Why the Riemann Hypothesis and Fluid Turbulence May Be the Same Problem

Tesla coils, Majorana towers, and the spectral fingerprint connecting two Millennium Prize problems.

In my free time, I used to build audio-modulated solid state Tesla coils as a hobbyist. The physics of the plasma streamers that came off those coils fascinated me (vacuum tube driven tesla coils produce long spearlike arcs without bufurcations and fractal patterns, what causes them?), and as it turns out, that physics is closely related to the physics of fluid turbulence. Plasmas obey a similar mathematical description. What is interesting is that two deep open problems: the Riemann hypothesis (a Clay Millennium Prize Problem) and the problem of intermittency in fluid turbulence (closely related to, though not identical with, the Clay problem on Navier–Stokes existence and smoothness) may be related to one another.

Presentation

https://www.youtube.com/watch?v=U9CuORupmS8

How can it be that two of the largest open problems in physics and mathematics are asking the same thing?

The Riemann Hypothesis as Physics

The Riemann zeta function encodes the distribution of prime numbers. Prime factorization is, of course, the bedrock of current cryptographic standards, which rely on the difficulty of factoring large primes. That difficulty can be sidestepped by quantum computers running Shor's algorithm, which factors integers in polynomial time.

The Riemann hypothesis asks whether all of the non-trivial zeros of this function lie along the so-called critical line, with real part 1/2. Numerical search has verified this for the first many billions of zeros. None has been found off the line. But the conjecture remains a conjecture.

What physicists noticed, and what is the doorway into everything that follows, is that this function looks suspiciously like it could be describing the energy levels of some quantum mechanical system. This is the Hilbert-Pólya conjecture: the idea that there exists a self-adjoint operator whose eigenvalues coincide with the imaginary parts of the non-trivial zeta zeros. Because self-adjoint operators have real eigenvalues, the existence of such an operator would automatically prove the Riemann hypothesis. The hypothesis would then be settled not by abstract mathematics but by physics.

This is not idle speculation. In 1973, Hugh Montgomery showed that the pair correlations of the non-trivial zeta zeros match the Gaussian Unitary Ensemble statistics of random matrix theory, the very same statistics that govern eigenvalue spacing in quantum chaotic Hamiltonians with broken time-reversal symmetry. Andrew Odlyzko verified this numerically to extraordinary precision for the first billion zeros. In 1999, Berry and Keating elaborated the picture: the underlying "Riemann dynamics" should be chaotic, with periodic orbits labeled by primes. They proposed the Berry-Keating Hamiltonian, H=(xp+px)/2H = (xp + px)/2 H=(xp+px)/2, as a candidate for the Hilbert-Pólya operator.

Then, in 2021, Tamburini and Licata showed the Majorana equation (the physicist Ettore Majorana famously "disappeared" in the 1930s) placed in Rindler spacetime which is the frame of a uniformly accelerated observer, reduces to a Hamiltonian of exactly the Berry-Keating form. The boundary conditions on the Majorana wavefunction involve Bessel functions which, through Mellin-Barnes integrals, connect directly to the Riemann zeta function.

Their 2025 preprint argues that the Majorana Hamiltonian in Rindler space is essentially self-adjoint and proposes a spectral correspondence with the non-trivial zeta zeros. This sits alongside other Hamiltonian constructions in the Hilbert–Pólya program (Berry–Keating, Connes, Sierra, Bender–Brody–Müller). None of these constructions has yet been accepted by the academic community as a proof, and the Tamburini–Licata version will need to clear the same hurdles, but given their compelling and rigorous constructions, the proposals deserve more attention. I have had many long conversations with Dr. Tamburini about this, and I think it deserves much more attention than it has received.

Why Fluid Turbulence Wears the Same Statistics

Here is where the physics widens. The Riemann zeta function does not just show up in random matrix theory and accelerated quantum field theory. It shows up in the statistics of nonlinear deterministic systems near phase transitions. Quantum chaos, fluid turbulence intermittency, magnetohydrodynamic instabilities. All of them.

Modeling fluid turbulence and atmospheric flows, or the plasma behavior in fusion tokamaks and coronal mass ejections, is largely intractable. The physics is not well understood. So if there really is a universal spectral signature governing these systems, and it really is the Riemann zeta function, then resolving this would do considerably more than collect two million dollars from the Clay Institute. It would help us understand magnetohydrodynamic instabilities, which are the bottleneck for plasma confinement in tokamaks. It would improve weather prediction. And, since the same physics sits at the intersection of nonlinear deterministic and quantum probabilistic regimes, it could illuminate quantum gravity itself. Quantum gravity is, in a sense, the missing physics of reconciling those two regimes. As I have discussed with Ed Witten, macroscopic quantum-like behavior is one direction for understanding it.

The Majorana Tower and the Cascade

So what is going on physically? One clue comes from Majorana fermions and Majorana zero modes. These are particles that are their own antiparticles, and they enjoy what is called topological protection, which makes them the leading candidate for noise-resilient scalable quantum computation. Microsoft is actively investing in this technology for that reason. There are also theories which link Majorana physics to how the brain processes information, something I have discussed at length with Dr. Tamburini.

The picture I find compelling is this. Fermionic spin systems can be entangled and pumped by Floquet drivers into a state of saturation. Beyond that point, the system undergoes a phase transition at which its statistics are described by the Riemann zeta function. The information stored in the entanglement structure is then bosonized into ultraweak superradiant Majorana-like vortex photon signals carrying orbital angular momentum (OAM). There may also be gravitational feedback at these phase transitions. Majorana-like vortex photons carry the necessary quantum information imprinted from these Majorana-like fermionic spin states necessary, and critically, in a way that can propagate through tissues but then also backpropagate which is necessary for adjusting dendritic weights and connections.

