Discrete Stochastic Cascade Dynamics on Finite Graphs

Part II: Necessity of Species Asymmetry for Structural Hierarchy

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

Abstract

The companion paper established a discrete stochastic cascade model with guaranteed termination, emergent structure, and a four-level hierarchy. A foundational question remained open: is the asymmetry between excitation species a necessary condition for this hierarchy, or an incidental modelling choice?

We prove that asymmetry is necessary, within the class of models defined in Part I, in a precise, theorem-level sense. Symmetric Collapse: when both species share identical self-interaction coefficients and frequencies, the two-species system degenerates into a single-species annihilation process, and the ordered bonding phase ceases to exist. Differential Depletion: the forbidden Y–Y self-interaction channel is independently necessary for bond formation; it creates an asymmetric consumption rate that concentrates the scarcer species at bond sites. Beat Frequency: the frequency mismatch $\omega_X \ne \omega_Y$ is independently necessary for sustained cascade dynamics; it drives the phase-interference factor $\Theta(n)$ to oscillate with beat period $2\pi/|\omega_X - \omega_Y|$ , generating a recurrent deterministic drive with explicit amplitude $A_{\mathrm{beat}} = (1-\vartheta)\sin(|\Delta\omega|/2)\,\gamma_1\alpha_{XY}\omega_X\omega_Y\bar{X}\bar{Y}$ ; at equal frequencies this drive vanishes and only decaying transients remain.

We introduce an asymmetry index $\mathcal{A}$ that reduces the combined deviation from species symmetry to a single scalar, prove a sharp critical threshold $\mathcal{A}_c$ below which no bonds form, and establish a monotone bound on hierarchy depth as a function of $\mathcal{A}$ .

Keywords: Species asymmetry · Symmetry breaking · Structural hierarchy · Phase transitions · Interacting particle systems · Necessity proofs

1Introduction

1.1 Context

Part I introduced a discrete stochastic cascade model with two asymmetric excitation species X and Y, a non-renewable capacity energy, threshold-driven regime switching, and Landau free-energy bond formation. The model was shown to possess three rigorous properties: energy monotonicity, finite total activity, and almost sure absorption.

Part I identified seven open problems. The seventh asked: what happens in the symmetric limit $\alpha_{XX} = \alpha_{YY}$ , $\omega_X = \omega_Y$ ? Does the structural hierarchy collapse? Are the forbidden Y–Y channel and frequency mismatch constitutive assumptions or relaxable choices?

This paper resolves that question completely. The answer is: yes, the hierarchy collapses, and both asymmetries are independently necessary within this model class.

1.2 Summary of Results

Symmetric Collapse (Theorem 3.2): When both species have identical self-interaction coefficients and identical intrinsic frequencies, the two-species model reduces to an effective single-species annihilation process. The Landau free energy degenerates and the ordered phase ceases to exist.
Differential Depletion (Theorem 4.2): The forbidden Y–Y channel ( $\alpha_{YY} = 0$ ) creates an asymmetric depletion rate: X is consumed faster than Y. This differential drives minority-species condensation at bond sites, the mechanism the Landau functional requires.
Beat Frequency (Theorem 5.3): The frequency mismatch $\omega_X \ne \omega_Y$ drives the phase-interference factor $\Theta(n)$ to oscillate with beat period $2\pi/|\omega_X - \omega_Y|$ , generating a recurrent deterministic ripple drive. At equal frequencies, $\Theta \equiv 1$ and only decaying transients remain: no sustained cascades.

Remark

All necessity results in this paper are proved for the specific model class defined in Part I. We show that within this class, both asymmetries are required for structural hierarchy; we do not claim that no alternative model could achieve similar structure by other mechanisms.

2Setup and Notation

All definitions, parameters, and notation are adopted from Part I. The system lives on a finite connected graph $\mathcal{G} = (\mathcal{V}, \mathcal{A})$ , $|\mathcal{V}| = P$ . Each vertex carries state $(X, Y, \mathcal{E}, S)$ . Update proceeds as: interaction → ripple → regime → bonds → energy → structure → diffusion.

