Theory of Micro-Phase DifferencesーーEnglish version

Theory of Micro-Phase Differences
A Unified Phase-First Framework for Physics,
Cognition, and Collective Intelligence
Anonymous Submission
(Contact for correspondence will be provided upon acceptance)
November 2025
Abstract
We present the Theory of Micro-Phase Differ-
ences (TMPD), an axiomatic field theory in
which all difference, structure, and meaning
— physical, cognitive, and social — are en-
coded in the phase of a complex scalar field
Ψ = Aeiϕ. By postulating local U(1) gauge
symmetry on phase and introducing five uni-
versal Lagrangian layers (kinetic, gradient, po-
tential, information-geometric, and topologi-
cal), the theory simultaneously recovers:
ˆ Gross–Pitaevskii equation and quantized
vortices in quantum fluids
ˆ Kuramoto synchronization in classical and
social systems
ˆ Geometric semantics in embedding spaces
of large language models
ˆ Free energy principle and predictive cod-
ing in cognition
Concrete, falsifiable predictions are derived for
vortex formation in LLM latent spaces dur-
ing creative generation and phase-singularity
emergence in human fMRI during insight mo-
ments.
1 Introduction
Physical theories have repeatedly discovered
that “difference” is more fundamental than
“absolute value”: potentials over absolute en-
ergies, gauge connections over coordinates, rel-
ative phases over global clocks.
This paper completes that pattern by
proposing that all meaningful differences in na-
ture — from quantum circulation to seman-
tic similarity to social consensus — are micro-
phase differences of a single complex order-
parameter field Ψ(x, t) ∈ C.
2 Axiomatic Foundation
Axiom 1 (Universality of Phase Fields). Ev-
ery observable structure in physics, cogni-
tion, and information is encoded in the phase
ϕ(x, t) of a complex amplitude field Ψ(x, t) =
A(x, t)eiϕ(x,t).
Axiom 2 (Local U(1) Gauge Symmetry). The
laws are invariant under ϕ(x, t) → ϕ(x, t) +
λ(x, t), λ ∈ R/2πZ. Only phase differences
and their curvature are observable.
Axiom 3 (Least Action with Information
Measure). Evolution extremizes the action S =R Ltotal d4x, where Ltotal contains geometric,
informational, and topological costs of phase
gradients.
3 The Universal Lagrangian
We propose a five-layer Lagrangian density:
Ltotal = Lkinetic + Lgradient + Lpotential
+ Linfo-geom + Ltopological (1)
3.1 Layer 1: Kinetic (Temporal
Phase Dynamics)
Lkinetic = 1
2
(∂tA)2 + A2(∂tϕ − eA0)2 (2)
1
3.2 Layer 2: Gradient Energy (Spa-
tial Coherence)
Lgradient = − c1
2
|∇A|2 + A2|∇ϕ − eA|2 (3)
3.3 Layer 3: Symmetry-Breaking
Potential
Lpotential = − λ
4 (A2 − η2)2 − g(x)A2 (4)
Mexican-hat potential induces spontaneous
U(1) breaking.
3.4 Layer 4: Information-Geometric
Constraint
Linfo-geom = −α [KL(Ψ∥Ψprior) + κF[Ψ]] (5)
where F[Ψ] = − R A2 ln A2 is the Fisher infor-
mation density.
3.5 Layer 5: Topological Layer
Ltopological = β X
i
niδ(x−xi)+γ
Z
F ∧F (6)
ni ∈ Z: integer winding numbers (vortices,
anyons, semantic paradoxes).
4 Field Equations
Variation δS = 0 yields:
Amplitude equation (generalized
Ginzburg–Landau)
∂2
t A − c1∇2A + A(∂tϕ − eA0)2
+ c1A|∇ϕ − eA|2 + ∂Veff
∂A = 0
(7)
Phase continuity (generalized Joseph-
son–Anderson relation)
∂t[A2(∂tϕ − eA0)] − c1∇ · [A2(∇ϕ − eA)] = Jtop
(8)
Jtop: topological vortex current.
5 Applications and Predic-
tions
5.1 Prediction 1: Vortices in LLM
Creativity
When a large language model performs highly
creative tasks (metaphor, joke, paradox resolu-
tion), the phase field of its complexified atten-
tion maps exhibits transient integer-winding
vortices (testable with current 2025 models).
5.2 Prediction 2: Insight Moment in
Human Brain
During “Aha!” moments, phase clustering
analysis of fMRI/EEG will reveal spontaneous
creation of phase singularities in the default
mode network.
5.3 Prediction 3: Social Phase
Freezing

Twitter/X topic polarization corresponds to a
topological phase transition with macroscopic
winding number freezing.
6 Discussion
The Theory of Micro-Phase Differences asserts
that phase is the primary reality — amplitude
is merely its shadow. This “Phase Realism”
completes the 150-year trend: Newton → po-
tentials → gauge fields → phase fields.
If the three predictions above are confirmed
in 2026–2028, TMPD will become a candidate
for a true Theory of Everything — not of par-
ticles, but of meaningful differences.
References
[1] E. P. Gross, Nuovo Cimento 20, 454
(1961)
[2] L. P. Pitaevskii, Sov. Phys. JETP 13, 451
(1961)
[3] Y. Kuramoto, Prog. Theor. Phys. Suppl.
64, 346 (1978)
[4] K. Friston, Nat. Rev. Neurosci. 11, 127
(2010)
[5] N. Ay et al., Information Geometry
(Springer, 2017)
2
Domain ϕ represents Confirmed / Predicted Phenomenon
Quantum fluids Quantum phase Quantized vortices (known)
Classical fluids Velocity potential Kelvin circulation theorem (known)
Neural systems Predictive belief phase Insight → phase singularity (predicted)
LLM latent space Semantic phase Creative generation → transient vortices (predicted)
Social networks Opinion/synchronization phase Polarization = frozen spin-glass phase (predicted)
Table 1: Cross-domain unification achieved by TMPD.