Ultraweak superradiant Majorana-like vortex biophotons have been measured in brain tissues, and light is known to modulate long term potentiation (LTP) and long term depression (LTD) which adjusts dendritic connections, weights, and arborization - much like plasma filamentation in tesla coils.

This sounds speculative. It is, at the edges. But the load-bearing pieces are not. There are now multiple publications reporting that the zeros of the Riemann zeta function have been reproduced experimentally by pumping or driving qubits, and the mathematical link to Majorana physics has appeared in respectable journals. OAM light is known to carry the kind of quantum information needed for these effects, and has been explored as a candidate explanation for the black hole information paradox and for intracellular communication in living tissue.

One way to read all this: the boundary between the right and left Rindler wedges, in collections of entangled Majorana particles, traces out the critical line of the Riemann zeta function. The split structure of these particles is how quantum information gets stored before being saturated and released as light and gravity. The mathematics here implicates Z2 orbifolds at the phase transition, the bosonization of information into light-like modes, and the resulting superradiant cascades.

Twistor theory, the theory of null geodesics that Penrose has long argued must be central to a complete theory of gravity, fits naturally into this picture. So does Einstein-Cartan theory, which introduces spacetime torsion to general relativity, or, equivalently, gravity to quantum theory through spin. Could these light-like signals be implicated in the transmission of the gravitational force? Beyond-standard-model physics has been investigated along exactly these lines as a direction for understanding dark matter and dark energy, through the seesaw mechanism and the unique state oscillations these particles allow.

The Original Majorana Tower

Most physicists know Majorana for the Majorana spinor, the equation, and the zero modes. Fewer know that Majorana's 1932 paper introduced an infinite-component relativistic wave equation, the first construction of infinite-dimensional unitary representations of the Lorentz group, predating Wigner and Bargmann by years. The equation yields an infinite tower of particle states with increasing spin and a characteristic mass spectrum. Unlike the Dirac equation, all energies are positive. Majorana's original motivation, in fact, was eliminating the negative-energy sea. Higher-spin states have lower effective mass, creating a natural hierarchy of states across scales.

Tamburini and collaborators showed that this extends to photons with orbital angular momentum propagating in structured plasmas via the Anderson-Higgs mechanism. The key point for turbulence is this: the Majorana tower describes an infinite set of states where higher angular momentum corresponds to lower effective mass, a natural hierarchy mirroring the multi-scale cascade structure of turbulence, where energy is transferred from large eddies to small. That is to say, eddies of low angular momentum at large scale, to eddies of high angular momentum at small scale.

Migdal's Loop Equation and the Zeta Zeros

The connection between the zeta function and turbulence has been made even more explicit by Alexander Migdal. His loop equation program provides a rigorous derivation showing that the non-trivial zeros of the Riemann zeta function directly govern the complexity decay exponents of turbulence. Migdal's strategy is to abandon the local point-wise description of turbulence in favor of integrated loop observables, a move borrowed directly from quantum chromodynamics.

In 1993, Migdal predicted that the probability distribution of circulation Γ\Gamma Γ around a large loop CC C depends on the minimal surface area AA A bounded by the loop, and not on the loop's detailed shape. This is the direct turbulence analog of the Wilson loop area law in confining gauge theories, where the Wilson loop expectation decays with area, signaling quark confinement.

The prediction went against every existing turbulence theory and was considered controversial. In 2019, a group at NYU confirmed it numerically via direct numerical simulations, finding that circulation is a bifractal quantity whose moments scale with loop area. This confirmation, twenty-six years after Migdal's original prediction, prompted him to return full-time to research in theoretical physics.

The area law also creates a structural parallel with the Ryu-Takayanagi formula from holographic gravity, where the entanglement entropy of a boundary region equals the area of the minimal bulk surface. All three contexts (QCD confinement, turbulence circulation, and holographic entropy) share a key physical observable computed from the area of a minimal surface bounded by specific boundary data. The shared minimal surface principle is, I think, the deepest geometric thread tying Migdal's turbulence to quantum gravity.

The most striking result of the Migdal program is how the non-trivial zeros of the Riemann zeta function appear as complex decay exponents of turbulence. This is not an assumption or a conjecture overlaid on the result. It emerges through a specific traceable chain of mathematical steps. The Euler ensemble involves star polygons with a co-primality constraint, and that constraint injects the Euler totient function into the statistical mechanics. When one computes the Mellin transform of the energy spectrum and averages over the ensemble, a Dirichlet series falls out. The physical consequence is log-periodic oscillations in turbulent correlation functions, oscillations on a logarithmic time scale generated by the imaginary parts of the zeta zeros. No phenomenological model of turbulence predicts them. Migdal reports emerging experimental evidence for them in high-precision wind tunnel data from the Max Planck turbulence group.

This result does not prove the Riemann hypothesis. But it creates a physical context in which the hypothesis has observable consequences. If all non-trivial zeros lie on the critical line, the complex exponents have a specific universal structure. An off-line zero would produce a detectable anomaly in turbulent decay measurements. The geometry of the Riemann zeta function becomes, in principle, measurable by experiment.

Cascades, Conformal Symmetry, and the Saturation Picture

The connection between the Majorana tower and turbulent cascades is currently at the level of conjecture rather than proven physical equivalence. But the parallels are detailed enough to constitute a serious research program.

