Arxe Theory

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Multidimensional ArXe table

Update notice (in effect until August 22, 2026): This article was updated on July 23, 2026 to correct numerical values and terminology superseded by later developments in the ArXe corpus.

n k T^k n(k) Arity number BC (c/o) Ontological Problem Logic Probability Physics Geometry Philosophy ArXe Terminology
1 0 T⁰ 1 – 0/0 Foundation without presuppositions Contradictory (S∧¬S) δ(x-a₀) contradictory certainty Tf, temporal quantum Self-negating point Pure act, encia istence, being, act, spark
2 1 T¹ 3 (2) 1/0 Distinction without third Binary, excluded middle P(A)+P(¬A)=1 Temporal flow, particle↔wave Segment 1D Parencia, formal identity parencia, alter, physical_one
3 -1 T⁻¹ 3 3 0/1 Connection without collapse Ternary, included middle Bayes P(A|B) Observer, emergent π, gauge field Triangle, plane, π Appearance, periencia, medium appearance, period, waiting
4 2 T² 5 (5) 2/0 Independent opposites 4-ary, structural frame Uniform 1/4, P(A,B) 2D space, U(1)×U(1) symmetries Square, quadrature Quadrature, frame, stability symmetry, space, position
5 -2 T⁻² 5 5 1/1 Persisting while changing 5-ary temporal Markov P(X_{t+1}|X_t) Curvature, inertia, hysteresis Pentagon, φ Experience, historicity experience, memory, temperature
6 3 T³ 7 (7) 3/0 Objectivity 6-ary epistemic Gaussian, frequentist Mass, 3D spatial, past/present/future Hexagon, packing Fact, objective reality identity, measurement, number
7 -3 T⁻³ 7 7 2/1 Complexity without collapse 7-ary paraconsistent Power law P(x)∝x^{-α} QCD color, confinement Heptagon non-constructible Holism, interiority complexity, interior, color
11 -5 T⁻⁵ 11 11 4/1 Self-limitation 11-ary fuzzy Extreme values (Gumbel) EM, α, renormalization 11-gon near-circle Kantian autonomy regulation, limit, ampere
13 -6 T⁻⁶ 13 13 5/1 The exceptional 13-ary exceptional Poisson (small λ) Weak interaction, β-decay 13-gon irregular Miracle, event singularity, event, decay
17 -8 T⁻⁸ 17 17 6/1 Hierarchy without collapse 17-ary spectral Multimodal, mixture Generations, scales 17-gon spectral Ontological hierarchy hierarchy, generation, levels
19 -9 T⁻⁹ 19 19 7/1 Hidden modulation 19-ary dark Dark distribution Dark matter (mass value withdrawn — see update notice) 19-gon modulated Dark degrees of being dark_matter, hidden
23 -11 T⁻¹¹ 23 23 8/1 Self-similar growth 23-ary fractal Exponential growth Inflation (~10¹⁶ GeV) 23-gon with gaps Emanation, unfolding inflation, fractal, expansion
29 -14 T⁻¹⁴ 29 29 10/1 Persistent substrate 29-ary background 1/f noise Dark energy, Λ 29-gon near-30 Substrate, foundation substrate, vacuum, background
31 -15 T⁻¹⁵ 31 31 11/1 Stable irregularity 31-ary chaotic Logistic map Phase transitions, turbulence 31-gon indist. circle Deterministic chaos chaos, transition, criticality

COMPACT TABLE (Reduced Version)

Level Problem Solution Main Physics Logic ArXe Term
T⁰ Foundation Contradictory act Tf Contradictory istence
T¹ Distinction Alternation Physical time Binary parencia
T⁻¹ Connection Mediation Observer, π, gauge Ternary appearance
T² Symmetry Structural frame 2D space 4-ary symmetry
T⁻² Persistence Memory Curvature, inertia 5-ary experience
T³ Objectivity Measurement Mass, 3D 6-ary identity
T⁻³ Complexity Internal organization QCD color 7-ary complexity
T⁻⁵ Self-limitation Regulation EM, α 11-ary regulation
T⁻⁶ Exceptionality Singularity Weak, β 13-ary singularity
T⁻⁸ Hierarchy Spectral separation Generations 17-ary hierarchy
T⁻⁹ Hidden modulation DARK Dark matter 19-ary dark_matter
T⁻¹¹ Growth Self-similarity Inflation 23-ary inflation
T⁻¹⁴ Substrate Persistent background Dark energy 29-ary vacuum
T⁻¹⁵ Stable irregularity Deterministic chaos Phase transitions 31-ary chaos

KEY PATTERNS

Positive k (k > 0): Closed BC → Can exist isolated

  • T¹: Physical time (1 closed BC)
  • T²: 2D space (2 closed BC)
  • T³: Mass, objectivity (3 closed BC)

Negative k (k < 0): Open BC → Cannot exist isolated

  • T⁻¹: Frequency, observer (1 open BC)
  • T⁻²: Curvature (1 open BC)
  • T⁻³: QCD color (1 open BC)
  • T⁻⁵: EM field (1 open BC)
  • T⁻⁶: Weak (1 open BC)

Twin Arity numbers in Critical Transitions

  • (5,7): Inertia ↔ Complexity
  • (11,13): EM ↔ Weak
  • (17,19): Generations ↔ Dark Matter

π Emerges in Ternary Levels

  • T⁻¹ (n=3): First ternary → geometric π (Buffon)
  • T³ (n=6=2×3): Second ternary → π in mass/objectivity

FUNDAMENTAL CORRESPONDENCES

Logical Indecidability ⟺ Spatial Simultaneity
Open BC ⟺ Gauge Freedom
Closed BC ⟺ Isolated Existence
Ternary Ambiguity ⟺ π (geometric constant)
Arity Encoding ⟺ Physical Structure