Ontology V8.4:Clifford Algebra Dimensional Upgrade, Action-Quantum Generator, Imaginary-Time Coordinate Interface and Dimensional-Reduction Core Formula

Building on the reconstruction of the Clifford algebra basis in \emph{Ontology V8.3}, this paper completes the following integration work:1.it gives the explicit mapping between the V8.3 dimensionless algebraic elements $\mathbf{i},\mathbf{j},\mathbf{k}$ and the Dirac algebra $\gamma^5,\gamma^1$;2.it introduces the action-quantum generator axiom A1 and, through the dimensional upgrade $\hat{\mathbf{j}}=\hbar\mathbf{j}$, gives $\hbar$ an algebraic origin;3.it introduces the imaginary-time coordinate interface axiom A2, defines $\tilde{t}=ict$, explains that the imaginary-time coordinate has dimension of length $[L]$, and bridges to the action quantum through the Planck momentum: $p_P\tilde{t}_P=i\hbar$;4.it clarifies that V8.3 is a 3+1 dimensional Clifford algebra while the core formula of V8.2 uses only one spatial direction of it, so the core formula is a \textbf{dimensional-reduction projection from the 3+1 dimensional algebra to 1+1 dimensions};5.it gives the complete reduction steps, the 1+1 dimensional Lorentz transformation, the time-quantization assumption, the back-solution for $dx$, and the substitution of the Planck time, finally obtaining the core formula $dx=\dfrac{1-\sqrt{1-v^2/c^2}}{v}\sqrt{\dfrac{\hbar G}{c}}$;6.it explains how A1, A2 and the reduction process connect with the physical assumptions of V8.2, and gives the complete 3+1 dimensional generalized form. This paper clearly distinguishes: $ict$ is an imaginary-time coordinate with dimension $[L]$; $\hbar$ is an action quantum with dimension $[ML^2T^{-1}]$; the two cannot be directly equated and must be bridged through momentum. This paper emphasizes: A1 and A2 are interface axioms, not theorems of V8.3; V8.3 alone cannot derive the core formula --- the core formula is a joint result of the V8.3 algebra, the V8.4 interface, and the V8.2 physical assumptions.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-26
DOI
https://doi.org/10.5281/zenodo.22972273
Primary Topic
Algebraic and Geometric Analysis
Type
preprint
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preprint

Ontology V8.4:Clifford Algebra Dimensional Upgrade, Action-Quantum Generator, Imaginary-Time Coordinate Interface and Dimensional-Reduction Core Formula

Shuai Wang
Zenodo (CERN European Organization for Nuclear Research)
Algebraic and Geometric Analysis
preprint

Ontology V8.4:Clifford Algebra Dimensional Upgrade, Action-Quantum Generator, Imaginary-Time Coordinate Interface and Dimensional-Reduction Core Formula

Shuai Wang
preprint en

Abstract

Building on the reconstruction of the Clifford algebra basis in \emph{Ontology V8.3}, this paper completes the following integration work:1.it gives the explicit mapping between the V8.3 dimensionless algebraic elements $\mathbf{i},\mathbf{j},\mathbf{k}$ and the Dirac algebra $\gamma^5,\gamma^1$;2.it introduces the action-quantum generator axiom A1 and, through the dimensional upgrade $\hat{\mathbf{j}}=\hbar\mathbf{j}$, gives $\hbar$ an algebraic origin;3.it introduces the imaginary-time coordinate interface axiom A2, defines $\tilde{t}=ict$, explains that the imaginary-time coordinate has dimension of length $[L]$, and bridges to the action quantum through the Planck momentum: $p_P\tilde{t}_P=i\hbar$;4.it clarifies that V8.3 is a 3+1 dimensional Clifford algebra while the core formula of V8.2 uses only one spatial direction of it, so the core formula is a \textbf{dimensional-reduction projection from the 3+1 dimensional algebra to 1+1 dimensions};5.it gives the complete reduction steps, the 1+1 dimensional Lorentz transformation, the time-quantization assumption, the back-solution for $dx$, and the substitution of the Planck time, finally obtaining the core formula $dx=\dfrac{1-\sqrt{1-v^2/c^2}}{v}\sqrt{\dfrac{\hbar G}{c}}$;6.it explains how A1, A2 and the reduction process connect with the physical assumptions of V8.2, and gives the complete 3+1 dimensional generalized form. This paper clearly distinguishes: $ict$ is an imaginary-time coordinate with dimension $[L]$; $\hbar$ is an action quantum with dimension $[ML^2T^{-1}]$; the two cannot be directly equated and must be bridged through momentum. This paper emphasizes: A1 and A2 are interface axioms, not theorems of V8.3; V8.3 alone cannot derive the core formula --- the core formula is a joint result of the V8.3 algebra, the V8.4 interface, and the V8.2 physical assumptions.

Zenodo (CERN European Organization for Nuclear Research)
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