Brusselator + ROP · MolFoundry

1 · Rules & state space

A, X, Y and E are C₄H₈ isomers; B and D are C₃H₆ isomers. Their graphs use ordinary atom and bond labels.

Cyclobutane graph — A · cyclobutane · C₄H₈

methylcyclopropane graph — Y · methylcyclopropane · C₄H₈

cyclopropane graph — B · cyclopropane · C₃H₆

Six DPO rule spans

The panels show L ← K → R. Matching atom IDs mark preserved vertices.

Source · A → X

Open-reactor feed contributes k₁aV to the X count. mapped rule GML

Source: complete DPO rule and application conditions

Autocatalysis · 2 X + Y → 3 X

Propensity k₂(N_X)₂N_Y/V²; this literal three-molecule span creates the infinite factorial-moment hierarchy. mapped rule GML

Autocatalysis: complete DPO rule and application conditions

Conversion · B + X → Y + D

Chemostatted B gives propensity k₃bN_X. mapped rule GML

Conversion: complete DPO rule and application conditions

Sink · X → E

Open-reactor removal channel with propensity k₄N_X. mapped rule GML

Sink: complete DPO rule and application conditions

Monomer inlet · boundary → boundary + M

An atom-creating boundary span supplies caprolactone at rate β; the water graph is unchanged context. mapped rule GML

Monomer inlet: complete DPO rule and application conditions

X-catalyzed growth · P(n) + M + X → P(n+1) + X

The caprolactone ring is opened and joined to the chain; the complete X molecule is preserved catalytic context. mapped rule GML

X-catalyzed growth: complete DPO rule and application conditions

One-step state expansion

Start with X=2, Y=1, one free caprolactone and an empty initiator chain. All six rule classes are enabled.

s₀ · all six channels enabled

X=2 · Y=1 · M=1 · N=0

×2X · selected Y molecule

×1Y · selected M molecule

×1M · support P(n) molecule

×1P(n) · n=0

atom-resolved representative GML

sourcek1*a*V · a_e(s₀)=1 · ΔX=+1literal L → literal R

s₁ · source

X=3 · Y=1 · M=1 · N=0

×3X · selected Y molecule

×1Y · selected M molecule

×1M · support P(n) molecule

×1P(n) · n=0

atom-resolved representative GML

autocatalysisk2/V^2 · a_e(s₀)=2 · ΔX=+1, ΔY=-1literal L → literal R

s₁ · autocatalysis

X=3 · Y=0 · M=1 · N=0

×3X · selected Y molecule

×0Y · selected M molecule

×1M · support P(n) molecule

×1P(n) · n=0

atom-resolved representative GML

conversionk3*b · a_e(s₀)=2 · ΔX=-1, ΔY=+1literal L → literal R

s₁ · conversion

X=1 · Y=2 · M=1 · N=0

×1X · selected Y molecule

×2Y · selected M molecule

×1M · support P(n) molecule

×1P(n) · n=0

atom-resolved representative GML

sinkk4 · a_e(s₀)=2 · ΔX=-1literal L → literal R

s₁ · sink

X=1 · Y=1 · M=1 · N=0

×1X · selected Y molecule

×1Y · selected M molecule

×1M · support P(n) molecule

×1P(n) · n=0

atom-resolved representative GML

monomer_inletbeta · a_e(s₀)=1 · ΔM=+1literal L → literal R

s₁ · monomer-inlet

X=2 · Y=1 · M=2 · N=0

×2X · selected Y molecule

×1Y · selected M molecule

×2M · support P(n) molecule

×1P(n) · n=0

atom-resolved representative GML

growthk_g · a_e(s₀)=2 · ΔM=-1, ΔN=+1literal L → literal R

s₁ · growth

X=2 · Y=1 · M=0 · N=1

×2X · selected Y molecule

×1Y · selected M molecule

×0M · support P(n) molecule

×1P(n) · n=1

atom-resolved representative GML

Nodes show molecular representatives; arrows link to the corresponding rule applications.

2 · Inputs & semantics

Queries: complete X and Y molecules. The event model also uses free monomer count M and incorporated-repeat count N.

The language permits arbitrary finite populations, one growing caprolactone chain and chemostatted A/B feeds. DPO rules preserve atoms and valence, apart from monomer input. These are hypothetical net reactions.

3 · Observable discovery

Generator images are expanded in falling-factorial counts. Each new coordinate corresponds to a pattern of complete X/Y molecules.

