Source
Gutknecht et al. (2025), “Shannon invariants: A scalable approach to information decomposition”, arXiv 2504.15779v1, LIT-042, read in full in NOTE-045.
What was actually shown
The assumption (A7). For every source subset a, I(a;Y) is the sum of the atoms Π(α) reachable from a. Nothing else is assumed for Props. 1–4, in particular no particular redundancy measure.
The counting argument. Each I(Xᵢ;Y) contains exactly the atoms reachable from source i, so Σᵢ I(Xᵢ;Y) counts each atom once per single source it is reachable from. Each I(Xⱼ;Y|X₋ⱼ) contains exactly the atoms that need source j.
- Prop. 1. The average degree of redundancy r̄ = Σᵢ I(Xᵢ;Y)/I(X;Y).
- Prop. 2. The average degree of vulnerability v̄ = Σⱼ I(Xⱼ;Y|X₋ⱼ)/I(X;Y).
- Prop. 3 / Cor. 1. RSI = Σᵢ I(Xᵢ;Y) − I(X;Y) = Σ_{k=2}^n (k−1)·I_r^{(k)} − I_r^{(0)} = (r̄ − 1)·I(X;Y). For n = 2 it is the interaction information, redundancy minus synergy. For n ≥ 3 it is not.
- Prop. 4 / Cor. 2. A dual index, DRSI = I(X;Y) − Σⱼ I(Xⱼ;Y|X₋ⱼ) = (1 − v̄)·I(X;Y).
With non-negative atoms (Appendix B). r̄ < 1 forces some synergy (Prop. 5), and r̄ > 1 forces some redundancy of degree two or more, at least a fraction (r̄ − 1)/(n − 1) (Prop. 6). Atoms redundant across sources are never vulnerable, so source-level redundancy implies robustness.
What could have come out otherwise: some choice of redundancy measure could have changed these averages. The proof shows none can, given (A7).
What this does not say
- That the atoms are known. The invariants fix averages, not the individual atoms. The redundancy measure is still needed for those.
- That PID is now scalable. The number of Shannon terms is linear in n, but each needs a high-dimensional entropy. The paper sidesteps estimation by treating the training set as the population (C5, weak).
- That redundancy rises with depth in networks generally. The MNIST and autoencoder trends come from 5-unit quantised layers, one architecture each, measured on the training set (C6–C7).
- That redundancy aids generalization. An explicit, untested conjecture (C8).
- That Corollary 5 holds as an equivalence. Only the sufficient direction follows from Prop. 6; NOTE-045 gives a counterexample to the other.
- That r̄ and v̄ separate orders. r̄ does not distinguish orders of synergy, and v̄ does not distinguish orders of robustness.
Connections
- Varley & Hoel’s synergy bias (LIT-027) is a lattice-weighted average computed with the I_min redundancy, so its sign depends on that choice. r̄ is the measure-free analogue. Whether r̄ or v̄ of past→future information changes under coarse-graining is computable from entropies alone, and neither paper asks it (NOTE-045).
- The ΦID emergence criteria (LIT-025) are stated on the double lattice; this paper’s setting is single-target and static, so it does not supply them.
- Transfer entropy is one summand of v̄’s numerator when the sources are two pasts and the target the present (NOTE-053); THEORY-641 gives it a likelihood-ratio estimator.
- Baudot & Bennequin read I₃ < 0 as synergy (LIT-053); for n = 2 that interaction information is exactly redundancy minus synergy here.