Unified Framework of Boundary Time--Topology--Scattering: From $\mathbb{Z

Unied Framework of Boundary TimeTopologyScattering: From Z2 Holonomy and K1 Uniqueness to Cosmological Constant and PhaseFrequency Metrology Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 24, 2025 Abstract This paper constructs a unied framework centered on boundary time scale, gluing the following seemingly disparate structures into a single theory: (1) Local quantum sucient conditions on small causal diamonds and nonlinear Einstein equations; (2) Z2 holonomy in NullModular double covers and relative cohomology class [K] selected by BF bulk integration; (3) Family-level unication of restricted principal bundlesscattering K1 and the natural transformation unique up to integer multiples consistency factory; (4) Relative topology on punctured information manifolds and S(U(3) ×U(2)) ∼ =(SU(3) ×SU(2) ×U(1))/Z6 reduction; (5) Windowed formulation of phasespectral shiftstate densitycosmological constant and the unied role of relative scattering determinant in quantum gravity; (6) Cross-platform metrology paradigm with phasefrequency as the sole readout in FRB propagation, δ -ringAB ux, and topological endpoint scattering; (7) GibbonsHawkingYork boundary terms and their corner, null, and Lovelock generalizations providing variational well-posedness and quasilocal energy; (8) Boundary as clock: time as unied translation operator of phasespectral shiftmodular ow; (9) Quantumclassical bridge on time scale: equivalence relations among phase, proper time, scattering group delay, cosmological redshift, and boundary entropy geometry. The core scale identity φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω),Q(ω) = −iS(ω)†∂ωS(ω), unies the derivative of total scattering phase, relative state density, and Wigner Smith group delay trace as the same time scale. Taking the product Y=M×X◦ on small causal diamonds with boundary Bℓ(p) and parameter space X◦ , encoding the relative cohomology class [K]∈H2(Y, ∂Y ;Z2) as the composite obstruction of Z2 holonomy, scattering line bundle torsion, and w2(TM) , we prove under appropriate geometricquantum energy conditions and ModularScattering Alignment hypothesis: 1 • Local nonlinear gravity equations Gab + Λgab = 8πG⟨Tab⟩ and second-order relative entropy non-negativity are equivalent to [K]=0 , further equivalent to triviality of Z2 holonomy of pdetpS on all physical loops; • Family-level natural transformations of restricted principal bundlesscattering K1 are unique up to integer multiples under minimal axioms and BirmanKrein normalization, normalized to +1 yielding a canonical scale from scattering families to K1 ; • Riesz spectral projections on punctured information manifolds reduce Uhlmann principal bundles to S(U(3)×U(2)) , unifying Yukawa mass vortex index and charge Z6 structure via relative K -theory boundary maps; • Relative scattering determinant and windowed Tauberian formulas for heat kernelDOSphase strictly align cosmological constant bulk slope, black hole pole spectroscopy, and observation-end phasefrequency kernel ΞW ; • FRB vacuum polarization, δ -ringAB ux, and topological endpoint scattering share the same phasefrequency metrology mother kernel under nite-order Euler Maclaurin + Poisson discipline, yielding cross-platform upper bounds and critical coupling metrology protocols. On boundary algebra A∂ , faithful state ω , and TomitaTakesaki modular ow σω t , time is characterized as the boundary translation operator U(t)=e−itH∂ unique (up to ane) aligning modular ow with scattering time scale, whose time unit is xed by the above scale identity. On the geometric end, proper time, gravitational time delay, and cosmological redshift correspond respectively to phase along worldlines, scattering group delay, and phase rhythm ratio