Dynamical-Lattice Regulator (DLR)

• Dynamical-Lattice Regulator (DLR) is a framework that unifies lattice geometry, probability, and dynamics to achieve symmetry, stability, and precise control in various systems.
• The method rigorously models phase transitions and symmetry restoration in statistical mechanics and quantum field theory through local specification and measure-based analysis.
• DLR also enables data-driven feedback control via techniques like Koopman-based EDMD, LQR, and KMPC, yielding improved performance in vibration attenuation and trajectory tracking.

The Dynamical-Lattice Regulator (DLR) encompasses a family of mathematical and engineering frameworks based on the concept that lattice geometry, probability, and dynamics can be regulated, controlled, or dynamically evolved to achieve specific technical or physical objectives. In theoretical statistical mechanics and quantum field theory, DLR refers to a measure- or geometry-based regulatory framework for modeling infinite-volume systems, symmetry-restoring mechanisms, and phase transitions on lattices. In control engineering, DLR denotes a fully data-driven feedback methodology for active control of reconfigurable, mass-efficient digital lattice structures. Across disciplines, the DLR formalism provides rigorous structures—ranging from probabilistic specifications to geometry-integrated field-theory regulators to operator-inferred control architectures—that ensure mathematical consistency, symmetry, stability, and practical implementability.

1. DLR Measures, Specifications, and Transfer Operators

DLR measures are defined on lattice-based dynamical systems as probability or $\sigma$-finite measures satisfying local specification kernels subject to conditional consistency. In one-dimensional or countable Markov shift settings, DLR measures formalize the thermodynamic limit or Gibbsian formalism via a local prescription.

Given a (possibly infinite) alphabet $S$, a shift space $\Sigma_A$ defined by a matrix $A$ (topologically mixing, with no zero rows/columns), and a real-valued potential $\varphi:\Sigma_A \rightarrow \mathbb{R}$, the $n$th-variation quantifies the sensitivity of $\varphi$ under initial segment coincidence: $$\mathrm{Var}_n(\varphi) = \sup_{\substack{x_i = y_i\ (0 \leq i < n)}} |\varphi(x) - \varphi(y)|.$$ Potentials with summable variation or satisfying Walters condition facilitate a comprehensive theory of existence and uniqueness of DLR measures (Beltrán et al., 2020, Cioletti et al., 2014).

The measure $v$ on the Borel $\sigma$-algebra $\mathcal{B}$ is called a $\varphi$-DLR measure if, for any admissible word $a=a_0\cdots a_{n-1}$,

$$\mathbb{E}_v[1_{[a]} \mid \mathscr{F}_n](x) = \frac{\sum_{y:\, y_i=x_i\, (i \geq n),\, y_0\cdots y_{n-1}=a}\, e^{\varphi_n(y)}}{\sum_{y:\, y_i=x_i\, (i \geq n)}\, e^{\varphi_n(y)}}$$

holds $v$-almost everywhere.

A key result is that under Walters regularity, the set of DLR measures coincides with conformal (Ruelle-eigen) measures associated with the Ruelle operator $\mathscr{L}_\varphi$, and is unique in the thermodynamic limit (Cioletti et al., 2014).

2. DLR in Quantum Field Theory: Dynamical Geometry Regulators

In Euclidean quantum field theory, the Dynamical-Lattice Regulator is constructed by promoting the lattice embedding $x:\Lambda \to \mathbb{R}^d$ (with $\Lambda \simeq \mathbb{Z}^d$) to a dynamical field integrated over admissible configurations subject to shape-regularity (no collapsed cells, bounded edge lengths, uniformly conditioned frame matrices) (Gantumur, 26 Dec 2025).

The total action,

$S[x,U,\Phi] = S_x[x] + S_\text{fields}[x,U,\Phi],$

includes $S_x[x]$ (a local geometry penalty, e.g., spring terms) and $S_\text{fields}$ for matter/gauge fields, constructed to be local, gauge invariant, and dependent only on Euclidean invariants of the geometry (edge vectors, metrics, volumes).

The partition function,

$$Z = \int_{X_\text{adm}} D x \int D U \int D \Phi\, e^{ - S_x[x] - S_\text{fields}[x, U, \Phi] }$$

is exactly covariant under the global Euclidean group SE($d$), ensuring rotational and translational invariance at finite lattice spacing.

A short-range geometry hypothesis (SR) posits exponential decay of connected correlators of local geometric observables (correlation length $O(1)$ lattice units). The intended symmetry-restoration mechanism is local twisting, encoded in fluctuating local orientation fields $Q(n) \in SO(d)$ via the polar decomposition of frame matrices, generating short-range orientation-mixing and suppressing axis-diagonal cutoff artifacts.

This construction enables reflection positivity (via Osterwalder–Schrader) and produces a local Symanzik effective action (upon integrating out $x$ under SR), ensuring universality and recovery of standard continuum results (e.g., the $\phi^4$ one-loop $\beta$-function in $d=4$) (Gantumur, 26 Dec 2025).

