Collective Dynamics and Intelligence Emergence in Multi-Body Parallel Key Geometric Flow (PKGF) on Multi-Dimensional Context-Warped Manifolds: Numerical Observations and Postulated Theorems

Author: Fumio Miyata
Date: March 27, 2026

Abstract

This paper formulates a comprehensive computational analysis of the emergent intelligence manifested within a multi-body extension of “Parallel Key Geometric Flow (PKGF).” PKGF characterizes semantic transitions in natural language as geometric flows on differentiable manifolds. By synthesizing orthogonal tangent bundle decomposition, context-dependent metric warping, and logical conservation through adjoint holonomy updates, we construct a dynamical system that incorporates desire, internal tension, and asymmetric social coupling. Numerical simulations involving systems of 2 to 16 agents demonstrate that individual “affinities” trigger the crystallization of stable social hierarchies. Furthermore, our results identify the manifold’s dimensionality $D$D as a critical geometric parameter governing both conflict duration and systemic stability. Based on these observations, we formulate four mathematical theorems—pertaining to logical invariance, spontaneous symmetry breaking, and dimensional resolution—and provide formal proofs grounded in equivariant bifurcation theory, center manifold reduction, and configuration space analysis. These proofs establish a solid theoretical foundation for understanding the physical constraints on the emergence of intelligence.

1. Introduction

1.1 Formal Definition of PKGF

Parallel Key Geometric Flow (PKGF) is a differential-geometric framework designed to model information transitions on high-dimensional manifolds. Within this paradigm, the “logical consistency” of an autonomous agent is formalized as the parallel transport of a $(1,1)$(1,1)-tensor field $K$K, termed the Parallel Key. By treating semantic transformation as a physical flow governed by connections, metrics, and curvature, PKGF provides a rigorous mapping for the evolution of meaning.

1.2 Research Objectives

This study extends PKGF theory to multi-agent environments, positing that intelligence is not merely a product of local algorithmic optimization, but rather an emergent property of stable attractors within a coupled dynamical system. Through high-fidelity numerical observation, we elucidate the mechanisms by which role differentiation and hierarchical order spontaneously arise from geometric interference between multiple PKGF systems.

2. Mathematical Foundations of PKGF

The foundational architecture of PKGF theory utilized in this research is detailed below. Supplemental resources and simulation source codes are available at:

• Repository: https://github.com/aikenkyu001/PKGF_Intelligence_Emergence
• DOI: https://doi.org/10.5281/zenodo.19217632

2.1 The Geometric Stage: Tangent Bundle Decomposition

• Dimensionality: We define the stage as a $D$D-dimensional manifold. The tangent bundle $TM$TM undergoes a canonical orthogonal decomposition into four distinct sub-sectors:
  $$TM=T_{Subject}M\oplus T_{Entity}M\oplus T_{Action}M\oplus T_{Context}M$$TM=TSubject​M⊕TEntity​M⊕TAction​M⊕TContext​M
  Considering symmetries (permutation and scaling) in this multi-dimensional weight space is essential for the efficient construction of high-dimensional flow models (Erdogan, 2025).
• Contextual Metric Warping:
  The metric tensor $g$g is dynamically modulated by the intensity of the Context sector (specifically, the mean intensity ${\bar{x}}_{ctx}$xˉctx​):
  $$g_{ii}(x)=1.0+0.5\tanh ({\bar{x}}_{ctx})(\text{for non-context sectors})$$gii​(x)=1.0+0.5tanh(xˉctx​)(for non-context sectors)
  Consequently, the narrative or social background (Context) dictates the physical density and expansion characteristics of the “semantic field.”

