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| Line 159: |
Line 159: |
| | <!--no latex--><span class="tex">Proj(π) β gauge equations</span><!--/no latex--><!--latex:\[ | | <!--no latex--><span class="tex">Proj(π) β gauge equations</span><!--/no latex--><!--latex:\[ |
| | \mathrm{Proj}(\mathcal{I}) \;\longrightarrow\; \text{gauge equations}\]--> | | \mathrm{Proj}(\mathcal{I}) \;\longrightarrow\; \text{gauge equations}\]--> |
| β |
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| β | == Forms ==
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| β |
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| β | No new fundamental equation
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| β |
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| β |
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| β | == The Primary Source of Reality ==
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| β |
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| β | === Fundamental Force (FF) ===
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| β |
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| β | <!--no latex-->Fβ<!--/no latex--><!--latex:$F_0$-->
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| β |
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| β | === Elementary Field (EF) ===
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| β |
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| β | <!--no latex--><span class="tex">π = Fβ(A, n, Ο, u; k<sub>A</sub>, k<sub>n</sub>, k<sub>Ο</sub>, k<sub>u</sub>)</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \mathcal{F} = F_{0}(A,n,\chi,u;\,k_{A},k_{n},k_{\chi},k_{u})
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| β | \]
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| β |
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| β | -->
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| β |
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| β | ==== Structure of resistances ====
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| β |
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| β | <!--no latex--><span class="tex">
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| β | k<sub>π²</sub> = k<sub>π²</sub><sup>(intrinsic)</sup> + k<sub>π²</sub><sup>(dynamic)</sup>, π² β {A,n,Ο,u}
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| β | </span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | k_{\mathcal{i}} = k_{\mathcal{i}}^{(\mathrm{intrinsic})} + k_{\mathcal{i}}^{(\mathrm{dynamic})},
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| β | \qquad
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| β | \mathcal{i} \in \{A,\,n,\,\chi,\,u\}
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| β | \]
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| β |
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| β | -->
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| β |
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| β | ==== Resistance parameters ====
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| β |
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| β | <!--no latex--><span class="tex">k<sub>A</sub>, k<sub>n</sub> β protoβstress resistances</span><br/>
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| β | <span class="tex">k<sub>Ο</sub>, k<sub>u</sub> β protoβvorticity resistances</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | k_{A},\,k_{n}\ \text{β protoβstress resistances}
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| β | \]
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| β |
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| β |
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| β |
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| β |
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| β | \[
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| β | k_{\chi},\,k_{u}\ \text{β protoβvorticity resistances}
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| β | \]
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| β |
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| β | -->
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| β |
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| β | ==== Operational hypotheses (to be refined) ====
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| β |
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| β | <!--no latex--><span class="tex">k<sub>Ο</sub> β Ξ΅<sub>0</sub></span><br/>
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| β | <span class="tex">k<sub>u</sub> β ΞΌ<sub>0</sub></span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | k_{\chi} \approx \varepsilon_{0}
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| β | \]
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| β |
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| β |
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| β |
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| β |
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| β | \[
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| β | k_{u} \approx \mu_{0}
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| β | \]
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| β |
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| β | -->
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| β |
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| β | == Event-point(s) (EP) ==
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| β |
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| β | <!--no latex--><span class="tex">
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| β | EP = infinitesimal analytical sampling locus of the Field at label π₯
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| β | </span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | EP = \text{infinitesimal analytical sampling locus of the Field at label }\mathit{x}
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| β | \]
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| β |
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| β | -->
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| β |
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| β | == Elementary Events (EE) ==
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| β |
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| β | <!--no latex--><span class="tex">EE(π₯,Ο)= Ξ΄π(π₯,Ο)</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | EE(\mathit{x},\tau)=\delta\mathcal{F}(\mathit{x},\tau)
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| β | \]
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| β |
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| β | -->
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| β |
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| β | === Response of the Field ===
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| β |
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| β | <!--no latex--><span class="tex">
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| β | EE(π₯,Ο)= Ξ΄π(π₯,Ο)
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| β | = Resp<sub>π</sub>(π₯,Ο; Fβ, k<sub>A</sub>, k<sub>n</sub>, k<sub>Ο</sub>, k<sub>u</sub>)
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| β | </span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | EE(\mathit{x},\tau)= \delta\mathcal{F}(\mathit{x},\tau)=
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| β | \mathrm{Resp}_{\mathcal{F}}\!\left(\mathit{x},\tau;\,F_{0},k_{A},k_{n},k_{\chi},k_{u}\right)
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| β | \]
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| β |
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| β | -->
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| β |
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| β | === Decomposition ===
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| β |
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| β | <!--no latex--><span class="tex">EE = {A, n, Ο, u}</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | EE = \{A,\,n,\,\chi,\,u\}
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| β | \]
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| β |
