Framework of Limit Unification and No-Observer Ontologization\\ \large Unified Time Scale, Boundary Time Geometry, and Consistent Variational Principle

Framework of Limit Unication and No-Observer Ontologization Unied Time Scale, Boundary Time Geometry, and Consistent Variational Principle Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 19, 2025 Abstract In previous works, we have characterized the Universe as a maximal, consistent, and complete ontological mathematical object, internally containing multiple layers of components such as causal manifolds, unied time scale, boundary time geometry, bulk quantum eld theory, scattering and spectral shift theory, Tomita Takesaki modular structure, and generalized entropy. Under this framework, an integrated description of time, causality, entropy, and observation can be achieved, but physical laws themselves and physical details (gauge groups, eld content, mass spectrum and couplings, uid and many-body eective equations, etc.) mostly appear as external additions. This paper proposes a framework of **limit unication completely independent of any observer ontological concept**: we take unied time scale and boundary time geometry as the sole fundamental geometricspectral structures, unifying all physical laws into necessary conditions of a single consistent variational principle. Specically: 1. Introducing the scale identity in scattering theory κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), (1) unifying scattering phase derivative, relative density of states, and WignerSmith group delay trace into a unique time scale mother ruler. 2. Introducing total connection in boundary time geometry Ω∂=ωLC ⊕AYM ⊕Γres, (2) unifying gravity, internal gauge elds, and resolution/renormalization group ow into a single geometric object on the boundary bundle. 3. Introducing generalized entropy on small causal diamonds Sgen(D) = A(∂D) 4Gℏ+Sbulk(D), (3) and constructing a global consistency functional I[U] = Igrav +Igauge +IQFT +Ihydro, (4) 1 where each term constrains consistency at geometricentropy, gaugetopological, quantumscattering, and coarse-grained uid levels respectively. The main result of this paper is: under natural assumptions of causality, unitarity, and entropy stability, applying the unied consistency principle δI[U] = 0 (5) to the Universe Ontology Object U , the necessary conditions are respectively equivalent to: 1. Geometric variation in the limit of small causal diamonds yields Einstein equations Gab + Λgab = 8πG⟨Tab⟩, (6) along with appropriate quantum energy conditions and focusing conditions; 2. Under xed K -theory class conditions, variation of boundary channel bundle and total connection yields YangMills equations and gauge eld anomaly cancellation conditions, thereby unifying eld content and gauge groups as consistency equations of boundary K class and scattering K1 class; 3. Under given geometric and gauge background, variation of relative entropy functional for bulk states and scattering data yields Wightman axioms, Euler Lagrange eld equations, and Ward identities of local quantum eld theory, meaning QFT is no longer an independent input but an inevitable result of the unied consistency principle at the quantumscattering level; 4. In long-wavelength and low-resolution limits, variation of resolution connection and macroscopic conserved currents yields generalized NavierStokes type uid equations and diusion equations, unifying macroscopic irreversible dynamics as gradient ows of generalized entropy on the unied time scale. Therefore, without introducing any observer ontology concept, this paper achieves limit unication in physics: General Relativity, Gauge Field Theory, Local Quantum Field Theory, Fluid Dynamics, and Many-Body Eective Dynamics are all necessary conditions of the same cosmic consistency variational principle under different resolutions and boundary conditions, while all physical details are uniedly encoded in boundary K -theory classes and scattering analytic invariants. 1 Introduction 1.1 Unication Problem and Residual Degrees of Freedom Major theories of modern physicsGeneral Relativity, Quantum Field Theory, Statistical Physics and Fluid Dynamics, Condensed Matter and Many-Body Systemshave been fully veried at their respective applicable scales. However, when attempting to provide a unied theory of the universe, a fundamental diculty persists: 1. We can unify spacetime and causality at the geometric level (causal manifolds, Lorentzian geometry); 2. We can unify various interactions at the quantum level into gauge eld theory systems (YangMills + Higgs + fermions); 3. We can link entropy, energy conditions, and time arrows at the information level (generalized entropy, relative