Null--Modular Double Cover and Overlapping Causal Diamond Chains: Total-Order Approximation Bridge for Quadratic Form Localization, Inclusion--Exclusion--Markov Splicing, and Parity Threshold for Distributional Scattering Calibration

NullModular Double Cover and Overlapping Causal Diamond Chains: Total-Order Approximation Bridge for Quadratic Form Localization, InclusionExclusionMarkov Splicing, and Parity Threshold for Distributional Scattering Calibration Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract We propose NullModular double cover carried by zero-measure boundaries of causal diamonds, decomposing modular Hamiltonians into local energy ux integrals on two null sheets in vacuum quadratic form sense. Through a total-order approximation bridge lemma , general diamonds are reduced to monotonic half-space family limits on the same zero-measure hyperplane, with quadratic form closedness and dominated convergence ensuring limit independence of approximation paths. We establish modular Hamiltonian inclusionexclusion identities and Markov splicing for overlapping causal diamond chains ; for non-totally-ordered cuts, we introduce Markov gap line density quantitatively characterizing failure with comparison inequalities versus stratication degree. On the scattering side, under distributional Birman KrenFriedelLloydWignerSmith calibration , we introduce windowed readout , providing visible constants and threshold inequalities via Toeplitz/Berezin compression, EulerMaclaurin and Poisson disciplines, thereby proving chain Z2 parity threshold stability with robustness conditions for weakly non-unitary perturbations . On the geometric side, half-sided modular inclusion constitutes one-parameter semigroup for chain advancement; in holographic limits, JLMS equality lifts boundary inclusionexclusionMarkov to bulk entanglement wedge normal modular ow, with dimensional upper bounds for subleading 1/N corrections. Finally, we explicitly compute GHY joint terms and Z2 ledger consistency of square-root splicing classes in minimal models in 1+1 and 2+1 dimensions, providing reproducible experimental parameter tables and verication checklists . 1 Introduction & Historical Context TomitaTakesaki modular theory endows von Neumann algebravector state pairs (A,Ω) with modular groups ∆it and modular conjugation J . BisognanoWichmann property geometrizes modular ow as Lorentz boosts on wedge regions. For zero-measure geometry, local modular Hamiltonians on half-spaces and their smooth deformations satisfy vacuum QNEC saturation, and vacuum Markovianity on light-cones/light-fronts with strong subadditivity saturation form solid foundations. Algebraically, half-sided modular inclusion (HSMI) provides algebraic skeleton of inclusionone-parameter semigroupBorchers commutation relations. Holographically, JLMS equality identies boundary and bulk relative entropies at leading order in large N . On scattering side, BirmanKren identies determinant phase with spectral shift function, FriedelLloyd and 1 WignerSmith unify density-of-states dierence with group delay trace; Toeplitz/Berezin compression with Szeg®/trace formulas provide operatorsymbol tools for windowed readout; Euler Maclaurin and Poisson disciplines yield exponential or algebraic decay error upper bounds. This paper systematically constructs integrated theory of NullModular double cover and overlapping diamond chains within this framework. 