Graph observable	Round	Exact coordinate	Degree	Pattern
X	0	N_X	1	GML
Y	0	N_Y	1	GML
ONE	1	1	0	GML
X2Y	1	(N_X)_2 N_Y	3	GML
XY	2	N_X N_Y	2	GML
X2Y2	2	(N_X)_2 (N_Y)_2	4	GML
X3Y	2	(N_X)_3 N_Y	4	GML
X3Y2	2	(N_X)_3 (N_Y)_2	5	GML
X4Y	2	(N_X)_4 N_Y	5	GML
X3	2	(N_X)_3	3	GML
X2	3	(N_X)_2	2	GML
XY2	3	N_X (N_Y)_2	3	GML
X2Y3	3	(N_X)_2 (N_Y)_3	5	GML
X3Y3	3	(N_X)_3 (N_Y)_3	6	GML
X4Y2	3	(N_X)_4 (N_Y)_2	6	GML
X5Y	3	(N_X)_5 N_Y	6	GML
X4	3	(N_X)_4	4	GML
X4Y3	3	(N_X)_4 (N_Y)_3	7	GML
X5Y2	3	(N_X)_5 (N_Y)_2	7	GML
X6Y	3	(N_X)_6 N_Y	7	GML
X5	3	(N_X)_5	5	GML
Y2	4	(N_Y)_2	2	GML
XY3	4	N_X (N_Y)_3	4	GML
X2Y4	4	(N_X)_2 (N_Y)_4	6	GML
X3Y4	4	(N_X)_3 (N_Y)_4	7	GML
X4Y4	4	(N_X)_4 (N_Y)_4	8	GML
X5Y3	4	(N_X)_5 (N_Y)_3	8	GML
X6Y2	4	(N_X)_6 (N_Y)_2	8	GML
X7Y	4	(N_X)_7 N_Y	8	GML
X6	4	(N_X)_6	6	GML
X5Y4	4	(N_X)_5 (N_Y)_4	9	GML
X6Y3	4	(N_X)_6 (N_Y)_3	9	GML

Linear closure unresolved. After 18 expansions, 14 coordinates remain pending.

Discovery trace

basis size: 2 → 4 → 10 → 21 → 32

4 · pJC

Linear closure unresolved. The first X equation already requires a third-order moment:

d E[N_X] / dt = k₁ aV + (k₂/V²) E[(N_X)₂ N_Y] − (k₃ b + k₄) E[N_X]

Next generator image

Expanding X2Y = (N_X)₂N_Y introduces moments up to degree five.

L₂[(N_X)₂N_Y] = V⁻²[−2 (N_X)₂N_Y − 4 (N_X)₃N_Y − (N_X)₄N_Y +4 (N_X)₂(N_Y)₂ +2 (N_X)₃(N_Y)₂]

pJC closed: 6 event classes, with activities of degree at most 3. This does not imply a finite expectation ODE.

All 6 derived EMGF event classes

δ is the coordinate jump; a_e is its activity.

Event class	Rule / rate	Computed δ	Activity a_e	exp(λ·δ) − 1
`source`	`source` k1aV	`ΔX=+1`	`1`	`exp(lambda_X) - 1`
`autocatalysis`	`autocatalysis` k2/V^2	`ΔX=+1, ΔY=-1`	`-XY + X^2Y`	`exp(lambda_X - lambda_Y) - 1`
`conversion`	`conversion` k3*b	`ΔX=-1, ΔY=+1`	`X`	`exp(-lambda_X + lambda_Y) - 1`
`sink`	`sink` k4	`ΔX=-1`	`X`	`exp(-lambda_X) - 1`
`monomer-inlet`	`monomer_inlet` beta	`ΔM=+1`	`1`	`exp(lambda_M) - 1`
`growth`	`growth` k_g	`ΔM=-1, ΔN=+1`	`X*M`	`exp(-lambda_M + lambda_N) - 1`

EMGF operator

\frac{\partial M}{\partial t} = \sum_{e} k_{e} (e^{λ δ_{e}} - 1) a_{e} (\partial_{λ}) M

5 · Initial states

The approximations start from x=0.4, y=2.5, M=4 and N=0. Higher factorial moments start from fixed populations; untracked moments use the stated boundary approximation.

The stochastic reference uses X=Vx and Y=Vy at V=200, with the same monomer and chain counts.

6 · Dynamics

Approximation. The selected moment equations are integrated with a factorization of untracked moments. Core rates k₁…k₄ are fixed at 1.

A feed, a 1.00

B feed, b 2.20

Monomer inlet, β 0.30

Growth rate, k_g 0.15

Approximation tier Mean field

System size and comparison settings

System size, V 200

Time horizon 80

Overlay mean-field tier

Mean field: limit-cycle side

X and Y over time

X · selectedY · selectedmean-field overlay

Phase plane · X versus Y

Instantaneous chain-growth rate n′

Cumulative chain coordinate n

Selected approximate ODE

Loading generated hierarchy…

Mean-field linearization and diagnostics

The mean-field threshold is b=1+a².

max Re λ*0.100

late X amplitude—

chain n(T)—

integrated dimension4

Stochastic and numerical reference runs

Stochastic and deterministic oscillatory trajectories — Scaled stochastic X/Y counts overlaid with the deterministic limit cycle. Noise perturbs phase and amplitude; it does not justify factorizing higher moments.

Selectable approximations built from the partial hierarchy

Tier	Expanded exact rows	Integrated dimension	Approximation
Mean field	2	4	Only X and Y are retained; X2Y is closed by x²y.
+ autocatalytic moment	4	5	X2Y receives its own inferred differential equation.
Nine-moment hierarchy	10	11	Nine nonconstant inferred rows are integrated.
Seventeen-moment hierarchy	18	19	Seventeen nonconstant inferred rows are integrated.

Verified damped and limit-cycle regimes with unbounded chain growth — High-accuracy SciPy reference runs. At a=1, b=1.8 damps and b=2.2 approaches a limit cycle; n keeps increasing in both.

Reference case	max Re λ	late X amplitude	n(300)
Damped, b=1.8	-0.100	2.10991e-08	92.000
Limit cycle, b=2.2	0.100	1.12845	91.973