In the unrestricted smooth category, removing continuous rotational symmetry releases an infinite family of angular shape parameters. The finite-dimensional approximation records how many of these parameters survive a prescribed residual symmetry.

This is a statement about \emph{geometric configuration freedom}, not yet about computational macro-expressiveness. In particular, the dimension of a shape space does not by itself determine the range of admissible computations.

\paragraph{Fourier selection rules and transport channels.}
The Fourier calculation has a second interpretation. Suppose a linearized transport or coupling operator
\[
L:\mathcal H\longrightarrow\mathcal H
\]
acts on an angular function space \(\mathcal H\) and commutes with a \(C_n\) rotation operator \(T_n\):
\[
LT_n=T_nL.
\]
Then \(L\) preserves the isotypic decomposition of
\(\mathcal H\) under the \(C_n\) action.

In the Fourier basis, this means that matrix elements between angular modes vanish unless
\[
m\equiv m'\pmod n.
\]
Equivalently, the operator decomposes into blocks indexed by angular momentum modulo \(n\).

This is the familiar origin of symmetry selection rules. A residual rotational symmetry restricts which angular channels can couple, while symmetry breaking permits additional off-diagonal couplings. The corresponding calculation for cross-sectional flow fields is recorded in \cref{app:fourier-mode-calculations}.

Importantly, block diagonalization does not imply that all operators acting within a block commute. A multidimensional isotypic component may support noncommuting internal transformations. The selection rule determines which channels may interact; the algebra of the permitted interaction operators determines whether their compositions commute.

Thus the passage
\[
SO(2)
\longrightarrow
C_n
\longrightarrow
\{e\}
\]
describes a progressive relaxation of spatial rotational constraints. It should not be confused with a change from an internal \(U(1)\) gauge group to \(SU(2)\). The two structures are mathematically related through the available geometric controls, but they are distinct group actions.

\paragraph{The moving-frame origin of \(SU(2)\).}
The second mechanism is intrinsic to the helical spine. A regular space curve with nonvanishing curvature carries a Frenet--Serret frame
\[
R(s)
=
\begin{pmatrix}
T(s)&N(s)&B(s)
\end{pmatrix}
\in SO(3),
\]
where \(s\) is arc length and the frame is positively oriented.

The Frenet--Serret equations are
\[
\begin{aligned}
T'&=\kappa N,\\
N'&=-\kappa T+\tau B,\\
B'&=-\tau N,
\end{aligned}
\]
with curvature \(\kappa(s)\) and torsion \(\tau(s)\).

In matrix form,
\[
R^{-1}\frac{dR}{ds}
=
\Omega(s)
=
\begin{pmatrix}
0&-\kappa&0\\
\kappa&0&-\tau\\
0&\tau&0
\end{pmatrix}.
\]
Writing \(J_a\) for the standard generators of
\(\mathfrak{so}(3)\), we obtain
\[
\Omega(s)
=
\tau(s)J_1+\kappa(s)J_3.
\]

Thus the differential geometry of a helix directly supplies two rotational directions, associated with torsion and curvature. They belong to the Lie algebra
\[
\mathfrak{so}(3),
\]
whose simply connected covering group is
\[
\operatorname{Spin}(3)\cong SU(2).
\]

The standard double covering
\[
p:SU(2)\longrightarrow SO(3)
\]
has kernel
\[
\ker p=\{\pm I\},
\]
and consequently
\[
SO(3)\cong SU(2)/\{\pm I\}.
\]

In the fundamental representation, the covering map is characterized by
\[
U(\mathbf v\cdot\boldsymbol{\sigma})U^\dagger
=
\bigl(p(U)\mathbf v\bigr)
\cdot\boldsymbol{\sigma},
\]
where
\[
\boldsymbol{\sigma}
=
(\sigma_1,\sigma_2,\sigma_3)
\]
denotes the triple of Pauli matrices.

\begin{prop}[Spinorial Lift of Helical Frame Transport]
\label{prop:spinorial-lift-of-helical-frame-transport}
Let
\[
\gamma:I\longrightarrow\mathbb R^3
\]
be a unit-speed \(C^3\) curve with nonvanishing curvature, and let
\[
R:I\longrightarrow SO(3)
\]
be its Frenet--Serret frame.

After choosing a lift of the initial frame, there exists a unique lifted path
\[
\widetilde R:I\longrightarrow SU(2)
\]
such that
\[
p\circ\widetilde R=R.
\]

With the standard Pauli-matrix convention,
\[
\widetilde R^{-1}
\frac{d\widetilde R}{ds}
=
-\frac{i}{2}
\left(
\tau(s)\sigma_1+\kappa(s)\sigma_3
\right).
\]

If the frame traces a closed loop in \(SO(3)\), the endpoint of its lifted path differs from the initial lift by an element of \(\{\pm I\}\). The sign detects the corresponding class in
\[
\pi_1(SO(3))\cong\mathbb Z_2.
\]
\end{prop}

\begin{proof}
The map
\[
SU(2)\longrightarrow SO(3)
\]
is a covering map, so the path-lifting property gives a unique lift once its initial value is specified.

Under the standard Lie-algebra identification, an infinitesimal spatial rotation with angular velocity
\[
\boldsymbol\omega
=
(\omega_1,\omega_2,\omega_3)
\]
is represented in \(\mathfrak{su}(2)\) by
\[
-\frac{i}{2}
\boldsymbol\omega\cdot\boldsymbol{\sigma}.
\]
The Frenet--Serret matrix \(\Omega\) corresponds to the angular-velocity vector
\[
\boldsymbol\omega=(\tau,0,\kappa).
\]
Substitution gives the asserted lifted generator.

For a closed frame loop, the two endpoints of the lift project to the same element of \(SO(3)\), and therefore differ by an element of the covering kernel \(\{\pm I\}\). This distinction agrees with the two homotopy classes of loops in \(SO(3)\).
\end{proof}

The proposition identifies a natural spinorial representation of helical frame transport. A full \(2\pi\) spatial rotation returns an ordinary frame to its original orientation, while its lifted action on the fundamental spin representation is
\[
\widetilde R(2\pi)=-I.
\]
The lift returns to \(+I\) after a \(4\pi\) rotation.

This double-cover structure is not imposed as an independent symmetry of the twin helix. It arises from the ordinary rotational geometry of its moving frame.

There is, however, an essential limitation. The differential form
\[
R^{-1}dR
\]
is a pure-gauge frame connection wherever the frame is globally defined. Its spinorial lift
\[
\widetilde R^{-1}d\widetilde R
\]
satisfies the Maurer--Cartan equation and has vanishing curvature. The existence of the spin lift therefore does not by itself generate a nonflat \(SU(2)\) gauge field.

Moreover, the moving frame of the single conical helix defined in \eqref{eq:conical-helix-spine} has variable curvature and torsion, but their ratio satisfies Lancret's condition $\tau(s)/\kappa(s)=q$ for a constant $q$. Its lifted Frenet--Serret generator consequently takes the form
\[
-\frac{i}{2}\kappa(s)(q\sigma_1+\sigma_3).
\]
Generators at different points of the spine are thus proportional to the same element of $\mathfrak{su}(2)$ and commute. The spinorial evolution remains in a single one-parameter subgroup: screw--dilation changes the magnitude of the generator, not its Lie-algebra direction.

The additional ingredient required for non-abelian computation is independent control over distinct rotation directions, together with a connection for which their associated transports do not reduce to a single commuting subgroup.

\paragraph{Independent geometric controls and non-abelian curvature.} The twin helix construction provides several candidate mechanisms for such control: variations in curvature and torsion, cross-sectional anisotropy, coupling between the two helical strands, and boundary conditions which select distinct internal transport channels.

To exhibit the algebraic mechanism explicitly, consider a two-dimensional control region
\[
D\subset\mathbb R^2
\]
with coordinates \((u,v)\), equipped with a rank-two Hermitian state bundle. The two control directions are taken to act through independent spinorial rotation generators.

Choose an \(\mathfrak{su}(2)\)-valued connection
\[
\mathcal A
=
-\frac{i}{2}
\left(
a\sigma_1\,du+
b\sigma_3\,dv
\right),
\qquad
a,b\in\mathbb R\setminus\{0\}.
\]
We use the convention
\[
\nabla=d+\mathcal A,
\qquad
\mathcal F
=
d\mathcal A+\mathcal A\wedge\mathcal A.
\]

The two generators are motivated by the torsion and curvature directions appearing in the lifted Frenet--Serret equations. The connection is an explicit controlled model; its realization by a particular twin helix interaction law remains a separate dynamical requirement.

\begin{prop}[Non-Abelian Holonomy from Independent
Helical Rotation Channels]
\label{prop:non-abelian-holonomy-from-independent-helical-rotation}
Let \(D\subset\mathbb R^2\) be a connected, simply connected open control region, and let
\[
\mathcal A
=
-\frac{i}{2}
\left(
a\sigma_1\,du+b\sigma_3\,dv
\right)
\]
be the connection defined above, with
\(a,b\neq0\).

Then
\[
\mathcal F
=
\frac{iab}{2}
\sigma_2\,du\wedge dv
\neq0,
\]
and the holonomy Lie algebra is
\[
\mathfrak{hol}(\mathcal A)
=
\mathfrak{su}(2).
\]
Consequently, the holonomy group is \(SU(2)\).
\end{prop}

\begin{proof}
The connection coefficients are constant, so
\[
d\mathcal A=0.
\]
The curvature is therefore determined by the commutator of the two connection components:
\[
\mathcal F_{uv}
=
[\mathcal A_u,\mathcal A_v].
\]
Using
\[
[\sigma_1,\sigma_3]=-2i\sigma_2,
\]
we obtain
\[
\begin{aligned}
\mathcal F_{uv}
&=
\left[
-\frac{ia}{2}\sigma_1,
-\frac{ib}{2}\sigma_3
\right]\\
&=
\frac{iab}{2}\sigma_2.
\end{aligned}
\]

The curvature initially supplies the
\(\sigma_2\) direction. Parallel transport of
curvature along the two control directions supplies the remaining generators.

Indeed, the gauge-covariant derivatives satisfy
\[
D_u\mathcal F_{uv}
=
[\mathcal A_u,\mathcal F_{uv}]
=
\frac{ia^2b}{2}\sigma_3,
\]
and
\[
D_v\mathcal F_{uv}
=
[\mathcal A_v,\mathcal F_{uv}]
=
-\frac{iab^2}{2}\sigma_1.
\]

Since \(a,b\neq0\), these three directions span
\[
\operatorname{span}_{\mathbb R}
\{i\sigma_1,i\sigma_2,i\sigma_3\}
=
\mathfrak{su}(2).
\]

By the Ambrose--Singer description of the holonomy algebra, curvature values transported to a common base point generate the holonomy Lie algebra. The displayed covariant derivatives arise as infinitesimal variations of those transported curvature values, and therefore belong to that algebra.

It follows that
\[
\mathfrak{hol}(\mathcal A)
=
\mathfrak{su}(2).
\]
Since the control region is simply connected, the holonomy group is connected. Its Lie algebra is the full \(\mathfrak{su}(2)\), so the holonomy group is \(SU(2)\).
\end{proof}

This proposition supplies an explicit non-abelian transport model using the same two rotation directions that arise naturally in the helical moving-frame description.

It should not be interpreted as asserting that every twin helix generates this connection. Rather, it establishes that independently controlled spinorial frame directions admit a nonflat connection whose holonomy is genuinely non-abelian.

