Axiomatic System of Parallel Key Geometric Flow (PKGF)

Author: Fumio Miyata
Date: April 8, 2026
DOI: 10.5281/zenodo.19481201

0. Purpose

This axiomatic system provides the minimal set of axioms required to establish
Parallel Key Geometric Flow (PKGF) as a self-contained mathematical structure.

PKGF is a new geometric framework that unifies
construction (Constructive PKGF), deconstruction (Destructive PKGF), and metabolism (Unified PKGF)
of intelligence within a single formal system.

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1. Fundamental Data of a PKGF Structure

Axiom A1 (Manifold)

$M$M is a finite-dimensional smooth Riemannian manifold.

Axiom A2 (Tangent Bundle Decomposition)

The tangent bundle $TM$TM decomposes into finitely many subbundles:
$$TM=\underset{\alpha \in I}{⨁}{E}_{\alpha }\mathrm{.}$$TM=α∈I⨁​Eα​.

Axiom A3 (Parallel Key)

The parallel key $K$K is a smooth endomorphism field on the tangent bundle:
$$K\in \mathrm{\Gamma }(\mathrm{E}\mathrm{n}\mathrm{d}(TM))\mathrm{.}$$K∈Γ(End(TM)).

Axiom A4 (Gauge Group)

The gauge group $\mathcal{G}$G is a smooth automorphism group acting on $TM$TM,
equipped with the adjoint action
$$K\mapsto HK{H}^{-1}\mathrm{.}$$K↦HKH−1.

(After gauge symmetry breaking in Unified PKGF, the stabilizer subgroup fixes KK.)

Axiom A5 (Connection)

$\mathrm{\nabla }$∇ is a connection on $TM$TM, with curvature
$$F=d\omega +\omega \wedge \omega \mathrm{.}$$F=dω+ω∧ω.

Axiom A6 (Semantic Potential)

$\mathrm{\Omega }$Ω is an endomorphism field depending on external information and internal representation:
$$\mathrm{\Omega }=\mathrm{\Omega }(\psi (\mathrm{\Phi }),x)\mathrm{.}$$Ω=Ω(ψ(Φ),x).

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2. Axioms of Constructive PKGF (Construction Theory)

Axiom C1 (Constructive Equation)

The fundamental equation of Constructive PKGF is
$$\mathrm{\nabla }K=[\mathrm{\Omega },K]\mathrm{.}$$∇K=[Ω,K].

Axiom C2 (Gauge Covariance)

For any $H\in \mathcal{G}$H∈G,
$$K\mapsto HK{H}^{-1},\mathrm{\Omega }\mapsto H\mathrm{\Omega }{H}^{-1},\mathrm{\nabla }\mapsto H\mathrm{\nabla }{H}^{-1}$$K↦HKH−1,Ω↦HΩH−1,∇↦H∇H−1
leave Axiom C1 formally invariant.

Axiom C3 (Sector Preservation)

If $[K,{\mathrm{\Pi }}_{\alpha }]=0$[K,Πα​]=0 holds at the initial time, then for all $t$t,
$$K(E_{\alpha })\subset E_{\alpha }\mathrm{.}$$K(Eα​)⊂Eα​.

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3. Axioms of Destructive PKGF (Deconstruction Theory)

Axiom D1 (Dissipative Operator)

The deconstruction operator $\mathcal{D}(K)$D(K) is a linear operator that is

• self-adjoint,
• negative definite (or with non-positive spectral half-plane),
• free of commutators.

Axiom D2 (Deconstruction Equation)

The fundamental equation of Destructive PKGF is
$$\dot{K}=-\lambda \mathcal{D}(K)\mathrm{.}$$K˙=−λD(K).

Axiom D3 (Monotonic Rank Reduction)

$$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(K(t+dt))\le \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(K(t))$$rank(K(t+dt))≤rank(K(t))
for all $t$t.