In classical turbulence, energy cascades from large eddies to small ones through a self-similar sequence of vortex breakups, each scale contributing to the overall energy spectrum. The intermediate inertial range is characterized by scale invariance: the same physics repeating at every scale. This is precisely the structure described by conformal field theories, the quantum field theories of critical points. The Majorana tower provides infinite-dimensional unitary representations of the Lorentz group, which contains the conformal group at the appropriate limit. The tower's infinite spectrum, with its mass-spin relation, creates a level spacing that maps onto the cascade's energy levels. And periodic driving of Majorana-like systems to high angular momentum states, climbing the tower, produces spectral statistics governed by the zeros of the Riemann zeta function. That ties the tower directly to the turbulence decay exponents.

The picture I keep coming back to is that the critical line of the Riemann zeta function is the statistical saturation point of these systems. It governs the tipping points in models of macroscopic quantum-like behavior, including quantum chaos and fluid turbulence. The Migdal program and the Tamburini results, taken together, suggest that both turbulence and the Majorana tower may realize precisely such a system, and that they may be the same system seen through different lenses.

The physics here looks, to me, similar to the physics of dendritic arborization in brain tissue, or to magnetohydrodynamic instabilities in plasmas, since magnetohydrodynamics is essentially the same as ordinary hydrodynamics with the addition of Maxwell's equations. One way to see it is through Tesla coil streamers.

Vacuum tube driven Tesla coils produce fractal streamers that seem to lack magnetohydrodynamic instabilities or streamer bifurcations. Streamer geometry can be modulated, with degrees of freedom pruned to a laminar setting or increased, in a way that resembles how dendritic connections are modulated in long-term depression and long-term potentiation. At the phase transition, there is a conformal fractal pattern across scales.

Vacuum tube driven tesla coils (VTTCs) produce characteristic streamers with fractal patterns and spearlike arcs where magnetohydrodynamic instabilities or bifurcations (MHDs) are pruned in a manner similar to dendritic arborization in brain tissues.

Migdal's 2025 extension to magnetohydrodynamic turbulence uses two coupled Euler ensembles, one for hydrodynamic circulation and one for magnetic circulation. The same zeta-zero structure persists, with an additional phase transition at magnetic Prandtl number Pm=1P_m = 1 Pm​=1 where the solution bifurcates. This robustness of the zeta structure across pure hydrodynamic and magnetohydrodynamic contexts strengthens the case that the zeta zeros represent a genuine universality class of cascade dynamics, and not an artifact of any one model.

Closing

Two Millennium Prize problems. One geometric structure. The Riemann zeta function turning up, again and again, at the saturation point of nonlinear deterministic systems, in the spectra of accelerated Majorana fields, in the loop statistics of turbulent fluids, and possibly in the bosonized output of driven entangled spin systems carrying orbital angular momentum. If this is right, then turbulence in a tokamak, the distribution of primes, the cascade of vortices in a wind tunnel, and the eigenvalues of the operator we have been searching for since Hilbert and Pólya are pieces of a single picture.

I do not claim this is settled. I claim it is the most interesting research direction I know of, and one where the experimental knobs are getting close enough to turn.

Backpropagation in the Brain Does Not Work Like AI Architectures

The data center arms race is a confidence trick that is doomed to fail due to the underlying physics which makes our brains work differently and orders of magnitude more efficiently.

In spite of attempts to undermine the credibility of theories of consciousness reliant on new physics, they are nonetheless credible enough to continue to demand the attention of the world's top academics, scientists, corporations, leaders, and governments.

What makes the human brain different from transformer based neural network architectures? Or, put another way, what gives us consciousness, and therefore the moral standing that flows from it, and can we ever approach this empirically rather than as a matter of faith or pure philosophical speculation?

This is a topic I've discussed with some of the top researchers in this area (Dr. Tuszynski, Dr. Murugan, Dr. Anirban, Dr. Craddock, James Tagg, Dr. Tamburini, Dr. Hameroff, etc). I was invited to present on this topic directly by Dr. Stuart Hameroff at the The Science of Consciousness (TSC) conference this year (though the TSC conference was allegedly shut down due to organizer affiliations to Jeffrey Epstein) and was also accepted to present at the APS/IPI conferences:

Backpropagation in the Brain: is the Transformer Architecture Viable? [REUPLOAD]

Why Current AI Architectures are Not Conscious: Neural Networks as Spinfoam Networks in a Theory of Quantum Gravity | IPI Letters

The singularity narrative is upside down

Large technology companies would like you to buy, on faith, the idea that we are approaching a so-called technological singularity at which machines will outwit the masses, after which we will, in some vague way, be enslaved by them. The narrative is then used to rationalize giving these systems agency, rights, and an ever-larger share of the world's energy and capital.

I want to make the opposite argument. The real singularity, if you want to use that word, is the point at which our collective intelligence (CI in the academic literature) catches up to and then surpasses the games being played on us by the people pushing this story. It is the moment we discover we have been outwitted the entire time, and not by the machines.

Information throughput in groups scales faster than per-unit-of-energy performance scales in transformer-style surveillance architectures. The latter is derivative and runs into asymptotic limits, a fact that Joseph Tainter described in The Collapse of Complex Societies without ever needing to mention GPUs. The former is what researchers studying interbrain synchrony are quietly documenting in lab after lab. Group cognition is not a metaphor. It is measurable, and it grows in a way that transformer architectures do not.

So why are we propping up the entire economy on the opposite assumption? Why are we taking seriously Mark Zuckerberg's stated ambition for data centers the size of Manhattan, or Google's reported plans for orbital data centers with their own dedicated power plants, when by many credible estimates the human brain is hundreds of thousands of times more efficient at compute than anything we are currently building, and when we cannot even reliably house, feed, or educate our own people?

