Target 3 — QIQT-H gives the Einstein field equations

GR field equations

Target 3 — QIQT-H gives the Einstein field equations

Einstein field equations for the free field (all geometric inputs discharged)

The maximally-discharged free-field QIQT→GR capstone. Einstein’s equations for the explicit free Klein–Gordon field, with the entropy/heat functionals built from a finite record law (T3-1, discharging hsat/hDnn/hD0) AND the wedge mode built from φ as ↑(∑ₐ vₐ ∂ₐφ)·gaussMode ℏ (T3-3-C3, discharging hTkk and the whole ff regularity block), on top of the hbridge/hFocus/hWarea-discharged ladder. Surviving labelled inputs: the dynamical FQ capacity bound hbound, the FQ reference identification hcap, the realization derivatives hS/hK/hA, the Raychaudhuri congruence setup, geometry scaffolding, and the matter EOM hKG. Axiom-free.

qiqt_gr_freefield_complete · capstone — there is $\Lambda$ such that

$a \cdot \href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-kgstress}{T({x})\,\mu\,\nu} = \href{/browser/qiqth-curvature#d-qiqth-curvature-einsteintensor}{G_{{\mu}{\nu}}({x})} + \Lambda \cdot g_{{\mu}{\nu}}({x})$

assuming

hKG $\href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-boxfield}{(\Box {\varphi})({x})} = {m}^{2} \cdot \varphi\,x$
hcap $\eta \cdot A({x},{v},{0}) = \log\,(\#\,\iota)$
hbound $\href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to \text{for }t\text{ near }0,\; \href{/browser/qiqth-branchledger#d-qiqth-branchledger-shannon}{S({p\,x\,v\,t})} \le \eta \cdot A({x},{v},{t})$

plus 25 routine conditions (17 regularity, 7 setup, 1 typeclass) — full list in the per-track PDF.

Einstein field equations on an explicit pp-wave spacetime

The instantiated showcase: QIQT→GR for the explicit pp-wave spacetime, floor laid bare. The Einstein equations a·kgStress = G + Λg with g = ppMetric H, with every geometric and analytic premise discharged — the pp-wave metric/tetrad (via qiqt_gr_ppwave), the area derivative hA (via area_hasDerivAt_of_covConst, the expansion-free congruence), and the entropy bound hbound (via shannon_le_log_card, the area = holographic capacity). The only remaining hypotheses are the irreducible floor: the matter EOM hKG, the FQ capacity hcap (η·c = log|R| = P4), and the localization map hS/hK (the field-coupled record law whose entropy rate is the stress flux — Gap-2), plus the covariantly-constant congruence and constants. …

qiqt_gr_ppwave_showcase · spine — there is $\Lambda$ such that

assuming

hCH $({H})\in C^{\infty}$
hKG $\href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-boxfield}{(\Box {\varphi})({x})} = {m}^{2} \cdot \varphi\,x$
hcap $\eta \cdot c = \log\,(\#\,\iota)$
hcov $\href{/browser/qiqth-curvature#d-qiqth-curvature-covderivvec}{(\nabla {W\,x\,v})_{{p}}{}^{{q}}({y})} = 0$

plus 14 routine conditions (9 regularity, 4 setup, 1 typeclass) — full list in the per-track PDF.

`qiqt_gr_freefield_gaussian`

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.

qiqt_gr_freefield_gaussian · spine — there is $\Lambda$ such that

assuming

when $\href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0$ :

hbound $\text{for }t\text{ near }0,\; \mathrm{Sf}\,x\,v\,t \le \eta \cdot A({x},{v},{t})$
hsat $\mathrm{Sf}\,x\,v\,0 = \eta \cdot A({x},{v},{0})$
hDnn $\forall (t : \mathbb{R}), 0 \le \mathrm{KE}({x},{v},{t}) - \mathrm{Sf}\,x\,v\,t$
hD0 $\mathrm{KE}({x},{v},{0}) - \mathrm{Sf}\,x\,v\,0 = 0$
hWarea $\dot{A}({x},{v}) = -\sum_{\nu} W\,x\,v\,x\,\nu \cdot \partial_{{\nu}}({\lambda y \mapsto \href{/browser/qiqth-raychaudhuri#d-qiqth-curvature-expansion}{\theta({y})}})({x})$

and

hKG $\href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-boxfield}{(\Box {\varphi})({x})} = {m}^{2} \cdot \varphi\,x$

plus 21 routine conditions (14 regularity, 7 setup) — full list in the per-track PDF.

