Spectral Windowing Unified Theory of Cosmological Constant and Dark Energy\\ \large Vacuum Energy Density in Unified Time Scale, Matrix Universe and QCA Universe

Spectral Windowing Unied Theory of Cosmological Constant and Dark Energy Vacuum Energy Density in Unied Time Scale, Matrix Universe and QCA Universe Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 19, 2025 Abstract Under the framework of unied time scale, phasespectral shiftdensity of states chain, and boundary time geometry, we provide a spectral windowing unied formulation for the cosmological constant and dark energy, implementing a discretized version in Matrix Universe and Quantum Cellular Automaton (QCA) Universe. First, on even-dimensional asymptotically hyperbolic or conformally compact geometries and static patch de Sitter backgrounds satisfying relative trace class and good scattering assumptions, we utilize BirmanKrein and LifshitsKrein trace formulas to unify generalized scattering phase derivative, spectral shift function derivative, and relative density of states (DOS) in frequency variable into a scale density κ(ω) = φ′(ω)/π = ∆ρω(ω) = (2π)−1tr Q(ω) , where Q(ω) is the WignerSmith group delay matrix. Provided the logarithmic frequency window kernel satises Mellin vanishing conditions, we establish a windowed Tauberian theorem: the nite part of the smalls heat kernel dierence is equivalent to the logarithmic window average of Θ′(ω) at scale µ∼s−1/2 , thereby completely rewriting vacuum energy density renormalization as a windowed integral of κ(ω) . Second, in the Matrix Universe representation, viewing FRW and de Sitter universes as scattering backgrounds containing horizon channels, we construct the cosmological scattering matrix Scos(ω) and its scale density κcos(ω) and window kernel Ξcos(ω) , proving that the eective cosmological constant increment Λeff(µ)−Λeff(µ0) can be expressed as the logarithmic frequency windowed spectral integral of the DOS dierence between Physical Universe and Reference Universe, thus reducing the cosmological constant problem to a relative spectral problem under unied time scale. Third, within the Universe QCA object UQCA = (Λ,Hcell,A, α, ω0) , replacing continuous spectrum ω with quasi-energy spectrum εj(k) , we dene the relative band density ∆ρj(k) of QCA Universe and Reference QCA, constructing the discrete windowed formula Λeff(µ) = PjRBZ Wµ(εj(k)) ∆ρj(k) ddk . Under conditions where high-energy bands satisfy symmetric pairing and inter-band harmonious sum rules, we prove that contributions to Λeff from high-energy regions are exponentially or power-law suppressed after windowing, leaving only nite residuals contributed by low-energy mass thresholds and topological modes within the redshift window, yielding a natural suppression estimate Λeff ∼m4 IR(mIR/MUV)γ with γ > 0 . 1 Finally, interfacing the spectral windowingband structure mechanism with running vacuum models in curved spacetime QFT, we construct the dark energy equivalent equation of state wde(z)≈ −1+δw(z) within the unied time scale framework, where δw(z) is controlled by the slow evolution of Ξ(ω) in the corresponding frequency band, compatible with current observational constraints of overall close to w=−1 allowing small amplitude evolution. Overall, this paper restates why the cosmological constant is far smaller than M4 Pl  as what kind of windowed sum rule the relative DOS satises under unied time scale, providing a self-consistent spectral theoretical framework for discussing the magnitude and running behavior of dark energy uniformly within Matrix Universe and QCA Universe. The cosmological constant problem is thus embedded into the broader program of unied time scalescatteringdiscrete universe. Keywords: Cosmological constant; Dark energy; Unied time scale; Scattering phase; Spectral shift function; Density of states; Heat kernel; Tauberian theorem; Matrix universe; Quantum cellular automaton; Vacuum energy density; Running vacuum model 1 Introduction & Historical Context 1.1 Cosmological Constant, Dark Energy, and Observational Picture In Einstein eld equations Gµν + Λgµν = 8πGTµν, (1) introducing