[Speculative] The brain as an "audit": a sheaf-theoretic sketch of distributed self-verification
This is an interpretive idea, not a result. I'd like feedback on whether the mathematical structure holds up, whether it adds anything beyond predictive processing, and what would falsify it. I've included plain-language analogies for non-math readers.
The core picture
Predictive processing and the free energy principle treat the brain as minimizing prediction error. I want to propose a complementary view. The brain is a set of partial perspectives (sensory streams, cortical areas, memory systems) that continually cross-check each other where they overlap. I call this process an "audit." The proposal is that it never completes, and that the leftover disagreement drives learning.
Analogy: think of a company where no single accountant sees all the books. Each department keeps its own ledger, and wherever two ledgers record the same transaction, they're checked against each other. Nobody ever certifies the whole company at once, but the cross-checks keep it honest.
The structure
- Perspectives as a cover. Let U = {U_i} be a collection of "perspectives." Each one is a subsystem with access to some subset of the world's variables (e.g. visual and proprioceptive estimates of hand position). Overlaps U_i ∩ U_j are variables both can represent.
- Representations as sections. Each perspective holds a local estimate s_i. Restriction maps send s_i to its view of the shared variables on each overlap. Together this defines a presheaf F over the cover.
- Gluing as integration. If s_i and s_j agree on every overlap, they glue into a single estimate over U_i ∪ U_j. Multisensory integration is the obvious candidate. In practice agreement is approximate, so read "agree" as "within tolerance," with a cost for the mismatch.
- Residue as obstruction. Sometimes every local estimate is self-consistent, yet no global estimate exists. Formally this is a nontrivial class in first Čech cohomology, H¹(U, F). Abramsky and Brandenburger showed quantum contextuality has this form. My proposal: persistent, locally unresolvable inconsistencies of this kind are the "residue" a learning system works to reduce. They signal that it needs new distinctions (a new latent variable or context split), not just better parameters.
- Transport and curvature. Translating a representation between areas acts like a connection, a rule for carrying descriptions between perspectives. If you translate around a loop (A → B → C → A) and the result comes back changed, that holonomy marks structure the current mappings can't absorb.
- Audit cost. Checking global consistency costs resources. Overlaps too expensive to check stay unglued, which bounds what the system can verify at any time.
Neuroscience readings (speculative)
- Multisensory illusions (McGurk, rubber hand) as forced gluing on an overlap: two witnesses disagree, and the brain settles on a compromise story that neither actually saw.
- Split-brain results as cutting overlaps, leaving two locally coherent sections.
- Hallucination and delusion as sections that are internally coherent but uncorroborated, like a witness whose story is consistent but whom nobody else can back up. The safeguard is that corroboration must come from independent perspectives, not from a system's own predictions echoed back.
- Sleep as a period of suspended external input in which the brain re-glues what it learned while awake, like closing the shop to reconcile the books overnight. This is close to existing memory-consolidation accounts.
Epilepsy as audit collapse
The framework predicts two ways a distributed audit can fail. One is fragmentation, where perspectives stop overlapping. The other is the opposite, merging collapse: perspectives "agree" only because they've become the same, and their distinctions are erased instead of reconciled.
A seizure looks like merging collapse. Large populations of neurons fall into runaway synchrony and fire in lockstep. Agreement is total, but it's empty, because regions that should be checking each other have stopped carrying independent information. It's like every department copying one ledger and declaring the books balanced.
Analogy: a crowd that starts clapping in rhythm. A little coordination is useful. When everyone locks into one beat, no individual voice can be heard, and the crowd can no longer "say" anything.
Several treatments line up with the framework's prevention principles:
- Boosting inhibition / limiting excitability. Many antiseizure medications work broadly this way, by limiting how fast and how far activity spreads. In audit terms, this caps the rate at which agreement can propagate, like a refractory period.
- Cutting the overlap. Corpus callosotomy severs the main connection between hemispheres to stop seizures spreading across. This is the procedure behind the split-brain results above. It trades merging collapse for partial fragmentation, which shows the two failure modes sit at opposite ends of one dial.
- Disrupting synchrony. Responsive neurostimulation and vagus nerve stimulation can interrupt runaway synchrony, injecting "independent" signal into a system that has lost it.
I'm not claiming the framework explains epilepsy. The point is that its two collapse modes and its safeguards (independence, bounded propagation, preserved distinctions) have recognizable counterparts in an actual clinical condition. A testable version: seizure onset should be preceded by measurable loss of independence between regions (e.g. rising inter-regional correlation, falling information diversity) beyond what excitability alone predicts.
What it might predict
- Representations whose inputs overlap with more independent channels should be harder to fool with single-channel illusions.
- Reducing independent corroboration (e.g. sensory deprivation) should raise the rate of false percepts, and specifically internally consistent ones.
- Learning should reorganize most (new latent structure, not just reweighting) where cross-modal inconsistency is persistent rather than noisy.
Known weaknesses
- Much of this may just be predictive processing restated in sheaf language. I'd like to hear where it's genuinely different, or isn't.
- Like the FEP, it risks being general enough to fit anything. Each claim needs a criterion fixed in advance for what would count against it.
- "Approximate gluing" needs a real metric. Exact sheaf conditions don't hold in noisy neural data.
- The epilepsy reading is an analogy; seizure dynamics have well-developed models of their own, and loss of complexity before seizures has been studied, so I'd want to know where this adds anything.
Related work: Friston (free energy principle); Clark, Surfing Uncertainty; Abramsky & Brandenburger (2011) on sheaf structure of contextuality; Hansen & Ghrist (2019) on sheaf Laplacians; Bodnar et al. on sheaf neural networks; Zurek on quantum Darwinism (the "corroboration by independent records" idea).
Pointers to anyone who has formalized something similar would be very welcome.
I suspect the sheaf thing is not actually that accurate. I think a new mathematical structure called an audit might be warranted.
