actual_phase_reversal_volume_laws

Plain-English statement

Independent science 2f2e2c01856c902b07406db87026c50639454d71. Under the original six analytic assumptions, the literal harmonic flow, state-dependent bounce and momentum reversal preserve canonical phase product volume. Joint Borel maps, both flow inverses, bounce/reversal involutions, R Phi_u R=Phi_-u, R S=S R, rate(R S z)=rate(z), and the real integrable-test A6 identity are proved. Covers rank0, zero normals and all real times without an unscaled isometry claim. Volume and regularity are derived, not provider assumptions. Finite prescribed endpoint/time reversal, stationary averaging, invariance, Markovness, hypocoercivity, implementation errors, expected costs, main results and actual-input composition remain open.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Lean statement

theorem actual_phase_reversal_volume_laws
    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : ℝ≥0} {η : ℝ}
    (hα : 0 < (α : ℝ)) (hαβ : α ≤ β) (hV : ContDiff ℝ 2 V)
    (hH : ∀ x v : E,
      (α : ℝ) * ‖v‖ ^ 2 ≤ (fderiv ℝ (fderiv ℝ V) x v) v ∧
      (fderiv ℝ (fderiv ℝ V) x v) v ≤ (β : ℝ) * ‖v‖ ^ 2)
    (hη : 0 < η) (hβη : (β : ℝ) * η ≤ 1) :
    actual_phase_reversal_volume_statement hα hαβ hV hH hη hβη := by
  let c : E → E → E := fun y r => y - η • gradient V r
  let h : E → E → E := fun r x => gradient V x - gradient V r
  let B : E → E → E := fun n p => p - (2 * inner ℝ p n / ‖n‖ ^ 2) • n
  let S : E → (E × E) → (E × E) := fun r z => (z.1, B (h r z.1) z.2)
  let R : (E × E) → (E × E) := fun z => (z.1, -z.2)
  let Φ : E → E → ℝ → (E × E) → (E × E) := fun y r u z =>
      (c y r + Real.cos u • (z.1 - c y r) + (Real.sqrt η * Real.sin u) • z.2,
       (-Real.sin u / Real.sqrt η) • (z.1 - c y r) + Real.cos u • z.2)
  let rate : E → (E × E) → ℝ := fun r z =>
      Real.sqrt η * max 0 (inner ℝ z.2 (h r z.1))
  let ell : Measure (E × E) := (volume : Measure E).prod (volume : Measure E)
  change
    Measurable (fun a : (E × E) × ℝ × (E × E) => Φ a.1.1 a.1.2 a.2.1 a.2.2) ∧
    Measurable (fun a : E × (E × E) => S a.1 a.2) ∧
    Measurable (fun a : E × (E × E) => rate a.1 a.2) ∧
    (∀ y r : E, ∀ u : ℝ, MeasurePreserving (Φ y r u) ell ell) ∧
    (∀ r : E, MeasurePreserving (S r) ell ell) ∧
    MeasurePreserving R ell ell ∧
    (∀ y r : E, ∀ u : ℝ, ∀ z : E × E,
      Φ y r (-u) (Φ y r u z) = z ∧ Φ y r u (Φ y r (-u) z) = z) ∧
    (∀ r : E, ∀ z : E × E, S r (S r z) = z) ∧
    (∀ z : E × E, R (R z) = z) ∧
    (∀ y r : E, ∀ u : ℝ, ∀ z : E × E, R (Φ y r u (R z)) = Φ y r (-u) z) ∧
    (∀ r : E, ∀ z : E × E, R (S r z) = S r (R z)) ∧
    (∀ r : E, ∀ z : E × E, rate r (R (S r z)) = rate r z) ∧
    (∀ r : E, ∀ F : (E × E) → ℝ, Integrable F ell →
      Integrable (fun z => F (S r z)) ell ∧
      (∫ z, F (S r z) ∂ell) = ∫ z, F z ∂ell)

  have hf := AutoSamplingTheory.ExampleCases.ProximalBPS.ActualHarmonicFlow.actual_harmonic_flow_laws
    hα hαβ hV hH hη hβη
  have hb := AutoSamplingTheory.ExampleCases.ProximalBPS.ActualBounceRate.actual_bounce_rate_energy_laws
    hα hαβ hV hH hη hβη
  have hSm : Measurable (fun a : E × (E × E) => S a.1 a.2) := hb.1
  have hSi (r : E) : Function.Involutive (S r) := fun z => (hb.2.2.2.2.2.1 r z).2
  have hSmp (r : E) : MeasurePreserving (S r) ell ell := by
    exact (MeasurePreserving.id (volume : Measure E)).skew_product
      (hSm.comp (measurable_const.prodMk measurable_id)).snd
      (Filter.Eventually.of_forall fun x => (reflect_fiber (h r x)).map_eq)
  refine ⟨hf.2.1, hSm, hb.2.2.1, ?_, hSmp, ?_, hf.2.2.2.2.1, hSi, ?_, ?_, ?_, ?_, ?_⟩
  · intro y r u
    exact harmonic_affine η u hη (c y r)
  · exact (MeasurePreserving.id (volume : Measure E)).prod
      (LinearIsometryEquiv.neg ℝ).measurePreserving
  · intro z
    simp [R]
  · intro y r u z
    apply Prod.ext <;> dsimp [R, Φ] <;>
      simp only [Real.cos_neg, Real.sin_neg, neg_neg, neg_div] <;> module
  · intro r z
    apply Prod.ext
    · rfl
    · dsimp [R, S, B]
      simp only [inner_neg_left, mul_neg, neg_div, neg_smul]
      module
  · intro r z
    have hflip : inner ℝ (B (h r z.1) z.2) (h r z.1) = -inner ℝ z.2 (h r z.1) :=
      (hb.2.2.2.2.1 (h r z.1) z.2).2.2
    dsimp [rate, R, S]
    rw [inner_neg_left, hflip, neg_neg]
  · intro r F hF
    refine ⟨(hSmp r).integrable_comp_of_integrable hF, ?_⟩
    exact (hSmp r).integral_comp
      (MeasurableEquiv.ofInvolutive (S r) (hSi r) (hSmp r).measurable).measurableEmbedding F

end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhaseReversalVolume

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualPhaseReversalVolume.lean:156published source at ed709a7f320d

Proof architecture

Actual PBPS phase-volume changes of variables; no SALD claim

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`intro` introduces quantified hypotheses into the local proof context.
`have` creates a named intermediate mathematical fact.
`rw` rewrites by an established identity.
`simp` normalizes through registered definitional and theorem rewrites.
`apply` reduces the goal to the hypotheses of a reusable theorem.
`refine` instantiates a reusable theorem while leaving explicit subgoals.
`exact` closes the current goal with an already typed term.

Why the statement has this shape

The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.

Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.
Integrability is an input or proved output; a displayed integral alone does not supply it.
Totalized `fderiv` values must not be read as a differentiability theorem.