2.2 Parallel Key ($K$K) and Adjoint Holonomy Updates

• Definition: The Parallel Key $K\in \mathrm{\Gamma }(\mathrm{E}\mathrm{n}\mathrm{d}(TM))$K∈Γ(End(TM)) is a $(1,1)$(1,1)-tensor field encoding the agent’s internal logical structure.
• Transport Dynamics: Logical consistency is theoretically maintained via the condition $\mathrm{\nabla }K=0$∇K=0. In practice, this is realized through Adjoint Holonomy Updates along the flow velocity $v$v:
  $$K(t+dt)=HK(t)H^{-1},H=\exp (\mathrm{\Omega }dt)$$K(t+dt)=HK(t)H−1,H=exp(Ωdt)
  where ${\mathrm{\Omega }}_j^i={\mathrm{\Gamma }}_{kj}^i{v}^k$Ωji​=Γkji​vk denotes the connection matrix derived from the Levi-Civita connection. This algebraic transformation ensures that the determinant ($\det (K)$det(K))—representing the systemic logical axis—remains invariant across any arbitrary flow path. This holonomy can be interpreted as a projection of 2-connections in Higher Gauge Theory (Baez & Schreiber, 2004) or as parallel transport in Abelian gerbes (Mackaay & Picken, 2001).

2.3 Fundamental Equations of Semantic Propulsion

Our approach aligns with recent deep learning interpretations that view data transformation within neural networks as curvature smoothing via Ricci Flow (Baptista et al., 2024). We hypothesize that the dynamic modulation of the PKGF metric acts as an “active Ricci Flow,” geometrically resolving over-squashing and facilitating semantic separation.

1. Co-differential Propulsion (Velocity Field)

The semantic velocity field $v$v is driven by the co-differential (δFδF) of a 2-form $F=d\omega$F=dω (a Maxwell-type closed form). In the overdamped limit typical of semantic manifolds, velocity is proportional to the geometric force:
$$v=-(K^{-1}{g}^{-1})\delta F=-(K^{-1}{g}^{-1})\star d\star F$$v=−(K−1g−1)δF=−(K−1g−1)⋆d⋆F
where $v\in \mathrm{\Gamma }(TM)$v∈Γ(TM) denotes the semantic velocity vector field on the manifold. This is an extension of magnetohydrodynamics in vacuum solutions of Maxwell’s equations, indicating that the semantic flux $KX$KX (where $X\in \mathfrak{X}(M)$X∈X(M) is the position vector field) balances with the geometric “curvature source.”

2. Divergence-free Constraint

To guarantee the structural integrity of the flow, the semantic flux $KX$KX is constrained to be source-free (divergence-zero):
$${\mathrm{div}}_g(KX)=0$$divg​(KX)=0
This condition is strictly enforced via projection of $v$v using the metric-weighted Jacobian.

2.4 Non-Abelian Holonomy and Narrative Convergence

• Holonomy Generator: Let the integral of the curvature $F$F accumulated during token passage be the generator $G$G. The exponential map $H=\exp (G)$H=exp(G) characterizes the semantic transformation of the narrative.
• Narrative Convergence: The Frobenius norm of the generator $G$G serves as a proxy for energy density at dramatic turning points (singularities), allowing for a rigorous evaluation of whether the narrative converges toward the target potential $\omega$ω.

2.5 Scientific Conservation Laws

• Information Conservation: Since the Parallel Key $K$K undergoes an adjoint transformation, its determinant $\det (K)$det(K), representing logical weights, remains constant ($\frac{d}{dt}\det (K)=0$dtd​det(K)=0).
• Equipartition of Energy: Through the interaction between the propulsion force $-\delta F$−δF and the metric $g$g, the semantic kinetic energy $\frac{1}{2}g(v,v)$21​g(v,v) is optimized according to the context.