Yann LeCun and others have proposed so-called world models, in which AI-endowed robotic agents are placed into society and taught to take human jobs. Years and oceans of training data later, we still do not have reliable self-driving cars, and we do not have the patience or resources to raise our own children. There is a tell in there, somewhere, if you are willing to look at it.

The 2024 Nobel Prize in Physics, awarded to John Hopfield and Geoffrey Hinton for work on machine learning, has come in for serious criticism along these lines. The complaint is not that the work is uninteresting. The complaint is that it does not teach us about the laws of nature in the way a physics prize is supposed to. It is computer science with a heuristic architecture imposed on it, evaluated by humans with attention in the loop, who are also the ones who decide what the simulation means. Nature has not been falsified. Nature has been bypassed. Some have gone further and argued that the prize is being used to rationalize the AI surveillance bubble that has been under construction for at least as long as I have been thinking about consciousness.

How I got here

I have long been drawn to the Penrose–Lucas argument that Gödel's incompleteness theorem points toward a non-algorithmic component of human reasoning. The argument has well-known claimed critiques (Putnam, Davis, Feferman, Chalmers), most centrally that it requires the human mathematician to know their own consistency, which is itself what Gödel's theorem says cannot be established. I find the argument suggestive rather than conclusive, and I treat it as motivation for looking at new physics, not as a derivation of it, and to inspire skepticism of any claims for cryptographic standards that are supposedly unbreakable because any conception a mathematician might have about the ultimate security of a standard is necessarily either incomplete or inconsistent.

Shortly after, I encountered the work of Roger Penrose, who had reached essentially the same conclusion from Gödel and gone further, implicating quantum gravity and macroscopic quantum-like effects in brain tissue. The standard objection is obvious. Wet, warm, noisy environments should decohere any such effect long before it could matter. I considered that a serious problem for the original formulation of the theory, but I was intrigued enough to keep going.

The next year I started studying quantum gravity and lattice mathematics under Richard Borcherds, the Fields medalist who proved the monstrous moonshine conjecture. In 2018, when I was in Boulder, NIST's post-quantum cryptography program leans on lattice problems whose security rests on a worst-case to average-case reduction (Ajtai, Regev). The average-case hardness used in the cryptosystem is reducible to the worst-case hardness of approximate lattice problems. Exact SVP is NP-hard, but the approximation factors used in cryptography are not currently known to be NP-hard, so the security claim is conjectural rather than proven.

Under known assumptions of either classical or quantum physics, they are intractable. That mattered to me for an old reason. It has always struck me as too hubristic a request of the universe that any class of unbreakable encryption should exist above scrutiny, available to a privileged group of elites for the hoarding of secrets.

Here is where it gets interesting.

Four problems, one shape

If you read the literature carefully, you find that four apparently distinct problems have all been framed as related, and in some cases as equivalent.

1. The black hole information paradox.

2. Post-quantum cryptography.

3. The shortest vector problem on a high-dimensional lattice.

4. The hard problem of consciousness, and the related binding problem.

   
   

https://www.mdpi.com/1099-4300/25/12/1645

https://ieeexplore.ieee.org/document/743432

https://www.researchgate.net/publication/242663551_Analyzing_Vision_at_the_Complexity_Level

This is not a fringe claim. It is in the literature. And it has practical consequences. The research of a physicist working on the black hole information paradox can, at least in principle, be quietly redirected toward developing cryptography. The work of a mathematician on string theory or lattice mathematics can be quietly redirected toward AI surveillance systems. The researchers and the public need not be told.

In each case, the underlying physics for resolution is not well-established or widely known. In the black hole case, information falls in but quantum mechanics insists unitarity is preserved. Several authors in the consciousness literature have proposed that the binding problem maps onto hard combinatorial-geometric problems, including lattice problems or noncommutative tori closely related to SVP, and there are real issues with alternative theories like predictive coding.

Both of those structures appear empirically in the brain. The Blue Brain Project found them. So did Edvard and May-Britt Moser, along with John O'Keefe, in their work on grid cells, for which they shared the 2014 Nobel Prize in Medicine. Related high-dimensional structures appear empirically in cortical tissue. The Blue Brain Project (Reimann, Nolte, Markram et al., 2017) reported high-dimensional simplicial complexes and rich directed-graph topology in reconstructed microcircuitry. This is algebraic-topological evidence rather than direct evidence of non-commutative tori, but it indicates that cortical connectivity sits naturally in a high-dimensional geometric setting and is one one study out of many in the literature.

When researchers actually went after the black hole information paradox, monstrous moonshine and macroscopic black hole entropy turned out to be linked through holographic descriptions of black hole microstates. Several modern theories of quantum gravity now treat gravity itself as an entropic or thermodynamic force. The Cardy formula gives the asymptotic density of states in a 2D conformal field theory, providing a microscopic derivation of the Bekenstein-Hawking entropy. To resolve the paradox of information that must be both publicly hidden and uniquely accessible, the theory of secret black hole information islands was introduced. These are regions inside the event horizon that, on recent quantum gravity accounts, are holographically encoded in, and possibly entangled with, the leaking Hawking radiation. They follow the unitarity-preserving Page curve.

Notice the shape of this. Information is trapped, and yet it escapes, encoded, in something that radiates out. That is a Cartesian duality if you want to read it that way. The hidden islands of entanglement entropy are where the mind is stored, apart from the body of the black hole. That same shape may be exactly what we need to understand the brain.