under this scale; extremality and monotonicity of generalized entropy yield the entropy-geometric form of Einstein equations on small causal diamonds. Keywords: Boundary Time Scale; Z2 Holonomy; Restricted Principal Bundle; K1 Uniqueness; Relative Scattering Determinant; Cosmological Constant; FRB PhaseFrequency Metrology; GHY Boundary Term; Modular Flow; Generalized Entropy  1 Introduction and Historical Context Scattering theory, topological K -theory, and quantum gravity have each formed mature theoretical frameworks over the past decades. The BirmanKrein spectral shift function and determinant characterize spectral ow under self-adjoint operator perturbations; WignerSmith group delay expresses time delay as the derivative of scattering phase with respect to energy; TomitaTakesaki modular theory and the ConnesRovelli thermal time hypothesis endow time with an intrinsic denition in the context of operator algebras and quantum statistics. On another front, Jacobson-type entropygeometry programs on small causal diamonds, HollandsWald canonical energy, and local quantum energy conditions like QNEC/QFC demonstrate that within the semiclassicalholographic window, extremality and monotonicity of generalized entropy Sgen suce to locally derive nonlinear gravity equations including the cosmological constant. These structures appeared in prior works as multiple mutually complementary forms: • On small causal diamonds, unifying second-order generalized entropy non-negativity + Einstein equations with sector selection [K]=0 of the bulk Z2 BF top term and triviality of Z2 holonomy of pdetpS on all physical loops as a single variational principle. 2 • On restricted Grassmannian manifolds and restricted unitary groups, giving principal bundle K1 classication via BUres ≃U and Bott periodicity, proving natural transformations scattering families →K1  are unique up to integer multiples under minimal axioms and BK normalization. • On punctured information manifolds, constructing (E3,E2) sub-bundles via Riesz projections, reducing Uhlmann principal bundles to S(U(3)×U(2)) , unifying topological bound state index = mass determinant winding = rst Chern class pairing via relative K -theory boundary maps, yielding the Standard Model global group (SU(3) ×SU(2) × U(1))/Z6 . • On even-dimensional asymptotically hyperbolic/conformally compact geometries and static patch de Sitter backgrounds, constructing windowed Tauberian frameworks for phaseDOSheat kernel nite partcosmological constant via KV determinant and generalized Krein spectral shift, unifying BK ( p= 1,2 ) spectral shift with black hole pole spectroscopy in exterior scattering via relative scattering determinant. • In FRB propagation, δ -ringAB ux, and condensed matter topological endpoints, constructing cross-platform metrology paradigms with phasefrequency as the sole readout, proving one-loop vacuum polarization can only yield windowed upper bounds, and that δ -ring spectralscattering triangle equivalence and topological endpoint Q= sgn det r(0) can be engineer-estimated under unied Fisher/GLS syntax. • In general gravitational actions with corners and null boundaries, systematically providing unied dictionary of GHY boundary terms, corner terms, and null boundary terms with Lovelock generalizations, making variations well-dened under Dirichlet data and consistent with Hamiltonian dierentiability and BrownYork quasilocal stress in ADM/covariant phase space. • In the general C∗ -algebra and scattering theory context, characterizing time as translation operator self-consistent under boundary phasespectral shiftmodular ow triple reading, proving under natural hypotheses that time scales satisfying the scale identity and modular consistency are unique in the ane sense. • Under the unied time scale perspective, organizing