\section{Abelian Gauge Dynamics and Phase Geometry}\label{sec:abelian-gauge-dynamics-and-phase-geometry} Recent work on the computational richness of fluid dynamics provides a useful jumping-off point. Tao's universality program for the incompressible Euler equations shows that apparently compact local differential laws may support a remarkably large class of embedded dynamical behavior~\cite{Tao2018EulerUniversality}. Cardona et al. sharpened this connection by constructing a Turing-complete stationary Euler flow on a Riemannian \(S^3\) \cite{Cardona_2021}. These results make an important distinction visible: the formal simplicity of a differential equation does not imply simplicity of its global solution space, nor does the existence of a solution theory by itself specify which features of those solutions should be regarded as computational states, transitions, or outputs. The present section develops the first continuous regime of GCT with this distinction in mind. We introduce the \emph{quasi-fluidic band}, in which a discrete geometric state is enlarged by compact abelian phase variables and quantized winding data. The resulting substrate is continuous, but its computational semantics remains organized by local interfaces, admissible transport, invariant structure, and specified readouts. In the simplest realization, the active continuous degree of freedom is a \(U(1)\)-valued phase, so that transport remains abelian and may be summarized by phase differences, winding ledgers, and curvature fluxes. The role of differential equations in this regime is therefore important but deliberately separated from the definition of computation. A differential law specifies which histories of the substrate are dynamically admissible. The GCM structure specifies which features of those histories carry information, which local histories may be composed, which quantities survive deformation, and which observables constitute a readout. Schematically, \[ \text{differential law} \longrightarrow \text{admissible histories} \longrightarrow \text{compositional geometric structure} \longrightarrow \text{readout}. \] The computation is identified with the latter structural organization of admissible histories, rather than with the task of reproducing the entire solution field. This separation will become increasingly important later. In the abelian regime, the relevant structural data can often be compressed into phases and integer ledgers. In the non-abelian regime of \cref{sec:non-abelian-geometric-computation}, the same principle is promoted to ordered transport, curvature, and holonomy. In \cref{sec:geometric-computational-field-theory}, histories themselves become inputs to a bordism- and field-theoretic semantics. As a motivating example, consider the classic experiment in which a hose filled with running water is placed across a vibrating speaker. The resulting jet forms a visible oscillating pattern: the driving frequency controls coil tightness, amplitude controls the spiral radius, and phase controls the initial offset. This is an instructive caricature of the quasi-fluidic regime: periodic forcing induces an effective $U(1)$-phase along a geometric substrate, and the observable shape provides a coarse readout of that phase. In the language adopted later (\ref{def:quasi-fluidic-regime}), the phase offset belongs to the $T$-layer ($S^1$ data), while winding/twist counts correspond to quantized $Z$-invariants; discrete readouts arise from coarse partitions of these quantities. Helical and twisted structures appear here not as an aesthetic choice, but as energetically natural carriers of phase and twist under periodic driving. A visual demonstration by~\cite{brusspup2013water} makes this phenomenon concrete.\footnote{Regarding an unconventional reference; the point is purely pedagogical: it is a compact way to visualize phase, winding, and coarse readout in a physical system.} \paragraph{Computational geometric bordisms.}\label{sec:computational-geometric-bordisms} The vibrating-hose experiment may be viewed as an analog of a \emph{computational geometric bordism}: a smooth evolution mediating between boundary conditions (fixed endpoints, imposed phase/frequency, initial and final configurations). Here, the bordism is ``internal'' to the geometry of the evolving flow, and the computable content is not the PDE solution itself but the stable, gauge-invariant features one can read from it---see~\cref {app:topological-kleene-field-theories-and-computational-geometric-bordisms} for cobordism formalisms. \begin{prop}\label{prop:geobit-encoding} A GCM has information-bearing degrees of freedom, but the \emph{effective} information content is relative to a chosen readout (coarse-graining). In particular, geometric computation is deterministic at the microstate level, while information is defined relative to an encoding. \end{prop} Unlike in classical computation---where the bit is treated as an absolute, context-free primitive---in geometric computation, the notion of a ``geobit'' is always specified with respect to a readout map (or partition) on the underlying state space. Concretely, one fixes an observable or coarse-graining \[ \rho:\Sigma \longrightarrow B, \] where $\Sigma$ is the (possibly structured) microstate space of the GCM and $B$ is a finite label set used for symbolic computation. Once $\rho$ is fixed, information becomes an objective property of the pair $(\Sigma,\rho)$: the dynamics may be fully deterministic on $\Sigma$, yet the induced evolution on $B$ depends on which features are chosen as readable and which are treated as hidden or irrelevant. This is directly analogous to physics, where measurements and reference choices enter through specified observables rather than through subjectivity. The boundary between symbolic and quasi-fluidic computation is therefore not a matter of taste. Still, of modeling choice: symbolic regimes arise when the operative degrees of freedom are purely discrete (a finite $B$ with static invariants), while the quasi-fluidic regime appears when one enlarges the local fiber to include compact abelian phases and quantized topological counters (the $B/T/Z$ structure of~\cref{def:quasi-fluidic-regime}). Geometric deformation then permits interpolation between discrete and continuous logic \emph{without} introducing probabilistic amplitudes. The quasi-fluidic regime should therefore not be understood as an attempt to remove differential dynamics from continuous computation. Rather, it separates the dynamics of the carrier from the semantics of the computation carried by it. Earlier symbolic constructions used externally prescribed pulses and local update schedules acting on finite geometric labels. The quasi-fluidic mode promotes part of this control structure into the geometry itself: compact phases evolve continuously, integer quantities record isolated winding events, and finite readouts extract stable information from the resulting histories. Differential equations may govern these histories, but the computational content is specified by the invariant structures and interface relations selected from them. \subsection{A New Kind of Computation}\label{sec:a-new-kind-of-computation} The distinction between a differential law and the computational semantics carried by its solutions can be stated geometrically. Let \[ \pi:E\longrightarrow \Sigma \] be a smooth field bundle. A regular finite-order differential equation may be represented by a differential relation \[ \mathcal{E}\hookrightarrow J^kE \] inside the \(k\)-jet bundle of \(E\). A section \[ \phi:\Sigma\longrightarrow E \] is a solution when its jet prolongation satisfies \[ j^k\phi(\Sigma)\subseteq\mathcal{E}. \] Higher prolongations \[ \mathcal{E}^{(r)} \subseteq J^{k+r}E \] encode the differential consequences of the original equation. Formal integrability asks, roughly, whether this tower of prolongations closes compatibly rather than producing new hidden differential constraints at successive orders; standard criteria for nonlinear systems were developed by~\cite{Goldschmidt1967}. This geometric viewpoint already moves substantially beyond identifying a PDE with a coordinate formula. More recently, Khavkine and Schreiber have reconstructed the formal theory of differential equations inside synthetic differential geometry, where the jet construction acquires a comonadic formulation and formally integrable PDEs may be treated intrinsically in a setting that also accommodates higher and stacky geometry~\cite{KhavkineSchreiber2017}. Their construction is motivated in part by the eventual treatment of locally variational field theories, including higher gauge fields. The lesson relevant here is not that PDEs disappear, but that differential equations themselves may be treated as geometric objects rather than as privileged coordinate syntax. GCT makes a further, orthogonal separation. The differential relation \(\mathcal{E}\) specifies the dynamically admissible histories, but it does not by itself specify their computational interpretation. For boundary data \(B_0,B_1\), write schematically \[ \operatorname{Sol}_{\mathrm{adm}} \bigl( \mathcal{E};B_0,B_1 \bigr) \] for the space of admissible solution histories connecting them. A GCM then equips such histories with additional structure: distinguished interfaces, allowed local transformations, equivalence relations, protected quantities, and readouts. A computational observable is a map of the form \[ R: \operatorname{Sol}_{\mathrm{adm}} \bigl( \mathcal{E};B_0,B_1 \bigr) \longrightarrow \mathcal{O}, \] where \(\mathcal{O}\) may be finite, phase-valued, integer-valued, or in later regimes, a gauge-invariant moduli or holonomy space. Thus, two dynamically distinct solutions need not represent distinct computations: \[ \phi_1\sim_R\phi_2 \qquad\Longleftrightarrow\qquad R(\phi_1)=R(\phi_2). \] Conversely, a computationally meaningful distinction need not be a pointwise property of the field. It may be carried by a winding number, a boundary transition, a defect event, an accumulated phase, or a global transport invariant. In this sense, the relevant computational object is not the full solution space alone, but its organization by admissible composition and observable quotient: \[ \operatorname{Sol}_{\mathrm{adm}}(\mathcal{E}) \longrightarrow \operatorname{Sol}_{\mathrm{adm}}(\mathcal{E})/{\sim_R} \longrightarrow \mathcal{O}. \] We refer to this separation as \emph{computation by structure}. The differential law answers \[ \text{\emph{Which histories are dynamically permitted?}} \] while the GCM semantics answers \[ \text{\emph{Which permitted histories compose as computations, and what information do they carry?}} \] Neither question replaces the other. This distinction is particularly important for continuous media. The results of Tao and of Cardona et al. show that Euler dynamics may support extremely rich embedded computation. Such results demonstrate that a compact differential law can generate a solution space with nontrivial computational content. They do not, however, imply that every variable of the fluid is itself a computational register, or that computation is best identified with reconstructing the complete Euler solution. A computational interpretation still requires a choice of stable states, admissible controls, interfaces, and readouts. \paragraph{The paradox of analog logic flow.} Historically, analog and fluidic devices exploited continuous physics to implement nontrivial transformations. Their central difficulty is not simply that a governing equation may possess singular solutions. Continuous computation also raises questions of well-posedness, stability, conditioning, sensitivity to initial data, and the robustness of the chosen readout. A physical trajectory may remain perfectly well defined while becoming computationally ambiguous because nearby states cease to be distinguishable under the intended encoding. If \[ \rho:\Sigma_{\mathrm{state}}\longrightarrow B \] is a finite readout, then the relevant stability question is not only whether nearby trajectories remain nearby in the full state space, but whether perturbations preserve their observable class. A small geometric displacement may cross a readout boundary and induce a large change in the symbolic output. Analog fragility is therefore often a failure of the map from continuous dynamics to stable computational semantics rather than a literal failure of the underlying physics. GCT responds by building the computational layer from invariant and compositional structure. A GCM update is a local admissible transformation at an interface, driven by specified boundary perturbations, pulses, or couplings. The ambient evolution may be described by ODEs or PDEs, but the computational semantics is carried by what may change, what must remain invariant, how local histories may be composed, and which quantities are retained by the readout. This is the sense in which the theory shifts emphasis from \emph{solving the whole equation} to \emph{controlling the structure of its admissible histories}. The analytic obligations nevertheless remain. A proposed realization must still possess whatever existence, uniqueness, regularity, stability, continuation, and boundary-compatibility properties are required for its update laws to be well defined. Recasting computation geometrically does not remove these problems; it prevents them from being conflated with the separate question of what the resulting dynamics computes. This structural viewpoint is also compatible with a broader movement in applied mathematics toward local interaction laws and networked descriptions of complex systems. Goriely and collaborators, for example, have modeled biological propagation through coupled local interaction mechanisms rather than treating the global phenomenon as a single undifferentiated state variable~\cite{Thompson2020Protein}. The relevance to GCT is methodological: complicated global behavior can be organized by local coupling structure even when continuous differential dynamics remain present underneath. The purpose is not to oppose PDE modeling, but to expose the compositional structure that a global equation alone does not identify. \paragraph{Boundaries in Quasi-Fluidic GCMs.} The passage from local differential dynamics to compositional computation requires boundary structure. A local differential law does not determine how independently specified regions may be assembled: one must also specify which boundary data are admissible, which traces of the fields are retained at an interface, and when two local histories are compatible under gluing. Let \(M_1\) and \(M_2\) be neighboring regions meeting along an interface \(Y\). For a differential relation \(\mathcal{E}\), denote by \[ \mathcal{B}_{\mathcal{E}}(Y) \] the relevant space of admissible boundary data on \(Y\). Depending on the equation and variational problem, these data may consist of field values, normal derivatives, finite jets, connection data, conserved fluxes, or some constrained subset thereof. Two local histories \[ \phi_1\in\operatorname{Sol}_{\mathrm{adm}}(\mathcal{E};M_1), \qquad \phi_2\in\operatorname{Sol}_{\mathrm{adm}}(\mathcal{E};M_2) \] may be glued only when their induced boundary data agree under the chosen interface identification. Schematically, compatible histories therefore form a fiber product \[ \operatorname{Sol}_{\mathrm{adm}}(\mathcal{E};M_1) \times_{\mathcal{B}_{\mathcal{E}}(Y)} \operatorname{Sol}_{\mathrm{adm}}(\mathcal{E};M_2). \] When the relevant