entropy, and quantum energy conditions). But these unications often still rely on numerous external laws and parameters, for example: 2 * Gravity derives Einstein equations through independently assumed EinsteinHilbert action; * Gauge eld theory gives the Standard Model through the externally added group SU(3) ×SU(2) ×U(1) and its representations; * Mass spectra and coupling constants of matter elds exist as experimental inputs; * Fluid and many-body eective equations (like NavierStokes and FokkerPlanck) are derived through independent approximations. In other words, even under highly unied structural frameworks, what are physical laws and what are values of detailed parameters still retain massive degrees of freedom, appearing not as unique consequences of some higher-level principle. 1.2 Limit Unication Approach without Observer Ontology Many unication schemes attempt to further constrain physical laws by introducing observers, computation, or information processing ontologies (e.g., viewing the universe as some observationcomputation network). However, such schemes often struggle to maintain formal objectivity and may introduce additional metaphysical assumptions. This paper deliberately **introduces no additional observer ontology concepts**, but treats observers only as a derived structure within the universe ontology object (e.g., local operator subalgebras and states on certain worldlines), and places the entire burden of unication on the following three types of no-observer ontology intrinsic structures: 1. **Unied Time Scale**: The scale mother formula κ(ω) given by scattering phase, relative density of states, and group delay trace, serving as the sole source of all physical time readings; 2. **Boundary Time Geometry**: Composed of spacetime boundary, induced metric, second fundamental form, and total connection Ω∂ , unifying gravity, gauge elds, and resolution ow into boundary bundle geometry; 3. **Generalized Entropy and Causal Structure**: Dening generalized entropy Sgen on small causal diamonds, and constraining geometry and quantum states using its extremum and monotonicity. On this basis, we construct a global consistency functional I[U] and propose the unied consistency principle δI[U] = 0 . This paper will prove: local and hierarchical expansions of this principle naturally yield all familiar physical laws. 1.3 Paper Structure Section 2 denes the Universe Ontology Object, unied time scale, boundary time geometry, and generalized entropy structure. Section 3 constructs the consistency functional I[U] and presents the unied consistency principle. Section 4 derives Einstein equations and quantum energy conditions at the level of small causal diamonds. Section 5 derives gauge equations and eld content constraints at the level of boundary K -theory and total connection. Section 6 derives local quantum eld theory and Ward identities at the quantumscattering level. Section 7 derives uid dynamics and many-body eective gradient ows in the coarse-grained limit. Appendix provides technical proofs of key formulas and theorems. 3 2 Universe Ontology Object and Unied Structure 2.1 Universe Ontology Object We characterize the universe as an ontological mathematical object U=Uevt, Ugeo, Umeas, UQFT, Uscat, Umod, Uent, (7) where: 1. Uevt = (M, g, ≺) is a globally hyperbolic Lorentzian manifold with causal partial order ≺ ; 2. Ugeo contains a family of small causal diamonds {Dp,r} , Brown York quasilocal stress tensor, and GibbonsHawkingYork boundary term; 3. Umeas = (A∂, ω∂) is the boundary observable algebra and state; 4. UQFT = (Abulk, ωbulk) is the bulk quantum eld theory algebra and state; 5. Uscat = (S(ω;ℓ), Q(ω;ℓ)) is the frequency resolution dependent scattering matrix and WignerSmith group delay matrix; 6. Umod is the TomitaTakesaki modular structure and modular ow induced by (A∂, ω∂) ; 7. Uent is generalized entropy Sgen , relative entropy, and quantum energy conditions on small causal diamonds. Note: No observer ontology is introduced in this denition; observers can be subsequently viewed as specic choices of local subalgebras and states in UQFT on worldlines. 2.2 Unied Time Scale In scattering theory, let S(ω;ℓ) be the scattering matrix at energy ω and resolution ℓ , its determinant phase is φ(ω;ℓ) = arg det S(ω;ℓ), (8) group delay matrix Q(ω;ℓ) = −iS(ω;ℓ)†∂ωS(ω;ℓ), (9) its trace tr Q(ω;ℓ) and spectral shift function derivative ρrel(ω;ℓ) satisfy the scale identity under natural regularity conditions κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω). (10) We call κ(ω) the unied scale density, and dene unied time scale τ (ane freedom omitted) via τ(E) = ZE −∞ κ(ω) dω. (11) The unied time scale axiom stipulates: all physical time readings (proper time, redshift time, modular ow parameter, etc.) in the universe belong to the same scale class [τ] . 