nucleation

CLAIM-343: For three or more parts, no ΦID can use a redundancy function that both satisfies the identity axiom and yields nonnegative atoms, because on the Williams–Beer lattice the two are inconsistent from three sources up; a ΦID verdict on three or more parts therefore rests on which axiom the analyst drops, and should name it

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Properties7
StatusActive
Rolethesis
Warrantdemonstrated
Tagsinformation-theory, complex-systems
Lineformal-apparatus
GroundsTHEORY-713 — No redundancy measure on the Williams–Beer partial-information lattice can satisfy symmetry, self-redundancy, monotonicity, the identity axiom and non-negative atoms once there are three sources: for two independent bits, their XOR and a target that copies all three, identity forces three one-bit atoms under a two-bit total, so one atom is at most −1 bit, and a multivariate decomposition that keeps identity must give up non-negativity or the lattice, THEORY-714 — Ince's I_ccs, the expected local co-information over only those events where every source, the joint source and the overlap change the target's surprisal in the same direction, under the maximum-entropy joint that preserves the source–target and joint-source marginals, is symmetric and self-redundant and satisfies the independent identity property; it fails Harder et al.'s identity and monotonicity, so its atoms can be negative, and, though its paper says otherwise, it fails subset equality
ComplementsCLAIM-335 — Transfer entropy measures directed predictive improvement, not information flow: as a conditional mutual information it is the source's unique information about the target's next value plus its synergy with the target's past, so it can credit a source with a bit the source carries only jointly and read zero from sources that jointly fix the target, and conditioning on more series does not remove the synergy

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The claim

ΦID decomposes the information a system’s past carries about its future over a product of two Williams–Beer redundancy lattices, one for the sources and one for the targets (THEORY-497, NOTE-010). With three or more parts, the source lattice is the three-source lattice or larger.

On that lattice, Rauh et al.’s Theorem 2 rules out any redundancy function satisfying symmetry, self-redundancy, monotonicity, the identity axiom and nonnegative atoms together (THEORY-713). The case is XORCOPY: Y₁, Y₂ independent bits, Y₃ = Y₁ ⊕ Y₂, target X = (Y₁, Y₂, Y₃). Each Yᵢ is a function of the other two, so identity gives each atom {i}{jk} 1 bit. The three come to 3 bits against I(X; Y₁Y₂Y₃) = 2, so {12}{13}{23} is at most −1. The arithmetic was checked on 2026-10-17 (NOTE-965). The proof also uses two premises the paper does not list as axioms: invariance under relabelling the target, and identity applied with a pair of variables treated as one source.

So the deficit lands somewhere, and the record’s measures show where:

  • I_min gives {1}{2}{3} = 1 and {12}{13}{23} = 1, all nonnegative, because it does not satisfy identity (THEORY-706, THEORY-708).
  • I_ccs gives {i}{jk} = 1 each and {12}{13}{23} = −1, exactly at Rauh et al.’s bound, keeping only an identity for independent sources (THEORY-714).
  • Griffith et al.’s I_∧ is defined for any number of sources and is nonnegative, but fails identity and has negative atoms already for two (THEORY-715).
  • I_red and Bertschinger et al.’s measure are bivariate and do not reach three parts (THEORY-708, THEORY-705).

A ΦID verdict on three or more parts therefore rests on which axiom the analyst gave up. Reported without that, it cannot be compared across studies that chose differently.

What it does not say

  • Anything against two-part ΦID. There the bivariate measures are consistent, and the record’s transfer-entropy examples give the same values under all four two-source measures (CLAIM-335).
  • That no multivariate decomposition can exist. Rauh et al. leave open what one off the Williams–Beer lattice would look like.
  • That negative atoms are meaningless. Ince reads them as mechanistic redundancy. The claim is only that some axiom has to give.

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