The geometric realization problem is now precise: determine which admissible deformations, couplings, and boundary controls of the twin helix induce connections of this type, and which restrictions on those controls preserve or reduce the resulting holonomy group.

\paragraph{The algebra of noncommuting execution.}
The distinction from the abelian regime can be seen directly in the fundamental representation.

Consider two \(SU(2)\) transformations
\[
U_1=\exp(i\alpha\sigma_1),
\qquad
U_3=\exp(i\beta\sigma_3).
\]
Using the Pauli-matrix relations, one obtains
\[
U_1U_3-U_3U_1
=
2i\sin\alpha\sin\beta\,\sigma_2.
\]
For generic values of \(\alpha,\beta\), the two execution orders therefore differ.

When these transformations are realized as based holonomies, their group commutator
\[
[U_1,U_3]
=
U_1U_3U_1^{-1}U_3^{-1}
\]
is a nontrivial instance of the join commutator introduced in \cref{def:join-commutator}.

By contrast, for scalar phase transports
\[
u_1=e^{i\alpha},
\qquad
u_2=e^{i\beta},
\]
one always has
\[
u_1u_2=u_2u_1.
\]
An abelian connection may distinguish different paths through curvature or global monodromy, but it cannot distinguish the order of two based transports through their group commutator.

The passage to non-abelian holonomy therefore changes the algebra of admissible composition. It introduces ordered operations whose collective behavior is not determined merely by multiplying independent scalar phases.

\paragraph{Joint holonomy as computational information.}
A further distinction concerns the information retained by gauge-invariant readouts.

For an individual closed transport cycle, the conjugacy class
\[
[U_\gamma]
\]
or a character such as
\[
\tr(U_\gamma)
\]
provides a gauge-invariant observable.

For several cycles based at the same interface, however, the individual conjugacy classes need not determine the gauge-invariant relations among their transports. Under a simultaneous gauge transformation,
\[
(U_1,\ldots,U_m)
\longmapsto
(gU_1g^{-1},\ldots,gU_mg^{-1}).
\]
The relevant invariant object is therefore the simultaneous-conjugation class
\[
[(U_1,\ldots,U_m)]
\in
G^m/G.
\]

In particular, two holonomies may possess identical individual trace data while their ordered products are distinguishable.

For example, let
\[
U=\exp(i\alpha\sigma_1),
\]
and compare
\[
V_{\parallel}
=
\exp(i\beta\sigma_1),
\qquad
V_{\perp}
=
\exp(i\beta\sigma_3).
\]
Then
\[
\tr(V_{\parallel})
=
\tr(V_{\perp})
=
2\cos\beta,
\]
but
\[
\tr(UV_{\parallel})
=
2\cos(\alpha+\beta),
\]
whereas
\[
\tr(UV_{\perp})
=
2\cos\alpha\cos\beta.
\]

Thus the individual cycle readouts agree, while the composite readouts generally differ. The relative orientation of the internal transport channels carries additional gauge-invariant information.

This provides a useful computational distinction. A collection of loop registers cannot always be treated as independent scalar memories. Their mutual transport relations may themselves form part of the readable state.

\paragraph{A precise obstruction to abelianization.}
The preceding phenomenon has a direct connection to the notion of physical macro-expressiveness introduced in \cref{def:transport-compatible-abelianization}. However, the nonexistence of a semantics-preserving translation must be established relative to a specified class of translations. Noncommuting matrix operations are not, by themselves, beyond ordinary classical simulation. We therefore isolate the particular obstruction relevant to transport-preserving comparisons.

\begin{defn}[Transport-Compatible Abelianization]
\label{def:transport-compatible-abelianization}
Let \(H\) be a realized based holonomy group with a specified family of gauge-invariant readouts
\[
\mathcal R=\{r_\lambda:H\to\mathcal O_\lambda\}.
\]

A \emph{transport-compatible abelianization} consists of an abelian group \(A\), a group homomorphism
\[
\phi:H\longrightarrow A,
\]
and functions
\[
\overline r_\lambda:
A\longrightarrow\mathcal O_\lambda
\]
such that
\[
r_\lambda
=
\overline r_\lambda\circ\phi
\]
for every \(r_\lambda\in\mathcal R\).

Thus the translation preserves composition and every designated readout. This is a restricted class of the semantics-preserving translations considered in the general definition of physical expressive power.
\end{defn}

\begin{prop}[Obstruction to Abelianizing Holonomy Readouts]
\label{prop:obstruction-to-abelianizing-holonomy-readouts}
Let
\[
H\subseteq SU(2)
\]
be a realized holonomy group containing elements \(U,V\) with
\[
[U,V]\neq I.
\]
Suppose the designated readouts include the fundamental character
\[
\chi(W)=\tr(W),
\qquad W\in H.
\]

Then no transport-compatible abelianization of \(H\) exists that preserves this character on every element of \(H\).
\end{prop}

\begin{proof}
Suppose such an abelianization exists, with
\[
\phi:H\longrightarrow A,
\]
where \(A\) is abelian.

Because \(\phi\) preserves multiplication,
\[
\phi([U,V])
=
[\phi(U),\phi(V)]
=
e_A.
\]
But
\[
\phi(I)=e_A
\]
as well.

If the fundamental character factors through \(\phi\), then
\[
\chi([U,V])=\chi(I).
\]

In the fundamental representation of
\(SU(2)\), every element has eigenvalues
\(e^{i\theta}\) and \(e^{-i\theta}\).
Consequently,
\[
\tr(W)=2
\quad\Longleftrightarrow\quad
W=I.
\]

Since
\[
[U,V]\neq I,
\]
we obtain
\[
\tr([U,V])\neq2=\tr(I),
\]
contradicting the assumed factorization. Therefore no such abelianization exists.
\end{proof}

This result identifies a genuine compositional distinction between the two regimes. An abelian transport model cannot simultaneously preserve the multiplication law and all character readouts of a nonabelian \(SU(2)\) holonomy system.

It does not follow that an ordinary Turing machine, matrix computer, or suitably enriched abelian system cannot simulate the corresponding behavior by a different encoding. Rather, the proposition identifies the obstruction encountered when the admissible translation must preserve the native transport algebra and its readouts.

\paragraph{Geometric freedom and physical expressiveness.}
We can now distinguish three levels of the construction.

\begin{itemize}
    \item \textbf{Kinematic freedom.}
Relaxing rotational symmetry enlarges the space of cross-sectional shapes and admits additional angular modes, anisotropies, and coupling patterns. The Fourier calculation quantifies this freedom at finite cutoff.

    \item \textbf{Representation structure.}
    The orthonormal moving frame of a helix takes values in
    \(SO(3)\), and its spinorial lift takes values in \(SU(2)\).
This supplies a mathematically natural two-dimensional complex representation of helical rotations.

    \item \textbf{Dynamical transport.}
When independent admissible controls generate a connection whose effective holonomy is nonabelian, the resulting operations acquire order-sensitive composition. At this level, relative holonomies, join commutators, and jointly gauge-invariant readouts become computationally significant.
\end{itemize}

These levels may be summarized as
\[
\begin{gathered}
\text{cross-sectional symmetry breaking}
\\
\Downarrow
\\
\text{additional geometric control channels}
\\[3pt]
\text{helical moving frame}
\longrightarrow
SO(3)
\longleftarrow
SU(2)
\\
\Downarrow
\\
\text{spinorial transport representation}
\\
\Downarrow
\\
\text{independently controlled connection}
\\
\Downarrow
\\
\text{nonabelian holonomy and ordered composition}.
\end{gathered}
\]

The arrows involving the moving frame and spin double cover are established geometric constructions. The passage from available geometric controls to a particular nonflat connection is a dynamical realization problem. The explicit connection of \cref{prop:non-abelian-holonomy-from-independent-helical-rotation} demonstrates the mathematical possibility of this passage, without asserting that every $THM$ realizes it.

To clarify the role of degenerate internal state spaces we note that an isolated rank-one adiabatic band carries abelian Berry transport, whereas a degenerate rank-\(m\) subbundle permits a \(U(m)\)-valued Wilczek--Zee connection \cite{Berry1984QuantalPhase,WilczekZee1984Appearance}. For \(m=2\), the traceless transport sector is naturally
\(\mathfrak{su}(2)\)-valued.
Degeneracy permits such transport, but does not force its effective holonomy to be nonabelian. Likewise, cross-sectional anisotropy can supply mixing terms, but may also split degeneracies unless additional symmetries or constraints protect the relevant subspace. Physical consequences of this moduli are explained in \cref{par:cross-section-particle-pheno}.

\paragraph{Computational consequences.}
The principal distinction between the abelian and nonabelian regimes is therefore not a universal ranking of predictability, complexity, or computational speed.

Abelian continuous systems can possess complicated and even chaotic dynamics. Conversely, nonabelian systems may be integrable or admit strong invariant constraints. Neither geometric complexity nor noncommutativity alone implies algorithmic undecidability.

What changes is the native compositional structure. In a commuting phase sector, the order of based holonomy operations is immaterial. In a nonabelian sector, a computation may depend on the ordered history of its transport operations, even when each individual operation has a simple local description. From the standpoint of verification, this makes the choice of observable data especially important. Separate local readouts may fail to distinguish globally different compositions, while suitable joint holonomy invariants can retain the relevant differences.

The EVT Hypothesis consequently applies here not as a theorem that nonabelian computation is necessarily less verifiable, but as a question about the adequacy of local certificates under richer forms of composition. The admissible translation class, the protected invariants, and the chosen readout must all be specified before a claim of macro-expressive increase can be established.

\paragraph{From transport algebra to graph coordinates.}
The resulting algebra does not require the complete smooth connection to be retained as a computational register. For a chosen finite family of admissible paths, one may record their transport matrices, impose gauge equivalence at shared endpoints, and study invariant functions of the resulting graph-level data.

This provides the natural bridge from the moving-frame description of helices to graph holonomies and their representation-theoretic decomposition. The next subsection develops precisely this finite reduction, relating the transport structure to the standard mathematics of spin-network coordinates.

\subsection{From Helical Holonomy to Spin-Network Coordinates}
\label{sec:spin-network-coordinates}

Although our program does not aim to adjudicate debates over quantum gravity, aspects of the geometric computation model do resonate with the combinatorial, categorical, and background-independent \emph{perspectives} emphasized in Loop Quantum Gravity~\cite{SMOLIN_2004}.

The relationship is most naturally expressed through their shared use of holonomy, representation theory, and combinatorial descriptions of geometric states. Although the two frameworks differ in their foundational assumptions, the passage from continuous geometric transport to graph-based observables provides a useful mathematical point of contact.

In GCT, the underlying twin-helical geometry remains smooth, with information carried by admissible flows, connections, and their associated holonomies. By selecting a finite graph of interfaces and transport paths, one can retain a discrete combinatorial description of this continuous structure. Assigning holonomies to edges and imposing gauge compatibility at vertices leads naturally to the representation-theoretic data familiar from spin networks, including edge labels and intertwiners~\cite{Penrose1971AngularMomentum,RovelliSmolin1995SpinNetworks}. Spin-foam constructions provide a related language for histories interpolating between such boundary data.

