Discrete Stochastic Cascade Dynamics on Finite Graphs

Part I: Construction, Guarantees, and Phase Structure

Shiv Goswami, Terraflock Pvt Ltd, Bihar, India, June 2026

Abstract

We construct a new class of discrete stochastic dynamical systems on finite graphs that generate rich, transient structure from irreversible resource consumption alone. Two fundamentally asymmetric excitation species, distinguished by intrinsic frequency and self-interaction rules, interact stochastically while drawing from a strictly non-renewable local energy field. The cross-species interaction efficiency is modulated by a phase-interference factor $\Theta(n) \in [\vartheta,\,1]$ with beat period $2\pi/|\omega_X - \omega_Y|$ , so that frequency mismatch directly controls the dynamics; same-species channels are always phase-aligned and unmodulated. Four coupled state variables (excitation counts, capacity energy, structural state, and a ripple measure) evolve through layered update rules producing three dynamical regimes: quiescent, dissipative leakage, and explosive pair creation.

We establish three rigorous guarantees. Energy Monotonicity: the total capacity energy is a non-negative supermartingale that decreases strictly in expectation whenever the system is active. Finite Activity Bound: the cumulative number of interactions across all space and time is almost surely finite. Almost Sure Absorption: the system reaches a frozen absorbing configuration in finite time with probability one, without requiring fine-tuning to a critical point.

Keywords: Interacting particle systems · Absorbing-state phase transitions · Stochastic cascades · Irreversible dynamics · Discrete dynamical systems · Landau phase transition

1Introduction

1.1 Motivation

The standard mathematical description of motion rests on a chain of continuum assumptions: space is $\mathbb{R}^d$ , time is $\mathbb{R}$ , trajectories are smooth, and evolution is governed by differential equations. This framework has been extraordinarily successful, yet it carries a foundational dependence on the completed infinite.

This work asks: can we build dynamics that do not require infinite subdivision? The approach replaces smooth trajectories with discrete evolution rules, fields on $\mathbb{R}^d$ with states on finite graphs, and conservation laws with irreversible resource budgets. The goal is a self-contained dynamical framework where:

Space is a finite graph with no notion of subdivision
Evolution proceeds in discrete steps with no limiting process
Two asymmetric excitation species interact stochastically
A strictly non-renewable capacity energy imposes irreversibility
Threshold-driven regime switching generates qualitative diversity from quantitative variation

The result is a minimal system that generates rich transient dynamics, including spatial structure, self-amplifying cascades, and metastable bonded configurations, yet is guaranteed to freeze into an absorbing state in finite time.

This work sits at the intersection of several classical threads, each of which it extends in a specific direction:

Interacting particle systems (contact process, voter model): multi-type extensions exist but retain infinite-time, stationary-measure analysis with symmetric interaction kernels. This model imposes a structurally forbidden channel ( $\alpha_{YY} = 0$ ) and a finite non-renewable energy field coupled to all transitions.
Absorbing-state phase transitions / Directed percolation: absorption requires tuning a control parameter below a critical value. This model guarantees absorption for all parameter values, because the driving resource is finite and strictly consumed.
Self-organised criticality / Sandpile models: require slow external driving. This model has no external drive; the transient complexity peak is inherently non-stationary and terminal.
Reaction-diffusion annihilation $(A+B \to \varnothing)$ : produces no persistent structure. This model adds energy budgets, threshold-driven pair creation, and irreversible structural accumulation.

1.3 Contributions

A new dynamical system class. Fully discrete, finite-resource stochastic system on arbitrary finite graphs, coupling four state variables through three dynamical regimes. No continuum limit, no external drive, no infinite-resource assumption.
Three rigorous guarantees. Energy decreases monotonically, total activity is almost surely finite, and the system absorbs in finite time with probability one.
Embedded phase transition. A Landau free-energy functional governs bond formation, embedding a local order-disorder transition within the cascade dynamics.
Phase diagram and transient complexity. Three principal dynamical regimes identified. Structural diversity, initially zero, becomes strictly positive in the cascade regime and eventually freezes; simulation consistently shows it passes through a peak before levelling off.

