feat: numeric per-stage correlation increment (Phase A) — #2965#2989
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Add correlation_pair_sub_inducedGraph_le_derivBound (new PerStageIncrement.lean):
for r,s ∈ S, r≠s, ferromagnetic, h=0, and r,s not on any cut (straddle) edge,
correlation G p {r,s} − correlation (inducedGraph G S) p {⟨r,_⟩,⟨s,_⟩}
≤ derivBound G (G.edgeFinset.filter straddle) p r s.
The full-model pair correlation exceeds the isolated induced-subgraph pair
correlation by at most the ball-boundary derivBound over the cut edges.
Composes the ball-boundary bond-deletion increment
correlation_sub_deleteEdges_le_derivBound (#2974) with the component-factorization
bridge correlation_deleteEdges_straddle_eq_inducedGraph (#2986), via
deleteEdges_filter_edgeFinset_eq (#2987) + correlation_congr_all +
triple_map_subtypeUnivEquiv_eq (#2988). The straddle predicate is @[reducible]
(to match the inline straddle set of the component lemmas under unification) and
the graph/observable bridge is a separate lemma (correlation_deleteEdges_filter_pair_eq)
to keep elaboration within the heartbeat budget.
Part of #2965
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
…s the composition) Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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* feat: nested induced-subgraph graph equality (Part of #2965) * feat: nested induced-subgraph graph equality Add nestedSubtypeEquiv and inducedGraph_induce_preimage_map_eq: for S ⊆ T, ((inducedGraph G T).induce {x : ↥T | x.val ∈ S}).map nestedSubtypeEquiv = inducedGraph G S — inducing box_k inside inducedGraph G box_{k+1} equals, up to the nested-subtype relabeling {x : ↥T // x.val ∈ S} ≃ ↥S, the direct induced subgraph on S. An edge survives iff its deep endpoints are G-adjacent and both in S. Graph-level foundation for instantiating the per-stage increment (correlation_pair_sub_inducedGraph_le_derivBound, #2989) on cubic exhaustion stages box_k ⊆ box_{k+1}. The correlation-level transport across this iso is a follow-up (it currently hits a whnf elaboration wall in correlation_map_equiv on the nested-subtype graph, needing a lighter sum-level encoding). Part of #2965 Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com> --------- Co-authored-by: Claude Opus 4.7 <noreply@anthropic.com>
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…#2992) * feat: two-box per-stage increment (Part of #2965) * Build two-box per-stage increment for cubic exhaustion stages Add `correlation_pair_two_box_le_derivBound`: for nested finsets T₁ ⊆ T₂ and a pair r,s interior to T₁, the pair correlation on the larger box exceeds the one on the smaller box by at most the ball-boundary `derivBound` over the cut edges of the T₁-slice. This is the form instantiated on cubic exhaustion stages box_k ⊆ box_{k+1} to bound the per-stage difference c_{k+1} − c_k (Issue #2965, Phase A). Supporting declarations: - `nestedFinsetEquiv`: the double-subtype relabeling ↥(T₁.subtype (· ∈ T₂)) ≃ ↥T₁. - `correlation_inducedGraph_nested_finset`: concrete-layer companion of `correlation_inducedGraph_induce_preimage` (#2991) using the `Finset.subtype` slice instead of a `Set`-predicate, so it matches the `S : Finset ↥T₂` region of the single-box increment directly and avoids the ↥S vs {x // x.val ∈ T₁} subtype-type mismatch. Composes the single-box increment `correlation_pair_sub_inducedGraph_le_derivBound` (#2989) on G' = inducedGraph G T₂ with the double-induce identification, recovering inducedGraph G T₁ on the inner box. Docs and proof-guide synced. Part of #2965 Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com> * Update module doc Main declarations list (codex cross-check) Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com> --------- Co-authored-by: Claude Opus 4.7 <noreply@anthropic.com>
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The weak derivBound carries a diagonal term ⟨σ_rσ_s⟩⟨σ_kσ_l⟩ whose shell
sum is ⟨σ_rσ_s⟩·O(|∂box_k|), which diverges with shell size and cannot
prove convergence of c_k. Re-derive the entire per-stage increment chain
with the tight ball-boundary Simon-Lieb bound derivBoundTight (cross
products only, via cor_4_3_3_scaled):
- Tight.lean: scaledCorrelation_one_sub_zero_le_derivBoundTight,
correlation_sub_deleteEdges_le_derivBoundTight,
derivBoundTight_le_of_correlation_le.
