Boundary as Unified Stage: Variational Completeness, Time Scale, Topological Branching

Boundary as Unied Stage: Variational Completeness, Time Scale, Topological Branching Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 24, 2025 Abstract From rst principles, elevate boundary from passive geometric appendage to unied physical stage. Propose axiomatic framework with boundary as fundamental object, gluing three seemingly separate structuresGibbonsHawkingYork (GHY) boundary term and BrownYork quasilocal quantities in gravitational variation, spectral shift function and WignerSmith time delay in scattering theory, modular ow and relative entropy monotonicity in operator algebrasas dierent projections of same boundary time geometry. Core viewpoint: boundary not merely separating bulk domains but compressing bulk continuous changes into nite measurable dierences (energy dierence, time dierence, topological class dierence, causal orientation). On geometricvariational side, review and rene complete variational structure of EinsteinHilbertGHYcornernull boundary terms, proving under xed induced metric condition, requiring well-dened variational principle uniquely selects variationally complete boundary geometry class, deriving BrownYork boundary stress energy tensor as boundary readout of bulk dierence. On spectralscattering side, under standard trace-class perturbation assumptions, starting from BirmanKrein formula, give scale identity relating total scattering phase derivative, relative state density, WignerSmith time delay trace, interpreting boundary scattering phase tiny variations uniformly as state number changes and residence time changes. On operator algebrainformation side, introduce modular ow under general boundary observable algebra and faithful state context, write modular Hamiltonian as energy ow integral along null boundary (or wedge boundary), characterize boundary time arrow unidirectionality using relative entropy monotonicity and quantum energy conditions. Moreover, dene discriminant boundary and Z2 branch index in parameter space, explaining after excluding spectral anomaly and topological phase transition hypersurfaces, spin structure dened by scattering matrix square root forms Null Modular double cover on parameter space. Double cover's non-triviality memorized by boundary as spectral ow parity and intersection number parity, compressing going around once continuous deformation into discrete topological class dierence. 1 Finally, give unied boundary time geometry denition: boundary carries geometricspectralinformationtopological data set making (i) variation well-dened, (ii) time scale identity holds, (iii) modular ow and generalized entropy monotonicity determine time arrow, (iv) Z2 index dened on discriminant boundary gives topological branching. Main theorem: under appropriate assumptions, can select unique (in ane rescaling sense) time parameter on boundary making these four structure time parameters belong to same scale equivalence class, thus precisely formalizing boundary generates dierence into testable, computable unied framework. Keywords: Boundary Geometry; Variational Completeness; Spectral Shift; Time Delay; Modular Flow; Relative Entropy; Topological Index; Z2 Branching  1 Introduction Boundaries appear ubiquitously in almost all corners of physics and mathematics: spatial innity and black hole horizons in general relativity, Cauchy surfaces and causal diamond boundaries in quantum eld theory, material interfaces and topological defects in condensed matter, incoming/outgoing asymptotic boundaries in scattering theory, even discriminant hypersurfaces of topological phase transitions in parameter space. Traditional treatments mostly view boundary as geometric appendage of bulk domain: in eld theory need to specify boundary conditions, in geometry need to supplement boundary terms to correct variation, in scattering theory boundary merely way of imposing asymptotic conditions at innity. This paper attempts to advance more radical viewpoint: in unied framework, **boundary should be viewed as true stage of physical structure**. Bulk continuous changes only become measurable, comparable, optimizable objects when translated on boundary into nite-dimensional scale dierences. More specically: • On **geometricvariational** level, requiring EinsteinHilbert action variation well-dened in boundary case forces us to introduce GibbonsHawking York term and corner/null boundary terms on boundary; BrownYork quasilocal stress energy tensor naturally appears as boundary ledger of how much bulk geometry and matter distribution dier. • On **spectralscattering** level, BirmanKrein spectral shift function and Wigner Smith time delay translate tiny phase changes with frequency into state density dierence and residence time dierence; this structure naturally is boundary structure since all scattering readouts measured on boundary (or at innity). • On **operator algebrainformation** level, TomitaTakesaki modular theory and relative entropy monotonicity show: given boundary observable algebra and faithful state, can dene modular ow and its generator (modular Hamiltonian), often writable as energy ow integral along boundary; ow parameter after appropriate normalization interpretable as intrinsic time on boundary. • On **topologicalparameter space** level, scattering matrix spectrum and phase in parameter space often have discriminant hypersurfaces; excluding these anomalous points, spin structure dened by square-root scattering matrix forms Z2 double cover on parameter space, non-triviality manifested on boundary as extra minus sign after going around once. 2 These seemingly scattered phenomena point to common structure: **boundary responsible for compressing invisible bulk changes into visible dierence scales**. This paper's goal: starting from this intuition, construct rigorous axiomatic boundary framework; give theorem series unifying variational completeness, time scale identity, modular ow time arrow, topological branching into boundary time geometry context.  2 Preliminaries and Notation 2.1 Geometry and Variation Let (M, g) be four-dimensional Lorentzian manifold with boundary ∂M . On non-null (spacelike or timelike) boundary, denote induced metric hab , outward normal na , extrinsic curvature Kab =hc ahd b∇cnd , trace K=habKab . EinsteinHilbert action dened as SEH(g) = 1 16πG ZM R(g)√−g d4x, where R is scalar curvature. Well-known: in boundary case, SEH alone not well-dened under variation xing hab ; variation produces boundary term depending on δ(∂g) . To correct, introduce GibbonsHawkingYork boundary term SGHY(g) = ε 8πG Z∂M Kp|h|d3x, where ε= +1 for spacelike boundary, ε=−1 for timelike boundary. For corner and null boundary cases, need introduce additional corner and null boundary terms; see Appendix A. 2.2 Scattering Theory and Time Delay Let H be complex Hilbert space, H0 and H=H0+V self-adjoint operators. Assume perturbation V is H0 -relative trace-class making wave operators W±= s - lim t→±∞ eiHte−iH0t exist and complete; then scattering operator dened as S=W∗ +W−. Under appropriate conditions, S writable in energy representation as ber decomposition S=Z⊕ S(ω)dµ(ω), where S(ω) is nite-dimensional (or separable) scattering matrix at energy ω . Dene WignerSmith time delay operator Q(ω) = −i S(ω)†∂ωS(ω). BirmanKrein spectral shift function ξ(λ) satises det S(λ) = exp(−2πi ξ(λ)). Under appropriate regularity assumptions, dierentiable; derivative ξ′(λ) related to state density dierence. Adopt notation Krein spectral shift density ρrel(ω) = ξ′(ω) . 3 2.3 Modular Theory, Modular Flow, Relative Entropy Let A be C∗ algebra or von Neumann algebra, ω faithful normal state on it. GNS construction gives triple (πω,Hω,Ωω) where Ωω is cyclic vector. Tomita operator Sω 's polar decomposition produces modular operator ∆ω and conjugation Jω . Modular ow dened as σω t(A)=∆it ωA∆−it ω, A ∈ A. If self-adjoint operator Kω exists satisfying ∆ω=e−Kω , formally σω t(A) = eiKωtAe−iKωt. For two states ω, φ , relative entropy dened as S(ω∥φ) = tr(ρω(log ρω−log ρφ)). Under appropriate generality satises monotonicity: non-increasing under restriction to subalgebra or subregion. In relativistic QFT double cone/wedge region cases, modular Hamiltonian Kω often writable as energymomentum tensor integral along boundary direction, giving modular time geometric meaning. 2.4 Spectral Flow and Z2 Index Let {At}t∈[0,1] be family of self-adjoint Fredholm operator paths; spectral ow Sf({At}) dened as oriented number of eigenvalues crossing zero. If only caring about parity, dene Z2 index νZ2({At}) = (−1)Sf({At})∈ {±1}. On parameter space X , can connect spectral ow parity with loop intersection number parity for discriminant set D⊂X , forming Z2 topological index. Will use this language to describe scattering square-root branch structure.  3 Axiomatic Denition of Physical Boundary System Give physical boundary system denition adopted in this paper, unifying geometric, scattering, modular ow, topological data into same boundary framework. 3.1 Boundary Data Quadruple Denition 3.1 (Physical Boundary System) . Physical boundary system consists of quadruple B= (∂M, A∂, ω∂,S∂) where: 1. ∂M is codimension-one boundary of four-dimensional spacetime M , equipped with induced metric hab and extrinsic curvature Kab , plus possible corners and null sheet segments; 2. A∂ is boundary observable algebra associated with ∂M (e.g., boundary-restricted eld operator algebra or scattering channel algebra); 4 3. ω∂ is faithful normal state on A∂ , giving modular ow σω∂ t ; 4. S∂ is set of boundary scattering and topological data, including: • Scattering matrix S(ω) in energy representation; • Frequency measure compatible with A∂ ; • Discriminant subset D⊂X on parameter space X and Z2 index dened from it. In concrete models, ∂M can be articial boundary of nite region, small causal diamond boundary, black hole horizon, AdS asymptotic boundary, material interface, even discriminant boundary in parameter space in abstract sense. 