Fixed ring · rules to exact ODE

1 · Rules & state space

Each of the eight sites has three typed vertices C/P/L, with cp, cl, next and bound edges. These labels describe an abstract binding model.

basal_bind

One free slot becomes occupied. NoEdge forbids an already present bound edge. Per-map rate: 1 × k_on.

Full-size rule View in MolFoundry

cooperative_bind

A free slot next to an occupied neighbour becomes occupied. Each occupied neighbour contributes a distinct match. Per-map rate: 1 × k_coop.

Full-size rule View in MolFoundry

unbind

An occupied slot becomes free. Per-map rate: 1 × k_off.

Full-size rule View in MolFoundry

Direct state-space expansion

Choose an initial graph and the number of rule applications. MolFoundry generates the successors; isomorphic states are merged. Depth counts events, not elapsed time.

Start Through depth 2

Empty · complete depth-3 expansion

00000000

Occupied: 0/8; orbit size 1.

All 24 vertices View graph in MolFoundry

Minimum depth 0

00000010

Occupied: 1/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 1

00010010

Occupied: 2/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 2

00001010

Occupied: 2/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 2

00000110

Occupied: 2/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 2

00100010

Occupied: 2/8; orbit size 4.

All 24 vertices View graph in MolFoundry

Minimum depth 2

00010110

Occupied: 3/8; orbit size 16.

All 24 vertices View graph in MolFoundry

Minimum depth 3

00100110

Occupied: 3/8; orbit size 16.

All 24 vertices View graph in MolFoundry

Minimum depth 3

00101001

Occupied: 3/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 3

00101010

Occupied: 3/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 3

00001110

Occupied: 3/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 3

Source state	Rule	Target state	Multiplicity/rate
g-16fbe06e041511	basal_bind	g-9d0019b1dc5b75	8
g-9d0019b1dc5b75	basal_bind	g-a611b88c54b79a	2
g-9d0019b1dc5b75	basal_bind	g-8426161b158ca6	2
g-9d0019b1dc5b75	basal_bind	g-4025d743f65523	2
g-9d0019b1dc5b75	basal_bind	g-a826c1634ae893	1
g-9d0019b1dc5b75	cooperative_bind	g-4025d743f65523	2
g-9d0019b1dc5b75	unbind	g-16fbe06e041511	1
g-a611b88c54b79a	basal_bind	g-64ba3311a86d37	2
g-a611b88c54b79a	basal_bind	g-ffa1f022b2f210	2
g-a611b88c54b79a	basal_bind	g-28ab1363db0948	2
g-a611b88c54b79a	cooperative_bind	g-64ba3311a86d37	2
g-a611b88c54b79a	cooperative_bind	g-ffa1f022b2f210	2
g-a611b88c54b79a	unbind	g-9d0019b1dc5b75	2
g-8426161b158ca6	basal_bind	g-699be0bb08ff4b	2
g-8426161b158ca6	basal_bind	g-64ba3311a86d37	2
g-8426161b158ca6	basal_bind	g-61b19ab04802bc	1
g-8426161b158ca6	basal_bind	g-28ab1363db0948	1
g-8426161b158ca6	cooperative_bind	g-64ba3311a86d37	2
g-8426161b158ca6	cooperative_bind	g-61b19ab04802bc	2
g-8426161b158ca6	unbind	g-9d0019b1dc5b75	2
g-4025d743f65523	basal_bind	g-64ba3311a86d37	2
g-4025d743f65523	basal_bind	g-61b19ab04802bc	2
g-4025d743f65523	basal_bind	g-ffa1f022b2f210	2
g-4025d743f65523	cooperative_bind	g-61b19ab04802bc	2
g-4025d743f65523	unbind	g-9d0019b1dc5b75	2
g-a826c1634ae893	basal_bind	g-699be0bb08ff4b	2
g-a826c1634ae893	basal_bind	g-ffa1f022b2f210	4
g-a826c1634ae893	cooperative_bind	g-ffa1f022b2f210	4
g-a826c1634ae893	unbind	g-9d0019b1dc5b75	2

Four consecutive occupied sites · complete depth-3 expansion

11110000

Occupied: 4/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 0

00111110

Occupied: 5/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 1

00111101

Occupied: 5/8; orbit size 16.

All 24 vertices View graph in MolFoundry

Minimum depth 1

00010110

Occupied: 3/8; orbit size 16.

All 24 vertices View graph in MolFoundry

Minimum depth 1

00001110

Occupied: 3/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 1

00111111

Occupied: 6/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 2

01111101

Occupied: 6/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 2

00110110

Occupied: 4/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 2

00101110

Occupied: 4/8; orbit size 16.

All 24 vertices View graph in MolFoundry

Minimum depth 2

01101111

Occupied: 6/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 2

00101101

Occupied: 4/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 2

00110101

Occupied: 4/8; orbit size 16.

All 24 vertices View graph in MolFoundry

Minimum depth 2

00111001

Occupied: 4/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 2

00010010

Occupied: 2/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 2

00001010

Occupied: 2/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 2

00000110

Occupied: 2/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 2

01111111

Occupied: 7/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 3

00110111

Occupied: 5/8; orbit size 16.

All 24 vertices View graph in MolFoundry

Minimum depth 3

01011101

Occupied: 5/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 3

01101101

Occupied: 5/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 3

00100110

Occupied: 3/8; orbit size 16.