What we know about the brain that does not fit the transformer story

Let me list the things we know empirically that the dominant AI story does not account for.

Information and memory in the brain are processed non-locally, distributed across tissues, and not stored in localized binary logic gates the way a von Neumann architecture stores them. The speed of behavior and information retrieval seems to outrun what standard electrochemical signaling across dendritic membranes can permit on its own. To make things even more interesting, single-celled organisms display Pavlovian learning which would seem implausible based on assumptions that this type of behavior comes from dendritic neural networks:

Biologists stunned after a brainless single cell showed Pavlovian learning

Backpropagation in the brain, also known as the weight transport problem or credit assignment problem, has no widely accepted, biologically plausible mechanism. In artificial networks we cheat. While useful in modeling economic systems which require one-way transactions, in biological tissue, no one has produced an obviously correct account of bidirectional feedforward and feedback signaling.

Hyperscanning studies show that brain activity synchronizes across people sharing a social environment, and that this synchronization correlates with shared understanding and empathy. Group performance, in many tasks, scales faster than the sum of individual performances. Transformer architectures do not have this feature.

Empirical studies of human decision-making show interference patterns that look more like the mathematics of non-classical physics than like classical probability. Psychedelics produce conformal fractal patterns across scales in the visual field, again more consistent with non-classical geometry than with classical signal processing. Single-celled organisms display behaviors complex enough that you would expect them to require a brain. The energy efficiency of the brain alone tells you that purely electrochemical signaling cannot account for perceptual binding.

Inside the cytoskeleton of cells are long cylindrical proteins called microtubules, which anesthetics selectively block. Xenon anesthetics have been tested with different xenon isotopes, and anesthetic potency varies with the isotope. That should stop you in your tracks. Different isotopes of the same element, with the same chemistry, produce different effects on consciousness. The natural reading is that consciousness is partly generated by non-classical means involving spin dynamics, consistent with the radical pair mechanism known from quantum biology.

Ultraviolet super-radiance has been measured in brain tissue in some newer (and admittedly contested) studies. Researchers including those I have spoken with at length suggest that microtubules act as time-crystalline optical waveguides. Light has been shown to modulate long-term potentiation and long-term depression. Ultra-weak photon emission from isolated neurons correlates with action potential firing. There are even more fringe studies suggesting superconductivity or near-superconductivity-induced effects in microtubules. Those last claims need much more experimental work, and I will not defend them past saying they should not be dismissed without that work.

Numerical simulations of information stored in fermionic spin states imprinted onto biophotons with orbital angular momentum.

The model

Penrose's original orchestrated objective reduction theory has had problems with experiment. But recent work in quantum biology suggests that macroscopic quantum effects in the warm wet noisy brain might be possible after all, through periodic driving into Fröhlich condensates, or through topological protection, both of which are under active investigation at the major tech companies. (Microsoft, where I once worked, has invested heavily in Majorana physics for exactly these reasons. The work of James Tagg and Dr. Kerskens ties this directly into the model I am about to sketch.)

Tegmark's 2000 decoherence criticism of Orch-OR was directly rebutted by Hagan, Hameroff, and Tuszyński in Phys Rev E (2002), who argued he modeled the wrong system (24 nm separations versus the smaller ones Orch-OR actually proposes) and ignored shielding mechanisms like Debye counterion layers, ordered water, and lattice-based error correction; their recalculation extended decoherence times by seven orders of magnitude, though still short of the 25 ms target. Subsequent experimental work, especially Babcock et al. 2024 demonstrating UV superradiance across tryptophan mega-networks in microtubules, plus xenon nuclear-spin anesthesia studies and the broader quantum biology field, has chipped further at the "warm-wet-noisy means impossible" framing without actually proving Orch-OR.

On the Penrose-Diósi side, Donadi et al. 2021 (Nature Physics) ruled out only the natural parameter-free version of the model using Gran Sasso germanium detectors, leaving regularized versions with larger R₀ alive; however, the 2024 follow-up by Figurato et al. showed that closing the remaining gap would require 18 orders of magnitude better experimental sensitivity, which is brutal but not strictly a falsification, and dissipative variants and alternative gravitational collapse models remain on the table. Additional studies on quantum chaos or even topological protection might close this gap. Some forms of structured OAM light like "Hopfions" could preserve properties at scale.

In our model, information stored in Majorana-like fermionic spin states hosted within microtubules is orchestrated to saturation. At a critical fixed point, a tipping point, the information bosonizes into light-like modes, manifesting as cascades of super-radiant ultra-weak Majorana-like vortex biophotons. These collapse the evolving superposition. The collapse is triggered gravitationally. The phase transition itself is captured mathematically by Z2 orbifolds.

I once discussed with Edward Witten his proposal that pure gravity in anti-de Sitter space (a spacetime of negative curvature) might be described by the monster conformal field theory, which describes massless bosons. Z2 orbifolds are how you transition from such a CFT to a fermionic spin system in de Sitter space (positive curvature), like the so-called baby monster CFT. These are real mathematical objects. In quantum gravity models, Z2 orbifolds are fundamental in constructing Israel junction conditions, the rules for gluing two spacetime geometries together.