2.1 Interaction Rates

R_{XY}(p,n) = \alpha_{XY}\,\omega_X\,\omega_Y\,\Theta(n)\,X(p,n)\,Y(p,n)

where $\Theta(n) = \vartheta + (1-\vartheta)\tfrac{1+\cos((\omega_X-\omega_Y)n)}{2} \in [\vartheta,\,1]$ is the phase-interference factor with coherence floor $\vartheta > 0$ . Same-species channels are always phase-aligned and unmodulated:

R_{XX}(p,n) = \alpha_{XX}\,\omega_X^2\,\frac{X(p,n)(X(p,n)-1)}{2}

R_{YY}(p,n) = \alpha_{YY}\,\omega_Y^2\,\frac{Y(p,n)(Y(p,n)-1)}{2}

In Part I, $\alpha_{YY} = 0$ (forbidden Y–Y channel). In this paper we consider the general case $\alpha_{YY} \ge 0$ to study what happens when the constraint is relaxed.

2.2 Depletion Rates

Definition 2.1 (Per-species expected depletion rate)

\mu_X(p,n) = \mathbb{E}[N_{XY}(p,n)] + 2\,\mathbb{E}[N_{XX}(p,n)] + \mathbb{E}[B(p,n)]

\mu_Y(p,n) = \mathbb{E}[N_{XY}(p,n)] + 2\,\mathbb{E}[N_{YY}(p,n)] + \mathbb{E}[B(p,n)]

Definition 2.2 (Depletion asymmetry)

\delta\mu(p,n) = \mu_X(p,n) - \mu_Y(p,n) = 2\,\mathbb{E}[N_{XX}(p,n)] - 2\,\mathbb{E}[N_{YY}(p,n)]

When $\alpha_{YY} = 0$ (Part I model), $\delta\mu = 2\,\mathbb{E}[N_{XX}] \ge 0$ : X always depletes at least as fast as Y.

3The Symmetric Limit

Definition 3.1 (Symmetric model)

The symmetric model is obtained by setting:

\alpha_{XX} = \alpha_{YY} = \alpha_s, \qquad \omega_X = \omega_Y = \omega, \qquad \gamma_{XX} = \gamma_{YY} = \gamma_s

3.1 Symmetric Collapse Theorem

Theorem 3.2 (Symmetric Collapse)

Under the symmetric conditions with symmetric initial data ( $X(p,0) \overset{d}{=} Y(p,0)$ for all $p$ ), the following hold:

(a) Species indistinguishability. The joint process $(X(p,n), Y(p,n))$ is exchangeable: $(X(p,n), Y(p,n)) \overset{d}{=} (Y(p,n), X(p,n))$ for all $p, n$ .
(b) Single-species reduction. The total excitation count $Z(p,n) = X(p,n) + Y(p,n)$ evolves as a single-species annihilation-diffusion process with effective rate:
$R_Z(p,n) = \frac{\alpha_{XY}\,\omega^2 + \alpha_s\,\omega^2}{2} \cdot Z(p,n)\bigl(Z(p,n) - 1\bigr)$
(c) Degenerate order parameter. The bond order parameter $\psi = \sqrt{X \cdot Y}$ becomes a function of the total density alone. The Landau free energy loses its order-disorder distinction: the system cannot distinguish a bond from a self-interaction.
(d) No structural hierarchy. The four-level hierarchy (excitation → pair → bond → clump) collapses to a single level (excitation → annihilation). Level 2 (bonds) is indistinguishable from self-interactions. Level 3 (clumps) reduces to generic density fluctuations.

Remark

The symmetric collapse is not a gradual degradation. It is a qualitative change: the four-level hierarchy reduces entirely to single-species annihilation. The entire structural emergence layer is absent.

