Difference between revisions of "Paper Symbols"
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=== 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</span><!--/no latex--><!--latex:\[\mathcal{F}\]--> |
| + | |||
| + | <!--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:\[k_{A},\,k_{n}\ \text{— proto‑stress resistances} | ||
| + | \] \[k_{\chi},\,k_{u}\ \text{— proto‑vorticity resistances}\]--> | ||
| − | = | + | <!--no latex--><span class="tex">k<sub>A</sub> — intensity resistance</span><!--/no latex--><!--latex:\[k_{A}\ \text{— intensity resistance}\]--> |
| − | <!--no latex--><span class="tex">k<sub> | + | <!--no latex--><span class="tex">k<sub>n</sub> — linear‑orientation resistance</span><!--/no latex--><!--latex:\[k_{n}\ \text{— linear‑orientation resistance}\] |
| − | + | --> | |
| + | <!--no latex--><span class="tex">k<sub>χ</sub> — chirality‑reorientation resistance</span><!--/no latex--><!--latex:\[k_{\chi}\ \text{— chirality‑reorientation resistance}\]--> | ||
| − | \[ | + | <!--no latex--><span class="tex">k<sub>u</sub> — intrinsic‑angular‑orientation resistance</span><!--/no latex--><!--latex:\[k_{u}\ \text{— intrinsic‑angular‑orientation resistance}\]--> |
| − | k_{ | + | |
| − | \] | + | |
| + | {<!--no latex--><span class="tex">k<sub>A</sub> — intensity‑resistance</span><!--/no latex--><!--latex:\[k_{A}\ \text{— intensity‑resistance}\]-->, <!--no latex--><span class="tex">k<sub>n</sub> — linear‑orientation resistance</span><!--/no latex--><!--latex:\[k_{n}\ \text{— linear‑orientation resistance}\]-->, <!--no latex--><span class="tex">k<sub>χ</sub> — chirality‑reorientation resistance</span><!--/no latex--><!--latex:\[k_{\chi}\ \text{— chirality‑reorientation resistance}\]-->, <!--no latex--><span class="tex">k<sub>u</sub> — intrinsic‑angular‑orientation resistance</span><!--/no latex--><!--latex:\[k_{u}\ \text{— intrinsic‑angular‑orientation resistance}\]-->} | ||
| + | {<!--no latex--><span class="tex">k<sub>A</sub></span><!--/no latex--><!--latex:\[k_{A}\]-->, <!--no latex--><span class="tex">k<sub>n</sub></span><!--/no latex--><!--latex:\[k_{n}\]-->, <!--no latex--><span class="tex">k<sub>χ</sub></span><!--/no latex--><!--latex:\[k_{\chi}\]-->, <!--no latex--><span class="tex">k<sub>u</sub></span><!--/no latex--><!--latex:\[k_{u}\]-->} | ||
| + | ==== Operational hypotheses (to be refined) ==== | ||
| + | |||
| + | <!--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:\[ | ||
| + | k_{\chi} \approx \varepsilon_{0}\] \[k_{u} \approx \mu_{0}\]--> | ||
| + | |||
| + | == Event-point(s) (EP) == | ||
| + | |||
| + | <!--no latex--><span class="tex"> | ||
| + | EP = infinitesimal analytical sampling locus of the Field at label 𝑥 | ||
| + | </span><!--/no latex--><!--latex: | ||
\[ | \[ | ||
| − | + | EP = \text{infinitesimal analytical sampling locus of the Field at label }\mathit{x} | |
\] | \] | ||
| − | |||
--> | --> | ||
| − | |||
| − | |||
| − | |||
| − | |||
== 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 === | ||
| − | <!--no latex--><span class="tex"> | + | <!--no latex--><span class="tex">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>)</span><!--/no latex--><!--latex:\[EE(\mathit{x},\tau)= \delta\mathcal{F}(\mathit{x},\tau) = \mathrm{Resp}_{\mathcal{F}}\!\left(\mathit{x},\tau;\,F_{0},k_{A},k_{n},k_{\chi},k_{u}\right)\]--> |
| − | 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>) | + | |
| − | </span><!--/no latex--><!--latex:\[EE(x,\tau)= \delta\mathcal{F}(x,\tau)= \mathrm{Resp}_{\mathcal{F}}\!\left(x,\tau;\,F_{0},k_{A},k_{n},k_{\chi},k_{u}\right)\]--> | + | |
=== 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> | + | <!--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 == | ||
| Line 72: | Line 77: | ||
<!--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:\[EA = (\Delta P_{A},\,\Delta P_{n},\,\Delta P_{\chi},\,\Delta P_{u})\]--> | <!--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:\[EA = (\Delta P_{A},\,\Delta P_{n},\,\Delta P_{\chi},\,\Delta P_{u})\]--> | ||
| − | <!--no latex--><span class="tex">ΔP<sub>𝓲</sub> = τ<sub>𝓲</sub><sup>actual</sup> − τ<sub>𝓲</sub><sup>stochastically | + | <!--no latex--><span class="tex"><b>EA = (ΔP<sub>A</sub>, ΔP<sub>n</sub>, ΔP<sub>χ</sub>, ΔP<sub>u</sub>)</b></span><!--/no latex--><!--latex:\[\mathbf{EA} = (\mathbf{\Delta P_{A}},\,\mathbf{\Delta P_{n}},\,\mathbf{\Delta P_{\chi}},\,\mathbf{\Delta P_{u}})\]--> |