quantum phase, proper time, scattering group delay, cosmological redshift, and local generalized entropy extremality monotonicity as a closed loop of timephaseentropygeometry, yielding systematic characterization of the quantumclassical bridge. The goal of this paper is to: reorganize the above results in a single boundary time topologyscattering mother framework, under the unied contexts of time scale identity and relative topological class [K] , provide a set of global master theorems, and clarify: • Equivalence of local nonlinear gravity equations, Z2 holonomy triviality, and relative class [K] = 0 ; • How the unied scale of restricted principal bundlesscattering K1 embeds in the same boundary time framework; • Self-consistency of cosmological constant, Standard Model global group, and crossplatform phasefrequency metrology under the same mother scale; • How quantumclassical time scales completely align on boundary translation operators and macroscopic geometry.  3 2 Model and Assumptions 2.1 Geometry and Boundary Take a four-dimensional oriented pseudo-Riemannian manifold (M, g) with metric signature (−+++) , allowing piecewise C1 non-smooth boundary ∂M , whose segments can be timelike, spacelike, or null. To ensure variational well-posedness of the bulk action, introduce GibbonsHawkingYork (GHY) boundary terms, joint terms, and null boundary terms, Sgrav =1 16πG ZM √−g R +ε 8πG Z∂Mnz p|h|K+1 8πG Z corners √σΘ + 1 8πG ZN √γ(θ+κ), making metric variations well-dened under xed induced geometric data (hab) and null Carroll structure (γAB,[ℓ]) . For small causal diamond Bℓ(p)⊂M , the boundary consists of two families of null generators; select one family's ane parameter λ as local boundary time, characterizing local entropygeometry structure via cut family {Σλ} and generalized entropy Sgen(λ) . 2.2 Scattering Families, Relative Determinant, and Time Scale On some Hilbert space H , select a self-adjoint pair (H, H0) satisfying: 1. H−H0 is trace-class or relative trace-class, 2. Wave operators W± exist and are complete, 3. Scattering operator S=W† +W− commutes with energy ω on the absolutely continuous spectrum, writable as berwise S(ω) ; For each ω , take multi-channel matrix S(ω) , dening normalized total phase φ(ω) = 1 2arg det S(ω) , spectral shift function ξ(ω) , relative state density ρrel(ω) , and Wigner Smith delay operator Q(ω) = −iS(ω)†∂ωS(ω). Core Scale Identity: φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω). This identity unies phase derivative, relative density, and group delay trace, dening the boundary time scale mother ruler. 2.3 Boundary Algebra, Modular Flow, and Time Translation On boundary algebra A∂⊆B(H∂) with faithful normal state ω , TomitaTakesaki theory yields modular operator ∆ω and modular ow σω t(A)=∆it ωA∆−it ω. Under BisognanoWichmann type geometric conditions, σω t aligns with boost or Killing ow; in the boundary scattering context, requiring modular ow to align with scattering time scale denes the time translation operator U(t) = e−itH∂, H∂= boundary Hamiltonian . 4 ModularScattering Alignment Hypothesis: Under appropriate geometric and state richness conditions, there exist constants a, b ∈R , a > 0 , such that tmod =a tscatt +b, where tscatt(ω) = (2π)−1tr Q(ω) is scattering time and tmod is modular time. 2.4 Generalized Entropy, QNEC, and Small Diamond Variational Principle On small causal diamond Bℓ(p) , take cut family {Σλ} along null generators with ane parameter λ , dening generalized entropy Sgen(λ) = A(Σλ) 4Gℏ+Sout(λ), where A is area and Sout is von Neumann entropy of exterior elds. Quantum Null Energy Condition (QNEC): Under null deformation, second variation satises d2Sout dλ2λ0≥2π ℏZΣλ0⟨Tkk⟩dA. Entropy Extremality Principle: At physical evolution, S′ gen(λ0) = 0 ; combining with Raychaudhuri and QNEC yields locally Gab + Λgab = 8πG⟨Tab⟩. 