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| β | -->
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| β |
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| β | === Instantaneous configuration ===
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| β |
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| β | <!--no latex--><span class="tex">Ο = (A, n, Ο, u)<sub>instantaneous</sub></span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \sigma = (A,\,n,\,\chi,\,u)_{\mathrm{instantaneous}}
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| β | \]
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| β |
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| β | -->
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| β |
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| β | === Statistical configuration ===
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| β |
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| β | <!--no latex--><span class="tex">(A(Ο), n(Ο), Ο(Ο), u(Ο))</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | (A(\tau),\,n(\tau),\,\chi(\tau),\,u(\tau))
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| β | \]
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| β |
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| β | -->
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| β |
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| β | == Elementary Action ==
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| β |
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| β | <!--no latex--><span class="tex">EA = (ΞP<sub>A</sub>, ΞP<sub>n</sub>, ΞP<sub>Ο</sub>, ΞP<sub>u</sub>)</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | EA = (\Delta P_{A},\,\Delta P_{n},\,\Delta P_{\chi},\,\Delta P_{u})
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| β | \]
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| β |
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| β | -->
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| β |
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| β | <!--no latex--><span class="tex">ΞP<sub>π²</sub> = Ο<sub>π²</sub><sup>actual</sup> β Ο<sub>π²</sub><sup>stochastically symmetric</sup></span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \Delta P_{\mathcal{i}} =
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| β | \tau_{\mathcal{i}}^{\mathrm{actual}} -
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| β | \tau_{\mathcal{i}}^{(\mathrm{stochastically\ symmetric})}
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| β | \]
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| β |
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| β | -->
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| β |
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| β | === Modes / Components ===
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| β |
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| β | * Perturbation: <!--no latex--><span class="tex">ΞP<sub>A</sub> = Ο<sub>A</sub><sup>actual</sup> β Ο<sub>A</sub><sup>stochastically symmetric</sup></span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \Delta P_{A} =
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| β | \tau_{A}^{\mathrm{actual}} -
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| β | \tau_{A}^{(\mathrm{stochastically\ symmetric})}
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| β | \]
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| β |
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| β | -->
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| β |
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| β | * Translation: <!--no latex--><span class="tex">ΞP<sub>n</sub> = Ο<sub>n</sub><sup>actual</sup> β Ο<sub>n</sub><sup>stochastically symmetric</sup></span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \Delta P_{n} =
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| β | \tau_{n}^{\mathrm{actual}} -
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| β | \tau_{n}^{(\mathrm{stochastically\ symmetric})}
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| β | \]
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| β |
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| β | -->
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| β |
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| β | * Rotation: Chirality: <!--no latex--><span class="tex">ΞP<sub>Ο</sub> = Ο<sub>Ο</sub><sup>actual</sup> β Ο<sub>Ο</sub><sup>stochastically symmetric</sup></span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \Delta P_{\chi} =
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| β | \tau_{\chi}^{\mathrm{actual}} -
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| β | \tau_{\chi}^{(\mathrm{stochastically\ symmetric})}
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| β | \]
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| β |
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| β | -->
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| β |
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| β | * Rotation: AxisOrientation: <!--no latex--><span class="tex">ΞP<sub>u</sub> = Ο<sub>u</sub><sup>actual</sup> β Ο<sub>u</sub><sup>stochastically symmetric</sup></span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \Delta P_{u} =
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| β | \tau_{u}^{\mathrm{actual}} -
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| β | \tau_{u}^{(\mathrm{stochastically\ symmetric})}
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| β | \]
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| β |
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| β | -->
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| β |
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| β | == Information / Energy (Potentials Ξ¦) ==
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| β |
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| β | === Equilibrium conditions ===
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| β |
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| β | <!--no latex--><span class="tex">π<sub>M</sub> = π(ΞP<sub>A</sub>) = 0</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \mathcal{C}_{M} = \mathcal{C}(\Delta P_{A}) = 0
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| β | \]
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| β |
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| β | -->
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| β |
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| β | <!--no latex--><span class="tex">π<sub>A</sub> = π(ΞP<sub>n</sub>) = 0</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \mathcal{C}_{A} = \mathcal{C}(\Delta P_{n}) = 0
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| β | \]
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| β |
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| β | -->
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| β |
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| β | <!--no latex--><span class="tex">π<sub>E</sub> = π(ΞP<sub>Ο</sub>) = 0</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \mathcal{C}_{E} = \mathcal{C}(\Delta P_{\chi}) = 0
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| β | \]
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| β |
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| β | -->
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| β |
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| β | <!--no latex--><span class="tex">π<sub>B</sub> = π(ΞP<sub>u</sub>) = 0</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \mathcal{C}_{B} = \mathcal{C}(\Delta P_{u}) = 0
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| β | \]
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| β |
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| β | -->
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| β |
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| β | === Diffuse solutions ===
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| β |
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| β | <!--no latex--><span class="tex">Ξ¦<sub>M</sub>, Ξ¦<sub>A</sub>, Ξ¦<sub>E</sub>, Ξ¦<sub>B</sub> β Sol(π)</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \Phi_{M},\;\Phi_{A},\;\Phi_{E},\;\Phi_{B} \in \mathrm{Sol}(\mathcal{C})