Axiom D4 (Entropy Increase)

For an information distribution $\mathrm{\Phi }$Φ,
$$S[\mathrm{\Phi }]=-\int \mathrm{\Phi }\log \mathrm{\Phi },$$S[Φ]=−∫ΦlogΦ,
the entropy satisfies
$${\mathrm{\partial }}_{t}S[\mathrm{\Phi }(t)]\ge 0.$$∂t​S[Φ(t)]≥0.

Axiom D5 (Minimum Residual Structure)

The fixed-point set of the dissipative operator,
$$\mathcal{F}=\{K:\mathcal{D}(K)=0\},$$F={K:D(K)=0},
is non-empty and compact, and the flow of Destructive PKGF converges to $\mathcal{F}$F in finite time.

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4. Axioms of Unified PKGF (Metabolic Theory)

Axiom U1 (Complex Parallel Key)

In Unified PKGF, the parallel key becomes complex:
$$K=K_{\text{core}}+i{K}_{\text{fluct}},$$K=Kcore​+iKfluct​,
where

• $K_{\text{core}}$Kcore​: deterministic, conservative structure
• $K_{\text{fluct}}$Kfluct​: fluctuation, creativity-generating component.

Axiom U2 (Orthogonality)

$$⟨K_{\text{core}},K_{\text{fluct}}⟩=0.$$⟨Kcore​,Kfluct​⟩=0.

Axiom U3 (Unified Equation)

The fundamental equation of Unified PKGF is
$$\mathrm{\nabla }K=[\mathrm{\Omega },K]-\lambda \mathcal{D}(K)\mathrm{.}$$∇K=[Ω,K]−λD(K).

Axiom U4 (Gauge Symmetry Breaking)

At some time $t_{SB}$tSB​, the gauge group spontaneously reduces:
$$\mathcal{G}⟶{\mathcal{G}}_{\text{broken}},$$G⟶Gbroken​,
where the stabilizer subgroup is defined by
$${\mathcal{G}}_{\text{broken}}=\{H\in \mathcal{G}:HK{H}^{-1}=K\}\mathrm{.}$$Gbroken​={H∈G:HKH−1=K}.

Axiom U5 (Dynamic Sectors)

Under Unified PKGF, the tangent bundle decomposition is not fixed;
sectors may emerge or vanish over time.

Axiom U6 (Dimensional Leap)

The effective dimension
$$d_{\text{eff}}(t)=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(K(t))$$deff​(t)=rank(K(t))
changes discontinuously when internal tension exceeds a critical threshold:
$$d_{\text{eff}}(t_c^{+})\mathrm{\ne }{d}_{\text{eff}}(t_c^{-})\mathrm{.}$$deff​(tc+​)=deff​(tc−​).

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5. Definition (PKGF Structure)

A PKGF structure is a quintuple
$$(M,K,\mathrm{\nabla },\mathrm{\Omega },\mathcal{G})$$(M,K,∇,Ω,G)
satisfying Axioms A1–A6, C1–C3, D1–D5, and U1–U6.

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6. Minimality of the Axiomatic System

This axiomatic system is the minimal set required to define PKGF.
No axiom can be derived from the others.

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7. Characteristics of the Axiomatic System

• Constructive PKGF (Conservative Structure)
  Generates order, coherence, and logical consistency.
• Destructive PKGF (Dissipative Structure)
  Produces collapse, degeneration, forgetting, and singularities.
• Unified PKGF (Metabolic Structure)
  Integrates both processes, enabling breathing dynamics, creativity, and phase transitions.

The three subsystems form a single closed axiomatic structure.

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8. Significance of Complete Axiomatization

• Establishes PKGF as an independent mathematical structure
• Allows Constructive, Destructive, and Unified PKGF to be treated independently and coherently
• Ensures all theorems are derivable from axioms
• Provides a rigorous foundation for dissipative geometry (Destructive PKGF)
• Formalizes creativity and conceptual transformation via complex PKGF

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