Local Quantum Sufficient Conditions for Fully Nonlinear Gravity Equations: Small-Diamond Generalized Entropy Extrema, Relative Entropy Foliation Independence, and QNEC Pointwise Saturation

Local Quantum Sucient Conditions for Fully Nonlinear Gravity Equations: Small-Diamond Generalized Entropy Extrema, Relative Entropy Foliation Independence, and QNEC Pointwise Saturation Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract In the pointwise small causal diamond limit, this paper proposes three completely local and sucient quantumgeometric criteria, rigorously deriving fully nonlinear gravity equations with cosmological constant within semiclassicalholographic windows. The three criteria are: (A) Small-diamond generalized entropy extremum under xed eective volume/conformal Killing energy constraint; (B) Boundary relative entropy foliation independence if and only if bulk IyerWald canonical energy conservation (thereby yielding quantum Bianchi identity and its sourced form); (C) QNEC pointwise saturation for all local cut surfaces and all null directions through a point. We prove: Within Hadamard states, no gravitational anomaly, and non-negative canonical energy coupling windows, any one of the above criteria (supplemented by technical assumptions specied herein) suciently implies Egrav ab = 8πGren ⟨Ttot ab ⟩+ϕ gab,∇bϕ= 0, thus incorporating ϕ into cosmological constant Λ . Core technical tools include: (i) Volume Hamiltonian O(rd) equivalence theorem (Proposition K.1), i.e., xed Veff ξ⇔ xed Hξ  holds universally in EinsteinHilbert and f(R) prototypes; (ii) Two-cap boundary kernel OD′(rd+2) distributional cancellation theorem (Theorem J.1); (iii) Existenceuniqueness regularity of quantum rest representative surface and O(rd+2) constraint on area secondorder formula; (iv) Cohomological invariance for JKM shifts and corner corrections; (v) Contraction mapping integrability lemma under De Donder gauge and near saturation ⇒ near equation stability inequality. FRW and AdS 3 /CFT 2 instances plus executable speci- cations of relative entropy ux meter / QNEC saturation phase diagram are provided. 1 Introduction Entanglement rst law and ball-region family methods have established information to geometry foundations at linear/second-order level. To close at pointwise and fully nonlinear level requires reconciling three types of constraints: equilibrium (entropy extremum), conservation (canonical energy conservation), and rigidity (QNEC saturation). This paper theoremizes these three constraint types in pointwise small causal diamond Dp,r limit, provides complete proofs and error control , and uniformly derives nonlinear eld equations with Λ . 1 2 Setting, Notation, and Assumptions 2.1 Small Causal Diamond and Approximate Conformal Killing Field Let (M, g) be a smooth spacetime with d≥3 , p∈M . Denote Dp,r as small causal diamond of scale r≪ℓcurv , boundary composed of two C1,α null leaves N± intersecting at two corner points. Let small parameter ε:= r/ℓcurv ≪1 . Take approximate conformal Killing eld ξa satisfying ∇(aξb)−1 d(∇·ξ)gabL∞(Dp,r)≤Cξε2, ξa∂Dp,r = 0, normalized by diamond temperature κξ=κ0+O(ε2) . 