the cosmological constant Λ allows describing de Sitter or antide Sitter solutions with constant curvature at the classical geometry level; in eective eld theory language, Λ corresponds to vacuum energy density ρΛ= Λ/(8πG) , with pressure satisfying pΛ=−ρΛ , i.e., equation of state parameter w=−1 . The standard Λ CDM model uses constant Λ as the simplest explanation for the universe's late-time accelerated expansion. Based on multiple observations including cosmic microwave background, Type Ia supernovae, baryon acoustic oscillations, and weak lensing, current joint analyses of dark energy density and equation of state indicate that nearly two-thirds of the universe's total energy density is contributed by dark energy, and the eective equation of state wde is highly close to −1 , with deviation δw =wde + 1 typically constrained within O(10−1) in joint ts. Latest reviews show multiple independent datasets remain compatible with w=−1 at 2σ level, but some analyses favor a slightly phantom-like equation of state w≲−1 , a trend potentially related to Hubble tension and σ8 tension. In a comprehensive review of dark energy equation of state, Escamilla et al. systematically compared constraints on various parameterized forms w(z) , noting that current data still provide only weak quantitative limits on redshift evolution of w(z) , but certain data combinations indeed favor evolution scenarios slightly deviating from −1 . This leaves observational room for whether the cosmological constant is strictly constant. 1.2 Cosmological Constant Problem and Running Vacuum Approach From a quantum eld theory perspective, free eld vacuum zero-point energy with UV cuto MUV formally yields ρvac ∼M4 UV . Taking MUV ∼MPl or typical particle physics 2 scales, compared to observed ρobs Λ∼10−120M4 Pl , there is a gap of 60 − −120 orders of magnitude, constituting the core of the cosmological constant problem. In curved spacetime QFT, vacuum energy density can be absorbed into the bare Λ term via renormalization, but this process does not provide an internal mechanism for why the total vacuum energy after renormalization leaves exactly a tiny positive number. Recent Running Vacuum Models (RVM) propose that in the context of cosmic expansion, vacuum energy density should be a function slowly evolving with curvature or Hubble scale, e.g., ρvac(H) = ρ0 vac +3ν 8πG(H2−H2 0) + ··· , (2) where ν is a small dimensionless coecient. By implementing asymptotic renormalization in curved spacetime, part of the dangerous m4 order terms can be eectively removed, making late-time universe vacuum energy exhibit weak running behavior without introducing extra scalar eld degrees of freedom. RVM-type models have shown potential in alleviating H0 and σ8 tensions. Nevertheless, most discussions remain at the level of assuming some functional form of ρvac(H) and tting with observations, lacking a unied way to naturally derive running vacuum behavior and explain the small value of vacuum energy starting from spectral and scattering structures. 1.3 Spectral and Scattering Perspective: Phase, Spectral Shift, and DOS In mathematical physics, a series of profound connections exist between pairs of selfadjoint operators (H, H0) and their scattering matrix S(ω) , spectral shift function ξ(ω) , and density of states dierence ∆ρ(ω) . The BirmanKrein formula links scattering determinant with spectral shift function, and LifshitsKrein trace formula expresses the trace of function f(H)−f(H0) as an integral over the spectral shift function. Guillarmou and collaborators further established generalized Krein formulas and spectral properties of scattering operators on asymptotically hyperbolic and conformally compact Einstein manifolds, allowing exact alignment of KV determinant phase with generalized spectral shift function. On open scattering manifolds, Dyatlov utilized classical escape rates and expansion rates to give rened estimates for Weyl-type asymptotics and remainder terms of scattering phase, showing high-energy information in scattering phase can be controlled by classical dynamical system invariants. Vasy developed