Difference between revisions of "Paper Symbols"
| Line 7: | Line 7: | ||
=== Elementary Field (EF) === | === Elementary Field (EF) === | ||
| − | <!--no latex--><span class="tex">𝓕 = F₀(A, n, χ, u; k<sub>A</sub>, k<sub>n</sub>, k<sub>χ</sub>, k<sub>u</sub>)</span><!--/no latex--><!--latex:\[\mathcal{F} = F_{0}(A,n,\chi,u;\,k_{A},k_{n},k_{\chi},k_{u})\]--> | + | <!--no latex--><span class="tex">𝓕 = F₀(A, n, χ, u; k<sub>A</sub>, k<sub>n</sub>, k<sub>χ</sub>, k<sub>u</sub>)</span><!--/no latex--><!--latex: |
| + | |||
| + | \[ | ||
| + | \mathcal{F} = F_{0}(A,n,\chi,u;\,k_{A},k_{n},k_{\chi},k_{u}) | ||
| + | \] | ||
| + | |||
| + | --> | ||
==== Structure of resistances ==== | ==== Structure of resistances ==== | ||
<!--no latex--><span class="tex"> | <!--no latex--><span class="tex"> | ||
| − | k<sub>𝓲</sub> = k<sub>𝓲</sub><sup>(intrinsic)</sup> + k<sub>𝓲</sub><sup>(dynamic)</sup>, 𝓲 ∈ {A,n,χ,u}</span><!--/no latex--><!--latex:\[k_{\mathcal{i}} = k_{\mathcal{i}}^{(\mathrm{intrinsic})} + k_{\mathcal{i}}^{(\mathrm{dynamic})}, \qquad | + | k<sub>𝓲</sub> = k<sub>𝓲</sub><sup>(intrinsic)</sup> + k<sub>𝓲</sub><sup>(dynamic)</sup>, 𝓲 ∈ {A,n,χ,u} |
| − | \mathcal{i} \in \{A,\,n,\,\chi,\,u\}\]--> | + | </span><!--/no latex--><!--latex: |
| + | |||
| + | \[ | ||
| + | k_{\mathcal{i}} = k_{\mathcal{i}}^{(\mathrm{intrinsic})} + k_{\mathcal{i}}^{(\mathrm{dynamic})}, | ||
| + | \qquad | ||
| + | \mathcal{i} \in \{A,\,n,\,\chi,\,u\} | ||
| + | \] | ||
| + | |||
| + | --> | ||
==== Resistance parameters ==== | ==== Resistance parameters ==== | ||
| − | <!--no latex--><span class="tex">k<sub>A</sub>, k<sub>n</sub> — proto‑stress resistances</span><br/><span class="tex">k<sub>χ</sub>, k<sub>u</sub> — proto‑vorticity resistances</span><!--/no latex--><!--latex:\[k_{A},\,k_{n}\ \text{— proto‑stress resistances}\] \[k_{\chi},\,k_{u}\ \text{— proto‑vorticity resistances}\]--> | + | <!--no latex--><span class="tex">k<sub>A</sub>, k<sub>n</sub> — proto‑stress resistances</span><br/> |
| + | <span class="tex">k<sub>χ</sub>, k<sub>u</sub> — proto‑vorticity resistances</span><!--/no latex--><!--latex: | ||
| + | |||
| + | \[ | ||
| + | k_{A},\,k_{n}\ \text{— proto‑stress resistances} | ||
| + | \] | ||
| + | |||
| + | |||
| + | |||
| + | |||
| + | \[ | ||
| + | k_{\chi},\,k_{u}\ \text{— proto‑vorticity resistances} | ||
| + | \] | ||
| + | |||
| + | --> | ||
==== Operational hypotheses (to be refined) ==== | ==== Operational hypotheses (to be refined) ==== | ||
| Line 23: | Line 51: | ||
<!--no latex--><span class="tex">k<sub>χ</sub> ≈ ε<sub>0</sub></span><br/> | <!--no latex--><span class="tex">k<sub>χ</sub> ≈ ε<sub>0</sub></span><br/> | ||
