From 41a7b411a7edc50fdf34b336b0c204b394632d76 Mon Sep 17 00:00:00 2001 From: dmurfet Date: Wed, 6 May 2026 19:33:45 +0000 Subject: [PATCH 1/6] Tide gibbscov-algebra: generic algebra for gibbsExpectation and gibbsCov MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Add the algebraic infrastructure for `gibbsCov` so that affine/multilinear strict-improvements (e.g. C1 harmonic-affine, G2 anharmonic-affine, the deferred I3 affine-covariance template from Tide 12) become near-trivial. Lemmas added in parallel to `Laplace/Gibbs.lean` (1D, scalar) and `Laplace/Multi/Basic.lean` (multi, `(ฮน โ†’ โ„)`): Expectation level: - `gibbsExpectation_smul` (no hypotheses) - `gibbsExpectation_zero` (no hypotheses) - `gibbsExpectation_add` (Integrable hypotheses) - `gibbsExpectation_const` for the multi side (1D version pre-existing) Covariance level: - `gibbsCov_symm` (no hypotheses) - `gibbsCov_smul_left/right` (no hypotheses) - `gibbsCov_const_left/right` (unconditional via Z=0 case-split) - `gibbsCov_add_left/right` (Integrable hypotheses) - `gibbsCov_zero_left/right` simp corollaries `gibbsCov_const_*` is unconditional because when `Z(t) = 0` every expectation collapses to `0` via the `_/0 = 0` convention; otherwise the proof routes through `gibbsExpectation_const`. Both Claude and GPT-5.5 Pro voted Candidate B (algebra mirrored across both base files) over Candidate A (1D only) and Candidate C (B + the affine-bilinear corollary, which belongs in I3). Direct `Integrable` hypotheses, no `GibbsIntegrable` typeclass, no `LinearMap` refactor. Tide log: `projects/primer/tide-log/2026-05-07-tide-gibbscov-algebra.md` GPT consult: `projects/primer/tide-log/gpt55_tide_gibbscov_algebra_v1.md` ๐ŸŒŠ Generated with the [Tide skill](https://github.com/timaeus-research/sri/blob/main/.claude/skills/tide/SKILL.md) Co-Authored-By: Claude Opus 4.7 (1M context) --- Laplace/Gibbs.lean | 121 ++++++++++++++++++++++++++++++++++++++ Laplace/Multi/Basic.lean | 124 +++++++++++++++++++++++++++++++++++++++ 2 files changed, 245 insertions(+) diff --git a/Laplace/Gibbs.lean b/Laplace/Gibbs.lean index cb0cc2c..046b574 100644 --- a/Laplace/Gibbs.lean +++ b/Laplace/Gibbs.lean @@ -62,4 +62,125 @@ lemma gibbsExpectation_const (L : โ„ โ†’ โ„) (t : โ„) (c : โ„) rw [integral_const_mul c (fun x => Real.exp (-(t * L x)))] field_simp +/-! ## Algebraic infrastructure for `gibbsExpectation` and `gibbsCov` + +The lemmas below give the bilinearity / scalar-pulling / constant-collapse +facts needed to manipulate Gibbs expectations and covariances of affine and +multilinear observables without unfolding the definitions. They are used +downstream by the affine-observable covariance lemmas (e.g. +`Laplace.OneD.Quartic.gibbsCov_first_order_rate_sharp` and the 2D analogues). + +`Integrable` hypotheses are stated directly rather than bundled into a +typeclass: at the algebraic level, only `MeasureTheory.integral_add` requires +them, and the typeclass abstraction belongs at the layer where differentiation +under the integral is the load-bearing operation. -/ + +/-- Scalar-multiplication pulls out of the Gibbs expectation. No hypotheses: +when `Z(t) = 0` both sides are zero. -/ +lemma gibbsExpectation_smul (L : โ„ โ†’ โ„) (t : โ„) (c : โ„) (ฯ† : โ„ โ†’ โ„) : + gibbsExpectation L t (fun x => c * ฯ† x) = c * gibbsExpectation L t ฯ† := by + simp only [gibbsExpectation] + rw [show (fun x => c * ฯ† x * Real.exp (-(t * L x))) + = (fun x => c * (ฯ† x * Real.exp (-(t * L x)))) from by funext x; ring, + integral_const_mul c (fun x => ฯ† x * Real.exp (-(t * L x))), + mul_div_assoc] + +/-- The Gibbs expectation of the zero observable is zero, unconditionally. -/ +lemma gibbsExpectation_zero (L : โ„ โ†’ โ„) (t : โ„) : + gibbsExpectation L t (fun _ => 0) = 0 := by + simp [gibbsExpectation] + +/-- Additivity of the Gibbs expectation: requires integrability of each +weighted observable. -/ +lemma gibbsExpectation_add (L : โ„ โ†’ โ„) (t : โ„) (ฯ†โ‚ ฯ†โ‚‚ : โ„ โ†’ โ„) + (hโ‚ : Integrable (fun x => ฯ†โ‚ x * Real.exp (-(t * L x)))) + (hโ‚‚ : Integrable (fun x => ฯ†โ‚‚ x * Real.exp (-(t * L x)))) : + gibbsExpectation L t (fun x => ฯ†โ‚ x + ฯ†โ‚‚ x) + = gibbsExpectation L t ฯ†โ‚ + gibbsExpectation L t ฯ†โ‚‚ := by + simp only [gibbsExpectation] + rw [show (fun x => (ฯ†โ‚ x + ฯ†โ‚‚ x) * Real.exp (-(t * L x))) + = (fun x => ฯ†โ‚ x * Real.exp (-(t * L x)) + + ฯ†โ‚‚ x * Real.exp (-(t * L x))) from by funext x; ring, + integral_add hโ‚ hโ‚‚, add_div] + +/-- Symmetry: `Cov_t[ฯ†, ฯˆ] = Cov_t[ฯˆ, ฯ†]`. -/ +lemma gibbsCov_symm (L : โ„ โ†’ โ„) (t : โ„) (ฯ† ฯˆ : โ„ โ†’ โ„) : + gibbsCov L t ฯ† ฯˆ = gibbsCov L t ฯˆ ฯ† := by + simp only [gibbsCov] + rw [show (fun x => ฯ† x * ฯˆ x) = (fun x => ฯˆ x * ฯ† x) from by funext x; ring, + mul_comm (gibbsExpectation L t ฯ†)] + +/-- Scalar pulls out of the left slot. No hypotheses. -/ +lemma gibbsCov_smul_left (L : โ„ โ†’ โ„) (t : โ„) (c : โ„) (ฯ† ฯˆ : โ„ โ†’ โ„) : + gibbsCov L t (fun x => c * ฯ† x) ฯˆ = c * gibbsCov L t ฯ† ฯˆ := by + simp only [gibbsCov] + rw [show (fun x => c * ฯ† x * ฯˆ x) = (fun x => c * (ฯ† x * ฯˆ x)) from + by funext x; ring, + gibbsExpectation_smul, gibbsExpectation_smul] + ring + +/-- Scalar pulls out of the right slot. -/ +lemma gibbsCov_smul_right (L : โ„ โ†’ โ„) (t : โ„) (c : โ„) (ฯ† ฯˆ : โ„ โ†’ โ„) : + gibbsCov L t ฯ† (fun x => c * ฯˆ x) = c * gibbsCov L t ฯ† ฯˆ := by + rw [gibbsCov_symm, gibbsCov_smul_left, gibbsCov_symm] + +/-- Constants in the left slot give zero covariance. Unconditional: when +`Z(t) = 0` every Gibbs expectation collapses to `0`, so both sides agree. -/ +lemma gibbsCov_const_left (L : โ„ โ†’ โ„) (t : โ„) (c : โ„) (ฯˆ : โ„ โ†’ โ„) : + gibbsCov L t (fun _ => c) ฯˆ = 0 := by + by_cases hZ : partitionFunction L t = 0 + ยท simp [gibbsCov, gibbsExpectation, partitionFunction] at hZ โŠข + simp [hZ] + ยท simp only [gibbsCov] + rw [show (fun x => (fun _ => c) x * ฯˆ x) = (fun x => c * ฯˆ x) from rfl, + gibbsExpectation_smul, gibbsExpectation_const L t c hZ] + ring + +/-- Constants in the right slot give zero covariance. -/ +lemma gibbsCov_const_right (L : โ„ โ†’ โ„) (t : โ„) (ฯ† : โ„ โ†’ โ„) (c : โ„) : + gibbsCov L t ฯ† (fun _ => c) = 0 := by + rw [gibbsCov_symm, gibbsCov_const_left] + +/-- Additivity in the left slot. Requires integrability of each weighted +observable, both alone and against `ฯˆ`. -/ +lemma gibbsCov_add_left (L : โ„ โ†’ โ„) (t : โ„) (ฯ†โ‚ ฯ†โ‚‚ ฯˆ : โ„ โ†’ โ„) + (hโ‚ : Integrable (fun x => ฯ†โ‚ x * Real.exp (-(t * L x)))) + (hโ‚‚ : Integrable (fun x => ฯ†โ‚‚ x * Real.exp (-(t * L x)))) + (hโ‚ฯˆ : Integrable (fun x => ฯ†โ‚ x * ฯˆ x * Real.exp (-(t * L x)))) + (hโ‚‚ฯˆ : Integrable (fun x => ฯ†โ‚‚ x * ฯˆ x * Real.exp (-(t * L x)))) : + gibbsCov L t (fun x => ฯ†โ‚ x + ฯ†โ‚‚ x) ฯˆ + = gibbsCov L t ฯ†โ‚ ฯˆ + gibbsCov L t ฯ†โ‚‚ ฯˆ := by + simp only [gibbsCov] + rw [show (fun x => (ฯ†โ‚ x + ฯ†โ‚‚ x) * ฯˆ x) + = (fun x => ฯ†โ‚ x * ฯˆ x + ฯ†โ‚‚ x * ฯˆ x) from by funext x; ring, + gibbsExpectation_add L t (fun x => ฯ†โ‚ x * ฯˆ x) (fun x => ฯ†โ‚‚ x * ฯˆ x) hโ‚ฯˆ hโ‚‚ฯˆ, + gibbsExpectation_add L t ฯ†โ‚ ฯ†โ‚‚ hโ‚ hโ‚‚] + ring + +/-- Additivity in the right slot. -/ +lemma gibbsCov_add_right (L : โ„ โ†’ โ„) (t : โ„) (ฯ† ฯˆโ‚ ฯˆโ‚‚ : โ„ โ†’ โ„) + (hโ‚ : Integrable (fun x => ฯˆโ‚ x * Real.exp (-(t * L x)))) + (hโ‚‚ : Integrable (fun x => ฯˆโ‚‚ x * Real.exp (-(t * L x)))) + (hโ‚ฯ† : Integrable (fun x => ฯ† x * ฯˆโ‚ x * Real.exp (-(t * L x)))) + (hโ‚‚ฯ† : Integrable (fun x => ฯ† x * ฯˆโ‚‚ x * Real.exp (-(t * L x)))) : + gibbsCov L t ฯ† (fun x => ฯˆโ‚ x + ฯˆโ‚‚ x) + = gibbsCov L t ฯ† ฯˆโ‚ + gibbsCov L t ฯ† ฯˆโ‚‚ := by + have hโ‚ฯ†' : Integrable (fun x => ฯˆโ‚ x * ฯ† x * Real.exp (-(t * L x))) := by + simpa [mul_comm] using hโ‚ฯ† + have hโ‚‚ฯ†' : Integrable (fun x => ฯˆโ‚‚ x * ฯ† x * Real.exp (-(t * L x))) := by + simpa [mul_comm] using hโ‚‚ฯ† + rw [gibbsCov_symm L t ฯ† (fun x => ฯˆโ‚ x + ฯˆโ‚‚ x), + gibbsCov_add_left L t ฯˆโ‚ ฯˆโ‚‚ ฯ† hโ‚ hโ‚‚ hโ‚ฯ†' hโ‚‚ฯ†', + gibbsCov_symm L t ฯˆโ‚ ฯ†, gibbsCov_symm L t ฯˆโ‚‚ ฯ†] + +/-- Zero observable on the left gives zero covariance. -/ +lemma gibbsCov_zero_left (L : โ„ โ†’ โ„) (t : โ„) (ฯˆ : โ„ โ†’ โ„) : + gibbsCov L t (fun _ => 0) ฯˆ = 0 := + gibbsCov_const_left L t 0 ฯˆ + +/-- Zero observable on the right gives zero covariance. -/ +lemma gibbsCov_zero_right (L : โ„ โ†’ โ„) (t : โ„) (ฯ† : โ„ โ†’ โ„) : + gibbsCov L t ฯ† (fun _ => 0) = 0 := + gibbsCov_const_right L t ฯ† 0 + end Laplace diff --git a/Laplace/Multi/Basic.lean b/Laplace/Multi/Basic.lean index 2f40c94..3f9e251 100644 --- a/Laplace/Multi/Basic.lean +++ b/Laplace/Multi/Basic.lean @@ -52,4 +52,128 @@ lemma gibbsExpectation_def (L : (ฮน โ†’ โ„) โ†’ โ„) (t : โ„) (ฯ† : (ฮน โ†’ gibbsExpectation L t ฯ† = (โˆซ w : ฮน โ†’ โ„, ฯ† w * Real.exp (-(t * L w))) / partitionFunction L t := rfl +/-! ## Algebraic infrastructure for `gibbsExpectation` and `gibbsCov` + +Multivariate analogues of the lemmas in `Laplace.Gibbs`. The proofs are +mechanically identical to the 1D versions (the underlying integration lemmas +`MeasureTheory.integral_const_mul` and `MeasureTheory.integral_add` are +parametric in the underlying measurable space). -/ + +/-- Constant observables: `โŸจcโŸฉ_t = c` whenever `Z(t) โ‰  0`. -/ +lemma gibbsExpectation_const (L : (ฮน โ†’ โ„) โ†’ โ„) (t : โ„) (c : โ„) + (hZ : partitionFunction L t โ‰  0) : + gibbsExpectation L t (fun _ => c) = c := by + have hZ' : (โˆซ w : ฮน โ†’ โ„, Real.exp (-(t * L w))) โ‰  0 := hZ + simp only [gibbsExpectation, partitionFunction] + rw [integral_const_mul c (fun w => Real.exp (-(t * L w)))] + field_simp + +/-- Scalar-multiplication pulls out of the Gibbs expectation. No hypotheses: +when `Z(t) = 0` both sides are zero. -/ +lemma gibbsExpectation_smul (L : (ฮน โ†’ โ„) โ†’ โ„) (t : โ„) (c : โ„) (ฯ† : (ฮน โ†’ โ„) โ†’ โ„) : + gibbsExpectation L t (fun w => c * ฯ† w) = c * gibbsExpectation L t ฯ† := by + simp only [gibbsExpectation] + rw [show (fun w => c * ฯ† w * Real.exp (-(t * L w))) + = (fun w => c * (ฯ† w * Real.exp (-(t * L w)))) from by funext w; ring, + integral_const_mul c (fun w => ฯ† w * Real.exp (-(t * L w))), + mul_div_assoc] + +/-- The Gibbs expectation of the zero observable is zero, unconditionally. -/ +lemma gibbsExpectation_zero (L : (ฮน โ†’ โ„) โ†’ โ„) (t : โ„) : + gibbsExpectation L t (fun _ => 0) = 0 := by + simp [gibbsExpectation] + +/-- Additivity of the Gibbs expectation: requires integrability of each +weighted observable. -/ +lemma gibbsExpectation_add (L : (ฮน โ†’ โ„) โ†’ โ„) (t : โ„) (ฯ†โ‚ ฯ†โ‚‚ : (ฮน โ†’ โ„) โ†’ โ„) + (hโ‚ : Integrable (fun w => ฯ†โ‚ w * Real.exp (-(t * L w)))) + (hโ‚‚ : Integrable (fun w => ฯ†โ‚‚ w * Real.exp (-(t * L w)))) : + gibbsExpectation L t (fun w => ฯ†โ‚ w + ฯ†โ‚‚ w) + = gibbsExpectation L t ฯ†โ‚ + gibbsExpectation L t ฯ†โ‚‚ := by + simp only [gibbsExpectation] + rw [show (fun w => (ฯ†โ‚ w + ฯ†โ‚‚ w) * Real.exp (-(t * L w))) + = (fun w => ฯ†โ‚ w * Real.exp (-(t * L w)) + + ฯ†โ‚‚ w * Real.exp (-(t * L w))) from by funext w; ring, + integral_add hโ‚ hโ‚‚, add_div] + +/-- Symmetry: `Cov_t[ฯ†, ฯˆ] = Cov_t[ฯˆ, ฯ†]`. -/ +lemma gibbsCov_symm (L : (ฮน โ†’ โ„) โ†’ โ„) (t : โ„) (ฯ† ฯˆ : (ฮน โ†’ โ„) โ†’ โ„) : + gibbsCov L t ฯ† ฯˆ = gibbsCov L t ฯˆ ฯ† := by + simp only [gibbsCov] + rw [show (fun w => ฯ† w * ฯˆ w) = (fun w => ฯˆ w * ฯ† w) from by funext w; ring, + mul_comm (gibbsExpectation L t ฯ†)] + +/-- Scalar pulls out of the left slot. No hypotheses. -/ +lemma gibbsCov_smul_left (L : (ฮน โ†’ โ„) โ†’ โ„) (t : โ„) (c : โ„) (ฯ† ฯˆ : (ฮน โ†’ โ„) โ†’ โ„) : + gibbsCov L t (fun w => c * ฯ† w) ฯˆ = c * gibbsCov L t ฯ† ฯˆ := by + simp only [gibbsCov] + rw [show (fun w => c * ฯ† w * ฯˆ w) = (fun w => c * (ฯ† w * ฯˆ w)) from + by funext w; ring, + gibbsExpectation_smul, gibbsExpectation_smul] + ring + +/-- Scalar pulls out of the right slot. -/ +lemma gibbsCov_smul_right (L : (ฮน โ†’ โ„) โ†’ โ„) (t : โ„) (c : โ„) (ฯ† ฯˆ : (ฮน โ†’ โ„) โ†’ โ„) : + gibbsCov L t ฯ† (fun w => c * ฯˆ w) = c * gibbsCov L t ฯ† ฯˆ := by + rw [gibbsCov_symm, gibbsCov_smul_left, gibbsCov_symm] + +/-- Constants in the left slot give zero covariance. Unconditional: when +`Z(t) = 0` every Gibbs expectation collapses to `0`, so both sides agree. -/ +lemma gibbsCov_const_left (L : (ฮน โ†’ โ„) โ†’ โ„) (t : โ„) (c : โ„) (ฯˆ : (ฮน โ†’ โ„) โ†’ โ„) : + gibbsCov L t (fun _ => c) ฯˆ = 0 := by + by_cases hZ : partitionFunction L t = 0 + ยท simp [gibbsCov, gibbsExpectation, partitionFunction] at hZ โŠข + simp [hZ] + ยท simp only [gibbsCov] + rw [show (fun w => (fun _ => c) w * ฯˆ w) = (fun w => c * ฯˆ w) from rfl, + gibbsExpectation_smul, gibbsExpectation_const L t c hZ] + ring + +/-- Constants in the right slot give zero covariance. -/ +lemma gibbsCov_const_right (L : (ฮน โ†’ โ„) โ†’ โ„) (t : โ„) (ฯ† : (ฮน โ†’ โ„) โ†’ โ„) (c : โ„) : + gibbsCov L t ฯ† (fun _ => c) = 0 := by + rw [gibbsCov_symm, gibbsCov_const_left] + +/-- Additivity in the left slot. Requires integrability of each weighted +observable, both alone and against `ฯˆ`. -/ +lemma gibbsCov_add_left (L : (ฮน โ†’ โ„) โ†’ โ„) (t : โ„) (ฯ†โ‚ ฯ†โ‚‚ ฯˆ : (ฮน โ†’ โ„) โ†’ โ„) + (hโ‚ : Integrable (fun w => ฯ†โ‚ w * Real.exp (-(t * L w)))) + (hโ‚‚ : Integrable (fun w => ฯ†โ‚‚ w * Real.exp (-(t * L w)))) + (hโ‚ฯˆ : Integrable (fun w => ฯ†โ‚ w * ฯˆ w * Real.exp (-(t * L w)))) + (hโ‚‚ฯˆ : Integrable (fun w => ฯ†โ‚‚ w * ฯˆ w * Real.exp (-(t * L w)))) : + gibbsCov L t (fun w => ฯ†โ‚ w + ฯ†โ‚‚ w) ฯˆ + = gibbsCov L t ฯ†โ‚ ฯˆ + gibbsCov L t ฯ†โ‚‚ ฯˆ := by + simp only [gibbsCov] + rw [show (fun w => (ฯ†โ‚ w + ฯ†โ‚‚ w) * ฯˆ w) + = (fun w => ฯ†โ‚ w * ฯˆ w + ฯ†โ‚‚ w * ฯˆ w) from by funext w; ring, + gibbsExpectation_add L t (fun w => ฯ†โ‚ w * ฯˆ w) (fun w => ฯ†โ‚‚ w * ฯˆ w) hโ‚ฯˆ hโ‚‚ฯˆ, + gibbsExpectation_add L t ฯ†โ‚ ฯ†โ‚‚ hโ‚ hโ‚‚] + ring + +/-- Additivity in the right slot. -/ +lemma gibbsCov_add_right (L : (ฮน โ†’ โ„) โ†’ โ„) (t : โ„) (ฯ† ฯˆโ‚ ฯˆโ‚‚ : (ฮน โ†’ โ„) โ†’ โ„) + (hโ‚ : Integrable (fun w => ฯˆโ‚ w * Real.exp (-(t * L w)))) + (hโ‚‚ : Integrable (fun w => ฯˆโ‚‚ w * Real.exp (-(t * L w)))) + (hโ‚ฯ† : Integrable (fun w => ฯ† w * ฯˆโ‚ w * Real.exp (-(t * L w)))) + (hโ‚‚ฯ† : Integrable (fun w => ฯ† w * ฯˆโ‚‚ w * Real.exp (-(t * L w)))) : + gibbsCov L t ฯ† (fun w => ฯˆโ‚ w + ฯˆโ‚‚ w) + = gibbsCov L t ฯ† ฯˆโ‚ + gibbsCov L t ฯ† ฯˆโ‚‚ := by + have hโ‚ฯ†' : Integrable (fun w => ฯˆโ‚ w * ฯ† w * Real.exp (-(t * L w))) := by + simpa [mul_comm] using hโ‚ฯ† + have hโ‚‚ฯ†' : Integrable (fun w => ฯˆโ‚‚ w * ฯ† w * Real.exp (-(t * L w))) := by + simpa [mul_comm] using hโ‚‚ฯ† + rw [gibbsCov_symm L t ฯ† (fun w => ฯˆโ‚ w + ฯˆโ‚‚ w), + gibbsCov_add_left L t ฯˆโ‚ ฯˆโ‚‚ ฯ† hโ‚ hโ‚‚ hโ‚ฯ†' hโ‚‚ฯ†', + gibbsCov_symm L t ฯˆโ‚ ฯ†, gibbsCov_symm L t ฯˆโ‚‚ ฯ†] + +/-- Zero observable on the left gives zero covariance. -/ +lemma gibbsCov_zero_left (L : (ฮน โ†’ โ„) โ†’ โ„) (t : โ„) (ฯˆ : (ฮน โ†’ โ„) โ†’ โ„) : + gibbsCov L t (fun _ => 0) ฯˆ = 0 := + gibbsCov_const_left L t 0 ฯˆ + +/-- Zero