`qiqt_gr_freefield_thermo`

The THERMODYNAMIC free-field QIQT→GR capstone. Einstein’s equations for the explicit free Klein–Gordon field, with the entropy/heat functionals CONSTRUCTED from a per-generator finite record law pp (a probability distribution for each deformation t, uniform at the equilibrium reference), and the saturation + relative-entropy premises (hsat/hDnn/hD0) DISCHARGED internally via clausius_package_from_finite_model (the axiom-free finite core: Gibbs/Jensen, uniform saturation, classical Klein). …

qiqt_gr_freefield_thermo · spine — there is $\Lambda$ such that

assuming

when $\href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0$ :

hbound $\text{for }t\text{ near }0,\; \href{/browser/qiqth-branchledger#d-qiqth-branchledger-shannon}{S({p\,x\,v\,t})} \le \eta \cdot A({x},{v},{t})$
hTkk $2 \cdot \pi / \hbar \cdot {(\sum_{b} v\,b \cdot \href{/browser/qiqth-curvature#d-qiqth-curvature-pd}{\partial_{{b}}({\varphi})({x})})}^{2} = (-(2 \cdot \pi \cdot \int (\theta : \mathbb{R}), (\mathrm{starRingEnd}\,\mathbb{C})\,(\mathrm{ff}\,x\,v\,\theta) \cdot \mathrm{ff}^{\prime}\,x\,v\,\theta)).\mathrm{im}$

and

hKG $\href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-boxfield}{(\Box {\varphi})({x})} = {m}^{2} \cdot \varphi\,x$
hcap $\eta \cdot A({x},{v},{0}) = \log\,(\#\,\iota)$

plus 29 routine conditions (21 regularity, 7 setup, 1 typeclass) — full list in the per-track PDF.

`qiqt_gr_freefield_geom`

Stage 2′ (T3-3, option b): hWarea discharged — ad defined geometrically. Identical to qiqt_gr_freefield_nullEnergy, but the area first-variation rate ad is no longer an abstract parameter paired with the labelled identity hWarea; it is defined as the congruence-expansion derivative ad x v := −∑ᵥ Wˣᵛ ∂ᵥ θ[Wˣᵛ], so hWarea becomes rfl. hA (the area functional’s rate) now reads in that explicit geometric form. …

qiqt_gr_freefield_geom · spine — there is $\Lambda$ such that

assuming

when $\href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0$ :

hbound $\text{for }t\text{ near }0,\; \mathrm{Sf}\,x\,v\,t \le \eta \cdot A({x},{v},{t})$
hsat $\mathrm{Sf}\,x\,v\,0 = \eta \cdot A({x},{v},{0})$
hDnn $\forall (t : \mathbb{R}), 0 \le \mathrm{KE}({x},{v},{t}) - \mathrm{Sf}\,x\,v\,t$
hD0 $\mathrm{KE}({x},{v},{0}) - \mathrm{Sf}\,x\,v\,0 = 0$
hTkk $2 \cdot \pi / \hbar \cdot {(\sum_{b} v\,b \cdot \href{/browser/qiqth-curvature#d-qiqth-curvature-pd}{\partial_{{b}}({\varphi})({x})})}^{2} = (-(2 \cdot \pi \cdot \int (\theta : \mathbb{R}), (\mathrm{starRingEnd}\,\mathbb{C})\,(\mathrm{ff}\,x\,v\,\theta) \cdot \mathrm{ff}^{\prime}\,x\,v\,\theta)).\mathrm{im}$

and

hKG $\href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-boxfield}{(\Box {\varphi})({x})} = {m}^{2} \cdot \varphi\,x$

plus 25 routine conditions (18 regularity, 7 setup) — full list in the per-track PDF.