analytic gluing problem is well-posed, such compatible data determine a history on \[ M_1\cup_Y M_2. \] This is the differential-geometric counterpart of the GCM interface law: composition is permitted only when the structures exposed at the common boundary are compatible. In finite-order problems, these compatibility conditions may be expressed through traces of finite jets. This parallels the jet-matching conditions already encountered in \cref{sec:twin-helical-manifolds} for smooth helical splices, but now the jets belong to fields or state sections rather than only to embedded centerlines. Formal prolongation provides the natural language for checking whether such local differential constraints remain compatible at higher order. The quasi-fluidic regime isolates a particularly controlled instance of this general picture. The boundary data retained by a cell consist of a finite symbolic sector, compact abelian phase variables, and quantized topological counters. Local evolution remains deterministic and nonsingular, while interfaces couple these data through equivariant maps. We now make this phase boundary precise. \begin{defn}[Quasi–Fluidic Computation and Regime Phases]\label{def:quasi-fluidic-regime} Given a GCM, we say it operates in the \textit{quasi–fluidic regime} if each region $R_i$ carries a finite–dimensional state fiber: \[ \mathcal{M} \;\cong\; B_i \times T_i \times Z_i, \] as in \cref{def:geometrically-computable-manifold} with the following structure: \begin{enumerate}[label=\textup{(QF\arabic*)}, leftmargin=*] \item \textbf{Discrete core (symbolic).} $B_i$ is finite, typically $B_i\cong (C_2)^{k_i}$; in the twin helix unit one has $k_i=4$ for chirality/orientation bits. \item \textbf{Phase layer.} $T_i$ is a continuous but compact abelian Lie group (a torus), $T_i\cong (S^1)^{m_i}\simeq U(1)^{m_i}$, encoding axial phase(s), orientation angles, or other bounded ``analog'' degrees of freedom. \item \textbf{Quantized invariants.} $Z_i$ is a finitely generated abelian group (e.g.\ $\mathbb Z^{p_i}$), recording twist/winding/linking numbers or other topological charges. \item \textbf{Abelian symmetry and equivariance.} The local symmetry acting on $\Sigma_i$ is \[ K_i \;\cong\; (C_2)^{k_i} \times (S^1)^{m_i}, \] and every interface map $I_{ij}\in\mathcal I$ and the induced flow on $\Sigma_i$ are $K$–equivariant, that is, they depend on phase differences and commute with flips/rotations. \item \textbf{Flow coupling.} The ambient flow $\{\mathsf{Ev}_t\}$ transports the $T$–coordinates continuously (phase evolution) and can modify $Z$ only by isolated, energy–threshold events (phase slips) prescribed by the interface rules; $B$ updates by the same local rules. No non-abelian monodromy is present. \item \textbf{Non–singularity and bounded energy.} Trajectories have no finite–time blow-ups; curvature/torsion remain bounded; state updates at interfaces are well defined for all $t$. \end{enumerate} A system is symbolic when $m_i=0$ and $Z_i$ is static; it is purely fluidic when $B_i$ is trivial and non-abelian holonomy appears. The quasi–fluidic band sits between these: it combines a finite discrete core with $U(1)$–type phases and integer topological counters, all governed by abelian symmetry. \end{defn} The \(B/T/Z\) decomposition should therefore be read not merely as a taxonomy of state variables, but as a choice of boundary data adapted to composition. The \(B\)-sector records finite locally readable labels, the \(T\)-sector records compact phase data that may be transported continuously across interfaces, and the \(Z\)-sector records discrete invariants that remain constant on smooth strata and change only at prescribed threshold events. The purpose of this decomposition is to retain enough information for neighboring histories to be glued without requiring the complete microscopic solution on either side of the interface. \paragraph{Composition of GCMs.} In the symbolic regime, interfaces exchange finite readouts and induce local discrete updates. In the quasi-fluidic regime, the same compositional principle persists, but the interface data are enlarged to include compact phases and quantized invariants. Composition is therefore controlled by compatibility of the relevant $B/T/Z$ boundary data rather than by a purely discrete transition table. Reconfiguration corresponds to changing the admissible interface maps, coupling parameters, or boundary conditions, while preserving the local gluing rules of the GCM. \subsection{Abelian Lattice Gauge Dynamics in the Quasi--Fluidic Mode}\label{sec:abelian-lattice-gauge-dynamics} In the QF band, the appropriate ``computational'' physics we should work with is not a discrete logic model as in the symbolic band but an \emph{abelian lattice gauge dynamics}: compact phases transported and coupled locally, with evolution governed by a bounded local energy and (optionally) a driven-damped relaxation law. This section is \emph{interpretive}; it provides a stable dynamical model for analyzing QF behavior, without elevating differential equations to the definition of geometric computability. \paragraph{Lattice variables (QF state).}Let $\Gamma=(V,E)$ be the interaction graph of a GL (cells/modules are vertices, interfaces are edges). For each cell $i\in V$, we take an internal phase (orientation layer) \[ \theta_i \in U(1)\cong \mathbb{R}/2\pi\mathbb{Z}, \] and for each interface $(i,j)\in E$ we take a \emph{link phase} (a discrete connection 1--form) \[ A_{ij}\in \mathbb{R}/2\pi\mathbb{Z},\qquad A_{ji}=-A_{ij}, \] equivalently $U_{ij}:=\exp(iA_{ij})\in U(1)$. (If one wishes to remain in the finite cyclic subband, replace $U(1)$ by $C_N$ and interpret all equations below as a continuous approximation to a stroboscopically driven $C_N$ update.) \paragraph{Gauge redundancy.}A $U(1)$ gauge transformation is a choice of angles $\alpha_i\in\mathbb{R}/2\pi\mathbb{Z}$ at each cell, \[ \theta_i \mapsto \theta_i+\alpha_i, \qquad A_{ij} \mapsto A_{ij}+\alpha_i-\alpha_j, \] under which the \emph{gauge--invariant phase difference} \[ \Delta_{ij} \;:=\; \theta_i-\theta_j-A_{ij}\quad (\mathrm{mod}\;2\pi) \] is unchanged. In this sense, ``independence of local modules'' is a choice of gauge/frame, while couplings are encoded invariantly by $\Delta_{ij}$. \paragraph{Local energy (interface functional).}A minimal gauge--invariant energy on $\Gamma$ is \begin{equation}\label{eq:qf-interface-energy} \mathcal{E}(\theta,A) \;=\; \sum_{(i,j)\in E} J_{ij}\,\bigl(1-\cos(\Delta_{ij})\bigr) \;+\; \sum_{p\in \mathsf{P}} \kappa_p\,\bigl(1-\cos((dA)_p)\bigr), \end{equation} where $J_{ij}\ge 0$ are interface couplings and $\mathsf{P}$ is a chosen set of plaquettes (2--cells) with discrete curvature \[ (dA)_p \;:=\; \sum_{(i,j)\in \partial p} s_{p,ij}\,A_{ij}\quad (\mathrm{mod}\;2\pi), \qquad s_{p,ij}\in\{\pm1\} \] the oriented sum around $\partial p$. The first term penalizes a mismatch between neighboring phases modulo the link variable; the second penalizes nonzero curvature (``flux'')~\cite{Kogut1979LatticeGauge}. \paragraph{Gradient--flow dynamics.