The Riemann zeta function and its generalizations, like the Epstein zeta function for high-dimensional lattices, are used in this setting to regularize divergent vacuum energy and define partition functions. In experiments, the zeros of the Riemann zeta function can be reproduced by periodically driving qubits. Mathematical physicists like Dr. Tamburini, with whom I have had many long conversations, have shown you can describe the behavior of particles that are their own antiparticles (Majorana fermions) on curved spacetime using the zeta function. The critical line marks the saturation point in the statistics of these systems. It also turns up in tipping points for macroscopic quantum-like behavior, including quantum chaos and fluid turbulence. This is more or less exactly what the Hilbert-Polya conjecture proposes, that the zeros of zeta could be the energy levels of some unknown quantum system. We may be looking at one.

In loop quantum gravity, the spacetime substrate is a spin foam network. Causal fermion systems theory uses similar graph structures to quantize spacetime. These look extraordinarily like the spin-state networks in the brain. Penrose's proposal is that quantum gravity introduces a non-computable element into physics. Dr. Scott Aaronson, by contrast, has been a sharp public critic of Orch-OR. Readers should weigh the Penrose proposal against Aaronson's critiques - but what is interesting is that although Dr. Aaronson expresses skepticism regarding Penrose's proposal, a similar proposal appears in one of his own publications "NP Problems and Physical Reality" where he suggests that spinfoam networks may possibly act as substrates for the noncomputable physics Penrose describes.

Penrose has further suggested that a complete theory of quantum gravity will be written in the mathematics of null light geodesics, sometimes called soft hair in twistor theory with similar structures in Einstein-Cartan theory. The monster vertex operator algebra, corresponding to the monster CFT, maps cleanly to twistors. Those light-like modes are analogous to the hidden islands of entanglement entropy I mentioned earlier in the black hole context. They are how the mind, in this picture, attaches to the body of the neural network.

If you have been keeping score, that is a single picture in which the black hole information paradox, the shortest vector problem, post-quantum cryptography, and the hard problem of consciousness are all aspects of the same physics.

Noncommutative tori and high dimensional lattices have been studied as representations of brain neural networks by credible research groups, where the perceptual binding problem is resolved through new physics - the equivalent to the shortest vector problem.

A falsifiable prediction

I do not currently have funding for this work, but have been in active discussions with research groups that have been interested in collaboration. It seems the available funding is mostly directed toward perpetuating the status quo. So what I can offer at the moment, in lieu of an experiment, is a numerical simulation and a falsifiable prediction.

If the model is correct, information stored in Majorana-like spin states within microtubules should imprint onto super-radiant ultra-weak biophotons.

Several studies have shown superradiant Majorana-like vortex biophotons propagate through brain tissue, and that light modulates long term potentiation (LTP) and long term depression (LTD). This light carries quantum information through the property of orbital angular momentum (OAM) with feedforward and feedbackward information necessary to carry out backpropagation.

Spectral analysis of those photon signatures could provide a method of post-quantum cryptanalysis. Specifically, the smallest eigenvalue of the Dirac-like operator spectrum over the relevant space corresponds to the shortest vector of the high-dimensional lattice, or the non-commutative torus, that any given neural network represents. There is already related work attacking the shortest vector problem with spin-glass and folded spectrum methods. The proposal here is to drive a neural network representing such a lattice problem to gravitational collapse and read out the geometry from the ultra-weak photon spectra at the phase transition.

If the photons originate from exotic Majorana-like states in the cell, they should carry a quantum fingerprint. A system with conserved parity is linked to the polarization of the photons it emits. From numerical simulations I predict three measurable signatures.

First, Floquet sidebands. Extra spectral lines from periodic driving.

Second, a magnetic field-dependent polarization bias.

Third, strong cross-correlations showing photons alternate polarization in sequence.

Detecting any of these would be strong evidence that biophotons are not metabolic noise but are carrying quantum information from deep inside the cell, and that this is the substrate through which the brain achieves the equivalent of backpropagation. It would also be evidence for what actually distinguishes mind from machine.

Why this matters now

The argument I am making is not anti-technology. It is anti-confidence-trick. We are being asked to spend astonishing amounts of money and energy on an architecture whose advocates cannot tell you, in physical terms, why it should approach what a three-pound piece of biological tissue does for twenty watts. We are being asked to grant moral standing to systems whose advocates have not solved, and in many cases have not seriously engaged with, the physics that would make such standing meaningful.

The serious answer to the consciousness question may not require Manhattan-sized data centers or orbital power plants. It may require something much harder for the present economic order to monetize, namely, a deep and patient interest in what makes us human, and in what allows one mind to connect with another.

Trevor Nestor trevor.nestor at berkeley dot edu

Possible Vulnerabilities of PostQuantum Cryptography to Attacks with Physics Beyond the Standard Model

What if our newest "quantum-resistant" cryptographic standards are vulnerable to neglected physics at the intersection of quantum and classical approaches and physics beyond the standard model?

I want to walk you through something I've been thinking about since I was a student of Fields Medalist Richard Borcherds at UC Berkeley. Borcherds specializes in lattice mathematics and string theory, and he is famous for solving the monstrous moonshine conjecture. Sitting in his classes planted a seed that has grown into what I think is one of the more uncomfortable questions in modern security research.

Presentation

https://www.youtube.com/watch?v=jSqeYz8Wh-Q

The setup

Most security professionals know the basic story. Quantum computers are coming. They will eventually break the cryptographic primitives we rely on today. In response, NIST has been standardizing what we call post-quantum cryptography, and back in 2018 (when I happened to be visiting Boulder, Colorado) they put out the first round of candidates supposed to be resilient to both quantum and classical attacks.

Here is the part people gloss over. That resilience has never been fully proven. It is a strong conjecture, not a theorem.

The cryptographic community got a sharp reminder of that when one of the candidates, SIKE (supersingular isogeny key encapsulation), was cracked in about 62 minutes on a standard Intel CPU. Not a quantum computer. A standard CPU.

