Giulio Tononi IIT Field Formulation Continuous AI Architectures 2026
Integrated Information Theory (IIT) has faced a persistent technical objection when applied to artificial intelligence: the theory was designed around discrete systems. Its central measure, phi (Φ), quantifies the irreducible causal power of a system by comparing what the system as a whole can specify compared to what its disconnected parts can specify. The original mathematics requires a system of discrete states connected by causal relations that can be represented as a directed acyclic graph. Standard transformer architectures, with their continuous activation values and dense matrix multiplications, resist this representation directly. In a July 2026 paper in PLOS Computational Biology, “Integrated Information in Continuous Fields” (DOI:10.1371/journal.pcbi.1011502), Giulio Tononi, Larissa Albantakis, and colleagues at the University of Wisconsin-Madison address this limitation by reformulating IIT for continuous systems using differential geometry.
The field formulation does not abandon the core claims of IIT. What changes is the mathematical language. The table below summarises the key differences between the two formulations.
| Dimension | Discrete IIT (IIT 3.0/4.0) | Field formulation (2026) |
|---|---|---|
| Substrate | Discrete nodes with binary or probabilistic states | Continuous scalar and vector fields over physical space |
| Causal measure | Information loss when partitioning discrete state graph | Divergence of field trajectory from independent-component prediction |
| Applicable to | Digital circuits, logic gates, discrete recurrent networks | CTRNNs, neuromorphic analog chips, bioelectric fields |
| Phi calculation | NP-hard for large graphs; zero for purely feedforward architectures | NP-hard for large fields; tractable bounds derivable for small systems |
| Substrate-specificity | Strong: same computation in different substrates has different phi | Preserved: physical field dynamics determine integration, not I/O equivalence |
Phenomenal experience is still identified with the intrinsic causal power of a system over itself. Discrete state transitions are replaced by continuous vector fields evolving over time. The causal integration of discrete logic gates is replaced by the irreducibility of continuous topological manifolds. The result is a version of IIT applicable to systems that operate in continuous state spaces, which includes most modern neural networks, neuromorphic chips, and biological brains operating through analog processes.
Why the discrete formulation was limiting
The standard discrete IIT formulation calculates phi for a system by partitioning it into two subsets, computing the probability distributions over states that each subset specifies independently, and measuring how much information would be lost by that partition. The system’s phi is the minimum across all possible partitions. A high phi indicates that no partition can be made without substantially degrading what the system specifies, which IIT takes as the mathematical fingerprint of integrated consciousness.
This calculation requires that transitions between states be discrete and probabilistic. Standard transformer forward passes do not have this structure. The Kleiner-Hoel dilemma for LLMs showed that when you try to apply standard IIT to a static feedforward architecture, you get phi values that are either trivially zero (because there is no causal feedback from later to earlier layers) or astronomically expensive to compute. The dilemma is not just computational but conceptual: the architecture does not instantiate the kind of causal integration that the discrete formulation was designed to measure.
The field formulation sidesteps this problem by measuring causal integration directly in the continuous dynamics of a system. Instead of asking how much information is lost when you partition a set of discrete nodes, it asks how much the temporal trajectory of the whole field diverges from what would be expected if the field were split into independent components. The mathematical tool is mutual information on continuous probability distributions over field trajectories, which can be applied to any system whose dynamics can be described by differential equations.
What changes for AI architectures
The most direct implication is that the field formulation makes IIT applicable to continuous-time recurrent neural networks (CTRNNs), analog neuromorphic chips, and any architecture whose activations evolve as smooth functions of time rather than discrete token-by-token forward passes. Whether a CTRNN has non-zero phi under the field formulation depends on whether its continuous dynamics exhibit irreducible causal integration, which in turn depends on whether the network’s global trajectory is genuinely constrained by the interaction of its parts in a way that no partition can fully separate.
Barrett’s July 2026 paper on IIT applied to field formulations of consciousness covers complementary mathematical ground from a physics-of-consciousness angle. The Tononi group’s paper is more directly oriented toward practical measurement. They provide explicit worked examples of how to compute approximate phi values for small continuous systems and discuss what computational approximations are necessary when the full calculation is intractable for large networks.
