feat(gj-17.5.1): conditional finite-region Lipschitz of m⁻(σ,A)^{2α+1} on the window — #4320 - #4332
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…} on the window — #4320 Upgrade the finite-region (distance-parametrized) pseudo-mass continuity to a Lipschitz estimate of its (2α+1)-power, for each fixed bounded region A, on the convergence window, conditionally on the faithful per-pair profile lower bounds. - pseudoMassFromParamsAtPairDist_eq_atPair_cubic: bridge (distance-radius = fixed-radius pseudo-mass at r=dist, cubic exhaustion → canonical Fintype instances; a general-Λ bridge is blocked by a pseudoMassExt dite defeq divergence on the synthesized-vs-passed instance mismatch, documented #4320). - pseudoMassFromParamsAtPairDist_pow_succ_lipschitz_on_window_of_profile_lower: per-pair distance Lipschitz = #4331 at ρ:=dist, constant (2α+1)K/dist. - finiteRegionPseudoMassDist_pow_succ_lipschitz_on_window_of_profile_lower: ∃ C>0, |m⁻(σ₂,A)^{2α+1} − m⁻(σ₁,A)^{2α+1}| ≤ C(β₂−β₁). (inf')^{2α+1} = inf'((·)^{2α+1}) (odd power monotone, Finset.comp_inf'_eq_inf'_comp) + inf' of finitely-many Lipschitz via choose! + Finset.sup' + achieved infimum. Axiom-free [propext, Classical.choice, Quot.sound]. Partial/conditional (per-pair hprofile faithful distance form, ∀-displacement false #4270; constant per-A, uniform-in-A/infinite-envelope continuity does not follow, #4320). Builds on #4330/#4331. docs/index.md §17.5 row + tex/proof-guide.tex updated. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
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…nd on the window — #4320 (#4333) Prove unconditionally the faithful per-pair correlation lower bound that gated the §17.5.1 conditional chain (#4330–#4332): pseudoMassG α (dist x z) (−log tanh(βJ)) ≤ ⟨φ_x φ_z⟩^∞ for every distinct pair x ≠ z on β ∈ ConvergenceRegion.window d J. Key: use the faithful inverse-correlation-length rate −log tanh(βJ) (not the slower Simon–Lieb rate −log(βJ·2d) whose ∀-displacement form is false, #4270). - pseudoMassG_le_exp_neg_of_one_le: 1 ≤ t·r ⇒ pseudoMassG α r t ≤ e^{−tr}. - tanh_betaJ_lt_exp_neg_one_of_window: on the window tanh(βJ) < R d ≤ e⁻¹ (R d = min(…) ≤ 1/(64((2d)²+1)e) ≤ e⁻¹). - one_le_neg_log_tanh_betaJ_of_window: hence −log tanh(βJ) ≥ 1. - so q·dist ≥ 1 ⇒ pseudoMassG α (dist) q ≤ e^{−q·dist} = tanh(βJ)^dist. - GKS direct-path twoPointFunction_ge_tanh_betaJ_pow_dist: tanh^dist ≤ ⟨φ₀φ_z⟩. - translation (correlationInfinite_latticeGraph_pair_eq_twoPointFunction, latticeDistance_translate_eq) lifts anchored → general pair. Theorems: pseudoMassG_dist_tanh_rate_le_correlationInfinite_cubic (general), ..._cubic_zero (anchored), pseudoMassG_le_exp_neg_of_one_le, tanh_betaJ_lt_exp_neg_one_of_window, one_le_neg_log_tanh_betaJ_of_window. Axiom-free [propext, Classical.choice, Quot.sound]. The per-pair hprofile that gated #4330–#4332 is now unconditional on the window; re-parametrizing that chain to the −log tanh(βJ) rate (next PR) yields the unconditional finite-region Lipschitz. docs/index.md §17.5 row + tex/proof-guide.tex updated. Co-authored-by: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
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…+1} — #4320 (#4334) Remove the hprofile hypothesis from the conditional finite-region Lipschitz (#4332): finiteRegionPseudoMassDist_pow_succ_lipschitz_on_window — for a fixed bounded region A and Icc β₁ β₂ ⊆ ConvergenceRegion.window d J, with NO profile hypothesis, ∃ C>0, |m⁻(σ₂,A)^{2α+1} − m⁻(σ₁,A)^{2α+1}| ≤ C·(β₂−β₁). Route: - pseudoMassFromParamsAtPair_pow_succ_lipschitz_on_window_of_ratio_lower: rate-agnostic engine (extracts #4331, taking the interval-uniform ratio lower bound as a hypothesis). - pseudoMassFromParamsAtPair_ratio_lower_of_pseudoMassG_le_corr: general-rate #4330 via pseudoMass_le_iff_pseudoMassG_le. - pseudoMassFromParamsAtPairDist_pow_succ_lipschitz_on_window: UNCONDITIONAL per-pair distance interval Lipschitz, discharging hprofile at the faithful rate −log tanh(βJ) via #4333; Lmin = pseudoMassG α (dist) q₁/q₁^{2α} interval-uniform by monotonicity of q(β)=−log tanh(βJ) (Real.tanh_strictMono) + pseudoMassG_antitoneOn. - finite Finset.inf' assembly (odd-power commutes with inf'; inf' of finitely-many Lipschitz via choose! + Finset.sup' + achieved infimum), as in #4332. Axiom-free [propext, Classical.choice, Quot.sound]. The §17.5.1 finite-region Lipschitz is now unconditional (conditional #4330–#4332 subsumed). Remaining: infinite-envelope globalPseudoMassDist continuity (#4320) and true-mass continuity (#4081). docs/index.md §17.5 row + tex/proof-guide.tex updated. Co-authored-by: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
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correlation length Three sites asserted that `−log tanh(βJ)`, or a finite-region infimum of per-pair pseudo-masses, *is* the inverse correlation length. The tree proves only bounds: `latticeMass_le_neg_log_tanh_betaJ`, `latticeMass_two_sided_bound`, and `onAxisInverseCorrelationLength_le_neg_log_tanh` are all inequalities, and no equality with that rate exists anywhere. * `UnconditionalFiniteRegionLipschitz.lean` header: "faithful inverse-correlation-length rate" -> "direct-path rate", the name the neighbouring `LatticeMassHighTemperature/UpperBound.lean` already uses. * `finiteRegionPseudoMassDist`: the claim that the finite infimum "is the genuine inverse correlation length restricted to `A`" is replaced by the faithful-radius statement plus an explicit note that no relation to `latticeMass` is proved here. Measured at the merge base: the file sets mentioning `finiteRegionPseudoMassDist` (24) and `latticeMass` (85) intersect in zero files. * `docs/index.md` #4333 row: it simultaneously claimed the `hprofile` of #4330-#4332 "is now an unconditional theorem" and, in parentheses, that the theorem is at a different rate. The row now states that the unconditional bound does not discharge that binder -- `#check` shows the binder is `pseudoMassG α ρ (−log(βJ·2d))` at a fixed radius while the theorem is `pseudoMassG α (latticeDistance d x z) (−log tanh(βJ))` -- and names the rate-agnostic engine as what it does discharge. Lean changes are comment-only. `citation_audit.py` is unaffected: the findings/ratchet counts are identical with the merge base's `docs/index.md`. Refs #5005 Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>
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…unds do exist Round-1 review follow-up on this branch's own prose fixes. docs/index.md, the #4333 row: * the title asserted it discharges the section 17.5 `hprofile` while the body denied it; retitle to the direct-path rate and to the engine's ratio hypothesis, matching the body; * "never by an equality" was false: `latticeMass_one_eq_correlationMass` proves `latticeMass 1 (cubicExhaustion 1) = ofReal(correlationMass (beta*J))` with `correlationMass a = -log tanh a`. State the true shape: upper bounds only for general `d` (`latticeMass_le_neg_log_tanh_betaJ`, bundled in `latticeMass_two_sided_bound`, plus the sharper on-axis inequality), equality in `d = 1` only; * the `hprofile` binder is a fixed radius in #4330/#4331 but the pair distance in #4332; the rate, not the radius, is what separates them, so replace the hedge with the source module's own ruling that this bound is strictly weaker; * the rate-agnostic engine's `hratio` carries no rate; #4333 discharges the profile bound that supplies it. docs/index.md, the #4334 row: `hprofile` named two different objects in