Information Nessconditional Independence Measure Blanket Projection

The blanket projection

A blanket projection is a mathematical way to divide a system into three parts: an internal part (the inside), a blanket part (the boundary or interface), and an external part (the outside). Formally, it is a structure that assigns to every possible state of the whole system a triple of values: one for the inside, one for the boundary, and one for the outside. This is a common idea in many fields, from thermodynamics to neuroscience, where you want to know how much the outside can influence the inside without directly looking at it.

The framework's declaration, BlanketProjection, establishes the precise condition under which the outside becomes irrelevant to the inside once the boundary is known. This condition is called conditional independence, and it is expressed as a factorization of probabilities. For any specific values of the inside, boundary, and outside, the probability of seeing that exact combination, multiplied by the probability of seeing the boundary alone, must equal the product of the probability of seeing the inside and boundary together and the probability of seeing the boundary and outside together. In plainer terms, once you know the boundary, knowing the outside gives you no additional information about the inside.

The declaration does not claim that this conditional independence always holds. It defines it as a property that a given probability measure may or may not have. The framework's library proves that this factorization condition is exactly equivalent to the standard definition of conditional independence, and it shows that if a measure satisfies this condition, then it is indeed conditionally independent. The library also proves that this condition is equivalent to the more familiar form where you divide by the probability of the boundary, but it keeps the multiplication form to avoid issues with dividing by zero.

In Recognition Science, this condition is given a specific name: ledger-boundary sparsity, which means the measure has the blanket factorization. The framework's library proves that this sparsity condition is exactly the same as the measure being conditionally independent. However, the library does not yet derive this condition from a more fundamental principle; it is currently stated as a definition, a hypothesis that can be assumed. The framework's documentation notes that in later work this condition might be derived from a concrete recognition-field generator, but that derivation is not yet complete.

What this means for a reader is that the framework has given a precise, machine-checked definition of what it means for a boundary to screen off the outside from the inside. This is a foundational step, not a finished theory. It tells you exactly what the condition is, and it proves the equivalence between two ways of writing it, but it does not yet tell you when or why this condition should hold in any particular physical system. That remains a target for future work.

/-- A measurable projection of a state space into the FEP partition. -/
structure BlanketProjection (Ω Internal Blanket External : Type*) where
  internal : Ω → Internal
  blanket : Ω → Blanket
  external : Ω → External

/-- Measure-theoretic conditional independence as blanket factorization. -/
def CondIndepGivenBlanket
    (P : ProbabilityMeasure Ω)
    (π : BlanketProjection Ω Internal Blanket External) : Prop :=
  ∀ (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)

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

What this page does not claim

The declaration does not prove that conditional independence always holds for any system. The declaration does not derive the blanket factorization from a more fundamental principle; it is a definition, not a theorem. The declaration does not specify how to construct a blanket projection for any particular physical system.

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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