<span class="tex">k<sub>u</sub> ≈ μ<sub>0</sub></span><!--/no latex--><!--latex: | <span class="tex">k<sub>u</sub> ≈ μ<sub>0</sub></span><!--/no latex--><!--latex: | ||
| − | |||
\[ | \[ | ||
| Line 35: | Line 62: | ||
k_{u} \approx \mu_{0} | k_{u} \approx \mu_{0} | ||
\] | \] | ||
| − | |||
--> | --> | ||
| Line 42: | Line 68: | ||
<!--no latex--><span class="tex"> | <!--no latex--><span class="tex"> | ||
| − | EP = infinitesimal analytical sampling locus of the Field at label 𝑥</span><!--/no latex--><!--latex:\[EP = \text{infinitesimal analytical sampling locus of the Field at label }\mathit{x} | + | EP = infinitesimal analytical sampling locus of the Field at label 𝑥 |
| − | ]--> | + | </span><!--/no latex--><!--latex: |
| + | |||
| + | \[ | ||
| + | EP = \text{infinitesimal analytical sampling locus of the Field at label }\mathit{x} | ||
| + | \] | ||
| + | |||
| + | --> | ||
== Elementary Events (EE) == | == Elementary Events (EE) == | ||
| − | <!--no latex--><span class="tex">EE(𝑥,τ)= δ𝓕(𝑥,τ)</span><!--/no latex--><!--latex:\[EE( | + | <!--no latex--><span class="tex">EE(𝑥,τ)= δ𝓕(𝑥,τ)</span><!--/no latex--><!--latex: |
| + | |||
| + | \[ | ||
| + | EE(\mathit{x},\tau)=\delta\mathcal{F}(\mathit{x},\tau) | ||
| + | \] | ||
| + | |||
| + | --> | ||
=== Response of the Field === | === Response of the Field === | ||
| Line 54: | Line 92: | ||
EE(𝑥,τ)= δ𝓕(𝑥,τ) | EE(𝑥,τ)= δ𝓕(𝑥,τ) | ||
= Resp<sub>𝓕</sub>(𝑥,τ; F₀, k<sub>A</sub>, k<sub>n</sub>, k<sub>χ</sub>, k<sub>u</sub>) | = Resp<sub>𝓕</sub>(𝑥,τ; F₀, k<sub>A</sub>, k<sub>n</sub>, k<sub>χ</sub>, k<sub>u</sub>) | ||
| − | </span><!--/no latex--><!--latex:\[EE(\mathit{x},\tau)= \delta\mathcal{F}(\mathit{x},\tau)= \mathrm{Resp}_{\mathcal{F}}\!\left(\mathit{x},\tau;\,F_{0},k_{A},k_{n},k_{\chi},k_{u}\right)\]--> | + | </span><!--/no latex--><!--latex: |
| + | |||
| + | \[ | ||
| + | EE(\mathit{x},\tau)= \delta\mathcal{F}(\mathit{x},\tau)= | ||
| + | \mathrm{Resp}_{\mathcal{F}}\!\left(\mathit{x},\tau;\,F_{0},k_{A},k_{n},k_{\chi},k_{u}\right) | ||
| + | \] | ||
| + | |||
| + | --> | ||
=== Decomposition === | === Decomposition === | ||
| − | <!--no latex--><span class="tex">EE = {A, n, χ, u}</span><!--/no latex--><!--latex:\[ | + | <!--no latex--><span class="tex">EE = {A, n, χ, u}</span><!--/no latex--><!--latex: |
| − | EE = \{A,\,n,\,\chi,\,u\}\]--> | + | |
| − | + | \[ | |
| + | EE = \{A,\,n,\,\chi,\,u\} | ||
| + | \] | ||
| + | |||
| + | --> | ||
| + | |||
=== Instantaneous configuration === | === Instantaneous configuration === | ||
| − | <!--no latex--><span class="tex">σ = (A, n, χ, u)<sub>instantaneous</sub></span><!--/no latex--><!--latex:\[\sigma = (A,\,n,\,\chi,\,u)_{\mathrm{instantaneous}}\]--> | + | <!--no latex--><span class="tex">σ = (A, n, χ, u)<sub>instantaneous</sub></span><!--/no latex--><!--latex: |