systematic microlocal analysis methods on conformally compact asymptotically hyperbolic spaces, achieving high-energy resolvent continuation and estimation for Laplace operators, providing control tools for linking heat kernel with scattering spectrum. These results indicate that under appropriate geometric assumptions, quantities on the spectral and scattering side (like scattering phase derivative) can be precisely connected to quantities on the geometry and heat kernel side (like heat kernel dierence and SeeleyDeWitt coecients), providing a natural entry point for rewriting the cosmological constant using spectral quantities. 3 1.4 Quantum Cellular Automata and Discrete Universe Quantum Cellular Automata (QCA), as discrete time unitary evolution models with locality, translation invariance, and nite propagation radius, have been systematically proven to recover free or interacting quantum eld theories in the continuous limit, including Dirac elds, scalar elds, and even (1+1) D and (3+1) D electrodynamics. These works show that under natural conditions like locality, Lorentz symmetry approximation, and gauge invariance, standard QFT dispersion relations and propagation properties can be recovered from simple discrete local rules. From this perspective, abstracting the universe as a whole into a QCA object UQCA satisfying certain axioms, recovering general relativity and eld theory in its continuous limit, and then embedding the cosmological constant and dark energy problem into QCA's spectral and band structure, becomes a natural unied approach: the universe itself is discrete, while continuous geometry and eld theory are merely its large-scale approximations. 1.5 Unied Time Scale and Spectral Windowing Strategy of This Paper Prior work has constructed the unied time scale mother scale κ(ω) = φ′(ω)/π =ρrel(ω) = (2π)−1tr Q(ω), (3) where φ(ω) is total scattering hemi-phase, ρrel(ω) relative density of states, and Q(ω) WignerSmith group delay matrix. This mother scale unies scattering phase gradient, relative DOS, and group delay trace into a single scale density, providing a unied source for dening unied time parameters across various geometric and physical scenarios. Based on this, the goals of this paper are: 1. On continuous geometries satisfying appropriate scattering and geometric assumptions, establish a windowed Tauberian theorem precisely corresponding the nite part of smalls heat kernel dierence to logarithmic window average of Θ′(ω) at scale µ∼s−1/2 , thereby rewriting vacuum energy renormalization as logarithmic frequency windowed spectral integral of κ(ω) . 2. In Matrix Universe THEMATRIX representation, view FRW and de Sitter universes as scattering backgrounds, construct cosmological scattering matrix Scos(ω) and relative DOS, proving eective cosmological constant Λeff(µ) is equivalent to windowed integral of DOS dierence between Physical Universe and Reference Universe. 3. In Universe QCA object, replace continuous spectrum with band structure εj(k) , dene QCA version relative band density and windowed vacuum energy, analyze under what band pairing symmetry and sum rule conditions high-energy contributions to Λeff naturally cancel, leaving only small residuals controlled by IR thresholds and topological modes. 4. At eective cosmological level, link windowed spectral expression to running vacuum models and dark energy equation of state wde(z) , demonstrating structural origin of wde(z)≈ −1 + δw(z) under unied time scale framework, and comparing with current observational constraints. These steps collectively form a multi-layer unied structure from scattering spectrum to dark energy, restating the cosmological constant problem as a relative spectral problem in Matrix Universe and QCA Universe. 