The starting assumption is what Werbos calls Hard Core Einsteinian realism, or HCER. The cosmos, on this view, is a dynamical system — a system whose state evolves according to exact laws — and at our level of experience it can be approximated extremely well by quantum electrodynamics as formulated in the Everett–Wheeler–Deutsch tradition, the interpretation associated with Hugh Everett, John Wheeler and David Deutsch of Oxford. In that formulation, the state of the universe at any time is a wave function psi, a complex-valued function of unit L2 norm defined over Fock space, and everything we observe is the emergent outcome of the dynamical law d psi/dt = iH psi, where H is the Hamiltonian operator over that space. The primary mission of fundamental physics, in this picture, is to find the exact algebraic specification of H — the Hamiltonian of the universe. Werbos does not argue that this equation is the final truth of the cosmos; he explicitly proposes giving equal attention to the Einsteinian alternative, in which we live in a flat or curved Minkowski space governed by Lagrange–Euler equations modified by general relativity and constrained by gauge fields, as developed for example in Moshe Carmeli’s Classical Fields. The point of the paper is not to adjudicate between these candidates but to build the mathematics that would let us extract their consequences either way.

The technical heart of the argument lies in a fact that Werbos insists is underappreciated: for the entire class of Everett–Wheeler–Deutsch systems, the allowed equilibrium solutions are already known exactly. They are given by the grand canonical ensemble, the equation familiar from authoritative texts on condensed matter physics such as Chaikin, Lubensky and Witten’s Principles of Condensed Matter Physics. It is essentially a generalized Boltzmann distribution, assigning probabilities of the form exp(−kE − k1C1 − …), where E is the energy of a state and C1 through Cn are the values of other conserved quantities. There is no randomness in the underlying equation and, contrary to popular myth, no metaphysical zero-point energy terms; the canonical form of quantum field theory predicted vacuum fluctuations without such barnacles. Probability is not fundamental to the law itself but fundamental to what emerges from it. If the state of the multiverse at time t is the wave function at time t, then the density operator rho — a certain function of the probability distribution over states — has known equilibrium forms for all choices of Hamiltonian H. The exact answer to whether patterns we would call life have significant probability of emerging is, Werbos argues, already embedded in that ensemble.

But knowing the equilibrium distribution exactly is not the same as understanding it. The grand canonical ensemble is an object over an infinite-dimensional space of states, and the probability of any interesting pattern — a cell, an organism, an ecosystem — is buried inside it in a form no human can read directly. This is where approximation enters, and where Werbos delivers his most provocative slogan: all life is approximation. Present theories of Darwinian evolution, he writes, are basically crude approximation systems for specific types of state, pattern or object which we usually call life. Ordinary Darwinian evolution can be represented approximately as the progression of a dynamical system toward an equilibrium probability distribution over states — for example, an infinite ocean idealized as a countable infinity of bodies embedded in a homogeneous soup. The mathematical task for question two, what is life, is therefore to develop approximation schemes not only for the state distribution of Everett–Wheeler–Deutsch systems over interesting choices of H, but also for Einsteinian and simpler systems, in uniform notation, so that the general principles and possibilities for emergent organization become visible across domains. Werbos points to the work of Vladimir Arnold probing dynamical systems between the extremes of heat death and frozen fixed points — the zone where turbulence, life and interesting structure live — as an important foundation, anticipating later work by Per Bak and Leo Kadanoff.

The same logic extends to question one, what is mind. Werbos follows the path initiated in discussions among Von Neumann, Wiener and McCulloch: define a mundane intelligent system as one which inputs an inborn cardinal utility function of the kind formalized in Von Neumann and Morgenstern’s Theory of Games and Economic Behavior, together with a universal learning ability to discover strategies of action that maximize the future expected value of that utility. Within neural networks and control theory, this program has already generated a huge mass of designs and mathematics under the banner of RLADP — reinforcement learning and approximate dynamic programming. Werbos, who funded much of this work at NSF, identifies three types of approximation task that structure the field. The first is to approximate a function f(x) by a general approximator F(x, W, C), where W is a set of continuous weights and C a set of discrete variables such as the presence or absence of a connection; the seminal theorem here is Andrew Barron’s 1994 result, which bounded how many weights are needed to approximate a large class of smooth functions to a given accuracy. The second is to predict or model a time series x(t) as a function of its own past, generally using time-lagged recurrent networks; the work of Siegelmann and Sontag showed that such networks have universal approximation abilities, a whole level beyond Solomonoff priors, which were limited to discrete finite systems without stochastic disturbance — Turing machines. The third is to approximate the value functions J or lambda, or the optimal policy of action, in the overall RLADP task; a paper by Kozma, Ilin and Werbos demonstrated that a simple decision task similar to what organisms must cope with already requires a more powerful class of approximation network than feedforward nets trained by backpropagation.

Werbos argues that the new mathematics must fill the obvious gaps in this landscape. It needs to consider all the important general classes of approximating networks, building from simple recurrent nets up through approximators that exploit Euclidean translational symmetry — the convolutional networks so important to today’s machine learning — to more general symmetries, graphical representations and object nets, integrated with issues of time. This ladder of ever more general approximation capability is the basis for his 2014 roadmap for building intelligence as high as that of a mouse brain. Empirical evidence, he notes, supports the fit: Werbos and Davis described regular cycles of forward and backward signal propagation in prefrontal cortex, integrating cognitive prediction approaches with the memory-like learning structures pioneered by Hebb, Grossberg and Kohonen, and argued that mammal brains fit this neural network mathematics better than old behaviorist models which assume no sense of purpose, utility or telos built into such brains. Minds, on this view, are embodied — as authors like Damasio have rightly said, they include sensory and motor connections and hard-wired responses to pain and pleasure — but the paper deliberately focuses on the intelligence component alone, as a way of maintaining mathematical focus.