3.2 Four Fundamental Postulates Postulate 1 (A1: Variational Completeness) . Exists action functional composed of bulk and boundary Stot =Sbulk[g, Φ] + Sbdy[h, K, Φ|∂M ], where specic boundary term Sbdy (including GHY, corner, null boundary terms) makes under variation xing boundary induced metric and matter boundary data (hab,Φ|∂M ) , rst-order variation of Stot depends only on bulk variation and equivalent to given eld equations (e.g., Einsteinmatter equations). Postulate 2 (A2: Scale Identity) . Exists frequency variable ω and corresponding scattering matrix S(ω) making following scale identity hold: φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), where φ(ω) = 1 2arg det S(ω) , ρrel(ω) is relative state density, Q(ω) = −iS(ω)†∂ωS(ω) is WignerSmith time delay matrix. Function dened from this κ(ω) := φ′(ω) π called boundary time scale density. Postulate 3 (A3: Modular Flow Orientation and Time Arrow) . Modular ow σω∂ t generator K∂ writable as integral of energymomentum tensor projection along boundary K∂=Z∂M f(x)Tab(x)χa(x)nb(x)dΣx, where χa is Killing-like or normalized boundary time translation vector eld, nb is normal, f is positive weight function. Relative entropy S(ω∂∥φ∂) monotonically non-decreasing along modular ow future direction, dening time arrow on boundary. Postulate 4 (A4: Topological Branching and Z2 Index) . Discriminant subset D⊂X exists in parameter space X such that on X◦=X\D can continuously select scattering matrix square root S1/2 . Any closed loop γ⊂X◦ lifting may return to opposite branch of original point, giving Z2 index ν(γ)∈ {±1}, this index equivalent to spectral ow parity of some self-adjoint family or intersection number parity with D .  5 4 Variational Completeness and Geometric Boundary Prove: Postulate A1's variational completeness requirement introduces GHY term and BrownYork boundary stressenergy tensor on non-null boundary; when corners and null boundaries exist, need additional corner and null boundary terms, completely compressing bulk dierence into readouts on boundary geometry and surface stress. Theorem 4.1 (Variational Completeness on Non-Null Boundary) . Under variation xing boundary induced metric hab , total action S[g] = SEH[g] + SGHY[g] rst-order variation is δS[g] = 1 16πG ZM (Gab + Λgab)δgab√−g d4x, i.e., all boundary terms completely cancel. Thus under given hab condition, variational principle well-dened, deriving Einstein equation Gab + Λgab = 8πGTab . BrownYork Quasilocal StressEnergy Tensor : After introducing matter action Smatter[g, Φ] , dene on boundary TBY ab =−2 p|h| δSGHY δhab =1 8πG(Kab −Khab). Integrating over spatial slice Σ⊂∂M gives BrownYork energy EBY(Σ) interpretable as quasilocal energy relative to reference background, acting as boundary time translation generator in Hamilton formalism. Corners and Null Boundaries : When timelike and spacelike boundaries intersect forming corners, or null boundaries (like horizons, null hypersurfaces) exist, GHY term alone insucient to ensure variational completeness. Need introduce corner term Scorner and null boundary term SN .  5 SpectralScattering Side Scale Identity and Boundary Time Realize Postulate A2; give scale identity sucient conditions; explain how it uniformly reads boundary phase tiny changes as state density dierence and time delay dierence. 5.1 BirmanKrein Spectral Shift and Scattering Phase Under Section 2.2 assumptions, Krein spectral shift function ξ(λ) dened making tr(f(H)−f(H0)) = Z+∞ −∞ f′(λ)ξ(λ)dλ hold for suciently many test functions f . BirmanKrein formula gives det S(λ) = exp(−2πi ξ(λ)). Dene total scattering phase Φ(λ) = Pjδj(λ) , half-phase φ(λ) = 1 2Φ(λ) . Then φ′(λ) π=ρrel(λ). 6 5.2 WignerSmith Time Delay Operator Recall WignerSmith operator Q(λ) = −iS(λ)†∂λS(λ). Taking trace yields tr Q(λ)=4φ′(λ). Combining with previous subsection relation, arranging notation, select unied convention: dene time scale density as κ(ω) := φ′(ω) π. Theorem 5.1 (Scale Identity) . Under standard assumptions (trace-class perturbation, wave operator completeness), normalization selection exists making for almost all ω κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω). Physical interpretation : Scale identity shows boundary phase perturbation δφ frequency change ∂ωφ readable by three equivalent ways: • As state density dierence ρrel(ω) per unit frequency; • As total residence time time delay density (2π)−1tr Q(ω) ; • As time scale density κ(ω) . This realizes Postulate A2, directly interfacing boundary scattering data with time scale.  6 Modular Flow, Generalized Entropy, Boundary Time Arrow Explain how Postulate A3 orients time scale on boundary, aligning with scattering scale identity. 