Information Nessconditional Independence Measure Nessmeasure Cert

The blanket certificate

Conditional independence is a standard idea in probability. Three random variables I, B, and E are conditionally independent given B when, after you learn B, knowing I gives you no extra information about E. The classical way to write this is that the joint probability of I and E, conditioned on B, factors into the product of the two separate conditioned probabilities.

The Recognition Science library formalizes this idea in a machine-checked collection of formal theorems. Its declaration NESSMeasureCert is a certificate: a packaged proof that a specific condition, called ledger-boundary sparsity, implies the standard conditional-independence factorization. The condition is written without division, as a product equality that avoids the side condition that the middle probability be nonzero, which the formal system would otherwise need to handle.

The certificate bundles two results. First, ledger-boundary sparsity implies conditional independence given the blanket, a theorem proved by direct application of the hypothesis. Second, the conditional-independence identity itself is restated in product form, again avoiding division. The certificate is a structure that holds these two proofs together, and a theorem confirms the certificate exists. It is a compact way to say: if the measure factors at the boundary, then the standard conditional-independence condition follows.

In Recognition Science, this is a bridge between a concrete sparsity condition on a probability measure and the classical probabilistic notion. The name "ledger-boundary sparsity" comes from the framework's picture of a ledger, a discrete record of events, where the blanket is the middle coordinate that screens off the internal from the external. The certificate does not claim that any particular physical system satisfies this sparsity. It only establishes the implication: sparsity, if it holds, gives conditional independence. The docstring notes that deriving sparsity from a concrete recognition-field generator is left for later work; here it is the exact hypothesis needed.

What the certificate does not do is just as important. It does not assert that any real system is conditionally independent. It does not define a new notion of independence; it uses the standard factorization. And it does not prove that the blanket is the unique or minimal set of variables that screens off I from E. The certificate is a formal implication, not an empirical claim about the world.

theorem nessMeasureCert_holds : NESSMeasureCert :=
{ conditional_independence := @ledger_sparsity_implies_measure_condIndep
  conditional_product := @conditional_product_form }

structure NESSMeasureCert where
  conditional_independence :
    ∀ {Ω I B E : Type*} [MeasurableSpace Ω]
      (P : ProbabilityMeasure Ω) (π : BlanketProjection Ω I B E),
      LedgerBoundarySparsity P π → CondIndepGivenBlanket P π
  conditional_product :
    ∀ {Ω I B E : Type*} [MeasurableSpace Ω]
      (P : ProbabilityMeasure Ω) (π : BlanketProjection Ω I B E)
      (h : CondIndepGivenBlanket P π)
      (i : I) (b : B) (e : E),
      (P : Measure Ω) (atomSet π i b e) * (P : Measure Ω) (blanketSet π b) =
      (P : Measure Ω) (internalBlanketSet π i b) *
        (P : Measure Ω) (blanketExternalSet π b e)

theorem ledger_sparsity_implies_measure_condIndep
    (P : ProbabilityMeasure Ω)
    (π : BlanketProjection Ω Internal Blanket External)
    (h : LedgerBoundarySparsity P π) :
    CondIndepGivenBlanket P π := by
  exact h

theorem ledger_sparsity_implies_measure_condIndep
    (P : ProbabilityMeasure Ω)
    (π : BlanketProjection Ω Internal Blanket External)
    (h : LedgerBoundarySparsity P π) :
    CondIndepGivenBlanket P π := by
  exact h

/-- Product form of conditional independence. This is the event-level
conditional-independence identity, kept in multiplication form to avoid
division side conditions in `ENNReal`. -/
theorem conditional_product_form
    (P : ProbabilityMeasure Ω)
    (π : BlanketProjection Ω Internal Blanket External)
    (h : CondIndepGivenBlanket P π)
    (i : Internal) (b : Blanket) (e : External) :
    (P : Measure Ω) (atomSet π i b e) * (P : Measure Ω) (blanketSet π b) =
    (P : Measure Ω) (internalBlanketSet π i b) *
      (P : Measure Ω) (blanketExternalSet π b e) := by
  exact h i b e

What this page does not claim

No real physical system is asserted to satisfy ledger-boundary sparsity. No new notion of conditional independence is introduced beyond the standard factorization. The blanket is not proved to be the unique or minimal screening variable.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Information/NESSConditionalIndependenceMeasure.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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