The numerical experiments on the complete graph K₄ reveal a remarkable and persistent structure that we refer to as the
κ₃-sector. This sector is characterized by two key observables:
- a stable holonomy deficit Δ ≈ 0.5 on the frustrated faces, insensitive to geometric tilts and linear SU(2) least-squares (LS) lifts,
- a topological stiffness κtop = 3, appearing as a robust amplification factor in the frustration energy.
The combination of robustness and insensitivity to local deformations suggests that these quantities are not numerical artifacts,
but rather reflect a topological structure in the space of SU(2) connections on K₄. In this section we clarify how
the holonomy deficit is defined, how κtop relates to the graph topology, and in what precise sense this structure can be
interpreted as emergent discrete Ricci curvature.
Contents
1. Holonomy Deficit and Target Flux
On each face f of the graph we prescribe a U(1) flux F_f. In the symmetric setup studied here, two faces are
“frustrated” with non-trivial flux
Ff = 2π / 3,
while the remaining two faces carry zero flux. The value 2π/3 (120°) is chosen so that the target configuration is a highly symmetric,
non-trivial curvature pattern compatible with the K₄ geometry: it imposes a fixed amount of “twist” on the frustrated faces while
leaving the other faces flat.
An SU(2) holonomy around face f can be parametrized as
Hf = cos(θf / 2) I + i sin(θf / 2) n̂f · σ⃗,
where θf is the rotation angle and n̂f is the rotation axis. The trace is then
Tr(Hf) = 2 cos(θf / 2).
The target U(1) flux Ff defines a target rotation angle θtarget = Ff, with corresponding
cos(θtarget / 2) = cos(Ff / 2).
For Ff = 2π/3 we have
cos(Ff / 2) = cos(π/3) = 0.5.
In the non-aligned LS solutions we find that the realized SU(2) holonomies on the frustrated faces are nearly trivial, with
Tr(Hf)/2 ≈ 1.0, corresponding to θf ≈ 0. Thus the system consistently converges to an SU(2) sector in which the
frustrated faces carry almost no visible rotation, in sharp contrast with the non-zero target curvature.
We therefore define the holonomy deficit on a frustrated face as
Δ = Tr(Hf) / 2 − cos(Ff / 2).
In our runs this gives
Δ ≈ 1.0 − 0.5 = 0.5,
up to small numerical fluctuations. The key observation is that Δ remains pinned near 0.5 under geometric tilts of the face normals,
axis permutations, and LS refinements. This persistent mismatch between realized and target holonomy motivates interpreting Δ as a
quantized unit of relational curvature: a stable “failure to realize” the prescribed loop curvature.
Physically, this trivialization represents a form of linear screening of non-abelian flux. The linear solver satisfies the
constraint B φ = F at the level of edge angles, but it can use the commutators [Ae, Ae’] (which are quadratic
in the fields) to cancel much of the effective rotation. In this way the system finds a “dark state” in which the linear constraints are
met, the non-abelian curvature is screened, and the residual gap Δ ≈ 0.5 is the visible signature of this screening mechanism.
2. Topological Stiffness and the Role of b₁(K₄)
The second observable is the topological stiffness κtop, defined as the ratio between the realized SU(2) stiffness
C* and an isotropic reference value Ciso built from the prescribed fluxes alone:
κtop = C* / Ciso.
In the K₄ setup we find κtop ≈ 3.0 with high numerical stability. This value is striking because the complete graph K₄
has three independent fundamental cycles, i.e. its first Betti number is
b₁(K₄) = 3.
This suggests a natural hypothesis: each independent cycle contributes an irreducible non-abelian constraint on the SU(2) field.
Since edges in K₄ are shared across multiple faces, frustration is non-local: “untwisting” one face inevitably “twists” its neighbors
through shared edges. The cycle space encodes the inter-dependency of these constraints.
We therefore formulate the following conjecture:
Conjecture (Topological Stiffness and Cycle Rank). For a graph Γ equipped with a symmetric flux pattern, the topological stiffness satisfies κtop ≈ b₁(Γ), where b₁(Γ) is the first Betti number (dimension of the cycle space). In this view, κtop measures the global connectivity of the frustration: how many independent loops must cooperate to accommodate the non-abelian curvature.
At present this conjecture is firmly supported by the K₄ experiments but has not yet been tested systematically on other graphs.
Work in progress includes K₅, Petersen-type graphs, and cycle graphs Cn, where b₁(Γ) takes different values. These tests
will determine whether κtop = b₁(Γ) holds in general or requires refinement by additional graph invariants such as girth or
valency distribution.
3. From Curvature Units to Relational Geometry
The claim that the κ₃-sector represents “emergent geometry” must be understood in a precise, relational sense. The microscopic variables
are SU(2) edge connections and their dynamical evolution; they do not come equipped with a predefined metric, distance, or time parameter.
Those notions arise only once certain invariants stabilize.
The holonomy deficit Δ and the stiffness κtop provide exactly such invariants. They allow us to define:
A characteristic length scale.
A non-zero deficit Δ on a face can be interpreted as a discrete curvature radius via
rf ∼ 1 / Δ. For Δ ≈ 0.5 this yields rf ≈ 2 (in natural units of the model),
giving a scale for how quickly parallel transport diverges around that face.