| + | |||
| + | \[ | ||
| + | \sigma = (A,\,n,\,\chi,\,u)_{\mathrm{instantaneous}} | ||
| + | \] | ||
| + | |||
| + | --> | ||
=== Statistical configuration === | === Statistical configuration === | ||
| − | <!--no latex--><span class="tex">(A(τ), n(τ), χ(τ), u(τ))</span><!--/no latex--><!--latex:\[ | + | <!--no latex--><span class="tex">(A(τ), n(τ), χ(τ), u(τ))</span><!--/no latex--><!--latex: |
| − | (A(\tau),\,n(\tau),\,\chi(\tau),\,u(\tau))\]--> | + | |
| + | \[ | ||
| + | (A(\tau),\,n(\tau),\,\chi(\tau),\,u(\tau)) | ||
| + | \] | ||
| + | |||
| + | --> | ||
== Elementary Action == | == Elementary Action == | ||
| − | <!--no latex--><span class="tex">EA = (ΔP<sub>A</sub>, ΔP<sub>n</sub>, ΔP<sub>χ</sub>, ΔP<sub>u</sub>)</span><!--/no latex--><!--latex: | + | <!--no latex--><span class="tex">EA = (ΔP<sub>A</sub>, ΔP<sub>n</sub>, ΔP<sub>χ</sub>, ΔP<sub>u</sub>)</span><!--/no latex--><!--latex: |
| − | <!--no latex--><span class="tex">ΔP<sub>𝓲</sub> = τ<sub>𝓲</sub><sup>actual</sup> − τ<sub>𝓲</sub><sup>stochastically symmetric</sup></span><!--/no latex--><!--latex:\[\Delta P_{\mathcal{i}} = \tau_{\mathcal{i}}^{\mathrm{actual}} - \tau_{\mathcal{i}}^{(\mathrm{stochastically\ symmetric})}\]--> | + | \[ |
| + | EA = (\Delta P_{A},\,\Delta P_{n},\,\Delta P_{\chi},\,\Delta P_{u}) | ||
| + | \] | ||
| + | |||
| + | --> | ||
| + | |||
| + | <!--no latex--><span class="tex">ΔP<sub>𝓲</sub> = τ<sub>𝓲</sub><sup>actual</sup> − τ<sub>𝓲</sub><sup>stochastically symmetric</sup></span><!--/no latex--><!--latex: | ||
| + | |||
| + | \[ | ||
| + | \Delta P_{\mathcal{i}} = | ||
| + | \tau_{\mathcal{i}}^{\mathrm{actual}} - | ||
| + | \tau_{\mathcal{i}}^{(\mathrm{stochastically\ symmetric})} | ||
| + | \] | ||
| + | |||
| + | --> | ||
=== Modes / Components === | === Modes / Components === | ||
| − | * Perturbation: <!--no latex--><span class="tex">ΔP<sub>A</sub> = τ<sub>A</sub><sup>actual</sup> − τ<sub>A</sub><sup>stochastically symmetric</sup></span><!--/no latex--><!--latex: | + | * Perturbation: <!--no latex--><span class="tex">ΔP<sub>A</sub> = τ<sub>A</sub><sup>actual</sup> − τ<sub>A</sub><sup>stochastically symmetric</sup></span><!--/no latex--><!--latex: |
| − | + | \[ | |
| + | \Delta P_{A} = | ||
| + | \tau_{A}^{\mathrm{actual}} - | ||
| + | \tau_{A}^{(\mathrm{stochastically\ symmetric})} | ||
| + | \] | ||
| − | + | --> | |
| − | * Rotation:AxisOrientation: <!--no latex--><span class="tex">ΔP<sub>u</sub> = τ<sub>u</sub><sup>actual</sup> − τ<sub>u</sub><sup>stochastically symmetric</sup></span><!--/no latex--><!--latex:\[\Delta P_{u} = \tau_{u}^{\mathrm{actual}} - \tau_{u}^{(\mathrm{stochastically\ symmetric})}\]--> | + | * Translation: <!--no latex--><span class="tex">ΔP<sub>n</sub> = τ<sub>n</sub><sup>actual</sup> − τ<sub>n</sub><sup>stochastically symmetric</sup></span><!--/no latex--><!--latex: |