All 24 vertices View graph in MolFoundry

Minimum depth 3

00101010

Occupied: 3/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 3

00101001

Occupied: 3/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 3

00000010

Occupied: 1/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 3

Source state	Rule	Target state	Multiplicity/rate
g-4c792a9a6a2f4e	basal_bind	g-d6a275e9e77dfc	2
g-4c792a9a6a2f4e	basal_bind	g-bc55abd890ef8f	2
g-4c792a9a6a2f4e	cooperative_bind	g-d6a275e9e77dfc	2
g-4c792a9a6a2f4e	unbind	g-64ba3311a86d37	2
g-4c792a9a6a2f4e	unbind	g-61b19ab04802bc	2
g-d6a275e9e77dfc	basal_bind	g-48f456409e67e8	2
g-d6a275e9e77dfc	basal_bind	g-3ae47080bdb9bf	1
g-d6a275e9e77dfc	cooperative_bind	g-48f456409e67e8	2
g-d6a275e9e77dfc	unbind	g-12db44f0311a4b	1
g-d6a275e9e77dfc	unbind	g-4c792a9a6a2f4e	2
g-d6a275e9e77dfc	unbind	g-b0175b382d00a2	2
g-bc55abd890ef8f	basal_bind	g-48f456409e67e8	1
g-bc55abd890ef8f	basal_bind	g-773d9784146cd6	1
g-bc55abd890ef8f	basal_bind	g-3ae47080bdb9bf	1
g-bc55abd890ef8f	cooperative_bind	g-48f456409e67e8	2
g-bc55abd890ef8f	cooperative_bind	g-773d9784146cd6	1
g-bc55abd890ef8f	cooperative_bind	g-3ae47080bdb9bf	1
g-bc55abd890ef8f	unbind	g-4c792a9a6a2f4e	1
g-bc55abd890ef8f	unbind	g-104f848faa6134	1
g-bc55abd890ef8f	unbind	g-b0175b382d00a2	1
g-bc55abd890ef8f	unbind	g-2e5b18b6ec0954	1
g-bc55abd890ef8f	unbind	g-ece84a5579394d	1
g-64ba3311a86d37	basal_bind	g-12db44f0311a4b	1
g-64ba3311a86d37	basal_bind	g-4c792a9a6a2f4e	1
g-64ba3311a86d37	basal_bind	g-104f848faa6134	1
g-64ba3311a86d37	basal_bind	g-b0175b382d00a2	1
g-64ba3311a86d37	basal_bind	g-2e5b18b6ec0954	1
g-64ba3311a86d37	cooperative_bind	g-12db44f0311a4b	1
g-64ba3311a86d37	cooperative_bind	g-4c792a9a6a2f4e	2
g-64ba3311a86d37	cooperative_bind	g-b0175b382d00a2	1
g-64ba3311a86d37	unbind	g-a611b88c54b79a	1
g-64ba3311a86d37	unbind	g-8426161b158ca6	1
g-64ba3311a86d37	unbind	g-4025d743f65523	1
g-61b19ab04802bc	basal_bind	g-4c792a9a6a2f4e	2
g-61b19ab04802bc	basal_bind	g-b0175b382d00a2	2
g-61b19ab04802bc	basal_bind	g-ece84a5579394d	1
g-61b19ab04802bc	cooperative_bind	g-4c792a9a6a2f4e	2
g-61b19ab04802bc	unbind	g-8426161b158ca6	1
g-61b19ab04802bc	unbind	g-4025d743f65523	2
g-48f456409e67e8	basal_bind	g-848af81403b953	2
g-48f456409e67e8	cooperative_bind	g-848af81403b953	2
g-48f456409e67e8	unbind	g-99123353db4d0a	2
g-48f456409e67e8	unbind	g-d6a275e9e77dfc	2
g-48f456409e67e8	unbind	g-bc55abd890ef8f	2
g-3ae47080bdb9bf	basal_bind	g-848af81403b953	2
g-3ae47080bdb9bf	cooperative_bind	g-848af81403b953	4
g-3ae47080bdb9bf	unbind	g-e6fa3fa70be64a	2
g-3ae47080bdb9bf	unbind	g-d6a275e9e77dfc	1
g-3ae47080bdb9bf	unbind	g-bc55abd890ef8f	2
g-3ae47080bdb9bf	unbind	g-a9205f763d92d9	1
g-12db44f0311a4b	basal_bind	g-99123353db4d0a	2
g-12db44f0311a4b	basal_bind	g-d6a275e9e77dfc	1
g-12db44f0311a4b	basal_bind	g-a9205f763d92d9	1
g-12db44f0311a4b	cooperative_bind	g-99123353db4d0a	2
g-12db44f0311a4b	cooperative_bind	g-d6a275e9e77dfc	2
g-12db44f0311a4b	unbind	g-64ba3311a86d37	2
g-12db44f0311a4b	unbind	g-ffa1f022b2f210	2
g-b0175b382d00a2	basal_bind	g-99123353db4d0a	1
g-b0175b382d00a2	basal_bind	g-e6fa3fa70be64a	1
g-b0175b382d00a2	basal_bind	g-d6a275e9e77dfc	1
g-b0175b382d00a2	basal_bind	g-bc55abd890ef8f	1
g-b0175b382d00a2	cooperative_bind	g-99123353db4d0a	1
g-b0175b382d00a2	cooperative_bind	g-d6a275e9e77dfc	2
g-b0175b382d00a2	cooperative_bind	g-bc55abd890ef8f	1
g-b0175b382d00a2	unbind	g-699be0bb08ff4b	1
g-b0175b382d00a2	unbind	g-64ba3311a86d37	1
g-b0175b382d00a2	unbind	g-61b19ab04802bc	1
g-b0175b382d00a2	unbind	g-ffa1f022b2f210	1
g-773d9784146cd6	basal_bind	g-848af81403b953	2
g-773d9784146cd6	cooperative_bind	g-848af81403b953	4
g-773d9784146cd6	unbind	g-99123353db4d0a	2
g-773d9784146cd6	unbind	g-bc55abd890ef8f	2
g-773d9784146cd6	unbind	g-a9205f763d92d9	2
g-104f848faa6134	basal_bind	g-bc55abd890ef8f	2
g-104f848faa6134	basal_bind	g-a9205f763d92d9	2
g-104f848faa6134	cooperative_bind	g-bc55abd890ef8f	4
g-104f848faa6134	cooperative_bind	g-a9205f763d92d9	2
g-104f848faa6134	unbind	g-64ba3311a86d37	2
g-104f848faa6134	unbind	g-28ab1363db0948	2
g-2e5b18b6ec0954	basal_bind	g-99123353db4d0a	1
g-2e5b18b6ec0954	basal_bind	g-e6fa3fa70be64a	1
g-2e5b18b6ec0954	basal_bind	g-bc55abd890ef8f	1
g-2e5b18b6ec0954	basal_bind	g-a9205f763d92d9	1
g-2e5b18b6ec0954	cooperative_bind	g-99123353db4d0a	2
g-2e5b18b6ec0954	cooperative_bind	g-e6fa3fa70be64a	1
g-2e5b18b6ec0954	cooperative_bind	g-bc55abd890ef8f	2
g-2e5b18b6ec0954	cooperative_bind	g-a9205f763d92d9	1
g-2e5b18b6ec0954	unbind	g-699be0bb08ff4b	1
g-2e5b18b6ec0954	unbind	g-64ba3311a86d37	1
g-2e5b18b6ec0954	unbind	g-ffa1f022b2f210	1
g-2e5b18b6ec0954	unbind	g-28ab1363db0948	1
g-ece84a5579394d	basal_bind	g-99123353db4d0a	2
g-ece84a5579394d	basal_bind	g-bc55abd890ef8f	2
g-ece84a5579394d	cooperative_bind	g-99123353db4d0a	2
g-ece84a5579394d	cooperative_bind	g-bc55abd890ef8f	2
g-ece84a5579394d	unbind	g-61b19ab04802bc	1
g-ece84a5579394d	unbind	g-ffa1f022b2f210	2
g-ece84a5579394d	unbind	g-28ab1363db0948	1
g-a611b88c54b79a	basal_bind	g-64ba3311a86d37	2
g-a611b88c54b79a	basal_bind	g-ffa1f022b2f210	2
g-a611b88c54b79a	basal_bind	g-28ab1363db0948	2
g-a611b88c54b79a	cooperative_bind	g-64ba3311a86d37	2
g-a611b88c54b79a	cooperative_bind	g-ffa1f022b2f210	2
g-a611b88c54b79a	unbind	g-9d0019b1dc5b75	2
g-8426161b158ca6	basal_bind	g-699be0bb08ff4b	2
g-8426161b158ca6	basal_bind	g-64ba3311a86d37	2
g-8426161b158ca6	basal_bind	g-61b19ab04802bc	1
g-8426161b158ca6	basal_bind	g-28ab1363db0948	1
g-8426161b158ca6	cooperative_bind	g-64ba3311a86d37	2
g-8426161b158ca6	cooperative_bind	g-61b19ab04802bc	2
g-8426161b158ca6	unbind	g-9d0019b1dc5b75	2
g-4025d743f65523	basal_bind	g-64ba3311a86d37	2
g-4025d743f65523	basal_bind	g-61b19ab04802bc	2
g-4025d743f65523	basal_bind	g-ffa1f022b2f210	2
g-4025d743f65523	cooperative_bind	g-61b19ab04802bc	2
g-4025d743f65523	unbind	g-9d0019b1dc5b75	2

Four alternating occupied sites · complete depth-3 expansion

10101010

Occupied: 4/8; orbit size 2.

