Appendix MM
Closure Susceptibility, Temporal Stability, and the End of Absolute Gravity
MM.1 Theorem — Divergence of χ C \chi_C χ C
Theorem (Gravitational Phase Transition by Closure Susceptibility).
Let a dynamical system be defined by the historical state
Ψ ( t ) = { ρ , v , ∇ Φ , γ , μ , ∫ 0 t K ( t − τ ) ρ ( τ ) d τ } , \Psi(t)=
\Big\{
\rho,\mathbf v,\nabla\Phi,\gamma,\mu,
\int_0^t K(t-\tau)\rho(\tau)\,d\tau
\Big\}, Ψ ( t ) = { ρ , v , ∇Φ , γ , μ , ∫ 0 t K ( t − τ ) ρ ( τ ) d τ } , with closure functional
C [ Ψ ] = Θ ( ⟨ ∣ L ∣ ⟩ − L c r i t [ Ψ h i s t o r y ] ) . C[\Psi]=
\Theta\!\big(
\langle|\mathbf L|\rangle
-
L_{\mathrm{crit}}[\Psi_{\mathrm{history}}]
\big). C [ Ψ ] = Θ ( ⟨ ∣ L ∣ ⟩ − L crit [ Ψ history ] ) . Define the closure susceptibility functional
χ C ( t ) ≡ δ δ Ψ ( ⟨ ∣ L ∣ ⟩ − L c r i t [ Ψ h i s t o r y ] ) . \chi_C(t)
\;\equiv\;
\frac{\delta}{\delta\Psi}
\Big(
\langle|\mathbf L|\rangle
-
L_{\mathrm{crit}}[\Psi_{\mathrm{history}}]
\Big). χ C ( t ) ≡ δ Ψ δ ( ⟨ ∣ L ∣ ⟩ − L crit [ Ψ history ] ) . If
lim t → t ∗ χ C ( t ) = + ∞ , \lim_{t\to t_\ast}\chi_C(t)=+\infty, t → t ∗ lim χ C ( t ) = + ∞ , then the system undergoes a gravitational phase transition at t = t ∗ t=t_\ast t = t ∗
C = 1 ↔ C = 0. C=1 \;\leftrightarrow\; C=0. C = 1 ↔ C = 0. Proof Sketch.
Define the phase distance
Δ ( t ) = ⟨ ∣ L ∣ ⟩ − L c r i t [ Ψ h i s t o r y ] . \Delta(t) = \langle|\mathbf L|\rangle - L_{\mathrm{crit}}[\Psi_{\mathrm{history}}]. Δ ( t ) = ⟨ ∣ L ∣ ⟩ − L crit [ Ψ history ] .
χ C = δ Δ / δ Ψ \chi_C=\delta\Delta/\delta\Psi χ C = δ Δ/ δ Ψ χ C → ∞ \chi_C\to\infty χ C → ∞ Δ \Delta Δ Since C = Θ ( Δ ) C=\Theta(\Delta) C = Θ ( Δ )
No force, metric, or additional dynamics are invoked.
□ \square □
MM.2 Explicit Two-Body Derivation of χ C \chi_C χ C
Setup.
v e f f ( t ) = r ˙ 1 − r ˙ 2 , L ( t ) = ( r 1 − r 2 ) × v e f f , ⟨ ∣ L ∣ ⟩ = 1 T ∫ 0 T ∣ L ( t ) ∣ d t . \begin{aligned}
\mathbf v_{\mathrm{eff}}(t) &= \dot{\mathbf r}_1-\dot{\mathbf r}_2,\\
\mathbf L(t) &= (\mathbf r_1-\mathbf r_2)\times\mathbf v_{\mathrm{eff}},\\
\langle|\mathbf L|\rangle &= \frac{1}{T}\int_0^T |\mathbf L(t)|\,dt.
\end{aligned} v eff ( t ) L ( t ) ⟨ ∣ L ∣ ⟩ = r ˙ 1 − r ˙ 2 , = ( r 1 − r 2 ) × v eff , = T 1 ∫ 0 T ∣ L ( t ) ∣ d t . Dynamics:
∂ t v e f f = − ∇ Φ − γ v e f f , ( ∇ 2 − μ 2 ) Φ = ρ − ⟨ ρ ⟩ . \begin{aligned}
\partial_t\mathbf v_{\mathrm{eff}} &= -\nabla\Phi-\gamma\mathbf v_{\mathrm{eff}},\\
(\nabla^2-\mu^2)\Phi &= \rho-\langle\rho\rangle.
\end{aligned} ∂ t v eff ( ∇ 2 − μ 2 ) Φ = − ∇Φ − γ v eff , = ρ − ⟨ ρ ⟩ . Damping contribution.
∂ ⟨ ∣ L ∣ ⟩ ∂ γ = − 1 T ∫ 0 T t ∣ L ( t ) ∣ d t < 0. \frac{\partial\langle|\mathbf L|\rangle}{\partial\gamma}
=
-\frac{1}{T}\int_0^T t\,|\mathbf L(t)|\,dt
<0. ∂ γ ∂ ⟨ ∣ L ∣ ⟩ = − T 1 ∫ 0 T t ∣ L ( t ) ∣ d t < 0. Spatial coherence contribution.
