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Basic Axioms (v0.4)

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Version 0.3 is quite similar to version 0.2. As a result, version 0.3 will be approved. - The universe we live in consists of a network of Qaethers, which are discrete spherical minimum spatial units at the Planck length scale. The diameter of a Qaether is equal to the Planck length \(\ell_p\). - Qaethers combine based on the most stable structure, the FCC (face-centered cubic) lattice structure, with the bonding directions discretized into the 12 nearest directions of the FCC lattice. - Qaethers possess only specific values for intrinsic spin: $$S \in \{+1, 0, -1\}$$ However, spins of +1 and -1 indicate that they have the same spin but opposite spin directions. Specifically, for a spin axis, 1 is defined as clockwise and -1 as counterclockwise. - When a Qaether attempts to bond with another Qaether, the bonding stability condition satisfies the criterion for minimal spin coupling energy, which is established when $$S_i = -S_j$$; only in this case, $$\delta^{\text{spin}}_{ij} = 1$$. - ...
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Why Version 0.2 Should Be Changed

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The Qaether model presented in version 0.2 was a promising first step towards constructing a discrete spacetime at the Planck scale. However, upon deeper analysis, several fundamental issues and logical leaps have been identified. I propose the following modifications to advance this model into version 0.4, outlining the core alterations I'd like to implement. 1. Coupling Mechanism: From Arbitrary Rules to Fundamental Principles *   Issue: The assumption that "coupling is only allowed at specific angles (θ∈{π/3, π/2, ...}) due to the rotational symmetry of the FCC lattice and the discrete changes in spin" is excessively arbitrary (ad-hoc). There lacks a fundamental physical principle explaining why it must be those angles. This reduces the phenomenon to a rule created after the fact. Additionally, the introduction of the "coupling tensor" complicates the model unnecessarily. *   Modification: I will discard the complex rules and tensors, opting instead to introd...
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Limitations of Existing Models (Fixed Spin Axis)

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The assumption that bonding occurs in a direction perpendicular to an axis with spin symmetry has the following issues. Limitations of Existing Models (Fixed Spin Axis) Problems:     *   Since the spin axis is fixed, bonding occurs only in specific directions of the FCC lattice.      *   This leads to anisotropy in space, resulting in a violation of Lorentz symmetry in the continuous limit (\(l_p \rightarrow 0\)).          Example:       If the spin axis is fixed along the z-axis, only the four neighbors in the xy-plane participate in bonding, while bonding in the z-axis direction is neglected. This issue persists even when scaled up and leads to violations of Lorentz symmetry.  Therefore, it seems necessary to modify the assumption to allow spins to rotate only in bondable directions. In such cases, Lorentz symmetry is certainly satisfied in the continuous limit.
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Physical phenomenon according to structure change by spin

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Depending on the actual rotation, the FCC structure can sometimes represent the HCP structure in certain regions and exhibit self-assembly patterns. For the case of Spin 0, it seems that the vibrational transmission due to collisions between Qaether should be calculated as its fundamental dynamics. 1. FCC ↔ HCP Structure Transformation and Self-Assembly      (1) Geometric Relationship of FCC and HCP      - FCC (Face-Centered Cubic): ABCABC stacking, 12 nearest directions, Oh symmetry.      - HCP (Hexagonal Close-Packed): ABABAB stacking, 12 directions (6+6), D6h symmetry.      - Transformation Mechanism:        - Stacking Fault: A change in spin state in part of the Qaether bonding network can induce sliding, leading to an FCC → HCP transition.        - Rotation-Induced Deformation: Changes in the bonding angle (θ) about a particular axis (e.g., [001] in FCC ...
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Basic Axioms (v0.2)

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The universe we live in is composed of a coupling network of Qaethers , which are discrete, sphere‑shaped minimal spatial units at the Planck length scale. The diameter of a Qaether equals the Planck length ℓ p \ell_p . Each Qaether intrinsically possesses a coupling tensor and a spin tensor, and since these tensors operate at the Planck scale, all of them are computable within discrete mathematics. The coupling tensor is the discrete mean coupling–direction tensor of the direction vectors d ⃗ i j \vec{d}_{ij} in which the Qaether actually couples and the corresponding spin states S j S_j . It is obtained via a discrete coupling function between direction and spin, yielding a direction function based on the coupled spin states for each pair. On an FCC lattice, coupling is allowed only along directions where the spin phase of each Qaether has discrete rotational angle differences: θ ∈ { π 3 , π 2 , 2 π 3 , π , 4 π 3 } \theta \in \left\{\tfrac{\pi}{3}, \tfrac{\pi}{2}, \tfrac{2...
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Key Enhancements in Version 0.2

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   After drafting version 0.1, it became clear that—while the conceptual foundations were in place—the specification lacked the precision and operational clarity needed to carry the theory forward into rigorous derivations or computer‑aided simulations. We realized that leaving key ingredients at an informal or purely descriptive level would create bottlenecks when attempting to: Formulate a well‑posed Lagrangian and Hamiltonian , because the “bonding interaction” was described conceptually but never pinned down as a concrete function of identifiable variables. Perform lattice path integrals , since there was no explicit finite set of allowed spin‑related rotation angles to sum over, forcing any numerical algorithm to guess or approximate a continuous range. Classify vibrational excitations consistently , as the criteria for which spin or phase differences gave rise to distinct energy modes were only hinted at, rather than spelled out. Embed curvature (deficit) effects into an...
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Why FCC structure?

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