| + | |||
| + | \[ | ||
| + | \Delta P_{n} = | ||
| + | \tau_{n}^{\mathrm{actual}} - | ||
| + | \tau_{n}^{(\mathrm{stochastically\ symmetric})} | ||
| + | \] | ||
| + | |||
| + | --> | ||
| + | |||
| + | * Rotation: Chirality: <!--no latex--><span class="tex">ΔP<sub>χ</sub> = τ<sub>χ</sub><sup>actual</sup> − τ<sub>χ</sub><sup>stochastically symmetric</sup></span><!--/no latex--><!--latex: | ||
| + | |||
| + | \[ | ||
| + | \Delta P_{\chi} = | ||
| + | \tau_{\chi}^{\mathrm{actual}} - | ||
| + | \tau_{\chi}^{(\mathrm{stochastically\ symmetric})} | ||
| + | \] | ||
| + | |||
| + | --> | ||
| + | |||
| + | * Rotation: AxisOrientation: <!--no latex--><span class="tex">ΔP<sub>u</sub> = τ<sub>u</sub><sup>actual</sup> − τ<sub>u</sub><sup>stochastically symmetric</sup></span><!--/no latex--><!--latex: | ||
| + | |||
| + | \[ | ||
| + | \Delta P_{u} = | ||
| + | \tau_{u}^{\mathrm{actual}} - | ||
| + | \tau_{u}^{(\mathrm{stochastically\ symmetric})} | ||
| + | \] | ||
| + | |||
| + | --> | ||
== Information / Energy (Potentials Φ) == | == Information / Energy (Potentials Φ) == | ||
| Line 90: | Line 197: | ||
=== Equilibrium conditions === | === Equilibrium conditions === | ||
| − | <!--no latex--><span class="tex">𝓒<sub>M</sub> = 𝓒(ΔP<sub>A</sub>) = 0</span><!--/no latex--><!--latex: | + | <!--no latex--><span class="tex">𝓒<sub>M</sub> = 𝓒(ΔP<sub>A</sub>) = 0</span><!--/no latex--><!--latex: |
| − | + | \[ | |
| + | \mathcal{C}_{M} = \mathcal{C}(\Delta P_{A}) = 0 | ||
| + | \] | ||
| − | + | --> | |
| − | <!--no latex--><span class="tex">𝓒<sub>B</sub> = 𝓒(ΔP<sub>u</sub>) = 0</span><!--/no latex--><!--latex:\[\mathcal{C}_{B} = \mathcal{C}(\Delta P_{u}) = 0\]--> | + | <!--no latex--><span class="tex">𝓒<sub>A</sub> = 𝓒(ΔP<sub>n</sub>) = 0</span><!--/no latex--><!--latex: |
| + | |||
| + | \[ | ||
| + | \mathcal{C}_{A} = \mathcal{C}(\Delta P_{n}) = 0 | ||
| + | \] | ||
| + | |||
| + | --> | ||
| + | |||
| + | <!--no latex--><span class="tex">𝓒<sub>E</sub> = 𝓒(ΔP<sub>χ</sub>) = 0</span><!--/no latex--><!--latex: | ||
| + | |||
| + | \[ | ||
| + | \mathcal{C}_{E} = \mathcal{C}(\Delta P_{\chi}) = 0 | ||
| + | \] | ||
| + | |||
| + | --> | ||
| + | |||
| + | <!--no latex--><span class="tex">𝓒<sub>B</sub> = 𝓒(ΔP<sub>u</sub>) = 0</span><!--/no latex--><!--latex: | ||
| + | |||
| + | \[ | ||
| + | \mathcal{C}_{B} = \mathcal{C}(\Delta P_{u}) = 0 | ||
| + | \] | ||
| + | |||
| + | --> | ||
=== Diffuse solutions === | === Diffuse solutions === | ||