2Model Definition

2.1 Graph Topology

Definition 2.1 (Spatial substrate)

Let $\mathcal{G} = (\mathcal{V}, \mathcal{A})$ be a finite, connected, undirected graph with vertex set $\mathcal{V}$ , $|\mathcal{V}| = P < \infty$ , and adjacency relation $\mathcal{A} \subseteq \mathcal{V} \times \mathcal{V}$ . For each vertex $p \in \mathcal{V}$ the neighbourhood is $\mathcal{N}(p) = \{q \in \mathcal{V} : (p,q) \in \mathcal{A}\}$ , with degree $\deg(p) = |\mathcal{N}(p)|$ .

The graph $\mathcal{G}$ may be a regular lattice, a random graph, or any finite connected graph. All results hold for arbitrary $\mathcal{G}$ .

2.2 State Variables

Definition 2.3 (Local state)

The state at vertex $p$ and time $n$ is the tuple:

\sigma(p,n) = \bigl(X(p,n),\; Y(p,n),\; \mathcal{E}(p,n),\; S(p,n)\bigr)

$X(p,n) \in \mathbb{N}_0$ : count of type-X excitations (species 1), intrinsic frequency $\omega_X > 0$
$Y(p,n) \in \mathbb{N}_0$ : count of type-Y excitations (species 2), intrinsic frequency $\omega_Y > 0$ , $\omega_Y \ne \omega_X$
$\mathcal{E}(p,n) \in \mathbb{R}_{\ge 0}$ : local capacity energy
$S(p,n) \in \mathbb{R}$ : structural state, encoding accumulated interaction history and bonded configurations

The two species are fundamentally asymmetric: different intrinsic frequencies, different self-interaction rules, and not interchangeable under any symmetry of the model.

The total capacity energy is defined as $\mathcal{E}_{\mathrm{tot}}(n) = \sum_{p \in \mathcal{V}} \mathcal{E}(p,n)$ .

2.3 Parameters

Symbol	Name	Constraint
`ω_X, ω_Y`	Intrinsic frequencies	`ω_X, ω_Y > 0, ω_X ≠ ω_Y`
`ϑ`	Coherence floor (phase-interference)	`ϑ ∈ (0, 1]`
`α_XY`	Cross-interaction coefficient	`α_XY > 0`
`α_XX`	X self-interaction coefficient	`α_XX ≥ 0`
`k_XY, k_XX`	Energy cost per interaction	`> 0`
`C`	Ripple threshold	`C > 0`
`Δ`	Overshoot margin	`Δ > 0`
`λ`	Leakage rate	`λ > 0`
`D_X, D_Y`	Diffusion coefficients	`[0, 1/deg_max)`
`γ₁, γ₂, γ_XX`	Structural coupling	`> 0`
`a₀, b`	Free-energy coefficients	`a₀, b > 0`
`T_c`	Critical temperature for bonding	`T_c > 0`
`κ, η`	Bond/explosion energy costs	`> 0`

2.4 Interaction Dynamics

Definition 2.3 (Interaction rules and asymmetry)

Three interaction channels operate at each vertex $p$ and time step $n$ :

(i) Cross-interaction (X–Y). Expected rate $R_{XY}(p,n) = \alpha_{XY}\,\omega_X\,\omega_Y\,\Theta(n)\, X(p,n)\, Y(p,n)$ , where the phase-interference factor

\Theta(n) = \vartheta + (1-\vartheta)\,\frac{1 + \cos((\omega_X - \omega_Y)\,n)}{2} \in [\vartheta,\,1]

modulates cross-channel efficiency with beat period $2\pi/|\omega_X - \omega_Y|$ ; the coherence floor $\vartheta > 0$ ensures the channel never closes entirely. Realised count $N_{XY}(p,n) \sim \operatorname{Poi}(R_{XY}(p,n))$ , capped at $\min\{X,Y\}$ . Each cross-interaction annihilates one X and one Y.

(ii) Self-interaction (X–X). Expected rate $R_{XX}(p,n) = \alpha_{XX}\,\omega_X^2\,\frac{X(p,n)(X(p,n)-1)}{2}$ . Same-species pairs are always phase-aligned; no interference factor. Each self-interaction annihilates two X excitations.

(iii) Y–Y interaction: forbidden. $N_{YY}(p,n) = 0$ for all $p, n$ . This is a constitutive property, not a consequence of dynamics.

2.5 Energy Depletion

Every interaction irreversibly consumes capacity energy:

\mathcal{C}_{\mathrm{int}}(p,n) = k_{XY} \cdot N_{XY}(p,n) + k_{XX} \cdot N_{XX}(p,n)

Energy is consumed by interactions, not released. The capacity field $\mathcal{E}$ acts as a finite fuel reserve that is irreversibly spent.