All 24 vertices View graph in MolFoundry

Minimum depth 0

01011101

Occupied: 5/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 1

00101010

Occupied: 3/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 1

01111101

Occupied: 6/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 2

01110111

Occupied: 6/8; orbit size 4.

All 24 vertices View graph in MolFoundry

Minimum depth 2

00101110

Occupied: 4/8; orbit size 16.

All 24 vertices View graph in MolFoundry

Minimum depth 2

00110101

Occupied: 4/8; orbit size 16.

All 24 vertices View graph in MolFoundry

Minimum depth 2

00001010

Occupied: 2/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 2

00100010

Occupied: 2/8; orbit size 4.

All 24 vertices View graph in MolFoundry

Minimum depth 2

01111111

Occupied: 7/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 3

00111110

Occupied: 5/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 3

00111101

Occupied: 5/8; orbit size 16.

All 24 vertices View graph in MolFoundry

Minimum depth 3

01101101

Occupied: 5/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 3

00110111

Occupied: 5/8; orbit size 16.

All 24 vertices View graph in MolFoundry

Minimum depth 3

00010110

Occupied: 3/8; orbit size 16.

All 24 vertices View graph in MolFoundry

Minimum depth 3

00001110

Occupied: 3/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 3

00100110

Occupied: 3/8; orbit size 16.

All 24 vertices View graph in MolFoundry

Minimum depth 3

00101001

Occupied: 3/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 3

00000010

Occupied: 1/8; orbit size 8.

All 24 vertices View graph in MolFoundry

Minimum depth 3

Source state	Rule	Target state	Multiplicity/rate
g-af0669e3baa5d6	basal_bind	g-e6fa3fa70be64a	4
g-af0669e3baa5d6	cooperative_bind	g-e6fa3fa70be64a	8
g-af0669e3baa5d6	unbind	g-699be0bb08ff4b	4
g-e6fa3fa70be64a	basal_bind	g-3ae47080bdb9bf	2
g-e6fa3fa70be64a	basal_bind	g-2908a9efa51c7e	1
g-e6fa3fa70be64a	cooperative_bind	g-3ae47080bdb9bf	4
g-e6fa3fa70be64a	cooperative_bind	g-2908a9efa51c7e	2
g-e6fa3fa70be64a	unbind	g-af0669e3baa5d6	1
g-e6fa3fa70be64a	unbind	g-b0175b382d00a2	2
g-e6fa3fa70be64a	unbind	g-2e5b18b6ec0954	2
g-699be0bb08ff4b	basal_bind	g-af0669e3baa5d6	1
g-699be0bb08ff4b	basal_bind	g-b0175b382d00a2	2
g-699be0bb08ff4b	basal_bind	g-2e5b18b6ec0954	2
g-699be0bb08ff4b	cooperative_bind	g-b0175b382d00a2	4
g-699be0bb08ff4b	cooperative_bind	g-2e5b18b6ec0954	2
g-699be0bb08ff4b	unbind	g-8426161b158ca6	2
g-699be0bb08ff4b	unbind	g-a826c1634ae893	1
g-3ae47080bdb9bf	basal_bind	g-848af81403b953	2
g-3ae47080bdb9bf	cooperative_bind	g-848af81403b953	4
g-3ae47080bdb9bf	unbind	g-e6fa3fa70be64a	2
g-3ae47080bdb9bf	unbind	g-d6a275e9e77dfc	1
g-3ae47080bdb9bf	unbind	g-bc55abd890ef8f	2
g-3ae47080bdb9bf	unbind	g-a9205f763d92d9	1
g-2908a9efa51c7e	basal_bind	g-848af81403b953	2
g-2908a9efa51c7e	cooperative_bind	g-848af81403b953	4
g-2908a9efa51c7e	unbind	g-99123353db4d0a	4
g-2908a9efa51c7e	unbind	g-e6fa3fa70be64a	2
g-b0175b382d00a2	basal_bind	g-99123353db4d0a	1
g-b0175b382d00a2	basal_bind	g-e6fa3fa70be64a	1
g-b0175b382d00a2	basal_bind	g-d6a275e9e77dfc	1
g-b0175b382d00a2	basal_bind	g-bc55abd890ef8f	1
g-b0175b382d00a2	cooperative_bind	g-99123353db4d0a	1
g-b0175b382d00a2	cooperative_bind	g-d6a275e9e77dfc	2
g-b0175b382d00a2	cooperative_bind	g-bc55abd890ef8f	1
g-b0175b382d00a2	unbind	g-699be0bb08ff4b	1
g-b0175b382d00a2	unbind	g-64ba3311a86d37	1
g-b0175b382d00a2	unbind	g-61b19ab04802bc	1
g-b0175b382d00a2	unbind	g-ffa1f022b2f210	1
g-2e5b18b6ec0954	basal_bind	g-99123353db4d0a	1
g-2e5b18b6ec0954	basal_bind	g-e6fa3fa70be64a	1
g-2e5b18b6ec0954	basal_bind	g-bc55abd890ef8f	1
g-2e5b18b6ec0954	basal_bind	g-a9205f763d92d9	1
g-2e5b18b6ec0954	cooperative_bind	g-99123353db4d0a	2
g-2e5b18b6ec0954	cooperative_bind	g-e6fa3fa70be64a	1
g-2e5b18b6ec0954	cooperative_bind	g-bc55abd890ef8f	2
g-2e5b18b6ec0954	cooperative_bind	g-a9205f763d92d9	1
g-2e5b18b6ec0954	unbind	g-699be0bb08ff4b	1
g-2e5b18b6ec0954	unbind	g-64ba3311a86d37	1
g-2e5b18b6ec0954	unbind	g-ffa1f022b2f210	1
g-2e5b18b6ec0954	unbind	g-28ab1363db0948	1
g-8426161b158ca6	basal_bind	g-699be0bb08ff4b	2
g-8426161b158ca6	basal_bind	g-64ba3311a86d37	2
g-8426161b158ca6	basal_bind	g-61b19ab04802bc	1
g-8426161b158ca6	basal_bind	g-28ab1363db0948	1
g-8426161b158ca6	cooperative_bind	g-64ba3311a86d37	2
g-8426161b158ca6	cooperative_bind	g-61b19ab04802bc	2
g-8426161b158ca6	unbind	g-9d0019b1dc5b75	2
g-a826c1634ae893	basal_bind	g-699be0bb08ff4b	2
g-a826c1634ae893	basal_bind	g-ffa1f022b2f210	4
g-a826c1634ae893	cooperative_bind	g-ffa1f022b2f210	4
g-a826c1634ae893	unbind	g-9d0019b1dc5b75	2

2 · Inputs & semantics

Exactly one supplied observable: P0 counts occupied slots. Initial bit strings are used only later to evaluate initial coordinates.

supplied query · depth —

M0 · word 1

Probability that a uniformly chosen site has word 1

z_M0=E[M0]/8. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M0 · L ← K → R

Conditions on L: No explicit input guard; ordinary DPO admissibility still applies.

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

Supplied assumption	Role
name	ring-eight-fixed-sites
sites	8
scaffold	Each of 8 sites has three graph vertices C/P/L: 24 vertices total; only 8 bound edges vary. Labels here are site types, not chemical elements.
reachable_states	256
dihedral_orbits	30
assumption	Exact fixed scaffold; rotation/reflection-invariant rules and observables.