`qiqt_gr_freefield_nullEnergy`

Stage 3 (T3-3): hTkk in transparent form — the single irreducible localization input. Identical to qiqt_gr_freefield_localized', but the one surviving Gap-2 input hTkk is stated in its physically transparent form via the Stage-0 null-stress identity BL(kgStress) v = (∑ₐ vₐ ∂ₐφ)²:

2π/ℏ · (∑ₐ vₐ ∂ₐφ(x))² = (−2π ∫ conj(ff x v)·ff' x v).im. …

qiqt_gr_freefield_nullEnergy · spine — there is $\Lambda$ such that

assuming

when $\href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0$ :

hbound $\text{for }t\text{ near }0,\; \mathrm{Sf}\,x\,v\,t \le \eta \cdot A({x},{v},{t})$
hsat $\mathrm{Sf}\,x\,v\,0 = \eta \cdot A({x},{v},{0})$
hDnn $\forall (t : \mathbb{R}), 0 \le \mathrm{KE}({x},{v},{t}) - \mathrm{Sf}\,x\,v\,t$
hD0 $\mathrm{KE}({x},{v},{0}) - \mathrm{Sf}\,x\,v\,0 = 0$
hTkk $2 \cdot \pi / \hbar \cdot {(\sum_{b} v\,b \cdot \href{/browser/qiqth-curvature#d-qiqth-curvature-pd}{\partial_{{b}}({\varphi})({x})})}^{2} = (-(2 \cdot \pi \cdot \int (\theta : \mathbb{R}), (\mathrm{starRingEnd}\,\mathbb{C})\,(\mathrm{ff}\,x\,v\,\theta) \cdot \mathrm{ff}^{\prime}\,x\,v\,\theta)).\mathrm{im}$
hWarea $\dot{A}({x},{v}) = -\sum_{\nu} W\,x\,v\,x\,\nu \cdot \partial_{{\nu}}({\lambda y \mapsto \href{/browser/qiqth-raychaudhuri#d-qiqth-curvature-expansion}{\theta({y})}})({x})$

and

hKG $\href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-boxfield}{(\Box {\varphi})({x})} = {m}^{2} \cdot \varphi\,x$

plus 25 routine conditions (18 regularity, 7 setup) — full list in the per-track PDF.

`qiqt_gr_freefield_localized'`

Stage 2 (T3-3): hFocus discharged via Raychaudhuri. The localized free-field capstone with the focusing identity hFocus (ad = R_kk) no longer assumed but DERIVED from the machine-checked Raychaudhuri equation (hFocus_of_raychaudhuri): per null generator (x,v) we supply a smooth geodesic congruence W x v through (x,v) (hWx : W x v x = v), at equilibrium (hWequil, the shear–expansion quadratic vanishes — Jacobson’s stationary/bifurcation horizon), with the area-vs-expansion identification hWarea. The Raychaudhuri focusing law ad = BL(Ric) v is then proved (no Einstein presupposed); christoffel smoothness is itself discharged (christoffel_contDiff). …

qiqt_gr_freefield_localized' · spine — there is $\Lambda$ such that

assuming

when $\href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0$ :

hbound $\text{for }t\text{ near }0,\; \mathrm{Sf}\,x\,v\,t \le \eta \cdot A({x},{v},{t})$
hsat $\mathrm{Sf}\,x\,v\,0 = \eta \cdot A({x},{v},{0})$
hDnn $\forall (t : \mathbb{R}), 0 \le \mathrm{KE}({x},{v},{t}) - \mathrm{Sf}\,x\,v\,t$
hD0 $\mathrm{KE}({x},{v},{0}) - \mathrm{Sf}\,x\,v\,0 = 0$
hTkk $2 \cdot \pi / \hbar \cdot ({\href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-kgstress}{T({x})}})({v},{v}) = (-(2 \cdot \pi \cdot \int (\theta : \mathbb{R}), (\mathrm{starRingEnd}\,\mathbb{C})\,(\mathrm{ff}\,x\,v\,\theta) \cdot \mathrm{ff}^{\prime}\,x\,v\,\theta)).\mathrm{im}$
hWarea $\dot{A}({x},{v}) = -\sum_{\nu} W\,x\,v\,x\,\nu \cdot \partial_{{\nu}}({\lambda y \mapsto \href{/browser/qiqth-raychaudhuri#d-qiqth-curvature-expansion}{\theta({y})}})({x})$

and

hKG $\href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-boxfield}{(\Box {\varphi})({x})} = {m}^{2} \cdot \varphi\,x$

plus 25 routine conditions (18 regularity, 7 setup) — full list in the per-track PDF.