| − | = \tau_{\mathcal{i}}^{\mathrm{actual}} - \tau_{\mathcal{i}}^{(\mathrm{stochastically symmetric})}\]--> | + | |
| + | <!--no latex--><span class="tex"><b>ΔP<sub>A</sub>, ΔP<sub>n</sub>, ΔP<sub>χ</sub>, ΔP<sub>u</sub></b></span><!--/no latex--><!--latex:\[\mathbf{\Delta P_{A}},\,\mathbf{\Delta P_{n}},\,\mathbf{\Delta P_{\chi}},\,\mathbf{\Delta P_{u}}\]--> | ||
| + | |||
| + | <!--no latex--><span class="tex">ΔP<sub>A</sub></span><!--/no latex--><!--latex:\[\Delta P_{A}\]--> | ||
| + | <!--no latex--><span class="tex"><b>ΔP<sub>A</sub></b></span><!--/no latex--><!--latex:\[\mathbf{\Delta P_{A}}\]--> | ||
| + | |||
| + | <!--no latex--><span class="tex">ΔP<sub>n</sub></span><!--/no latex--><!--latex:\[\Delta P_{n}\]--> | ||
| + | <!--no latex--><span class="tex"><b>ΔP<sub>n</sub></b></span><!--/no latex--><!--latex:\[\mathbf{\Delta P_{n}}\]--> | ||
| + | |||
| + | <!--no latex--><span class="tex">ΔP<sub>χ</sub></span><!--/no latex--><!--latex:\[\Delta P_{\chi}\]--> | ||
| + | <!--no latex--><span class="tex"><b>ΔP<sub>χ</sub></b></span><!--/no latex--><!--latex:\[\mathbf{\Delta P_{\chi}}\]--> | ||
| + | |||
| + | <!--no latex--><span class="tex">ΔP<sub>u</sub></span><!--/no latex--><!--latex:\[\Delta P_{u}\]--> | ||
| + | <!--no latex--><span class="tex"><b>ΔP<sub>u</sub></b></span><!--/no latex--><!--latex:\[\mathbf{\Delta P_{u}}\]--> | ||
| + | |||
| + | <!--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:\[\Delta P_{\mathcal{i}} = \tau_{\mathcal{i}}^{\mathrm{actual}} - \tau_{\mathcal{i}}^{(\mathrm{stochastically\ symmetric})}\]--> | ||
| + | |||
| + | Stochastically symmetric => Stochastically symmetric reference τ(σ) distribution | ||
| + | |||
| + | When symmetry holds: | ||
| + | <!--no latex--><span class="tex">ΔP<sub>𝓲</sub> = 0</span><!--/no latex--><!--latex:\[\Delta P_{\mathcal{i}} = 0\]--> | ||
| + | |||
| + | When symmetry is broken: | ||
| + | <!--no latex--><span class="tex">ΔP<sub>𝓲</sub> ≠ 0</span><!--/no latex--><!--latex:\[\Delta P_{\mathcal{i}} \neq 0\]--> | ||
| + | |||
| + | <!--no latex--><span class="tex">(A, n, χ, u)</span><!--/no latex--><!--latex:\[(A,\,n,\,\chi,\,u)\]--> | ||
=== 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:\[\Delta P_{A} = \tau_{A}^{\mathrm{actual}} - \tau_{A}^{(\mathrm{stochastically\ symmetric})}\]--> | + | * 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:\[\Delta P_{A} = \tau_{A}^{\mathrm{actual}} - |
| + | \tau_{A}^{(\mathrm{stochastically\ symmetric})}\]--> | ||
| − | * 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:\[\Delta P_{n} = \tau_{n}^{\mathrm{actual}} - \tau_{n}^{(\mathrm{stochastically\ symmetric})}\]--> | + | * 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: 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:\[\Delta P_{u} = \tau_{u}^{\mathrm{actual}} - \tau_{u}^{(\mathrm{stochastically\ symmetric})}\]--> | + | * 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 | + | == Information / Energy (Potentials) == |
=== Equilibrium conditions === | === Equilibrium conditions === | ||
| Line 123: | Line 157: | ||
=== 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 131: | Line 171: | ||
==== 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 145: | Line 192: | ||
==== 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 159: | Line 213: | ||
==== 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 171: | Line 224: | ||
<!--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 | ||
Latest revision as of 23:44, 14 July 2026
Contents
The Primary Source of Reality
Fundamental Force (FF)
F₀
Elementary Field (EF)
𝓕
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
kA — intensity resistance
kn — linear‑orientation resistance
kχ — chirality‑reorientation resistance
ku — intrinsic‑angular‑orientation resistance
{kA — intensity‑resistance, kn — linear‑orientation resistance, kχ — chirality‑reorientation resistance, ku — intrinsic‑angular‑orientation resistance}
{kA, kn, kχ, ku}
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)
EA = (ΔPA, ΔPn, ΔPχ, ΔPu)
ΔPA, ΔPn, ΔPχ, ΔPu
ΔPA ΔPA
ΔPn ΔPn
ΔPχ ΔPχ
ΔPu ΔPu
ΔP𝓲 = τ𝓲actual − τ𝓲stochastically symmetric
Stochastically symmetric => Stochastically symmetric reference τ(σ) distribution
When symmetry holds: ΔP𝓲 = 0
When symmetry is broken: ΔP𝓲 ≠ 0
(A, n, χ, u)
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)
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