| − | <!--no latex--><span class="tex">Φ<sub>M</sub>, Φ<sub>A</sub>, Φ<sub>E</sub>, Φ<sub>B</sub> ∈ Sol(𝓒)</span><!--/no latex--><!--latex:\[\Phi_{M},\;\Phi_{A},\;\Phi_{E},\;\Phi_{B} \in \mathrm{Sol}(\mathcal{C})\]--> | + | <!--no latex--><span class="tex">Φ<sub>M</sub>, Φ<sub>A</sub>, Φ<sub>E</sub>, Φ<sub>B</sub> ∈ Sol(𝓒)</span><!--/no latex--><!--latex: |
| + | |||
| + | \[ | ||
| + | \Phi_{M},\;\Phi_{A},\;\Phi_{E},\;\Phi_{B} \in \mathrm{Sol}(\mathcal{C}) | ||
| + | \] | ||
| + | |||
| + | --> | ||
== InfoStructures == | == InfoStructures == | ||
| Line 108: | Line 245: | ||
==== Derived from ==== | ==== Derived from ==== | ||
| − | <!--no latex--><span class="tex">ΔP<sub>n</sub>, ΔP<sub>u</sub></span><!--/no latex--><!--latex:\[\Delta P_{n},\;\Delta P_{u}\]--> | + | <!--no latex--><span class="tex">ΔP<sub>n</sub>, ΔP<sub>u</sub></span><!--/no latex--><!--latex: |
| + | |||
| + | \[ | ||
| + | \Delta P_{n},\;\Delta P_{u} | ||
| + | \] | ||
| + | |||
| + | --> | ||
==== Equilibrium condition ==== | ==== Equilibrium condition ==== | ||
| − | <!--no latex--><span class="tex">𝓒<sub>wave</sub>(ΔP<sub>n</sub>, ΔP<sub>u</sub>) = 0</span><!--/no latex--><!--latex:\[\mathcal{C}_{wave}(\Delta P_{n},\,\Delta P_{u}) = 0\]--> | + | <!--no latex--><span class="tex">𝓒<sub>wave</sub>(ΔP<sub>n</sub>, ΔP<sub>u</sub>) = 0</span><!--/no latex--><!--latex: |
| + | |||
| + | \[ | ||
| + | \mathcal{C}_{wave}(\Delta P_{n},\,\Delta P_{u}) = 0 | ||
| + | \] | ||
| + | |||
| + | --> | ||
==== Proto‑wave equation ==== | ==== Proto‑wave equation ==== | ||
| − | <!--no latex--><span class="tex">𝓦[Ψ; ΔP<sub>n</sub>, ΔP<sub>u</sub>] = 0</span><!--/no latex--><!--latex:\[\mathcal{W}[\Psi;\,\Delta P_{n},\,\Delta P_{u}] = 0\]--> | + | <!--no latex--><span class="tex">𝓦[Ψ; ΔP<sub>n</sub>, ΔP<sub>u</sub>] = 0</span><!--/no latex--><!--latex: |
| + | |||
| + | \[ | ||
| + | \mathcal{W}[\Psi;\,\Delta P_{n},\,\Delta P_{u}] = 0 | ||
| + | \] | ||
| + | |||
| + | --> | ||
=== Vortex‑like InfoStructures === | === Vortex‑like InfoStructures === | ||
| Line 122: | Line 277: | ||
==== Derived from ==== | ==== Derived from ==== | ||
| − | <!--no latex--><span class="tex">ΔP<sub>A</sub>, ΔP<sub>n</sub>, ΔP<sub>χ</sub>, ΔP<sub>u</sub></span><!--/no latex--><!--latex:\[\Delta P_{A},\;\Delta P_{n},\;\Delta P_{\chi},\;\Delta P_{u}\]--> | + | <!--no latex--><span class="tex">ΔP<sub>A</sub>, ΔP<sub>n</sub>, ΔP<sub>χ</sub>, ΔP<sub>u</sub></span><!--/no latex--><!--latex: |
| + | |||
| + | \[ | ||
| + | \Delta P_{A},\;\Delta P_{n},\;\Delta P_{\chi},\;\Delta P_{u} | ||
| + | \] | ||
| + | |||
| + | --> | ||
==== Equilibrium condition ==== | ==== Equilibrium condition ==== | ||