Most of the surviving post-quantum schemes lean on what are called lattice problems, and specifically on the difficulty of the Shortest Vector Problem over a high-dimensional lattice, or its close geometric cousin, the non-commutative torus. The bet is that finding the shortest vector in that kind of structure is computationally infeasible for any classical or quantum attacker.

My own bias has always been that any claim of truly unbreakable encryption is too hubristic a request to make of the universe. Nature has a long track record of collapsing our strongest assumptions when we get too confident. So I have been looking for where the next surprise might come from.

The bridge nobody is crossing

What I have come to believe is that there is an entire class of physics, sitting at the intersection of classical and quantum (sometimes called physics beyond the standard model) that has gone almost completely unexamined as an attack surface.

To see why this matters, you have to notice some strange coincidences in the literature.

The NP-hard Shortest Vector Problem turns out to be deeply related to the so-called Learning With Errors problem, which is the same mathematical structure researchers use to understand how the brain efficiently performs the equivalent of backpropagation. Approximate-SVP and Learning With Errors are tightly linked through Regev's reduction. Some neuroscientists have proposed that LWE-like structures appear in cortical computation. It is also related to the perceptual binding problem, which is the question of how the brain stitches sensory features into a unified, coherent experience.

The perceptual binding problem has been mapped to the Shortest Vector Problem in published work by researchers like Tsotsos, where alternative theories like predictive coding have their own issues. Brain neural networks have been mapped to high-dimensional lattices and non-commutative tori in academic literature by groups like the Blue Brain Project. And here is where it gets stranger. The Shortest Vector Problem has also been tied to the black hole information paradox, because that paradox is essentially a cryptographic question. Information flows one way across the event horizon, but quantum theory demands unitarity, so the information must somehow escape in scrambled form.

So you have three NP-hard problems that look identical at the level of mathematical structure in the same universality class. Post-quantum cryptography. The brain's binding of conscious experience. The black hole information paradox. If they really are equivalent, then any physics that resolves one of them is in principle physics that breaks the other two.

What if we used neurons?

In theory, you could take a culture of biological neurons, stimulate them to encode a particular lattice problem (which Hamiltonian engineering can in fact do), and then read out the shortest vector by spectral analysis. Studying the physics of how this works might also shed light on the black hole information paradox.

How do you actually retrieve the shortest vector from a culture of neurons? The answer involves a layer of biology underneath the neural network itself.

Microtubules, biophotons, and the spectral readout

Inside neuronal cytoskeletons there are long cylindrical proteins called microtubules. Mounting evidence suggests they host topologically protected fermionic spin states. This matters because the brain runs on roughly 20 W of power, which is preposterously efficient compared to the supercomputers that some of our tech leaders now want to power with their own dedicated nuclear power plants. Pure electrochemical signaling cannot account for that efficiency. Something more is going on.

The picture I find compelling is that microtubules host entangled networks of these fermionic spin states distributed across the tissue. These networks are driven toward saturation, and at a critical phase transition the information stored in those entanglements gets bosonized into light-like modes. In experiments this looks like superradiant cascades of ultra-weak Majorana-like vortex biophotons, biophotons that carry a quantum property called orbital angular momentum, and that orbital angular momentum is the carrier of the information previously held in the spin states.

Light with angular orbital momentum (OAM) carries the necessary information required to encode high-dimensional lattice problems. Majorana-like vortex photons carry this OAM property in a feedforward and feedbackwards way that makes decoding lattice problems possible.

At critical points, those superradiant cascades might broadcast error backpropagation across the brain tissue, which would account for perceptual binding. There are experiments showing the cascades exist, and that light can indeed modulate long-term potentiation and long-term depression in neuron cells.

The picture I am sketching would imply that the lattice geometry is in some sense readable from the spectrum of the emitted photons. Whether that readout is computationally efficient, that is, whether it actually solves SVP rather than just exposing it, is a separate and much harder question. Standard quantum complexity (BQP not believed to contain NP) suggests that no physical readout, classical or quantum, gets SVP for free. An efficient attack along these lines would require either new complexity-theoretic franeworks or physics genuinely outside the standard model.

Frameworks in math, physics, and computer science have required modifications before to accommodate new discoveries, however, and the suggestions here do warrant further study.

If this picture is right, then in principle you should be able to extract the shortest vector over your encoded lattice space by doing spectral analysis on this light. The shortest vector should show up as the smallest non-zero eigenvalue. Recent work has also shown that orbital angular momentum light is capable of storing information about exactly the kind of high-dimensional lattice geometries you would need.

Parity-Sector Signatures in Ultraweak Photon Emission from Driven-Dissipative Majorana Spin Systems | IPI Letters

The evidence has been quietly piling up

Experiments with xenon anesthetics that block microtubule channels showed that the specific isotope of xenon used modulated anesthetic potency. Different isotopes differ in nuclear spin, not chemistry. That is a strong hint that the quantum property of spin is implicated in the way the brain processes information, via what is called the radical pair mechanism.

We also know, fairly confidently at this point, that the brain's speed and efficiency cannot be fully accounted for by electrochemical signaling alone, and that information is non-locally distributed across the tissue in a way very unlike a von Neumann architecture.

Studies of microtubules have found resonance frequency peaks across scales that are consistent with conformal field theories, and even time-crystalline behaviors, both of which could be implicated in how microtubules facilitate backpropagation. The picture is that fermionic, possibly Majorana-like spin states are hosted within the hydrophobic pockets of microtubules, the information gets bosonized into superradiant cascades, and the microtubules themselves act as optical waveguides.