2.3 Boundary Time Geometry and Total Connection Consider bulk region MR⊂M and its boundary ∂MR . Dene induced metric hab , outward normal na , second fundamental form Kab on ∂MR , and GibbonsHawkingYork boundary term SGHY =1 8πG Z∂MR Kp|h|dd−1x. (12) 4 All geometry and interactions on the boundary are uniedly encoded into the total connection Ω∂=ωLC ⊕AYM ⊕Γres, (13) where: 1. ωLC is LeviCivita connection, characterizing spacetime gravitational geometry; 2. AYM is YangMills connection on internal gauge group, characterizing electroweak, strong interactions, etc.; 3. Γres is connection on resolution space, characterizing renormalization group ow and observational resolution change. Its curvature decomposes as F(Ω∂) = R⊕FYM ⊕Fres, (14) corresponding to curvature of spacetime, gauge eld strength, and resolution ow. 2.4 Generalized Entropy and Causal Structure On (M, g, ≺) , select a point p and parameter r , construct small causal diamond Dp,r =J+γ(−r)∩J−γ(r), (15) where γ(τ) is a timelike geodesic passing through p . Dene generalized entropy Sgen(Dp,r) = A(∂Dp,r) 4Gℏ+Sbulk(Dp,r), (16) where Sbulk is the von Neumann entropy of bulk quantum elds on Dp,r . The unication requirement is: under appropriate constraints, the rst-order variation extremum condition of Sgen on small causal diamond family {Dp,r} yields gravitational eld equations, second-order variation yields quantum energy conditions and focusing conditions, while the nested family {Dτ} along unied scale parameter τ satises generalized entropy monotonicity, thereby dening macroscopic time arrow. 3 Consistency Functional and Unied Consistency Principle 3.1 Structure of Consistency Functional Dene cosmic consistency functional on the above structure I[U] = Igrav +Igauge +IQFT +Ihydro, (17) corresponding to consistency constraints at geometricentropy, gaugetopological, quantum scattering, and coarse-grained levels respectively. 1. GeometricEntropy Term Igrav =1 16πG ZM (R−2Λ)p|g|ddx+1 8πG Z∂M Kp|h|dd−1x−λent X D∈DmicroSgen(D)−S∗ gen(D), (18) where S∗ gen(D) is entropy extremum under xed external conditions, Dmicro is the family of small causal diamonds covering M . 5 2. GaugeTopological Term Igauge =Z∂M×ΛhtrFYM ∧⋆FYM+µtop CS(AYM) + µKIndex(D[E])i, (19) where [E]∈K(∂M ×Λ) is the K -class of channel bundle, CS is ChernSimons term, Index(D[E]) is the index of Dirac operator coupled to E . 3. QuantumScattering Term IQFT =X D∈Dmicro SωD bulk∥ωD scat, (20) where ωD bulk is restriction of bulk state on D , ωD scat is reference state predicted by scattering data and unied scale, S(·∥·) is relative entropy. 4. Coarse-Grained Fluid Term Ihydro =ZMhζ(∇µuµ)2+η σµνσµν +X k Dk(∇µnk)2ip|g|ddx, (21) where uµ is macroscopic velocity eld, σµν shear tensor, nk densities of conserved quantities, coecients ζ, η, Dk determined by Γres and microscopic scattering data. 3.2 Unied Consistency Principle **Unied Consistency Principle** The Universe Ontology Object U must satisfy: under all allowed variations δgab, δE, δΩ∂, δωbulk, δ(Γres, uµ, nk) (22) we have δI[U]=0. (23) In other words, the real universe is a structure that makes the consistency functional I[U] achieve stable extremum. Subsequent sections will demonstrate: EulerLagrange conditions of this unied consistency principle at dierent levels are precisely the physical laws we know. 4 Geometric Level: Small Causal Diamonds and Einstein Equations 4.1 Expansion of Small Causal Diamonds Introduce Riemann normal coordinates in neighborhood of p∈M , such that gab(p) = ηab, ∂cgab(p) = 0. (24) Take timelike unit vector ua , let γ(τ) be geodesic satisfying γ(0) = p, ˙γ(0) = u . For suciently small r , causal diamond Dp,r =J+(γ(−r)) ∩J−(γ(r)) (25) has volume and boundary area expansions V(Dp,r) = αdrdh1 + c1Rab(p)uaubr2+O(r3)i, (26) A(∂Dp,r) = βdrd−1h1 + c2Rab(p)uaubr2+O(r3)i. (27) 6 4.2 First-Order Variation of Generalized Entropy Generalized entropy is Sgen(Dp,r) = A(∂Dp,r) 4Gℏ+Sbulk(Dp,r). (28) Variation with respect to metric δgab gives δSbulk(Dp,r) = −1 2ZDp,r p|g| ⟨Tab⟩δgab ddx. (29) Substituting δA(∂Dp,r) and δSbulk into δIgrav ∼X Dp,rh1 4GℏδA(∂Dp,r) + δSbulk(Dp,r)−λentδ(Sgen −S∗ gen)i. (30) In limit r→0 , requiring δIgrav = 0 for any local δgab yields Gab + Λgab = 8πG⟨Tab⟩. (31) Thus, Einstein equations are necessary conditions of unied consistency principle at geometricentropy level, not independent axioms. 4.3 Second-Order Variation and Quantum Energy Conditions Further considering deformation along light ray directions, analyzing second-order variation of Sgen , and utilizing quantum information inequality δ2Sgen ≥0 (32) can derive local forms of quantum energy conditions and quantum focusing conjecture. This ensures stability of geometricentropy structure and consistency of time arrow under unied scale. 