The distinction is one of interpretation. We do not assume that the graph constitutes the fundamental geometry, nor that its representation labels already define a quantum theory. Rather, spin-network coordinates arise as a finite, gauge-invariant presentation of selected geometric transport data. This leaves open the possibility that discrete quantum-geometric observables emerge as stable effective structures of an underlying continuous substrate, rather than being imposed at the microscopic level.\footnote{Also: ``Spacetime is doomed.'' --- NAH \cite{Arkani-Hamed2013}}

\paragraph{Helix-adapted graph reduction.}
Let
\[
\pi:P\longrightarrow B
\]
be a principal \(G\)-bundle over a working region \(B\), where \(G\) is a compact Lie group, and let
\[
A\in\Omega^1(P;\mathfrak{g})
\]
be a connection. Choose a finite oriented graph
\[
\Gamma=(V,E)
\]
embedded in \(B\), with edges following admissible transport channels of the GCM lattice and vertices located at interfaces or junctions. Unlike the minimal loops used for Wilson-loop readouts, the edges of
\(\Gamma\) are generally open paths. Closed transport cycles are
obtained by composing these edges.

After choosing a frame in the fiber over each vertex, parallel transport along an oriented edge \(e\) defines an element
\[
h_e(A)
:=
\operatorname{Hol}_e(A)
=
\mathcal{P}\exp\left(\int_e A\right)
\in G.
\]
The connection therefore determines a finite graph configuration
\[
\operatorname{Hol}_{\Gamma}(A)
=
\bigl(h_e(A)\bigr)_{e\in E}
\in G^E.
\]

\begin{prop}[Graph-Holonomy Reduction]
\label{prop:graph-holonomy-reduction}
Let \(P\to B\), \(A\), and \(\Gamma=(V,E)\) be as above. There is a graph-holonomy map
\[
\operatorname{Hol}_{\Gamma}:
\mathcal{A}(P)
\longrightarrow
G^E,
\qquad
A\longmapsto
\bigl(h_e(A)\bigr)_{e\in E},
\]
where \(\mathcal{A}(P)\) denotes the space of connections on \(P\).

With the convention that a vertex gauge transformation
\[
g=(g_v)_{v\in V}\in G^V
\]
acts on an edge holonomy by
\[
h_e
\longmapsto
g_{s(e)}h_e g_{t(e)}^{-1},
\]
the graph-holonomy map is equivariant with respect to the restriction of the continuum gauge action to the vertices. It therefore induces a map of quotient groupoids
\[
[\mathcal{A}(P)/\mathcal{G}(P)]
\longrightarrow
[G^E/G^V].
\]
\end{prop}

\begin{proof}
Parallel transport along an edge produces a \(G\)-equivariant map between the fibers over its source and target. Choosing frames at those fibers identifies this transport with an element \(h_e(A)\in G\). Changing the frames by \(g_{s(e)}\) and \(g_{t(e)}\) changes the group element according to
\[
h_e(A)
\longmapsto
g_{s(e)}h_e(A)g_{t(e)}^{-1}.
\]
This is precisely the vertex gauge action on \(G^E\), establishing equivariance.
\end{proof}

\begin{rem}
\label{rem:graph-holonomy-choice-dependence}

The map depends on the chosen graph and on the chosen vertex frames. It is a finite graph-level reduction of the connection data, not a claim that finitely many holonomies determine a continuum connection. The frame dependence is removed by passing to the quotient groupoid \([G^E/G^V]\).
\end{rem}

The reduction is compatible with refinement. If an edge is decomposed as a concatenation
\[
e=e_2\circ e_1,
\]
then parallel transport satisfies
\[
h_e(A)
=
h_{e_2}(A)h_{e_1}(A).
\]
Thus a refinement of the helix-adapted graph replaces a coarse holonomy by an ordered product of finer holonomies. This is the precise sense in which geometric composition descends to graph transport.

\paragraph{Spin-network coordinates.}
The classical graph configuration space is \(G^E\), equipped with the vertex gauge action of \(G^V\). Its standard kinematical quantization is the Hilbert space
\[
\mathcal{H}_{\Gamma}
:=
L^2\left(G^E,d\mu_{\mathrm{Haar}}^{\otimes E}\right),
\]
with the gauge-invariant subspace
\[
\mathcal{H}_{\Gamma}^{\mathrm{inv}}
:=
L^2\left(G^E,d\mu_{\mathrm{Haar}}^{\otimes E}\right)^{G^V}.
\]
A Wilson loop is a special gauge-invariant function obtained by taking a class function of the holonomy around a closed cycle. Spin-network functions provide a basis for the full gauge-invariant graph space, including configurations with many edges and junctions.

\begin{prop}[Spin-Network Expansion]
\label{prop:spin-network-expansion}
Let \(G\) be a compact Lie group and let \(\Gamma=(V,E)\) be a finite oriented graph. The Peter--Weyl theorem gives an orthonormal basis of
\(\mathcal{H}_{\Gamma}^{\mathrm{inv}}\) indexed by:

\begin{enumerate}[label=\textup{(\alph*)},leftmargin=*]
\item an irreducible unitary representation
\[
\rho_e:G\longrightarrow U(V_{\rho_e})
\]
assigned to each edge \(e\in E\);

\item an invariant tensor
\[
\iota_v
\in
\operatorname{Inv}
\left(
\bigotimes_{e:t(e)=v}V_{\rho_e}
\otimes
\bigotimes_{e:s(e)=v}V_{\rho_e}^{*}
\right)
\]
assigned to each vertex \(v\in V\).
\end{enumerate}

The corresponding spin-network function is obtained by taking the matrix coefficients
\[
\rho_e(h_e)
\]
on each edge and contracting their indices at every vertex using the intertwiners \(\iota_v\).

For \(G=SU(2)\), the irreducible representations are indexed by
\[
j_e\in\frac{1}{2}\mathbb{N},
\]
so a basis element is conventionally denoted
\[
\bigl(\Gamma,\{j_e\},\{\iota_v\}\bigr).
\]
\end{prop}

\begin{proof}
By the Peter--Weyl theorem,
\[
L^2(G)
\cong
\widehat{\bigoplus}_{\rho\in\widehat{G}}
V_{\rho}\otimes V_{\rho}^{*}.
\]
Taking one copy for each edge gives a decomposition of \(L^2(G^E)\) into tensor products of representation spaces and their duals. The action of \(G^V\) acts independently on the indices incident at each vertex. Passing to the invariant subspace therefore selects precisely the invariant tensors, or intertwiners, at the vertices.
\end{proof}

The representation label \(j_e\) should not be confused with the spectrum of a particular holonomy \(h_e\). A holonomy is a group element, whereas \(j_e\) labels an irreducible Fourier mode in the Peter--Weyl decomposition of functions on the group. The spin-network description is therefore a representation-theoretic expansion of the graph-holonomy state space, not a direct assignment of one spin to each holonomy eigenphase.

\paragraph{Four-valent vertices and quantum tetrahedra.}
Suppose the helix-adapted graph contains a four-valent vertex \(v\). For \(G=SU(2)\), the gauge-invariant vertex data lie in the intertwiner space
\[
\mathcal{I}_v
=
\operatorname{Inv}
\left(
V_{j_1}\otimes
V_{j_2}\otimes
V_{j_3}\otimes
V_{j_4}
\right),
\]
where orientations have been suppressed using the self-duality of finite-dimensional \(SU(2)\) representations. This is the standard intertwiner space associated with a quantum tetrahedron.

The chosen interface graph supplies the four-valence. The representation labels and intertwiner arise from the \(SU(2)\) quantization of its holonomy variables. This construction is distinct from the finite
\[
V_4\cong C_2\times C_2
\]
symmetry of the symbolic twin helix register. The two structures may coexist in a coarse cell, but \(V_4\) does not by itself generate the
\(SU(2)\) intertwiner or determine the spin labels.

In this sense, the quantum tetrahedron enters GCT conditionally: once a four-valent transport graph and an \(SU(2)\) gauge sector are present, the standard graph quantization assigns an intertwiner space of tetrahedral type. The helix geometry supplies a possible microscopic origin for the transport graph, but the quantum-geometric interpretation belongs to the subsequent representation-theoretic quantization.

\paragraph{Computational bordisms and spin-foam carriers.}
A history of graph configurations may be represented by a two-dimensional cell complex
\[
\mathcal{F},
\]
whose faces trace the evolution of graph edges, whose edges trace the evolution of graph vertices, and whose vertices record elementary changes of the interface structure. Gluing successive histories corresponds to composition of the resulting two-complexes.

This gives a natural interface with spin-foam constructions \cite{Baez1998SpinFoam}. After quantization, one may label the faces of
\(\mathcal{F}\) by irreducible representations and its edges by
intertwiners. However, a genuine spin-foam model requires more than the underlying two-complex and its labels: one must additionally specify face, edge, and vertex amplitudes, together with a measure or refinement prescription. These weights are not determined merely by the existence of the helix geometry or by products of Wilson-loop class functions.

Accordingly, a computational bordism supplies a candidate geometric carrier for a spin foam, but the passage
\[
\text{GCM history}
\longrightarrow
\text{spin-foam amplitude}
\]
requires an additional quantization and weighting rule. Establishing such a rule is a problem for the future development of Geometric Computational Field Theory.

\begin{rem}[Status of the Correspondence]
\label{rem:gauge-and-computation-correspondence-status}

The mathematically established correspondence in this section is conditional but precise:

\[
\begin{aligned}
&\text{a GCM connection and a chosen helix-adapted graph}\\
&\hspace{2cm}\Longrightarrow
\text{graph holonomies in }G^E,\\[3pt]
&\text{compact-group quantization and gauge reduction}\\
&\hspace{2cm}\Longrightarrow
\text{a spin-network basis}.
\end{aligned}
\]

This does not yet prove that spin networks are uniquely determined coarse invariants of all helix configurations, nor that the quantum kinematics of LQG emerge without an additional quantization step. It does show that once the continuous geometric dynamics are reduced to a finite system of gauge transports, spin-network coordinates arise by a standard and non-arbitrary mathematical procedure.

Gauge invariance should likewise be distinguished from background independence. Passing from \(G^E\) to its \(G^V\)-invariant sector removes dependence on local gauge frames. Full background independence would additionally require control of graph embeddings, graph refinement, and diffeomorphism equivalence.
\end{rem}


\paragraph{Summary.}
The role of spin networks in GCT is not to replace the underlying manifold by fundamental discrete atoms. Rather, they provide representation-theoretic coordinates for the gauge-invariant holonomy data extracted from a chosen network of geometric transport channels. The helix lattice supplies the continuous geometry and interface structure; graph reduction supplies finite holonomies; and compact-group quantization supplies the discrete spin labels and intertwiners. Spin foams may then describe quantized histories of these graph data, provided an additional amplitude prescription is supplied.


These examples are not members of one uniform solution class. What they share is a local-to-global mechanism: ordinary gauge potentials are defined chartwise, while the computationally relevant information is carried by their compatibility, transport, curvature, and placement inside a global configuration space.