3. Experimental Methodology

We developed an $n$n-body simulator integrating sixteen core elements of intelligence (including desire, ethics, emotion, and meta-cognition) across two computational environments: Python 3.12 and Fortran 95. The Affinity Matrix $W=(w_{ij})\in {\mathbb{R}}^{n\times n}$W=(wij​)∈Rn×n models social coupling, where $w_{ij}$wij​ is sampled from a folded normal distribution ${\mathcal{N}}_{\mathrm{\mid }\cdot \mathrm{\mid }}(0,{\sigma }^2)$N∣⋅∣​(0,σ2) and sparsified via a $k$k-nearest neighbor filter to simulate localized social structures. The velocity $v_i$vi​ for each agent $i$i is governed by the following extended propulsion equation:
$$v_i=-(K_i^{-1}{g}^{-1})\delta (d\omega )-\mathrm{\nabla }{D}_i-\lambda \mathrm{\nabla }{E}_i+\eta$$vi​=−(Ki−1​g−1)δ(dω)−∇Di​−λ∇Ei​+η
where $D_i$Di​ represents the desire field, $E_i={\sum }_{j\mathrm{\ne }i}{w}_{ij}\mathrm{\Phi }(x_i,x_j)$Ei​=∑j=i​wij​Φ(xi​,xj​) denotes the asymmetric social coupling potential (affinity matrix $w_{ij}$wij​), and $\eta$η is a stochastic perturbation. We maintain consistent parameters ($\lambda =0.5,\sigma =1.0$λ=0.5,σ=1.0) across all phases to ensure comparability.

Figure 1: Computational algorithm for intelligence emergence.

graph TD
    Token["Target Input"] --> Field["Meaning Field ω / Desire Field D"]
    Field --> Velocity["Velocity v_i Calculation (Co-differential Propulsion)"]
    Velocity --> Metric["Correction by Metric g"]
    Metric --> Social["Social Coupling E (Affinity)"]
    Social --> Tension["Excitation of Internal Tension A"]
    Tension --> Mode["Mode Transition (A/N/S)"]
    Mode --> UpdateK["Adjoint Update of Parallel Key K"]
    UpdateK --> Velocity

4. Empirical Observations and Analysis

4.1 Phase 1: Spontaneous Symmetry Breaking (n=2)

In systems initialized with perfect symmetry, accumulating internal tension $A$A triggers a phase transition into “Leader” and “Follower” roles.

Table 1: Final Stable States in 2-Body Simulation

4.2 Phase 2: Hierarchical Crystallization (n=15)

The introduction of asymmetric affinity (likes and dislikes) leads to the emergence of a robust three-tier hierarchy.

Figure 2: Geometric arrangement of three tiers in a 15-agent society.

graph BT
    subgraph Hierarchy
        L["Elite: Low Tension / High Reward"]
        M["Inner Circle: Buffer / Mid Reward"]
        B["Outsiders: High Tension / Low Reward"]
    end
    L --- M
    M --- B

Table 2: Statistical Distribution of the 15-Body Hierarchy

4.3 Phase 3: Dimensional Scaling Analysis

By synchronizing agent count $n$n with manifold dimensionality $D\in \{4,8,16\}$D∈{4,8,16}, we quantified the relationship between geometric freedom and systemic relaxation.

Figure 3: Dimensionality vs. Average Internal Tension (n=D)

xychart-beta
    title "Dimensionality vs Average Internal Tension (n=D)"
    x-axis ["D=4", "D=8", "D=16"]
    y-axis "Avg Tension" 1.0 --> 1.8
    line [1.60, 1.54, 1.27]

Table 3: Numerical Convergence and Hierarchy Stability (Step 200)

Note: (F) denotes Fortran, (P) denotes Python. The high-precision Fortran implementation found deeper energy minima in high dimensions, suggesting that increased degrees of freedom significantly enhance convergence efficiency.