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| β | \]
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| β |
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| β | -->
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| β |
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| β | == InfoStructures ==
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| β |
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| β | === Waveβlike InfoStructures ===
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| β |
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| β | ==== Derived from ====
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| β |
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| β | <!--no latex--><span class="tex">ΞP<sub>n</sub>, ΞP<sub>u</sub></span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \Delta P_{n},\;\Delta P_{u}
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| β | \]
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| β |
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| β | -->
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| β |
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| β | ==== Equilibrium condition ====
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| β |
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| β | <!--no latex--><span class="tex">π<sub>wave</sub>(ΞP<sub>n</sub>, ΞP<sub>u</sub>) = 0</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \mathcal{C}_{wave}(\Delta P_{n},\,\Delta P_{u}) = 0
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| β | \]
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| β |
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| β | -->
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| β |
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| β | ==== Protoβwave equation ====
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| β |
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| β | <!--no latex--><span class="tex">π¦[Ξ¨; ΞP<sub>n</sub>, ΞP<sub>u</sub>] = 0</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \mathcal{W}[\Psi;\,\Delta P_{n},\,\Delta P_{u}] = 0
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| β | \]
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| β |
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| β | -->
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| β |
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| β | === Vortexβlike InfoStructures ===
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| β |
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| β | ==== Derived from ====
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| β |
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| β | <!--no latex--><span class="tex">ΞP<sub>A</sub>, ΞP<sub>n</sub>, ΞP<sub>Ο</sub>, ΞP<sub>u</sub></span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \Delta P_{A},\;\Delta P_{n},\;\Delta P_{\chi},\;\Delta P_{u}
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| β | \]
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| β |
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| β | -->
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| β |
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| β | ==== Equilibrium condition ====
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| β |
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| β | <!--no latex--><span class="tex">π<sub>vortex</sub>(ΞP<sub>A</sub>, ΞP<sub>n</sub>, ΞP<sub>Ο</sub>, ΞP<sub>u</sub>) = 0</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \mathcal{C}_{vortex}(\Delta P_{A},\,\Delta P_{n},\,\Delta P_{\chi},\,\Delta P_{u}) = 0
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| β | \]
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| β |
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| β | -->
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| β |
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| β | ==== Protoβvortex equation ====
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| β |
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| β | <!--no latex--><span class="tex">π₯[Ξ; ΞP<sub>A</sub>, ΞP<sub>n</sub>, ΞP<sub>Ο</sub>, ΞP<sub>u</sub>] = 0</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \mathcal{V}[\Xi;\,\Delta P_{A},\,\Delta P_{n},\,\Delta P_{\chi},\,\Delta P_{u}] = 0
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| β | \]
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| β |
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| β | -->
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| β |
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| β | === Interactionβmediating InfoStructures ===
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| β |
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| β | ==== Relational equilibrium ====
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| β |
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| β | <!--no latex--><span class="tex">π<sub>int</sub>(Ξ¨, Ξ) = 0</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \mathcal{C}_{int}(\Psi,\,\Xi) = 0
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| β | \]
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| β |
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| β | -->
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| β |
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| β | ==== Protoβinteraction equation ====
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| β |
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| β | <!--no latex--><span class="tex">π[Ξ¨, Ξ] = 0</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \mathcal{I}[\Psi,\,\Xi] = 0
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| β | \]
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| β |
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| β | -->
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| β |
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| β | == Geometric Projection ==
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| β |
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| β | === Wave projection ===
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| β |
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| β | <!--no latex--><span class="tex">Proj(π¦) β QM wave equations</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \mathrm{Proj}(\mathcal{W}) \longrightarrow \text{QM wave equations}
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| β | \]
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| β |
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| β | -->
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| β |
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| β | === Vortex projection ===
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| β |
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| β | <!--no latex--><span class="tex">Proj(π₯) β spinor equations</span><!--/no latex--><!--latex:
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| β |
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| β | \[
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| β | \mathrm{Proj}(\mathcal{V}) \longrightarrow \text{spinor equations}
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| β | \]
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| β |
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| β | -->
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| β |
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| β | === Interaction projection ===
| |
| β |
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| β | <!--no latex--><span class="tex">Proj(π) β gauge equations</span><!--/no latex--><!--latex:
| |
| β |
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| β | \[
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| β | \mathrm{Proj}(\mathcal{I}) \longrightarrow \text{gauge equations}
| |
| β | \]
| |
| β |
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| β | -->
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| | | | |
| | == Forms == | | == Forms == |
| | | | |
| | No new fundamental equation | | No new fundamental equation |