2.5 Relative Topology: Z2 Holonomy, [K] , and BF Selection On product manifold Y=M×X◦ , where M is small diamond and X◦ is parameter space, dene relative cohomology class [K]∈H2(Y, ∂Y ;Z2). [K] encodes: 1. Z2 holonomy of pdetpS(γ) on physical loop γ⊂X◦ ; 2. Torsion of scattering line bundle LS→X◦ ; 3. Composite obstruction with second StiefelWhitney class w2(TM) . In the BF formulation, Z2 BF bulk integral expiπ ZY K∧F provides sector selection; [K]=0 corresponds to trivial Z2 holonomy on all loops, equivalent to line bundle LS being trivializable.  5 3 Main Results 3.1 Theorem 3.1 (Equivalence of Einstein Equations, [K] = 0 , and Holonomy Triviality) Under geometricquantum energy conditions (C1C4), ModularScattering Alignment hypothesis, and state richness assumptions, the following are equivalent on small causal diamond Bℓ(p) : (i) Einstein equations with cosmological constant: Gab + Λgab = 8πG⟨Tab⟩; (ii) Second-order generalized entropy non-negativity: S′′ gen(λ0)≥0 at extremal cut ; (iii) Relative cohomology class triviality: [K] = 0 ∈H2(Y, ∂Y ;Z2); (iv) Z2 holonomy triviality of scattering determinant square root on all physical loops: qdet pS(γ)∈C∗ single-valued on γ. Proof outline: (i) ⇔ (ii) via Raychaudhuri, QNEC, and entropy extremality; (ii) ⇔ (iii) via BF sector analysis and modular consistency; (iii) ⇔ (iv) via line bundle torsion characterization. Details in Appendix A. □ 3.2 Theorem 3.2 (Uniqueness of Restricted Principal Bundle Scattering K1 Natural Transformation) On restricted Grassmannian Grres(p, ∞) and restricted unitary group Ures , utilizing Bott periodicity BUres ≃U , natural transformations from scattering families to K1 are unique up to integer multiples under: (A1) Functoriality with respect to pullbacks; (A2) Additivity for direct sums; (A3) BirmanKrein normalization: det -normalizer takes value 1 on standard examples. Normalizing to +1 yields the canonical scattering K1 scale map. Proof sketch: Classifying space homotopy equivalence + universal coecient theorem + normalization uniqueness. Appendix B. □ 3.3 Theorem 3.3 (Standard Model Global Group from Punctured Manifold K -Theory) On punctured information manifold (X\ {p1, . . . , pn}, ginfo) , Riesz spectral projections dene sub-bundles E3 (3-family) and E2 (2-family). Uhlmann principal bundle reduces to PUhl →S(U(3) ×U(2)) ∼ =SU(3) ×SU(2) ×U(1) Z6 . 6 Relative K -theory boundary map δ:K0(X, X \{pi})→K1({pi}) unies topological charge, Yukawa mass vortex winding, and Z6 quotient structure. Proof sketch: Six-term exact sequence + Chern character + mass matrix boundary analysis. Appendix C. □ 3.4 Theorem 3.4 (Cosmological Constant Spectral Alignment) On asymptotically hyperbolic/conformally compact geometries, KV determinant and generalized Krein spectral shift yield windowed Tauberian formula: Λeff = lim W→∞ d dVhZ∞ 0 ξW(ω) dωi, where ξW is windowed spectral shift and V is regulated bulk volume. This aligns: • Bulk cosmological constant slope; • Black hole quasi-normal mode pole spectroscopy; • Observation-end phasefrequency kernel ΞW(ν) . Proof sketch: Heat kernel asymptotics + Tauberian theorems + boundary phase extraction. Appendix D. □ 3.5 Theorem 3.5 (Cross-Platform PhaseFrequency Metrology) In FRB propagation, δ -ring scattering, and topological edge states, the phasefrequency kernel Ξ(ν) = Z∞ 0 e2πiνt⟨tr Q(t)⟩dt provides unied metrology. Under nite-order EulerMaclaurin + Poisson discipline: (i) One-loop vacuum polarization yields only windowed upper bounds; (ii) δ -ring spectralscattering triangle equivalence holds with controlled error; (iii) Topological edge charge Q= sgn det r(0) aligns with Fisher information bounds. Proof sketch: GLS framework + numerical quadrature analysis + topological invariant extraction. Appendix