`qiqt_gr_freefield_localized`

Stage 1 (T3-3): hbridge discharged. The free-field QIQT→GR capstone with the heat coefficient FIXED to the boost flux kd x v := (2π/ℏ)·BL(kgStress) v and the modular-localization hypothesis hbridge DERIVED internally from freeField_oneParticle_hFlux (the axiom-free +2π one-particle Bisognano–Wichmann machinery) given hTkk. The thermodynamic premise hK now reads HasDerivAt (KE x v) ((2π/ℏ)·T_kk) 0 — the genuine Clausius statement that the heat-functional rate IS the boost-energy flux (correctly kept labelled). Of the Gap-2 localization map only hTkk (Stage 3) and the focusing identity hFocus (Stage 2) survive as inputs. Axiom-free.

qiqt_gr_freefield_localized · spine — there is $\Lambda$ such that

assuming

when $\href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0$ :

hbound $\text{for }t\text{ near }0,\; \mathrm{Sf}\,x\,v\,t \le \eta \cdot A({x},{v},{t})$
hsat $\mathrm{Sf}\,x\,v\,0 = \eta \cdot A({x},{v},{0})$
hDnn $\forall (t : \mathbb{R}), 0 \le \mathrm{KE}({x},{v},{t}) - \mathrm{Sf}\,x\,v\,t$
hD0 $\mathrm{KE}({x},{v},{0}) - \mathrm{Sf}\,x\,v\,0 = 0$
hTkk $2 \cdot \pi / \hbar \cdot ({\href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-kgstress}{T({x})}})({v},{v}) = (-(2 \cdot \pi \cdot \int (\theta : \mathbb{R}), (\mathrm{starRingEnd}\,\mathbb{C})\,(\mathrm{ff}\,x\,v\,\theta) \cdot \mathrm{ff}^{\prime}\,x\,v\,\theta)).\mathrm{im}$
hFocus $\dot{A}({x},{v}) = ({\lambda i j \mapsto \href{/browser/qiqth-curvature#d-qiqth-curvature-ricci}{R_{{i}{j}}({x})}})({v},{v})$

and

hKG $\href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-boxfield}{(\Box {\varphi})({x})} = {m}^{2} \cdot \varphi\,x$

plus 21 routine conditions (18 regularity, 3 setup) — full list in the per-track PDF.

`qiqt_gr_freefield`

THE FREE-FIELD QIQT→GR CAPSTONE. Einstein’s equations for the explicit free Klein–Gordon field, with the wedge-KMS modular flux supplied entirely by the axiom-free +2π one-particle Bisognano–Wichmann machinery — NOT a labelled WedgeKMSFlux_complete bundle. Identical to qiqt_gr_explicit_kg (geometry hC/hric_symm/hreg, matter conserv, and hT_symm all discharged internally for kgStress), but the modular input is the per-null-generator localization datum (mw, f, f', …, hTkk, hbridge) feeding freeField_kd_conclusion. …

qiqt_gr_freefield · spine — there is $\Lambda$ such that

assuming

when $\href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0$ :

