QiqtGrGaussian · section of the QIQT-H book

QIQTH.QiqtGrGaussian

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QiqtGrGaussian · entries 556–556 of 1000

Theorem 556 (qiqt_gr_freefield_gaussian).  source ↗

The free-field QIQT→GR capstone with the localization mode constructed from the field. Identical to qiqt_gr_freefield_nullEnergy, but the wedge mode ff and its derivative ff', all their regularity, and the localization identity hTkk are no longer inputs — they are BUILT from φ: ff x v θ := ↑(∑ₐ vₐ ∂ₐφ(x))·gaussMode ℏ θ. hTkk is discharged by localized_mode_hTkk + gaussMode_calibration. Only the Clausius/area physics (hbound/hsat/hDnn/hD0/hK) and the Raychaudhuri congruence setup (hWx/hWC/hWgeo/hWequil/hWarea) remain labelled. Axiom-free.

(∀(y:M4)(ab:Fin 4),gab(y)=gba(y))→(∀(y:M4)(ab:Fin 4),gab(y)=gba(y))→(∀(y:M4)(ab:Fin 4),∑σgaσ(y)⋅gσb(y)=δab)→(∀(ab:Fin 4),(λy↦gab(y))∈C∞)→(∀(ab:Fin 4),(λy↦gab(y))∈C∞)→∀(φ:M4→R)(mηℏa:R),ℏ≠0→0<ℏ→η≠0→a=2⋅π/(ℏ⋅η)→(φ)∈C∞→(∀(x:M4),(□φ)(x)=m2⋅φ x)→∀(PPinv:M4→Fin 4→Fin 4→R),(∀(x:M4)(ij:Fin 4),∑kPik(x)⋅(P−1)kj(x)=δij)→(∀(x:M4)(ij:Fin 4),∑k(P−1)ik(x)⋅Pkj(x)=δij)→(∀(x:M4)(ij:Fin 4),gij(x)=∑k∑lPki(x)⋅ηkl⋅Plj(x))→∀(SfKEA:M4→(Fin 4→R)→R→R)(sdad:M4→(Fin 4→R)→R),(∀(x:M4)(v:Fin 4→R),(g x)(v,v)=0→(Sf x v)′(0)=S˙(x,v))→(∀(x:M4)(v:Fin 4→R),(g x)(v,v)=0→(KE x v)′(0)=2⋅π/ℏ⋅(T(x))(v,v))→(∀(x:M4)(v:Fin 4→R),(g x)(v,v)=0→(A x v)′(0)=A˙(x,v))→(∀(x:M4)(v:Fin 4→R),(g x)(v,v)=0→for t near 0,  Sf x v t≤η⋅A(x,v,t))→(∀(x:M4)(v:Fin 4→R),(g x)(v,v)=0→Sf x v 0=η⋅A(x,v,0))→(∀(x:M4)(v:Fin 4→R),(g x)(v,v)=0→∀(t:R),0≤KE(x,v,t)−Sf x v t)→(∀(x:M4)(v:Fin 4→R),(g x)(v,v)=0→KE(x,v,0)−Sf x v 0=0)→∀(mw:M4→(Fin 4→R)→R),(∀(x:M4)(v:Fin 4→R),0<mw x v)→∀(W:M4→(Fin 4→R)→M4→Fin 4→R),(∀(x:M4)(v:Fin 4→R),(g x)(v,v)=0→W x v x=v)→(∀(x:M4)(v:Fin 4→R)(μ:Fin 4),(λy↦W x v y μ)∈C∞)→(∀(x:M4)(v:Fin 4→R)(y:M4)(μ:Fin 4),∑νW x v y ν⋅(∇W x v)νμ(y)=0)→(∀(x:M4)(v:Fin 4→R),(g x)(v,v)=0→∑μ∑ν(∇W x v)μν(x)⋅(∇W x v)νμ(x)=0)→(∀(x:M4)(v:Fin 4→R),(g x)(v,v)=0→A˙(x,v)=−∑νW x v x ν⋅∂ν(λy↦θ(y))(x))→∃Λ,∀(x:M4)(μν:Fin 4),a⋅T(x) μ ν=Gμν(x)+Λ⋅gμν(x)(\forall (y : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (a b : \mathrm{Fin}\,4), g_{{a}{b}}({y}) = g_{{b}{a}}({y})) \to (\forall (y : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (a b : \mathrm{Fin}\,4), g^{{a}{b}}({y}) = g^{{b}{a}}({y})) \to (\forall (y : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (a b : \mathrm{Fin}\,4), \sum_{\sigma} g_{{a}{\sigma}}({y}) \cdot g^{{\sigma}{b}}({y}) = \delta_{ab}) \to (\forall (a b : \mathrm{Fin}\,4), ({\lambda y \mapsto g_{{a}{b}}({y})})\in C^{\infty}) \to (\forall (a b : \mathrm{Fin}\,4), ({\lambda y \mapsto g^{{a}{b}}({y})})\in C^{\infty}) \to \forall (\varphi : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}} \to \mathbb{R}) (m \eta \hbar a : \mathbb{R}), \hbar \ne 0 \to 0 < \hbar \to \eta \ne 0 \to a = 2 \cdot \pi / (\hbar \cdot \eta) \to ({\varphi})\in C^{\infty} \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}), \href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-boxfield}{(\Box {\varphi})({x})} = {m}^{2} \cdot \varphi\,x) \to \forall (P \mathrm{Pinv} : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}} \to \mathrm{Fin}\,4 \to \mathrm{Fin}\,4 \to \mathbb{R}), (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (i j : \mathrm{Fin}\,4), \sum_{k} P_{{i}{k}}({x}) \cdot (P^{-1})_{{k}{j}}({x}) = \delta_{ij}) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (i j : \mathrm{Fin}\,4), \sum_{k} (P^{-1})_{{i}{k}}({x}) \cdot