| − | <!--no latex--><span class="tex">𝓒<sub>vortex</sub>(ΔP<sub>A</sub>, ΔP<sub>n</sub>, ΔP<sub>χ</sub>, ΔP<sub>u</sub>) = 0</span><!--/no latex--><!--latex:\[\mathcal{C}_{vortex}(\Delta P_{A},\,\Delta P_{n},\,\Delta P_{\chi},\,\Delta P_{u}) = 0\]--> | + | <!--no latex--><span class="tex">𝓒<sub>vortex</sub>(ΔP<sub>A</sub>, ΔP<sub>n</sub>, ΔP<sub>χ</sub>, ΔP<sub>u</sub>) = 0</span><!--/no latex--><!--latex: |
| + | |||
| + | \[ | ||
| + | \mathcal{C}_{vortex}(\Delta P_{A},\,\Delta P_{n},\,\Delta P_{\chi},\,\Delta P_{u}) = 0 | ||
| + | \] | ||
| + | |||
| + | --> | ||
==== Proto‑vortex equation ==== | ==== Proto‑vortex equation ==== | ||
| − | <!--no latex--><span class="tex">𝓥[Ξ; ΔP<sub>A</sub>, ΔP<sub>n</sub>, ΔP<sub>χ</sub>, ΔP<sub>u</sub>] = 0</span><!--/no latex--><!--latex:\[\mathcal{V}[\Xi;\,\Delta P_{A},\,\Delta P_{n},\,\Delta P_{\chi},\,\Delta P_{u}] = 0\]--> | + | <!--no latex--><span class="tex">𝓥[Ξ; ΔP<sub>A</sub>, ΔP<sub>n</sub>, ΔP<sub>χ</sub>, ΔP<sub>u</sub>] = 0</span><!--/no latex--><!--latex: |
| + | |||
| + | \[ | ||
| + | \mathcal{V}[\Xi;\,\Delta P_{A},\,\Delta P_{n},\,\Delta P_{\chi},\,\Delta P_{u}] = 0 | ||
| + | \] | ||
| + | |||
| + | --> | ||
=== Interaction‑mediating InfoStructures === | === Interaction‑mediating InfoStructures === | ||
| Line 136: | Line 309: | ||
==== Relational equilibrium ==== | ==== Relational equilibrium ==== | ||
| − | <!--no latex--><span class="tex">𝓒<sub>int</sub>(Ψ, Ξ) = 0</span><!--/no latex--><!--latex:\[ | + | <!--no latex--><span class="tex">𝓒<sub>int</sub>(Ψ, Ξ) = 0</span><!--/no latex--><!--latex: |
| − | \mathcal{C}_{int}(\Psi,\,\Xi) = 0\]--> | + | |
| + | \[ | ||
| + | \mathcal{C}_{int}(\Psi,\,\Xi) = 0 | ||
| + | \] | ||
| + | |||
| + | --> | ||
==== Proto‑interaction equation ==== | ==== Proto‑interaction equation ==== | ||
| − | <!--no latex--><span class="tex">𝓘[Ψ, Ξ] = 0</span><!--/no latex--><!--latex:\[\mathcal{I}[\Psi,\,\Xi] = 0\]--> | + | <!--no latex--><span class="tex">𝓘[Ψ, Ξ] = 0</span><!--/no latex--><!--latex: |
| + | |||
| + | \[ | ||
| + | \mathcal{I}[\Psi,\,\Xi] = 0 | ||
| + | \] | ||
| + | |||
| + | --> | ||
== Geometric Projection == | == Geometric Projection == | ||
| Line 148: | Line 332: | ||
<!--no latex--><span class="tex">Proj(𝓦) → QM wave equations</span><!--/no latex--><!--latex: | <!--no latex--><span class="tex">Proj(𝓦) → QM wave equations</span><!--/no latex--><!--latex: | ||
| − | \[\mathrm{Proj}(\mathcal{W}) | + | |
| + | \[ | ||
| + | \mathrm{Proj}(\mathcal{W}) \longrightarrow \text{QM wave equations} | ||
| + | \] | ||
| + | |||
| + | --> | ||
=== Vortex projection === | === Vortex projection === | ||
