Quantizing Lacan
How Recursive Prediction Gives Shape to the Divided Mind
Lacanians and Fristonians are alike, in that they are infamous for using abstruse, nigh-impenetrable thickets of abstract signifiers to make sweeping generalizations about the nature and operation of the mind.
Hold my beer.
A Confusion of Tongues
If you spend time around neuroscientists and psychoanalysts, you eventually notice that they talk about the same problems in entirely different dialects.
Both camps want to understand how a living being keeps its inner world coherent when the outer world never stops surprising it. Each describes, in their own language, the precarious trick by which an organism maintains order in the face of entropy.
This crackpot essay tries to lash those two stories together, with hubris and math. The bridge is the machinery of predictive processing and active inference developed by Karl Friston and collaborators—and the destination is a formal version of Jacques Lacan’s three great “registers” of the psyche: the Imaginary, the Symbolic, and the Real.
The idea is that the mind is a hierarchy of predictions. At one level it predicts sensations; at a higher level it predicts patterns of prediction; and at a still higher level it predicts the very transformations by which prediction happens.
The miracle is that this never spirals out of control. The errors converge. That convergence, I merrily speculate, is nothing less than Lacan’s Real—the built-in remainder that keeps sense-making from consuming itself.
Free Energy (they’ve been framed!)
To understand how prediction becomes a theory of the self, start with the simplest version of the Free-Energy Principle (FEP).
Every self-maintaining system—your body, a bacterium, a thermostat—can be described as trying to avoid states that would destroy it. Because it cannot inspect the world directly, it keeps an internal model that guesses which external causes x give rise to its sensory inputs o.
The model’s variational free energy is
where p(o,x) encodes what the system believes about the world and q(x) its current best estimate of hidden causes.
Minimizing F reduces the gap between what the system expects and what it encounters; this is a minimal description of self-correction.
Active inference extends this principle into the future. Instead of passively updating beliefs, an agent selects actions—policies, π—that minimize the expected free energy G(π).
That expectation folds together two motives: the pragmatic wish to reach preferred states (hunger drives you to food) and the epistemic wish to reduce uncertainty (curiosity drives you to information). To act is to make the world confirm your model while letting the world correct it.
If this sounds eerily psychological, that is the point. The Freudian “pleasure principle” and the Fristonian “free-energy principle” both describe a creature trying to keep surprise within tolerable bounds.
Imaginary, Symbolic, and Real
Lacan divided experience into three intertwined domains. The Imaginary is the realm of images and bodily wholeness—the sense that your reflection in the mirror is you. The Symbolic is the network of words, laws, and social expectations into which you are born and through which you must define yourself. And the Real is what neither pictures nor words can capture: the raw noise of existence that always exceeds representation.
For Lacan, the self is not a single substance but the dynamic tension among these registers. The Imaginary gives coherence, the Symbolic gives meaning, and the Real punctures both. To be human is to live at the intersection of what we can sense, what we can say, and what we can never quite grasp.
Or to put it less poetically, each of us is a layered hierarchy of generative models that continually predict, correct, and mis-predict one another.
It’s only a model
Let the lowest layer—a computational Imaginary—represent the body’s continuous states: interoception, proprioception, the felt sense of being an organism.
Here xₜᴵ) are hidden bodily causes, oₜᵇᵒᵈ are sensations, and wₜ, vₜ are noise terms.
The precision matrix Πᵢ=Σᵥ⁻¹ measures how confidently the system trusts its own bodily model—too high and the body image becomes rigid, too low and coherence dissolves.
The Symbolic layer sits above, operating on discrete states sₜ: words, categories, social cues:
This level captures the big Other of culture, the probabilistic grammar of meaning.
Its precision Πₛ reflects the firmness of one’s social expectations: high precision corresponds to dogmatism; low to confusion or psychosis.
Between them is the crucial coupling—language naming the body:
The mapping ϕ(sₜ) converts signifiers into bodily adjustments; M controls how strongly words reshape sensations. Out of this dance between top-down naming and bottom-up feeling emerges the ego.