Microtubule theories of consciousness or brain function have historically been criticized because they sometimes invoke what seem like bizarre ideas of quantum gravity or macroscopic quantum entanglement, and they do not appear to be viable based on the physics most of us were taught. Newer investigations push back on those assumptions. A whole emerging field called quantum biology now points to under-examined quantum effects being necessary to explain things like cellular signaling, photosynthesis, avian navigation, olfaction, and even patterns in human decision-making that look more like quantum interference than classical probability.

Tegmark's 2000 decoherence criticism of Orch-OR was directly rebutted by Hagan, Hameroff, and Tuszyński in Phys Rev E (2002), who argued he modeled the wrong system (24 nm separations versus the smaller ones Orch-OR actually proposes) and ignored shielding mechanisms like Debye counterion layers, ordered water, and lattice-based error correction; their recalculation extended decoherence times by seven orders of magnitude, though still short of the 25 ms target. Subsequent experimental work, especially Babcock et al. 2024 demonstrating UV superradiance across tryptophan mega-networks in microtubules, plus xenon nuclear-spin anesthesia studies and the broader quantum biology field, has chipped further at the "warm-wet-noisy means impossible" framing without actually proving Orch-OR.

On the Penrose-Diósi side, Donadi et al. 2021 (Nature Physics) ruled out only the natural parameter-free version of the model using Gran Sasso germanium detectors, leaving regularized versions with larger R₀ alive; however, the 2024 follow-up by Figurato et al. showed that closing the remaining gap would require 18 orders of magnitude better experimental sensitivity, which is brutal but not strictly a falsification, and dissipative variants and alternative gravitational collapse models remain on the table. Some forms of structured OAM light like what are called "Hopfions" can preserve quantum properties at scale. Additional work on quantum chaos or topological protection might close the gap.

The mathematics behind all of this lives in some surprisingly familiar places. Twistor theory describes null light geodesics. Einstein-Cartan theory describes spacetime torsion along those geodesics. In string theory, the information carried by light with orbital angular momentum is sometimes called "soft hair" and shows up as one theoretical angle for resolving the black hole information paradox.

The transition of information from fermionic spin entanglements (sometimes called hidden islands of entanglement entropy in the literature) into light-like modes in superradiant cascades can be described with Z2 orbifolds. The behavior at the phase transitions can be described using the Riemann zeta function. There have been recent studies that managed to replicate the zeros of the Riemann zeta function by periodically driving qubits, in models that explicitly link the zeta function to the behavior of Majorana spin states in curved spacetimes, which can be simulated in those environments. Both of these point at a possible resolution to the Hilbert-Polya conjecture, which speculates that the Riemann zeta zeros might eventually be observed as the energy levels of a real quantum physical system. Separately, the Riemann zeta function has been linked to macroscopic quantum-like physics in fluid turbulence, quantum chaos, and phase transitions in nonlinear systems.

There are also adjacent experimental approaches that get at the Shortest Vector Problem from a different angle, including spin glasses and folded spectrum methods.

Penrose and Hameroff's model says the phase transition I keep describing is facilitated by gravity itself. At the critical point, macroscopic quantum superpositions and entanglements of those spin states saturate a complexity bound, possibly forming what are called Frohlich condensates, after which the information stored in those entanglement islands or entanglement wedges is discharged in what they call an objective reduction event, broadcast through the superradiant cascade. That cascade is the macroscopic quantum-like behavior, and the gravitational feedback adjusts dendritic weights, which is to say, it facilitates learning in the neural network. This is supposed to be the resolution to the measurement problem and an explanation for why we do not observe the world around us in superposition. Whether that holds up is going to take more experiments.

It is also conceivable that the information about a black hole interior may escape encoded in a similar fashion, printed on light with orbital angular momentum, in which case spectral analysis is the right experimental tool there too.

Here is what I want every security professional reading this to sit with for a moment.

Your trust in post-quantum cryptography rests on the assumption that nobody can find an efficient way to solve the Shortest Vector Problem. The mathematical literature already links that problem to two phenomena (the brain's binding of experience, and the black hole information paradox) that are subjects of active investigation by top scientists, major corporations, and governments. They are taking it seriously. You might think it is a fringe theory. You are free to think that. But you might also be left behind.

Even if every specific claim I have made above turns out to be wrong, the underlying point stands. There is a road map here for a class of attack on post-quantum cryptography that does not look anything like a classical attack or a standard quantum attack. That alone should be fuel for healthy skepticism toward the marketing language around "quantum-resistant" anything, and toward the viability of the AI architectures we are currently betting the global economy on.

A closing thought on AI

I will end somewhere that might surprise you. The same physics has implications for the current AI boom. If the brain's efficiency relative to our silicon really is a story about microtubule-hosted spin states and bosonized light, then the next breakthroughs in artificial intelligence may not come from scaling up data centers powered by Manhattan-sized nuclear plants and launched into orbit, which some of our tech leaders are now openly proposing. Sort of a crazy idea when you say it out loud. It also may not come from gold-plated nanowires at milliKelvin temperatures.

It might come from a deeper understanding of what makes us human, of how our brains work, and of how that physics extends through our relationships and into our communities.

It might, in other words, be more economical to invest directly in people than in data centers.

That is the conclusion I keep arriving at. I think it is worth taking seriously.

Theoretical Approaches to Solving the Shortest Vector Problem in NP-Hard Lattice-Based Cryptography with Post-SUSY Theories of Quantum Gravity in Polynomial Time by Orch-Or | IPI Letters