Then comes the most speculative and, for many readers, the most electrifying section: time-symmetric approximators. Ordinary simultaneous recurrent networks implemented in time-forwards electronics may take enormous time to settle into equilibrium, especially when designed to solve NP-hard problems. But a new class of quantum system, exploiting time-symmetric device models for which empirical data exists but is not yet published, can reach equilibrium between a starting time and a readout time by exploiting quantum mechanics, without requiring an actual passage of time. Werbos and Dolmatova laid out the case for analog quantum computing and the need for time-symmetric physics in Quantum Information Processing, and in 2025 a patent was granted to a new type of quantum computing that applies quantum superposition directly to the optimization calculations underlying deep learning, with detailed papers in press on thermal Quantum Annealing. The formal structure is captured in the distinction between a local Markov process, where the probability of a field value at a point in spacetime depends only on its neighbors in the recent past, and a local Markov Random Field, where it depends on neighbors in past, present and future. The LMP class is a proper subset of the LMRF class, in the same way that feedforward networks with finite time lags are a proper subset of time-lagged recurrent networks. Werbos’s LMRF models over continuous Minkowski space predict that certain components of optical computers may have properties different from what traditional LMP models predict, and that we can build computers with powers classical ones — and traditional quantum computers — cannot have.

The implications for question two are profound. Classical earth-based Darwinian evolution is a feedforward process which takes many, many generations to converge toward an equilibrium thermodynamic distribution. But approximation theorems relevant to the more general time-symmetric class might allow much faster convergence or evolution — the emergence of lifelike patterns — in ecologies or universes where time symmetry comes into play. Werbos raises the possibility that some types of cosmos, such as what certain models of dark matter and energy might allow, could permit the evolution of lifelike patterns like solar system noospheres without the requirement for trillions of years of evolution. He goes further: many people staring at the modern Schrödinger equation assume nothing brain-like could emerge from such a system, but the Boltzmann operator of modern thermodynamics shares interesting properties with the Bellman operator of dynamic programming, whose approximation Werbos has spent decades studying. The story of chaos theory warns that simple-looking systems can generate properties far more interesting than anyone imagined; his claim is that nonlinear Lagrange–Euler systems over Minkowski space, and psi-dot equals iH psi, sometimes generate much more interesting emergent behavior — interesting in ways which may even mirror the kind of complexity, reflection and synchronicity phenomena we see in complex intelligent systems. To what extent, he asks, could gating and object mapping, present even in mouse brains, be reflected in the resonances of large-scale patterns in our cosmos?

A practical example from energy system modeling shows how partial gating works. From 1979 to 1989, Werbos worked at the US Energy Information Administration, charged with developing and using the main official models of the US and world energy economy. The team studied not only how to build good predictive models of systems described by streams of time-series data, but also integration — how to combine multiple models of multiple subsystems so they work together effectively. The Electric Power Research Institute created a vision of a future modeling system entailing a whole ecology of modules: why spend hours of expensive computer time running the most precise detailed model of X, when for a given application a reduced-form mirror of that best model would suffice? The mathematics of deciding whether two modules are congruent — whether one can effectively approximate another — is, Werbos argues, very fundamental, and it is the same mathematics the brain uses when cerebral cortex predicts or models the environment of the organism as it appears on the movie screen of the brain, the thalamus. More general approximation theorems with partial gating would fit both the matching seen in the mundane brain and the capabilities of fifth-generation analog quantum computing learning systems.

The paper closes with a call in the spirit of Hilbert: the next milestone is a more complete, multi-author paper which clearly enumerates the basic types of dynamical system and approximation scheme, states theorems and conjectures in uniform notation, and identifies the gaps that need filling to provide a unified picture. Werbos frames the opportunity against a backdrop of fragmenting human understanding, invoking the integrative tradition of Von Neumann — whose Grundlagen of Quantum Mechanics still anchors the field, and whose three great questions structure the paper — and Hilbert’s articulation of well-defined problems as the fertilizer of solid progress. Whether or not the Everett–Wheeler–Deutsch equation is the total dynamic law of the cosmos or multiverse, Werbos writes, it is the best basis we have right now to make real sense of the emergent phenomena around us on all scales of human experience. To ignore what it can tell us is, in his words, neurotic avoidance behavior. The mathematics of approximation — of mirrors and reflections, as he puts it, echoing the deepest literature — may be the thread that ties physics to life to mind, and the hall of mirrors we live in may yet be straightened out by theorems rather than dogma.

Subject of Research: Mathematics

Article Title: Mathematical foundations of life, mind and all emergent phenomena

Article References: Werbos, P. J. (2026). Mathematical foundations of life, mind and all emergent phenomena. Frontiers in Systems Neuroscience, 20, Article 1803434. https://doi.org/10.3389/fnsys.2026.1803434

Image Credits: AI Generated

DOI: 10.3389/fnsys.2026.1803434

Keywords: dynamical systems, approximation theory, emergence, quantum field theory, neural networks, life, mind, Von Neumann, thermodynamics, evolution, Mathematical, foundations