one row; the second occurrence is now described as the engine's ratio hypothesis. `MagnetizationInfiniteSusceptibility.lean`: the claim that no stage-uniform bound exists and that `susceptibility_nonneg` is the only sign/size fact is refuted by `susceptibilityAlongExhaustion_le_of_high_temp`, `susceptibilityAlongExhaustion_bddAbove_latticeGraph_of_high_temp` (which discharges the `BddAbove` hypothesis this same docstring points at) and `susceptibilityInfinite_J_zero`. Also align "grows" with the module header's "nondecreasing". `FiniteRegionPseudoMassDistContinuity.lean`: the global envelope reaches `latticeMass` from below with no constant (`globalPseudoMassDist_le_latticeMass`); only the reverse direction carries one. Terminology: "faithful" named the radius elsewhere in the tree but was also used for the tanh rate; the rate is now uniformly "direct-path" across `UnconditionalFiniteRegionLipschitz.lean`, `MassContinuityPairMassUpperIcc.lean` and their `docs/index.md` rows. Shorthand legends: cover `vdSum_tanh`, the parenthesised-activity forms and the section comments that use them, and introduce the forms in the two Mayer modules that used them without one. Comment-only in Lean; `lake build` clean. Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>
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Part of #4320. Builds on PR #4330 (pointwise hcomp) + #4331 (per-pair interval Lipschitz). Unblocks the finite-region step previously reported blocked in #4320 (the AtPairDist↔AtPair bridge defeq wall — resolved by specializing the bridge to the cubic exhaustion so the
Fintype edgeSetinstances are canonical).Summary
Upgrades the finite-region (distance-parametrized) pseudo-mass continuity to a Lipschitz estimate of its (2α+1)-power, for each fixed bounded region A, on the convergence window, conditionally on the faithful per-pair profile lower bounds. This is the GJ §17.5 Lemma 17.5.2(a) / Theorem 17.5.1 intermediate-Lipschitz claim restricted to a finite region.
finiteRegionPseudoMassDist_pow_succ_lipschitz_on_window_of_profile_lower: forIcc β₁ β₂ ⊆ ConvergenceRegion.window d Jand per-pairhprofile(one per distinct pair of A),∃ C>0, |m⁻(σ₂,A)^{2α+1} − m⁻(σ₁,A)^{2α+1}| ≤ C·(β₂−β₁).Route (3 lemmas)
pseudoMassFromParamsAtPairDist_eq_atPair_cubic: distance-radius = fixed-radius pseudo-mass atr = dist, cubic exhaustion (canonicalFintype edgeSetinstances; a general-Λ bridge diverges inpseudoMassExt'sditedefeq on the synthesized-vs-passed instance mismatch — documented in GJ Theorem 17.5.1: mass continuity — uniform-in-(σ,A) Lipschitz of m⁻^{2α+1} (proof content; final m-continuity transfer-principle-blocked) #4320).pseudoMassFromParamsAtPairDist_pow_succ_lipschitz_on_window_of_profile_lower: PR feat(gj-17.5.1): conditional interval Lipschitz of (m⁻)^{2α+1} on the convergence window — #4320 #4331 instantiated at ρ:=dist, constant (2α+1)K/dist.(inf')^{2α+1} = inf'((·)^{2α+1})(odd power monotone,Finset.comp_inf'_eq_inf'_comp); inf' of finitely-many Lipschitz functions is Lipschitz with constantFinset.sup'of the per-pair constants (choose!+ achieved infimumFinset.exists_mem_eq_inf'+Finset.inf'_le).Status: Partial / conditional (honest)
Per-pair
hprofileis the faithful distance form (∀-displacement false, #4270). The constant is per-A — uniform-in-A / infinite-envelopeglobalPseudoMassDistcontinuity does not follow (per-pair constant (2α+1)K/dist uncontrolled as diam A→∞). Unconditional headline =globalPseudoMassDist_fullSandwich(#4317).Verification
lake build✓ (5541 jobs),lake exe GKSTest✓#print axioms = [propext, Classical.choice, Quot.sound](both new public theorems); linter zerodocs/index.md§17.5 row +tex/proof-guide.texupdated; TeX compiles (492 pp), no Japanese🤖 Generated with Claude Code