2.6 Ripple Measure

Definition 2.9 (Ripple intensity)

For $n \ge 2$ , the ripple intensity at vertex $p$ is the discrete second-order temporal variation of the structural state:

F(p,n) = \bigl|S(p,n) - 2\,S(p,n{-}1) + S(p,n{-}2)\bigr| = |\Delta^2 S(p,n)|

Equivalently, $F$ is the discrete second difference of $S$ in time, a proxy for local dynamical volatility.

2.7 Regime Classification

Definition 2.10 (Dynamical regimes)

At vertex $p$ , time $n$ , the ripple intensity classifies into one of three regimes:

Quiescent: $F(p,n) \le C$ . No leakage, no explosion.
Leakage: $C < F(p,n) < C + \Delta$ . Slow dissipation: $L(p,n) = \lambda\,F(p,n)$ .
Explosive: $F(p,n) \ge C + \Delta$ . An explosion event creates $m(p,n) = \lfloor(F(p,n) - C)/\Delta\rfloor$ new XY pairs at energy cost $M(p,n) = \eta \cdot m(p,n)$ .

2.8 Bond Formation via Free-Energy Minimisation

Bond formation is modelled as a local phase transition using a Landau free-energy functional. The bond order parameter is:

\psi(p,n) = \sqrt{X(p,n) \cdot Y(p,n)}

The effective temperature (normalised ripple intensity) is $T_{\mathrm{eff}}(p,n) = F(p,n)/C$ . High ripple = high temperature (disordered); low ripple = low temperature (ordered).

Definition 2.16 (Landau free energy)

\mathcal{F}(\psi) = a_0\bigl(T_{\mathrm{eff}} - T_c\bigr)\,\psi^2 + b\,\psi^4

Bond formation is thermodynamically favourable when $\partial\mathcal{F}/\partial\psi < 0$ , i.e. when:

T_{\mathrm{eff}}(p,n) < T_c - \frac{2b}{a_0}\,\psi^2(p,n)

This requires sufficiently low ripple (ordered local state) and sufficiently high co-density.

The quartic term $b\,\psi^4$ prevents unbounded bond formation. The condition $T_{\mathrm{eff}} < T_c$ defines a Landau-type order-disorder transition at the local level.

2.9 Complete Update Rule

The full time-step $n \to n+1$ proceeds in deterministic order:

Interaction: sample $N_{XY}, N_{XX}$ for all vertices, annihilate excitations
Ripple computation: compute $F(p,n)$ for all $p$
Regime classification: apply leakage and explosion rules
Bond formation: compute $\psi, T_{\mathrm{eff}}, \mathcal{F}$ and form bonds
Energy update
Structural update: $S(p,n{+}1) = S(p,n) + \gamma_1 N_{XY} + \gamma_{XX} N_{XX} + \gamma_2 B$
Diffusion: nearest-neighbour excitation diffusion

Definition 2.19 (Energy update)

\mathcal{E}(p,n{+}1) = \mathcal{E}(p,n) - k_{XY} N_{XY} - k_{XX} N_{XX} - L(p,n) - M(p,n) - \kappa B(p,n)

clamped to $\max(\cdot,\, 0)$ . Summing over all vertices: $\mathcal{E}_{\mathrm{tot}}(n+1) \le \mathcal{E}_{\mathrm{tot}}(n)$ .

3Main Results

3.1 Well-Posedness

Proposition 3.1 (Well-posedness)

For any initial configuration with $X(p,0), Y(p,0) \in \mathbb{N}_0$ , $\mathcal{E}(p,0) \ge 0$ , and $S(p,0) \in \mathbb{R}$ for all $p \in \mathcal{V}$ , the update rule preserves these domains for all $n \ge 0$ .

3.2 Energy Monotonicity

Theorem 3.3 (Energy Monotonicity)

The total capacity energy $\mathcal{E}_{\mathrm{tot}}(n)$ satisfies:

$\mathcal{E}_{\mathrm{tot}}(n{+}1) = \mathcal{E}_{\mathrm{tot}}(n) - \mathcal{D}(n)$ for all $n \ge 0$
$\mathcal{D}(n) \ge 0$ almost surely
$\mathbb{E}[\mathcal{D}(n) \mid \mathcal{F}_n] > 0$ whenever the system is not absorbing

Consequently, $(\mathcal{E}_{\mathrm{tot}}(n))_{n \ge 0}$ is a non-negative supermartingale. This provides an intrinsic thermodynamic arrow: energy never increases.