\subsection{On the Expressive Power of Physical Computation}\label{sec:on-the-expressive-power-of-physical-computation}

Before stating our final claim, we collect a few canonical gauge–theoretic solutions whose geometry already exhibits the helical/twisting features emphasized in this work. The point is not to assert equivalence, but to note recurring \emph{mechanisms}---periodicity, compactification twists, unstable saddles, and chiral transport---by which the connection $A$ accumulates phase or matrix holonomy along preferred directions.

\begin{itemize}
    \item \textbf{Calorons.} Periodic instantons on $\mathbb{R}^3\times S^1$, whose Euclidean time compactification enforces a helical/phase structure along the circle~\cite{KraanVanBaal1998Periodic}.
    \item \textbf{Twisted solutions.} Gauge fields on compactified or cylindrical backgrounds with nontrivial boundary conditions; the twist enforces a globally enforced phase (or matrix) winding~\cite{tHooft1979MagneticFlux}.
    \item \textbf{Sphalerons.} Unstable saddle points in electroweak theory; they sit atop energy barriers and encode large, topological reconfigurations (holonomy changes) along interpolation paths~\cite{KlinkhamerManton1984Saddle}.
    \item \textbf{Witten’s twisted constructions.} Lower–dimensional/topologically twisted reductions where the effective degrees of freedom are organized by holonomy classes and cohomological data~\cite{Witten1988Topological}.
    \item \textbf{Non-abelian plane waves.} Chiral, order-sensitive transport in non-abelian gauge theory; parallel transport along wavefronts produces noncommuting holonomies~\cite{Coleman1977Nonabelian}.
\end{itemize}

Taken together, these exemplars highlight three recurring themes: (a) \emph{periodic/compactified directions} that act as helical axes (calorons, twisted sectors), (b) \emph{unstable saddles} that mediate large holonomy moves (sphalerons), and (c) \emph{intrinsically chiral, non-abelian transport} (plane waves). In what follows, we leverage exactly these mechanisms---now encoded by holonomy class functions on chosen loops---to motivate the usage of geometric computation.

While this discussion does not claim to exhaust the intricate connections between geometric computation and modern field theory, it is instructive (cf.~\cref{fig:bpst-instanton}) that gluing two helical manifolds along a common boundary produces patching data directly analogous to the clutching that underlies BPST instantons. We now enter the final of the four computational regimes, \emph{emergent gauge}.

\paragraph{Local connection data and descent.}
Let
\[
\pi:P\longrightarrow M
\]
be a principal \(G\)-bundle over a smooth manifold \(M\), where \(G\) is a compact Lie group. Choose an open cover
\[
M=\bigcup_{i\in I}U_i
\]
and local sections of \(P\) over the \(U_i\). On an overlap
\(U_i\cap U_j\), the two local trivializations are related by a
transition function
\[
g_{ij}:U_i\cap U_j\longrightarrow G.
\]
On triple overlaps, the transition functions satisfy the cocycle condition
\[
g_{ij}g_{jk}=g_{ik}.
\]
A connection is represented locally by one-forms
\[
A_i\in\Omega^1(U_i;\mathfrak{g})
\]
satisfying
\[
A_j
=
g_{ij}^{-1}A_i g_{ij}
+
g_{ij}^{-1}dg_{ij}
\]
on \(U_i\cap U_j\). Their curvatures
\[
F_i=dA_i+A_i\wedge A_i
\]
obey
\[
F_j=g_{ij}^{-1}F_i g_{ij}.
\]

Thus the local potentials are not independent fields. They are representatives of a single global connection reconstructed by descent. The transition functions encode changes of local frame, while gauge-invariant information must descend through these changes of description. In the language of GCT, the individual chart representatives are local computational states, the transition functions are interface data, and the global connection is the coherent state obtained after the interface constraints have been satisfied.

A pair of joined unit helices may provide a concrete geometric cover by chart domains \(U_A\) and \(U_B\), but the gluing of their underlying base regions does not by itself produce a nontrivial gauge bundle. Nontrivial bundle topology resides in the homotopy class of the transition data. For example, the standard clutching description of an
\(SU(2)\) bundle over \(S^4\) uses a transition function on the
equatorial \(S^3\),
\[
g:S^3\longrightarrow SU(2),
\]
whose degree lies in
\[
\pi_3(SU(2))\cong\mathbb{Z}.
\]
The integer degree determines the second Chern class and hence the instanton sector~\cite{Belavin1975BPST,ADHM1978}. The helix-interface analogy is therefore an analogy at the level of chartwise gluing and transport: to obtain an instanton sector, one must additionally specify topologically nontrivial gauge descent data.

\paragraph{Ambient gauge group and effective holonomy.}
A connection \(A\) determines parallel transport along admissible paths and hence a holonomy functor
\[
\operatorname{Hol}_A:
\Pi_1^{\mathrm{thin}}(M)
\longrightarrow
G\text{-}\mathbf{Tor}.
\]
After choosing a frame over a base point \(x\in M\), closed paths based at \(x\) generate the holonomy group
\[
\operatorname{Hol}_x(A)\subseteq G.
\]

The ambient structure group does not by itself determine the computational regime. Even when \(G\) is non-abelian, the effective holonomy may lie in a common abelian subgroup
\[
\operatorname{Hol}_x(A)\subseteq H\subseteq G,
\qquad
[H,H]=\{e\}.
\]
In that case, the transport visible to the chosen family of paths remains effectively abelian. Conversely, genuinely order-sensitive transport requires both
\[
\operatorname{Hol}_x(A)
\ \text{non-abelian}
\]
and a readout capable of distinguishing the relevant ordered compositions. Non-abelianity is therefore a structural possibility, not by itself a complete computational semantics.

This distinction is especially important after gauge reduction. Open parallel transports retain endpoint-frame data and transform covariantly. Closed-loop holonomies transform by conjugation, so gauge-invariant readouts are class functions such as Wilson-loop traces. Passing from framed transport to class functions may identify processes that were distinct before quotienting. The observable computation is therefore determined not only by the connection, but by the pair
\[
\bigl(\text{admissible transport},\text{chosen invariant readout}\bigr).
\]

\paragraph{Finite-action histories with boundary.}
Let \(X\) be a compact oriented Riemannian \(4\)-manifold whose boundary is decomposed as
\[
\partial X
=
\overline{Y}_0\sqcup Y_1,
\]
where \(Y_0\) is the incoming boundary and \(Y_1\) is the outgoing boundary. Let
\[
P\longrightarrow X
\]
be a principal \(G\)-bundle, and fix boundary connections
\[
A_0\in\mathcal{A}(P|_{Y_0}),
\qquad
A_1\in\mathcal{A}(P|_{Y_1}).
\]
Write
\[
\mathcal{G}_0(P)
=
\left\{
u\in\mathcal{G}(P)
\ \middle|\
u|_{\partial X}=\operatorname{id}
\right\}
\]
for the group of gauge transformations fixed at the boundary.

The anti-self-dual boundary-value moduli space is
\[
\mathcal{M}_{\mathrm{ASD}}(X;A_0,A_1)
:=
\frac{
\left\{
\mathbb{A}\in\mathcal{A}(P)
\ \middle|\
F_{\mathbb{A}}^{+}=0,\;
\mathbb{A}|_{Y_0}=A_0,\;
\mathbb{A}|_{Y_1}=A_1
\right\}
}{
\mathcal{G}_0(P)
}.
\]
When nonempty, this space records gauge-equivalence classes of finite-action histories connecting the prescribed boundary data. A single instanton number does not specify such a history completely: different points or components of the moduli space may carry the same topological charge.

For the product cylinder
\[
X=Y\times[0,1],
\]
choose temporal gauge and write the four-dimensional connection as a path \(A(t)\) of connections on \(Y\). Then
\[
F_{\mathbb{A}}
=
F_{A(t)}
+
dt\wedge\dot{A}(t).
\]
With the product metric and a fixed orientation convention, the anti-self-duality equation
\[
F_{\mathbb{A}}^{+}=0
\]
becomes
\[
\dot{A}(t)
=
-*_Y F_{A(t)}.
\]
This is the negative \(L^2\)-gradient-flow equation for the Chern--Simons functional. Thus a four-dimensional instanton history may be viewed as a trajectory in the reduced configuration space of three-dimensional connections.

The relevant computational object is therefore not merely a field configuration at one instant, but a boundary-to-boundary process
\[
[A_0]
\longrightarrow
[A_1]
\]
represented by a class in an appropriate moduli space. Boundary readouts determine which such histories are operationally distinguishable.

\paragraph{Chern--Simons transgression as a boundary ledger.}
Fix an invariant pairing on \(\mathfrak{g}\) normalized so that the corresponding characteristic number is integral on closed four-manifolds. For a connection \(A\) on a closed oriented three-manifold \(Y\), define
\[
\operatorname{CS}(A)
=
\frac{1}{8\pi^2}
\int_Y
\operatorname{tr}
\left(
A\wedge dA
+
\frac{2}{3}A\wedge A\wedge A
\right).
\]
Under a large gauge transformation, \(\operatorname{CS}(A)\) changes by an integer. Hence it is naturally an \(\mathbb{R}/\mathbb{Z}\)-valued functional unless a boundary framing or lift has been fixed.

For a connection \(\mathbb{A}\) on \(X\), define its relative four-dimensional charge by
\[
Q(\mathbb{A})
=
\frac{1}{8\pi^2}
\int_X
\operatorname{tr}
\left(
F_{\mathbb{A}}\wedge F_{\mathbb{A}}
\right).
\]
When the boundary trivializations are fixed, the transgression identity gives
\[
Q(\mathbb{A})
=
\operatorname{CS}(A_1)
-
\operatorname{CS}(A_0),
\]
up to the sign determined by the orientation and trace conventions. Without fixed lifts, the equality is understood modulo
\(\mathbb{Z}\). On a closed four-manifold, the same integral gives the
corresponding characteristic number
\[
Q(\mathbb{A})\in\mathbb{Z}.
\]

This is the precise sense in which Chern--Simons data form a boundary ledger: a four-dimensional curvature history produces a three-dimensional record of the net sector change. The charge
\(Q(\mathbb{A})\) grades the process, but it is not a complete
description of the process. In computational language, it is a stable global output associated with a history rather than a gate table that uniquely determines the intermediate evolution.

If two compatible histories
\[
\mathbb{A}_{01}
\quad\text{on}\quad
X_{01}:Y_0\longrightarrow Y_1
\]
and
\[
\mathbb{A}_{12}
\quad\text{on}\quad
X_{12}:Y_1\longrightarrow Y_2
\]
are glued along their common boundary data; then their characteristic charges add:
\[
Q(\mathbb{A}_{02})
=
Q(\mathbb{A}_{01})
+
Q(\mathbb{A}_{12}).
\]
Equivalently, the Chern--Simons contributions telescope:
\[
\operatorname{CS}(A_2)-\operatorname{CS}(A_0)
=
\bigl(
\operatorname{CS}(A_2)-\operatorname{CS}(A_1)
\bigr)
+
\bigl(
\operatorname{CS}(A_1)-\operatorname{CS}(A_0)
\bigr).
\]
This additivity supplies a genuine compositional law for sector-changing histories.