Angular relations between faces.
Given two faces f₁ and f₂ with holonomies Hf₁ and Hf₂, we can define a discrete “angle”
between them by
φf₁,f₂ = arccos(Tr(Hf₁ Hf₂†) / 2).
Differences in Δ across faces then translate into curvature gradients, providing a combinatorial analogue
of Ricci flow on the graph.
A proto-time parameter.
The relaxation dynamics toward the κ₃-sector can be monitored by the deviation
|Tr(Hf)/2 − cos(Ff/2)|as a function of iteration steps. If this deviation decreases
monotonically (or in a controlled fashion) under the chosen dynamics, it defines an emergent “arrow” of time in terms
of approach to the κ₃ fixed point.
In this way Δ and κtop do not merely “decorate” an existing geometry; they create a network of relational
lengths, angles, and temporal ordering on a configuration space that initially has none. This is the precise sense in which
we interpret the κ₃-sector as a proto-geometric stage of the model.
4. Gauge Invariance and Wilson Loops
All physically meaningful quantities in a gauge theory must be gauge-invariant. In our construction the basic observable on
each face is the trace of the holonomy, Tr(Hf), which is invariant under local SU(2) gauge transformations at the vertices.
Thus both the deficit Δ and the stiffness κtop are gauge-invariant by construction.
From the perspective of lattice gauge theory, Hf is precisely a Wilson loop around face f:
Wf = Tr( P exp(i ∮f A) ),
where A is the SU(2) connection and P denotes path-ordering. The deficit Δ therefore measures the failure of these Wilson loops to match
target values imposed by the prescribed U(1) fluxes. In this language, the κ₃-sector is a phase in which Wilson loops are locked into a
particular non-trivial pattern that cannot be unwound by small gauge-invariant deformations.
5. The κ₃-Sector as a Non-Abelian Bifurcation
The stability of the κ₃-sector under perturbations suggests that it is a fixed point in the space of SU(2) connections on K₄.
This has a natural parallel with the notion of curvature rank in bifurcation theory. In earlier work on one-dimensional maps, we defined a
curvature rank as the order of the first non-vanishing derivative at a critical point, and showed how it controls universality classes and
scaling exponents.
In the present non-abelian setting, κtop plays an analogous role:
- it counts how many independent cycles contribute to non-commuting holonomies (via b₁(K₄) = 3),
- it quantifies the amplification of frustration energy relative to a flat reference,
- it labels a topological sector (the κ₃-sector) that acts as an attractor under relaxation dynamics.
We may therefore view the transition into the κ₃-sector as a non-abelian bifurcation in the space of SU(2) connections:
the system moves from a trivial sector, where holonomies can in principle realize the target fluxes, to a frustrated sector in which
the holonomy deficit Δ is locked at a non-zero value enforced by the global topology.
6. The Energy Barrier as the Cost of Geometry
The separation between the κ₃-sector (where Δ ≈ 0.5) and a hypothetical target configuration (where Δ ≈ 0 on frustrated faces) is not
merely a numerical curiosity; it reflects an energetic barrier imposed by the non-abelian algebra. We can quantify this barrier in terms
of the energy functional C* as follows:
Ebarrier = minA ∈ Target C*(A) − minA ∈ κ₃ C*(A).
The first term corresponds to minimizing the energy while forcing the non-linear holonomies Hf(A) to match the target
values implied by the flux Ff; the second term is the relaxed energy found by the linear solver in the κ₃-sector, where the
holonomies are nearly trivial on frustrated faces. The difference Ebarrier is the energy cost of enforcing the geometric law.
In this picture, the LS solution performs a kind of BCH screening: it rearranges the edge fields Ae so that the linear
constraints are satisfied as cheaply as possible, while the Baker–Campbell–Hausdorff commutator terms cancel much of the effective
non-abelian rotation. The κ₃-sector is then a deep local minimum—a “false vacuum” that is linearly stable but topologically misaligned
with the target flux configuration. Escaping this sector to realize the true target geometry requires paying the energy penalty
Ebarrier.
A detailed numerical determination of Ebarrier would require constructing approximate target-like configurations, performing
non-linear refinement to enforce the holonomy constraints, and comparing their C* values to those of the relaxed κ₃-sector. This is left
as an important next step, but the existing LS results already imply that Ebarrier > 0: the system never crosses into the
target sector under linear relaxation alone.
7. Summary
To summarize, the κ₃-sector is best understood as:
- a discrete Ricci curvature invariant encoded in stable holonomy deficits on frustrated faces,
- a stiffness factor κtop naturally linked to the cycle rank b₁(Γ) and the global connectivity of frustration,
- a non-abelian bifurcation point and topological attractor in the space of SU(2) connections,
- a proto-geometric structure that supports emergent notions of length, angle, and time within a purely relational model,
- and a false vacuum stabilized by BCH screening, with a finite energy barrier Ebarrier representing the cost of enforcing the full geometric law.
In this sense, the κ₃-sector does not merely sit inside a pre-existing geometry; it is the mechanism by which a first layer of
relational geometry crystallizes out of an underlying SU(2) dynamical system.