5 GaugeTopological Level: Boundary K -Class and Field Content Unication 5.1 Boundary Channel Bundle and K -Class On ∂M ×Λ , frequencyresolution dependent scattering matrix S(ω;ℓ) acts on channel space Hchan(ω, ℓ) at each (ω, ℓ) . These channel spaces glue into ber bundle E→∂M ×Λ, (33) with structure group restricted unitary group Ures . Its stable equivalence class [E]∈K(∂M ×Λ) (34) uniedly encodes: * Gauge groups and representations (determined by structure group and associated bundle); * Fermi/Bose statistics and chirality (determined by Z2 grading and spin structure); * Topological phases and protected boundary modes (determined by K -class invariants). 7 5.2 Consistency Variation and YangMills Equations Varying AYM under xed [E] , we have δIgauge =Z∂M×Λ trδAYM ∧⋆∇µFµν YMdd−1x+· · · , (35) requiring δIgauge = 0 for any δAYM yields ∇µFµν YM =Jν YM, (36) i.e., YangMills equations with source, where Jν YM comes from momentum conservation and boundarybulk coupling. Thus, gauge eld equations are EulerLagrange conditions of unied consistency principle at gaugetopological level. 5.3 Index Constraint and Field Content Selection There is a natural pairing between index of Dirac operator D[E] coupled to E Index(D[E])∈Z (37) and scattering K1 class [S]∈K1(∂M ×Λ) ⟨[E],[S]⟩= Index(D[E]). (38) Index term in consistency functional µKIndex(D[E]) (39) requires invariance under allowed variations, imposing conditions similar to anomaly cancellation: only those [E] are allowed that make index and scattering class pairing satisfy specic constraints. In other words, which gauge group and eld content the universe chooses is not an external assumption in this framework, but a solution to K -theory index and scattering consistency equations. 6 QuantumScattering Level: Relative Entropy Fixed Point and Local QFT 6.1 Relative Entropy Consistency On each small causal diamond D , actual bulk state restriction is ωD bulk , reference state ωD scat is given by scattering data S(ω;ℓ) , unied scale κ(ω) , and boundary time geometry via some scattering-to-state construction. Consistency term IQFT =X D SωD bulk∥ωD scat (40) has variational properties: * First-order variation is zero at ωD bulk =ωD scat ; * Second-order variation gives Fisher information metric, ensuring stability of minimum. Unied consistency principle requires ωD bulk =ωD scat to be a minimum for all D . 8 6.2 Wightman Axioms and QFT Reconstruction Under unied time scale and causality axioms, n -point functions Wn(x1, . . . , xn) of scattering scale reference state ωD scat satisfy: 1. Lorentz covariance and locality; 2. Microcausality (commutativity of operators at spacelike separation); 3. Spectrum condition (energy spectrum has lower bound); 4. Positivity and cluster decomposition. By Wightman reconstruction theorem, one can construct Hilbert space H , vacuum state Ω , local algebra family {A(O)} , and unitary representation U(a, Λ) , thereby obtaining local quantum eld theory Q= (H,Ω,{A(O)}, U). (41) Therefore, local QFT is not an independent assumption, but an inevitable result of unied consistency principle at quantumscattering level. 6.3 Field Equations and Ward Identities Further, varying scattering matrix S(ω;ℓ) and corresponding Green functions, requiring IQFT stability under xed unied scale and boundary K class, leads to: 1. Existence of local eld operators ϕa(x) satisfying EulerLagrange equations Ea(ϕ, ∂ϕ, . . . ) = 0, (42) whose mass spectrum and coupling constants are uniquely determined by scattering analytic invariants; 2. Internal symmetries correspond to Noether currents Jµ , whose Ward identities are given by invariance of IQFT under symmetry variation; 3. LSZ limit and LehmannSymanzikZimmermann structure ensure equivalence between eld theory and scattering descriptions. Thus, eld equations and symmetry structures are also unied back into consistent variational principle. 7 Coarse-Grained Limit: Unication of Fluid Dynamics and Many-Body Gradient Flow 7.1 Resolution Connection and Macroscopic Conserved Currents In long-wavelength and low-resolution limits, resolution connection Γres induces projection from microscopic degrees of freedom to macroscopic conserved quantities. For example, for energymomentum and particle number conservation, there are macroscopic currents Tµν hydro, Jµ a, (43) satisfying ∇µTµν hydro = 0,∇µJµ a= 0. (44) Consistency functional Ihydro =ZMhζ(∇µuµ)2η σµνσµν +X k Dk(∇µnk)2ip|g|ddx (45) minimization condition gives viscosity and diusion controlled macroscopic dynamics, i.e., generalized NavierStokes equations and diusion equations. 9