Information Nessconditional Independence Measure Ness Measure Cert Holds

The blanket certificate

In probability theory, two events are conditionally independent given a third when knowing the third makes learning one of them tell you nothing new about the other. The standard definition writes this as a factorization: the joint probability of all three equals the product of the two pairwise joint probabilities, divided by the probability of the conditioning event, whenever that probability is nonzero. The Recognition Science library formalizes this definition in a machine-checked collection of formal theorems, using a multiplication form that avoids division side conditions.

The declaration nessMeasureCert_holds is a certificate: a single theorem that packages two results. The first says that a condition called ledger boundary sparsity, which means the probability measure factors exactly as the blanket identity requires, implies conditional independence in the standard sense. The second restates the same factorization as the product form. Together they establish that a sparsity condition on the measure surface is sufficient for the classical conditional-independence property.

In Recognition Science, the framework models a system as a discrete record of events with three coarse-grained coordinates: internal, blanket, and external. The blanket coordinate is the boundary that separates the internal from the external. The certificate proves that when the measure satisfies the blanket factorization, the internal and external coordinates are conditionally independent given the blanket. This is the event-level form of the Markov blanket condition used in the free energy principle.

The certificate does not claim that the sparsity condition holds for any particular system. It does not derive the condition from a physical generator or from the recognition-field dynamics. It only proves the implication: if the sparsity holds, then conditional independence follows. The library notes that deriving the sparsity from a concrete generator is later work, not part of this theorem.

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

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

The certificate does not assert that ledger boundary sparsity actually holds for any physical system. The certificate does not derive the sparsity condition from a recognition-field generator or from the framework's forcing chain. The certificate does not claim to prove the free energy principle itself, only its event-level Markov blanket condition.

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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