Add Mayer K_n recurrence collapse toward closed form (GJ §18.4) - #3498
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Completes the root-component recurrence programme for the complete-graph
connected-spanning signed sum:
- mayer_surviving_set: the C ∋ 0 with |Cᶜ| ≤ 1 are exactly univ and the n-1
cofinite singletons {j}ᶜ (j ≠ 0).
- alternatingConnectedSubgraphSum_completeGraph_recurrence: c_n + (n-1)·c_{n-1} = 0
for n ≥ 2, collapsing the root-component recurrence via D(K_{Cᶜ}) = 0 unless
|Cᶜ| ≤ 1, cardinality-invariance of c, and D_n = 0.
- alternatingConnectedSubgraphSum_completeGraph_closed_form: the general-n Mayer
coefficient identity alternatingConnectedSubgraphSum K_n = (-1)^(n-1)(n-1)! for
n ≥ 1, by induction from the recurrence (base c_1 = 1).
This is the general-n Mayer/Ursell coefficient of the complete-graph cluster
expansion, obtained through the root-component recurrence without any
chromatic-polynomial or matrix-tree machinery (previously marked DEFERRED /
research-level in #1499).
Part of #1499 (Phase B). docs/index.md §18.4 row and tex/proof-guide.tex updated.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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Connects the Mayer K_n closed form (#3498) to the actual Ursell coefficient: - ursellCoefficient_eq_alternatingConnectedSubgraphSum_div: ϕ^T(ω) = c(G(ω))/n!. - polymerSeqIncompatibilityGraph_eq_top_of_pairwise: pairwise incompatibility ⟹ the incompatibility graph is ⊤. - alternatingConnectedSubgraphSum_polymerSeq_complete: the complete cluster's connected-spanning signed sum is (-1)^(n-1)(n-1)!, via the identity iso G(ω) ≃g ⊤ and the closed form. - ursellCoefficient_complete / ursellCoefficient_complete_eq: the classic single-cluster Mayer value ϕ^T(ω) = (-1)^(n-1)(n-1)!/n! = (-1)^(n-1)/n for n pairwise incompatible polymers. Part of #1499. docs/index.md §18.4 row and tex/proof-guide.tex updated. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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) * WIP: Ursell coefficient of a fully-incompatible cluster (GJ §18.4) * Add Ursell coefficient of a fully-incompatible cluster (GJ §18.4) Connects the Mayer K_n closed form (#3498) to the actual Ursell coefficient: - ursellCoefficient_eq_alternatingConnectedSubgraphSum_div: ϕ^T(ω) = c(G(ω))/n!. - polymerSeqIncompatibilityGraph_eq_top_of_pairwise: pairwise incompatibility ⟹ the incompatibility graph is ⊤. - alternatingConnectedSubgraphSum_polymerSeq_complete: the complete cluster's connected-spanning signed sum is (-1)^(n-1)(n-1)!, via the identity iso G(ω) ≃g ⊤ and the closed form. - ursellCoefficient_complete / ursellCoefficient_complete_eq: the classic single-cluster Mayer value ϕ^T(ω) = (-1)^(n-1)(n-1)!/n! = (-1)^(n-1)/n for n pairwise incompatible polymers. Part of #1499. docs/index.md §18.4 row and tex/proof-guide.tex updated. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com> --------- Co-authored-by: Claude Opus 4.8 <noreply@anthropic.com>
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Part of #1499 (Phase B). Collapses the root-component recurrence
D_n = ∑_{C ∋ r} c(K_C)·D(K_{Cᶜ})(#3497) usingD(K_{Cᶜ}) = 0unless|Cᶜ| ≤ 1, leaving the|C| = n(C = univ) and|C| = n-1terms, towardc_n + (n-1)c_{n-1} = 0and the closed formc_n = (-1)^(n-1)(n-1)!.WIP / draft.