2 Model & Assumptions 2.1 Quadratic Form Framework and Natural Domain Take Minkowski spacetime R1,d−1 ( d≥2 ). Let D0 be dense domain of energy-bounded vectors in vacuum. Notation and measure convention : Zero-measure boundary decomposes into two sheets e E=E+⊔E− ; notation REσ(···)dλ dd−2x⊥ refers to standard measure integration on this sheet by ane parameter λ and transverse coordinate x⊥ . Assume for any region R there exists lower bounded closed quadratic form kR[ψ] := X σ=±ZEσ g(R) σ(λ, x⊥)⟨ψ, Tσσ(λ, x⊥)ψ⟩dλ dd−2x⊥, ψ ∈ D0, thus there exists self-adjoint operator KR satisfying ⟨ψ, KRψ⟩=kR[ψ] . CFT's spherical regions/wedges and their conformal images yield exact geometric equalities. Let kR have lower bound aR∈R , i.e., kR[ψ]≥aR|ψ|2 . Take any cR>−aR , dene shifted graph norm |ψ|2 kR,cR:= |ψ|2+kR[ψ] + cR|ψ|2, then (D(kR),|·|kR,cR) is complete, compatible with representation theorem for self-adjoint operator KR . 2.2 Zero-Measure Localization and QNEC In zero-measure half-space RV={u= 0, v ≥V(x⊥)} ( V∈C2 ), KV= 2πZdd−2x⊥Z∞ V(x⊥) (v−V)Tvv(v, x⊥)dv holds as quadratic form identity; its second-order variational kernel is 2π Tvv , consistent with vacuum QNEC saturation. 2.3 Double Cover and Splicing, Square-Root Cover and Ledger Zero-measure boundary decomposes into two sheets e E=E+⊔E− . Modular conjugation J exchanges two sheets and reverses orientation, modular group generates integrable ow along ane parameter λ in geometrizable cases. Seam splicing accounted by ϵi∈ {±1} . On scattering side introduce square-root cover P√S={(E, σ) : σ2= det S(E)} as Z2 principal bundle structure; splicing class of closed chain loops shares same Z2 ledger with joint term orientation signs. 2 2.4 ScatteringInformation Calibration and Windowing Unitary scattering matrix S(E) piecewise C2m with S(E)−I trace-class within energy band; dene Q(E) := −iS†∂ES, φ(E) := 1 2arg det S(E), ρrel(E) := 1 2πtr Q(E). Employ window function h∈ S(R) (e.g., Gaussian), or h∈C2m+1 c(R) with endpoint jets up to 2m order vanishing ( m≥1 ). In this case b h(ω) = O(|ω|−(2m+1)) . If h only piecewise C2m with compact support (endpoints allow corners, including KaiserBessel), adopt corner tail bound (at least O(|ω|−2) ), whereby Theorem G's Poisson aliasing series converges. Corresponding Toeplitz/Berezin compression and trace formulas follow error decomposition in 3.5, where endpoint remainder REM : for C∞ c windows take O(ℓ−(m−1)) ; for piecewise C2m compact support windows (including KaiserBessel) adopt corner version estimate (order generally drops to O(ℓ−1) ), incorporated into total error budget Eh(γ) . Additional assumption (Toeplitz commutator integrability) : On any examined energy band I , ∂ES(E)∈S2 and ZI|∂ES(E)|2dE < ∞ . Thus RT≤CTℓ−1/2RI|∂ES|2dE is bounded. Global convention (window and tail term) : Set RRh= 1 and h≥0 , scale hℓ(E) = ℓ−1h(E/ℓ) . Dene