3.3 Finite Activity Bound

Theorem 3.4 (Finite Activity Bound)

The total number of interactions, leakage events, explosion events, and bond formations across all space and time is almost surely finite. Specifically:

\sum_{n=0}^{\infty} \sum_{p \in \mathcal{V}} N_{XY}(p,n) \;\le\; \frac{\mathcal{E}_{\mathrm{tot}}(0)}{k_{XY}} \qquad \text{a.s.}

More generally:

\sum_{n=0}^{\infty} \mathcal{D}(n) \le \mathcal{E}_{\mathrm{tot}}(0) \qquad \text{a.s.}

The system generates at most $\lfloor \mathcal{E}_{\mathrm{tot}}(0) / \min(k_{XY}, k_{XX}, \eta, \kappa) \rfloor$ total events across its entire history.

3.4 Almost Sure Absorption

Theorem 3.6 (Almost Sure Absorption)

There exists an almost surely finite random time $\tau < \infty$ such that $\Sigma(n)$ is absorbing for all $n \ge \tau$ :

\mathbb{P}\bigl(\exists\, \tau < \infty : \Sigma(n) \text{ is absorbing } \forall\, n \ge \tau\bigr) = 1

This holds for all parameter values, without fine-tuning to a critical point. The proof proceeds by showing that if the system were non-absorbing for infinitely many steps, the total depletion $\sum_n \mathcal{D}(n)$ would be infinite, contradicting the Finite Activity Bound.

Remark

At absorption, the frozen structural state $S(p,\tau)$ encodes the cumulative history of all interactions and bonds, a unique stochastic record of the system's entire biography, imprinted when the energy ran out.

3.5 Convergence Rate

Proposition 3.9 (Expected absorption time)

Let $k_{\min} = \min(k_{XY}, k_{XX}, \lambda C, \eta, \kappa)$ and let $\delta$ be the minimum expected depletion rate while active. Then:

\mathbb{E}[\tau] \;\le\; \frac{\mathcal{E}_{\mathrm{tot}}(0)}{\delta}

4Phase Structure and Transient Complexity

4.1 Phase Diagram

The qualitative behaviour depends on two dimensionless control parameters: the initial co-density $\rho_0 = \frac{1}{P}\sum_p X(p,0)Y(p,0)$ and the initial energy density $\varepsilon_0 = \mathcal{E}_{\mathrm{tot}}(0)/P$ .

Theorem 4.2 (Phase classification)

The model exhibits three qualitative phases:

Immediate absorption ( $\varepsilon_0 < k_{\min}$ or $\rho_0 = 0$ ). Insufficient energy or excitations. System absorbs within $O(1)$ steps.
Dissipative decay ( $\varepsilon_0 \ge k_{\min}$ , $\rho_0 > 0$ , initial $F < C + \Delta$ everywhere). Interactions occur but no explosions trigger. Excitation density decays monotonically.
Cascade regime ( $\varepsilon_0 \gg k_{\min}$ , sufficient $\rho_0$ ). Explosions create new excitation pairs, sustaining and amplifying activity in transient bursts. The system passes through a complexity peak before eventual absorption.

Figure 1 — Conceptual flow of cascade dynamics

Energy fuels interactions, which build structure. Structure drives the ripple measure, which governs regime switching and bond formation. Bonds consume energy, creating a feedback loop that terminates at absorption.

4.2 Transient Complexity Peak

Definition

The structural complexity is the spatial variance of the structural state:

\mathcal{K}(n) = \frac{1}{P} \sum_{p \in \mathcal{V}} \left(S(p,n) - \bar{S}(n)\right)^2

Proposition 4.4 (Complexity peak)

In the cascade regime, the structural complexity $\mathcal{K}(n)$ satisfies:

$\mathcal{K}(0) = 0$ if $S(p,0) = 0$ for all $p$
There exists $n^* > 0$ such that $\mathcal{K}(n^*) > 0$
$\mathcal{K}(n)$ is eventually constant, frozen at $\mathcal{K}(\tau)$

Consequently, $\mathcal{K}$ has a non-trivial transient profile: it first increases, may fluctuate, and ultimately freezes. The complexity peak is a transient phenomenon bounded by the initial energy endowment.