3.2 Reduction Corollary

Corollary 3.4 (Reduction to single-species annihilation)

Under symmetric conditions with symmetric initial data, the dynamics of $Z(p,n) = X(p,n) + Y(p,n)$ are equivalent in law to the classical $A + A \to \varnothing$ annihilation-diffusion system on the same graph with finite energy budget. In particular:

The energy monotonicity, finite activity, and almost sure absorption guarantees of Part I still hold.
The Landau bond-formation mechanism is inoperative.
No structural hierarchy exists.

4Necessity of the Forbidden Y–Y Channel

4.1 Depletion Gap Lemma

Lemma 4.1 (Depletion gap)

In the Part I model ( $\alpha_{YY} = 0$ , $\alpha_{XX} > 0$ ), for any vertex $p$ with $X(p,n) \ge 2$ and $Y(p,n) \ge 1$ :

\delta\mu(p,n) = 2\,\mathbb{E}[N_{XX}(p,n)] = \alpha_{XX}\,\omega_X^2\,X(p,n)\bigl(X(p,n)-1\bigr) > 0

X depletes strictly faster than Y.

4.2 Differential Depletion Theorem

Theorem 4.2 (Differential Depletion)

In the Part I model, the depletion gap produces:

(a) Species ratio divergence. The ratio $Y(p,n)/X(p,n)$ increases in expectation over time at any active vertex. Y becomes the majority species.
(b) Minority bottleneck. The bond order parameter is bounded by the minority species:
$\psi = \sqrt{X \cdot Y} \le \min(X,Y) \le X$
Bond formation is X-limited: bonds form only where X persists.
(c) Spatial concentration. X excitations that survive are concentrated at vertices where self-interaction is locally low (due to stochastic fluctuation). This concentrates the order parameter $\psi$ at sparse, specific sites.
(d) Bond formation as condensation. At sites where X survives (sparse), Y is abundant. The co-density $XY$ is maximised here, bonds form preferentially: condensation of the minority species at bond sites.

Corollary 4.3 (Necessity of forbidden Y–Y)

If $\alpha_{YY} > 0$ , then $\delta\mu = 2(\mathbb{E}[N_{XX}] - \mathbb{E}[N_{YY}])$ , which can be zero or negative. If $\alpha_{YY} = \alpha_{XX}$ , the differential depletion mechanism is destroyed: the species ratio does not diverge, no minority bottleneck forms, and the spatial concentration that drives bond formation vanishes.

5Necessity of Frequency Mismatch

The cross-channel rate depends on $\Theta(n)$ , the phase-interference factor introduced in Part I. At equal frequencies, $\Theta(n) \equiv 1$ identically and the cross-channel structural increment is simply a decaying multiple of the species counts. At unequal frequencies, $\Theta(n)$ oscillates, injecting a recurrent modulation into $\Delta S(p,n)$ and thereby into the ripple $F$ .

The cross-channel rate is modulated by the phase-interference factor $\Theta(n)$ (Definition 2.1 of Part I). When $\omega_X = \omega_Y$ , $\Theta(n) \equiv 1$ for all $n$ ; the rate becomes a fixed multiple of the species counts and the structural increment loses its time-varying character. When $\omega_X \ne \omega_Y$ , $\Theta(n)$ oscillates with beat period $2\pi/|\omega_X-\omega_Y|$ , producing a recurrent modulation of $\Delta S$ .

5.3 Beat Frequency Theorem

Theorem 5.3 (Beat Frequency)

Let $\Delta\omega = \omega_X - \omega_Y$ .