Supplied reduction choice	Meaning
method	exact evaluation on exhaustive dihedral-orbit representatives
motif_max_length	8
initial_motif	1
basis	not preloaded; retain linearly independent jump count vectors and choose shortest equivalent binary word
discovery_order	unit-rate aggregate coefficients first; any additional rate-separated direction is also retained
evidence	Every rate channel is checked independently against the finite orbit generator.

Counting observable does not always mean a strict identity span. A strict identity is I ← I → I. A stored DPO jump observable can instead be I ← K → I with K a proper subgraph: the input and output graphs agree, but some edges are deleted and recreated. On graph-isomorphism classes it acts diagonally, weighted by the number of admissible matches. For the vertex-preserving diagonal spans shown here, the equivalent strict counting identity has the same input graph and every original application condition. We show both, without altering the computed source.

First composition step

Observable	Reaction	Before terms	After terms	Reduced coefficients (if stored)
P0	basal_bind	5	7	See exact rate matrix
P0	cooperative_bind	12	14	See exact rate matrix
P0	unbind	7	5	See exact rate matrix

After signed jump projection and language reduction, the first derivative needs the free-slot count P1 and a mixed-neighbour count P2. They are discovered from the occupied-site query.

presentation coordinate · depth —

M1 · word 0

Probability that a uniformly chosen site has word 0

z_M1=E[M1]/8. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M1 · L ← K → R

Conditions on L: NoEdge(1,2)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M2 · word 01

Probability that a uniformly chosen oriented cyclic start has word 01

z_M2=E[M2]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M2 · L ← K → R

Conditions on L: NoEdge(1,2)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

3 · Observable discovery

1 → 3 → 4 → 6 → 8 → 12 → 17 → 25 → 29 → 30 → 30

The displayed motif coordinates M₀,…,M₂₉ are an invertible linear transformation of the computed basis P₀,…,P₂₉. Both share the occupied-slot query M₀=P₀.

Round 0: 1 computed coordinates; added P0

Round 1: 3 computed coordinates; added P1, P2

Round 2: 4 computed coordinates; added P3

Round 3: 6 computed coordinates; added P4, P5

Round 4: 8 computed coordinates; added P6, P7

Round 5: 12 computed coordinates; added P8, P9, P10, P11

Round 6: 17 computed coordinates; added P12, P13, P14, P15, P16

Round 7: 25 computed coordinates; added P17, P18, P19, P20, P21, P22, P23, P24

Round 8: 29 computed coordinates; added P25, P26, P27, P28

Round 9: 30 computed coordinates; added P29

Round 10: 30 computed coordinates; added

No new independent count direction: CLOSED.

All 30 count patterns

supplied query · depth —

M0 · word 1

z_M0=E[M0]/8. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M0 · L ← K → R

Conditions on L: No explicit input guard; ordinary DPO admissibility still applies.

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M1 · word 0

z_M1=E[M1]/8. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M1 · L ← K → R

Conditions on L: NoEdge(1,2)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M2 · word 01

z_M2=E[M2]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M2 · L ← K → R

Conditions on L: NoEdge(1,2)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M3 · word 101

Probability that a uniformly chosen oriented cyclic start has word 101

z_M3=E[M3]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M3 · L ← K → R

Conditions on L: NoEdge(4,5)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M4 · word 1001

z_M4=E[M4]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M4 · L ← K → R

Conditions on L: NoEdge(4,5); NoEdge(7,8)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M5 · word 111

Probability that a uniformly chosen oriented cyclic start has word 111

z_M5=E[M5]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M5 · L ← K → R

Conditions on L: No explicit input guard; ordinary DPO admissibility still applies.

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M6 · word 1011

z_M6=E[M6]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M6 · L ← K → R

Conditions on L: NoEdge(4,5)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M7 · word 10001

z_M7=E[M7]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M7 · L ← K → R

Conditions on L: NoEdge(4,5); NoEdge(7,8); NoEdge(10,11)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M8 · word 1111

z_M8=E[M8]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M8 · L ← K → R

Conditions on L: No explicit input guard; ordinary DPO admissibility still applies.

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M9 · word 10101

z_M9=E[M9]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M9 · L ← K → R

Conditions on L: NoEdge(4,5); NoEdge(10,11)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M10 · word 10011

z_M10=E[M10]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M10 · L ← K → R

Conditions on L: NoEdge(4,5); NoEdge(7,8)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M11 · word 100001

z_M11=E[M11]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M11 · L ← K → R

Conditions on L: NoEdge(4,5); NoEdge(7,8); NoEdge(10,11); NoEdge(13,14)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M12 · word 10111

z_M12=E[M12]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M12 · L ← K → R

Conditions on L: NoEdge(4,5)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M13 · word 100101

z_M13=E[M13]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M13 · L ← K → R

Conditions on L: NoEdge(4,5); NoEdge(7,8); NoEdge(13,14)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M14 · word 11011

z_M14=E[M14]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M14 · L ← K → R

Conditions on L: NoEdge(7,8)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M15 · word 100011

z_M15=E[M15]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M15 · L ← K → R

Conditions on L: NoEdge(4,5); NoEdge(7,8); NoEdge(10,11)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M16 · word 1000001

z_M16=E[M16]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M16 · L ← K → R

Conditions on L: NoEdge(4,5); NoEdge(7,8); NoEdge(10,11); NoEdge(13,14); NoEdge(16,17)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M17 · word 11111

z_M17=E[M17]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M17 · L ← K → R

Conditions on L: No explicit input guard; ordinary DPO admissibility still applies.

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M18 · word 100111

z_M18=E[M18]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M18 · L ← K → R

Conditions on L: NoEdge(4,5); NoEdge(7,8)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M19 · word 101101

z_M19=E[M19]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M19 · L ← K → R

Conditions on L: NoEdge(4,5); NoEdge(13,14)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M20 · word 101011

z_M20=E[M20]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M20 · L ← K → R

Conditions on L: NoEdge(4,5); NoEdge(10,11)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M21 · word 1000101

z_M21=E[M21]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M21 · L ← K → R

Conditions on L: NoEdge(4,5); NoEdge(7,8); NoEdge(10,11); NoEdge(16,17)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M22 · word 1001001

z_M22=E[M22]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M22 · L ← K → R

Conditions on L: NoEdge(4,5); NoEdge(7,8); NoEdge(13,14); NoEdge(16,17)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M23 · word 110011

z_M23=E[M23]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M23 · L ← K → R

Conditions on L: NoEdge(7,8); NoEdge(10,11)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M24 · word 1000011

z_M24=E[M24]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M24 · L ← K → R

Conditions on L: NoEdge(4,5); NoEdge(7,8); NoEdge(10,11); NoEdge(13,14)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M25 · word 00000011

z_M25=E[M25]/16. A closed eight-site motif.

Exact typed graph, span and NoEdge conditions

M25 · L ← K → R

Conditions on L: NoEdge(1,2); NoEdge(4,5); NoEdge(7,8); NoEdge(10,11); NoEdge(13,14); NoEdge(16,17)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M26 · word 110111

z_M26=E[M26]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M26 · L ← K → R

Conditions on L: NoEdge(7,8)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M27 · word 1010101

z_M27=E[M27]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M27 · L ← K → R

Conditions on L: NoEdge(4,5); NoEdge(10,11); NoEdge(16,17)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M28 · word 1001011

z_M28=E[M28]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M28 · L ← K → R

Conditions on L: NoEdge(4,5); NoEdge(7,8); NoEdge(13,14)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

presentation coordinate · depth —

M29 · word 1101011

z_M29=E[M29]/16. A local path motif, not a complete ring state.