}A simple well-behaved evolution law is driven by gradient flow on the compact state space: \begin{equation}\label{eq:qf-gradient-flow} \dot{\theta}_i \;=\; -\gamma_\theta \,\frac{\partial \mathcal{E}}{\partial \theta_i} \;+\; \eta_i(t), \qquad \dot{A}_{ij} \;=\; -\gamma_A\,\frac{\partial \mathcal{E}}{\partial A_{ij}} \;+\; \eta_{ij}(t), \end{equation} with mobilities $\gamma_\theta,\gamma_A>0$ and external drives $\eta_i(t),\eta_{ij}(t)$ (global pulses, boundary forcing, or localized injections). Concretely, the Euler--Lagrange components are \begin{align} \frac{\partial \mathcal{E}}{\partial \theta_i} &=\sum_{j\sim i} J_{ij}\,\sin(\Delta_{ij}), \label{eq:qf-phase-energy-derivative}\\ \frac{\partial \mathcal{E}}{\partial A_{ij}} &=-J_{ij}\,\sin(\Delta_{ij}) \;+\; \sum_{p\ni (i,j)} \kappa_p\, s_{p,ij}\,\sin((dA)_p), \label{eq:qf-link-energy-derivative} \end{align} so \eqref{eq:qf-gradient-flow} relaxes toward gauge--invariant consistency ($\Delta_{ij}\to 0$) and low curvature ($(dA)_p\to 0$) unless forced otherwise. Because the variables are compact and $\mathcal{E}$ is bounded below, this regime is dynamically stable and aligns with the QF ``no blow-ups '' philosophy. \begin{rem}[Scope and Interpretation] \label{rem:scope-and-interpretation} Equations \eqref{eq:qf-interface-energy}--\eqref{eq:qf-gradient-flow} are not proposed as fundamental laws of GCT; they are a \emph{normal form} for the abelian corridor: local couplings, compact phases, gauge redundancy, and dissipative relaxation. In particular, continuous flows of this type do \emph{not} implement jump--style updates by themselves; any discrete ``computational'' layer is induced later by readout/partitioning (return maps). \end{rem} \paragraph{Auxiliary geometric ODE variables.}To retain an explicit connection to helix geometry, one may attach slowly varying, bounded geometric parameters to each cell, \begin{align} \label{eq:qf-plaquette-phase-slip} r_i(t) &\in \mathbb{R}_{>0} && \text{(effective radius/scale),} \\ \tau_i(t) &\in \mathbb{R} && \text{(effective twist rate),} \\ \mathcal{L}_i(t) &\in \mathbb{R}_{>0} && \text{(effective length)} \end{align} and couple them to \emph{local energy density} and/or \emph{local phase current}. A simple choice is: \begin{align} \dot r_i &= -\lambda_r\,(r_i-r_0)\;+\;\alpha_r \sum_{j\sim i}\bigl(1-\cos(\Delta_{ij})\bigr), \label{eq:qf-radius-relaxation}\\ \dot \tau_i &= -\lambda_\tau\,(\tau_i-\tau_0)\;+\;\alpha_\tau \sum_{j\sim i}\sin(\Delta_{ij}), \label{eq:qf-twist-relaxation}\\ \dot{\mathcal{L}}_i &= -\lambda_L\,(\mathcal{L}_i-\mathcal{L}_0)\;+\;\alpha_L\, r_i\,|\tau_i|, \label{eq:qf-length-relaxation} \end{align} with $\lambda_\bullet>0$ ensuring relaxation to baseline values $(r_0,\tau_0,\mathcal{L}_0)$ and $\alpha_\bullet$ setting coupling strength. These equations encode the informal idea that mismatch/coupling stress can thicken the effective geometry \eqref{eq:qf-radius-relaxation}, phase transport can bias twist \eqref{eq:qf-twist-relaxation}, and geometry co-scales \eqref{eq:qf-length-relaxation}. One may also let the mobilities in \eqref{eq:qf-gradient-flow} depend on $\tau_i$ (twist as a local ``clock'' factor), e.g.\ $\gamma_\theta=\gamma_\theta(\tau_i)$, without changing the gauge structure. \paragraph{Synchronization limit: Kuramoto reduction.}A useful special case of the QF phase dynamics is obtained by restricting to nearest-neighbor coupling on the $U(1)$ phase layer and passing to a co-rotating frame. In that case, the local phase evolution reduces to a Kuramoto-type system~\cite{Acebron2005, Kuramoto1984} \[ \frac{d\theta_i}{dt} = \Omega_i + K\sum_{j\sim i}\sin(\theta_j-\theta_i), \] where $\theta_i\in S^1$ is the phase of cell $i$, $\Omega_i$ is its intrinsic frequency, and $K$ is the coupling strength. This should be understood as a \emph{synchronization limit} of the abelian gauge dynamics, not as the full semantics of quasi--fluidic computation. Physically, it describes the tendency of neighboring helices to phase-lock under local coupling; computationally, it captures the emergence of a shared clock or coherent phase carrier across a region of the lattice. In particular, when the $U(1)$ corridor is used only as a transport layer, Kuramoto synchronization provides a natural mechanism by which continuous phase variables may stabilize before a stroboscopic readout is applied. Equivalently, if the local energy is taken to be of XY type, \[ E(\theta)=\sum_{(i,j)\in E(\Lambda)}\kappa_{ij}\bigl(1-\cos(\theta_i-\theta_j)\bigr), \] then the dissipative phase flow \[ \dot{\theta}_i=-\frac{\partial E}{\partial \theta_i} \] recovers the Kuramoto form above up to the choice of sign convention and the addition of the drift terms $\Omega_i$. Thus, the Kuramoto system is a familiar, physically interpretable subcase of the general QF lattice dynamics. \begin{ex}[Plaquette Phase Slip and Integer Counter Update] \label{ex:plaquette-phase-slip-and-integer-counter-update} Let $p$ be an oriented plaquette in the lattice, and let \[ \Delta_e = \theta_{s(e)}-\theta_{t(e)}-A_e \qquad (\mathrm{mod}\ 2\pi) \] denote the gauge--invariant phase difference on each oriented boundary link $e\in\partial p$. Along a continuous trajectory, choose continuous real lifts $\widetilde{\Delta}_e(t)\in\mathbb{R}$ and define the lifted circulation \[ \widetilde{\Gamma}_p(t) := \sum_{e\in\partial p} s_{p,e}\,\widetilde{\Delta}_e(t), \qquad s_{p,e}\in\{\pm1\}. \] Because each $\Delta_e$ is gauge invariant, so is $\widetilde{\Gamma}_p$ once its initial lift has been fixed. At every time, decompose the lifted circulation uniquely as \[ \widetilde{\Gamma}_p(t) = \Gamma_p(t)+2\pi z_p(t), \] where \[ \Gamma_p(t)\in(-\pi,\pi], \qquad z_p(t)\in\mathbb{Z}. \] The variable $\Gamma_p$ is the principal compact $U(1)$ flux observed on the plaquette, while $z_p$ records the number of complete $2\pi$ windings that have passed through the chosen branch. As long as $\widetilde{\Gamma}_p(t)$ remains within a single branch interval, $z_p$ is constant and the evolution lies entirely in the smooth phase layer. Suppose, however, that an applied drive causes $\widetilde{\Gamma}_p$ to cross a branch boundary at a time $t_\ast$. Then the principal phase wraps by $2\pi$, while the integer counter changes by one: \[ z_p(t_\ast^+)-z_p(t_\ast^-) = \begin{cases} +1, & \dot{\widetilde{\Gamma}}_p(t_\ast)>0,\\[4pt] -1, & \dot{\widetilde{\Gamma}}_p(t_\ast)<0. \end{cases} \] The compact quantity \[ \exp\!\bigl(i\widetilde{\Gamma}_p(t)\bigr) = \exp\!\bigl(i\Gamma_p(t)\bigr) \] remains continuous throughout the event; the discrete change occurs in the lifted winding ledger $z_p$. This is a \emph{phase slip}. In the phase--only reduction $A_e=0$, the same construction measures the winding of neighboring cell phases around the plaquette. In a continuum physical realization, the event corresponds to the passage or nucleation of a vortex through the plaquette; the phase-only description may then fail briefly at the defect core while the surrounding $U(1)$ phase remains well-defined. Thus, the $Z$--coordinate is not introduced independently of the continuous dynamics: it records isolated threshold events generated by the evolution of the $T$-- layer. \end{ex} \begin{rem}[Smooth Core and Thresholded Boundary Events] \label{rem:smooth-core-and-thresholded-boundary-events} The strict smooth QF core is obtained by restricting to trajectories for which every $z_p$ remains fixed. Allowing the isolated events of \cref{ex:plaquette-phase-slip-and-integer-counter-update} enlarges this core by adjoining transitions between integer-labeled strata. Such phase slips do not by themselves introduce non-abelian holonomy: the structure group remains $U(1)$, but the evolution is no longer confined to a single smooth lift of the compact phase space. This thresholded extension supplies the natural bridge from abelian phase transport to vortex and defect dynamics. \end{rem} \paragraph{Recovery of discrete readouts.