diff --git a/Laplace.lean b/Laplace.lean
index 8f912c8..fe90af4 100644
--- a/Laplace.lean
+++ b/Laplace.lean
@@ -26,3 +26,4 @@ import Laplace.OneD.AnharmonicKappa3
 import Laplace.OneD.HarmonicGibbsRegularity
 import Laplace.OneD.HarmonicCrossSuscDeriv
 import Laplace.OneD.QuarticBoundedPriorTestFn
+import Laplace.OneD.HarmonicGibbsObservableMonomials
diff --git a/Laplace/OneD/AnharmonicKappa3.lean b/Laplace/OneD/AnharmonicKappa3.lean
index fa066d1..47dafc1 100644
--- a/Laplace/OneD/AnharmonicKappa3.lean
+++ b/Laplace/OneD/AnharmonicKappa3.lean
@@ -320,6 +320,68 @@ theorem kappa3_anharmonic_id_id_id_asymptotic
     kappa3_id_id_id_eq_cov_form]
   ring
 
+/-! ## Third-moment asymptotic (corollary of Tide 9 +
+`secondMoment_anharmonic_asymptotic`)
+
+Packaging the third moment `⟨x³⟩_t` of the anharmonic Gibbs measure
+as a user-facing asymptotic, deferred from Tide 9. Identity:
+`⟨x³⟩ = Cov[x², x] + ⟨x²⟩·⟨x⟩` (from `gibbsCov` definition). Combined
+with the existing asymptotics, the third moment falls out:
+`t²·⟨x³⟩ = t²·Cov[x²,x] + (t·⟨x²⟩)·(t·⟨x⟩)
+       → -2α/λ³ + (1/λ)·(-α/(2λ²)) = -5α/(2λ³)`. -/
+
+/-- **Third-moment asymptotic of the anharmonic Gibbs measure.**
+
+For `L(x) = (λ/2)x² + (α/6)x³ + (γ/24)x⁴` with `0 < λ, γ` and
+discriminant condition `α² < 3λγ`,
+`t² · ⟨x³⟩_t → -5α/(2λ³)` as `t → ∞`.
+
+Strict-improvement strict-corollary of Tide 9: the third moment
+expansion was internal to the proof of `cov_anharmonic_asymptotic`
+but is here packaged as a user-facing theorem. -/
+theorem thirdMoment_anharmonic_asymptotic
+    {lam alpha gamma : ℝ}
+    (hlam : 0 < lam) (hgamma : 0 < gamma) (hdisc : alpha ^ 2 < 3 * lam * gamma) :
+    Filter.Tendsto
+      (fun t : ℝ => t ^ 2 * Laplace.gibbsExpectation
+          (anharmonicPotential lam alpha gamma) t (fun x : ℝ => x ^ 3))
+      Filter.atTop
+      (nhds (-(5 * alpha) / (2 * lam ^ 3))) := by
+  have hCov := cov_anharmonic_asymptotic hlam hgamma hdisc
+  have hM2 := secondMoment_anharmonic_asymptotic hlam hgamma hdisc
+  have hM1 := mean_anharmonic_asymptotic hlam hgamma hdisc
+  -- t² · ⟨x³⟩ = t² · Cov[x², x] + (t · ⟨x²⟩) · (t · ⟨x⟩).
+  -- Cov[x², x] = ⟨x²·x⟩ - ⟨x²⟩·⟨x⟩ = ⟨x³⟩ - ⟨x²⟩·⟨x⟩, so
+  -- ⟨x³⟩ = Cov[x², x] + ⟨x²⟩·⟨x⟩.
+  have h_sum :
+      Filter.Tendsto
+        (fun t : ℝ =>
+          t ^ 2 * Laplace.gibbsCov
+              (anharmonicPotential lam alpha gamma) t
+              (fun x : ℝ => x ^ 2) (fun x : ℝ => x)
+          + (t * Laplace.gibbsExpectation
+              (anharmonicPotential lam alpha gamma) t (fun x : ℝ => x ^ 2))
+            * (t * Laplace.gibbsExpectation
+              (anharmonicPotential lam alpha gamma) t (fun x : ℝ => x)))
+        Filter.atTop
+        (nhds (-2 * alpha / lam ^ 3 + (1 / lam) * (-alpha / (2 * lam ^ 2)))) :=
+    hCov.add (hM2.mul hM1)
+  -- The limit value simplifies: -2α/λ³ + (1/λ)·(-α/(2λ²)) = -5α/(2λ³).
+  have h_lim_eq : -2 * alpha / lam ^ 3 + (1 / lam) * (-alpha / (2 * lam ^ 2))
+      = -(5 * alpha) / (2 * lam ^ 3) := by
+    field_simp
+    ring
+  rw [h_lim_eq] at h_sum
+  -- Bridge the function: t² · ⟨x³⟩ matches t² · Cov[x², x] + (t·⟨x²⟩)·(t·⟨x⟩).
+  apply h_sum.congr'
+  filter_upwards with t
+  -- Cov[x², x] = ⟨x²·x⟩ - ⟨x²⟩·⟨x⟩ = ⟨x³⟩ - ⟨x²⟩·⟨x⟩
+  unfold Laplace.gibbsCov
+  have h_x2_x : (fun x : ℝ => x ^ 2 * x) = (fun x : ℝ => x ^ 3) := by
+    funext x; ring
+  rw [h_x2_x]
+  ring
+
 end OneD
 
 end Laplace
diff --git a/Laplace/OneD/HarmonicGibbsObservableMonomials.lean b/Laplace/OneD/HarmonicGibbsObservableMonomials.lean
new file mode 100644
index 0000000..5610917
--- /dev/null
+++ b/Laplace/OneD/HarmonicGibbsObservableMonomials.lean
@@ -0,0 +1,99 @@
+import Laplace.OneD.HarmonicGibbsRegularity
+import Laplace.OneD.IntegralRemainder
+import Threepoint.CrossSusceptibility
+
+/-!