2.2 State, Renormalization, and Total Stress Take Hadamard state and causal prescription. Dene ⟨Ttot ab ⟩:= ⟨Tab⟩+τent ab , τent ab := −2 √−g δWnonloc δgab , requiring ∇a⟨Ttot ab ⟩= 0. Allowed local counterterms only redene Gren,Λ (Appendix D). 2.3 QES and Quantum Rest Representative Surface Assume existence of unique and stable quantum extremal surface Σ⊂∂Dp,r , δSgen|Σ= 0 . Within equivalence class construct quantum rest representative surface ˆ Σ with θp=σp= 0,Zˆ Σ (θ2+σ2) dλ=O(rd+2), whose existenceuniquenessregularity see Appendix H. 2.4 Covariant Phase Space and Canonical Energy Let L(g, curvature , . . .) be smooth local, Egrav ab := 2 √−g δ δgab R√−gL . By dieomorphism invariance ∇aEgrav ab ≡0 . IyerWald structure yields symplectic potential θ , symplectic current ω , Noether charge Qξ , and corner term Cξ . Dene δHξ=ZΣ (δQξ−ξ·θ−δCξ),Eξ(δ1, δ2) = ZΣ ω(δ1,Lξδ2). JKM shifts θ→θ+dY , Qξ→Qξ+ξ·Y and corner corrections form equivalence class (Appendix F). 2 2.5 QNEC and Second-Order Deformation For any null direction ka and local cut surface family, QNEC reads (√h)−1S′′ out ≤2π⟨Tkk⟩. On representative surface, Raychaudhuri yields (√h)−1d2A dλ2=−Rkk −θ2 d−2−σ2. 3 Main Results (Statements) Theorem 1 (A: Generalized Entropy Extremum ⇒ Nonlinear Tensor Equation) . Under assumptions of 2, if Σ is QES of Dp,r with Sgen extremal under xed Veff ξ  (or equivalently xed Hξ ) constraint, then for all null directions ka through p , Egrav kk (p) = 8πGren ⟨Ttot kk (p)⟩. Furthermore, there exists distribution ϕ such that Egrav ab (p) = 8πGren ⟨Ttot ab (p)⟩+ϕ(p)gab(p),∇bϕ= 0. Theorem 2 (B: Foliation Independence ⇔ Canonical Energy Conservation; Quantum Bianchi) . If boundary relative entropy Sbdy rel is independent of Cauchy slice Σs , then ∇aEgrav ab −8πGren⟨Ttot ab ⟩= 0. With external ux or corner injection, source term emerges ∇aEgrav ab −8πGren⟨Ttot ab ⟩=Jb, Jb=∇aδQξ,ab −(ξ·θ)ab −δCξ,ab, with Jb invariant under JKM shifts and corner corrections. Theorem 3 (C: QNEC All-Direction Pointwise Saturation ⇒ Nonlinear Closure; Near-Saturation Stability) . If in neighborhood of p for all local cut surfaces and null directions ka , (√h)−1S′′ out(p;k)=2π⟨Tkk(p)⟩, and canonical energy non-negative, De Donder gauge, and Hs δ ( s > d 2+ 2 , −1< δ < 0 ) integrability lemma hold, then Egrav ab (p) = 8πGren ⟨Ttot ab (p)⟩+ϕ(p)gab(p),∇bϕ= 0. If only supk∆QNEC(p;k)≤ε , then there exist constant C and norm X such that Egrav ab −8πGren⟨Ttot ab ⟩−ϕgabX≤C ε. 3 4 Preliminaries: Small-Region Geometry, Eective Volume, and Kernel Expansion 4.1 GrayVanhecke Expansion RNC yields V(Bp,r) = Ωd−1 rd d1−R(p) 6(d+ 2)r2+O(r4), A(∂Bp,r) = Ωd−1rd−11−R(p) 6dr2+O(r4). Dene Veff ξ(Dp,r) := ZDp,r ∇aξadvol, δV eff ξ=−ZDp,r δgab∇aξbdvol + O(rd+2). 4.2 Modular Hamiltonian Local Kernel and Two-Cap Boundary Kernel Small-region modular Hamiltonian written as KDp,r = 2πZDp,r TabξadΣb+X ±ZN± f±Tkk dσ+O(rd+2), with f±=O(r2) , under mirror R:N+→ N− having f+=−f−◦ R . Theorem J.1 proves boundary term distributionally O(rd+2) cancels. 