Exact typed graph, span and NoEdge conditions

M29 · L ← K → R

Conditions on L: NoEdge(7,8); NoEdge(13,14)

Full-size span View in MolFoundry

This span is already a strict identity

Equivalent context-only identity GML. All application conditions are preserved.

4 · ODE & pJC

zᵢ = E[Mᵢ]/sᵢ, with s₀=s₁=8 and sᵢ=16 otherwise. These are oriented local-motif probabilities. They are not probabilities that a motif exists somewhere in the ring. All 30 equations are shown below.

\frac{d z_{M0}}{d t} = 2 k_{coop} z_{M2} - k_{off} z_{M0} + k_{on} z_{M1}

\frac{d z_{M1}}{d t} = - 2 k_{coop} z_{M2} + k_{off} z_{M0} - k_{on} z_{M1}

\frac{d z_{M2}}{d t} = - 2 k_{coop} z_{M3} + k_{off} z_{M0} - 2 k_{off} z_{M2} + k_{on} z_{M1} - 2 k_{on} z_{M2}

\frac{d z_{M3}}{d t} = - 2 k_{coop} z_{M3} + 2 k_{coop} z_{M4} - 2 k_{off} z_{M3} + k_{off} z_{M5} + 2 k_{on} z_{M2} - 3 k_{on} z_{M3}

\frac{d z_{M4}}{d t} = - 2 k_{coop} z_{M4} + 2 k_{coop} z_{M7} - 2 k_{off} z_{M4} + 2 k_{off} z_{M6} + 2 k_{on} z_{M2} - 2 k_{on} z_{M3} - 4 k_{on} z_{M4}

\frac{d z_{M5}}{d t} = 2 k_{coop} z_{M0} - 2 k_{coop} z_{M2} + 2 k_{coop} z_{M3} - 2 k_{coop} z_{M5} + 2 k_{coop} z_{M6} - 3 k_{off} z_{M5} + 2 k_{on} z_{M0} - 2 k_{on} z_{M2} + k_{on} z_{M3} - 2 k_{on} z_{M5}

\frac{d z_{M6}}{d t} = k_{coop} z_{M10} + k_{coop} z_{M3} + k_{coop} z_{M4} - 3 k_{coop} z_{M6} + k_{coop} z_{M9} - 3 k_{off} z_{M6} + k_{off} z_{M8} + k_{on} z_{M0} - k_{on} z_{M2} + k_{on} z_{M3} + k_{on} z_{M4} - k_{on} z_{M5} - 3 k_{on} z_{M6}

\frac{d z_{M7}}{d t} = 2 k_{coop} z_{M11} - 2 k_{coop} z_{M7} + 2 k_{off} z_{M10} - 2 k_{off} z_{M7} + k_{off} z_{M9} + 2 k_{on} z_{M2} - 2 k_{on} z_{M3} - 2 k_{on} z_{M4} - 5 k_{on} z_{M7}

\frac{d z_{M8}}{d t} = 2 k_{coop} z_{M12} + 2 k_{coop} z_{M5} + 4 k_{coop} z_{M6} - 2 k_{coop} z_{M8} - 4 k_{off} z_{M8} + 2 k_{on} z_{M5} + 2 k_{on} z_{M6} - 2 k_{on} z_{M8}

\frac{d z_{M9}}{d t} = 2 k_{coop} z_{M13} - 4 k_{coop} z_{M9} + 2 k_{off} z_{M12} - 3 k_{off} z_{M9} + 2 k_{on} z_{M3} - 2 k_{on} z_{M6} + k_{on} z_{M7} - 4 k_{on} z_{M9}

\frac{d z_{M10}}{d t} = - 3 k_{coop} z_{M10} + k_{coop} z_{M13} + k_{coop} z_{M15} + k_{coop} z_{M4} + k_{coop} z_{M7} - 3 k_{off} z_{M10} + k_{off} z_{M12} + k_{off} z_{M14} + k_{on} z_{M0} - 4 k_{on} z_{M10} - k_{on} z_{M2} + k_{on} z_{M4} - k_{on} z_{M5} - k_{on} z_{M6} + k_{on} z_{M7}

\frac{d z_{M11}}{d t} = - 2 k_{coop} z_{M11} + 2 k_{coop} z_{M16} - 2 k_{off} z_{M11} + 2 k_{off} z_{M13} + 2 k_{off} z_{M15} - 6 k_{on} z_{M11} + 2 k_{on} z_{M2} - 2 k_{on} z_{M3} - 2 k_{on} z_{M4} - 2 k_{on} z_{M7}

\frac{d z_{M12}}{d t} = k_{coop} z_{M10} - 3 k_{coop} z_{M12} + k_{coop} z_{M18} + k_{coop} z_{M19} + k_{coop} z_{M6} + 2 k_{coop} z_{M9} - 4 k_{off} z_{M12} + k_{off} z_{M17} + k_{on} z_{M10} - 3 k_{on} z_{M12} + k_{on} z_{M5} + k_{on} z_{M6} - k_{on} z_{M8} + k_{on} z_{M9}

\frac{d z_{M13}}{d t} = - 4 k_{coop} z_{M13} + k_{coop} z_{M21} + k_{coop} z_{M22} - 3 k_{off} z_{M13} + k_{off} z_{M18} + k_{off} z_{M19} + k_{off} z_{M20} - k_{on} z_{M10} + k_{on} z_{M11} - 5 k_{on} z_{M13} + k_{on} z_{M3} + k_{on} z_{M4} - k_{on} z_{M6} - k_{on} z_{M9}

\frac{d z_{M14}}{d t} = 2 k_{coop} z_{M10} - 4 k_{coop} z_{M14} + 2 k_{coop} z_{M20} + 2 k_{coop} z_{M6} - 4 k_{off} z_{M14} + k_{off} z_{M17} + 2 k_{on} z_{M10} - 3 k_{on} z_{M14} + 2 k_{on} z_{M6}

\frac{d z_{M15}}{d t} = k_{coop} z_{M11} - 3 k_{coop} z_{M15} + k_{coop} z_{M21} + k_{coop} z_{M24} + k_{coop} z_{M7} - 3 k_{off} z_{M15} + k_{off} z_{M18} + k_{off} z_{M20} + k_{off} z_{M23} + k_{on} z_{M0} - k_{on} z_{M10} + k_{on} z_{M11} - 5 k_{on} z_{M15} - k_{on} z_{M2} - k_{on} z_{M5} - k_{on} z_{M6} + k_{on} z_{M7}

\frac{d z_{M16}}{d t} = - 2 k_{coop} z_{M16} + 2 k_{coop} z_{M25} - 2 k_{off} z_{M16} + 2 k_{off} z_{M21} + k_{off} z_{M22} + 2 k_{off} z_{M24} - 2 k_{on} z_{M11} - 7 k_{on} z_{M16} + 2 k_{on} z_{M2} - 2 k_{on} z_{M3} - 2 k_{on} z_{M4} - 2 k_{on} z_{M7}

\frac{d z_{M17}}{d t} = - 2 k_{coop} z_{M10} - 2 k_{coop} z_{M11} + 2 k_{coop} z_{M12} - 4 k_{coop} z_{M13} - 2 k_{coop} z_{M15} - 4 k_{coop} z_{M17} - 2 k_{coop} z_{M18} - 2 k_{coop} z_{M19} - 2 k_{coop} z_{M20} + 2 k_{coop} z_{M4} + 4 k_{coop} z_{M6} + 4 k_{coop} z_{M8} - 5 k_{off} z_{M17} + 2 k_{on} z_{M12} + k_{on} z_{M14} - 2 k_{on} z_{M17} + 2 k_{on} z_{M8}