Rtail(ℓ, I, E0) := ZR\I(γ)|hℓ(E−E0)|dE ∈[0,1]. Note : In this case Rtail = 1 −RI(γ)hℓ(E−E0)dE . Notation convention (Poisson step size) : Denote ∆>0 as energy band segmentation/frequency sampling step size (grid spacing); in Poisson resummation estimate take ZI|RP|dE ≤ChX |q|≥1b h(2πq ℓ/∆) , consistently using this ∆ as in 3.5's identically named term. 2.5 Chain and Overlap, Algebraic Assumptions Chain {Dj} adjacent overlaps on same surface; for each transverse point x⊥ total-order cut is default assumption. Algebraically adopt standard assumptions of split property and strong additivity; HSMI as algebraic realization of chain advancement. 3 Main Results (Each Result Labeled with Signicance/Domain) 3.1 Double-Sheet Geometric Decomposition and Total-Order Approximation Bridge Theorem 1 (A: Double-Sheet Geometric Decomposition) . KD= 2πX σ=±ZEσ gσ(λ, x⊥)Tσσ(λ, x⊥)dλ dd−2x⊥, where T++ =Tvv , T−− =Tuu . In CFT spherical diamonds gσ(λ) = λ(1 −λ) . [ Quadratic form; domain: vacuum, CFT exact equality ] 3 Assumption 2 (A ′ : Null Energy Flux Uniform Integrability) . For any ψ∈ D0 and geometrically bounded monotonic approximation family {R± Vα} , there exists Hσ∈L1 loc(Eσ×Rd−2) such that g(α) σ(λ, x⊥)⟨ψ, Tσσ(λ, x⊥)ψ⟩≤Hσ(λ, x⊥) holds almost everywhere, and supαRKHσ<∞ for any compact set K ⊂ Eσ×Rd−2 . Lemma 3 (A: Ordered Cut Approximation) . There exists monotonic half-space family {R± Vα} along E± such that ⟨ψ, KDψ⟩= lim α→∞ X σ=± 2πZEσ g(α) σ⟨ψ, Tσσψ⟩, g(α) σ→gσ in L1 loc, and limit is independent of chosen ordered approximation. [ Quadratic form convergence; domain: vacuum, vacuum QNEC saturation ] Exclusion remark : Without BW/HSMI or boundary roughness breaking vacuum QNEC saturation, above decomposition may not hold. Assumption 4 (A ′′ : Quadratic Form Lower Bound and Closedness Threshold) . Assume all participating regions R have quadratic forms kR with uniform lower bound a∈R , i.e., kR[ψ]≥a|ψ|2 . Take any c > −a dening shifted graph norm |ψ|2 kR,c =|ψ|2+ (kR[ψ] + c|ψ|2) , then kR closed and D(kR) complete under |·|kR,c . Proposition 5 (A.1: Necessary and Sucient Condition for Limit Path Independence) . Under Assumptions A ′ and A ′′ , if along any two monotonic approximation families {RVα} , {Re Vβ} we have g(α)→g , eg(β)→g in L1 loc , then for each ψ∈ D0 , lim α→∞X σZg(α) σ⟨ψ, Tσσψ⟩= lim β→∞X σZeg(β) σ⟨ψ, Tσσψ⟩. Reason: Dominated convergence identies each approximation's limit with g ; closedness and lower bound yield quadratic form continuity, thus independent of approximation path. 3.2 InclusionExclusion and Closedness Theorem 6 (B: InclusionExclusion Identity) . For {RVi}N i=1 on same zero-measure surface, K∪iRVi= N X k=1 (−1)k−1X 1≤i1<···<ik≤N KRVi1∩···∩RVik . Derived from pointwise identity (v−miniVi)+=Pk≥1(−1)k−1P|I|=k(v−maxi∈IVi)+ . [ Quadratic form; domain: vacuum, Vi piecewise smooth ] Proposition 7 (B: Closedness) . Denote k:= k∪iRVi as container domain closed quadratic form, with lower bound a∈R . Take any c > −a . If ψn, ψ ∈ D(k)∩\ I=∅D(kRVI), ψn→ψ under shifted graph norm |·|k,c, then inclusionexclusion identity's both sides for quadratic form values on ψn converge simultaneously to values on ψ ; thus identity closes on above form domain. Where 4 |ψ|2 k,c := |ψ|2+k[ψ] + c|ψ|2. [ Quadratic form closedness ] Operational domain remark : Above closedness holds on common form domain D∗:= D(k)∩ TI=∅D(kRVI) ; for chain applications, taking Vi piecewise C1 with uniform Lipschitz constant ensures D∗ non-empty and dense. 