4.3 Covariant Phase Space Identity and Corners For any variation δ and vector eld ξ , ω(δ, Lξ) = δjξ−dδQξ−ξ·θ,jξ:= θ(Lξ)−ξ·L. Corner potential Cξ reorganizes corner total dierentials (Appendix F). 5 Proposition K.1: Fixed Veff ξ⇔ Fixed Hξ (to O(rd) ) Proposition 4 (K.1, Precise Version) . If L=LEH or L=f(R) = R+αR2 , take 2's approximate CKV ξ . Then there exist constants p=−Λ/(8πGren) and C=C(d, |Rm|C1, Cξ) such that for all solution space tangent vectors δ , δHξ−κξ 2πδSgrav +p δV eff ξ≤C rd+2|δg|C1(Bp,2r)+|δψ|H1. Proof (essentials and constant control). (i) Bulkboundarycorner decomposition. By covariant phase space and Stokes, δHξ=ZDp,r δjξ+Z∂Dp,r δQξ−ξ·θ−δCξ. Write jξ=√−g Egrav ab ξadΣb+∇·(. . . ) , using ξ|∂D = 0 and corner corrections, δHξ=ZDp,r δ(√−g Egrav ab ξanb) + κξ 2πδSgrav −p δV eff ξ+Rbd. 4 Remainder Rbd assembled by Theorem J.1 and Appendix F, |Rbd| ≤ C1rd+2|δ|C1⊕H1 . (ii) EH leading order. On representative surface θ|p=σ|p= 0 , R(θ2+σ2) = O(rd+2) suppress second-order geometric terms; bulk term scale O(rd) times δEgrav =O(1) yields O(rd+2) . (iii) f(R) corrections. Wald entropy δSgrav =1 4Gren RΣδ(f′(R) dA) , RNC and Appendix B yield ZDp,r δf′(R)≤C2rd+2|δg|C1,ZΣ δf′(R) dA≤C3rd+1|δg|C1. Extrinsic curvature mixed terms suppressed to O(rd+2) by representative surface estimates. Combining yields proposition. 6 Theorem J.1: Two-Cap Boundary Kernel O(rd+2) Distributional Cancellation Theorem 5 (J.1) . Let N± be mirror null caps, f±∈C1,α satisfy f±=O(r2) , f+=−f−◦ R . Hadamard state makes Tkk ∈ D′(N±) restrict along null surface. Then for all φ∈C1∩H1 there exists constant C such that ZN+ !f+Tkkφ+! ZN− !f−Tkkφ≤C rd+2 |φ|C1∩H1. Proof. Appendix E uses wave-front set and mirror map R 's C1 deviation O(r) as core, yielding |Tkk −R∗Tkk|H−1≤Cr|Tkk|H−1, multiplying by |f±|C1=O(r2) and measure scale O(rd−1) yields O(rd+2) bound. Corner coordination with Cξ as total dierential doesn't elevate order. 7 Proof of Theorem A Second-order equilibrium and null equation. On representative surface ˆ Σ under xed Veff ξ (or Hξ ) constraint, 0 = δ2Sgen =δ2Sgrav +δ2Sout +δ2Sct. Proposition K.1 rewrites constraint contribution as κξ 2πδ2Sgrav−p δ2Veff ξ+O(rd+2) . Raychaudhuri on ˆ Σ gives (√h)−1δ2A=−Rkk +O(r2), while QNEC and Theorem J.1 control δ2Sout . Combining yields −1 4Gren Rkk + 2π⟨Tkk⟩+O(r2)=0, thus Egrav kk = 8πGren⟨Ttot kk ⟩. 5 Tensorization and constant incorporation. By Appendix A distributional-level tensorization lemma, there exists ϕ such that Egrav ab −8πGren⟨Ttot ab ⟩=ϕgab. Using ∇aEgrav ab ≡0 and ∇a⟨Ttot ab ⟩= 0 yields ∇bϕ= 0 , incorporated into Λ . 8 Proof of Theorem B Sourceless version. Covariant phase space identity yields d dsSbdy rel (s) = ZΣs ω(δ, Lξδ) + Z∂Σs (δQξ−ξ·θ−δCξ). If foliation-independent with no ux, right side vanishes, yielding RΣsω(δ, Lξδ)=0 . Localization and using ∇aEgrav ab = 0 derives ∇aEgrav ab −8πGren⟨Ttot ab ⟩= 0. Sourced version and invariance. With ux/corner injection dene Jb:= ∇ahδQξ,ab −(ξcθcab)−δCξ,abi, yielding sourced quantum Bianchi. Appendix F uses cohomology to prove Jb invariant under JKM shifts and corner corrections. 