P_{{k}{j}}({x}) = \delta_{ij}) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (i j : \mathrm{Fin}\,4), g_{{i}{j}}({x}) = \sum_{k} \sum_{l} P_{{k}{i}}({x}) \cdot \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-gm}{\eta_{{k}{l}}} \cdot P_{{l}{j}}({x})) \to \forall (\mathrm{Sf} \mathrm{KE} A : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}} \to (\mathrm{Fin}\,4 \to \mathbb{R}) \to \mathbb{R} \to \mathbb{R}) (\mathrm{sd} \mathrm{ad} : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}} \to (\mathrm{Fin}\,4 \to \mathbb{R}) \to \mathbb{R}), (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to ({\mathrm{Sf}\,x\,v})'({0})={\dot{S}({x},{v})}) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to ({\mathrm{KE}\,x\,v})'({0})={2 \cdot \pi / \hbar \cdot ({\href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-kgstress}{T({x})}})({v},{v})}) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to ({A\,x\,v})'({0})={\dot{A}({x},{v})}) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to \text{for }t\text{ near }0,\; \mathrm{Sf}\,x\,v\,t \le \eta \cdot A({x},{v},{t})) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to \mathrm{Sf}\,x\,v\,0 = \eta \cdot A({x},{v},{0})) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to \forall (t : \mathbb{R}), 0 \le \mathrm{KE}({x},{v},{t}) - \mathrm{Sf}\,x\,v\,t) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to \mathrm{KE}({x},{v},{0}) - \mathrm{Sf}\,x\,v\,0 = 0) \to \forall (\mathrm{mw} : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}} \to (\mathrm{Fin}\,4 \to \mathbb{R}) \to \mathbb{R}), (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), 0 < \mathrm{mw}\,x\,v) \to \forall (W : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}} \to (\mathrm{Fin}\,4 \to \mathbb{R}) \to \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}} \to \mathrm{Fin}\,4 \to \mathbb{R}), (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to W\,x\,v\,x = v) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}) (\mu : \mathrm{Fin}\,4), ({\lambda y \mapsto W\,x\,v\,y\,\mu})\in C^{\infty}) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}) (y : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (\mu : \mathrm{Fin}\,4), \sum_{\nu} W\,x\,v\,y\,\nu \cdot \href{/browser/qiqth-curvature#d-qiqth-curvature-covderivvec}{(\nabla {W\,x\,v})_{{\nu}}{}^{{\mu}}({y})} = 0) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to \sum_{\mu} \sum_{\nu} \href{/browser/qiqth-curvature#d-qiqth-curvature-covderivvec}{(\nabla {W\,x\,v})_{{\mu}}{}^{{\nu}}({x})} \cdot \href{/browser/qiqth-curvature#d-qiqth-curvature-covderivvec}{(\nabla {W\,x\,v})_{{\nu}}{}^{{\mu}}({x})} = 0) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to \dot{A}({x},{v}) = -\sum_{\nu} W\,x\,v\,x\,\nu \cdot \partial_{{\nu}}({\lambda y \mapsto \href{/browser/qiqth-raychaudhuri#d-qiqth-curvature-expansion}{\theta({y})}})({x})) \to \exists \Lambda, \forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (\mu \nu : \mathrm{Fin}\,4), a \cdot \href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-kgstress}{T({x})\,\mu\,\nu} = \href{/browser/qiqth-curvature#d-qiqth-curvature-einsteintensor}{G_{{\mu}{\nu}}({x})} + \Lambda \cdot g_{{\mu}{\nu}}({x})

Proof. By gaussC, gaussMode, gaussMode', gaussMode_calibration, gaussMode_hasDerivAt, gaussMode'_continuous, gaussMode'_norm_le, gaussMode_memLp, gaussMode_integrable_fn, localized_mode_hTkk, qiqt_gr_freefield_nullEnergy. □\square

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