6.1 Geometric Expression of Modular Hamiltonian Let A∂ be local operator algebra associated with wedge region or causal diamond boundary, ω∂ vacuum or KMS state on it. In many models, modular Hamiltonian K∂ writable as K∂= 2πZ∂M ξaTabnbdΣ, where ξa is appropriately normalized timelike Killing vector or diamond-like boost vector, Tab is energymomentum tensor. Modular ow σω∂ t(A) = eiK∂tAe−iK∂t thus interpretable as thermal time or modular time along boundary direction. 7 6.2 Relative Entropy Monotonicity and Time Arrow Let ω∂, φ∂ be two boundary states; corresponding relative entropy S(ω∂∥φ∂) . In local QFT framework, provable under restriction to nested region family, relative entropy monotonically non-increasing with region expansion. Translating to boundary geometry, this monotonicity governs generalized entropy growth along certain future directions. Proposition 6.1 (Modular Time Arrow) . Assume for nested boundary cross-section family {∂Mt} (e.g., sections advancing along null direction) d dtSgen(∂Mt)≥0, where Sgen is generalized entropy; then parameter t selectable as time arrow parameter on boundary. If rescaling t proportionally to align with scattering time scale κ(ω) , can simultaneously view on boundary as modular time and scattering time.  7 Topological Branching, Discriminant Boundary, Z2 Index Realize Postulate A4, connecting discriminant boundary in parameter space, spectral ow parity, scattering matrix square root branch structure. 7.1 Discriminant Boundary and Parameter Space Consider parameter space X ; each point x∈X corresponds to scattering system with scattering matrix S(ω;x) . Discriminant subset D⊂X exists such that if and only if x∈D , S(ω;x) has eigenvalue −1 near some energy, degenerate eigenvalues, or other spectral anomalies. Dene X◦=X\D . On X◦ , spectral anomalies excluded; can select principal branch square root of scattering matrix S1/2(ω;x) satisfying (S1/2(ω;x))2=S(ω;x), S1/2(ω;x0) given . 7.2 Z2 Index and Spectral Flow Parity Take any closed loop γ: [0,1] →X◦ ; parallel transporting square root S1/2 along γ may have overall sign ip S1/2(ω;γ(1)) = ±S1/2(ω;γ(0)). Dene ν(γ) = (+1, S1/2 no ip , −1, S1/2 ips . Proposition 7.1 ( Z2 Index and Spectral Flow Parity) . Under appropriate dierentiability and spectral gap assumptions, ν(γ) equals spectral ow parity of some self-adjoint family: ν(γ) = (−1)Sf({At}), where {At} is related self-adjoint operator family constructed along γ ; spectral ow Sf({At}) records number of eigenvalues crossing zero. 8 Geometric interpretation : Discriminant D as parameter space topological boundary divides X◦ into dierent sectors; ν(γ) records whether closed loop passes around boundary odd number of times. Thus going around once continuous deformation compressed by boundary into simple discrete label ±1 .  8 Unied Theorem of Boundary Time Geometry Glue above geometric, spectralscattering, modular ow, topological structures into uni- ed boundary time geometry framework; prove time scale uniqueness result. 8.1 Denition of Boundary Time Geometry Denition 8.1 (Boundary Time Geometry) . Boundary time geometry consists of data G∂= (∂M, hab, Kab;A∂, ω∂;S(ω); D, ν) satisfying: 1. (∂M, hab, Kab) makes gravitational and matter action variation well-dened under xed hab condition, giving BrownYork boundary stressenergy tensor; 2. (A∂, ω∂) gives modular ow σω∂ t and modular Hamiltonian K∂ ; denes time arrow through generalized entropy monotonicity; 3. Scattering matrix S(ω) and spectral shift data satisfy scale identity; time scale density κ(ω) well-dened; 4. Discriminant D and ν dene topological branching Z2 index. Call time parameter t unied scale parameter of this boundary time geometry if simultaneously scales three time structures: • Gravitygeometric side : t is parameter along boundary time translation vector eld χa making BrownYork energy change rate under t consistent with bulk energy ow; • Scattering side : t related to frequency ω via bijection t=t(ω) making κ(ω) interpretable as dt density; • Modular ow side : t is modular ow parameter making modular Hamiltonian K∂ and BrownYork energy generator dier only by constant factor. Theorem 8.2 (Boundary Time Scale Uniqueness Theorem) . Let B= (∂M, A∂, ω∂,S∂) be physical boundary system satisfying Postulates A1A4 with technical assumptions: 1. BrownYork boundary energy EBY(t) change with parameter t writable as boundary energy ow integral; 2. Modular Hamiltonian K∂ compatible with time translation generated by EBY ; positive constant β > 0 exists making K∂=βEBY + constant . 3. Scattering matrix S(ω) frequency dependence reparametrizable as ω=ω(t) ; scale identity maintains form under this reparametrization. Then unique (in ane rescaling sense) time parameter t exists making: • Gravity geometric side boundary time translation; • Scattering side time delay scale; • Modular ow side modular time ow belong to same scale equivalence class. 9