- PerStageIncrement.lean: correlation_pair_sub_inducedGraph_le_derivBoundTight,
correlation_pair_two_box_le_derivBoundTight.
- CubicPerStageIncrement.lean:
correlationAlongExhaustion_cubic_succ_sub_le_derivBoundTight.
- CubicShellInfiniteVolumeBound.lean:
derivBoundTight_inducedGraph_cubic_le_infiniteVolume_sum.
Net diagonal-free bound:
c_{k+1} − c_k ≤ β·J·∑_{⟨a,b⟩∈shell}[g{r,a}g{s,b} + g{r,b}g{s,a}]
(g = infinite-volume correlation), the summable form on which Phase B
spatial decay yields a geometric rate.
Each tight lemma mirrors its weak counterpart (#2972/#2974/#2989/#2992/
#2993/#2994/#2995); the component-factorization and observable-matching
infrastructure is bound-independent and reused unchanged.
Docs and proof-guide synced.
Part of #2965
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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* feat: tight bond-deletion increment (Part of #2965) * Build tight (diagonal-free) per-stage increment chain The weak derivBound carries a diagonal term ⟨σ_rσ_s⟩⟨σ_kσ_l⟩ whose shell sum is ⟨σ_rσ_s⟩·O(|∂box_k|), which diverges with shell size and cannot prove convergence of c_k. Re-derive the entire per-stage increment chain with the tight ball-boundary Simon-Lieb bound derivBoundTight (cross products only, via cor_4_3_3_scaled): - Tight.lean: scaledCorrelation_one_sub_zero_le_derivBoundTight, correlation_sub_deleteEdges_le_derivBoundTight, derivBoundTight_le_of_correlation_le. - PerStageIncrement.lean: correlation_pair_sub_inducedGraph_le_derivBoundTight, correlation_pair_two_box_le_derivBoundTight. - CubicPerStageIncrement.lean: correlationAlongExhaustion_cubic_succ_sub_le_derivBoundTight. - CubicShellInfiniteVolumeBound.lean: derivBoundTight_inducedGraph_cubic_le_infiniteVolume_sum. Net diagonal-free bound: c_{k+1} − c_k ≤ β·J·∑_{⟨a,b⟩∈shell}[g{r,a}g{s,b} + g{r,b}g{s,a}] (g = infinite-volume correlation), the summable form on which Phase B spatial decay yields a geometric rate. Each tight lemma mirrors its weak counterpart (#2972/#2974/#2989/#2992/ #2993/#2994/#2995); the component-factorization and observable-matching infrastructure is bound-independent and reused unchanged. Docs and proof-guide synced. Part of #2965 Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com> * Weaken convergence wording to convergence-rate bound (codex cross-check) Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com> --------- Co-authored-by: Claude Opus 4.7 <noreply@anthropic.com>
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Part of #2965. Closes the numeric per-stage increment — the culmination of the Phase A component-factorization bridge.
Adds
correlation_pair_sub_inducedGraph_le_derivBound(newPerStageIncrement.lean, importsWeakBound+InducedUnion): forr,s ∈ S,r≠s, ferromagnetic,h=0, andr,snot on any cut edge,correlation G p {r,s} − correlation (inducedGraph G S) p {⟨r,_⟩,⟨s,_⟩} ≤ derivBound G (G.edgeFinset.filter straddle) p r s.Composes #2974 (ball-boundary bond-deletion increment) with #2986 (component-factorization bridge) via #2987 (Finset↔Set deletion) +
correlation_congr_all+ #2988 (observable identity).Elaboration-performance resolution (the earlier monolithic
calctimed out): the straddle predicate is@[reducible](matches the inline straddle set of the component lemmas under unification), and the graph/observable bridge is a separate lemma to keep each elaboration within the heartbeat budget.lake build IsingModel.AmbientLatticeSum.PerStageIncrementgreen, sorry 0, file linter-clean, tex compiles.🤖 Generated with Claude Code