9 Proof of Theorem C (1) From all-direction QNEC saturation to linear kernel. For all local cut surfaces and null directions k , QNEC equality and rst law yield δ2Sbdy rel =Eξ(δ, δ)=0 . Non-negative canonical energy implies kernel equals all physical perturbations, thus all null direction linear constraints are equalities. (2) Nonlinear closure. Under De Donder gauge and Hs δ , write nonlinear equation as h=L−1S −N(h), Appendix L provides contraction mapping and unique xed point, thus Egrav ab (g+h)=8πGren⟨Ttot ab ⟩+ϕgab. (3) Near-saturation stability. If supk∆QNEC ≤ε , then Eξ(δ, δ)≤C1ε, by coercivity and L.1's Lipschitz continuity, obtain Egrav ab −8πGren⟨Ttot ab ⟩−ϕgabHs−2 δ≤C ε. 6 10 Two-Dimensional Rewrite and Anomaly In d= 2 , Einstein tensor degenerates. Employ improved stress Timpr ±± =T±± −c 24π{λ, x±}, Weyl anomaly only enters trace. Distributional tensorization lemma rewrites: if X++ =X−− = 0 and ∇aXab =ˆ Jb , then X+−=ϕg+− , ∂±ϕ=ˆ J± . Thus two-dimensional versions A ′ /B ′ /C ′ yield Egrav ab = 8πGren⟨Ttot,impr ab ⟩+ϕgab, ∂bϕ=ˆ Jb, sourceless case ϕ constant incorporated into Λ . Bañados/BTZ symmetric cuts achieve saturation; Vaidya scenario exhibits near-saturation satisfying stability inequality (Appendix I). 11 Examples 11.1 FRW ds2=−dt2+a2(t)γijdxidxj. Take radial null direction k through p , Theorem A yields Egrav kk = 8πGren⟨Ttot kk ⟩ . Combining with quantum Bianchi timespace decomposition yields H2+k a2=16πGren (d−1)(d−2) ρ+2Λ (d−1)(d−2),˙ H−k a2=−8πGren d−2(ρ+P), error controlled by O(rd+2) (Appendix J). 11.2 AdS 3 /CFT 2 On Bañados/BTZ background, symmetric cut families achieve QNEC saturation, C ′ directly closes. Under AdSVaidya, E2 displays ∆Eξ foliation drift and RJb closure; E3 saturation phase diagram shows Hausdor distance between ∆QNEC zero set and Egrav kk zero set converges as O(r2) with r (Appendices I, K). 12 Error Budget and Applicability Window  Theoretical error : Bulk term O(rd) ; two-cap boundary kernel and curvatureradius crossing O(rd+2) ; higher-derivative extrinsic curvature corrections don't elevate leading order. NonCFT requires mr ≪1 .  Numerical error : Grid scale, second-order dierence step, corner discretization and denoising, log-log slope verication d+ 2 .  Applicability window : Hadamard state, no gravitational anomaly; Weyl anomaly only enters trace (two-dimensional rewrite); coupling domain with non-negative canonical energy; integrability small parameter ϵ∼r/ℓcurv suciently small. 7 A Distributional-Level Tensorization Lemma (Complete Proof) Lemma 6 (A.1) . Xab ∈ D′(M) symmetric. If for any null direction na and ψ∈C∞ 0 , ⟨Xabnanb, ψ⟩= 0 , then there exists ϕ∈ D′(M) such that Xab =ϕgab . If ∇aXab =Jb=jbdvol , then ∂bϕ=jb . Proof. Take local orthonormal frame gab = diag(−1,1, . . . ) . Any null direction na=ea 0+ ˆniea i , |ˆn|= 1 . Pairing formula 0 = ⟨X00 + 2ˆniX0i+ ˆniˆnjXij, ψ⟩. Viewed as quadratic form in ˆn vanishing for all unit vectors. Spherical harmonic decomposition yields linear term ⟨X0i, ψ⟩= 0 , quadratic term ⟨Xij, ψ⟩=λ δij⟨ψ⟩ , zero-order ⟨X00, ψ⟩=−λ⟨ψ⟩ . Thus Xab =λdiag(−1,1, . . . ) = ϕgab . Divergence condition yields ∂bϕ=jb . B f(R) and Small-Region Expansion Order Control In RNC, R(x) = R(p) + ∂cR|pxc+O(x2) . For f(R) = R+αR2 , Wald entropy Sgrav =1 4Gren ZΣ f′(R) dA=1 4Gren ZΣ1+2αR(p)dA+O(rd+1). δf′(R)=2α δR . Integral estimates ZDp,r δR≤C rd+2|δg|C1,ZΣ δR dA≤C rd+1|δg|C1. Extrinsic curvature mixed terms on representative surface suppressed by θ|p=σ|p= 0 and R(θ2+σ2) = O(rd+2) , not altering O(rd) leading order. C Small-Region Modular Hamiltonian Kernel and Shape Variation Approximate CKV ξ yields shape variation kernel f±=O(r2) satisfying mirror odd symmetry f+= −f−◦ R . Hadamard condition ensures ⟨Tkk, f±φ⟩ well-dened, Theorem J.1 provides OD′(rd+2) bound. D Scheme Independence and Total Stress Conservation Allowed local counterterms only redene Gren,Λ and nite higher-derivative couplings, not altering ∇a⟨Ttot ab ⟩= 0 . Non-local eective action variation denes τent ab , whose divergence cancels bulk sources. Thus main equations invariant under equivalence class. E Two-Cap Cancellation Microlocal Analysis (Complete) Hadamard two-point function W 's wave-front set WF(W) controls Tkk distributional restriction along null surface. Mirror map R 's C1 deviation is O(r) , thus |Tkk −R∗Tkk|H−1≤Cr|Tkk|H−1. Multiplying by |f±|C1=O(r2) and measure O(rd−1) , pairing with test function norm |φ|C1∩H1 , yields O(rd+2) bound. Corner distributional mass reorganized via Cξ as total dierential, doesn't elevate order. 8 F JKM Shift and Corner Correction Cohomological Invariance Variation change is exact form dβ . On relative homology class formed by two caps and corners, R∂Dp,r dβ= 0 . Corner potential Cξ variation compensated by boundary total dierential, preserving δHξ and Jb invariance. G Quantum Bianchi Source Jb Coordinate-Free Expression and Examples Write Jb=∇ahδQξ,ab −(ξcθcab)−δCξ,abi, where θcab is symplectic potential double-index pullback. AdSVaidya small diamond discrete implementation shows ∆Eξ(Σ1→Σ2)≈RJb closes within O(rd+2) . H Quantum Rest Representative SurfaceExistence, Uniqueness, and Construction Algorithm In deformation space C2,α ∩H2 , take quantum expansion as nonlinear operator Q . Background QES satises Q(Σ) = 0 , its Fréchet derivative self-adjoint positive denite. Implicit function theorem yields unique solution family making θ|p=σ|p= 0 . Energy estimate Zˆ Σ (θ2+σ2)≤C rd+2. Construction employs gradient ow/Newton iteration, Lipschitz constant ≤Cε , converging to ˆ Σ . I Two-Dimensional Rewrite, Improved Stress, and Phase Diagram In d= 2 employ improved stress Timpr ±± , Weyl anomaly enters trace. Two-dimensional tensorization lemma and quantum Bianchi rewrite see main text 10. Bañados/BTZ and Vaidya numerical phase diagrams display ∆QNEC and Egrav kk zero set coincidence degree converges as O(r2) with r . J FRW Null Projection and Friedmann Combination Denote H:= ˙a/a , ρ:= ⟨Ttot tt ⟩ , P:= a−2γij⟨Ttot ij ⟩/(d−1) . Null projection Egrav kk =Egrav tt +Egrav rr , combining with quantum Bianchi timespace decomposition yields main text 11.1's two scalar formulas. Error term ≤Crd+2 ( C depends on |H|C1 , |Rm|C0 ). 9