The Subgraph Lattice Theorem and Geometric Partition Counting

The Core Idea

Classical combinatorics counts the number of ways to partition a set of n elements into non-empty blocks. That count is the Bell number B(n), and it depends only on how many elements there are, never on how they relate to each other.

This manuscript introduces a strict refinement. Given a connected graph G on n labeled vertices, define a connected set-partition as a partition of V(G) in which every block induces a connected subgraph. The geometric partition function is

p₍𝔾₎(G) = |{ {V₁, …, Vₖ} : each G[Vᵢ] is connected }|

This count depends on G, not just n. Two graphs on the same number of vertices can have wildly different geometric partition counts. The entire theory follows from taking that dependence seriously.

Main Theorems

Theorem 1 (Subgraph Lattice Theorem). If G and H share the same vertex set and E(G) ⊆ E(H), then p₍𝔾₎(G) ≤ p₍𝔾₎(H). Adding edges can only increase the connected partition count, never decrease it.

Theorem 2 (Tree Evaluation). For any tree T on n vertices, p₍𝔾₎(T) = 2ⁿ⁻¹. Every tree on n vertices, regardless of shape, has exactly the same geometric partition count. This is the universal floor for connected graphs.

Theorem 3 (Complete-Graph Evaluation). For Kₙ, p₍𝔾₎(Kₙ) = B(n), the n-th Bell number. When every possible edge is present, every set-partition is automatically connected, so the geometric count recovers the classical count. This is the universal ceiling.

Theorem 4 (Cycle Closed Form). For the cycle graph Cₙ with n ≥ 3, p₍𝔾₎(Cₙ) = 2ⁿ − n. This matches OEIS sequence A000325.

Corollary (Sandwich Hierarchy). For every connected graph G on n vertices with n ≥ 4:

2ⁿ⁻¹ = p₍𝔾₎(Tₙ) < p₍𝔾₎(Cₙ) = 2ⁿ − n < p₍𝔾₎(Kₙ) = B(n)

Both inequalities are strict. Trees sit at the floor, complete graphs at the ceiling, and the cycle sits strictly between them.

What the Numbers Look Like

For cycles versus trees versus complete graphs:

• n = 5: Tree = 16, Cycle = 27, Complete = 52
• n = 6: Tree = 32, Cycle = 58, Complete = 203
• n = 7: Tree = 64, Cycle = 121, Complete = 877
• n = 8: Tree = 128, Cycle = 248, Complete = 4,140
• n = 10: Tree = 512, Cycle = 1,014, Complete = 115,975

The gap between floor and ceiling grows super-exponentially. At n = 10, the complete graph has 227× more connected partitions than a tree on the same vertices.

At fixed n, different topologies produce sharply different counts. At n = 6 for example:

P₆ = 32 < C₆ = 58 < Ladder = 74 < Fan = 89 < K₂,₄ = 96 < Prism = 114 < W₆ = 118 < K₆ = 203

The ordering tracks graph density with Spearman ρ = +0.955 (p < 10⁻⁶) across all tested sizes n = 4 through n = 10.

The Connection Curvature

Define the partition connection on cycles as Γₙ = p₍𝔾₎(Cₙ₊₁) / p₍𝔾₎(Cₙ). The curvature κₙ = Γₙ − 2 measures deviation from the flat tree-growth rate:

κₙ = (n − 1) / (2ⁿ − n)

This is strictly positive and monotonically decaying to zero:

• n = 3: κ = 0.4000
• n = 5: κ = 0.1481
• n = 7: κ = 0.0496
• n = 10: κ = 0.0089

The cycle approaches tree-like growth exponentially fast. The successive ratio κₙ / κₙ₋₁ converges toward ½ from above.

Empirical Validation Program

The deposit includes a preregistered validation suite with predeclared falsifiers, frozen metrics, and external ground truth. All outcomes, including failures, are retained as first-class results.

Test 1 (OEIS Path Recursion). Brute-force enumeration confirms p₍𝔾₎(Pₙ) = 2ⁿ⁻¹ for n = 1 through 20. The sequence matches OEIS A000079 (powers of 2) exactly. Algebraic DP and direct bitmask enumeration agree at every tested value.