} The QF dynamics above are continuous (compact phases with local energy and gradient flow). As such, they do \emph{not} natively implement jump--style updates. The bridge back to classical symbolic computation is therefore \emph{stroboscopic}: we sample the system once per pulse and apply a \emph{discretized readout} (a quantizer) to obtain an induced update on a finite alphabet. In this sense, discrete computation in the QF band is a \emph{return--map phenomenon}, not an ODE phenomenon. \begin{defn}[Pulse Times and Stroboscopic Map] \label{def:pulse-times-and-stroboscopic-map} Fix a pulse period $\Delta t>0$ and define $t_n:=n\Delta t$. Let $P:L\to L$ denote the time--$\Delta t$ map on the QF state space $L$ induced by one pulse: \[ P \;:=\; \mathsf{Ev}_{\Delta t}. \] We write $x^n\in L$ for the post--pulse state at time $t_n$, so $x^{n+1}=P(x^n)$. \end{defn} Because the abelian corridor carries gauge redundancy, any robust readout should be phrased in terms of \emph{gauge--invariant local observables}. In the lattice--gauge normal form, the basic invariants are: \[ \Delta_{ij}=\theta_i-\theta_j-A_{ij}\;\;(\mathrm{mod}\;2\pi) \qquad\text{and}\qquad (dA)_p=\sum_{(i,j)\in\partial p}s_{p,ij}A_{ij}\;\;(\mathrm{mod}\;2\pi). \] We therefore define a cellwise observable $y_i=y_i(x)$ as a finite list of such invariants in a bounded neighborhood (e.g.\ the $\Delta_{ij}$ on links incident to $i$, and optionally nearby plaquette fluxes $(dA)_p$). \begin{defn}[Discretized Readout/Quantizer] \label{def:discretized-readout-quantizer} Fix a finite alphabet $B$ and a (piecewise constant) quantizer \[ Q:\;\mathcal{Y}\longrightarrow B, \] where $\mathcal{Y}$ is the range of local invariants $y_i$. The induced discrete configuration for a state $x\in L$ is \[ b=\mathrm{Read}(x)\in B^{V}, \qquad b_i:=Q\bigl(y_i(x)\bigr). \] For example, one may partition $S^1$ into arcs and threshold a chosen invariant angle, or partition the flux $(dA)_p$ into bins. The specific partition is a modeling choice. \end{defn} \begin{defn}[Local QF Return Maps and Induced Symbolic Dynamics]\label{def:local-qf-return-maps-and-induced-symbolic-dynamics} Let \[ P:\mathcal L\longrightarrow\mathcal L \] be the time-\(\Delta t\) stroboscopic map introduced above. Finite-range coupling of the infinitesimal evolution law does not, by itself, imply that the exact finite-time map \(P\) has strictly finite propagation. We therefore say that a pulse is \emph{finite-propagating} if, for every cell \(i\), there exists a finite neighborhood \(N(i)\) such that the post-pulse state at \(i\) depends only on the pre-pulse states in \(N(i)\): \[ x_i^{n+1} = U_i\!\left( \{x_j^n\}_{j\in N(i)} \right), \qquad x_i^n\in X, \] where \(X\) is the local compact state space. If the lattice and update law are translation invariant, one may take \(U_i=U\). Under this additional finite-propagation hypothesis, the stroboscopic dynamics is a continuous-alphabet cellular automaton. Now let \[ \operatorname{Read}:\mathcal L\longrightarrow B^V \] be the discretized readout. An autonomous symbolic return map \[ F_B:B^V\longrightarrow B^V \] exists precisely when the stroboscopic evolution preserves the fibers of the readout: \[ \operatorname{Read}(x)=\operatorname{Read}(y) \quad\Longrightarrow\quad \operatorname{Read}(Px)=\operatorname{Read}(Py). \] When this condition holds, \(F_B\) is uniquely determined by \[ F_B\circ\operatorname{Read} = \operatorname{Read}\circ P. \] Equivalently, if \[ \operatorname{Lift}:B^V\longrightarrow\mathcal L \] is any section selecting one representative of each readable state, then \[ F_B = \operatorname{Read}\circ P\circ\operatorname{Lift}, \] and the result is independent of the chosen section precisely because the readout fibers are preserved. \end{defn} \begin{rem}[What is ``Computed''?] \label{rem:what-is-computed} In the QF corridor, the continuous dynamics are the physical substrate, and symbolic computation appears only after stroboscopic sampling and a specified discretized readout. The infinitesimal dynamics may be local even when the exact finite-time flow is not strictly finite-propagating. A genuine cellular-automaton description therefore requires either an explicitly local pulse scheme or the finite-propagation hypothesis of~\cref{def:local-qf-return-maps-and-induced-symbolic-dynamics}. More generally, whenever the stroboscopic map preserves the readout fibers, it induces a well-defined symbolic return map on \(B^V\). \end{rem} \paragraph{The $U(1)$ Corridor and an E/V Paradox.}\label{sec:the-u1-corridor-and-an-e-v-paradox} It is at this point necessary to issue some disclaimers regarding the use of the full $S^1$ state spaces, that is, the $U(1)$ corridor. As established, within the quasi–fluidic band, the $\omega$-logic layer may be taken either as a finite cyclic alphabet $C_N$ or as a continuous phase $U(1)$. Passing to the full $S^1$ limit is mathematically tidy (abelian, compact), but it also oddly backfires, creating a regime with atypical E/V trade-off behavior. The following remarks record the current hypothesis of this paper: \begin{itemize} \item \textbf{Low geometric expressiveness.} Pure $U(1)$ phase composition is commutative. As a result, sequencing power collapses: one cannot exploit non-commuting compositions to build richer path-dependent behavior, and the phase layer alone tends to support only limited families of smooth, energetically bounded dynamics. In this corridor, most ``computational complexity'' must therefore be carried either by the discrete core $B$ (via readout and interface thresholds) or by leaving the abelian setting (e.g.\ via threshold events/topological defects or non-abelian holonomy). \item \textbf{Low algebraic verifiability.} Any \emph{finite} readout must discretize phase (explicitly or implicitly) by thresholds. In the $U(1)$ corridor, near-resonant phase evolution makes this discretization brittle: small phase perturbations can move trajectories across readout boundaries, producing significant changes in the inferred discrete state. Moreover, phase-locking reduces redundancy (many degrees of freedom share a common ``clock''), weakening independent cross-checks.\footnote{In essence, pure $U(1)$-logic has \textit{too many} microstates, and this points toward a concrete mathematical reason why large-scale analog computers were mostly abandoned by the late 1950s.} \end{itemize} \paragraph{Transition, not destination.} For these reasons, the $U(1)$ phase limit is best regarded as a \emph{critical corridor}: visually orderly and analytically tractable, yet brittle under discretized readout and shallow in purely geometric expressiveness unless supplemented by discrete structure. It serves as a staging layer for robust computation on $B$ and as a boundary marker before the onset of genuinely non-abelian (fluidic/gauge) behavior. \subsection{Application of Lie Groupoids in Geometric Computation} \label{sec:application-of-lie-groupoids-in-geometric-computation} In gauge theory, geometric data are assigned to fibers over a base manifold, and a connection determines how these data are transported and compared along paths. Such comparisons have specified source and target points, with composition defined only when endpoints agree. For a principal $G$-bundle $P\to M$, this structure is naturally organized by the gauge groupoid \[ \mathcal{G}(P)=(P\times P)/G\rightrightarrows M, \] whose arrows represent equivariant identifications between fibers, while a connection supplies parallel transport and its associated holonomy. Lie groupoids provide the corresponding smooth framework, originating in the differential-geometric work of Ehresmann~\cite{Ehresmann1959} and developed further by Pradines~\cite{Pradines1966}; see also~\cite{Mackenzie2005LieGroupoids}. Informally, a Lie groupoid consists of smooth manifolds of objects and morphisms, with smooth structural maps (source, target, identity, inversion, and composition), where the source and target maps are submersions. Unlike a single global symmetry group, a Lie groupoid allows the available transformations to depend on their source and target, and morphisms compose only when their endpoints match. This makes Lie groupoids a natural framework for the reversible transport and local symmetry of GCMs. The additional admissibility