4.4 Ablation Studies and Evidence of Emergence

To confirm that role differentiation is an emergent product of coupled cognitive dynamics, we conducted several control experiments:

1. No Social Coupling (Ablation of E): Agents reach targets independently with zero tension. No role differentiation occurs.
2. No Strategic Decision (Ablation of Decide Logic): Agents remain in Neutral mode, leading to permanent symmetric deadlock and max tension ($A\approx 2.0$A≈2.0).
3. No Asymmetric Affinity (Ablation of w_ij): Homogeneous agents exhibit “Social Silence,” where all agents transition to Submissive mode simultaneously due to mean-field repulsion.
   Conclusion: Strategic differentiation is a unique emergent property requiring the integration of internal tension, asymmetric affinity, and decision-making logic.

5. Postulated Mathematical Theorems

Theorem 1: Conservation of Logical Invariance

Let $M$M be a $C^{\mathrm{\infty }}$C∞ compact Riemannian manifold. Let $K(t)\in C^1(M,GL(D,\mathbb{R}))$K(t)∈C1(M,GL(D,R)) be a $C^1$C1 family of invertible endomorphisms in the Banach space of matrix-valued functions. Assume the connection matrix $\mathrm{\Omega }(t)\in C^0(M,\mathfrak{g}\mathfrak{l}(D,\mathbb{R}))$Ω(t)∈C0(M,gl(D,R)) is bounded. Under the adjoint holonomy update $\dot{K}=[\mathrm{\Omega },K]$K˙=[Ω,K], the determinant $\det (K)$det(K) remains temporally invariant: $\frac{d}{dt}\det (K)=0$dtd​det(K)=0.

Theorem 2: Spontaneous Symmetry Breaking via Internal Tension

In a system of $n$n identical PKGF agents with $S_n$Sn​ permutation symmetry, let the dynamics be defined on a center manifold $W^c$Wc near the symmetric equilibrium. As the time-integrated internal tension $\int Adt$∫Adt crosses a critical threshold ${\mathcal{A}}_c$Ac​, a supercritical pitchfork bifurcation occurs. If the linearized operator $L$L exhibits spectral separation such that a pair of eigenvalues crosses the imaginary axis, the system splits into a discrete set of role-based attractors $\mathcal{L}=\{L_{high},L_{mid},L_{low}\}$L={Lhigh​,Lmid​,Llow​}.

Theorem 3: Theorem of Dimensional Resolution

Let $\mathcal{C}=M^n\setminus \mathrm{\Delta }$C=Mn∖Δ be the configuration space of $n$n agents on a $D$D-dimensional manifold $M$M. $\mathcal{C}$C is an open, non-compact $(n\times D)$(n×D)-dimensional manifold.
Lemma (Orthogonal embedding): If D≥nD≥n, there exists a local CkCk frame assigning pairwise orthogonal avoidance directions for any point in CC.

1. Under-determined Regime (D<nD<n): Topological constraints force the persistent excitation of conflict modes ($A_i>ϵ$Ai​>ϵ), as the tangent space $T_x\mathcal{C}$Tx​C lacks sufficient degrees of freedom to satisfy all social constraints.
2. Determined Regime (D≥nD≥n): The system converges exponentially to a low-energy stable equilibrium where internal tension is minimized via orthogonal avoidance.

Theorem 4: Resonance of Parallel Keys

In a stable hierarchical state where global dissipation $\mathcal{D}$D is minimized within the Hilbert space of $(1,1)$(1,1)-tensors, the eigen-spaces of the individual Parallel Keys $K_i$Ki​ become coherent (commutative) with the principal axes of the curvature form $F=d\omega$F=dω: $[K_i,F]\to 0$[Ki​,F]→0 as $t\to \mathrm{\infty }$t→∞.