3.3 Markov Splicing, Petz Recovery, and Non-Total-Order Gap Theorem 8 (C: Markov Splicing) . Under same-surface total order, vacuum satises I(Dj−1:Dj+1 |Dj)=0, KDj−1∪Dj+KDj∪Dj+1 −KDj−KDj−1∪Dj∪Dj+1 = 0. [ Information equivalence; domain: vacuum, split/strong additivity ] Theorem 9 (C ′ : Markov Gap for Non-Total-Order) . Denition (stratication degree) : Let V± i(x⊥) be thresholds on E± respectively, dene κ(x⊥) := #{(a, b) : a < b, (V+ a−V+ b)(V− a−V− b)<0}. Note : Under total-order cut κ≡0 . Thus ι monotonically non-decreasing in κ yields comparison inequality. To bound ι(v, x⊥) 's v domain, denote v−(x⊥) := min iV+ i(x⊥), v+(x⊥) := max iV+ i(x⊥), i.e., eective support interval endpoints covered by chain on E+ sheet; below statements about v understood within [v−(x⊥), v+(x⊥)] . Markov gap line density ι(v, x⊥)≥0 dened by relative entropy density kernel satises I(Dj−1:Dj+1 |Dj) = ZZ ι(v, x⊥)dv dd−2x⊥, ι monotonically non-decreasing in κ. Particularly, under total order κ≡0 and I(Dj−1:Dj+1 |Dj) = 0 (Markov saturation). [ Inequality; domain: vacuum ] Lemma 10 (C.1: Stratication DegreeGap Comparison) . Assume V± i piecewise C1 with only nitely many crossings at each x⊥ . Then there exists constant c∗>0 (depending on sup |∂V ± i| and crossing number upper bound) such that in distributional sense ι(v, x⊥)≥c∗κ(x⊥)1{v∈[v−(x⊥),v+(x⊥)]}. Combined with FawziRenner lower bound, yields quantitative gap lower bound under non-totalorder. Fidelity convention : This paper uniformly takes Uhlmann delity (not squared) F(ρ, σ) := √ρ√σ1∈[0,1]. Accordingly, FawziRenner inequality writes I(A:C|B)≥ −2 ln F, equivalently F≥e−I(A:C|B)/2. 5 Theorem 11 (D: Petz Recovery and Stability  Self-Consistent Version) . Denote A=Dj−1 , B=Dj , C=Dj+1 . Take forgetting channel ΦBC→B(XBC ) = TrC[XBC],Φ∗(YB) = YB⊗IC. With σBC =ρBC as reference state (thus σB=ρB ), Petz recovery map RB→BC dened as RB→BC(XB) = σ1/2 BCσ−1/2 BXBσ−1/2 B⊗ICσ1/2 BC , where inverse takes pseudo-inverse on supp(σB) . If and only if I(A:C|B) = 0 perfect recovery exists (idA⊗RB→BC)(ρAB) = ρABC. Generally there exists rotationally averaged Petz recovery Rrot B→BC such that I(A:C|B)≥ −2 ln FρABC,(idA⊗Rrot B→BC)(ρAB), equivalently F≥e−I(A:C|B)/2. Above inequality generally not guaranteed for unrotated RB→BC ; this paper uniformly adopts Rrot B→BC for stability propositions. [ Perfect recovery/stability; domain: Markov saturation ] 3.4 Half-Sided Modular Inclusion and Chain Advancement Theorem 12 (E: HSMI Advancement) . If (A(Dj)⊂ A(Dj+1),Ω) is right HSMI, then there exists positive-energy one-parameter semigroup covariant with ∆it A(Dj+1) , intrinsically advancing A(Dj) to A(Dj+1) . [ Algebraic structure; domain: HSMI ] 3.5 Distributional KFLWS Calibration and Windowed Parity Threshold Non-smooth window transition and error incorporation : If window h∈C0 c piecewise C2m within support (endpoints allow corners), take standard smoothing kernel ρδ and dene hℓ,δ := hℓ∗ρδ . Then for each xed ℓ > 0 , hℓ,δ −hℓL1(R)=O(δ), and Theorem F, Toeplitz/Berezin compression and trace formula rst apply to hℓ,δ ; by triangle inequality Rsmooth(δ) := ZI(γ)|hℓ,δ −hℓ|dE incorporates into total error budget Eh(γ) . Under Theorem G threshold conditions, choose δ=δ(ℓ, m) making Rsmooth(δ)≤1 2δ∗(γ) , preserving same parity threshold conclusion as hℓ . Theorem 13 (F: Distributional Calibration Identity) . For h∈C∞ c(R) (or h∈ S(R) ), Z∂Earg det S(E)h(E)dE =Ztr Q(E)h(E)dE =−2πZξ′(E)h(E)dE, 6 where ξ is spectral shift function. (Convention: BirmanKren takes det S(E) = e−2πiξ(E) .) Energy band thresholds and embedded eigenstates avoided by choosing supp h ; long-range potentials require corresponding generalized KFL. [ Distributional equality; domain: S−I∈S1 , piecewise smooth ] Proposition 14 (F ′ : Relative/Modied Calibration) . If S0(E) is reference scattering co-analytically segment-wise within energy band, without zeros/poles, and U(E) := S(E)S0(E)−1, U(E)−I∈S2, ∂EU∈S2,ZI|∂EU|2<∞, then Carleman determinant satises Z∂Earg det 2U(E)h(E)dE =Ztr Q(E)−Q0(E)h(E)dE, where Q=−iS†∂ES , Q0=−iS† 0∂ES0 . If S unitary and S0=I , above reduces to Theorem F. This proposition yields phasegroup delayspectral shift consistency under non-trace-class but relatively second-order traceable window. Note ( π/2 buer origin) : In parity determination, (−1)⌊Θ/π⌋ only ips when Θ crosses odd multiples of π . Converging perturbation total to < π/2 ensures not crossing nearest integer multiple of π , thus consistent with unperturbed parity; taking δ∗(γ) = min{π/2, δgap(γ)} − ε is explicit formulation of this buer. Branch convention (arg regularization) : Take continuous branch of arg det S dened within energy band except countable discrete set; its distributional derivative ∂Earg det S independent of branch's 2π jump choice, as h∈C∞ c annihilates jumps and matches tr Q via DOI/HelerSjöstrand representation. Theorem 15 (G: Windowed Parity Threshold; With-Gap Threshold) . Let Θh(γ) := 1 2ZI(γ) tr Q(E)hℓ(E−E0)dE, νchain(γ) := (−1)⌊Θh(γ)/π⌋. Dene unwindowed limit Θgeom(γ) := 1 2ZI(γ) tr Q(E)dE =ZI(γ) φ′(E)dE =φ(E2)−φ(E1), where I(γ)=[E1, E2] , φ(E) = 1 2arg det S(E) . Dene gap δgap(γ) := dist Θgeom(γ), πZ. Under RRh= 1 , and setting Eh(γ) := ZI|REM|dE | {z } EM endpoint +ZI|RP|dE | {z } Poisson aliasing +CTℓ−1/2ZI|∂ES|2dE |{z } Toeplitz commutator +Rtail(ℓ, I, E0) | {z } out-of-interval tail ≤δ∗(γ), where Rtail(ℓ, I, E0) := ZR\I(γ)|hℓ(E−E0)|dE ∈[0,1]. 7 Note : If h≥0 and RRh= 1 , then Rtail = 1−RI(γ)hℓ(E−E0)dE . Set δ∗(γ) := min π 2, δgap(γ)− ε . If there exist ℓ > 0 , ∆>0 , m∈N and ε∈(0, δgap(γ)) making above inequality hold, then for any window center E0 satisfying above window quality conditions, νchain(γ) = (−1)⌊Θh(γ)/π⌋= (−1)⌊Θgeom(γ)/π⌋. Here •REM is EulerMaclaurin endpoint remainder, satisfying R|REM| ≤ Cmℓ−(m−1) ; •RP is Poisson aliasing, satisfying ZI|RP|dE ≤ChX |q|≥1b h2πq ℓ ∆, where ∆>0 is energy sampling step size (energy band lattice spacing) used in Poisson summation; •RT