conditions governing computational updates and boundary interfaces are specified by the chosen geometric and compositional structures. We begin with a general decorated formalism, and then specialize it to the quasi-fluidic abelian regime. \begin{defn}[S--decorated Groupoid] \label{def:s-decorated-groupoid} Let $(S,*)$ be a monoid with unit $e$, and let $\mathbf{B}S$ denote the one-object category whose endomorphism monoid is $S$. An \emph{$S$--decorated groupoid} is a pair $(\mathcal{G},d)$ consisting of a small groupoid $\mathcal{G}$ together with a functor \[ d:\;\mathcal{G}\longrightarrow \mathbf{B}S. \] Equivalently, each morphism $f\in \mathrm{Mor}(\mathcal{G})$ is assigned a label $d(f)\in S$ such that \[ d(g\circ f)=d(g)*d(f), \qquad d(\mathrm{id}_x)=e \] for all composable $f,g$ and all objects $x$. Since functors preserve isomorphisms, every decoration of an arrow lies in the unit group $S^\times\subseteq S$. \end{defn} \begin{defn}[Morphisms of S--decorated Groupoids] \label{def:morphisms-of-s-decorated-groupoids} A morphism \[ (\mathcal{G},d)\longrightarrow (\mathcal{H},\delta) \] of $S$--decorated groupoids is a functor $F:\mathcal{G}\to\mathcal{H}$ such that \[ \delta\circ F=d. \] This defines the category $\mathrm{DecGrpd}_S$ of $S$--decorated groupoids. \end{defn} \begin{rem} \label{rem:decorated-groupoid-terminology} The use of the term ``decorated'' follows modern geometric and categorical usage~\cite{Penner1987Decorated,fong2015decoratedcospans}. The point is to enrich a base compositional structure with auxiliary data while preserving functorial composition. In the present context, the decorations record geometric or computational quantities associated with paths, interfaces, or local transformations. \end{rem} \paragraph{General decorated path data.} Let $M$ be a smooth manifold (or a stratified manifold when needed), and let $\mathcal{S}$ denote a specified collection of geometric or computational structures on $M$---for example, twist counts, phase accumulation, chirality labels, or local field values. The classical fundamental groupoid remembers only path homotopy classes. In geometric computation, however, one often needs a refinement that also records such auxiliary data. \begin{defn}[Decorated Fundamental Groupoid] \label{def:decorated-fundamental-groupoid} Let $M$ be a smooth manifold and let $S$ be a monoid. An \emph{$S$--decorated fundamental groupoid} of $M$ is an $S$--decorated groupoid \[ (\Pi_1^\ast(M),d) \] whose underlying groupoid $\Pi_1^\ast(M)$ is chosen from a path groupoid naturally associated to $M$ (for example, the ordinary fundamental groupoid, or the thin fundamental groupoid when differential invariants are to be preserved), and whose decoration functor \[ d:\Pi_1^\ast(M)\to \mathbf{B}S \] records the specified geometric data along paths. Composition is induced by path concatenation together with the monoid law in $S$. \end{defn} \begin{rem} \label{rem:ordinary-versus-thin-path-groupoids} When the decoration depends only on ordinary homotopy classes, one may work with the usual fundamental groupoid $\Pi_1(M)$. When the decoration depends on differential data such as connection holonomy or thin geometric transport, one should instead use the thin fundamental groupoid $\Pi_1^{\mathrm{thin}}(M)$, since such invariants are constant under thin homotopy but need not factor through ordinary homotopy classes. \end{rem} \paragraph{A geometric model for the twin helix.} For geometric computation, a finite twin helix module is best modeled as a regular neighborhood of an embedded arc. Topologically, this gives \[ THM \;\cong\; D^2\times I \;\cong\; B^3. \] Thus, the underlying space is contractible: its ordinary homotopy type is trivial, and the undecorated fundamental groupoid carries no nontrivial loop information. It means that the computationally relevant structure of a twin helix module is not encoded by the bare topology of the carrier space, but by the additional geometric data placed on it: phase fields, twist laws, chirality assignments, framings, and interface rules. In the symbolic and general decorated setting, one may attach labels such as chirality or twist to paths in this contractible space. In the quasi-fluidic abelian reduction developed later in this chapter, chirality is no longer treated as an actively updated path label, but instead as a choice of orientation convention for link variables. \begin{ex}[Decorated Loops in a Finite Module] \label{ex:decorated-loops-in-a-finite-module} Let \[ THM \;\cong\; D^2_{(r,\varphi)}\times I_z, \] and choose a smooth phase 1-form on the cross-sectional angular direction, \[ A \;=\; \Omega(r,\varphi,z)\,d\varphi. \] For fixed $(r,z)$, let $\gamma_{r,z}$ denote the loop obtained by traversing once around the circle of radius $r$ in the cross-section $D^2\times \{z\}$. As a loop in the contractible manifold $THM$, $\gamma_{r,z}$ is null-homotopic in the ordinary sense. Nevertheless, it may carry nontrivial geometric data. For example, taking \[ S = U(1)\times C_2, \] one may define a decoration on an appropriate refined path groupoid (for example, the thin fundamental groupoid) by \[ d([\gamma_{r,z}]) \;:=\; \left( \exp\!\Bigl(i\!\int_{\gamma_{r,z}} A\Bigr),\; \chi(\gamma_{r,z}) \right), \] where the first component records the accumulated phase and the second records a chirality label determined by the chosen geometric structure~\cite{Barrett1991Holonomy}. The resulting decorated class \[ ([\gamma_{r,z}],\,d([\gamma_{r,z}])) \] is computationally nontrivial even though the underlying ordinary homotopy class of $\gamma_{r,z}$ is trivial. Thus, in a finite twin helix module, the computational content lives in the decoration rather than in the bare fundamental groupoid. \end{ex} \begin{rem} \label{rem:general-versus-abelian-decorations} This is precisely why the general decorated picture and the later quasi-fluidic abelian reduction must be kept distinct. In the symbolic/general setting, chirality may legitimately appear as part of the decoration target $S$, and one works with a refined path groupoid because ordinary homotopy forgets too much. In the specialized $U(1)$ phase regime, by contrast, chirality is frozen into an orientation convention, and the active structure reduces to abelian phase transport. \end{rem} \paragraph{Why Lie groupoids appear in the QF regime.} The preceding decorated picture is intentionally broad. In the quasi-fluidic regime, however, the relevant structure simplifies dramatically. The active degree of freedom is no longer a freely manipulated chirality/twist label, but an abelian phase field. The correct object is therefore not a general decorated path groupoid, but a Lie groupoid encoding \emph{gauge redundancy} of local $U(1)$ phase assignments on a lattice. To provide a rigorous setting for this maximal abelian regime, we explicitly construct the corresponding Lie groupoid. \begin{defn}[S--decorated Lie Groupoid] \label{def:s-decorated-lie-groupoid} If $\mathcal{G}\rightrightarrows M$ is a Lie groupoid and $S$ is a Lie group, an \emph{$S$--decorated Lie groupoid} is a smooth functor \[ d:\mathcal{G}\to \mathbf{B}S, \] where $\mathbf{B}S$ is the one--object Lie groupoid with endomorphism group $S$. \end{defn} \begin{defn}[Underlying Lattice/Graph Datum]\label{def:underlying-lattice-graph-datum} Fix a finite connected graph \[ \Gamma=(V,E), \] for example, a finite patch of a $GL$, where $V$ is the set of sites and $E$ the set of undirected adjacency links. Choose an orientation on each undirected edge, written as a subset \[ E^{+}\subseteq \vec E, \] so that every edge $\{i,j\}\in E$ appears exactly once as either $e:i\to j$ or $\bar e:j\to i$ in $E^+$. Write $s(e),t(e)\in V$ for the source and target of an oriented edge $e$. \end{defn} \begin{defn}[Core Configuration Space in the QF Regime]\label{def:core-configuration-space-in-the-qf-regime} The quasi-fluidic core state of the lattice is a \emph{link phase assignment} \[ M_{\mathrm{QF}} \;:=\; U(1)^{E^{+}} \;\cong\; (S^1)^{|E^{+}|}. \] An element $U\in M_{\mathrm{QF}}$ is a family $U=(U_e)_{e\in E^{+}}$ with $U_e\in U(1)$. The value on the oppositely oriented edge is determined by inversion: \[ U_{\bar e} \;:=\; U_e^{-1}. \] \end{defn} \begin{rem}[Chirality as Orientation Convention]\label{rem:chirality-as-orientation-convention} In this quasi-fluidic interpretation, chirality is \emph{not} a separate state coordinate. Instead, the handedness of the underlying helix selects (or suggests) a preferred orientation for reading the link phase; reversing that convention replaces $U_e$ with $U_e^{-1}$. Thus, the earlier symbolic chirality label is absorbed into the orientation convention on each link and disappears from gauge-invariant observables. \end{rem} \begin{defn}[Gauge Group and Action on Link Phases]\label{def:gauge-group-and-action-on-link-phases} Let the site gauge group be \[ G_{\mathrm{QF}} \;:=\; U(1)^{V}. \] For $g=(g_i)_{i\in V}\in G_{\mathrm{QF}}$ and $U=(U_e)_{e\in E^{+}}\in M_{\mathrm{QF}}$, define the action \[ g\cdot U\in M_{\mathrm{QF}} \] by the standard link transformation law \[ (g\cdot U)_e \;:=\; g_{s(e)}\,U_e\,g_{t(e)}^{-1}, \qquad e\in E^{+}. \] \end{defn} \begin{defn}[QF Core Lie Groupoid: the Gauge Action Groupoid]\label{def:qf-core-lie-groupoid-the-gauge-action-groupoid} The quasi-fluidic core Lie groupoid is the action groupoid \[ \mathcal{G}_{\mathrm{QF}} \;:=\; G_{\mathrm{QF}} \ltimes M_{\mathrm{QF}} \rightrightarrows M_{\mathrm{QF}}. \] An arrow is a pair $(g,U)$ with $g\in G_{\mathrm{QF}}$ and $U\in M_{\mathrm{QF}}$. The source and target maps are \[ s(g,U)=U, \qquad t(g,U)=g\cdot U, \] and composition is \[ (h,\,g\cdot U)\circ(g,U)=(hg,\,U), \qquad h,g\in G_{\mathrm{QF}}. \] For finite $\Gamma$, this is a finite-dimensional Lie groupoid. \end{defn} \begin{rem}[Gauge Invariants Live on the Quotient Stack]\label{rem:gauge-invariants-live-on-the-quotient-stack} The physically meaningful quantities in the quasi-fluidic abelian regime are gauge--invariant functions on $M_{\mathrm{QF}}$ (equivalently, functions on the quotient stack \[ [M_{\mathrm{QF}}/G_{\mathrm{QF}}]). \] The groupoid $\mathcal{G}_{\mathrm{QF}}$ records \emph{gauge redundancy}, that is, equivalences of description rather than physical time evolution. Dynamics---for example, the gradient flow of a local gauge-invariant energy built from plaquette terms---is additional structure: a $G_{\mathrm{QF}}$--equivariant flow on $M_{\mathrm{QF}}$ that descends to the quotient. We treat this dynamical layer separately from the gauge groupoid itself; see also~\cite{costa2018liegroupoidsclassicalfield} as a reference. \end{rem} The viewpoint adopted here is therefore two-tiered. At the general level, decorated groupoids package the local geometric and combinatorial data relevant to symbolic or hybrid computation. At the quasi-fluidic abelian level, this broad picture reduces to a concrete lattice $U(1)$ gauge groupoid whose arrows encode only local phase-frame redundancy. This distinction is intentional: transcendental or geometric data may decorate computation in general, whereas the $U(1)$ corridor isolates the simplest smooth abelian core on which the later return-map and phase-ledger constructions are built. \paragraph{A warning on the Homotopy Hypothesis.} It would be amiss not to mention the Grothendieck Homotopy Hypothesis~\cite{grothendieck2021pursuingstacks}, but equally important not to overstate what follows from it. The classical principle concerns the equivalence between homotopy types and suitable \(\infty\)-groupoidal models of those homotopy types. It does \emph{not} imply that arbitrary differential, geometric, dynamical, or computational decorations are themselves determined by the underlying homotopy type. This distinction is essential in GCT. A connection, framing, phase field, chirality assignment, holonomy, interface law, or local computational constraint may distinguish two geometric systems whose underlying spaces have the same ordinary homotopy type. When such data factor through homotopy, the usual homotopy-theoretic classification is sufficient. When they do not, the correct object is not a new ``homotopy type'' in the classical sense, but a homotopy type equipped with \emph{additional structure}---for example a thin, Lie, decorated, or higher groupoid, a bundle with connection, or an appropriate higher-geometric stack. Accordingly, decorated groupoids in this paper should not be read as a replacement for regular homotopy types. \begin{defn}[Homotopy Computation]\label{def:homotopy-computation} Fix a GCM lattice in the quasi-fluidic regime with configuration space \(M_{\mathrm{QF}}\), pulse map \[ P:M_{\mathrm{QF}}\to M_{\mathrm{QF}}, \] and readout \[ \mathrm{Read}:M_{\mathrm{QF}}\to B^{V}. \] A \emph{homotopy computation} is a finite or continuous state evolution \[ \gamma:[0,1]\longrightarrow M_{\mathrm{QF}} \] such that \(\gamma\) is generated by admissible local dynamics of the model (for example, gradient flow segments, pulse updates, or their hybrid composition), together with its induced readout trajectory \[ \mathrm{Read}\circ \gamma:[0,1]\to B^{V}. \] Two homotopy computations \(\gamma_0,\gamma_1:[0,1]\to M_{\mathrm{QF}}\) with the same endpoints are said to be \emph{computationally equivalent} if there exists a homotopy \[ H:[0,1]\times[0,1]\longrightarrow M_{\mathrm{QF}} \] such that: \begin{enumerate} \item for each fixed \(u\in[0,1]\), the path \(t\mapsto H(t,u)\) is again an admissible computation, \item \(H(t,0)=\gamma_0(t)\) and \(H(t,1)=\gamma_1(t)\), \item the relevant observable data are preserved along the deformation (for example, the same decorated invariants, or the same induced symbolic readout up to the chosen notion of equivalence). \end{enumerate} \end{defn} \paragraph{Interpretation.} This definition is intentionally regime-dependent. In the symbolic regime, a homotopy computation reduces to a path through a finite transition graph. In the quasi-fluidic regime, it is a path in the continuous phase configuration space \(M_{\mathrm{QF}}\), typically observed only through the stroboscopic return map and the readout \(\mathrm{Read}\). Thus, the same underlying idea---computation as an admissible path modulo invariant-preserving deformation---specializes to both discrete and continuous settings. \begin{rem} \label{rem:homotopy-computation-interpretation} The point of the terminology is not that every computation is literally a topological homotopy, but that admissible evolutions in configuration space form path-like objects whose equivalence is controlled by the invariants one chooses to preserve. In the abelian quasi-fluidic corridor, these are typically gauge-invariant phase observables and their induced symbolic readouts. \end{rem} \paragraph{Lie algebroid chart and Atiyah's linearization of flow.} Readers from differential geometry may reasonably ask why we do not take Lie algebras and Lie brackets as the primary algebraic language throughout. The short answer is that our computational semantics is organized by \emph{global, composable transport data} (paths, holonomy, winding/phase, and gluing across interfaces), and these are most naturally encoded at the groupoid level rather than by infinitesimal generators. Nothing here rejects the Lie-algebraic viewpoint; rather, it appears as a \emph{linearization chart}. Given a Lie groupoid $\mathcal{G}\rightrightarrows M$, its infinitesimal object is the Lie algebroid \[ A(\mathcal{G}) \;:=\; \ker(ds)\big|_{M}\ \longrightarrow\ M, \qquad \rho := dt\big|_{A(\mathcal{G})}:A(\mathcal{G})\to TM, \] whose bracket on sections encodes first-order commutator information of local bisections. This is the many-object analogue of ``Lie algebra = infinitesimal Lie group''~\cite{Weinstein1996Groupoids}. Thus, whenever one restricts to a sufficiently smooth regime where a genuine local symmetry pseudogroup governs evolution, one can pass from the groupoid semantics to an algebroid bracket calculus. \paragraph{Atiyah’s gauge-theoretic instance.