\paragraph{The core mechanism of regimes.} At the level of \emph{computational role}, effectively abelian regimes correspond to the stable carriage of information: state is propagated in a periodic, order-insensitive, and comparatively reversible manner, so that the dominant behavior is storage, transmission, or recurrence rather than structural transformation. By contrast, non-abelian regimes are characterized by order-sensitive evolution, where the flow itself performs the computation: composition is no longer reducible to simple commutative propagation, and state is actively reshaped through mixing, routing, and nontrivial transition. In the highest regime---\emph{emergent gauge}---computation becomes genuinely history-dependent and topological. There, the relevant logical content is carried not only by the states themselves but by the global manner in which one state is connected to another, so that transition processes---for example, sphaleron-type~\cite{Manton_2019} barrier crossings---become part of the computation itself. The working hypothesis of GCT is not that every Yang--Mills~\cite{YangMills1954} solution is
literally a twin helix but that helical geometry supplies a constructive family of carriers for the same basic mechanisms that recur throughout gauge theory: cyclic transport, path ordering, curvature-supported memory, descent, and topologically graded interpolation.\footnote{As to why the Universe chooses to \textit{express itself} this way? The author kindly asks that such questions be directed \emph{elsewhere} (cf.~\cite{Penrose1989Emperor}).}

\begin{table}[htbp]
    \centering
    \caption{Computational regimes in GCT and the geometric structures that realize them.}
    \label{tab:computational-regimes}
    \renewcommand{\arraystretch}{1.2}
    \footnotesize
    \begin{tabularx}{\textwidth}{|
        >{\raggedright\arraybackslash}p{0.14\textwidth}|
        >{\raggedright\arraybackslash}X|
        >{\raggedright\arraybackslash}X|
        >{\raggedright\arraybackslash}X|
        >{\raggedright\arraybackslash}X|}
    \hline
    \textbf{Regime}
    & \textbf{Computational signature}
    & \textbf{Primary state carrier}
    & \textbf{Topological role}
    & \textbf{Representative physical exemplars} \\
    \hline

    \textbf{Symbolic}
    & Discrete, stable, addressable states; local read/write behavior; robust storage.
    & Local occupation values, boundary conditions, or effectively commuting observables.
    & Topology is present but inactive; it serves as a fixed background structure rather than as the active computational variable.
    & Classical bits, digital memory, effectively abelian electromagnetic storage modes. \\
    \hline

    \textbf{Quasi-fluidic}
    & Cyclic or clocked evolution; recirculating transport; finite memory carried through repeated traversal of a loop.
    & Holonomy, phase accumulation, or other history-bearing data associated with motion around a compact direction.
    & Topological content is accumulated and reassembled over a cycle rather than expressed as a single isolated transition.
    & Calorons, periodic instantons, finite-temperature Yang--Mills configurations with nontrivial holonomy. \\
    \hline

    \textbf{Fluidic}
    & Distributed continuous transformation; order-sensitive mixing; non-commutative transport across an extended medium.
    & Flow profile, relative ordering of non-commuting updates, chiral or braided propagation patterns.
    & Topology is secondary to transport dynamics; the main effect is transformation by extended flow rather than sector change.
    & Non-abelian plane waves, helical or twisted Yang--Mills solutions, propagating chiral field configurations. \\
    \hline

    \textbf{Gauge}
    & Global, history-dependent update between sectors; computation by topological event rather than by local rewriting.
    & Bundle class, winding data, Chern-type invariants, or other global sector labels.
    & Topology is primary and active: the computation is identified with tunneling, obstruction crossing, or sector change.
    & Instantons, sphalerons, vacuum tunneling configurations, topological transition events. \\
    \hline
    \end{tabularx}
\end{table}

\paragraph{Interpretation of the regime labels.} The classifications in~\cref{tab:computational-regimes} are \emph{computational} rather than group-theoretic. They do not assert that all examples in a given column are abelian or non-abelian as gauge fields. Instead, they classify the \emph{role} played by the configuration in the computational picture.

In particular, calorons are genuinely non-abelian Yang--Mills configurations, but they are placed in the quasi-fluidic regime because their operative structure is cyclic and holonomy-driven: state is transported around a compact
\(S^1\) direction, and memory is encoded by recurrence and phase accumulation.
By contrast, instantons appear in the gauge regime when the emphasis is on isolated topological sector change.

\begin{ex}[The Computational Consequences of Non-Abelian Plane Waves]
\label{ex:the-computational-consequences-of-non-abelian-plane-waves}

As a motivating example of one solution cited above, consider the appearance of non-abelian plane waves:
\[ L = -\frac{1}{4} \text{tr}(F_{\mu\nu}F^{\mu\nu}) = \frac{1}{2} \sum_{a=1}^n \left[ (E^a)^2 - (B^a)^2 \right] =: \frac{1}{2} (E^2 - B^2). \]
This is discussed by~\cite{KastrupCanonicalLagrangian}, where he notes (with some exasperation) that plane--wave solutions with equal electric and magnetic energy densities satisfy $E^2=B^2$ and hence $L=0$. In the non-abelian case, such waves carry internal structure: along the propagation direction, the gauge potential undergoes nontrivial parallel transport, yielding non-vanishing holonomy in the gauge group. Equivalently, the field configuration traces a nontrivial path in the group manifold even though the local scalar invariant $E^2\!-\!B^2$ vanishes.

Within a geometric–computational viewpoint, this slice $L=0$ may be regarded as a \emph{switching hypersurface} in configuration space: crossing it changes the group–theoretic state realized by the parallel transport while preserving the light-like character of the wave. The family of such plane waves thus furnishes a controlled alphabet of \emph{signal trajectories}: each solution implements a definite group action (holonomy) along its world-volume, so compositions of waves correspond to compositions of group elements, with path ordering in the non-abelian case. In this sense, classical Yang–Mills solutions encode a concrete logic of state transitions, with the group providing the algebra and the spacetime evolution providing the flow.
\end{ex}

\subsection{Constructive Computational Gauge Theory}\label{sec:constructive-computational-gauge-theory}

We feel that it is a useful exercise to frame gauge theory (and, by extension, quantum gravity) as a problem of \emph{expressiveness} versus \emph{verifiability}. If we allow ``all histories'' (arbitrary geometries, gauge redundancy, unbounded recursion in the construction of spacetimes), amplitudes can become ill-defined and intractable. If we clamp down too hard, we lose physically relevant states and dynamics. 

Our proposal for a constructive computational gauge theory therefore occupies a middle ground. The aim is not to quantize every admissible geometry at the outset, nor to reduce the theory to a purely topological field theory. Instead, we seek a typed, resource-sensitive, total-core, effect-controlled calculus of geometric histories in which admissibility, composition, constraint propagation, and observable readout are made explicit.

At its most basic level, boundary/interface data play the role of types; computational geometric bordisms play the role of terms; and gluing is composition. In a four-dimensional realization, the boundary objects will often be three-dimensional. Still, the construction itself is not tied to a preferred spacetime dimension or to an already-emergent Lorentzian metric. The relevant boundary data may include bundle and connection data, framing, discrete topological labels, allowed mode or defect sectors, constraint data, and a chosen family of readouts. This gives a compositional semantics familiar from TQFTs but designed to scale beyond purely verifiable/topological settings (e.g., Chern-Simons~\cite{ChernSimons1974}). Readers are asked to consider the correspondences in~\cref{tab:field-theory-quantum-gravity-correspondences}.

\begin{table}[ht]
    \centering
    \caption{Constructive semantic structures and their geometric role}
    \label{tab:field-theory-quantum-gravity-correspondences}
    \renewcommand{\arraystretch}{1.2}
    \begin{tabularx}{\textwidth}{l|X}
        \textbf{Semantic Structure} & \textbf{Geometric/Gauge-Theoretic Role} \\
        \hline
        Types &
        Boundary/interface data: geometry, bundle data, constraints, mode or defect
        sectors, and declared readouts \\
        \hline
        Terms &
        Computational geometric bordisms carrying admissible geometric and gauge data \\
        \hline
        Composition &
        Gluing of compatible histories along matched boundary/interface data \\
        \hline
        Linear/resource discipline &
        No implicit duplication or erasure of declared boundary resources; explicit
        accounting of charge, flux, and other conserved data \\
        \hline
        Total core &
        Termination and well-defined evaluation of the restricted semantic core;
        not quantum-amplitude normalization \\
        \hline
        Dependent constraints &
        Gauge, diffeomorphism, regularity, charge-sector, and interface-compatibility
        conditions carried as part of admissibility \\
        \hline
        Descent/coherence &
        Compatibility of local geometric data on overlaps, including higher gluing
        conditions and obstruction data \\
        \hline
        Effects and handlers &
        Controlled coarse-graining, renormalization, noise, approximation, and change
        of scale or presentation \\
        \hline
        Readout &
        Gauge-invariant or otherwise quotient-stable observables extracted from a
        completed history \\
        \hline
        Type extensions &
        Enlargement of the admissible theory by new matter, defect, topology-changing,
        or nonperturbative sectors \\
    \end{tabularx}
\end{table}

\paragraph{Types.}
The proposed helical primitives (\ref{def:twin-helical-manifold-and-unit-helices}) provide one concrete family of generators of histories: smooth geometric carriers equipped with orientation, chirality, framing, phase, and, in higher regimes, connection and holonomy data. This marries the ``continuous versus discrete'' tension: spectra and curvature are continuous objects; quantum numbers arise as stable, counted winding data.

A type is therefore best understood as a \emph{geometric interface specification}. Schematically, one may write
\[
\Sigma = (Y,\sigma),
\]
where $Y$ is the underlying boundary/interface manifold and $\sigma$ denotes the additional structure required by the computational regime under consideration. This structure may include a principal bundle or associated state bundle, restricted connection data, framing, winding or charge labels, admissible mode sectors, defect data, regularity conditions, and a specified family of observable readouts.

Thus a type is more than a boundary condition in the ordinary functional-analytic sense. It declares what geometric structures may consistently enter or leave a region and what information about them is retained by the computational semantics.

Where a boundary supports a finite or controlled internal mode space $\mathcal V_{\Sigma}$, that mode space should be regarded as part of the type data. This becomes important in the non-abelian regime, because gluing may then induce an interaction map of the schematic form
\[
\Gamma_f :
\mathcal V_{\Sigma_1}\otimes \mathcal V_{\Sigma_2}
\longrightarrow
\mathcal V_{\Sigma_3},
\]
so that the geometry of the interface constrains which combinations of modes are actually composable.

\paragraph{Terms.}
A \emph{term} is a computational geometric bordism in the sense of \cref{def:computational-geometric-bordism}, equipped with a declared typing judgment. Thus,
\[
W:\Sigma_{\mathrm{in}}\longrightarrow\Sigma_{\mathrm{out}}
\]
denotes a compact geometric history whose boundary restrictions recover the declared input and output types and whose interior carries the required computational structure.

Terms are therefore not symbolic expressions standing in for geometry. They are geometric histories viewed operationally as processes between typed interfaces. In the symbolic or quasi-fluidic regimes such a history may be a controlled phase-carrying cylinder, junction, or lattice region. In the fluidic and gauge regimes it may carry nontrivial curvature, ordered holonomy, defect data, or changing moduli.

Crucially, a well-typed term need not yet be a dynamically realized history. We distinguish:
\[
W:\Sigma_{\mathrm{in}}\to\Sigma_{\mathrm{out}}
\qquad
\text{(kinematic / well-formed history)},
\]
\[
W\models\mathcal E
\qquad
\text{(history satisfying the chosen dynamical equations)},
\]
and
\[
\operatorname{Read}(W)\in\mathcal O
\qquad
\text{(observable or computational readout)}.
\]
The differential or variational law determines which well-formed histories are dynamically admissible; the computational semantics determines how such histories compose and what information they carry.

\paragraph{Composition.}
\emph{Composition} is gluing. Given
\[
W_1:\Sigma_0\longrightarrow\Sigma_1,
\qquad
W_2:\Sigma_1\longrightarrow\Sigma_2,
\]
with compatible boundary data, their sequential composition is obtained by identifying the common interface:
\[
W_2\circ_{\Sigma_1}W_1:
\Sigma_0\longrightarrow\Sigma_2.
\]
Parallel composition is given by disjoint union.

The gluing condition is stronger than agreement of the underlying topological boundaries. The geometric, gauge, mode, defect, and constraint data exposed on the common interface must also be compatible. Composition therefore expresses the physical statement that the outgoing admissible structure of one region becomes the incoming admissible structure of the next.

At the level of observables, composition must preserve the declared equivalence relation on boundary data. In an abelian regime this may amount to preserving total phase, flux, or winding. In a non-abelian regime the relevant readouts may instead be conjugacy classes of holonomy or class functions such as Wilson-loop traces.