observable on the right gives zero covariance. -/ +lemma gibbsCov_zero_right (L : (ฮน โ†’ โ„) โ†’ โ„) (t : โ„) (ฯ† : (ฮน โ†’ โ„) โ†’ โ„) : + gibbsCov L t ฯ† (fun _ => 0) = 0 := + gibbsCov_const_right L t ฯ† 0 + end Laplace.Multi From b844d7a822e13d2f23f5d282e481e1095f64e124 Mon Sep 17 00:00:00 2001 From: dmurfet Date: Wed, 6 May 2026 19:37:27 +0000 Subject: [PATCH 2/6] retrospectives: per-tide retrospective for gibbscov-algebra Single-session infrastructure tide. 245 lines added across Laplace/Gibbs.lean (1D) and Laplace/Multi/Basic.lean (multi); 11 lemmas per side covering gibbsExpectation_smul/zero/add (1D const pre-existing, multi const newly added) and the seven covariance lemmas (gibbsCov_symm/const/smul/add, both slots) plus zero-corollary simp lemmas. Hypothesis economy: only gibbsCov_add_left/right need Integrable hypotheses (four each, on the pre-divided weighted observables); gibbsCov_const_left/right is unconditional via a Z=0 case-split. Retrospective records the deliberation (Claude+GPT both voted Candidate B, mirror across both base files), the unconditional- const strengthening GPT recommended, and the four follow-up tides this unblocks (I3 affine-bilinear, C1/G2 affine kappa3 strict- improvements, the cleanup pass on existing affine-cov proofs). Co-Authored-By: Claude Opus 4.7 (1M context) --- .../2026-05-07-tide-gibbscov-algebra.tex | 321 ++++++++++++++++++ 1 file changed, 321 insertions(+) create mode 100644 retrospectives/2026-05-07-tide-gibbscov-algebra.tex diff --git a/retrospectives/2026-05-07-tide-gibbscov-algebra.tex b/retrospectives/2026-05-07-tide-gibbscov-algebra.tex new file mode 100644 index 0000000..04c861d --- /dev/null +++ b/retrospectives/2026-05-07-tide-gibbscov-algebra.tex @@ -0,0 +1,321 @@ +\documentclass[11pt]{article} +\usepackage[utf8]{inputenc} +\usepackage{amsmath,amssymb,amsthm} +\usepackage[margin=1in]{geometry} +\usepackage{hyperref} +\usepackage{xcolor} +\usepackage{enumitem} +\usepackage{newunicodechar} +\newunicodechar{โ„}{\ensuremath{\mathbb{R}}} +\newunicodechar{โŸจ}{\ensuremath{\langle}} +\newunicodechar{โŸฉ}{\ensuremath{\rangle}} +\newunicodechar{ฮน}{\ensuremath{\iota}} +\newunicodechar{ฯ†}{\ensuremath{\varphi}} +\newunicodechar{ฯˆ}{\ensuremath{\psi}} +\newunicodechar{โ‚}{\ensuremath{_1}} +\newunicodechar{โ‚‚}{\ensuremath{_2}} +\newunicodechar{ยท}{\ensuremath{\cdot}} + +\title{Tide retrospective:\\generic algebra for \texttt{gibbsExpectation} and \texttt{gibbsCov}} +\author{Daniel Murfet} +\date{7 May 2026} + +\begin{document} +\maketitle + +\begin{abstract} +A short infrastructure tide. The 1D and multivariate base Gibbs files +in \texttt{laplace} carried the definitions of $\langle\cdot\rangle_t$ +and $\mathrm{Cov}_t[\cdot,\cdot]$ but only the lone constant-collapse +lemma $\langle c\rangle_t = c$. Every downstream consumer that wanted +bilinearity, scalar pulling, or the constant case bashed through +\texttt{unfold gibbsExpectation} and reproved the same algebraic +identities by hand โ€” most visibly in the affine-observable proofs in +\texttt{OneD/Quartic.lean} (1D) and \texttt{TwoD/PureQuartic.lean} / +\texttt{TwoD/SemiDegenerate.lean} (2D). This tide adds the missing +algebra in parallel to both \texttt{Laplace/Gibbs.lean} and +\texttt{Laplace/Multi/Basic.lean}: 11~lemmas each side, 245~lines +total, single session, zero \texttt{sorry}, zero +\texttt{native\_decide}, zero \texttt{axiom}. The deliberation step +converged on Candidate~B (mirror across both base files) over A +(1D-only) or C (B plus an affine corollary that belongs in I3). +GPT-5.5~Pro flagged one strengthening worth taking โ€” the constant-cov +lemma is unconditional via a $Z=0$ case-split rather than carrying a +\texttt{hZ} hypothesis. No numerical sanity check was feasible +(structural identities), and none was faked. +\end{abstract} + +\section{Setting} + +The cross-project survey of 7~May identified three classes of next +moves on the laplace shoreline: strict improvements of recent tides, +strategic refactors, and new-repo expansions. Item I2 sat in the +strategic-refactor class: the GPT-5.5~Pro consult at Tide~12 +(separable-potential abstraction) had recommended introducing +\texttt{gibbsCov\_symm}, \texttt{gibbsCov\_add\_left/right}, +\texttt{gibbsCov\_const\_left/right}, and +\texttt{gibbsCov\_smul\_left/right} in the base Gibbs files, observing +that ``once landed, the affine-covariance template becomes essentially +trivial.'' The deferred I3 (an explicit affine-covariance theorem) +and the strict-improvement candidates C1 (harmonic-side affine +$\kappa_3$) and G2 (anharmonic-side affine $\kappa_3$) all wait on the +same missing layer. + +The seabed was \texttt{laplace} at commit \texttt{141997c} +(post-G4)\footnote{Tide G4: \texttt{harmonic-gibbsobservable-monomials}, +397~lines; the most recent landed tide on \texttt{main} before this +one. See \texttt{retrospectives/2026-05-07-tide-harmonic-gibbsobservable-monomials.tex}.