5Emergent Properties and Interpretation

5.1 Layered Architecture

The model decomposes into five conceptual layers, each emergent from the one below:

Micro-field layer. Stochastic X–Y and X–X interactions at individual vertices. The fundamental asymmetry ( $\alpha_{YY}=0$ ) and distinct frequencies $\omega_X \ne \omega_Y$ are the irreducible source of all dynamics.
Energy layer. Capacity energy $\mathcal{E}$ consumed per interaction. Provides the irreversible resource constraint.
Instability control layer. The ripple measure $F$ and threshold $C$ classify dynamics into quiescent, dissipative, and explosive regimes.
Structural emergence layer. Bond formation and structural state $S$ create persistent, spatially heterogeneous configurations: the “fossils” of past activity.
Thermodynamic arrow layer. The monotonic decrease of $\mathcal{E}_{\mathrm{tot}}$ imposes a global directionality. No time-reversal symmetry exists.

5.2 Structural Hierarchy

Level	Entity	Characteristics
0	Free excitation (X or Y)	Single species, mobile, reactive
1	Co-located pair (X + Y at same p)	Can interact, annihilate, or bond
2	Bond (X–Y composite in S)	Metastable, reduced ℰ, obeys same rules
3	Clump (cluster of bonds on 𝒢)	Spatially extended, collective ripple

Each level emerges from the dynamics of the level below. The hierarchy is transient: clumps form, persist while local energy supports them, and ultimately freeze when the system absorbs. The frozen final state is not optimised, not symmetric, and not predictable from initial conditions alone.

5.3 Thermodynamic Arrow

Corollary 5.1 (Irreversibility)

The dynamics are time-irreversible. For any non-absorbing state $\Sigma$ :

\mathbb{P}\bigl(\mathcal{E}_{\mathrm{tot}}(n{+}1) > \mathcal{E}_{\mathrm{tot}}(n)\bigr) = 0 \qquad \forall\; n \ge 0

The system possesses a strict thermodynamic arrow without invoking entropy or ergodic assumptions.

5.4 Stochastic Memory

The final frozen structural state $S(p,\tau)$ depends on the entire realisation of the stochastic process. Different runs from the same initial conditions produce generically different final configurations. This “stochastic memory” is a permanent record of the system's particular history, a discrete analogue of symmetry breaking in equilibrium phase transitions, except irreversible and path-dependent.

6Discussion

6.1 Comparison with Existing Frameworks

Feature	Contact Process	Sandpile	A+B Annihilation	Multi-type IPS	This Model
Multi-species dynamics	—	—	✓	✓	✓
Species asymmetry (forbidden channel)	—	—	—	partial	✓
Finite, non-renewable energy	—	—	—	—	✓
Guaranteed absorption (no tuning)	tuned	—	✓	tuned	✓
Structural memory (S field)	—	—	—	—	✓
Threshold-driven regime switching	—	—	—	—	✓
Landau bond formation	—	—	—	—	✓
Structural hierarchy depth	1	1	1	1–2	4

6.2 Open Problems

Does the absorbing-state transition exhibit directed percolation universality, or does the energy constraint define a new universality class?
What is the distribution of cascade sizes? Preliminary analysis suggests a truncated power law.
How does the peak structural complexity $\max_n \mathcal{K}(n)$ scale with graph size $P$ and energy density $\varepsilon_0$ ?
Can the model be extended to $\mathbb{Z}^d$ with appropriate density constraints?
Under appropriate rescaling, does the model converge to a stochastic PDE?
What is the expected lifetime of a bonded configuration before disruption by a nearby explosion?
What happens in the symmetric limit $\alpha_{XX} = \alpha_{YY}$ , $\omega_X = \omega_Y$ ? (Resolved in Part II.)

6.3 Scope and Limits

No continuum claim. The model is defined entirely on a finite graph with discrete state variables. No convergence to a PDE is claimed.
No physical identification. Simulations exhibit dynamics qualitatively reminiscent of expansion-cooling and cascade phenomena. These are observations about the dynamical system, not claims of physical equivalence.
Finite graphs only. All results assume $|\mathcal{V}| < \infty$ . Finiteness of the graph combined with finiteness of energy drives the central guarantees.
Landau functional as decision rule. The Landau free energy governs bond formation as a static threshold criterion, not as a dynamical equation of motion.

7Conclusion

We have introduced a discrete stochastic cascade model on finite graphs coupling four state variables through layered update rules with three dynamical regimes. The model is fully specified by twenty parameters and a choice of graph topology.