Why recursion matters
So far this is an ordinary hierarchical predictor: the Symbolic layer guesses which bodily states the Imaginary will produce and corrects itself when it is wrong. But language and culture do more than anticipate events—they learn how their own rules change. Children learn transformations like “add –ed for past tense” and later learn when that transformation fails (“go → went”). To model such self-rewriting, the Symbolic must predict not just the next state but the mapping that generates states. That is recursive prediction.
Formally, let gᵢ denote the Imaginary’s generative mapping from hidden causes to bodily observations.
The Symbolic now carries an operator R that predicts that mapping itself:
The update rule
lets the Symbolic adjust its internal grammar for translating between words and sensations.
So that’s my crackpot idea: the mind learns how to learn itself.
This recursive trick yields qualitatively different dynamics. In a normal hierarchy, small mismatches can amplify exponentially down the chain: each layer over- or under-corrects the next. In a recursive hierarchy, higher layers actively predict how the lower mappings should adapt, driving the amplification factor Aₗ toward 1.
Errors no longer blow up; they plateau. The system achieves bounded error growth.
There is always some remainder that cannot be corrected away. And that remainder, that irreducible surprise, is the Real.
Desire, enjoyment, and the unlearnable remainder
Lacan’s three key affects—desire, jouissance, and the objet a—are different ways of living with persistent error.
Desire is the gradient of misfit between the Imaginary and Symbolic:
Because the mapping between word and sensation never perfectly aligns, there is always a slope left to descend, always something left to explain. Desire is that slope—the motivation to keep updating when closure would mean death.
If the update gain κ on this gradient becomes too large, the system overshoots and oscillates:
It repeats the same prediction-error pattern, deriving a perverse satisfaction from the repetition. This is jouissance: pleasure in the persistence of displeasure, the system enjoying the very signal of its failure.
And lurking behind both is objet a, the unattainable target that keeps inference in motion. Mathematically, it can be treated as a latent variable a that never produces observations yet continually promises information gain:
The system hunts a phantom cause it can never resolve. The chase itself becomes its equilibrium.
When recursion breaks
Different psychic structures correspond to different precision regimes—different ways of weighting prediction and correction.
When symbolic precision Πₛ is too high, the rules never bend; reality must contort to fit language. This rigidity characterizes neurosis. When Πₛ collapses to zero, the Symbolic can no longer predict the Imaginary’s transformations; words lose their anchor and the psychotic world of private coherence emerges. When coupling M and drive gain κ dominate, the system loops through transgression and satisfaction—perversion as a computational limit cycle.
These are not moral categories but dynamic regimes of recursive inference.
Some of us are still empiricists you know
Although the language here is metaphorical, the hypotheses are testable. Experiments that perturb the link between linguistic labels and bodily ownership—say, altering pronouns in a rubber-hand illusion—could reveal changes in the coupling parameters M, Πₛ, Πᵢ. Simulations in which two active-inference agents exchange signifiers could model transference, the synchronization of mutual prediction. And in psycholinguistics, it is feasible to test whether comprehension of deeply nested clauses (“the cat that the dog that the boy saw chased ran”) follows the bounded-error curve predicted by recursive inference: human accuracy should degrades linearly or saturate, whereas non-recursive networks should fail exponentially.
The subject as the act of updating
When all the equations are stripped back, what remains is a simple but profound claim: the subject is not a variable inside the model but the operation of revision itself.
If free energy F measures the mismatch between model and world, then the subject is the derivative:
–the act of correcting the error. Each update divides the self into what it expected and what it must now become. The gap never closes.
Closing thoughts
This is an attempt to turn a philosophical topology into a working computational scheme; you should take it with a huge grain of salt. Nevertheless: the Imaginary corresponds to first-order inference over embodied states, and the Symbolic to second-order inference over the mappings that relate signifiers to sensations. The Real emerges as the residual bound on that recursion—the ineradicable noise that both limits and sustains learning.
From this perspective, the mind is not an engine that seeks perfect accuracy but a system that lives off its own error, constantly rewriting the rules by which it predicts itself. Psychoanalysis intuited this long ago in the language of lack and desire; predictive neuroscience rediscovers it in the calculus of free energy. Their convergence delights me.
We are bounded by what we can never fully predict—the Real, the noise, the remainder.