The theory
The data field
on a manifold.
Data Field Theory starts with a single object, a continuous, vector-valued field on a geometric substrate, and follows the observable consequences of treating that object as the carrier of cognitive activity. Three axioms, one equation, four cardinal observables.
The data field
Intelligence manifests in the dynamics of a data field, a
time-dependent, vector-valued function
φ: M × ℝ≥0 → ℝk
defined on a compact Riemannian manifold (M, g),
called the substrate.
Four interpretive commitments
-
01
Continuum. The state is continuous, not a discrete set. A neural network's parameter vector is a discretization of the underlying field, not the field itself.
-
02
Substrate is active. The metric gij carries the inductive bias and relational structure of the system. Different substrates produce different physics on the same equation.
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03
Multi-channel. The field is vector-valued: overlapping features coexist at the same location, each in its own component of φ.
-
04
The field is the cognition. Not a representation of inputs. A concept is a stable configuration of the field, not something the field stands for.
From the axioms chapter (Axiom I), the geometric-foundations paper, and the substrate chapter.
Explore
Walk the basis.
On the simplest non-trivial substrate (the sphere S²) the field decomposes into spherical harmonics Yℓm. Each harmonic is an eigenfunction of the Laplace–Beltrami operator with eigenvalue ℓ(ℓ+1)/R². Drag the sliders to walk the basis, and see why the brand mark sits at (ℓ, m) = (2, ±2).
Free energy and Ginzburg–Landau dynamics
Learning is the gradient flow of a free energy functional. The canonical functional is the Ginzburg–Landau form, with gradient (kinetic) energy on the manifold, a mass term, a quartic self-interaction, and a source coupling:
The gradient flow gives the Time-Dependent Ginzburg–Landau (TDGL) equation:
η
is stochastic forcing tied to fluctuation–dissipation.
The equation is not a metaphor. TDGL is a parabolic PDE on a
manifold whose solutions exist and are unique under standard
regularity. The free energy F[φ] is the Lyapunov function, meaning the DFT-Solver preserves dF/dt ≤ 0 at the discrete level by construction.
The two control knobs
- m² sets the spectral gap, controls the correlation length ξ ~ 1/m, and parametrizes distance from criticality.
- λ drives nonlinear self-interaction: pattern formation, symmetry breaking, multi-modal landscape of stable configurations.
Where you clicked
- m²
- —
- T
- —
- phase
- —
- ξ ~ 1/|m|
- —
Click anywhere in the plane. The marker shows where you are; the readout gives the operative parameters and the regime DFT would predict at that point.
Play
The field, evolving.
The same equation, live. Pull m² below zero and sign-broken patches form (the broken-symmetry phase). Push it well above zero and the field collapses toward zero (symmetric). Near m² ≈ 0 patches grow large: this is the operational fingerprint of the critical point.
From the axioms chapter (Axiom II), the dynamical-laws chapter, the geometric-foundations paper, and the solver chapter.
Emergence
Cognitive phenomena emerge as physical properties of stable field configurations. A concept is a long-lived metastable attractor of the dynamics. Robustness, generalisation, and finite-speed propagation are measurable spectral quantities, not architectural side-effects.
Mass gap: the smallest non-zero eigenvalue of the Hessian of the free energy at a learned configuration, measuring the depth of the energy minimum. Large gap means robustly anchored; small gap means shallow and sensitive.
Mass-Robustness Law: εgen ≤ K / mgap². Empirical correlation ρ ≈ −0.81 across vision (N=200), language (N=156), and reinforcement control (N=189).
From the observables chapter, the mass-robustness paper, and the empirical-signatures chapter.
The substrate
The manifold is an active component of the theory.
The substrate (M, g) is not a passive coordinate system. The metric encodes the inductive bias of the system; the Laplace–Beltrami operator on it determines how information diffuses; the eigenfunctions of that operator are the natural basis for the field. Different applications instantiate M differently: data manifold, weight manifold, representation manifold, agent state manifold, but the mathematical machinery is the same.
Eigenfunctions of Δg on S², the simplest substrate
Peter–Weyl decomposition on S², developed in the substrate chapter.
Methodological discipline
What we borrow from physics, and what we don't.
DFT borrows machinery from condensed matter physics extensively and from quantum field theory selectively. The structural-analogies paper articulates the four-step framework that licenses each borrow and names the inherited limits: formal isomorphism with an explicit interpretational gap.
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Identify the DFT object.
Name the mathematical object on our side: free energy F[φ], generating functional Z[J], scaling exponent ν.
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Identify the physics analogue.
Name the object on the QFT side: Euclidean action S[φ], QFT generating functional with external source, critical exponent at an RG fixed point.
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State when the borrow applies.
RG language requires a meaningful scale and a fixed point. Universality language requires the fixed point to be identified. Partition-function language requires a positive measure on configurations. Apply each only where its prerequisite is met.
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Name the gap that remains.
DFT's φ is not an operator-valued quantized field. DFT has no Hilbert space of states. DFT's particles, if any, are not excitations of operator-valued fields. Every borrowed term carries its interpretational gap with it.
From the structural-analogies paper and the symmetry-and-equivariance chapter.
Continue
From theory to evidence.
The four cardinal signatures: diverging correlation length, 1/fβ temporal spectra, mass-gap–robustness scaling, and finite effective causal speed, are the empirical content of the theory. Reported across vision, language, and reinforcement control within shared uncertainty bands.