Prime-Modulus Laplacian Cellular Automata

• Prime-modulus Laplacian cellular automata are discrete, linear dynamical systems defined on finite fields of prime order using convolution with the Moore neighborhood.
• They exhibit prolonged chaotic transients and Frobenius-driven replica revivals, where initial seeds reappear as exact, spatially separated copies.
• Their inherent reversible encoding and error-correcting mechanisms connect algebraic combinatorics with applications in information theory and cryptographic systems.

Prime-modulus Laplacian cellular automata (CA) are a class of discrete, linear dynamical systems defined over a finite field of prime order, distinguished by their algebraic structure and the presence of exact, spatially redundant pattern revivals at prescribed times. Their dynamics are governed by convolution with the Moore neighborhood and an identity term, resulting in unique phenomena such as extended chaotic transients, Frobenius-driven replica formation, and robust, reversible encoding capabilities. These automata connect algebraic combinatorics with cellular automaton dynamics and have implications for information theory, reversible computation, and structured randomness (Nowak-Kępczyk, 21 Nov 2025).

1. Formal Construction and Evolution Rule

Let $p$ be a prime and $\mathbb{F}_p$ the finite field of integers modulo $p$. The CA evolves on the two-dimensional lattice $\mathbb{Z}^2$, with the configuration at discrete time $t$ given by $u_t: \mathbb{Z}^2 \to \mathbb{F}_p$.

The update rule is defined using the Moore neighborhood: $$N(i, j) = \{(i+\delta_i, j+\delta_j) : \delta_i, \delta_j \in \{-1, 0, 1\}\}$$ excluding the center $(i, j)$. The discrete convolution operator $B$ is: $$(Bu)(i, j) = \sum_{(k, \ell) \in N(i, j) \setminus \{(i, j)\}} u(k, \ell), \quad \text{with all operations in } \mathbb{F}_p.$$ The evolution operator is then $T = I + B$, so the CA updates by: $u_{t+1} = T u_t = (I + B)u_t \pmod{p}.$ This linear rule repeatedly disperses and mixes the initial data, enabling algebraically controlled global dynamics that crucially depend on the field structure and the inclusion of the identity term (Nowak-Kępczyk, 21 Nov 2025).

2. Frobenius Identity and Replica Revival Phenomenon

A central feature of these automata is the collapse of seemingly chaotic patterns to exact multi-tile replicas of the initial seed at specific revival times. The underpinning mechanism is the Frobenius identity: $$(I+B)^{p^m} = I + B^{p^m} \quad \text{in } \mathbb{F}_p, \quad \forall m \ge 0.$$ This collapse is due to the fact that binomial coefficients $\binom{p^m}{k}$ vanish modulo $p$ for $0<k<p^m$, so only the $k=0$ and $k=p^m$ terms remain. For a finite “seed” $u_0$ supported within a square of size $N\times N$, the evolution at time $t=p^m$ obeys: $u_{p^m} = T^{p^m} u_0 = u_0 + B^{p^m} u_0.$ If $p^m \geq N$, the two summands have disjoint support, producing non-overlapping replicas. Continuing up to $p^{2m}$ time steps yields a tiling of $p^m$-shifted copies in each spatial direction, resulting in $p^{2m}$ total replicas. This deterministic phenomenon is termed "Frobenius-driven revival" (Nowak-Kępczyk, 21 Nov 2025).

3. Dynamical Properties: Chaos, Entropy, and Stability

Chaotic Transients and Entropy Dynamics

Despite linearity, the CA displays prolonged high-entropy “chaotic” stages between revivals, quantifiable via Shannon entropy: $$\rho_t^{(c)} = \frac{|\{ x \in r_t : u_t(x) = c \}|}{|r_t|},\quad c \in \mathbb{F}_p$$

$$H_t = -\sum_{c \in \mathbb{F}_p} \rho_t^{(c)} \log \rho_t^{(c)}$$

with $r_t$ the minimal bounding box of activity. Typically, $H_t$ quickly reaches a plateau near $\log p$ for $1 \ll t \ll p^m$, abruptly drops at each $t=p^k$ revival time, then returns to the plateau afterwards.

Spatial Organization and Correlation Structure

At revival times, the pattern consists of a union of disjoint shifted seeds, confirming the tiling effect. The spatial two-point correlation function,

$$C_t(r) = \frac{1}{|r_t|} \sum_{x \in r_t} u_t(x) u_t(x+r) - \left(\frac{1}{|r_t|} \sum_{x \in r_t} u_t(x)\right)^2,$$

is featureless during chaotic phases but shows periodic peaks consistent with replica spacing at revivals.

