What Finite Experiments Can Certify About Integrated Information Under Identical Behavior

If two neural networks produce the exact same input-output behavior across every possible stimulus, can a finite laboratory experiment ever certify their internal integrated information ($\varphi$)? In their November 2026 paper Same Behavior, Different Mechanism: What Finite Experiments Can Certify About Integrated Information (arXiv:2611.08469), Linqian Wu (University of Waterloo and Perimeter Institute for Theoretical Physics) and Diaoxuan He (University of Debrecen) provide a rigorous mathematical audit of mechanism-level consciousness metrics under finite, partial, and noisy experimental interventions.

For researchers tracking how scientists race to define and measure AI consciousness in 2026, Wu and He’s theorems cut to the heart of the methodological clash between Integrated Information Theory (IIT 4.0, Giulio Tononi and Larissa Albantakis), functionalist indicator checklists, and behavioral Turing-style evaluations. While philosophers have long debated the Unfolding Argument and rapid-plasticity responses against IIT at the level of idealized transition probability matrices (TPMs), real neuroscience and AI interpretability audits never have infinite noise-free access to every hidden unit. Wu and He prove exactly where point estimates of $\varphi$ fail and how one-sided statistical lower-bound certificates can survive.

Four Mathematical Findings on Auditing Integrated Information

In Integrated Information Theory, system integrated information ($\varphi$) is a mechanism-level functional defined over a network’s causal transition law (its state-by-state interventional transition probability matrix) rather than its external input-output distribution. Measuring $\varphi$ requires identifying a candidate substrate, perturbing its units into all possible states, recording one-step state transitions, and searching over discrete bipartitions and cause-effect states to locate the minimum information partition (MIP).

Wu and He analyze what happens to this pipeline when an experimenter must estimate the causal transition law from finite trial counts, partial unit recordings, or learned surrogate models rather than reading it from an exact mathematical oracle. Their proofs and numerical experiments establish four quantitative results.

1. Two stochastic networks can exhibit identical input-output behavioral laws at their response delay while possessing drastically different internal $\varphi$ values (for example, an integrated recurrent circuit versus a decoupled or feedforward circuit). Pure black-box behavioral testing has zero statistical power to distinguish them, whereas full internal one-step interventional recordings separate their mechanisms within a modest number of trials.
2. If an experimenter leaves even a small subset of network units unobserved during one-step recordings, marginalizing over the hidden units imposes a strict precision floor on $\varphi$ bounds that no amount of additional sample size ($N \to \infty$) can eliminate. In addition, structural symmetries in the network can mask hidden units unless broken by targeted multi-node interventions.
3. Because the definition of $\varphi$ involves discrete minimization and maximization selections over partitions and cause-effect repertoires, $\varphi$ is non-smooth across mechanism space. Across a vast majority of random network transitions that cross a partition-selection boundary, an infinitesimal shift in transition probabilities causes the true $\varphi$ value to jump or shift slope abruptly (changing by 25 percent to 100 percent across edge crossings). Consequently, surrogate models that fit interventional data to near-perfect statistical accuracy can still misreport point estimates of $\varphi$ by wide margins.
4. Despite the failure of point estimation near partition boundaries, one-sided lower bounds remain mathematically certifiable. By propagating uniform confidence regions over the estimated transition probability matrix through the discrete partition optimizations, Wu and He construct rigorous statistical certificates (demonstrating, for example, a certified bound of $\varphi > 1.004$ bits across multi-dimensional mechanism spaces) with a strictly bounded false-certificate probability.

Implications for IIT 4.0, Toy Models, and AI Interpretability

Wu and He’s results connect directly to Larissa Albantakis’s analysis of toy models in theories of consciousness and Giulio Tononi, Matteo Grasso, and Jeremiah Hendren’s QSTR framework for intrinsic structure. Proponents of IIT have long insisted that behavioral equivalence does not imply phenomenal equivalence, arguing that a lookup table or feedforward network reproducing human behavior has $\varphi = 0$.

Wu and He confirm IIT’s claim that internal interventional access can distinguish behaviorally identical mechanisms, while simultaneously exposing a severe vulnerability in how $\varphi$ is computed in practice. In empirical neuroscience (such as calcium imaging or high-density Neuropixels recordings) and in mechanistic interpretability of large language models, researchers never observe 100 percent of neurons at single-step causal resolution. Instead, they fit reduced dynamical systems or sparse autoencoders to a subsample of units and compute $\varphi$ on the surrogate model.

Wu and He prove that this surrogate workflow is mathematically unsafe for point claims. Because partition-switching ridges pervade the space of stochastic networks, a surrogate model whose transition probabilities differ from the true circuit by less than one percent can land on the opposite side of a minimum-information partition edge and report a $\varphi$ value that is off by 50 percent or more. To make a scientifically valid claim about integrated information from experimental data, researchers must abandon point estimates of $\varphi$ in favor of confidence-bounded one-sided inequalities computed over full interventional ensembles.

Comparison to The Consciousness AI

This paper provides independent mathematical confirmation of a central engineering lesson learned inside The Consciousness AI. As documented on our Core Architecture page and in our November 2026 research inventory (tlcdv/tcai-planning#285), Phi is turned off by default on main in our research codebase, and 0 of our 21 scalar consciousness instruments are currently classified as trusted.

During earlier evaluations of our scalar Phi-1 proxy and related complexity metrics, small changes in sampling noise, hidden-layer subsampling, or partition approximations produced volatile swings that did not track whether the global workspace was actually causally coupled to agent behavior. Wu and He’s theorem explains why scalar $\varphi$ estimators behave that way, since partial observability creates an irreducible precision floor and discrete partition selection creates 25 to 100 percent sensitivity cliffs across mechanism space. When our scalar instrument suite returns to evaluation after the workspace-to-behavior link repair, Wu and He’s one-sided confidence-certificate method provides the exact statistical test needed to verify whether any integrated-information proxy clears a non-zero lower bound under finite trial noise.

Practical Takeaways for Consciousness Auditing

For experimentalists and AI auditors building consciousness evaluation suites alongside Patrick Butlin and Robert Long’s indicator validation roadmap, Wu and He’s study establishes three operational rules. First, never treat a high-accuracy behavioral or next-token surrogate as mechanism-faithful for integrated information. Second, report explicit sensitivity analyses showing how close an estimated network sits to the nearest partition-switching boundary. Third, replace uncalibrated scalar $\varphi$ point numbers with certified lower bounds accompanied by their false-certificate probability and observability assumptions.

Researchers covered here

• LW Linqian WuUniversity of Waterloo and Perimeter Institute for Theoretical PhysicsFinite-experiment certifiability of integrated information, partial-observability precision floors, mechanism auditing 1 article
• DH Diaoxuan HeUniversity of DebrecenCertifiability of integrated information under finite noisy interventions, partition discontinuity analysis in stochastic networks 1 article