GJ §17.5 Lemma 17.5.2 upper side: finite-volume β-derivative convergence-rate provider (volume coupling)

[phasetr]

Purpose

Discharge the single remaining genuine analytic input of GJ §17.5 Lemma 17.5.2
upper side: the derivative-limit provider Lemma_17_5_2_DerivativeLimitProvider,
i.e. local uniform convergence of the finite-volume β-derivative profiles
F_n(β) = ∂_β⟨σ_xσ_z⟩_{Λ_n} to the infinite-volume derivative g'(β).

After #2931 (decay/clustering/susceptibility, now complete) and the conditional
reduction (#2929–#2942), this is the only remaining input: it is the
finite-volume → infinite-volume convergence rate of the β-derivatives, distinct
from the (already-built) spatial decay of the infinite-volume correlation.

Background

• The conditional capstone lemma_17_5_2_capstone_of_poly_geometric_increments_on_covered_stages_and_pseudoMass_le_rate
  (Add poly-geometric fully-concrete capstone (rate infra Phase 3) #2942) reduces the full sandwich to: a summable/geometric bound on the
  consecutive β-derivative increments |F_{k+1}(β) − F_k(β)|, plus
  m⁻(β₂) ≤ -log(β₂J·2d).
• The increment F_{k+1} − F_k is a difference of finite Lebowitz edge sums over
  the different induced graphs G_{k+1} and G_k; the naive finite-volume
  distance decay carries a divergent 2^(edge count) prefactor.
• The mechanism is a finite-volume Simon–Lieb / coupling difference identity:
  corr_∞{x,z} − corr_{Λ_n}{x,z} is controlled by boundary terms
  ∑_{b ∈ ∂Λ_n} corr_∞{x,b}·corr_∞{b,z}, which the (now-built) infinite-volume
  spatial decay makes summable and small.

Tracking (PRs)

• Phase A (foundation, feat: scaledCorrelation difference bound (boundary MVT step, no separation) — #2965 #2972–feat: correlation invariance under graph isomorphism — #2965 #2976): ball-boundary MVT difference bound, s=0 = bond-deleted model, concrete bond-deletion increment, disjoint-sum correlation factorization, correlation iso-invariance. Remaining: deleteEdges(cross)≅inducedGraph(Λ)⊕inducedGraph(Λᶜ) graph-iso bridge to exhaustion stages.
• Phase A (orig): finite-volume vs infinite-volume correlation difference identity /
  bound via a separating-surface (ball-boundary) Simon–Lieb step on the box Λ_n.
• Phase B (partial, PR Add unconditional high-temp convolution summability (Issue #2965 Phase B) #2966 boundary-sum summability): bound the boundary sum by the infinite-volume spatial decay
  (correlationInfinite_latticeGraph_le_contractionFactor_pow_dist_pair,
  _susceptibility_summable) → geometric/summable volume-convergence rate of the
  correlations.
• Phase C: transfer the correlation convergence rate to the β-derivative
  increments (the Lebowitz edge-sum difference), giving the
  |F_{k+1}−F_k| ≤ M·(2k+3)^d·ratio^k input.
• Phase D: feed
  lemma_17_5_2_capstone_of_poly_geometric_increments_on_covered_stages_and_pseudoMass_le_rate
  and close Lemma 17.5.2 Step 117l-e/f.

References

• Glimm–Jaffe, Quantum Physics (2nd ed.), §5.1 pp. 74–79, §17.5 Lemma 17.5.2
  pp. 311–312; §17.8.
• Part of GJ §17.5 Lemma 17.5.2 (full): substantive bridge from pseudoMass to latticeMass #1645. Builds on GJ §17.5 Lemma 17.5.2: finite-volume β-derivative convergence-rate infrastructure (Step 117l-e core) #2931.