Difference between revisions of "Paper Symbols"

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<!--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}) \;\longrightarrow\; \text{gauge equations}\]-->
 
\mathrm{Proj}(\mathcal{I}) \;\longrightarrow\; \text{gauge equations}\]-->
 +
 +
== Forms ==
 +
 +
No new fundamental equation
 +
 +
 +
== The Primary Source of Reality ==
 +
 +
=== Fundamental Force (FF) ===
 +
 +
<!--no latex-->F₀<!--/no latex--><!--latex:$F_0$-->
 +
 +
=== 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:
 +
 +
\[
 +
\mathcal{F} = F_{0}(A,n,\chi,u;\,k_{A},k_{n},k_{\chi},k_{u})
 +
\]
 +
 +
-->
 +
 +
==== Structure of resistances ====
 +
 +
<!--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:
 +
 +
\[
 +
k_{\mathcal{i}} = k_{\mathcal{i}}^{(\mathrm{intrinsic})} + k_{\mathcal{i}}^{(\mathrm{dynamic})},
 +
\qquad
 +
\mathcal{i} \in \{A,\,n,\,\chi,\,u\}
 +
\]
 +
 +
-->
 +
 +
==== 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:
 +
 +
\[
 +
k_{A},\,k_{n}\ \text{— proto‑stress resistances}
 +
\]
 +
 +
 +
 +
 +
\[
 +
k_{\chi},\,k_{u}\ \text{— proto‑vorticity resistances}
 +
\]
 +
 +
-->
 +
 +
==== 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) ==
 +
 +
<!--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 ===
 +
 +
<!--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:
 +
 +
\[
 +
EE(\mathit{x},\tau)= \delta\mathcal{F}(\mathit{x},\tau)=
 +
\mathrm{Resp}_{\mathcal{F}}\!\left(\mathit{x},\tau;\,F_{0},k_{A},k_{n},k_{\chi},k_{u}\right)
 +
\]
 +
 +
-->
 +
 +
=== Decomposition ===
 +
 +
<!--no latex--><span class="tex">EE = {A, n, χ, u}</span><!--/no latex--><!--latex:
 +
 +
\[
 +
EE = \{A,\,n,\,\chi,\,u\}
 +
\]
 +
 +
-->
 +
 +
=== Instantaneous configuration ===
 +
 +
<!--no latex--><span class="tex">σ = (A, n, χ, u)<sub>instantaneous</sub></span><!--/no latex--><!--latex:
 +
 +
\[
 +
\sigma = (A,\,n,\,\chi,\,u)_{\mathrm{instantaneous}}
 +
\]
 +
 +
-->
 +
 +
=== Statistical configuration ===
 +
 +
<!--no latex--><span class="tex">(A(τ), n(τ), χ(τ), u(τ))</span><!--/no latex--><!--latex:
 +
 +
\[
 +
(A(\tau),\,n(\tau),\,\chi(\tau),\,u(\tau))
 +
\]
 +
 +
-->
 +
 +
== 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:
 +
 +
\[
 +
EA = (\Delta P_{A},\,\Delta P_{n},\,\Delta P_{\chi},\,\Delta P_{u})
 +
\]
 +
 +
-->
 +
 +
<!--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 ===
 +
 +
* 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:
 +
 +
\[
 +
\Delta P_{n} =
 +
\tau_{n}^{\mathrm{actual}} -
 +
\tau_{n}^{(\mathrm{stochastically\ symmetric})}
 +
\]
 +
 +
-->
 +
 +
* 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) ==
 +
 +
=== Equilibrium conditions ===
 +
 +
<!--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>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 ===
 +
 +
<!--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 ==
 +
 +
=== Wave‑like InfoStructures ===
 +
 +
==== Derived from ====
 +
 +
<!--no latex--><span class="tex">ΔP<sub>n</sub>, ΔP<sub>u</sub></span><!--/no latex--><!--latex:
 +
 +
\[
 +
\Delta P_{n},\;\Delta P_{u}
 +
\]
 +
 +
-->
 +
 +
==== Equilibrium condition ====
 +
 +
<!--no latex--><span class="tex">𝓒<sub>wave</sub>(ΔP<sub>n</sub>, ΔP<sub>u</sub>) = 0</span><!--/no latex--><!--latex:
 +
 +
\[
 +
\mathcal{C}_{wave}(\Delta P_{n},\,\Delta P_{u}) = 0
 +
\]
 +
 +
-->
 +
 +
==== Proto‑wave equation ====
 +
 +
<!--no latex--><span class="tex">𝓦[Ψ; ΔP<sub>n</sub>, ΔP<sub>u</sub>] = 0</span><!--/no latex--><!--latex:
 +
 +
\[
 +
\mathcal{W}[\Psi;\,\Delta P_{n},\,\Delta P_{u}] = 0
 +
\]
 +
 +
-->
 +
 +
=== Vortex‑like InfoStructures ===
 +
 +
==== 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:
 +
 +
\[
 +
\Delta P_{A},\;\Delta P_{n},\;\Delta P_{\chi},\;\Delta P_{u}
 +
\]
 +
 +
-->
 +
 +
==== 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:
 +
 +
\[
 +
\mathcal{C}_{vortex}(\Delta P_{A},\,\Delta P_{n},\,\Delta P_{\chi},\,\Delta P_{u}) = 0
 +
\]
 +
 +
-->
 +
 +
==== 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:
 +
 +
\[
 +
\mathcal{V}[\Xi;\,\Delta P_{A},\,\Delta P_{n},\,\Delta P_{\chi},\,\Delta P_{u}] = 0
 +
\]
 +
 +
-->
 +
 +
=== Interaction‑mediating InfoStructures ===
 +
 +
==== Relational equilibrium ====
 +
 +
<!--no latex--><span class="tex">𝓒<sub>int</sub>(Ψ, Ξ) = 0</span><!--/no latex--><!--latex:
 +
 +
\[
 +
\mathcal{C}_{int}(\Psi,\,\Xi) = 0
 +
\]
 +
 +
-->
 +
 +
==== Proto‑interaction equation ====
 +
 +
<!--no latex--><span class="tex">𝓘[Ψ, Ξ] = 0</span><!--/no latex--><!--latex:
 +
 +
\[
 +
\mathcal{I}[\Psi,\,\Xi] = 0
 +
\]
 +
 +
-->
 +
 +
== Geometric Projection ==
 +
 +
=== Wave projection ===
 +
 +
<!--no latex--><span class="tex">Proj(𝓦) → QM wave equations</span><!--/no latex--><!--latex:
 +
 +
\[
 +
\mathrm{Proj}(\mathcal{W}) \longrightarrow \text{QM wave equations}
 +
\]
 +
 +
-->
 +
 +
=== Vortex projection ===
 +
 +
<!--no latex--><span class="tex">Proj(𝓥) → spinor equations</span><!--/no latex--><!--latex:
 +
 +
\[
 +
\mathrm{Proj}(\mathcal{V}) \longrightarrow \text{spinor equations}
 +
\]
 +
 +
-->
 +
 +
=== Interaction projection ===
 +
 +
<!--no latex--><span class="tex">Proj(𝓘) → gauge equations</span><!--/no latex--><!--latex:
 +
 +
\[
 +
\mathrm{Proj}(\mathcal{I}) \longrightarrow \text{gauge equations}
 +
\]
 +
 +
-->
  
 
== Forms ==
 
== Forms ==
  
 
No new fundamental equation
 
No new fundamental equation

Revision as of 21:14, 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


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

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