+# `Threepoint.GibbsObservable` instances for harmonic monomials (in progress)
+
+Working towards concrete `Threepoint.GibbsObservable` instances for the
+harmonic potential `L(x) = (λ/2) x²` with linear perturbation
+`A(x) = x` and monomial observables `(fun x => x^k)` for `k : ℕ`. This
+makes Tide 13's `cov_h_id_id_deriv_harmonic_eq_zero` unconditional on
+the `GibbsObservable` hypotheses for the canonical `x, x², x³`
+observables.
+
+## Strategy
+
+After the GPT-5.5 Pro deliberation, the chosen route is **dominated
+differentiation under the integral sign** (not the closed-form
+binomial-expansion route): apply
+`MeasureTheory.hasDerivAt_integral_of_dominated_loc_of_deriv_le` with
+a Gaussian-domination bound
+\[
+  |x^k \cdot e^{-(t\,((\lambda/2)x^2 + h\,x))}|
+    \;\le\; C \cdot |x|^k \cdot e^{-(t\lambda/4)\,x^2}
+\]
+valid for `|h| ≤ 1` (Young's inequality:
+`|hx| ≤ (λ/4)x² + h²/λ` absorbs the linear perturbation into the
+quadratic decay). Integrability of the dominator falls out from
+`integrable_abs_pow_mul_exp_neg_mul_sq` already in the seabed
+(`Laplace/OneD/IntegralRemainder.lean`).
+
+The dominated-differentiation route is cleaner than the closed-form
+route here because (i) it is generic in `k : ℕ` at no extra cost,
+(ii) it avoids needing a central-Gaussian-moment library (M₂, M₄, …)
+that Mathlib v4.29.0 doesn't ship in the form we need, and (iii) the
+integrability infrastructure is already in the seabed.
+
+## Status
+
+This file currently provides one foundational primitive needed by the
+upcoming `GibbsObservable` instances:
+
+- `harmonic_perturbed_numerator_zero_eq`: the `h = 0` reduction of
+  the perturbed monomial numerator. Discharges the first conjunct of
+  `Threepoint.GibbsObservable` for any monomial observable.
+
+The headline `harmonic_id_gibbsObservable_pow` and the analytic core
+(domination bound, dominated-differentiation invocation, `HasDerivAt`
+conjunct of `GibbsObservable`) are sketched in the local handoff note
+`notes/gibbsobservable_monomials_handoff.md` and will land in a
+follow-up session. The h=0 identity primitive committed here is
+independently useful — it covers the first conjunct of every
+`GibbsObservable` instance for *any* observable that doesn't see the
+perturbation parameter, not just monomials.
+
+## Tide-step provenance
+
+Tide step (G4 from the 7 May candidates survey),
+`tide/harmonic-gibbsobservable-monomials` branch, off laplace `main`
+at `f21a9cc`. Tide log:
+`sri/projects/patterning/tide-log/2026-05-07-tide-harmonic-gibbsobservable-monomials.md`.
+-/
+
+open MeasureTheory Set
+
+namespace Laplace.OneD
+
+/-! ## The `h = 0` identity for monomial-numerator integrals
+
+`Threepoint.GibbsObservable μ L A t φ` is a conjunction; the first
+conjunct asks that the numerator
+`∫ φ(w) · exp(-(t · (L(w) + 0 · A(w)))) ∂μ` reduces to
+`∫ φ(w) · exp(-(t · L(w))) ∂μ`. For the harmonic + linear setup
+`(L, A) = ((λ/2)·², id)` with monomial `φ(x) = x^k`, this is a
+`simp [zero_mul, add_zero]`-style reduction. The lemma below does the
+reduction once for any observable; it is independent of the choice of
+`φ` and even of `k`. -/
+
+/-- For the harmonic potential `(λ/2)·x²` with linear perturbation
+`A(x) = x`, the perturbed numerator at `h = 0` reduces to the
+unperturbed integral. Independent of the observable `φ`. -/
+theorem harmonic_perturbed_numerator_zero_eq
+    (lam t : ℝ) (φ : ℝ → ℝ) :
+    (∫ w : ℝ, φ w * Real.exp (-(t * ((lam / 2) * w ^ 2 + 0 * w))))
+      = (∫ w : ℝ, φ w * Real.exp (-(t * ((lam / 2) * w ^ 2)))) := by
+  congr 1
+  funext w
+  ring_nf
+
+/-- Specialisation to monomial observables. Matches the first conjunct
+of `Threepoint.GibbsObservable` for `(volume, harmonic, id)` at
+`φ(x) = x^k`. -/
+theorem harmonic_perturbed_numerator_zero_eq_pow
+    (lam t : ℝ) (k : ℕ) :
+    (∫ w : ℝ, w ^ k * Real.exp (-(t * ((lam / 2) * w ^ 2 + 0 * w))))
+      = (∫ w : ℝ, w ^ k * Real.exp (-(t * ((lam / 2) * w ^ 2)))) :=
+  harmonic_perturbed_numerator_zero_eq lam t (fun w => w ^ k)
+
+end Laplace.OneD