Local Confinement of Perturbations

Perturbations applied at time $s$ propagate inside a finite light-cone, affecting at most a region of radius $r(p^m-s)$, with $r=1$ for Moore neighborhood. If $r(p^m - s) < \frac{1}{2}p^m$, deviations remain confined to a single replica at the next revival, enabling spatially localized error correction by majority consensus among tiles (Nowak-Kępczyk, 21 Nov 2025).

4. Multi-Prime Compositions: Prolonged Periodicity and Reversibility

Significant extension of the orbit period and increased cryptographic complexity arise when composing Laplacian operators modulo multiple distinct primes $p_1 < p_2 < \cdots < p_m$. Given revival times $T_{p_i} = p_i^{m_i}$ and offsets $x_i$, the composite evolution is: $$\mathcal{C} = L_{p_1}^{T_{p_1} - x_1} L_{p_2}^{T_{p_2} - x_2} \cdots L_{p_m}^{T_{p_m} - x_m} \;\times\; L_{p_m}^{x_m} \cdots L_{p_2}^{x_2} L_{p_1}^{x_1}$$ where $L_{p_i}$ is the Laplacian CA over $\mathbb{F}_{p_i}$.

The exact period is

$$T_{\rm global} = \operatorname{lcm}(T_{p_1}, T_{p_2}, \ldots, T_{p_m}),$$

with full reversibility in the noiseless regime. Non-commutativity of operators implies that both the sequence of primes and the offsets act as secret parameters, relevant for information-hiding applications (Nowak-Kępczyk, 21 Nov 2025).

5. Reversible Encoding Scheme and Error Correction

The replica structure inherent in revival phases enables a robust, explicit reversible encoding/decoding protocol:

• Encoding: From a chosen initial seed $u_0$, apply composite CA evolution up to a time $s < T_{\rm global}$; transmit the high-entropy configuration $u_s$.
• Decoding: Apply the remaining steps in reverse order, yielding a spatial tiling of $M$ disjoint $N \times N$ replicas of $u_0$.
• Reconstruction: Extract replicas and recover $u_0$ via pixelwise majority voting,

$$\hat{u}_0(x) = \operatorname{mode}_{1 \leq i \leq M} u^{(i)}(x).$$

This construction affords strong noise tolerance. If each pixel at each step is independently corrupted at rate $p_{\rm noise}$, the per-replica marginal error rate $q$ yields a final error after voting as

$$q_{\rm maj}(M) = \sum_{k = \lceil (M+1)/2 \rceil}^{M} \binom{M}{k}\,q^k (1-q)^{M-k}$$

decaying exponentially in $M$ (the replica number). Empirical studies for binary and ternary rules with $M \lesssim 100$ confirm error rates tolerable up to $p_{\rm noise} \approx 10^{-5}$–$10^{-4}$ per step (Nowak-Kępczyk, 21 Nov 2025).

Temporal redundancy is possible: by choosing multiple encoding times $S = \{s_1, \dots, s_L\}$, and performing consensus voting over independent reconstructions, one achieves further robustness.

6. Principal Results and Implications

The key algebraic identities and dynamical lemmas characterizing prime-modulus Laplacian CA are:

• Definition (Laplacian CA):

$u_{t+1} = (I+B)u_t \pmod p$

• Frobenius Identity:

$(I+B)^{p^m} = I + B^{p^m} \text{ in }\mathbb{F}_p$

• Seed Revival Mechanism:

$T^{p^m}u_0 = u_0 + B^{p^m}u_0$

• Global Periodicity under Multi-Prime Composition:

$$T_{\rm global} = \operatorname{lcm}(T_{p_1}, \ldots, T_{p_m})$$

• Spatial Confinement of Perturbations: Local modifications are constrained to non-overlapping tiles if the propagation bound

$r(p^m - s) < \tfrac{1}{2}p^m$

is satisfied at revival.

These properties yield CA orbits characterized by lengthy, entropy-maximizing transients, deterministic multi-revival patterns, intrinsic spatial and temporal data redundancy, and robust error correction. Applications include reversible steganography, pseudorandom pattern generation, and error-tolerant information representation, exploiting the superposition of chaos and order engineered by the algebraic structure of the evolution (Nowak-Kępczyk, 21 Nov 2025).