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

Source

Ince (2017), “Measuring multivariate redundant information with pointwise common change in surprisal”, Entropy 19(7): 318, arXiv 1602.05063v3, LIT-1201, read in NOTE-968: definitions, properties and examples in full, the Gaussian section and discussion at lower resolution.

What was actually shown

The measure. Under P̂, the maximum-entropy joint with every Q(Ai, S) and Q(A1…An) fixed, I_ccs is the expectation of the local co-information on events where all local informations and the co-information share a sign, and zero elsewhere (Defs. 4.1–4.2).

Properties proved (§4.4). Symmetry, self-redundancy and subset equality, so the lattice can be used; the independent identity property, so two independent bits copied into the target share nothing. The subset-equality proof does not hold: it cancels the co-information terms but not the sign rule (Note of 2026-10-20).

Properties that fail. Harder et al.’s identity, whenever there are negative local information terms. Monotonicity, by Table 7: redundancy 0.77 bit exceeds I(S; X2) = 0.61, giving unique information −0.16. Hence non-negative atoms.

Examples, recomputed. REDUCEDOR 0/0.31/0.31/0.38; SUM 0/0.5/0.5/0.5; AND 0.104/0.208/0.208/0.293; XORCOPY with {12}{13}{23} = −1; DBLXOR with {123} = −1; giant bit, parity and XORUNQ as built (NOTE-968).

What could have come out otherwise: on XORCOPY the measure could have violated independent identity to keep the atoms non-negative, as I_min does. It takes the negative atom instead, exactly at the bound THEORY-713 proves.

What this does not say

• That negative atoms are meaningful. “Misinformation” and “mechanistic redundancy” are readings the paper offers, not results.
• That I_broja is wrong. The REDUCEDOR game shows that a game-theoretic operationalisation of unique information disagrees with the decision-theoretic one (THEORY-705). Which one is right is not shown.
• That I_ccs is continuous. Conjectured, and checked only locally in the examples.
• That it is the redundancy ΦID uses. The ΦID work held here uses the minimum-mutual-information redundancy (NOTE-799).

Connections

• THEORY-713 proves that some axiom must go once there are three sources. I_ccs is the measure that gives up non-negativity and keeps an independent-sources identity.
• CLAIM-335’s examples and THEORY-712’s coupled pair keep their values under I_ccs. Harder et al.’s bystander example (THEORY-708) gets a third split, with a negative unique atom at d = 0.25 (NOTE-968).
• THEORY-706 (I_min) and THEORY-708 (I_red) are the measures it argues against on the two-bit copy and on identity respectively.

Note of 2026-10-20: the properties re-derived, and subset equality

The promotion audit re-derived each property from Definitions 4.1–4.2 of the paper. Write i(x; s) = log p(s|x)/p(s) and c for the local co-information, all under P̂. I_ccs keeps an event only when every source’s i, the joint source’s i and c share a sign.

• Symmetry. c, the sign rule and P̂’s constraints are all invariant under permuting the sources.
• Self-redundancy. For one source c = i(a; s), the sign rule is met trivially, and P̂ preserves Q(A, S), so I_ccs = I(S; A).
• Independent identity. For S = (A1, A2) with I(A1; A2) = 0, every admissible joint is supported on s = (a1, a2), so P̂ = P. Then i(a1; s) + i(a2; s) = i(a1a2; s) pointwise, c = 0 and I_ccs = 0.
• Harder et al.’s identity fails. For the same target with dependent sources, c = i(a1; a2), the local mutual information, while all three local informations are non-negative. Events with negative local mutual information drop out, so I_ccs = Σ p max(i(a1; a2), 0) ≥ I(A1; A2). For p(a1, a2) = (0.4, 0.1; 0.2, 0.3) that is 0.3415 bit against 0.1245.
• Monotonicity fails. Table 7 recomputes to I_ccs = 0.7685 bit against I(S; X2) = 0.6100, a unique atom of −0.1585.
• Subset equality fails. The paper’s proof (§4.4, Eqs. 39–41) shows the co-information cancels when A_{l−1} ⊆ A_l, which is right. But the sign rule still requires sgn i(a_{l−1}; s) = sgn i(a_l; s), and a coarse and a fine source can disagree. Take A1 = X1 and A2 = (X1, X2) with p(x1, x2, s) ∝ (0,0,0) 0.2, (0,0,1) 0.05, (0,1,0) 0.05, (0,1,1) 0.2, (1,0,0) 0.3, (1,0,1) 0.2. Here P̂ = P, because A2 determines A1. The events (0,0,0) and (0,0,1) drop out (i(x1; s) = −0.138 against i(x1x2; s) = +0.541, and +0.152 against −1.170). So I_ccs({1}{12}) = 0.0272 bit, while I_ccs({1}) = I(S; X1) = 0.0073 bit.

The recipe, kept in the audit’s decision file, fits P̂ by iterative proportional fitting from the uniform distribution. Every number above was recomputed with it.

Subset equality is the property Ince argues licenses restricting the redundancy function to the lattice’s antichains, so its failure bears on the measure’s rationale. It leaves the lattice computation and every value in the record (the −1-bit XORCOPY atom, the examples above, the values CLAIM-343 and THEORY-712 use) unchanged. Whether a modified sign rule could restore the property is QUESTION-223.

Note of 2026-10-20: the wider claim, filed Proposed

Under ADR-047 v4 the narrowing-loss sweep (batch 069) filed THEORY-804 as Proposed, extending this entry. It carries Ince’s thesis that I_ccs is a correct redundancy measure and that its negative atoms are local misinformation and mechanistic redundancy, which this entry’s title carried before batch 056 corrected and narrowed it. This entry is its evidence and is not changed by the filing.