4 2 Model & Assumptions 2.1 Scattering Pair, Spectral Shift Function, and Unied Scale Density Consider a pair of self-adjoint operators (H, H0) dened on the same Hilbert space H . Assume H−H0 is a relative trace class perturbation in appropriate sense, ensuring welldened xed-energy scattering theory, and existence of LifshitsKrein trace formula for all Borel bounded measures f tr(f(H)−f(H0)) = ZR f′(λ)ξE(λ) dλ, (4) where ξE(λ) is spectral shift function in energy variable λ . In frequency variable ω≥0 , let λ=ω2 , dene ξ(ω) = ξE(ω2), (5) then relative DOS in energy and frequency representations are respectively ∆ρE(λ) = −ξ′ E(λ), (6) ∆ρω(ω)=2ω∆ρE(ω2) = −∂ωξ(ω). (7) Let S(ω) be xed-energy scattering matrix, Q(ω) = −iS(ω)†∂ωS(ω) WignerSmith group delay matrix. BirmanKrein formula gives det S(ω) = exp−2πiξ(ω), (8) from which dene scattering determinant phase Θ(ω) = (2π)−1arg det S(ω), (9) and have Θ′(ω) = −∂ωξ(ω) = ∆ρω(ω) = (2π)−1tr Q(ω). (10) Unied time scale density is dened as κ(ω) = φ′(ω) π= ∆ρω(ω) = 1 2πtr Q(ω), (11) where φ(ω) is total scattering hemi-phase. In this paper, all constructions regarding unied time scale use κ(ω) as the sole scale source. 2.2 Heat Kernel Dierence, Geometric Assumptions, and Scattering Operator Let H be generalized Laplace-type operator on non-compact Riemann manifold (X, g) , H0 on corresponding reference background (X0, g0) , both having same geometric and potential structure at innity. Assume (X, g) and (X0, g0) are even-dimensional asymptotically hyperbolic or conformally compact Einstein manifolds, satisfying GuillarmouVasy even metric condition, such that resolvent (H−λ)−1 has good meromorphic continuation 5 on appropriate Riemann surface of λ plane, and scattering operator S(λ) is classical pseudodierential operator near critical line. Under these assumptions, Guillarmou proved Krein-type formula between KV determinant phase and generalized spectral shift function, while diagonal trace of heat kernel KH(s, x, y) = exp(−sH)(x, y) tr(exp(−sH)−exp(−sH0)) = ∆K(s) (12) has standard SeeleyDeWitt asymptotic expansion as s→0+ , whose nite part can be expressed via Mellin transform of scattering phase. This paper specically focuses on symmetric even-dimensional cases and static patch de Sitter backgrounds, where connection between scattering phase and heat kernel difference is most concise, suitable for constructing logarithmically windowed Tauberian theorems. 2.3 Logarithmic Frequency Window Kernel and Tauberian Conditions To convert nite part of smalls heat kernel dierence into logarithmic average in frequency domain, introduce a family of logarithmic frequency window kernels W(ln(ω/µ)), (13) where µ > 0 is spectral scale, W∈C∞ 0(R) is smooth compact support function, satisfying following Mellin vanishing conditions: 1. ZR W(u) du= 0 ; 2. ZR e2nuW(u) du= 0 for several low integers n . These conditions ensure window kernel cancels corresponding dominant singular terms in heat kernel as ω→0 and ω→ ∞ , extracting nite geometric information. Dene logarithmic average ⟨Θ′⟩W(µ) = ZR Θ′(ω)W(ln(ω/µ)) d ln ω, (14) and further dene spectral window kernel ΞW(µ) = ∂ln µ⟨Θ′⟩W(µ). (15) In Tauberian theory, if ∆K(s) and Θ′(ω) satisfy appropriate regularity and growth conditions, equivalence can be established between nite part of smalls heat kernel and ΞW(µ) under hyperbolic scaling µ∼s−1/2 . This paper will provide a nite-order version sucient for cosmological applications. 2.4 Cosmological Geometry and Reference Background In cosmological applications, H and H0 correspond to wave operators on geometric backgrounds of Physical Universe and Reference Universe respectively. For example: 1. (M, g) is FRW with curvature perturbations or de Sitter universe with structure, H is LaplaceBeltrami operator plus potential for scalar or tensor perturbations. 2. (M0, g0) is smooth, same-topology structureless FRW or pure de Sitter background, H0 corresponding unperturbed operator. 6 Reference background should possess physical reasonableness, e.g., same topology, same cosmological constant, but no complex structure. Spectral dierence ∆ρω of physical universe relative to reference universe characterizes spectral deviation of real structure on background, its windowed integral gives eective increment of cosmological constant. 