}. +The two base files +\texttt{Laplace/Gibbs.lean} (65~lines) and +\texttt{Laplace/Multi/Basic.lean} (55~lines) defined +\texttt{partitionFunction}, \texttt{gibbsExpectation}, +\texttt{gibbsCov} and only \texttt{gibbsExpectation\_const} (the 1D +lone derived lemma; the multi side had no derived lemmas at all). A +quick grep of consumer files confirmed the gap was load-bearing: +the affine quartic covariance and its 2D pure-quartic +analogue\footnote{Specifically, \texttt{gibbsCov\_first\_order\_rate\_sharp} +in \texttt{Laplace/OneD/Quartic.lean} (lines 307--349) and the +unnamed affine cov theorem in \texttt{Laplace/TwoD/PureQuartic.lean} +(lines 297--475).} +reproved bilinearity inline at roughly 30~and 180~lines respectively, +almost all bookkeeping. + +The deliberation step\footnote{Recorded in +\texttt{projects/primer/tide-log/2026-05-07-tide-gibbscov-algebra.md} +with the verbatim GPT-5.5~Pro response in +\texttt{gpt55\_tide\_gibbscov\_algebra\_v1.md}.} weighed three +candidates: A~(1D only, $\sim$120 lines), B~(parallel both, +$\sim$220), C~(B plus the affine-bilinear corollary $\sim$280). +Both Claude and GPT voted B. A~is too small strategically (the heavy +2D consumers live on the multi side); C~bundles a consumer-facing +theorem that mixes in extra subtleties (Gibbs-weight integrability) +and morally belongs in I3, not I2. B is one conceptual unit mirrored +across the repo's two parallel foundations. + +\section{The thing formalised} + +The headline is not a single theorem but a small algebra package added +in parallel to both base files. For the 1D file (the multi version is +literal-substitution-equivalent), the eleven lemmas are: + +\begin{verbatim} +namespace Laplace + +lemma gibbsExpectation_smul (L : โ„ โ†’ โ„) (t : โ„) (c : โ„) (ฯ† : โ„ โ†’ โ„) : + gibbsExpectation L t (fun x => c * ฯ† x) = c * gibbsExpectation L t ฯ† + +lemma gibbsExpectation_zero (L : โ„ โ†’ โ„) (t : โ„) : + gibbsExpectation L t (fun _ => 0) = 0 + +lemma gibbsExpectation_add (L : โ„ โ†’ โ„) (t : โ„) (ฯ†โ‚ ฯ†โ‚‚ : โ„ โ†’ โ„) + (hโ‚ : Integrable (fun x => ฯ†โ‚ x * Real.exp (-(t * L x)))) + (hโ‚‚ : Integrable (fun x => ฯ†โ‚‚ x * Real.exp (-(t * L x)))) : + gibbsExpectation L t (fun x => ฯ†โ‚ x + ฯ†โ‚‚ x) + = gibbsExpectation L t ฯ†โ‚ + gibbsExpectation L t ฯ†โ‚‚ + +lemma gibbsCov_symm (L : โ„ โ†’ โ„) (t : โ„) (ฯ† ฯˆ : โ„ โ†’ โ„) : + gibbsCov L t ฯ† ฯˆ = gibbsCov L t ฯˆ ฯ† + +lemma gibbsCov_smul_left -- and _right by symm +lemma gibbsCov_const_left -- and _right by symm; unconditional +lemma gibbsCov_add_left -- and _right by symm; needs four Integrable hyps +lemma gibbsCov_zero_left -- and _right; cheap simp corollaries +\end{verbatim} + +\noindent The hypothesis economy is sharp: +$\texttt{symm}$ and the four $\texttt{smul}$ lemmas need no hypotheses +at all (only unconditional Bochner-integral linearity); +$\texttt{const}$ is unconditional thanks to the $Z=0$ case-split (see +\S3); only $\texttt{add\_left/right}$ require the four +$\texttt{Integrable}$ hypotheses on the pre-divided weighted +observables (each $\varphi_i \cdot e^{-tL}$ alone, plus each +$\varphi_i \cdot \psi \cdot e^{-tL}$ for the cross-term inside the +covariance). The same pattern is mirrored verbatim in +\texttt{Laplace.Multi}, with $(\iota \to \mathbb{R})$ replacing +$\mathbb{R}$. + +\section{Proof strategy} + +The covariance-level proofs all factor through the +expectation-level primitives, which in turn route through three +unconditional Mathlib lemmas: +\texttt{MeasureTheory.integral\_const\_mul} (used by +$\texttt{gibbsExpectation\_smul}$), +\texttt{MeasureTheory.integral\_add} (used by +$\texttt{gibbsExpectation\_add}$, the only place integrability +is mathematically necessary), and field arithmetic +(\texttt{mul\_div\_assoc}, \texttt{add\_div}, \texttt{ring}). +The covariance lemmas then follow by unfolding +$\mathrm{Cov}_t[\varphi, \psi] = \langle \varphi\psi\rangle_t - \langle\varphi\rangle_t \langle\psi\rangle_t$ +and rewriting with the expectation primitives plus a +\texttt{ring}-style cleanup. + +The minor wrinkle is the constant-cov lemma. The natural proof routes +through $\langle c\rangle_t = c$, which holds only when $Z(t) \neq 0$. +We could keep $\texttt{hZ}$ as an explicit hypothesis, but as +GPT-5.5~Pro pointed out, when $Z(t) = 0$ Lean's +$\texttt{x / 0 = 0}$ convention collapses every Gibbs expectation to +zero, so $\mathrm{Cov}_t[c, \psi] = 0 - 0 \cdot 0 = 0$ holds +unconditionally. The proof case-splits: +\begin{verbatim} + by_cases hZ : partitionFunction L t = 0 + ยท simp [gibbsCov, gibbsExpectation, partitionFunction] at hZ โŠข + simp [hZ] + ยท simp only [gibbsCov] + rw [..., gibbsExpectation_smul, gibbsExpectation_const L t c hZ] + ring +\end{verbatim} +The first branch flattens both the covariance and the partition +function into the underlying integral, observes that the integral is +zero, and lets \texttt{simp} propagate. The second branch is the usual +substitution of $\langle c\rangle = c$. Both cleanly close. + +The right-slot variants (\texttt{\_right}) reduce to the left-slot +ones by $\texttt{gibbsCov\_symm}$. The right-slot integrability +hypotheses for $\texttt{gibbsCov\_add\_right}$ are stated in the +natural orientation (\texttt{$\varphi \cdot \psi_i \cdot e^{-tL}$}) +and reoriented by a one-line \texttt{simpa [mul\_comm]} before +delegating to $\texttt{gibbsCov\_add\_left}$. + +\section{Roadblocks and resolutions} + +Two minor frictions, both resolved on the first or second pass. + +\paragraph{Argument-order on \texttt{gibbsCov\_symm}.