| − | <!--no latex--><span class="tex">Proj(𝓥) → spinor equations</span><!--/no latex--><!--latex:\[ | + | <!--no latex--><span class="tex">Proj(𝓥) → spinor equations</span><!--/no latex--><!--latex: |
| − | \mathrm{Proj}(\mathcal{V}) | + | |
| + | \[ | ||
| + | \mathrm{Proj}(\mathcal{V}) \longrightarrow \text{spinor equations} | ||
| + | \] | ||
| + | |||
| + | --> | ||
=== Interaction projection === | === Interaction projection === | ||
| − | <!--no latex--><span class="tex">Proj(𝓘) → gauge equations</span><!--/no latex--><!--latex:\[ | + | <!--no latex--><span class="tex">Proj(𝓘) → gauge equations</span><!--/no latex--><!--latex: |
| − | \mathrm{Proj}(\mathcal{I}) | + | |
| + | \[ | ||
| + | \mathrm{Proj}(\mathcal{I}) \longrightarrow \text{gauge equations} | ||
| + | \] | ||
| + | |||
| + | --> | ||
== Forms == | == Forms == | ||
No new fundamental equation | No new fundamental equation | ||
Revision as of 21:16, 13 July 2026
Contents
The Primary Source of Reality
Fundamental Force (FF)
F₀
Elementary Field (EF)
𝓕 = F₀(A, n, χ, u; kA, kn, kχ, ku)
Structure of resistances
k𝓲 = k𝓲(intrinsic) + k𝓲(dynamic), 𝓲 ∈ {A,n,χ,u}
Resistance parameters
kA, kn — proto‑stress resistances
kχ, ku — proto‑vorticity resistances
Operational hypotheses (to be refined)
kχ ≈ ε0
ku ≈ μ0
Event-point(s) (EP)
EP = infinitesimal analytical sampling locus of the Field at label 𝑥
Elementary Events (EE)
EE(𝑥,τ)= δ𝓕(𝑥,τ)
Response of the Field
EE(𝑥,τ)= δ𝓕(𝑥,τ) = Resp𝓕(𝑥,τ; F₀, kA, kn, kχ, ku)
Decomposition
EE = {A, n, χ, u}
Instantaneous configuration
σ = (A, n, χ, u)instantaneous
Statistical configuration
(A(τ), n(τ), χ(τ), u(τ))
Elementary Action
EA = (ΔPA, ΔPn, ΔPχ, ΔPu)
ΔP𝓲 = τ𝓲actual − τ𝓲stochastically symmetric
Modes / Components
- Perturbation: ΔPA = τAactual − τAstochastically symmetric
- Translation: ΔPn = τnactual − τnstochastically symmetric
- Rotation: Chirality: ΔPχ = τχactual − τχstochastically symmetric
- Rotation: AxisOrientation: ΔPu = τuactual − τustochastically symmetric
Information / Energy (Potentials Φ)
Equilibrium conditions
𝓒M = 𝓒(ΔPA) = 0
𝓒A = 𝓒(ΔPn) = 0
𝓒E = 𝓒(ΔPχ) = 0
𝓒B = 𝓒(ΔPu) = 0
Diffuse solutions
ΦM, ΦA, ΦE, ΦB ∈ Sol(𝓒)
InfoStructures
Wave‑like InfoStructures
Derived from
ΔPn, ΔPu
Equilibrium condition
𝓒wave(ΔPn, ΔPu) = 0
Proto‑wave equation
𝓦[Ψ; ΔPn, ΔPu] = 0
Vortex‑like InfoStructures
Derived from
ΔPA, ΔPn, ΔPχ, ΔPu
Equilibrium condition
𝓒vortex(ΔPA, ΔPn, ΔPχ, ΔPu) = 0
Proto‑vortex equation
𝓥[Ξ; ΔPA, ΔPn, ΔPχ, ΔPu] = 0
Interaction‑mediating InfoStructures
Relational equilibrium
𝓒int(Ψ, Ξ) = 0
Proto‑interaction equation
𝓘[Ψ, Ξ] = 0
Geometric Projection
Wave projection
Proj(𝓦) → QM wave equations
Vortex projection
Proj(𝓥) → spinor equations
Interaction projection
Proj(𝓘) → gauge equations
Forms
No new fundamental equation