\frac{d z_{M18}}{d t} = - k_{coop} z_{M11} - 3 k_{coop} z_{M12} - k_{coop} z_{M13} - 2 k_{coop} z_{M16} - 5 k_{coop} z_{M18} - k_{coop} z_{M19} - 2 k_{coop} z_{M21} - k_{coop} z_{M22} - k_{coop} z_{M24} + 2 k_{coop} z_{M3} + k_{coop} z_{M4} + 2 k_{coop} z_{M5} - k_{coop} z_{M6} - 2 k_{coop} z_{M8} - 2 k_{coop} z_{M9} - k_{off} z_{M10} - k_{off} z_{M11} - k_{off} z_{M12} - 2 k_{off} z_{M13} - k_{off} z_{M14} - k_{off} z_{M15} - k_{off} z_{M17} - 5 k_{off} z_{M18} - k_{off} z_{M19} - k_{off} z_{M20} + k_{off} z_{M26} + k_{off} z_{M4} + 2 k_{off} z_{M6} + k_{off} z_{M8} + k_{on} z_{M10} - k_{on} z_{M12} + k_{on} z_{M13} + k_{on} z_{M15} - 4 k_{on} z_{M18} + k_{on} z_{M5} - k_{on} z_{M8}

\frac{d z_{M19}}{d t} = - 2 k_{coop} z_{M10} - 2 k_{coop} z_{M11} - 4 k_{coop} z_{M12} - 2 k_{coop} z_{M13} - 2 k_{coop} z_{M15} - 2 k_{coop} z_{M16} - 2 k_{coop} z_{M18} - 6 k_{coop} z_{M19} - 2 k_{coop} z_{M21} - 2 k_{coop} z_{M22} + 2 k_{coop} z_{M3} + 2 k_{coop} z_{M4} + 2 k_{coop} z_{M5} - 2 k_{coop} z_{M8} - 2 k_{coop} z_{M9} - 2 k_{off} z_{M10} - 2 k_{off} z_{M11} - 2 k_{off} z_{M12} - 4 k_{off} z_{M13} - 2 k_{off} z_{M14} - 2 k_{off} z_{M15} - 2 k_{off} z_{M17} - 2 k_{off} z_{M18} - 6 k_{off} z_{M19} - 2 k_{off} z_{M20} + 2 k_{off} z_{M4} + 4 k_{off} z_{M6} + 2 k_{off} z_{M8} - 2 k_{on} z_{M12} + 2 k_{on} z_{M13} - 4 k_{on} z_{M19} + 2 k_{on} z_{M6}

\frac{d z_{M20}}{d t} = k_{coop} z_{M13} - 5 k_{coop} z_{M20} + k_{coop} z_{M27} + k_{coop} z_{M28} + k_{coop} z_{M9} - k_{off} z_{M10} - k_{off} z_{M11} - k_{off} z_{M12} - 2 k_{off} z_{M13} - k_{off} z_{M14} - k_{off} z_{M15} - k_{off} z_{M17} - k_{off} z_{M18} - k_{off} z_{M19} - 5 k_{off} z_{M20} + k_{off} z_{M26} + k_{off} z_{M4} + 2 k_{off} z_{M6} + k_{off} z_{M8} + k_{on} z_{M13} - k_{on} z_{M14} + k_{on} z_{M15} - 4 k_{on} z_{M20} + k_{on} z_{M6} + k_{on} z_{M9}

\frac{d z_{M21}}{d t} = 2 k_{coop} z_{M0} - 3 k_{coop} z_{M10} - k_{coop} z_{M11} - 3 k_{coop} z_{M13} - 2 k_{coop} z_{M15} - 2 k_{coop} z_{M16} - 2 k_{coop} z_{M2} - 6 k_{coop} z_{M21} - k_{coop} z_{M22} - k_{coop} z_{M24} - 2 k_{coop} z_{M25} + 2 k_{coop} z_{M3} + k_{coop} z_{M4} - 2 k_{coop} z_{M5} - 4 k_{coop} z_{M6} - 2 k_{coop} z_{M9} + k_{off} z_{M10} - 3 k_{off} z_{M12} - k_{off} z_{M13} + k_{off} z_{M14} - k_{off} z_{M15} - 2 k_{off} z_{M16} - 2 k_{off} z_{M18} - k_{off} z_{M19} + 2 k_{off} z_{M20} - 4 k_{off} z_{M21} - k_{off} z_{M22} - k_{off} z_{M23} - 2 k_{off} z_{M24} + 2 k_{off} z_{M27} + 2 k_{off} z_{M3} + 2 k_{off} z_{M5} - 2 k_{off} z_{M6} - 2 k_{off} z_{M8} - 3 k_{off} z_{M9} - k_{on} z_{M13} - k_{on} z_{M15} + k_{on} z_{M16} - 6 k_{on} z_{M21} + k_{on} z_{M3} - k_{on} z_{M6} + k_{on} z_{M7} - k_{on} z_{M9}

\frac{d z_{M22}}{d t} = 2 k_{coop} z_{M0} - 4 k_{coop} z_{M10} - 2 k_{coop} z_{M11} - 4 k_{coop} z_{M13} - 2 k_{coop} z_{M15} - 2 k_{coop} z_{M16} - 2 k_{coop} z_{M2} - 2 k_{coop} z_{M21} - 6 k_{coop} z_{M22} - 2 k_{coop} z_{M25} + 2 k_{coop} z_{M3} + 2 k_{coop} z_{M4} - 2 k_{coop} z_{M5} - 4 k_{coop} z_{M6} - 2 k_{coop} z_{M9} - 2 k_{off} z_{M10} - 2 k_{off} z_{M11} - 4 k_{off} z_{M12} - 4 k_{off} z_{M13} - 2 k_{off} z_{M15} - 2 k_{off} z_{M16} - 2 k_{off} z_{M18} - 2 k_{off} z_{M19} - 2 k_{off} z_{M21} - 5 k_{off} z_{M22} + 2 k_{off} z_{M28} + 2 k_{off} z_{M3} + 2 k_{off} z_{M4} + 2 k_{off} z_{M5} - 2 k_{off} z_{M8} - 2 k_{off} z_{M9} - 2 k_{on} z_{M10} - 2 k_{on} z_{M13} + k_{on} z_{M16} - 6 k_{on} z_{M22} + 2 k_{on} z_{M4}

\frac{d z_{M23}}{d t} = 6 k_{coop} z_{M10} + 2 k_{coop} z_{M11} + 4 k_{coop} z_{M13} + 2 k_{coop} z_{M14} + 2 k_{coop} z_{M15} + 4 k_{coop} z_{M20} + 2 k_{coop} z_{M21} - 6 k_{coop} z_{M23} - 2 k_{coop} z_{M24} + 2 k_{coop} z_{M27} - 2 k_{coop} z_{M4} - 2 k_{coop} z_{M6} - 2 k_{coop} z_{M9} - 4 k_{off} z_{M23} + 2 k_{off} z_{M26} + 2 k_{on} z_{M10} + 2 k_{on} z_{M15} - 4 k_{on} z_{M23}