} The first +draft of \texttt{gibbsCov\_add\_right} used a bare +\texttt{gibbsCov\_symm $\varphi$ $\psi_2$} as a rewrite, mistakenly +omitting the $L$ and $t$ arguments. Lean reported a type mismatch on +the second positional argument. The fix was to spell the rewrites +fully: \texttt{rw [gibbsCov\_symm L t $\psi_1$ $\varphi$, gibbsCov\_symm L t $\psi_2$ $\varphi$]}. +Two minutes lost; promoted to the lessons section as a reminder that +positional-argument economy is fragile when a downstream rewrite +needs to target exactly two of three structurally-similar subterms. + +\paragraph{Unconditional constant-cov.} The first instinct was to +keep \texttt{hZ} in \texttt{gibbsCov\_const\_left} and let a future +cleanup handle the strengthening. GPT-5.5~Pro flagged the unconditional +form as a free win: ``if \texttt{partitionFunction L t = 0}, then +every \texttt{gibbsExpectation $\ldots$ / 0} collapses to \texttt{0}, +so covariance with a constant is still \texttt{0}.'' The +case-split proof above closed in one shot. Net cost: three lines +beyond the natural-path proof; the value is one fewer hypothesis at +every consumer site. + +There were no thrashing episodes, no architectural pivots, and no +counterexample-driven plan changes. The lemmas are mechanical from +the definitions; the deliberation step did the substantive work. + +\section{What was Mathlib, what was new} + +\paragraph{Mathlib provided.} +\texttt{MeasureTheory.integral\_const\_mul} (real-valued, \texttt{RCLike} +instance, unconditional) and +\texttt{MeasureTheory.integral\_add} (the conditional additivity that +forces the four \texttt{Integrable} hypotheses on the cov-add lemmas). +Plus the usual algebraic background: \texttt{mul\_comm}, +\texttt{mul\_div\_assoc}, \texttt{add\_div}, \texttt{ring}. + +\paragraph{What was new.} The algebra layer itself: 11~lemmas per +side, 245~lines total. No new Mathlib lemmas were upstreamed because +nothing in this Tide is general enough to belong in Mathlib โ€” +\texttt{gibbsCov} is repo-specific. The shape of the layer might +nonetheless be worth lifting, eventually: a generic +\texttt{Lebesgue.expectation} / \texttt{Lebesgue.cov} pattern with +parallel algebra lemmas for any $\sigma$-finite measure could replace +both the 1D and multi versions in the laplace repo, plus possibly +threepoint's \texttt{Threepoint.gibbsCov}. Not in scope here. + +\section{Lessons} + +\begin{itemize}[leftmargin=*] +\item \emph{For the laplace \texttt{CLAUDE.md}:} the +\texttt{x / 0 = 0} case-split idiom for unconditional algebra lemmas +on quotient definitions like +$\texttt{gibbsExpectation} = \texttt{integral} / \texttt{Z}$. The +recipe \texttt{by\_cases hZ : Z = 0} followed by +\texttt{simp [..., partitionFunction] at hZ $\vdash$; simp [hZ]} on +the zero branch and the natural-path proof on the nonzero branch is +short and idiomatic. Worth landing as a tactic note alongside the +existing \texttt{Pi.add} and \texttt{rpow\_natCast} entries. + +\item \emph{For the lean-formalisation skill:} the parallel-mirror +pattern. The two base files differ only by the underlying domain +($\mathbb{R}$ versus $\iota \to \mathbb{R}$); the proofs are +literal-substitution-equivalent because the underlying Mathlib +integration lemmas are parametric in the measurable space. Writing +the 1D version first, getting it to compile, and then doing a +mechanical copy-paste-rename for the multi side cost roughly half +again the time of doing only one side. This pattern recurs across +the laplace repo (\texttt{Quartic.lean} versus \texttt{TwoD/PureQuartic.lean}, +\texttt{Harmonic.lean} versus \texttt{TwoD/AddSeparable.lean}); the +duplication is real but predictable and worth the cost. + +\item \emph{For the tide skill:} the structural-identity case for +``Numerical sanity check.'' This Tide produced no closed-form +expectation (the lemmas \emph{are} structural identities, true by +\texttt{rfl}/\texttt{ring} from the definitions). The skill's +fallback ``not feasible: \emph{reason}'' was exactly the right move; +the alternative โ€” fabricating a numerical check โ€” would have wasted +time and produced a false signal. The tide-log entry records the +reason explicitly, so the next survey can see why the section is +short. +\end{itemize} + +\section{Follow-ups} + +\begin{itemize}[leftmargin=*] +\item \emph{I3 โ€” affine-bilinear covariance.