3. Volume-Type Phase Transitions in DLR Frameworks

DLR formalism on infinite or countable Markov shifts yields a rigorous approach to describing volume-type phase transitions driven by the first variation of the potential. The renewal shift (specific $A$ matrix) provides an explicit setting, where the set of eigenmeasures jumps from finite to infinite depending on the temperature parameter $\beta$.

The critical inverse temperature

$$\beta_c = \sup\left\{ \beta > 0 :\, \limsup_{n \to \infty} \frac{\beta\, \varphi_n(I_n)}{n} < P_G(\beta\varphi) \right\},$$

with $I_n = (1, 2, \ldots, n)$ and $$\varphi_n(I_n) = \sum_{k=0}^{n-1} \varphi(\sigma^k x)$$, quantifies this transition. Potentials with finite first variation yield $\beta_c = \infty$ (no transition), while those with infinite first variation manifest a volume-type phase transition (Beltrán et al., 2020).

4. DLR-Based Feedback Control: Data-Driven Lattice Regulation

In control engineering, DLR refers to a data-driven pipeline for real-time feedback regulation of flexible digital lattice structures (Fischer et al., 2024). Key technical features include:

• Koopman-based Extended Dynamic Mode Decomposition (EDMD) for linear predictor identification from input-output data (${y_k}$, ${u_k}$).
• Synthesis of an LQR (Linear Quadratic Regulator) for stabilization and disturbance rejection, optimizing

$J = \sum_{k=0}^\infty (z_k^T Q z_k + u_k^T R u_k)$

with state-feedback law $u_k = -K z_k$.

• Koopman Model Predictive Control (KMPC) for reference trajectory tracking, solving a quadratic program each sampling instant subject to state/input constraints and reference tracking objectives.

The DLR hardware architecture leverages localized actuated/sensing voxels with brushless DC motor actuation and inertial (IMU) sensing, enabling closed-loop feedback with minimal resources.

Empirical results demonstrate significantly reduced vibration amplitudes (>50% attenuation), improved settling times (from 3 s to 1 s in experimental towers), and sub-degree tracking RMS error under torque bounds, validating the effectiveness of EDMD+LQR/KMPC control without explicit physics-based modeling (Fischer et al., 2024).

5. Practical Algorithms and Symmetry Restoration in Lattice QFT

In the lattice QFT context, DLR enables practical local Monte Carlo update schemes with $O(1)$ local cost per lattice site. For periodic $L \times L$ lattices, each sweep applies geometry and scalar field updates affecting only a local neighborhood and ensuring admissibility (shape-regularity) (Gantumur, 26 Dec 2025). This preserves exact gauge invariance and reflection positivity.

Empirical $d=2$ simulations reveal well-behaved geometric sector ensembles (edge length and angle distributions away from pathologies), exponential decay in geometry correlators (evidence for SR), and strong suppression of axis/diagonal anisotropy in observable statistics, confirming substantial cutoff artifact reduction via local twisting.

Universality is maintained, as critical cumulants and finite-size scaling curves coincide for baseline and DLR-regulated ensembles, ensuring the continuum universality class is unchanged.

6. Applications and Open Problems

DLR formalisms are pivotal in:

• The rigorous analysis of thermodynamic limits, equilibrium states, and specification-based probabilities in statistical mechanics and dynamical systems (Cioletti et al., 2014, Beltrán et al., 2020).
• Symmetry-restoring regulation of lattice QFTs, achieving exact Euclidean invariance and reflection positivity at finite cutoff, controlling irrelevant operators, and streamlining universality proofs (Gantumur, 26 Dec 2025).
• Embedded, data-driven control of digital material lattice structures, acquiring high-performance feedback strategies via operator-theoretic identification with minimal sensing/actuation (Fischer et al., 2024).

Open challenges include dictionary selection and dimensionality reduction for high-complexity models in EDMD-based control, and scaling dynamical-geometry regulators to large-scale 2D/3D lattices with multiple actuators and sensors while maintaining admissibility and local update cost. Further theoretical investigations are needed regarding universality beyond moderate cutoff regimes and higher-order corrections induced by geometry fluctuations.

7. Summary Table: DLR Manifestations

Aspect	QFT Regulator (Gantumur, 26 Dec 2025)	Statistical Mechanics (Beltrán et al., 2020, Cioletti et al., 2014)	Control Engineering (Fischer et al., 2024)
Lattice	Embedding $x(n)$ dynamical	Shift space $\Sigma_A$, Markov/renewal shifts	Beam/voxel network
Dynamics/Measure	Integration over $x,U,\Phi$	DLR/conformal $\sigma$-finite measures	Koopman predictor, LQR/KMPC
Symmetry	Exact SE(d) invariance	Invariant under shift/reparameterization	Task/trajectory invariance
Regulation Goal	Restore physical symmetry	Equilibrium state, phase transition	Stabilization/tracking
Implementation	Local Metropolis updates	Operator, kernel-based, limit constructions	Data-driven feedback pipeline

The Dynamical-Lattice Regulator is a rigorous tool for symmetry restoration, phase-transition analysis, probabilistic specification, and real-time control on spatial lattices, unifying key principles from mathematics, physics, and engineering.