hbound $\text{for }t\text{ near }0,\; \mathrm{Sf}\,x\,v\,t \le \eta \cdot A({x},{v},{t})$
hsat $\mathrm{Sf}\,x\,v\,0 = \eta \cdot A({x},{v},{0})$
hDnn $\forall (t : \mathbb{R}), 0 \le \mathrm{KE}({x},{v},{t}) - \mathrm{Sf}\,x\,v\,t$
hD0 $\mathrm{KE}({x},{v},{0}) - \mathrm{Sf}\,x\,v\,0 = 0$
hTkk $2 \cdot \pi / \hbar \cdot ({\href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-kgstress}{T({x})}})({v},{v}) = (-(2 \cdot \pi \cdot \int (\theta : \mathbb{R}), (\mathrm{starRingEnd}\,\mathbb{C})\,(\mathrm{ff}\,x\,v\,\theta) \cdot \mathrm{ff}^{\prime}\,x\,v\,\theta)).\mathrm{im}$
hbridge $({\lambda t \mapsto \langle {\mathrm{toLp}\,(\mathrm{ff}\,x\,v)\,\cdots },{(\href{/browser/qiqth-standardsubspacemodularflow#d-qiqth-standardsubspacemodular-modunitary}{\Delta}\,(\href{/browser/qiqth-fock-boostkms#d-qiqth-fock-boostkms-nicewedgestandardsubspace}{\mathcal{K}}\,(\mathrm{mw}\,x\,v)\,\cdots \,\cdots )\,t)\,(\mathrm{toLp}\,(\mathrm{ff}\,x\,v)\,\cdots )}\rangle})'({0})={i \cdot (\dot{K}({x},{v}))}$
hFocus $\dot{A}({x},{v}) = ({\lambda i j \mapsto \href{/browser/qiqth-curvature#d-qiqth-curvature-ricci}{R_{{i}{j}}({x})}})({v},{v})$

and

hKG $\href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-boxfield}{(\Box {\varphi})({x})} = {m}^{2} \cdot \varphi\,x$

plus 21 routine conditions (18 regularity, 3 setup) — full list in the per-track PDF.

`qiqt_gr_explicit_kg`

THE QIQT→GR EINSTEIN EQUATIONS FOR THE EXPLICIT FREE KLEIN–GORDON FIELD, axiom-free. Specialising the abstract qiqt_gr_from_wedge_kms_complete to T = kgStress (the concrete KG stress tensor): the matter-conservation input conserv is discharged INTERNALLY (kg_conserv_of_contDiff, from ContDiff smoothness of φ, g, gi + the equation of motion □φ = m²φ), and the stress-tensor symmetry is proved from metric symmetry. …

qiqt_gr_explicit_kg · spine — there is $\Lambda$ such that

assuming

when $\href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0$ :

hbound $\text{for }t\text{ near }0,\; \mathrm{Sf}\,x\,v\,t \le \eta \cdot A({x},{v},{t})$
hsat $\mathrm{Sf}\,x\,v\,0 = \eta \cdot A({x},{v},{0})$
hDnn $\forall (t : \mathbb{R}), 0 \le \mathrm{KE}({x},{v},{t}) - \mathrm{Sf}\,x\,v\,t$
hD0 $\mathrm{KE}({x},{v},{0}) - \mathrm{Sf}\,x\,v\,0 = 0$
hFocus $\dot{A}({x},{v}) = ({\lambda i j \mapsto \href{/browser/qiqth-curvature#d-qiqth-curvature-ricci}{R_{{i}{j}}({x})}})({v},{v})$

and

hKG $\href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-boxfield}{(\Box {\varphi})({x})} = {m}^{2} \cdot \varphi\,x$
hKMS $\href{/browser/qiqth-wedgekmstogr#d-qiqth-wedgekmstogr-wedgekmsflux-complete}{\mathrm{WedgeKMSFlux\_complete}}\,g\,(\href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-kgstress}{\mathrm{kgStress}}\,m\,\varphi\,g\,\mathrm{gi})\,\mathrm{kd}\,\hbar$

plus 15 routine conditions (12 regularity, 3 setup) — full list in the per-track PDF.

`qiqt_gr_from_flux_complete`

THE GOAL THEOREM, taking the per-generator flux EQUATION directly. Identical to qiqt_gr_from_wedge_kms_complete, but the modular input is the bare conclusion hflux : kd x v = (2π/ℏ)·BL(T x)v per null generator — exactly what qiqt_bekenstein_gives_gr consumes — instead of the WedgeKMSFlux_complete (−2π/wedgeGenSet) bundle. This is the convention-agnostic GR entry point: the bundle supplies hflux via hFlux_of_wedgeKMS_complete, and the free-field +2π route supplies it via freeField_component_hFlux — both land here. Axiom-free, no sorry.