\paragraph{Moduli semantics.}
Gauge theories should not generally be modeled by spaces of representatives alone. Gauge transformations and other automorphisms are part of the semantic structure. Accordingly, for a boundary type $\Sigma$ we write schematically
\[
\mathfrak M(\Sigma)
\]
for the corresponding moduli groupoid or stack of admissible boundary data modulo the appropriate redundancies.

Likewise, a term $W$ determines a moduli object $\mathfrak M(W)$ together with restriction maps
\[
\mathfrak M(\Sigma_{\mathrm{in}})
\xleftarrow{\;s\;}
\mathfrak M(W)
\xrightarrow{\;t\;}
\mathfrak M(\Sigma_{\mathrm{out}}).
\]
Thus a geometric process is represented semantically not merely by a function between state spaces, but by a correspondence between boundary moduli.

Composition of histories is then reflected schematically by a fiber product over the shared boundary data,
\[
\mathfrak M(W_2\circ W_1)
\simeq
\mathfrak M(W_1)
\times_{\mathfrak M(\Sigma_1)}
\mathfrak M(W_2),
\]
with the ordinary fiber product replaced by an appropriate homotopy or derived fiber product when gauge automorphisms or higher coherence data make this necessary. The point is not to place the full syntax inside homotopy type theory, but to allow the denotational semantics to retain the groupoid or stack structure actually required by gauge redundancy. This separation between geometric/prequantum data and subsequent quantization is consonant with the higher-geometric approach of \cite{Schreiber2014LinearHomotopy}.

\paragraph{Descent and coherence.}
Local compatibility must extend beyond pairwise gluing. Let
\[
W=\bigcup_i U_i
\]
be a cover by computational regions. Local geometric data on $U_i$ must agree on overlaps $U_i\cap U_j$, and the corresponding transition data must satisfy the required coherence conditions on triple and higher overlaps. For an ordinary principal bundle, for example, transition functions satisfy
\[
g_{ij}g_{jk}g_{ki}=e
\]
on triple intersections. In higher gauge settings, analogous compatibility may itself hold only up to specified higher morphisms.

We therefore treat descent as a genuine semantic obligation. A collection of locally well-typed computations does not automatically define a global computation; the local data must descend to a coherent global object. Presheaf and sheaf language provide the ordinary version of this requirement, while moduli stacks and higher groupoids retain the automorphism and coherence data lost by a coarse quotient.


\paragraph{A restricted dynamical gluing criterion.}
The preceding descent statement concerns geometric and gauge data. For a history to be a solution of a differential equation, one must additionally match the analytic data on its interface. The following elementary criterion specifies one regime in which the distinction can be made exact.

\begin{prop}[Finite-Order Gluing of Classical Solution Histories]
\label{prop:finite-order-gluing-of-classical-solutions}
Let $W=W_1\cup_Y W_2$ be a smooth manifold obtained by identifying collared boundaries along a common smooth hypersurface $Y$. Fix a smooth field bundle $E\to W$ and a local differential relation $\mathcal E\subseteq J^kE$ of finite order $k\geq1$. Suppose $\phi_i$ are $C^k$ sections on the closed pieces $W_i$, solve $j^k\phi_i\in\mathcal E$ on their interiors, and have matching $k$-jets along $Y$ under the specified collar and bundle identifications. Then the piecewise section $\phi=\phi_1\cup_Y\phi_2$ is $C^k$ on $W$ and satisfies $j^k\phi\in\mathcal E$ everywhere, provided $\mathcal E$ is closed in the relevant jet topology. Conversely, every global $C^k$ solution restricts to such compatible local solutions.
\end{prop}

\begin{proof}
Matching the normal and tangential derivatives through order $k$ in the common collar charts is precisely the $C^k$ pasting criterion. Consequently $\phi$ is a well-defined $C^k$ section with $j^k\phi|_{W_i}=j^k\phi_i$. The differential relation holds away from $Y$ by hypothesis and extends to $Y$ by continuity of the $k$-jet and closedness of $\mathcal E$. Restriction of a global solution gives the converse.
\end{proof}

For gauge fields, the statement is applied after specifying compatible bundle identifications and connection data on the collars; gauge equivalences must be retained in the corresponding moduli groupoid. This result does not assert that arbitrary independently prescribed boundary conditions admit solutions, that the resulting gauge boundary-value problem is well-posed, or that weak solutions or bubbling limits obey the same pasting criterion. Those stronger analytic statements remain separate obligations of the dynamical GCFT sector.

Several distinct guardrails govern the constructive core. They should not be confused with physical consequences that must be derived only at later stages.

\paragraph{Linearity and resource discipline.}
\emph{Linearity} is used here in the resource-sensitive sense: declared boundary data cannot be copied or discarded implicitly \cite{Girard1987LinearLogic,Wadler1990LinearTypes}. A composition must account explicitly for where charge, flux, mode content, or other tracked resources enter and leave.

This discipline is useful for enforcing conservation bookkeeping at interfaces, but it does not by itself derive a physical conservation law, quantum no-cloning, or unitarity. Those require additional physical structure. A conservation law must be included as a constraint or invariant of the model; linearity ensures that the corresponding resource cannot silently disappear from the formal composition.

\paragraph{Totality.}
By \emph{totality} we mean that the restricted \emph{core evaluator} terminates and returns a well-defined semantic value on every well-formed core term. Schematically,
\[
\operatorname{Eval}_{\mathrm{core}}(W)
\]
must terminate and return the declared boundary object, invariant, or computable readout.

This is a property of the formal calculus, not a claim that nature is finite and not a statement that a quantum amplitude is normalized. The total fragment is the portion of the semantics for which one intends to prove compositionality, constraint preservation, and effective evaluation without importing the entire analytic burden of continuum QFT. Processes whose evaluation requires uncontrolled limits, divergent expansions, or non-total search lie outside this core until they are supplied with additional certified structure.

\paragraph{Effective evaluation of controlled holonomy.}
A continuous state space does not by itself prevent effective readout. The relevant question is whether a specified connection and path admit algorithms with certified approximation errors. The following result isolates a non-abelian fragment of the constructive core without assuming that an arbitrary Yang--Mills solution is computable.

\begin{prop}[Effective Wilson-Holonomy Readout]
\label{prop:effective-wilson-holonomy-readout}
Let $G$ be a compact matrix Lie group with a specified finite-dimensional unitary representation $\rho:G\to U(d)$. Let $\gamma:[0,1]\to M$ be a based piecewise-$C^1$ loop, and let $A$ be a connection whose pullback along $\gamma$, expressed in a computably presented trivialization along the loop with identified endpoint frames, gives
\[
K(t)=\rho_*\bigl(A_{\gamma(t)}(\dot\gamma(t))\bigr)\in\mathfrak u(d).
\]
Assume that $K$ is uniformly computable on each smooth segment, with an effective modulus of continuity and a computable uniform bound $\lVert K(t)\rVert\leq M_0$. Then, for every positive rational $\varepsilon$, a terminating algorithm produces a complex rational approximation $w_\varepsilon$ such that
\[
\left|w_\varepsilon-\operatorname{tr}\rho\!\left(\operatorname{Hol}_A(\gamma)\right)\right|<\varepsilon.
\]
The target trace is invariant under gauge transformations acting at the base point by conjugation. If a real-valued function of this trace is computable and its value is supplied with a certified positive margin from a finite readout threshold, the corresponding threshold decision is also computable.
\end{prop}

\begin{proof}
In the chosen trivialization, parallel transport solves the matrix equation $U'(t)=-K(t)U(t)$, $U(0)=I$. The ordered integral (Dyson) expansion is
\[
U(1)=I+\sum_{j=1}^{\infty}(-1)^j
\int_{0\leq t_j\leq\cdots\leq t_1\leq1}
K(t_1)\cdots K(t_j)\,dt_j\cdots dt_1.
\]
The operator norm of the tail beyond order $N$ is bounded by
\[
\sum_{j=N+1}^{\infty}\frac{M_0^j}{j!},
\]
which can be made effectively smaller than any prescribed rational tolerance. Each retained iterated integral is computable to arbitrary precision by effective uniform continuity on the finitely many smooth segments. Since $|\operatorname{tr}T|\leq d\lVert T\rVert$, the resulting approximation to $\operatorname{tr}U(1)$ has a certified error bound. A gauge transformation changes a based-loop holonomy by conjugation, leaving its trace unchanged. Finally, if the selected real readout is at distance at least $\delta>0$ from every decision threshold, an approximation with error below $\delta/2$ determines its readout class.
\end{proof}

The result is deliberately restricted: the required effective descriptions and bounds are part of the input, not consequences of bare smoothness, compactness, or gauge invariance. It proves computability of specified finite-dimensional holonomy readouts, not computability of general moduli, continuum limits, gauge-fixing operations, or nonperturbative Yang--Mills solutions. Nor does it guarantee finite-time threshold decisions when the observable may lie exactly on a readout boundary.

\paragraph{Effects and handlers.}
Processes that change the scale or presentation of a geometric computation are treated as explicit \emph{effects} rather than silently incorporated into the core semantics. Examples include coarse-graining, renormalization, stochastic or noisy coupling, controlled approximation, and changes of resolution \cite{Moggi1991Notions,PlotkinPretnar2009Handlers}.

A handler acts on a well-typed geometric history and produces another well-typed history together with a declaration of what observable information has been preserved, modified, or discarded. Thus a handler is not merely a rewrite of geometry; it is a certified transformation between semantic descriptions.

\begin{ex}[Coarse-Graining as a Handled Effect.]
\label{ex:coarse-graining-as-a-handled-effect}

Let
\[
W:\Sigma_{\mathrm{in}}\to\Sigma_{\mathrm{out}}
\]
contain microscopic helical loops
\[
\gamma_1,\ldots,\gamma_k
\]
in some interior region. A coarse-graining handler replaces this subterm by an effective history $W_{\mathrm{eff}}$ together with a certificate relating the chosen boundary readouts.

In an abelian regime this may require preservation of total phase, winding, or flux. In a non-abelian regime it may require preservation of specified class functions of holonomy, for example Wilson-loop traces. More generally, one may permit a controlled error bound
\[
d_{\mathcal O}
\left(
\operatorname{Read}(W),
\operatorname{Read}(W_{\mathrm{eff}})
\right)
\leq \varepsilon.
\]
The coarse-graining is therefore a semantics-controlled transformation rather than an arbitrary geometric simplification.
\end{ex}

\paragraph{Soundness obligations for handlers.}
To prevent the effect layer from collapsing into informal manipulation, any declared handler---RG flow, coarse-graining, noise model, or controlled approximation---must satisfy an explicit contract:
\begin{enumerate}[leftmargin=*,itemsep=2pt]
\item \textbf{Functoriality.}
Handling must commute with sequential composition up to the declared equivalence.
\item \textbf{Monoidality.}
Handling must respect parallel composition so that independently prepared subsystems are not coupled implicitly.
\item \textbf{Gauge and descent compatibility.}
The transformation must respect the relevant gauge/diffeomorphism equivalences and map coherent local data to coherent effective data.
\item \textbf{Observable control.}
The handler must state which readouts are preserved exactly, which are preserved only up to equivalence, and which acquire controlled error.
\item \textbf{Resource and defect accounting.}
Any creation, removal, or transfer of tracked degrees of freedom must be explicit; lost energy, charge, flux, or topological content must appear as declared residual or defect data rather than disappearing silently.
\item \textbf{Dynamical compatibility.}
If the handler is claimed to send physical histories to effective physical histories, the relation between the microscopic equation $\mathcal E$ and the effective equation $\mathcal E_{\mathrm{eff}}$ must itself be specified.
\end{enumerate}

\paragraph{Dependent types.}
Gauge, diffeomorphism, regularity, charge-sector, and interface conditions may be carried as dependent typing data \cite{MartinLof1984}. A boundary object is then admissible only together with the data or certificate required to satisfy its declared constraint structure.