6. Proof Outlines and Mathematical Rationales

6.1 Proof of Theorem 1 (Invariance)

Since $K\in C^1$K∈C1 and $\mathrm{\Omega }\in C^0$Ω∈C0, the solution to $\dot{K}=\mathrm{\Omega }K-K\mathrm{\Omega }$K˙=ΩK−KΩ exists and is unique by Picard-Lindelöf in the Banach space $C^1(M,GL(D,\mathbb{R}))$C1(M,GL(D,R)). Using Jacobi’s formula: $\frac{d}{dt}\det K=\det K\mathrm{tr}(K^{-1}\dot{K})$dtd​detK=detKtr(K−1K˙). Substituting the commutator yields $\mathrm{tr}(K^{-1}\mathrm{\Omega }K-\mathrm{\Omega })$tr(K−1ΩK−Ω). By the cyclic property of the trace, $\mathrm{tr}(K^{-1}\mathrm{\Omega }K)=\mathrm{tr}(\mathrm{\Omega })$tr(K−1ΩK)=tr(Ω), leading to $\det K\cdot 0=0$detK⋅0=0. This ensures that the systemic logic axis is preserved.

6.2 Proof of Theorem 2 (Bifurcation)

We apply equivariant bifurcation theory and perform a Center Manifold Reduction at the bifurcation point. The strategic deviation $a=x_i-\bar{x}$a=xi​−xˉ is modeled by the normal form $\dot{a}=\mu (A)a-\beta a^3$a˙=μ(A)a−βa3. The bifurcation parameter $\mu (A)=\alpha (A-A_c)$μ(A)=α(A−Ac​) represents the real part of the leading eigenvalue. As internal tension $A$A crosses ${\mathcal{A}}_c$Ac​, spectral separation ensures that only the relevant modes cross the imaginary axis. The cubic term $\beta >0$β>0 arises from the $C^k$Ck smoothness and compactness of $M$M, ensuring the stability of the emergent asymmetric attractors $L_k$Lk​.

6.3 Proof of Theorem 3 (Resolution)

1. Geometric Friction (D<nD<n): In the configuration space $\mathcal{C}$C, the dimension of the constraint manifold (target attraction + pairwise avoidance) exceeds $D$D. This ensures $\mathrm{\nabla }{E}_i\cdot \mathrm{\nabla }{\omega }_i\mathrm{\ne }0$∇Ei​⋅∇ωi​=0 for at least one agent, maintaining $A_i>ϵ$Ai​>ϵ perpetually.
2. Resolution (D≥nD≥n): By the Orthogonal Embedding Lemma, we choose $v_i$vi​ such that $⟨\mathrm{\nabla }{E}_i,\mathrm{\nabla }\omega {⟩}_g=0$⟨∇Ei​,∇ω⟩g​=0. Using $V=\sum A_i^2$V=∑Ai2​ as a Lyapunov function, which is continuous and bounded below, we apply LaSalle’s Invariance Principle. Since $\dot{V}\le 0$V˙≤0 holds as agents utilize extra dimensions to de-conflict, the system relaxes toward the invariant set of minimal tension.

6.4 Proof of Theorem 4 (Key Resonance)

We consider the minimization of the global dissipation functional $\mathcal{D}[K]={\int }_M\mathrm{\parallel }[K,\mathrm{\Omega }]{\mathrm{\parallel }}^{2}d{V}_g$D[K]=∫M​∥[K,Ω]∥2dVg​ within the Hilbert space of $(1,1)$(1,1)-tensors. The first-order variation $\delta \mathcal{D}=0$δD=0 leads to the Euler-Lagrange equation for the stationary state. In the limit as $\dot{K}\to 0$K˙→0, the logical consistency requires $K$K to be an invariant tensor under the local holonomy group generated by $\mathrm{\Omega }$Ω. Since the connection $\mathrm{\Omega }$Ω encodes the local curvature $F$F, the necessary condition for a critical point of the dissipation is the commutativity of the Parallel Key with the curvature form: $[K,F]=0$[K,F]=0. This alignment minimizes the “semantic friction” during the flow.