qiqt_gr_from_flux_complete · spine — there is $\Lambda$ such that

$a \cdot T_{{\mu}{\nu}}({x}) = \href{/browser/qiqth-curvature#d-qiqth-curvature-einsteintensor}{G_{{\mu}{\nu}}({x})} + \Lambda \cdot g_{{\mu}{\nu}}({x})$

assuming

when $\href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0$ :

hbound $\text{for }t\text{ near }0,\; \mathrm{Sf}\,x\,v\,t \le \eta \cdot A({x},{v},{t})$
hsat $\mathrm{Sf}\,x\,v\,0 = \eta \cdot A({x},{v},{0})$
hDnn $\forall (t : \mathbb{R}), 0 \le \mathrm{KE}({x},{v},{t}) - \mathrm{Sf}\,x\,v\,t$
hD0 $\mathrm{KE}({x},{v},{0}) - \mathrm{Sf}\,x\,v\,0 = 0$
hflux $\dot{K}({x},{v}) = 2 \cdot \pi / \hbar \cdot \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({T\,x})({v},{v})}$
hFocus $\dot{A}({x},{v}) = ({\lambda i j \mapsto \href{/browser/qiqth-curvature#d-qiqth-curvature-ricci}{R_{{i}{j}}({x})}})({v},{v})$

plus 17 routine conditions (14 regularity, 3 setup) — full list in the per-track PDF.

Einstein field equations from a finite-information bound (Jacobson route)

THE END-TO-END THEOREM — QIQT-H + (cited Bisognano–Wichmann & Raychaudhuri) ⇒ the Einstein field equations. Assembles the whole chain into one theorem. Along each local null generator (x, v) (with v metric-null), QIQT-H’s content — the capacity bound S ≤ η·A (shannon_le_log_card), saturation at the reference (shannon_uniform_eq_log_card), and relative-entropy positivity (Klein, relEntropy_nonneg) — DERIVES the differential area law / modular relation, which with the two cited inputs (hFlux = Bisognano–Wichmann boost flux, hFocus = Raychaudhuri focusing) gives Jacobson’s per-null premise; jacobson_einstein_equation_of_state then yields a·T = G + Λ·g with genuine Einstein tensor and constant Λ. …

qiqt_bekenstein_gives_gr · spine — there is $\Lambda$ such that

$a \cdot T_{{\mu}{\nu}}({x}) = \href{/browser/qiqth-curvature#d-qiqth-curvature-einsteintensor}{G_{{\mu}{\nu}}({x})} + \Lambda \cdot g_{{\mu}{\nu}}({x})$

assuming

when $\href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0$ :

hbound $\text{for }t\text{ near }0,\; S({x},{v},{t}) \le \eta \cdot A({x},{v},{t})$
hsat $S({x},{v},{0}) = \eta \cdot A({x},{v},{0})$
hDnn $\forall (t : \mathbb{R}), 0 \le \mathrm{KE}({x},{v},{t}) - S({x},{v},{t})$
hD0 $\mathrm{KE}({x},{v},{0}) - S({x},{v},{0}) = 0$
hFlux $\dot{K}({x},{v}) = 2 \cdot \pi / \hbar \cdot \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({T\,x})({v},{v})}$
hFocus $\dot{A}({x},{v}) = ({\lambda i j \mapsto \href{/browser/qiqth-curvature#d-qiqth-curvature-ricci}{R_{{i}{j}}({x})}})({v},{v})$

plus 17 routine conditions (14 regularity, 3 setup) — full list in the per-track PDF.

`oneParticleBW_niceWedge_unconditional`

THE free-field one-particle Bisognano–Wichmann — FULLY UNCONDITIONAL, axiom-free. For every mass m > 0 and every candidate boost representation V t = boostUnitary(2πt), the modular flow of the nice-core wedge standard subspace equals the boost: modUnitary S t = V t, with NO Reeh–Schlieder hypotheses whatsoever. BOTH analytic inputs are now discharged internally and unconditionally: niceWedgeSeparating_pos_mass (Pauli–Jordan symplectic non-degeneracy, via the KMS uniqueness argument) and niceWedgeCyclic_pos_mass (wedge-totality, via the Wiener–Tauberian theorem). …

oneParticleBW_niceWedge_unconditional · spine — we have

$\href{/browser/qiqth-standardsubspacemodularflow#d-qiqth-standardsubspacemodular-modunitary}{\Delta}\,(\href{/browser/qiqth-fock-boostkms#d-qiqth-fock-boostkms-nicewedgestandardsubspace}{\mathcal{K}}\,m\,\cdots \,\cdots )\,t = V\,t$

plus 2 routine conditions (2 regularity) — full list in the per-track PDF.