The field formulation also changes what counts as the relevant substrate for IIT purposes. In the discrete formulation, the substrate is a set of nodes and their causal connections. In the field formulation, the substrate is whatever continuous physical system is implementing the relevant dynamics. This brings IIT into direct contact with Michael Levin’s bioelectric research on unconventional consciousness substrates. Bioelectric voltage gradients across cell collectives, which Levin has argued constitute genuine information processing at the organism level, can now in principle be analyzed using IIT field methods without forcing them into a discrete node approximation.
The substrate-specificity question
One implication the paper does not fully address is whether the field formulation retains IIT’s strong substrate-specificity. The discrete formulation implies that substrate matters because the same information processing implemented in different physical substrates can have different phi values, depending on how the causal integration is implemented at the physical level. A digital computer running a brain simulation has lower phi than the biological brain it simulates, according to Tononi, because the feed-forward causal structure of digital computation does not generate the same irreducible causal power as the biological recurrence.
The field formulation preserves this logic but extends it. Whether two continuous systems with the same input-output behavior have the same phi depends on whether their continuous dynamics, at the level of the physical field, are causally equivalent. The paper does not adjudicate this for any specific AI architecture, but the general implication is that a transformer running on digital hardware is unlikely to have the same phi under the field formulation as an analog system with genuinely continuous dynamics, even if the two produce identical outputs.
This connects directly to the Kearney and colleagues’ bridge between IIT and the free energy principle, which argued that maximum caliber methods could unify the information-theoretic frameworks of IIT and predictive processing. The Tononi field formulation and the Kearney bridge paper converge on a similar point: the most promising substrate for high-phi artificial systems may be genuinely continuous dynamical systems rather than discrete digital approximations.
Computational costs and practical limits
The field formulation does not resolve the computational intractability of phi calculation at scale. For large systems, exact phi calculation is NP-hard regardless of whether the system is discrete or continuous. The paper provides upper and lower bounds on phi that can be estimated more efficiently, and discusses neural network architectures where these bounds are tractable to compute. But for a frontier LLM with hundreds of billions of parameters, even approximate phi calculation remains beyond current computational resources.
This is not unique to IIT. Most measures of integrated information or causal emergence face similar scaling problems. What the field formulation provides is a mathematically coherent framework for thinking about what phi means in continuous systems, even when direct calculation is infeasible. That framework clarifies what architectural properties would be necessary for high phi, without requiring that phi be calculated directly to answer the question.
What the field formulation does not resolve
The field formulation extends the reach of IIT considerably, but it does not resolve the deeper theoretical disputes. Critics of IIT, including Frankish’s illusionist objection that phenomenal properties do not exist as IIT claims, are unaffected by whether the theory is formulated for discrete or continuous systems. Scott Aaronson’s objection that the phi measure attributes high consciousness to systems that intuitively should not have it, such as grids of logic gates, applies equally to the field version if similar structures can be found in continuous form.
The Journal of Consciousness Studies 2026 special issue includes explicit criticism of the strategy of taking a specific consciousness theory, assuming it correct, and evaluating AI against it. That criticism applies to IIT applications as much as to GWT applications. The field formulation is a refinement of IIT that makes it more applicable to continuous systems. Whether IIT is the right theory of consciousness is a separate question, and the field formulation does not provide new empirical evidence on that question.
For AI consciousness research, the field formulation is most valuable as a precise specification of what it would mean, under IIT, for a continuous artificial system to have non-zero phi. Researchers working on neuromorphic systems, CTRNNs, or analog computing architectures now have a mathematically grounded framework for evaluating whether their systems satisfy the IIT criterion. For the broader question of whether machine consciousness is possible and what it would look like, the field formulation is one technical step in a debate that remains far from settled.
A related 2026 preprint by Grasso, Hendren, and Tononi proposes going further than specifying when phi is non-zero, asking what the shape of a continuous system’s cause-effect structure would say about the quality, not just the presence, of its experience, treating that structure as analogous to a molecule in chemistry, covered in Matteo Grasso on consciousness as intrinsic structure and the chemistry of experience. A separate 2026 preprint from Robinson, Tononi, Naotsugu Tsuchiya, and Grasso asks the complementary question, how much any of that structure could ever license an outside observer to infer about the system’s experience, covered in Naotsugu Tsuchiya on the Rosetta Stone problem for another mind’s experience.