The central structural result is that the total capacity energy is a strict Lyapunov function, a non-negative supermartingale, for the dynamics. This single invariant implies finite total activity, almost sure absorption, and a strict thermodynamic arrow, all as rigorous consequences, not assumptions.

The framework provides a minimal, internally consistent setting for studying the interplay between stochastic interactions, irreversible resource depletion, threshold-driven instabilities, and emergent structure. It raises several open questions regarding universality, scaling, and continuum limits that merit further investigation.

Remark

More broadly, the results suggest that discrete asymmetry combined with irreversible resource constraints may be a general mechanism for producing transient structural complexity in stochastic systems. Whether analogous hierarchy-forming dynamics arise in ecological competition, distributed computing with finite energy budgets, or catalytic chemistry with irreversible substrate depletion is an open and fertile direction.

AAppendix

A.1 Simulation Pseudocode

The update rule proceeds as follows at each vertex $p$ at time step $n$ :

// LocalVertex(p, n)

Θ ← ϑ + (1−ϑ)·(1 + cos((ω_X−ω_Y)·n)) / 2

R_XY ← α_XY · ω_X · ω_Y · Θ · X(p) · Y(p)

N_XY ← min(Poi(R_XY), X(p), Y(p))

X(p) ← X(p) − N_XY ; Y(p) ← Y(p) − N_XY

R_XX ← α_XX · ω_X² · X(p)(X(p)−1)/2

N_XX ← min(Poi(R_XX), ⌊X(p)/2⌋)

X(p) ← X(p) − 2·N_XX

F ← |S(p,n) − 2·S(p,n−1) + S(p,n−2)| // ripple

if F ≥ C + Δ: // explosive regime

m ← ⌊(F − C)/Δ⌋ ; M ← η·m

if M > ℰ(p): m ← ⌊ℰ(p)/η⌋ ; M ← η·m

X(p) ← X(p)+m ; Y(p) ← Y(p)+m

ψ ← √(X(p)·Y(p)) ; T_eff ← F/C

∂F/∂ψ ← 2·a₀·(T_eff−T_c)·ψ + 4·b·ψ³

if ∂F/∂ψ < 0 and ψ > 0: // bond formation

B ← min(⌊ψ²·|∂F/∂ψ|/b⌋, ⌊ℰ(p)/κ⌋, X(p), Y(p))

X(p)−=B ; Y(p)−=B

cost ← k_XY·N_XY + k_XX·N_XX + L + M + κ·B

ℰ(p) ← max(ℰ(p) − cost, 0)

S(p) ← S(p) + γ₁·N_XY + γ_XX·N_XX + γ₂·B

A.2 Parameter Sensitivity

Ratio	Controls	Effect of increasing
`α_XY·ω_X·ω_Y·ρ₀ / C`	Cascade onset	Easier to trigger explosions
`α_XX / α_XY`	Self- vs. cross-interaction	More X depletion, fewer bonds
`ω_X / ω_Y`	Frequency asymmetry	Stronger species differentiation
`ℰ_tot(0) / (k_XY·P·ρ₀)`	System lifetime	Longer active phase
`η / k_XY`	Explosion cost ratio	Fewer but costlier explosions
`T_c`	Bond formation ease	More bonds at higher T_c

A.3 Key Equations Summary

Interaction rates: $R_{XY} = \alpha_{XY}\omega_X\omega_Y\,\Theta(n)\,XY$ where $\Theta(n) = \vartheta + (1-\vartheta)\frac{1+\cos((\omega_X-\omega_Y)n)}{2}$ ; $R_{XX} = \alpha_{XX}\omega_X^2 X(X-1)/2$ ; Y–Y forbidden.
Ripple: $F(p,n) = |S(p,n) - 2S(p,n-1) + S(p,n-2)|$
Energy update: $\mathcal{E}(p,n+1) = \mathcal{E}(p,n) - k_{XY}N_{XY} - k_{XX}N_{XX} - L - M - \kappa B$ , clamped to $\ge 0$
Structural update: $S(p,n+1) = S(p,n) + \gamma_1 N_{XY} + \gamma_{XX} N_{XX} + \gamma_2 B$
Global invariant: $\mathcal{E}_{\mathrm{tot}}(n+1) = \mathcal{E}_{\mathrm{tot}}(n) - \mathcal{D}(n)$ , $\mathcal{D}(n) \ge 0$

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