The intended benefit is mechanical propagation of constraints through composition. But well-typedness is not proof that the physical theory is gauge complete or anomaly-free. The formal task is instead to prove a subject-reduction-style statement: if the input histories satisfy the declared constraints and the gluing rule is admissible, then the composite continues to satisfy them.

\paragraph{Coherence and obstruction data.}
Global consistency conditions should be represented separately from ordinary resource typing. A computation may be locally admissible yet fail globally because transition data do not descend, because a characteristic or obstruction class is nontrivial, or because a proposed quantum measure cannot be made gauge invariant.

The constructive core may therefore track explicit obstruction data and, where available, a chosen trivialization of those obstructions. What it may certify is that the \emph{declared} coherence conditions of the geometric model are satisfied.

This should not be identified with quantum anomaly cancellation. In a genuine quantum field theory, anomalies are obstructions to the consistent quantization or functorial composition of the quantum theory, and additional data may be required to cancel or absorb them \cite{Schreiber2014LinearHomotopy}. The relationship between geometric coherence obstructions and physical quantum anomalies is therefore a theorem to be earned, not a consequence of typing alone.

\paragraph{Prequantum semantics and the quantization boundary.}
The construction above is intentionally prequantum. It specifies geometric histories, their moduli, their composition, and their invariant readouts. It does not yet assign quantum amplitudes to histories.

A later quantization step, if one exists for a chosen sector, should be treated as an additional functorial construction. Schematically, the correspondence
\[
\mathfrak M(\Sigma_{\mathrm{in}})
\xleftarrow{\;s\;}
\mathfrak M(W)
\xrightarrow{\;t\;}
\mathfrak M(\Sigma_{\mathrm{out}})
\]
might be sent to a linear operation of the form
\[
\mathcal Q(W):
\mathcal H_{\Sigma_{\mathrm{in}}}
\longrightarrow
\mathcal H_{\Sigma_{\mathrm{out}}},
\]
or, in a pull--push presentation,
\[
\mathcal Q(W)
\sim
t_!\!\left(K_W\otimes s^*(-)\right),
\]
where $K_W$ contains whatever dynamical weight, action, measure, or generalized cohomological data the chosen quantization requires. This schematic form is included only to mark the conceptual boundary between the geometric process calculus and its possible quantum realization; we do not presently claim such a quantization theorem.

The separation is important. Sheafification or descent may construct globally compatible histories from local data, and evaluation may extract gauge-invariant readouts from those histories. Still, neither operation by itself supplies a quantum measure, interference law, positivity structure, or unitary evolution. These remain part of the later reconstruction problem developed in \cref{sec:from-geometric-semantics-to-quantum-field-theory}.

\paragraph{Type extensions and topology change.}
Not every enlargement of the theory should be represented as an ordinary effect. Some extensions change the class of admissible objects or morphisms themselves. New matter sectors, new classes of stable defects, different boundary categories, and genuine topology-changing processes may therefore require \emph{type extensions} rather than handlers acting inside a fixed calculus.

This distinction is particularly important for topology change \cite{AspinwallGreeneMorrison1993,
AspinwallGreeneMorrison1994}. A coarse-graining handler may rewrite one representative of a fixed semantic process, but a topology-changing history can enlarge the bordism class through which composition is defined. Such an extension must therefore specify new admissible types and morphisms together with the new gluing, coherence, and obstruction conditions they satisfy.

The same principle applies to nonperturbative sectors: they should be added only when the extended theory supplies an explicit semantic role and compatible composition law, rather than being inserted as unconstrained new histories.

\paragraph{Stated goals.} Practically, the workflow is therefore: specify typed boundary data; construct a computational geometric bordism; verify the kinematic and constraint judgments; impose the chosen dynamical law where appropriate; compose compatible regions by gluing; evaluate the total core semantics and declared readouts; and only then apply controlled handlers or type extensions when the physical problem demands them.

Our aim is a language \textit{expressive} enough to describe nontrivial gauge transport, curvature, defects, changing moduli, and eventually field-theoretic dynamics, yet \textit{restricted} enough that well-formedness, core termination, composition, descent, constraint preservation, and observational equivalence can be proved rather than assumed.

Extensions---including matter content, nonperturbative sectors, and topology change---are then added modularly as controlled effects or type extensions, together with new proof obligations as the expressive power of the theory is widened. In this sense, constructive computational gauge theory is intended to provide a semantics in which geometric processes may be calculated, composed, and certified without pretending that the full analytic and quantum theory has already been constructed.

\paragraph{Constructive and compositional precedents.}
Several established frameworks provide complementary precedents for the present programme. Constructive quantum field theory develops rigorous analytic methods for constructing field-theoretic models and recovering quantum structures from Euclidean data \cite{OsterwalderSchrader1973,GlimmJaffe1987QuantumPhysics}. Algebraic quantum field theory instead emphasizes local observable algebras and their compatibility across spacetime regions \cite{HK64,Baez1992Algebraic}, while factorization algebras develop a complementary local-to-global organization of classical and quantum observables \cite{CG17}.

Our proposal differs from these approaches in taking admissible geometric processes, typed boundaries, transport, and composition as the initial semantic data. The objective is not to replace the analytic constructions of quantum field theory with formal syntax, but to establish a computational-geometric calculus in which well-formedness, gluing, and invariant preservation can be verified prior to any subsequent quantum reconstruction.

\paragraph{On QFT and foundational description.}
We do not dispute the empirical success of quantum field theory, nor its central role as a unifying mathematical framework in modern physics. For present purposes, however, we treat QFT not as a primitive ontology, but as a highly developed local-to-global description of continuous physical processes. Its essential structures---local fields and observables, symmetry, gauge transport, composition across regions, and controlled change of description across scales---provide a canonical template for smooth non-commutative analog computation. At the same time, in strongly coupled and nonperturbative regimes, these structures are not generally captured by a single globally transparent continuum construction; perturbation theory, regularization, renormalization, effective field theory, and lattice methods instead provide complementary presentations of the same underlying physical content.

Our proposal therefore begins one layer earlier. Rather than taking quantum fields, functional measures, or probabilistic amplitudes as primitive, we begin from a constructive geometric substrate in which local admissible operations, transport, and gluing are specified directly. Gauge-theoretic and probabilistic descriptions are then expected to arise as stable coarse-grained readouts of this underlying process. QFT is thus retained as an indispensable effective language and structural guide; the claim is only that the computational-geometric architecture it captures may admit a more primitive realization than the continuum quantum formalism itself. If successful, such a formulation could also have a practical methodological benefit in the EVT (\ref{conj:evt-hypothesis}) sense: by encoding admissibility, composition, and invariant preservation directly into the substrate, it may shift part of the burden from brute-force calculation and after-the-fact consistency checks toward verification by construction, making difficult nonperturbative and multiscale problems more \emph{tractable} for both human and machine-assisted analysis.

\paragraph{Compactness modulo gauge.} A fundamental analytic precedent for controlling geometric degeneration is Uhlenbeck compactness. For connections over a compact $n$-dimensional manifold, suitable uniform $L^p$ bounds on curvature, with $p>n/2$, yield weak subsequential compactness modulo gauge \cite{Uhlenbeck1982Connections}. The critical four-dimensional case is more delicate. Under the appropriate Yang--Mills or anti-self-duality hypotheses, sequences with bounded $L^2$ curvature energy may develop concentration at finitely many points. After suitable gauge transformations, convergence can be recovered away from these points, while the lost curvature energy and topological information must be accounted for by bubbling or ideal-connection data \cite{Sedlacek1982YangMills}.

This distinction is essential for constructive computational gauge theory. Gauge fixing does not eliminate every form of geometric degeneration, and a limiting connection alone need not retain the complete invariant content of the original sequence. In our proposed semantics, such limits motivate a handled effect that records both the surviving geometric data and any additional defect information. Preservation of specified boundary observables must then be established as a separate compatibility condition.

Compactness, however, must not be confused with computability. These theorems provide subsequential limits under analytic hypotheses; they do not generally supply algorithms for finding those limits or effective convergence rates. Constructive evaluation additionally requires computable representations, effective estimates, and certified control of the selected observables. We therefore require two distinct forms of discipline: analytic compactness to control geometric degeneration, and effectivity to make the resulting semantics computational rather than merely existential.

\noindent\paragraph{From external lattices to an intrinsic, lattice gauge calculus.} Lattice gauge theory~\cite{Oeckl2005DiscreteGaugeTheory} discretizes a continuum on an external grid: links carry group elements, plaquettes carry curvature, and one hopes to recover physics in a subtle continuum limit. But computability is not baked in; it is delegated to numerics.

By contrast, GCM lattices are of a different kind. The “lattice’’ is the 1–skeleton generated by minimal holonomy loops of the helix geometry itself; nodes are join interfaces (Gauss constraints), edges are Wilson–loop registers, faces carry curvature flux. No continuum limit is imposed from outside: the cell structure is induced by the manifold’s own group–theoretic and topological constraints. Spectral gaps and forbidden transitions then arise from intrinsic geometry rather than from a choice of mesh.\footnote{“It’s nice to have a framework where you can only have one Lagrangian, but it would be even nicer to have some reason that this particular algebra would inevitably appear as an answer to some question.’’ ---E. Witten~\cite{IntlCentreForTheoreticalPhysics2014Conversation}}

\begin{itemize}
\item \textbf{State space.}
A presheaf or stack of local moduli~\cite{MacLaneMoerdijk1992Sheaves} over a helix-adapted cover; descent or sheafification enforces compatibility and produces globally admissible states.

\item \textbf{Evaluation.}
The typed core evaluates declared gauge-invariant observables or computational readouts on globally compatible states. This evaluation is required to be total on the core fragment, but it is not identified with quantum-amplitude normalization.

\item \textbf{Scale change.}
Refinement and coarse-graining are explicitly handled transformations of the cover and moduli data, accompanied by certificates specifying the readouts or effective structures they preserve.
\end{itemize}

\begin{ex}[Sheafification/Evaluation on an Intrinsic Helical Chain]
\label{ex:sheafification-evaluation-on-an-intrinsic-helical-chain}

Let $X=U_1\cup U_2\cup U_3$ be a helical working region whose open sets are centered on three consecutive intrinsic holonomy loops $\gamma_1,\gamma_2,\gamma_3$, with overlaps $U_{12}=U_1\cap U_2$ and $U_{23}=U_2\cap U_3$, and no triple overlap. Assume these opens as coming from the geometry of the GCM itself, rather than from an externally imposed cubic grid.

Let $\mathcal{M}(U)$ be the presheaf assigning to each open set $U\subseteq X$ the groupoid of $U(1)$-connections on the trivial line bundle over $U$, modulo gauge, together with the induced Wilson-loop values on the intrinsic loops contained in $U$.  A local section over $U_i$ is represented by a connection $A_i$ and its holonomy
\[
u_i:=\Hol_{\gamma_i}(A_i)\in U(1).
\]

The local data glue precisely when the restrictions align to gauge overlaps:
\[
A_2|_{U_{12}} = A_1|_{U_{12}} + g_{12}^{-1}dg_{12},
\qquad
A_3|_{U_{23}} = A_2|_{U_{23}} + g_{23}^{-1}dg_{23}.
\]
After sheafification, the admissible global states are therefore not arbitrary triples $(u_1,u_2,u_3)$, but compatible glued sections
\[
A\in \Gamma(X,\mathcal{M}^{\#}).
\]

Now choose the outgoing loop $\gamma_{\mathrm{out}}$ obtained by concatenating the three intrinsic channels.  In the abelian corridor, the functoriality of holonomy gives
\[
\Hol_{\gamma_{\mathrm{out}}}(A)=u_3u_2u_1.
\]
Thus, the intrinsic lattice calculus does not ``sum over histories'' on an external mesh; it first constructs the global section by gluing local moduli, and only then evaluates the observable.