\frac{d z_{M24}}{d t} = 2 k_{coop} z_{M0} - 2 k_{coop} z_{M10} + k_{coop} z_{M11} - k_{coop} z_{M13} - 2 k_{coop} z_{M15} - 2 k_{coop} z_{M2} - k_{coop} z_{M21} - 5 k_{coop} z_{M24} - 2 k_{coop} z_{M25} + k_{coop} z_{M3} - 2 k_{coop} z_{M5} - 3 k_{coop} z_{M6} - k_{coop} z_{M9} + 3 k_{off} z_{M10} + 2 k_{off} z_{M11} - k_{off} z_{M12} + 2 k_{off} z_{M13} + 2 k_{off} z_{M14} + k_{off} z_{M15} - k_{off} z_{M16} - k_{off} z_{M18} + 3 k_{off} z_{M20} + k_{off} z_{M22} - k_{off} z_{M23} - 6 k_{off} z_{M24} + k_{off} z_{M27} + 2 k_{off} z_{M28} + k_{off} z_{M3} - 2 k_{off} z_{M4} + k_{off} z_{M5} - 3 k_{off} z_{M6} - k_{off} z_{M8} - 2 k_{off} z_{M9} + k_{on} z_{M0} - k_{on} z_{M10} + k_{on} z_{M11} - k_{on} z_{M15} + k_{on} z_{M16} - k_{on} z_{M2} - 6 k_{on} z_{M24} - k_{on} z_{M5} - k_{on} z_{M6}

\frac{d z_{M25}}{d t} = - 2 k_{coop} z_{M11} - 2 k_{coop} z_{M16} + 2 k_{coop} z_{M2} - 4 k_{coop} z_{M25} - 2 k_{coop} z_{M3} - 2 k_{coop} z_{M4} - 2 k_{coop} z_{M7} + 6 k_{off} z_{M0} - 8 k_{off} z_{M10} - 2 k_{off} z_{M11} - 6 k_{off} z_{M13} - 6 k_{off} z_{M15} - 4 k_{off} z_{M16} - 6 k_{off} z_{M2} - 4 k_{off} z_{M21} - 2 k_{off} z_{M22} - 4 k_{off} z_{M24} - 8 k_{off} z_{M25} + 4 k_{off} z_{M3} + 2 k_{off} z_{M4} - 6 k_{off} z_{M5} - 10 k_{off} z_{M6} - 4 k_{off} z_{M9} - 2 k_{on} z_{M11} - 2 k_{on} z_{M16} + 2 k_{on} z_{M2} - 8 k_{on} z_{M25} - 2 k_{on} z_{M3} - 2 k_{on} z_{M4} - 2 k_{on} z_{M7}

\frac{d z_{M26}}{d t} = - 2 k_{coop} z_{M10} - k_{coop} z_{M11} + 3 k_{coop} z_{M12} - 2 k_{coop} z_{M13} + k_{coop} z_{M18} + k_{coop} z_{M19} - 3 k_{coop} z_{M20} - 3 k_{coop} z_{M21} + 2 k_{coop} z_{M23} + k_{coop} z_{M24} - 6 k_{coop} z_{M26} - 3 k_{coop} z_{M27} - k_{coop} z_{M28} + k_{coop} z_{M4} + k_{coop} z_{M6} + 3 k_{coop} z_{M9} + 2 k_{off} z_{M10} + k_{off} z_{M11} + 2 k_{off} z_{M12} + 2 k_{off} z_{M13} + 2 k_{off} z_{M14} + 2 k_{off} z_{M17} + k_{off} z_{M19} - k_{off} z_{M23} - 7 k_{off} z_{M26} - k_{off} z_{M4} - 2 k_{off} z_{M6} - k_{off} z_{M8} + k_{on} z_{M12} + k_{on} z_{M14} + k_{on} z_{M18} + k_{on} z_{M20} + k_{on} z_{M23} - 3 k_{on} z_{M26}

\frac{d z_{M27}}{d t} = - 2 k_{coop} z_{M0} + 2 k_{coop} z_{M10} + 2 k_{coop} z_{M13} - 2 k_{coop} z_{M14} + 2 k_{coop} z_{M15} + 2 k_{coop} z_{M16} + 2 k_{coop} z_{M2} - 4 k_{coop} z_{M20} + 2 k_{coop} z_{M24} + 2 k_{coop} z_{M25} - 8 k_{coop} z_{M27} - 2 k_{coop} z_{M28} - 2 k_{coop} z_{M3} + 2 k_{coop} z_{M5} + 6 k_{coop} z_{M6} + 4 k_{coop} z_{M9} - 2 k_{off} z_{M10} + 6 k_{off} z_{M12} - k_{off} z_{M14} + 2 k_{off} z_{M15} + k_{off} z_{M16} + k_{off} z_{M17} + 2 k_{off} z_{M18} + 2 k_{off} z_{M19} - 6 k_{off} z_{M20} - 4 k_{off} z_{M21} + k_{off} z_{M22} + 2 k_{off} z_{M23} + 2 k_{off} z_{M24} - 2 k_{off} z_{M26} - 9 k_{off} z_{M27} - k_{off} z_{M3} - k_{off} z_{M5} + 6 k_{off} z_{M9} - 2 k_{on} z_{M20} + 2 k_{on} z_{M21} - 5 k_{on} z_{M27} + 2 k_{on} z_{M9}

\frac{d z_{M28}}{d t} = - 2 k_{coop} z_{M0} + 2 k_{coop} z_{M10} + 3 k_{coop} z_{M13} - 2 k_{coop} z_{M14} + 2 k_{coop} z_{M15} + 2 k_{coop} z_{M16} + 2 k_{coop} z_{M2} - 3 k_{coop} z_{M20} + k_{coop} z_{M21} + k_{coop} z_{M22} + 2 k_{coop} z_{M24} + 2 k_{coop} z_{M25} - k_{coop} z_{M27} - 7 k_{coop} z_{M28} - 2 k_{coop} z_{M3} + 2 k_{coop} z_{M5} + 6 k_{coop} z_{M6} + 3 k_{coop} z_{M9} + k_{off} z_{M10} + k_{off} z_{M11} + 3 k_{off} z_{M12} + 3 k_{off} z_{M13} + k_{off} z_{M15} + k_{off} z_{M16} + 2 k_{off} z_{M18} + 2 k_{off} z_{M19} - 2 k_{off} z_{M20} - k_{off} z_{M26} - k_{off} z_{M27} - 6 k_{off} z_{M28} + k_{off} z_{M29} - k_{off} z_{M3} - k_{off} z_{M4} - k_{off} z_{M5} + k_{off} z_{M8} + 2 k_{off} z_{M9} + k_{on} z_{M13} - k_{on} z_{M14} - k_{on} z_{M20} + k_{on} z_{M22} + k_{on} z_{M24} - 5 k_{on} z_{M28} + k_{on} z_{M6}

