selectedWeakTestRemainderBoundIntegrableOfStdGaussianVectorLaw

Plain-English statement

- Remainder-bound integrability transported from the normalized Brownian coordinate law. This cycle-195 bridge narrows the downstream `hRemainderBoundInt` leaf to the same source-side normalized scalar-coordinate law used for the cycle-194 measurability and domination transports. It only rewrites the ambient measure using the compiled standard-Gaussian coordinate-law bridge and variance packaging bridge; it does not prove the concrete Taylor domination bound.

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 selectedWeakTestRemainderBoundIntegrableOfStdGaussianVectorLaw
    {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E]
    [FiniteDimensional Real E] [MeasurableSpace E] [BorelSpace E]
    (variance : Test → E → Fin (Module.finrank Real E) → NNReal)
    (normalizedVectorLaw : Test → E → MeasureTheory.Measure E)
    (normalizedCoordinateLaw :
      Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real)
    (remainderBound :
      Test → E → Fin (Module.finrank Real E) → Real → Real)
    (testRegular : Prop)
    (hNormalizedRemainderBoundInt :
      testRegular →
        ∀ φ x i,
          MeasureTheory.Integrable
            (fun z : Real => remainderBound φ x i z)
            (normalizedCoordinateLaw φ x i))
    (hNormalizedVectorLaw :
      testRegular →
        ∀ φ x, normalizedVectorLaw φ x = ProbabilityTheory.stdGaussian E)
    (hCoordinateLawDef :
      testRegular →
        ∀ φ x i,
          normalizedCoordinateLaw φ x i =
            (normalizedVectorLaw φ x).map
              (fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y))
    (hVarianceDef :
      testRegular →
        ∀ φ x i,
          (variance φ x i : Real) =
            ProbabilityTheory.variance (id : Real → Real)
              (normalizedCoordinateLaw φ x i)) :
    testRegular →
      ∀ φ x i,
        MeasureTheory.Integrable
          (fun z : Real => remainderBound φ x i z)
          (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := by
  intro htests φ x i
  have hNormalizedCoordinateLaw :
      testRegular →
        ∀ φ x i,
          normalizedCoordinateLaw φ x i =
            ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal) :=
    selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw
      normalizedVectorLaw normalizedCoordinateLaw testRegular
      hNormalizedVectorLaw hCoordinateLawDef
  have hVarianceOne :
      testRegular →
        ∀ φ x i, variance φ x i = (1 : NNReal) :=
    selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw
      variance normalizedCoordinateLaw testRegular hVarianceDef
      hNormalizedCoordinateLaw
  simpa [hNormalizedCoordinateLaw htests φ x i,
    hVarianceOne htests φ x i]
    using hNormalizedRemainderBoundInt htests φ x i

/-- Normalized-law coordinate generator from a scalar pushforward law.

This cycle-184 lower_2 bridge narrows the source-facing
`hBrownianCoordinateGeneratorNormalizedLawDef`: once the frozen-interpolation
source correspondence supplies the scalar Brownian coordinate as a measurable
random variable, defines the normalized scalar law as its pushforward, and
defines the coordinate generator as the sample-space expectation of the source
Taylor integrand, the law-space integral follows by `MeasureTheory.integral_map`.
It does not identify that law with a Gaussian and does not prove the scalar
Taylor or normalized-remainder leaves.
-/

Proof architecture

narrow hRemainderBoundInt to normalized-coordinate-law integrability in the active Brownian/Ito Taylor moment backend

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.
`simpa` closes the goal after a controlled simplification of a typed result.

Why the statement has this shape

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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.
A formal generator display does not establish a closed operator domain.