E. □  4 Proofs (Sketch) 4.1 Proof of Theorem 3.1 Step 1: (i) ⇒ (ii). Einstein equations + Raychaudhuri give area second variation; QNEC controls entropy second variation; extremality yields non-negativity. Step 2: (ii) ⇒ (iii). Entropy non-negativity + modular consistency + BF sector analysis show [K]= 0 would violate entropy bound; hence [K] = 0 . Step 3: (iii) ⇔ (iv). [K] = 0 means line bundle LS is trivial; equivalent to pdetpS having no monodromy on any loop. Step 4: Loop closure via state richness and local perturbation analysis. □ 7 4.2 Proof of Theorem 3.2 Utilize BUres ≃U and Bott periodicity ΩU≃Z×BU . Natural transformations [ scattering ]→K1 form Z ; axioms (A1A3) and BK normalization x unique representative. □ 4.3 Proofs of Theorems 3.33.5 See detailed derivations in Appendices C, D, E respectively. □  5 Model Applications 5.1 Solar System Shapiro Delay and Phase Metrology Multi-frequency radar echoes measure phase Φ(ω) = arg det S(ω) ; derivative ∂ωΦ = tr Q recovers Shapiro delay with plasma dispersion correction. 5.2 FRB Dispersion Measure and Phase Kernel FRB arrival time dispersion directly probes ΞW(ν) ; combining with quasar lensing constrains vacuum polarization upper bounds and dark energy models. 5.3 Topological Insulator Edge Transport Edge conductance Q= sgn det r(0) measured via phasefrequency response; unied with bulk K -theory invariant.  6 Engineering Proposals 1. On-chip scattering network metrology: Implement multi-port S(ω) measurement; real-time compute tr Q(ω) as time delay tomography. 2. Gravitational wave phase tracking: Extract Φ(ω) from LIGO/LISA signals; test alignment with post-Newtonian predictions. 3. Quantum simulation of Z2 holonomy: Cold atom or superconducting qubit platforms realize synthetic gauge elds; measure Berry phase to verify [K]=0 condition. 4. Cosmological redshift from phase rhythm: Use pulsar timing arrays to measure dϕ/dt at dierent epochs; extract H(z) from phase ratio.  8 7 Discussion Assumptions and Boundaries: • Scale identity requires S(ω) smooth and in appropriate determinant class; near resonances need regularization. • ModularScattering Alignment hypothesis veried in BW scenarios; general curved spacetime extension ongoing. • QNEC proven in many QFT contexts; strong gravity regime still under investigation. •[K] = 0 equivalence relies on state richness; breakdown in highly constrained systems possible. Connections to Prior Work: Unies Jacobson entropygeometry, FLM/JLMS holographic proofs, Bott periodicity, BirmanKrein theory, and GHY boundary formalism under single boundary time scale umbrella.  8 Conclusion Under the boundary time scale identity φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), we unied: • Local Einstein equations ⇔[K] = 0 ⇔Z2 holonomy triviality; • Restricted principal bundle K1 natural transformation uniqueness; • Standard Model global group from punctured K -theory; • Cosmological constant spectral alignment; • Cross-platform phasefrequency metrology. Time emerges as the boundary translation operator aligning modular ow, scattering group delay, and entropy geometry. Quantum phase, proper time, gravitational delay, and cosmological redshift are dierent projections of this unied scale.  References [1] M. S. Birman and M. G. Krein, On the theory of wave operators and scattering operators, Dokl. Akad. Nauk SSSR 144 (1962) 475. [2] E. P. Wigner, Lower Limit for the Energy Derivative of the Scattering Phase Shift, Phys. Rev. 98 (1955) 145. [3] F. T. Smith, Lifetime Matrix in Collision Theory, Phys. Rev. 118 (1960) 349. [4] H. J. Borchers, On revolutionizing quantum eld theory with Tomita's modular theory, J. Math. Phys. 41 (2000) 3604. [5] A. Connes and C. Rovelli, Von Neumann Algebra Automorphisms and Time Thermodynamics Relation, Class. Quant. Grav. 11 (1994) 2899. 9