`freeField_oneParticle_hFlux`

The free-field one-particle hFlux, FULLY ASSEMBLED in the satisfiable +2π convention. For any smooth wedge state ξ = f.toLp and the nice-wedge standard subspace S, the modular-energy derivative is i·(2π/ℏ)·T_kk: HasDerivAt (t ↦ ⟪ξ, modUnitary S t ξ⟫) (i·(2π/ℏ·T_kk)) 0, with EVERYTHING operator/analytic discharged axiom-free — the Bisognano–Wichmann identification (oneParticleBW_niceWedge_unconditional) and the boost-charge derivative (hasDerivAt_inner_boostUnitary_imaginary_pos) are both supplied internally. The ONLY labelled input is the single scalar physics identification hTkk : (2π/ℏ)·T_kk = (−(2π·∫ conj(f)·f')).im (the conserved boost Killing charge = stress-tensor flux, in the +2π orientation). …

freeField_oneParticle_hFlux · spine — we have

$({\lambda t \mapsto \langle {\mathrm{toLp}\,f\,\mathrm{hf2}},{(\href{/browser/qiqth-standardsubspacemodularflow#d-qiqth-standardsubspacemodular-modunitary}{\Delta}\,(\href{/browser/qiqth-fock-boostkms#d-qiqth-fock-boostkms-nicewedgestandardsubspace}{\mathcal{K}}\,m\,\cdots \,\cdots )\,t)\,(\mathrm{toLp}\,f\,\mathrm{hf2})}\rangle})'({0})={i \cdot (2 \cdot \pi / \hbar \cdot T_{kk})}$

assuming

hTkk $2 \cdot \pi / \hbar \cdot T_{kk} = (-(2 \cdot \pi \cdot \int (\theta : \mathbb{R}), (\mathrm{starRingEnd}\,\mathbb{C})\,(f\,\theta) \cdot f^{\prime}\,\theta)).\mathrm{im}$

plus 6 routine conditions (6 regularity) — full list in the per-track PDF.

`freeField_component_hFlux`

The free-field per-generator flux equation kd = (2π/ℏ)·T_kk (the +2π/nice-wedge analog of component_hFlux_of_wedgeKMS_complete). For the nice-wedge standard subspace S and smooth wedge state ξ = f.toLp, given (i) hbridge — that the abstract per-generator modular-energy coefficient kd IS the derivative of t ↦ ⟪ξ, modUnitary S t ξ⟫ — and (ii) hTkk — the localization identification of the horizon stress component T_kk with the mode’s rapidity stress flux — derivative uniqueness pins kd = (2π/ℏ)·T_kk. …

freeField_component_hFlux · spine — we have

$\mathrm{kd} = 2 \cdot \pi / \hbar \cdot T_{kk}$

assuming

hTkk $2 \cdot \pi / \hbar \cdot T_{kk} = (-(2 \cdot \pi \cdot \int (\theta : \mathbb{R}), (\mathrm{starRingEnd}\,\mathbb{C})\,(f\,\theta) \cdot f^{\prime}\,\theta)).\mathrm{im}$
hbridge $({\lambda t \mapsto \langle {\mathrm{toLp}\,f\,\mathrm{hf2}},{(\href{/browser/qiqth-standardsubspacemodularflow#d-qiqth-standardsubspacemodular-modunitary}{\Delta}\,(\href{/browser/qiqth-fock-boostkms#d-qiqth-fock-boostkms-nicewedgestandardsubspace}{\mathcal{K}}\,m\,\cdots \,\cdots )\,t)\,(\mathrm{toLp}\,f\,\mathrm{hf2})}\rangle})'({0})={i \cdot \mathrm{kd}}$

plus 6 routine conditions (6 regularity) — full list in the per-track PDF.