\frac{d z_{M29}}{d t} = 2 k_{coop} z_{M0} - 2 k_{coop} z_{M10} - 2 k_{coop} z_{M13} + 2 k_{coop} z_{M14} - 2 k_{coop} z_{M15} - 2 k_{coop} z_{M16} - 2 k_{coop} z_{M2} + 8 k_{coop} z_{M20} - 2 k_{coop} z_{M24} - 2 k_{coop} z_{M25} + 2 k_{coop} z_{M27} + 4 k_{coop} z_{M28} - 8 k_{coop} z_{M29} + 2 k_{coop} z_{M3} - 2 k_{coop} z_{M5} - 6 k_{coop} z_{M6} - 4 k_{coop} z_{M9} - 2 k_{off} z_{M10} - 2 k_{off} z_{M11} - 4 k_{off} z_{M12} - 4 k_{off} z_{M13} - 2 k_{off} z_{M14} - 2 k_{off} z_{M15} - 2 k_{off} z_{M17} - 2 k_{off} z_{M18} - 2 k_{off} z_{M19} + 2 k_{off} z_{M20} + 2 k_{off} z_{M21} + 2 k_{off} z_{M26} + 2 k_{off} z_{M27} - 7 k_{off} z_{M29} + 2 k_{off} z_{M4} + 4 k_{off} z_{M6} + 2 k_{off} z_{M8} - 2 k_{off} z_{M9} + k_{on} z_{M10} + k_{on} z_{M11} + k_{on} z_{M13} + k_{on} z_{M14} + k_{on} z_{M15} + 3 k_{on} z_{M20} + k_{on} z_{M22} - k_{on} z_{M24} + 3 k_{on} z_{M28} - 4 k_{on} z_{M29} - k_{on} z_{M4} - k_{on} z_{M6}

Plain-text 30D ODE

There are 256 labelled bit states and 30 rotation/reflection classes. This basis has rank 30: it spans the full symmetry-invariant function space, not a compression below that quotient dimension. Special zero-rate choices may admit a smaller minimal subsystem.

Polynomial jump closure (pJC) and EMGF

Generated event model. The ODE uses the displayed motif basis M₀,…,M₂₉; the event model uses the computed basis P₀,…,P₂₉. Rule applications are grouped by their complete coordinate jump.

pJCCLOSED196 automatically derived jump classes

Function algebra30all functions on 30 signatures

Operator span144nested-commutator worklist

EMGF order1linear propensity differential order

D = \sum_{e} k_{e} (e^{λ δ_{e}} - 1) a_{e} (\partial_{λ})

Each event groups rule applications with the same coordinate jump. Its activity is interpolated in the finite function algebra.

Derived event classes · grouped view

The 196 exact event classes are grouped by rule and rate.

Rule / rate	Classes	Distinct δ	Max order	Example class IDs
`basal_bind` basal_bind	72	72	1	`basal_bind-jump-1, basal_bind-jump-2, …`
`cooperative_bind` cooperative_bind	52	52	1	`cooperative_bind-jump-1, cooperative_bind-jump-2, …`
`unbind` unbind	72	72	1	`unbind-jump-1, unbind-jump-2, …`

5 · Initial states

The same symbolic ODE applies to all three starts. Match the displayed motifs, then divide by their fixed scales.

Raw pattern expectation at t=0	Empty	Four consecutive occupied sites	Four alternating occupied sites
M0	0	4	4
M1	8	4	4
M2	0	2	8
M3	0	0	8
M4	0	0	0
M5	0	4	0
M6	0	0	0
M7	0	0	0
M8	0	2	0
M9	0	0	8
M10	0	0	0
M11	0	2	0
M12	0	0	0
M13	0	0	0
M14	0	0	0
M15	0	0	0
M16	0	0	0
M17	0	0	0
M18	0	0	0
M19	0	0	0
M20	0	0	0
M21	0	0	0
M22	0	0	0
M23	0	0	0
M24	0	2	0
M25	0	0	0
M26	0	0	0
M27	0	0	8
M28	0	0	0
M29	0	0	0

Displayed ODE coordinate at t=0	Empty	Four consecutive occupied sites	Four alternating occupied sites
z_M0	0	1/2	1/2
z_M1	1	1/2	1/2
z_M2	0	1/8	1/2
z_M3	0	0	1/2
z_M4	0	0	0
z_M5	0	1/4	0
z_M6	0	0	0
z_M7	0	0	0
z_M8	0	1/8	0
z_M9	0	0	1/2
z_M10	0	0	0
z_M11	0	1/8	0
z_M12	0	0	0
z_M13	0	0	0
z_M14	0	0	0
z_M15	0	0	0
z_M16	0	0	0
z_M17	0	0	0
z_M18	0	0	0
z_M19	0	0	0
z_M20	0	0	0
z_M21	0	0	0
z_M22	0	0	0
z_M23	0	0	0
z_M24	0	1/8	0
z_M25	0	0	0
z_M26	0	0	0
z_M27	0	0	1/2
z_M28	0	0	0
z_M29	0	0	0

For a probability distribution over these single-system experiments, use the same weighted combination of their initial vectors. This is not a physical mixture of several catalysts, chains or rings; that would require a different state language.

6 · Dynamics

Numerical solutions of the exported ODE. Adjust the three rates or select a preset.

Rate program Initial experiment Compare all three starts

Rates from t=0Basal binding · k_on 0.4Cooperative binding · k_coop 0.0Unbinding · k_off 0.6

Time horizon 12

All ODE coordinates for the selected scenario

Rates are illustrative. Curves use numerical matrix exponentials at the selected rates.

Reference plots and parameters

Independent sites

k_on=2/5, k_coop=0, k_off=3/5

Neighbour-enhanced binding

k_on=2/5, k_coop=6/5, k_off=3/5

Binding switched off

k_on=0, k_coop=0, k_off=3/5

Block-four and alternating-four both begin with four occupied sites. With neighbour-enhanced binding their initial mean-occupancy slopes are 1.6 and 8.8; independent-site rates give −0.8 for both. The difference shows the effect of spatial correlations.

Raw pattern expectation at t=0	Empty	Four consecutive occupied sites	Four alternating occupied sites
M0	0	4	4
M1	8	4	4
M2	0	2	8
M3	0	0	8
M4	0	0	0
M5	0	4	0
M6	0	0	0
M7	0	0	0
M8	0	2	0
M9	0	0	8
M10	0	0	0
M11	0	2	0
M12	0	0	0
M13	0	0	0
M14	0	0	0
M15	0	0	0
M16	0	0	0
M17	0	0	0
M18	0	0	0
M19	0	0	0
M20	0	0	0
M21	0	0	0
M22	0	0	0
M23	0	0	0
M24	0	2	0
M25	0	0	0
M26	0	0	0
M27	0	0	8
M28	0	0	0
M29	0	0	0

Raw pattern expectation at t=0	Empty	Four consecutive occupied sites	Four alternating occupied sites
M0	0	4	4
M1	8	4	4
M2	0	2	8
M3	0	0	8
M4	0	0	0
M5	0	4	0
M6	0	0	0
M7	0	0	0
M8	0	2	0
M9	0	0	8
M10	0	0	0
M11	0	2	0
M12	0	0	0
M13	0	0	0
M14	0	0	0
M15	0	0	0
M16	0	0	0
M17	0	0	0
M18	0	0	0
M19	0	0	0
M20	0	0	0
M21	0	0	0
M22	0	0	0
M23	0	0	0
M24	0	2	0
M25	0	0	0
M26	0	0	0
M27	0	0	8
M28	0	0	0
M29	0	0	0

Raw pattern expectation at t=0	Empty	Four consecutive occupied sites	Four alternating occupied sites
M0	0	4	4
M1	8	4	4
M2	0	2	8
M3	0	0	8
M4	0	0	0
M5	0	4	0
M6	0	0	0
M7	0	0	0
M8	0	2	0
M9	0	0	8
M10	0	0	0
M11	0	2	0
M12	0	0	0
M13	0	0	0
M14	0	0	0
M15	0	0	0
M16	0	0	0
M17	0	0	0
M18	0	0	0
M19	0	0	0
M20	0	0	0
M21	0	0	0
M22	0	0	0
M23	0	0	0
M24	0	2	0
M25	0	0	0
M26	0	0	0
M27	0	0	8
M28	0	0	0
M29	0	0	0