∂ ⟨ ∣ L ∣ ⟩ ∂ μ = 1 T ∫ 0 T L ∣ L ∣ ⋅ ( r × ∂ μ v e f f ) d t < 0. \frac{\partial\langle|\mathbf L|\rangle}{\partial\mu}
=
\frac{1}{T}\int_0^T
\frac{\mathbf L}{|\mathbf L|}
\cdot(\mathbf r\times\partial_\mu\mathbf v_{\mathrm{eff}})\,dt
<0. ∂ μ ∂ ⟨ ∣ L ∣ ⟩ = T 1 ∫ 0 T ∣ L ∣ L ⋅ ( r × ∂ μ v eff ) d t < 0. Memory contribution.
∂ ⟨ ∣ L ∣ ⟩ ∂ K = 1 T ∫ 0 T L ∣ L ∣ ⋅ ( r × ∫ 0 t F [ ρ ( τ ) ] d τ ) d t > 0. \frac{\partial\langle|\mathbf L|\rangle}{\partial K}
=
\frac{1}{T}\int_0^T
\frac{\mathbf L}{|\mathbf L|}
\cdot
\left(
\mathbf r\times
\int_0^t \mathcal F[\rho(\tau)]\,d\tau
\right)dt
>0. ∂ K ∂ ⟨ ∣ L ∣ ⟩ = T 1 ∫ 0 T ∣ L ∣ L ⋅ ( r × ∫ 0 t F [ ρ ( τ )] d τ ) d t > 0. Operational form.
χ C = − 1 T ∫ 0 T t ∣ L ∣ d t + I μ + I K \boxed{
\chi_C
=
-\frac{1}{T}\int_0^T t\,|\mathbf L|\,dt
+\mathcal I_\mu
+\mathcal I_K
} χ C = − T 1 ∫ 0 T t ∣ L ∣ d t + I μ + I K MM.3 Closure Time Functional
τ C ( t ) = inf { Δ t > 0 ∣ C [ Ψ ( t + Δ t ) ] ≠ C [ Ψ ( t ) ] } \boxed{
\tau_C(t)
=
\inf\Big\{
\Delta t>0
\;\big|\;
C[\Psi(t+\Delta t)]\neq C[\Psi(t)]
\Big\}
} τ C ( t ) = inf { Δ t > 0 C [ Ψ ( t + Δ t )] = C [ Ψ ( t )] } First-order approximation:
τ C ( 1 ) ( t ) = ∣ Δ ( t ) ∣ ∣ d d t Δ ( t ) ∣ \boxed{
\tau_C^{(1)}(t)
=
\frac{|\Delta(t)|}
{\left|\dfrac{d}{dt}\Delta(t)\right|}
} τ C ( 1 ) ( t ) = d t d Δ ( t ) ∣Δ ( t ) ∣ MM.4 Closure Entropy Production Rate
σ C ( t ) = d d t ln ( ⟨ ∣ L ∣ ⟩ + ε ) ( ε > 0 ) \boxed{
\sigma_C(t)
=
\frac{d}{dt}
\ln\!\big(\langle|\mathbf L|\rangle+\varepsilon\big)
}
\qquad(\varepsilon>0) σ C ( t ) = d t d ln ( ⟨ ∣ L ∣ ⟩ + ε ) ( ε > 0 ) Equivalent form:
σ C ( t ) = d d t ⟨ ∣ L ∣ ⟩ ⟨ ∣ L ∣ ⟩ + ε . \sigma_C(t)
=
\frac{\dfrac{d}{dt}\langle|\mathbf L|\rangle}
{\langle|\mathbf L|\rangle+\varepsilon}. σ C ( t ) = ⟨ ∣ L ∣ ⟩ + ε d t d ⟨ ∣ L ∣ ⟩ . MM.5 Gravitational Closure Phase Space
P G = { Δ , χ C , τ C , σ C } \boxed{
\mathcal P_G
=
\{\Delta,\;\chi_C,\;\tau_C,\;\sigma_C\}
} P G = { Δ , χ C , τ C , σ C } Hierarchy:
Δ → χ C → ( τ C , σ C ) . \Delta
\;\rightarrow\;
\chi_C
\;\rightarrow\;
(\tau_C,\sigma_C). Δ → χ C → ( τ C , σ C ) . MM.6 Reinterpretation of Gravitational Collapse
χ C → ∞ , τ C → 0 , \chi_C\to\infty,
\qquad
\tau_C\to0, χ C → ∞ , τ C → 0 , defines a phase-boundary event, not a spacetime singularity.
MM.7 End of Absolute Gravity
T ( t ) = 1 χ C ( t ) \boxed{
\mathcal T(t)=\frac{1}{\chi_C(t)}
} T ( t ) = χ C ( t ) 1
T ≪ 1 \mathcal T\ll1 T ≪ 1 T ≫ 1 \mathcal T\gg1 T ≫ 1
Final Statement.
Gravity is not a force nor a universal constant, but a metastable historical
closure phase characterized by susceptibility, lifetime, and directional evolution.
End of Appendix MM
Source:
Gravity as a Temporally Closed Dynamical Phase/47_Appendix_MM_Closure_Susceptibility_and_End_of_Absolute_Gravity.tex in the verified
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