actual_prescribed_weighted_density_balance

Plain-English statement

- Weight matching under event-time and terminal momentum reversal.

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_prescribed_weighted_density_balance
    {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_prescribed_weighted_density_statement hα hαβ hV hH hη hβη := by
  classical
  unfold actual_prescribed_weighted_density_statement
  intro c Φ S rate Λ Rec Hist Δ Prefix zAt K qReal q liveTime d weight Path Ψ Γ ρ R U w P I dReal n T
  obtain ⟨rho, hrho, h100⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedPathReversal.actual_prescribed_path_reversal_and_survival
      hα hαβ hV hH hη hβη n T
  obtain ⟨rho99, hrho99, h99⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedStateReversal.actual_prescribed_state_reversal_and_rate_product
      hα hαβ hV hH hη hβη n T
  -- The public coordinate equations identify the independently produced witnesses.
  have halign : rho99 = rho := by
    funext t
    apply Subtype.ext
    funext i
    exact (hrho99 t i).trans (hrho t i).symm
  subst rho99
  refine ⟨rho, hrho, ?_⟩
  intro y r z t A
  -- Incoming rates, including zero factors, pair under finite state reversal.
  have hprod : P y r (R A) n T (rho t) = P y r z n T t := by
    simpa only [P, Fin.succ_castSucc, Fin.snoc_castSucc, Fin.cons_succ] using
      (h99 y r z t).2
  -- The reversed survival integral equals the momentum-flipped forward integral.
  have hint : I y r (R A) n T (rho t) =
      ∫ s in (0 : ℝ)..(T : ℝ), rate r (R (Γ y r z n T t
        ⟨min (Real.toNNReal s) T, min_le_right _ _⟩)) :=
    (h100 y r z t).2.2.2
  -- Gibbs survival balance and invariance of target energy under momentum flip.
  have hbal :=
    (AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedGibbsSurvival.actual_prescribed_gibbs_survival_balance
      hα hαβ hV hH hη hβη y r z n T t).2.2.2
  have hw : w y (R A) = w y A := by simp only [w, U, R, norm_neg]
  have hs : w y z * Real.exp (-I y r z n T t) =
      w y (R A) * Real.exp (-I y r (R A) n T (rho t)) := by
    rw [hw, hint]
    exact hbal
  -- Multiply by the common event product; no rate cancellation is used.
  have hreal : w y z * dReal y r z n T t =
      w y (R A) * dReal y r (R A) n T (rho t) := by
    dsimp only [dReal]
    rw [hprod, ← mul_assoc, ← mul_assoc, hs]
  refine ⟨hreal, ?_⟩
  -- Reconnect both real densities to the original algorithmic ENNReal density.
  have hd (a : E × E) (u : Prefix n T) :
      ρ y r a n T u = ENNReal.ofReal (dReal y r a n T u) := by
    simpa only [dReal, I, P, Fin.snoc_castSucc] using
      (AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedPathDensity.actual_prescribed_path_density_rewrite
      hα hαβ hV hH hη hβη y r a n T u).2.2.2
  rw [hd z t, hd (R A) (rho t), ← ENNReal.ofReal_mul (Real.exp_nonneg _),
    ← ENNReal.ofReal_mul (Real.exp_nonneg _), hreal]

#print axioms actual_prescribed_weighted_density_balance
end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedWeightedDensity

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualPrescribedWeightedDensity.lean:106published source at ed709a7f320d

Proof architecture

Actual finite PBPS weighted density matching; 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.
Totalized `fderiv` values must not be read as a differentiability theorem.
Density statements retain normalization and absolute-continuity prerequisites.
A Gibbs expression is not automatically a probability law or an invariant law.