Difference between revisions of "Paper Symbols"

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(→‎Resistance parameters)
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=== Elementary Field (EF) ===
 
=== Elementary Field (EF) ===
 +
 +
<!--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:\[\mathcal{F} = F_{0}(A,n,\chi,u;\,k_{A},k_{n},k_{\chi},k_{u})\]-->
 
<!--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:\[\mathcal{F} = F_{0}(A,n,\chi,u;\,k_{A},k_{n},k_{\chi},k_{u})\]-->
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{<!--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> — 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:\[k_{u}\]-->}
 
{<!--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:\[k_{u}\]-->}
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=== 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 ===  
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== 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:\[EA = (\Delta P_{A},\,\Delta P_{n},\,\Delta P_{\chi},\,\Delta P_{u})\]-->
  
−
\[
+
<!--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:\[\mathbf{EA} = (\mathbf{\Delta P_{A}},\,\mathbf{\Delta P_{n}},\,\mathbf{\Delta P_{\chi}},\,\mathbf{\Delta P_{u}})\]-->
−
EA = (\Delta P_{A},\,\Delta P_{n},\,\Delta P_{\chi},\,\Delta P_{u})
+
−
\]
+
  
−
-->
+
<!--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>𝓲</sub> = τ<sub>𝓲</sub><sup>actual</sup> − τ<sub>𝓲</sub><sup>stochastically symmetric</sup></span><!--/no latex--><!--latex:
+
<!--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:\[\Delta P_{n}\]-->
−
\Delta P_{\mathcal{i}} =
+
<!--no latex--><span class="tex"><b>ΔP<sub>n</sub></b></span><!--/no latex--><!--latex:\[\mathbf{\Delta P_{n}}\]-->
−
\tau_{\mathcal{i}}^{\mathrm{actual}} -
+
−
\tau_{\mathcal{i}}^{(\mathrm{stochastically\ symmetric})}
+
−
\]
+
  
−
-->
+
<!--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:\[\mathbf{\Delta P_{\chi}}\]-->
  
−
=== Modes / Components ===
+
<!--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:\[\mathbf{\Delta P_{u}}\]-->
  
−
* 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:
+
<!--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:\[\Delta P_{\mathcal{i}} = \tau_{\mathcal{i}}^{\mathrm{actual}} - \tau_{\mathcal{i}}^{(\mathrm{stochastically\ symmetric})}\]-->
  
−
\[
+
Stochastically symmetric => Stochastically symmetric reference τ(σ) distribution
−
\Delta P_{A} =
+
−
\tau_{A}^{\mathrm{actual}} -
+
−
\tau_{A}^{(\mathrm{stochastically\ symmetric})}
+
−
\]
+
  
−
-->
+
When symmetry holds:
 +
<!--no latex--><span class="tex">ΔP<sub>𝓲</sub> = 0</span><!--/no latex--><!--latex:\[\Delta P_{\mathcal{i}} = 0\]-->
  
−
* 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:
+
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:\[(A,\,n,\,\chi,\,u)\]-->
−
\Delta P_{n} =
+
−
\tau_{n}^{\mathrm{actual}} -
+
−
\tau_{n}^{(\mathrm{stochastically\ symmetric})}
+
−
\]
+
  
−
-->
+
=== Modes / Components ===
  
−
* 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:
+
* 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:\[\Delta P_{n} =
−
\Delta P_{\chi} =
+
\tau_{n}^{\mathrm{actual}} -\tau_{n}^{(\mathrm{stochastically\ symmetric})}\]-->
−
\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:
+
* 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})}\]-->
−
\[
+
−
\Delta P_{u} =
+
−
\tau_{u}^{\mathrm{actual}} -
+
−
\tau_{u}^{(\mathrm{stochastically\ symmetric})}
+
−
\]
+
−
 
+
−
-->
+
  
 
== Information / Energy (Potentials) ==
 
== Information / Energy (Potentials) ==

Latest revision as of 23:44, 14 July 2026

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

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