Test 2b (Same-n Topology Comparison). At each n from 4 to 10, connected graphs on the same vertex count were compared. Spearman rank correlation between vertex connectivity and p₍𝔾₎ is ρ ≥ 0.83 at every tested n. Pooled edge-density correlation: ρ = 0.955 with p < 10⁻⁶. Negative control (max degree): ρ ≈ 0.35, not significant at any n. The lattice theorem's monotonicity prediction holds empirically, and counterexamples to vertex-connectivity ordering exist only between spanning-subgraph-incomparable pairs, exactly as the theory permits.

Test 4 (Internet Autonomous Systems, 2010–2024). Using 180 monthly CAIDA BGP snapshots: AS count grew from 33,778 to 77,495 while edges grew from 93,998 to 501,473. The edge-succession to vertex-succession ratio Sₑ/Sᵥ increased with a linear slope of 3.68 per year, R² = 0.88, p = 2.23 × 10⁻⁷, Spearman ρ = 0.968. The pre-registered threshold was a slope above 0.05. The observed slope is 73× that threshold. Edge growth dominates vertex growth in a real infrastructure graph, consistent with the lattice theorem's prediction that denser graphs have exponentially more connected partitions.

Test 5 (OpenAlex Citation Subgraphs). Among 10,000 co-citation subgraphs from 2020–2024: only 3 out of 10,000 were path graphs (0.03%). Among connected subgraphs, paths were 0.47%. For n ≥ 7, zero path graphs appeared among 527 connected subgraphs. Real citation networks avoid the tree floor almost entirely.

Test 6/6b (Universal Dependencies Treebanks). The primary test fired its falsifier: English had 19.63% path-topology sentences, exceeding the 10% threshold. However, the pre-registered follow-up (Test 6b) conditioning on n ≥ 5 tokens found all six languages below 10%, with a maximum of 1.57% (Russian). At n ≥ 7 tokens, all languages drop below 0.15%. Short sentences are path-like; longer sentences are not.

Test 7 (C. elegans Connectome). 279 neurons, 1,961 synapses. The biological network sits near the tree floor: 81% of 4-vertex connected induced subgraphs achieve p₍𝔾₎ = 2ⁿ⁻¹ exactly. KS tests showed no significant difference from Erdős–Rényi at matched density. Pre-registered hypothesis not supported. The sandwich bound is tight in sparse biological networks.

Test 10 (Crystallographic Space Groups). Commutation graphs of all 230 space groups tested against a Ramsey proxy bound. The bound holds for 230/230 groups (100%). Saturation target (≥ 3 groups at n = 5 with zero triangles and zero independent triples) was not achieved. Verdict: partial.

Tests 2, 8 (Falsifier Fired). Cycloalkane ring strain shows no monotonic observable matching p₍𝔾₎ (Pearson r = −0.11, p = 0.79). KEGG metabolic pathways achieved 14/18 hits for the Sₑ ≥ 2 × Sᵥ criterion versus a target of 15. Both failures are preserved in the record.

Validation Ledger

• Supported: Tests 2b, 4, 5, 6b (4 of 10 completed tests)
• Falsifier fired: Tests 1 (by design), 2, 6, 8 (4 of 10)
• Not supported: Test 7 (1 of 10)
• Partial: Test 10 (1 of 10)
• Blocked/Deferred: Tests 9, 11

Negative and null results are first-class outcomes, not omissions. The program-level verdict is assessed across the full ledger, not cherry-picked from successes.

Limitations

Some tests depend on external APIs (OpenAlex, CAIDA, KEGG) whose upstream data evolves. Test 9 is blocked by endpoint observability constraints. This archive captures a dated snapshot; future reruns under changed data conditions should be treated as independent replications, not as invalidations.

Citation

Cite the Zenodo DOI for this archived package and the manuscript title. If reusing specific test scripts or results, cite the relevant script and JSON artifact in your methods section.