}In particular, for a principal $G$--bundle $P\to M$, the gauge groupoid $(P\times P)/G\rightrightarrows M$ has associated Lie algebroid \[ A \;\cong\; TP/G \;\to\; M, \] fitting into the short exact ``Atiyah sequence'' \[ 0\ \to\ \mathrm{ad}(P)\ \to\ TP/G\ \to\ TM\ \to\ 0, \] introduced by Atiyah in his study of connections. This is precisely the setting in which the Lie bracket formalism is maximally effective: it governs infinitesimal deformations of connections and local curvature evolution~\cite{Atiyah1957}. \paragraph{Why we do not build the theory around brackets.}The point of the ``Algebra of Flow'' emphasized here (\ref{sec:algebra-of-flow}, \ref{sec:euclidean-gauge-computation-and-emergent-spacetime}) is that many of the invariants relevant to computation are inherently global and persist under operations for which infinitesimal linearization is either insufficient or not stable. Accordingly, we treat Lie algebroids (and brackets) as a \emph{derived} local chart available in the upcoming fluidic/gauge regimes, while keeping the groupoid/decorated-path level as the primitive semantic layer. Unlike Lie algebras, Lie algebroids need not integrate to global Lie groupoids without additional conditions; integrability is controlled by explicit obstructions~\cite{CrainicFernandes2003}. This reinforces the methodological choice: our semantics is formulated on the global/compositional side, with infinitesimal calculus used when (and only when) the regime admits it. \paragraph{Boundary of the groupoid formalism: CJS and Floer flow categories.} The decorated Lie groupoids introduced above encode reversible transport, gauge equivalence, and path-dependent geometric data. Their arrows are invertible, as required for a groupoid, and the passage to the thin fundamental groupoid permits connection-dependent quantities such as holonomy to be retained even when ordinary homotopy is trivial. This should be distinguished from the categorical structure appearing in Morse--Floer theory. There the relevant arrows are not equivalences but directed trajectories between critical configurations. If \(x\) and \(y\) are critical points of a Floer functional, the corresponding morphism object is a compactified moduli space \(\overline{\mathcal M}(x,y)\) of flow trajectories, whose boundary strata are assembled from broken trajectories, \[ \partial\overline{\mathcal M}(x,z) \supseteq \bigcup_y \overline{\mathcal M}(x,y)\times \overline{\mathcal M}(y,z). \] Such data are naturally organized by a \emph{flow category}, rather than by a groupoid. The ordinary Floer chain complex retains only a severe algebraic compression of the trajectory geometry: in particular, zero-dimensional moduli spaces supply the coefficients of the Floer differential. Cohen, Jones, and Segal proposed that, when the compactified trajectory spaces admit sufficiently coherent stable framings, the entire flow category may instead determine a stable homotopy type whose ordinary homology recovers Floer homology \cite{CohenJonesSegal1995, Cohen2019Floer}. This should be regarded as a \emph{homotopy refinement problem}, rather than as a universally missing object (in particular, see \cref{app:higher-dimensional-algebra}). Stable homotopy refinements have been constructed in important settings, notably Seiberg--Witten Floer theory \cite{Manolescu2003}, and recent work of Abouzaid and Blumberg gives a stable \(\infty\)-categorical formulation in which flow categories and flow bimodules produce Floer homotopy types, including an implementation for Hamiltonian Floer theory \cite{AbouzaidBlumberg2024}. For the present construction, the dividing line is therefore explicit. The decorated Lie groupoid records \emph{transport through a geometric configuration}; it does not by itself encode the full moduli spaces of dynamically admissible histories between critical configurations. A Floer-type refinement of geometric computation would require an additional directed layer in which admissible histories form compactified moduli spaces, broken histories supply their boundary strata, and suitable orientation, framing, or bordism data survive gluing. Only at that stage would a comparison with the Cohen--Jones--Segal construction, or with a corresponding Floer homotopy type, become mathematically substantive. \subsection{Connections and Phase Ledgers on a Trivial Helical Bundle}\label{sec:connections-and-phase-ledgers} The finite twin--helix module used in the basic construction is locally modeled by a regular neighborhood of an embedded arc, and is therefore topologically \[ D^2\times I \;\cong\; B^3. \] Thus, as a $D^2$--bundle over the interval $I$, the local helical model is topologically trivial. In particular, there is no intrinsic bundle monodromy in the $D^2$ fibers. This does \emph{not} mean that the geometry is computationally trivial. Rather, it means that the bundle class itself does not carry the information relevant to computation, but that additional geometric structure is placed on this trivial scaffold: chosen phase fields, connections, deformations, and local interface rules. This is the correct abelian analogue of the Aharonov--Bohm (AB) viewpoint~\cite{aharonov1959significance}. In the AB effect, the underlying space may be locally simple while the physically meaningful information is carried by phase accumulation of a $U(1)$ connection around loops. The same lesson applies here: even on a topologically trivial helical module, one may impose a phase field on the transverse circular directions in each cross-section, and the resulting loop integral defines a nontrivial $U(1)$ response. Thus, the relevant invariant in the present setting is not topological twisting of the bundle, but phase accumulation determined by the chosen geometric control data. We therefore use the above product structure to isolate the abelian structure that will persist in later sections: an $S^1$--valued phase accumulated around transverse circles in a fixed cross-section. In modern language, this is most naturally described by a $U(1)$ connection on a trivial circle bundle over such loops (equivalently, a Hermitian line bundle with unitary connection). However, in the present discussion only the elementary notions of phase, curvature, and holonomy are required. As an organizational comparison, the product structure also means that local differential data may be written globally in coordinates; in this limited sense, one avoids the extra cocycle bookkeeping that would arise on a genuinely nontrivial fibration. Fix a longitudinal parameter value $z_0\in I$, and work in the transverse disk \[ D^2\times \{z_0\}. \] For each radius $r$ for which the circle \[ \gamma_r(\theta) \;:=\; (r,\theta,z_0),\qquad \theta\in[0,2\pi], \] lies in the interior of the disk, let \[ A \;:=\; \Omega(r,\theta)\,d\theta \] be a $U(1)$ connection one--form on that transverse slice. Define the associated phase ledger by \[ U(r) \;:=\; \exp\!\Bigl(i\!\int_{\gamma_r} A\Bigr) \;=\; \exp\!\Bigl(i\!\int_0^{2\pi}\Omega(r,\theta)\,d\theta\Bigr) \;\in\; U(1). \] \begin{prop}[Radial Shear Identity] \label{prop:radial-shear-identity} Let \[ A:=\Omega(r,\theta)\,d\theta \qquad\text{and}\qquad F:=dA=\partial_r\Omega(r,\theta)\,dr\wedge d\theta \] on a fixed transverse slice $D^2\times\{z_0\}$. For $r_1