(a) Frequency matching removes the deterministic drive. When $\omega_X = \omega_Y$ , $\Theta(n) \equiv 1$ for all $n$ . The cross-channel structural increment $\gamma_1 N_{XY}$ then decays monotonically as species are depleted; no recurrent ripple drive exists.
(b) Frequency mismatch generates a persistent beat. When $\Delta\omega \ne 0$ , $\Theta(n)$ oscillates between $\vartheta$ and $1$ with period $2\pi/|\Delta\omega|$ . Each oscillation cycle drives a recurrent increase then decrease in $\Delta S$ , producing sustained non-zero ripple $F$ as long as species remain.
(c) Explicit beat amplitude.
$A_{\mathrm{beat}} = (1-\vartheta)\sin\!\left(\frac{|\Delta\omega|}{2}\right)\gamma_1\,\alpha_{XY}\,\omega_X\,\omega_Y\,\bar{X}\,\bar{Y}$
When $\Delta\omega = 0$ , $A_{\mathrm{beat}} = 0$ : no oscillation, no sustained ripple drive, no cascades.

5.4 No Cascades Without Frequency Mismatch

Corollary 5.4

In the absence of frequency mismatch ( $\omega_X = \omega_Y$ ), $A_{\mathrm{beat}} = 0$ . The system cannot enter the explosive regime ( $F \ge C + \Delta$ ). No pair creation occurs, no cascades develop, and the system decays monotonically to absorption without passing through the complexity peak described in Part I.

6Asymmetry Index and Hierarchy Depth

6.1 Asymmetry Index

Definition 6.1 (Asymmetry index)

\mathcal{A} = \underbrace{\frac{\alpha_{XX} - \alpha_{YY}}{\alpha_{XX} + \alpha_{YY} + \epsilon}}_{\text{channel asymmetry } \mathcal{A}_c} \;\times\; \underbrace{\frac{|\omega_X - \omega_Y|}{\omega_X + \omega_Y}}_{\text{frequency asymmetry } \mathcal{A}_f}

$\mathcal{A} \ge 0$ , with $\mathcal{A} = 0$ if and only if the model is symmetric
$\mathcal{A} \le 1$ , with maximum at full channel asymmetry and maximal frequency separation
$\mathcal{A}$ is the product of two independent factors: either factor being zero kills $\mathcal{A}$ , encoding that both asymmetries are needed simultaneously

Asymmetry Index Structure

Channel Asymmetry 𝒜_c

Forbidden Y–Y channel

α_XX − α_YY

Drives minority bottleneck

Frequency Asymmetry 𝒜_f

Intrinsic frequency mismatch

|ω_X − ω_Y|

Drives beat oscillation

Asymmetry Index 𝒜

Combined measure

0 ≤ 𝒜 ≤ 1

Must exceed 𝒜_crit for full hierarchy

6.3 Critical Asymmetry Threshold

Theorem 6.3 (Critical asymmetry)

There exists a critical value $\mathcal{A}_{\mathrm{crit}} > 0$ such that:

(a) If $\mathcal{A} < \mathcal{A}_{\mathrm{crit}}$ : no explosions, no cascades. Maximum hierarchy depth: 2 (excitation → pair → bond, no clumps).
(b) If $\mathcal{A} \ge \mathcal{A}_{\mathrm{crit}}$ : with positive probability, cascades occur, spatial concentration develops, and the full four-level hierarchy emerges.

6.5 Hierarchy Depth Bound

Proposition 6.5 (Hierarchy depth bound)

The peak hierarchy depth $H^* = \max_n H(\Sigma(n))$ satisfies $H^* \le h(\mathcal{A})$ , where $h : [0,1] \to \{0,1,2,3,4\}$ is a non-decreasing step function with:

h(\mathcal{A}) = \begin{cases} 1 & \mathcal{A} = 0 \text{ (symmetric)} \\ \le 2 & 0 < \mathcal{A} < \mathcal{A}_{\mathrm{crit}} \\ \le 4 & \mathcal{A} \ge \mathcal{A}_{\mathrm{crit}} \end{cases}

The transitions are sharp, governed by the critical asymmetry. There are no fractional hierarchy levels.

7Complete Phase–Asymmetry Diagram

Combining the phase results of Part I with the present paper, the full behaviour is classified by two control parameters: initial energy density $\varepsilon_0$ and asymmetry index $\mathcal{A}$ .

