feat(gj-18.4): high-temperature convergence of the Ising cluster expansion — #3954 - #3997
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…nsion — #3954 Specialise the Mayer absolute convergence (#3996) to the Ising activity t = tanh(β·J): - summable_mayerExpansionTerm_tanh: Summable (fun n => mayerExpansionTerm G n (tanh βJ)) for e·|allPolymers G|·|tanh(βJ)| < 1 (via |tanh| < 1 and #3996's _of_card_mul_lt); - summable_mayerExpansionTerm_tanh_ferro: ferromagnetic 0 ≤ β·J form; - summable_mayerExpansionTerm_tanh_of_activity: activity-sum form; - tanh_activity_sum_le_card_mul_tanh: ∑_P tanh(βJ)^|P| ≤ |allPolymers G|·tanh(βJ) for βJ ≥ 0. Since tanh is continuous with tanh 0 = 0, the criterion holds at high temperature (β·J near 0): the Ising cluster expansion converges absolutely at high temperature. New file MayerCore/MayerTanhConvergence.lean (4 theorems). Updates docs/index.md and tex/proof-guide.tex. Part of #3954. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
This was referenced Jun 14, 2026
GJ §18.4–18.5: general interacting cluster-expansion convergence (Kotecký–Preiss / tree-graph)
#3954
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Part of #3954 (milestone M2 — general interacting cluster-expansion convergence).
Summary
Specialise the Mayer absolute convergence (#3996) to the Ising activity
t = tanh(β·J), giving the high-temperature convergence of the Ising cluster expansion.Declarations (
MayerCore/MayerTanhConvergence.lean)summable_mayerExpansionTerm_tanh—Summable (fun n => mayerExpansionTerm G n (tanh(β·J)))fore·|allPolymers G|·|tanh(β·J)| < 1(via|tanh| < 1and feat(gj-18.4): absolute convergence of the Mayer expansion — #3954 #3996'ssummable_mayerExpansionTerm_of_card_mul_lt).tanh_activity_sum_le_card_mul_tanh— for0 ≤ β·J,∑_P tanh(β·J)^|P| ≤ |allPolymers G|·tanh(β·J).summable_mayerExpansionTerm_tanh_ferro— ferromagnetic form (0 ≤ β·J,e·|allPolymers G|·tanh(β·J) < 1).summable_mayerExpansionTerm_tanh_of_activity— activity-sum form.Since
tanhis continuous withtanh 0 = 0, the criterion holds at high temperature (β·Jnear0). The remaining step is the Mayer–Montroll identitylog Ξ = ∑ₙ mayerExpansionTerm(Issue #1499 Phase C).Verification
lake buildclean (linter warnings zero);lake exe GKSTestpasses; nosorry.tex/proof-guide.texcompiles (lualatex), no Japanese.docs/index.md(§18.4 row) andtex/proof-guide.texupdated.🤖 Generated with Claude Code