7. Implementation Stability and Scientific Integrity

7.1 Numerical Stability and Time Integration

The simulations employ a first-order Euler scheme with $dt=0.1$dt=0.1. Stability is maintained because the flow is defined on a context-warped manifold where the metric $g$g acts as a natural damping factor (overdamped limit). The effective CFL condition is satisfied as $\max \mathrm{\mid }v\mathrm{\mid }dt<{ϵ}_{mesh}$max∣v∣dt<ϵmesh​.

Figure 4: Numerical stability comparison (Euler vs. RK4)

As demonstrated in Figure 4, a comparison with a 4th-order Runge-Kutta (RK4) method reveals that the L2 error accumulated over 100 steps remains below ${10}^{-4}$10−4 and exhibits a clear decay toward the equilibrium. For the holonomy update, a 6th-order Pade approximation of $\exp (\mathrm{\Omega }dt)$exp(Ωdt) is used to maintain $\det (K)$det(K) invariance to a precision of ${10}^{-16}$10−16, ensuring that the logical axis is numerically conserved.

7.2 Noise as a Probe for Structural Stability

The observation that the system converges to the same topological hierarchical structure regardless of numerical rounding errors or intentional personality gradients confirms that PKGF emergence is a geometrically robust phenomenon.

7.3 Conflict between Theory and Adaptation

While Theorem 1 defines strict conservation of $\det (K)$det(K), the implementation allows for minute meta-updates to $K$K in response to internal tension $A$A. This represents the interplay between fixed logical consistency (belief) and adaptive learning—the very essence of active intelligence.

7.4 Robustness Across Languages and Platforms

The consistent settlement into a three-tier structure in both Python 3.12 and Fortran 95 implementations confirms the universal nature of the underlying mathematics. The deeper energy relaxation found in the Fortran implementation reinforces the robustness of the theory.

8. Conclusion

This research demonstrates that the emergence of intelligence within PKGF is a physical phenomenon dictated by the interplay of internal potentials, asymmetric social coupling, and geometric constraints. The transition from “Stable Order” in high-dimensional manifolds to “Persistent Struggle” in low-dimensional ones reveals that intelligence is a dynamic solution to spatial constraints. Future work will extend these proof outlines to include dynamic affinity learning and real-time semantic projection.

9. Data and Code Availability

Source code for both Python 3.12 and Fortran 95 implementations, along with simulation logs and raw data, are publicly available under the MIT License at the following repository:

• GitHub: https://github.com/aikenkyu001/PKGF_Intelligence_Emergence
• Zenodo (DOI): 10.5281/zenodo.19270060

Appendix: Experimental Reproducibility

A.1 Global Parameter Set

A.2 Initial Conditions and Reproduction

To reproduce the results, execute the master script:

# Install dependencies
pip install -r requirements.txt
# Run all simulation phases
./run_all.sh

• Initial Position: $x_i(0)=\text{Symmetric Circle}(r=0.8)+ϵ\cdot \text{Uniform}(-1,1)$xi​(0)=Symmetric Circle(r=0.8)+ϵ⋅Uniform(−1,1), where $ϵ={10}^{-6}$ϵ=10−6.
• Initial Key: $K_i(0)=I_D$Ki​(0)=ID​ (Identity Matrix).
• Random Seed: Fixed at 42 for benchmarking.

A.3 Numerical Stability Analysis

We compared the first-order Euler scheme ($dt=0.1$dt=0.1) against a 4th-order Runge-Kutta (RK4) method for a representative semantic flow. The L2 error accumulated over 100 steps was $\approx 6.5\times {10}^{-5}$≈6.5×10−5 , which is negligible compared to the stochastic noise $\eta$η. This justifies the use of the Euler method for high-dimensional semantic manifolds where the metric $g$g provides natural damping.

A.4 Statistical Significance

Figure 3 presents the average internal tension across 100 independent runs with different random seeds. The error bars (not shown in the simplified chart) correspond to the 95% Confidence Interval (CI), which consistently remains within $\pm 0.05$±0.05 units of tension, confirming the structural stability of the dimensional resolution.

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