Energy density ε₀	Asymmetry 𝒜	Regime	Max Hierarchy H*
Low	`Any`	Immediate absorption	H* = 1
High	`= 0 (symmetric)`	Symmetric decay	H* = 1
High	`0 < 𝒜 < 𝒜_crit`	Dissipative	H* ≤ 2
High	`𝒜 ≥ 𝒜_crit`	Full cascade	H* ≤ 4

The only regime that produces the full structural hierarchy is high energy and sufficient asymmetry. Neither condition alone is sufficient.

8Implications

8.1 Asymmetry as Constitutive, Not Contingent

The results establish that the two asymmetries in Part I are not modelling conveniences. They are constitutive requirements for the structural hierarchy. Removing either one collapses the hierarchy entirely.

This has a conceptual consequence: the minimum viable model for discrete cascade dynamics with emergent structure requires at least two non-interchangeable species. A single species, or two interchangeable species, cannot produce the four-level hierarchy.

8.2 Asymmetry Creates Timescale Separation

The frequency mismatch $\omega_X \ne \omega_Y$ creates two timescales:

A fast timescale $\tau_{\mathrm{fast}} \sim 1/\omega_X^2$ governing X–X self-interaction (X depletes quickly)
A slow timescale $\tau_{\mathrm{slow}} \sim 1/(\omega_X\omega_Y)$ governing X–Y cross-interaction (Y persists longer)

This timescale separation is analogous to the fast-slow decomposition in singularly perturbed systems. The “fast” X dynamics create the fluctuations; the “slow” Y dynamics provide the substrate. Without separation, both species fluctuate identically and no structured behaviour emerges.

8.3 Differential Depletion as Symmetry Breaking

The forbidden Y–Y channel is a form of explicit symmetry breaking: it removes one interaction channel from species Y, making X and Y non-interchangeable. The differential depletion that follows is a dynamical consequence of this explicit breaking.

The spatial concentration of bond sites is then a form of spontaneous symmetry breaking: the initial spatial distribution of X may be uniform, but the dynamics create non-uniform X density through stochastic fluctuation amplified by differential depletion. Bond sites “condense” at locations selected by noise, a discrete analogue of nucleation.

9Conclusion

We have resolved the asymmetry question posed in the companion paper. The answer is definitive:

The forbidden Y–Y self-interaction channel is necessary. It creates a differential depletion rate between species that drives minority-species condensation at bond sites. Without it, the depletion gap vanishes and the spatial concentration that underlies bond formation dissolves.
The frequency mismatch $\omega_X \ne \omega_Y$ is necessary. It drives the phase-interference factor $\Theta(n)$ to oscillate, generating a recurrent deterministic ripple drive with explicit amplitude $A_{\mathrm{beat}}$ . At equal frequencies, $\Theta \equiv 1$ and the deterministic drive vanishes; only decaying transients remain, no cascades develop.
The two asymmetries are independently necessary and jointly sufficient for the structural hierarchy. The asymmetry index $\mathcal{A}$ must exceed a critical threshold $\mathcal{A}_{\mathrm{crit}}$ for the full four-level hierarchy to emerge.

The structural hierarchy of the discrete cascade model (excitation → pair → bond → clump) is not an incidental feature. Within this model class, it is a direct, necessary consequence of species asymmetry, and it vanishes the moment that asymmetry is removed.

Remark

In equilibrium statistical mechanics, symmetry breaking is spontaneous: a symmetric Hamiltonian gives rise to an asymmetric ground state. Here, the asymmetry is constitutive: it is built into the interaction rules as a design requirement. Moreover, the system is finite, transient, and non-equilibrium; the standard machinery of infinite-volume limits and ergodic measures does not apply. The asymmetry index $\mathcal{A}$ and its critical threshold provide a quantitative bridge between this constitutive asymmetry and the structural complexity it enables, showing that the relationship admits a sharp, measurable transition.

← Part I: Construction, Guarantees, and Phase Structure Back to Research →