is Toeplitz commutator term, under assumption ∂ES∈S2 and RI|∂ES|2dE < ∞ satisfying RT≤CTℓ−1/2RI|∂ES|2dE ; • Out-of-interval tail term : Rtail(ℓ, I, E0) := ZR\I(γ)|hℓ(E−E0)|dE ∈[0,1]. Note : If h≥0 and RRh= 1 , then Rtail = 1 −RI(γ)hℓ(E−E0)dE . Note : For piecewise smooth compact support windows (e.g., Kaiser), above REM 's Cmℓ−(m−1) should be replaced with corner estimate (e.g., O(ℓ−1) ), other three terms RP, RT, Rtail unchanged. By above decay orders, RP≤ChX |q|≥1b h(2πq ℓ/∆)  nite , maintaining same order as corner estimate. [ Windowed distributional equality + explicit threshold; domain: unitary scattering, h∈C∞ c or h∈ S ] Lemma 16 (T: Toeplitz/Berezin Compression Error) . Let Tℓ be windowed compression operator on energy axis (kernel is convolution with hℓ(E−E′) ), let Q(E) = −iS(E)†∂ES(E) , with ∂ES∈S2 satisfying RI|∂ES|2dE < ∞ . Then there exists constant CT>0 such that tr Q∗hℓ−ZQ(E)hℓ(E−E0)dE≤CTℓ−1/2ZI|∂ES|2dE. Proof essentials: Write compression error as [Tℓ,·] commutator, do one mean value estimate on energy derivative; use HilbertSchmidttrace Hölder with window expansion scale R(E−E0)2hℓ∼ ℓ−1 to obtain ℓ−1/2 decay. Lemma 17 (P: Poisson/EM Window Conditions) . If h∈C2m+1 c with endpoint ≤2m order jets vanishing, then b h(ω) = O(|ω|−(2m+1)) , thus X |q|≥1b h2πq ℓ ∆<∞,Z|REM| ≤ Cmℓ−(m−1). For piecewise C2m compact support windows with endpoint corners, use corner estimate to replace REM order, maintaining Poisson series convergence. Lemma 18 (G: Windowed Phase Perturbation) . If two scattering groups S , ˜ S satisfy on energy region IZI|S−˜ S|2|∂ES|2+|∂ES−∂E˜ S|1dE ≤η, 8 then Θh[S]−Θh[˜ S]≤Chη, Ch= sup EZ|hℓ(E−E′)|dE′. Corollary 19 (G: Weakly Non-Unitary Stability) . Dene ∆nonU(E) = |S†S−I|1 . Let δgap(γ) := dist Θgeom(γ), πZ . If ZI(γ) ∆nonU(E)dE ≤ε, Eh(γ)≤δ∗(γ) := min π 2, δgap(γ)−ε, where ε∈(0, δgap(γ)) , Eh(γ) := RI(γ)|REM|dE +RI(γ)|RP|dE +CTℓ−1/2RI(γ)|∂ES|2dE + Rtail(ℓ, I, E0) , then νchain(γ)=(−1)⌊Θh(γ)/π⌋ invariant, consistent with unwindowed limit (−1)⌊Θgeom(γ)/π⌋ . (Threshold fully aligned with Theorem G.) [ Stability; domain: weak dissipation ] Lemma 20 (N: Weakly Non-Unitary Phase Dierence Bound) . Write polar decomposition S= U(I−A) , U unitary, A≥0 . If RI|S†S−I|1dE ≤ε , then there exists constant CN such that ZI tr Q(S)hℓ−ZI tr Q(U)hℓ≤CNε. Proof essentials: Q(S) = Im tr(S−1∂ES) , for near-unitary S have ∥S−1∥ ≤ (1 − ∥A∥)−1 ; use ∥∂ES∥1≤ ∥∂EU∥1+∥∂EA∥1 with ∥A∥1≲∥S†S−I∥1 to control dierence and integrate. 3.6 Joint Terms and Z2 Ledger Theorem 21 (H: Ledger Consistency and Gauge Transformation) . At nullnull and nullspacelike corners, Ijoint =εJ 8πG Z√γΞdd−2x, where Ξ = ln |k1·k2| 2 (nullnull) or Ξ = ln |n·k| (nullspacelike). Under independent rescaling ki→αiki , n→βn , Ξ7→ Ξ + ln |α1α2| (nullnull), Ξ7→ Ξ + ln |α|+ ln |β| (nullspacelike). Only when normal ips k→ −k (or n→ −n ), εJ changes sign while Ξ unchanged. Thus single corner's Ijoint not purely sign invariant; but after closing along chain with square-root splicing class ϵi accounting, net eect only depends on Qiϵi parity, consistent with ⌊Θh/π⌋ parity. [ Gauge transformation; domain: anely parametrized null boundaries ] 9