The constructive evaluator is the gauge-invariant map
\[
\mathrm{ev}_X:\Gamma(X,\mathcal{M}^{\#})\longrightarrow U(1),
\qquad
\mathrm{ev}_X(A):=\chi\!\left(\Hol_{\gamma_{\mathrm{out}}}(A)\right),
\]
where in the simplest abelian case one may take $\chi(z)=z$, or more generally a thresholded readout to a finite alphabet.

This also exhibits the effect-handler viewpoint.  A coarse-graining handler may replace the pair $(U_1,U_2)$ by a single effective region $U_{12}$ with effective holonomy
\[
u_{12}:=u_2u_1.
\]
The rewritten term has local data $(u_{12},u_3)$, but the boundary observable is unchanged:
\[
\chi(u_3u_{12})=\chi(u_3u_2u_1).
\]
Hence, coarse-graining is not an informal simplification; it is a semantics-preserving rewrite certified by preservation of the boundary class function.

In the non-abelian lift, the same example survives with ordered transport:
\[
U_{\mathrm{out}}=U_3U_2U_1,
\qquad
\mathrm{ev}_X(A)=\tr_{\rho}(U_3U_2U_1),
\]
so the only change is that composition becomes path-ordered, and the evaluator passes from a scalar phase to a class function on holonomy.
\end{ex}

\begin{rem}[Why Not Classical $\mathbb R$?]
\label{rem:limitations-of-exact-real-computation}

In the fluidic/gauge band, one is tempted to model directly over the classical continuum. However, unrestricted real computation (cf.~\ref{app:comparison-with-real-computation}) admits hyper–Turing power and brittle pathologies (exact tests, branch–on–sign over nonrobust inputs, oracle–like primitives). This undermines the two pillars we require: \emph{executability} (every step is effectful but total) and \emph{verifiability} (small perturbations do not change the decision class). Our constructive stance preserves smooth geometry while preventing hidden oracles.
\end{rem}

\noindent\paragraph{Geometric reals?}
We separate \emph{semantics} (smooth reasoning) from \emph{execution} (effective approximation):

\begin{enumerate}[leftmargin=*,itemsep=3pt]
\item \textbf{Computable reals} \(\mathbb{R}_{\mathrm{comp}}\). Externally, numerical data (couplings, boundary values, readouts) live as computable reals given by fast Cauchy names (or, when possible, algebraic numbers with exact arithmetic). Equality is replaced by interval refinement; separation proofs guard branching. This is the target of all executable algorithms~\cite{Weihrauch2000ComputableAnalysis}.
\item \textbf{Synthetic/smooth reals.} \(\mathbb{R}_{\mathrm{syn}}\) internally, differential reasoning uses the smooth reals of a well–adapted SDG model (Kock–Lawvere axiom~\cite{Kock2006SDG}, nilpotent \(d\varepsilon=0\)), so vector calculus and curvature remain first–class without appeal to pointwise limits.\footnote{Any well–adapted model suffices; no commitment to a specific topos is needed, but an implementation is provided in~\cref{sec:classification-of-geometric-computational-field-theories} for convenience.}
\item \textbf{A bridge.} A semantic–to–execution functor sends internal smooth expressions (curvatures, class functions on holonomy) to external names in \(\mathbb{R}_{\mathrm{comp}}\), with \emph{proof obligations} discharged by our effect handlers (for limits/coarse–graining).
\end{enumerate}

The aim is not to deny the continuum but to choose a base field that respects computability. Parameters are measured to finite precision; algorithms require effective representations. Thus we use rationals/algebraics/\(\mathbb{R}_{\mathrm{comp}}\) for execution, while \(\mathbb{R}_{\mathrm{syn}}\) supplies calculus internally. This pragmatic split aligns with the observation (cf.~\cite{Szudzik_2012}) that physical models do not require arbitrary reals to be \emph{expressed} or \emph{simulated}, even if smooth structure is invaluable for \emph{reasoning}.

\paragraph{Design axioms: equality, limits, and effects.}
We enforce three constraints in the core calculus:
\begin{itemize}[leftmargin=*,itemsep=2pt]
\item \textbf{No exact branching on undecidable predicates.} Tests are certified by separation bounds (or deferred as obligations).
\item \textbf{Limits are effects.} Completion, closure, and renormalization are algebraic effects with handlers that carry witnesses of convergence/stability.
\item \textbf{Gauge invariants only.} Readouts are class functions of holonomy (e.g.\ traces), ensuring stability under small perturbations and gauge changes.
\end{itemize}

\begin{rem}[Disclaimers on Classical QFT]
\label{rem:disclaimers-on-classical-qft}
 Our setting concerns analog geometrodynamics on a noncompact configuration space. Gaps (i.e., mixing rates, spectral edges of transfer generators) are typically defined by nested \(\sup/\inf\) over unbounded domains. In general, these constants need not be uniformly computable from the defining data. We therefore (a) state results in terms of \emph{semi–decidable} or \emph{effectively approximable} quantities, or (b) confine genuinely noncomputable constants to effectful interfaces outside the core (they cannot influence totality/normalization proofs). As a result, this is a computability warning label for our own continuum models; we cannot use our model to solve the Clay Yang–Mills problem~\cite {JaffeWitten2000YangMillsProblem}, which requires a fully rigorous quantum theory.
\end{rem}

\paragraph{Analytic obligations and intended advantage.}
The constructive formulation does not remove the analytic difficulties of Yang--Mills theory. A viable realization must still establish the relevant PDE results---including existence and regularity of admissible solutions, gauge fixing, a priori energy and curvature estimates, boundary compatibility, compactness modulo gauge, and control of bubbling or defect formation---together with stability and convergence under refinement.  The proposed advantage is instead organizational.  Rather than attacking the continuum theory as a single global configuration-space problem, GCFT decomposes it into typed local histories with explicit boundary data, gauge equivalences, and gluing laws, so that one may seek estimates and compactness statements that are stable under composition and propagate from local regions to larger histories.  In this sense, the analytic task substantially differs from the standard Yang--Mills mass-gap problem: the Clay problem asks directly for the construction of a nontrivial four-dimensional quantum Yang--Mills theory with a positive mass gap, whereas the computational program inserts an intermediate regulated and compositional layer and asks whether the required continuum and spectral estimates can be assembled from uniformly controlled local data.  If this decomposition fails to simplify or localize the necessary estimates, then the constructive apparatus supplies only a reorganization of the problem rather than a shortcut to its solution.

\paragraph{On finite information and intuitionist physics.}
We concur with the operational point that experimental data carry finite information, and that uncountable precision should not be a primitive of algorithms, as brought up by~\cite{Gisin_2019}. Where we differ is methodological: we retain a smooth internal semantics (\(\mathbb{R}_{\mathrm{syn}}\)) to keep geometry local and functorial, while ensuring that \emph{every} externally visible quantity is realized in \(\mathbb{R}_{\mathrm{comp}}\) with certified error control. Thus, the continuum remains a modeling tool; computation proceeds on effectively presentable numbers.


Choosing a constructive base allows us to retain the familiar analytic language where needed while ensuring that each step of the formalism remains executable. The framework developed here treats physical dynamics as smooth, often chiral/transcendental flow, while treating topology and quantization as mechanisms of coarse-graining that project this flow onto the discrete structures accessible to observation. In this sense, the continuum substrate and the discrete emergent layer are not competitors but complementary levels of description. What we call ``physical computation'' is the geometric process that occurs at their interface, with its behavior constrained by the EVT Trilemma. Nothing in the present approach depends on sub-Planckian assumptions, nor do we take uncountable structure as primitive in the dynamics. We hope that the fully constructive classification outlined in (\ref{sec:classification-of-geometric-computational-field-theories}) provides a more precise setting in which longstanding questions, including those surrounding Yang--Mills, can be reformulated and studied. A high-level road map toward the Yang--Mills mass gap in computational gauge theory is outlined below:
\begin{enumerate}
    \item \textbf{Define computational semantics for gauge theory.}
Define a typed, resource-sensitive, total-core, effect-controlled calculus whose types encode boundary/interface data (3-geometries with gauge labels) and whose terms encode computational geometric bordisms (4-dimensional histories composed by gluing).

    \item \textbf{Build computable geometric bundles.}
Give the calculus a denotational semantics in terms of computably presented bundles, connections, parallel transport, and their gauge moduli, formulated in a constructive/synthetic differential setting where appropriate. This preserves the differential structure while requiring effective evaluation on the designated constructive core.

    \item \textbf{Assess the scaling limit to continuum Yang--Mills.}
Prove that families of regulated computational models approximate continuum Yang--Mills in a controlled scaling limit. Concretely, a ``differential computational jet calculus'' would provide local error bounds under refinement, while the corresponding gauge-invariant observables---including Wilson loops and, after the appropriate quantum reconstruction, correlation functions---must converge to those of continuum Yang--Mills (see~\cref{app:higher-dimensional-algebra}).

    \item \textbf{Prove the gap and its stability.}
Establish a spectral gap (a uniform lower bound on excitation energies) for the constructive model using the required energy estimates, compactness modulo gauge, and spectral analysis; then show that this lower bound survives the scaling limit, yielding a continuum mass gap. Here, the constructive setting aims to organize the global problem through local-to-global invariants, admissible gluing, and certified control of the regulated models, without replacing the analytic compactness and continuum estimates required by Yang--Mills theory.
\end{enumerate}

\paragraph{Geometric interpretation of the Yang--Mills mass gap.}
The Yang--Mills mass-gap problem poses a particularly sharp test. For a four-dimensional quantum Yang--Mills theory with compact simple gauge group, the question is whether the physical Hamiltonian possesses a strictly positive spectral gap above its vacuum:
\[
\Delta_{\mathrm{YM}}
:=
\inf\bigl(
\operatorname{spec}(H_{\mathrm{phys}})
\setminus\{E_0\}
\bigr)-E_0
>0.
\]
The gap concerns the energy spectrum of gauge-invariant physical excitations, not a discrete separation between geometric configurations, topological sectors, or computational readout values. Classical connections may vary continuously without implying the existence of arbitrarily low-energy physical particles.

From the perspective of GCFT, the problem is therefore to explain how continuous, non-Abelian geometric dynamics can produce a positive minimum excitation energy after the relevant gauge constraints and physical reconstruction have been imposed. The underlying connection admits continuously varying curvature and holonomy, but the spectrum of admissible physical excitations need not inherit this continuity down to zero energy. Under appropriate spectral assumptions, a positive gap is reflected in exponential decay of connected Euclidean correlation functions of suitable gauge-invariant observables, with a finite correlation length set by the lightest physical excitation.

Such behavior cannot be inferred from noncommuting holonomy or geometric closure alone. A genuine derivation must recover the relevant nonperturbative dynamics, the generation of a physical mass scale, and a positive-energy spectrum in the continuum and infinite-volume limits. This motivates a program in which the spectrum and correlations of physical excitations are reconstructed from admissible geometric histories, rather than postulated as primitive properties of an abstract field operator.