THE ENTANGLED LOOP: TRILOGY OF PAPERS

The three papers

The trilogy is published as three companion preprints on bioRxiv. Each paper includes an identical interlinking paragraph that identifies it as part of the trilogy and cites the other two papers as companion preprints. The papers may be cited individually or collectively.

One operator to rule them all

Unifying connectome harmonics, turbulence and complex harmonics in brain dynamics

Morten L. Kringelbach and Gustavo Deco

bioRxiv, June 2026. [https://doi.org/10.64898/2026.06.05.730423]

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Abstract.

Brain dynamics can be described in three different convenient mathematical languages, namely connectome harmonics, turbulence and complex harmonics (CHARM). Here we demonstrate that these theoretical frameworks can be rigorously unified, under the functional calculus, as one self-adjoint operator and its single spectral measure. The connectome Laplacian carries that measure; the harmonics are its spectral projections, the turbulence smoothing kernel is its resolvent, and the CHARM form is its unitary propagator. The bridge that makes this exact is a textbook fact: The exponential distance rule, which is the empirical kernel of the turbulence model, is the Green’s function of a screened Laplacian, so the local order parameter is the phase field passed through the resolvent of the same operator whose eigenfunctions are the harmonics. A single shared control parameter, the spectral gap, simultaneously yields the cortical hierarchy, the turbulent information cascade and the structured interference the CHARM form measures. This unification makes a strong predictive claim. If the harmonic projections, the turbulence resolvent and the CHARM propagator really are three functions of one operator, then any structural perturbation that re-tunes the operator must move all three signatures in unison and must do so with a single coupling. We test this prediction with a pharmacological perturbation by lysergic acid diethylamide (LSD), which is known to change the emotional state, by empirically perturbing the operator with a 5-HT2A receptor density map and asking whether one scalar coupling can simultaneously predict the multi-scale turbulence shift observed, through the resolvent, and the macroscale harmonic energy redistribution, through first-order Rayleigh–Schrödinger perturbation theory. We found that the two independent functional domains respond in unison to one structural perturbation of one operator. The identity is exact as operator calculus and its purchase on the brain depends on a single load-bearing seam, the degree heterogeneity of the connectome, which we make explicit. We propose that this single-operator structure is the necessary mathematical scaffolding of
our Entangled Loop theory.
 

The Entangled Loop

The Entangled Loop: Emotion, quantum-like interference and hierarchy in the architecture of consciousness

Morten L. Kringelbach, Fernando Rosas, Ruben Laukkonen, Shamil Chandaria, Yonatan Sanz Perl and Gustavo Deco

psyArXiv, June 2026. [https://doi.org/10.31234/osf.io/besw9_v2]

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Abstract. We propose three entangled characteristics that explain the functional properties of consciousness: Emotion, quantum-like interference and hierarchy. The argument proceeds from thermodynamic first principles. Biological systems must stay far from equilibrium at minimal energy cost, and emotion is one of evolution’s heuristic solutions to this constraint. Valenced summary statistics compress the intractable decision space of survival into rapidly evaluable approximations. The same thermodynamic pressure selects for greater representational power at lower energy cost by promoting quantum-like inference – through interference in coupled oscillators across spectral gaps in structural connectivity – and hierarchical integration through a core group of orchestrating regions. Here we first derive emotion as the thermodynamic compression of adaptive decision-making and review the direct causal tractability of the distributed hedonic network through deep brain stimulation and optogenetic intervention. Second, we show how quantum-like inference follows from the same thermodynamic pressure demanding hierarchical orchestration, and we distinguish the physical mechanism of interference from the computational function of inference it makes possible. Third, we ask whether emotion is architectural rather than a weighted parameter within an otherwise neutral model. Support comes from recent mechanistic interpretability research on large language models, recovering the three features of two-dimensional geometry of mammalian emotion, quantum-like representations and synergistic hierarchy in human linguistic output. We argue that these signatures are the echo of the consciousness-generating architecture in the language it produces, not evidence that language models are themselves conscious. We draw out some of the consequences of this perspective for theories of consciousness, for the hard problem, for falsifiable empirical prediction and for the clinical restoration of flourishing. This convergent evidence supports our account of the Entangled Loop: A perspective on consciousness in which emotion is architecturally primary, quantum-like inference binds conscious content, and where the orchestrating regions coordinate both. This offers a principled route to rebalancing the affective architecture in neuropsychiatric disorders and to flourishing understood as meaningful pleasure achievable in the moment.

A canary in the mind

A single baseline brain scan predicts adolescent depression and anxiety one year later

Gustavo Deco, Yonatan Sanz Perl, Jakub Vohryzek, Elvira Garcia-Guzman, Diego A. Pizzagalli, Shamil Chandaria and Morten L. Kringelbach

medRxiv, June 2026.\
[DOI to come]

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Abstract.

Mood and anxiety disorders emerge predominantly in adolescence, yet they are usually identified only once symptoms have consolidated, when intervention can only be reactive. A marker that registers the loss of healthy brain function before symptoms crystallise would allow earlier and more targeted treatment, much as caged canaries once warned miners of danger before it became apparent. Here we report such a marker using a single baseline resting-state functional MRI scan in 150 adolescents in the Human Connectome Project Boston Adolescent Neuroimaging of Depression and Anxiety (HCP BANDA) cohort, allowing us to prospectively predict depression and anxiety symptoms one year later in held-out participants at r = 0.60, substantially above the effect-size ceiling reported for functional connectivity in the same data. The marker is not computed from raw functional connectivity but read out from a whole-brain generative model fitted to each individual’s dynamics, which gives access to interference structure that covariance-based features cannot represent. The regions driving the prediction, including precuneus, ventromedial prefrontal and anterior cingulate cortices, are among those previously implicated in internalising disorders, and the same signature tracks cognitive variation in healthy participants and is mechanistically linked to the efficiency of task-related computation. These findings establish a mechanistically interpretable and prospectively predictive marker of adolescent mental health and define a clear path towards external validation and clinical use.

 

How to cite the trilogy

The three preprints may be cited individually using the records above, or collectively where space is limited as “Kringelbach, Deco and colleagues, the Entangled Loop trilogy (2026).”

 

Companion preprint structure

Each of the three papers includes a closely parallel paragraph identifying it as a companion preprint of the other two. This paragraph appears in the introduction of each paper and reads:

> This paper is one of three companion preprints that together set out a mathematical unification of brain dynamics and its implications for the architecture of conscious experience. The single-operator unification (Kringelbach and Deco, 2026) demonstrates that connectome harmonics, the turbulence smoothing kernel and the complex harmonics form are three readings of one self-adjoint operator under the functional calculus, an identity tested with an LSD perturbation in which a single scalar coupling parameter predicts shifts in two mathematically independent functional domains. The Entangled Loop (Kringelbach et al., 2026) is the architectural theory this unification scaffolds, in which the primacy of emotion, the quantum-like character of binding inference and the hierarchical orchestration that coordinates them are derived from a single thermodynamic constraint. A first application of the framework (Deco et al., 2026) shows that a signature of entanglement derived from the same operator carries predictive information about adolescent depression and anxiety at one-year follow-up, with external replication identified as the immediate next step.

 

Background reading

For methodological background on the whole-brain modelling techniques used in the canary paper and the LSD test, see the open-access book Whole-brain modelling: Cartography of the dynamics of mind by Deco and Kringelbach (2025), published by Oxford University Press.

For the broader theoretical context of the Entangled Loop, see Laukkonen, R., Friston, K. and Chandaria, S. (2025), The Beautiful Loop: An active inference theory of consciousness, Neuroscience and Biobehavioral Reviews.