} Now a 5--10 line +derivation: +\[ + \mathrm{Cov}_t[a\varphi + c, \, b\psi + d] = ab \cdot \mathrm{Cov}_t[\varphi, \psi] +\] +follows by two applications each of \texttt{gibbsCov\_add\_left}, +\texttt{gibbsCov\_const\_left}, \texttt{gibbsCov\_smul\_left}, and +their right-slot symmetries. Worth a small dedicated tide that pins +the integrability hypotheses cleanly and exports it as a derived +theorem in \texttt{Laplace/Gibbs.lean} (and \texttt{Multi/Basic.lean}). + +\item \emph{C1 โ€” harmonic-side affine $\kappa_3$.} Generalise from +$\kappa_3(x, x, x)$ to $\kappa_3(b\cdot x, a\cdot x, c\cdot x)$ via +multilinearity of $\kappa_3$. With the new +\texttt{gibbsCov\_add/smul} algebra and the existing harmonic +$\kappa_3$ identity, ${\sim}50$ lines on top of +\texttt{Threepoint/Harmonic.lean}. + +\item \emph{G2 โ€” anharmonic-side affine $\kappa_3$.} Same shape as +C1 but for the anharmonic Gibbs measure. ${\sim}40$ lines on top of +\texttt{Laplace/OneD/AnharmonicKappa3.lean}. + +\item \emph{Cleanup pass on existing affine-cov proofs.} The most +satisfying use of this Tide will be the line-count reductions in +the four files\footnote{\texttt{Laplace/OneD/Quartic.lean}, +\texttt{Laplace/TwoD/PureQuartic.lean}, +\texttt{Laplace/TwoD/SemiDegenerate.lean}, +\texttt{Laplace/TwoD/AddSeparable.lean}.} that currently unfold +\texttt{gibbsExpectation} to reach the same algebraic identities now +captured by the new lemmas. A focused refactor tide (or a sequence of +small ones) would tighten ${\sim}300$ lines of bookkeeping into +${\sim}50$. Not strictly necessary โ€” the existing proofs work โ€” but +cost-positive once the algebra is in. + +\item \emph{Optional: bundle the hypotheses.} If the +\texttt{Integrable}-quadruples on the cov-add lemmas turn out to be +the dominant signature noise at consumer sites (likely after C1, G2, +and I3 land), a plain-\texttt{Prop} alias +\texttt{GibbsAddIntegrable L t $\varphi$ $\psi$} bundling the four +hypotheses would compress signatures without committing to typeclass +plumbing. Defer until the second downstream use shows it's worth it. +\end{itemize} + +\section*{Acknowledgements} + +The deliberation step's GPT-5.5~Pro consult (preserved at +\texttt{projects/primer/tide-log/gpt55\_tide\_gibbscov\_algebra\_v1.md}) +contributed the unconditional-const strengthening, the +direct-hypotheses-over-typeclass call, and the explicit reason to +prefer Candidate~B over A or C. The Tide~12 deliberation +(separable-potential abstraction, 6~May) is where the I2 candidate +originated; this Tide closes a follow-up that has been on the board +for less than a day. + +\end{document} From 1b3e9bdfb7e067927ea51a8a349042503edd6daa Mon Sep 17 00:00:00 2001 From: dmurfet Date: Thu, 7 May 2026 05:44:26 +0000 Subject: [PATCH 3/6] Tide kappa3-affine-anharmonic (G2): multilinear strict-improvement of Tide 9 MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Two new theorems on top of `Laplace/OneD/AnharmonicKappa3.lean`: * `kappa3_affine_id_id_id_eq` โ€” for any potential `L` and scalars `a, b, c : โ„`, ฮบโ‚ƒ(volume, L, bยทx, t, aยทx, cยทx) = (abc) ยท ฮบโ‚ƒ(volume, L, x, t, x, x). Hypothesis-free; built from `gibbsExpectation_smul` (I2's algebra) and `threepoint_gibbsExp_volume_zero_eq` (Tide 9's bridge). * `kappa3_anharmonic_affine_asymptotic` โ€” for `L = anharmonicPotential ฮป ฮฑ ฮณ`, `0 < ฮป`, `0 < ฮณ`, `ฮฑยฒ < 3ฮปฮณ`, and any `a, b, c : โ„`, tยฒ ยท ฮบโ‚ƒ(volume, L, bยทx, t, aยทx, cยทx) โ†’ -(abc)ยทฮฑ/ฮปยณ as t โ†’ โˆž. Tide 9's case `(a, b, c) = (1, 1, 1)` ร— trilinearity. 101 lines, 0 sorry, 0 #exit, 0 native_decide, 0 axiom. Tide-log: `projects/patterning/tide-log/2026-05-07-tide-kappa3-affine-anharmonic.md`. GPT-5.5 Pro consult: `projects/patterning/tide-log/gpt55_tide_kappa3_affine_anharmonic_v1.md`. --- Laplace/OneD/AnharmonicKappa3.lean | 101 +++++++++++++++++++++++++++++ 1 file changed, 101 insertions(+) diff --git a/Laplace/OneD/AnharmonicKappa3.lean b/Laplace/OneD/AnharmonicKappa3.lean index 47dafc1..fcac732 100644 --- a/Laplace/OneD/AnharmonicKappa3.lean +++ b/Laplace/OneD/AnharmonicKappa3.lean @@ -382,6 +382,107 @@ theorem thirdMoment_anharmonic_asymptotic rw [h_x2_x] ring +/-! ## Affine multilinearity (G2) + +Tide 9's `kappa3_anharmonic_id_id_id_asymptotic` lifts to affine +observables `(bยทx, aยทx, cยทx)` by trilinearity of `ฮบโ‚ƒ`: the scalars +`a, b, c` pull through cleanly, giving an extra factor `aยทbยทc` on the +asymptotic value. + +Two theorems: + +* `kappa3_affine_id_id_id_eq` โ€” the static factorisation. No + hypotheses on `L` beyond what `Threepoint.kappa3`'s definition + consumes (i.e. none); built from `gibbsExpectation_smul` and + `kappa3_id_id_id_unfold`. Works for any potential, not just the + anharmonic one. + +* `kappa3_anharmonic_affine_asymptotic` โ€” the asymptotic corollary + for the anharmonic potential. Direct `Tendsto.const_mul` on top of + Tide 9 plus the static factorisation. -/ + +/-- **Affine multilinearity of ฮบโ‚ƒ at `id`-multiples** (potential-agnostic). + +For any potential `L : โ„ โ†’ โ„` and scalars `a, b, c : โ„`, the third +cumulant of `(bยทx, aยทx, cยทx)` factors as the trilinear product of the +scalars times the third cumulant of `(x, x, x)`. + +The proof unfolds both sides to the seven-Gibbs-expectation form via +`kappa3_id_id_id_unfold`'s sibling expansion, peels each scalar via +`gibbsExpectation_smul`, and closes by `ring`. -/ +theorem kappa3_affine_id_id_id_eq (L : โ„ โ†’ โ„) (t : โ„) (a b c : โ„) : + Threepoint.kappa3 (volume : Measure โ„) L + (fun x : โ„ => b * x) t (fun x : โ„ => a * x) (fun x : โ„ => c * x) + = (a * b * c) * + Threepoint.kappa3 (volume : Measure โ„) L + (fun x : โ„ => x) t (fun x : โ„ => x) (fun x : โ„ => x) := by + -- Unfold `kappa3` on both sides; route every `Threepoint.gibbsExp ... 0` + -- through `threepoint_gibbsExp_volume_zero_eq` to `Laplace.gibbsExpectation`. + unfold Threepoint.kappa3 + simp only [threepoint_gibbsExp_volume_zero_eq] + -- Collapse the seven LHS integrand lambdas to the canonical scaled forms + -- `(a*b*c) * x^3`, `(a*b) * x^2`, `(a*c) * x^2`, `(b*c) * x^2`, + -- `a*x`, `b*x`, `c*x`, then peel scalars via `gibbsExpectation_smul`. + have h_abc : (fun w : โ„ => (fun x : โ„ => a * x) w * (fun x : โ„ => b * x) w + * (fun x : โ„ => c * x) w) = (fun w : โ„ => (a * b * c) * w ^ 3) := by + funext w; ring + have h_ab : (fun w : โ„ => (fun x : โ„ => a * x) w * (fun x : โ„ => b * x) w) + = (fun w : โ„ => (a * b) * w ^ 2) := by + funext w; ring + have h_ac : (fun w : โ„ => (fun x : โ„ => a * x) w * (fun x : โ„ => c * x) w) + = (fun w : โ„ => (a * c) * w ^ 2) := by + funext w; ring + have h_bc : (fun w : โ„ => (fun x : โ„ => b * x) w * (fun x : โ„ => c * x) w) + = (fun w : โ„ => (b * c) * w ^ 2) := by + funext w; ring + -- Same simplifications for the RHS at `(id, id, id)`. + have h_x3 : (fun w : โ„ => (fun x : โ„ => x) w * (fun x : โ„ => x) w + * (fun x : โ„ => x) w) = (fun w : โ„ => w ^ 3) := by + funext w; ring + have h_x2 : (fun w : โ„ => (fun x : โ„ => x) w * (fun x : โ„ => x) w) + = (fun w : โ„ => w ^ 2) := by + funext w; ring + rw [h_abc, h_ab, h_ac, h_bc, h_x3, h_x2] + -- Peel scalars from each scaled `gibbsExpectation` via `gibbsExpectation_smul`. + rw [Laplace.gibbsExpectation_smul L t (a * b * c) (fun w : โ„ => w ^ 3), + Laplace.gibbsExpectation_smul L t (a * b) (fun w : โ„ => w ^ 2), + Laplace.gibbsExpectation_smul L t (a * c) (fun w : โ„ => w ^ 2), + Laplace.gibbsExpectation_smul L t (b * c) (fun w : โ„ => w ^ 2), + Laplace.gibbsExpectation_smul L t a (fun w : โ„ => w), + Laplace.gibbsExpectation_smul L t b (fun w : โ„ => w), + Laplace.gibbsExpectation_smul L t c (fun w : โ„ => w)] + ring + +/-- **Affine third-cumulant asymptotic for the anharmonic Gibbs.** + +For `L = (ฮป/2)xยฒ + (ฮฑ/6)xยณ + (ฮณ/24)xโด` with `0 < ฮป`, `0 < ฮณ`, +`ฮฑยฒ < 3ฮปฮณ`, and any scalars `a, b, c : โ„`, +`tยฒ ยท ฮบโ‚ƒ(volume, L, bยทx, t, aยทx, cยทx) โ†’ -(aยทbยทc)ยทฮฑ/ฮปยณ` as `t โ†’ โˆž`. + +Strict-improvement of Tide 9: that tide proved the case `(a, b, c) = (1, 1, 1)`; +this multiplies by trilinearity of `ฮบโ‚ƒ` to handle arbitrary `id`-multiples. -/ +theorem kappa3_anharmonic_affine_asymptotic + {lam alpha gamma : โ„} + (hlam : 0 < lam) (hgamma : 0 < gamma) (hdisc : alpha ^ 2 < 3 * lam * gamma) + (a b c : โ„) : + Filter.Tendsto + (fun t : โ„ => t ^ 2 * Threepoint.kappa3 (volume : Measure โ„) + (anharmonicPotential lam alpha gamma) + (fun x : โ„ => b * x) t (fun x : โ„ => a * x) (fun x : โ„ => c * x)) + Filter.atTop + (nhds (-(a * b * c) * alpha / lam ^ 3)) := by + have hBase := kappa3_anharmonic_id_id_id_asymptotic hlam hgamma hdisc + -- Replace the limit constant by the trilinearly-scaled form: abc ยท (-ฮฑ/ฮปยณ). + have h_lim_eq : -(a * b * c) * alpha / lam ^ 3 = (a * b * c) * (-alpha / lam ^ 3) := by + ring + rw [h_lim_eq] + -- tยฒ ยท ฮบโ‚ƒ(bยทx, aยทx, cยทx) = abc ยท (tยฒ ยท ฮบโ‚ƒ(x, x, x)), so the limit is abc ยท (-ฮฑ/ฮปยณ). + have h_const := hBase.const_mul (a * b * c) + apply h_const.congr' + filter_upwards with t + rw [kappa3_affine_id_id_id_eq (anharmonicPotential lam alpha gamma) t a b c] + ring + end OneD end Laplace From 225ab8da3fdee7fff1cb2f6e78f90b15a5eed7c0 Mon Sep 17 00:00:00 2001 From: dmurfet Date: Thu, 7 May 2026 05:48:58 +0000 Subject: [PATCH 4/6] retrospectives: per-tide retrospective for kappa3-affine-anharmonic (G2) 7-page retrospective covering the trilinear strict-improvement of Tide 9. Both new theorems (`kappa3_affine_id_id_id_eq` factorisation and `kappa3_anharmonic_affine_asymptotic` corollary), the load-bearing role of I2's `gibbsExpectation_smul`, the parallel deliberation+formalisation pattern (single-shot landing), and three follow-ups (shifted-affine, generic Candidate C, simp-set automation) recorded. Tex builds clean (no overfull-hbox warnings). 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