2.5 Matrix Universe and Cosmological Scattering Matrix In Matrix Universe THEMATRIX framework, consider channel Hilbert space decomposed by frequency Hchan =M v∈VHv, (16) where v labels dierent directions, polarizations, cosmological modes, and horizon channels. Cosmological scattering matrix Scos(ω)∈ B(Hchan) (17) encodes scattering process from Reference Universe modes to Physical Universe modes. Its determinant phase Θcos(ω) , spectral shift function ξcos(ω) , and relative DOS ∆ρcos(ω) satisfy same chain relations as general scattering pairs. Unied time scale density in cosmological case is written as κcos(ω) = 1 2πtr Qcos(ω), (18) where Qcos(ω) = −iScos(ω)†∂ωScos(ω) . This paper will use κcos(ω) to control spectral windowed expression of cosmological constant. 2.6 QCA Universe Object and Band Structure Universe QCA object is dened as UQCA = (Λ,Hcell,A, α, ω0), (19) where Λ is countable connected graph (usually Zd ), Hcell nite-dimensional Hilbert space per cell, A quasilocal C∗ algebra, α:Z→Aut(A) time evolution automorphism with nite propagation radius, ω0 initial universe state. In momentum space k∈BZ , single-step evolution operator U ber decomposes as U(k)∈U(N), (20) with eigendecomposition U(k)|ψj,k⟩=e−iεj(k)∆t|ψj,k⟩, (21) where N= dim Hcell , quasi-energy εj(k)∈(−π/∆t, π/∆t] . In continuous limit ∆t→0 , dispersion relation εj(k)≈Ej(k)∆t , where Ej(k) is energy spectrum of corresponding continuous eld theory. Dene single-band DOS ρj(E) = ZBZ δ(E−Ej(k)) ddk (2π)d, (22) total DOS ρ(E) = Pjρj(E) . Selecting Reference QCA U0(k) and its spectrum {Ej,0(k)} , dene relative DOS ∆ρ(E) = ρ(E)−ρ0(E), (23) and its band decomposition ∆ρj(k) . These quantities correspond to continuous scattering theory in continuous limit. 7 3 Main Results This section gives main theorems of this paper, proof details in Section 4 and Appendix. 3.1 Windowed Tauberian Theorem and Scale Density Formulation Theorem 3.1 (Windowed Tauberian Theorem) . Let (H, H0) satisfy geometric and scattering assumptions of Section 2.2, let ∆K(s) = tr(e−sH −e−sH0) (24) be heat kernel dierence, Θ(ω) scattering determinant phase, Θ′(ω)=∆ρω(ω) = (2π)−1tr Q(ω) . Take a family of logarithmic window kernels W satisfying Mellin vanishing conditions of Section 2.3, dene ΞW(µ) = ZR ωΘ′′(ω)W(ln(ω/µ)) d ln ω. (25) Then there exists hyperbolic scaling µ∼s−1/2 and constants C, γ > 0 such that as s→0+ , FPs→0∆K(s) = κΛΞW(µ) + O(sγ), (26) where FP denotes nite part, κΛ normalization constant depending only on window kernel choice. In other words, nite part of smalls heat kernel dierence is Tauberian equivalent to logarithmic window average of unied time scale density κ(ω) . 3.2 Windowed Unied Expression of Cosmological Constant Theorem 3.2 (Windowed Cosmological Constant Mother Formula) . Under conditions of Theorem 3.1, let Λeff(µ) be renormalized value of cosmological constant parameter in eective eld theory action at observation scale µ , then there exists spectral window kernel Ξ(µ) such that ∂ln µΛeff(µ) = κΛΞ(µ), (27) explicitly Ξ(µ) = ZR ωΘ′′(ω)W(ln(ω/µ)) d ln ω, (28) and for any reference scale µ0 , Λeff(µ)−Λeff(µ0) = Zµ µ0 Ξ(ω) d ln ω. (29) Here Ξ(ω) has dimension L−2 , viewable as cosmological constant spectral window kernel, fully determined by unied time scale density κ(ω) and scattering phase. 3.3 Cosmological Scattering and Relative DOS in Matrix Universe Theorem 3.3 (Relative DOS Formulation in Matrix Universe) . In Matrix Universe THEMATRIX representation, take Physical Universe (M, g) and Reference Universe 8 (M0, g0) and their corresponding scattering matrices Scos(ω) and Scos,0(ω) . Dene relative cosmological scattering matrix Srel(ω) = Scos(ω)Scos,0(ω)−1, (30) and relative DOS ∆ρcos(ω) = ∆ρω(ω;H, H0). (31) Under conditions of Theorem 3.2, Λeff(µ)−Λeff(µ0) = Zµ µ0 Ξcos(ω) d ln ω, (32) where Ξcos(ω) = ZR ωΘ′′ rel(ω)W(ln(ω/µ)) d ln ω, (33) Θrel(ω) = (2π)−1arg det Srel(ω) , and Θ′ rel(ω) = ∆ρcos(ω) = (2π)−1tr Qrel(ω). (34) Thus Λeff(µ) can be interpreted as logarithmic frequency windowed integral of DOS difference between cosmological scattering matrices of Physical and Reference Universes. 3.4 Discrete Windowed Formula and High-Energy Suppression in QCA Universe Theorem 3.4 (Windowed Vacuum Energy Formula in QCA Universe) . In Universe QCA object UQCA , let Physical QCA and Reference QCA have energy spectra {Ej(k)} and {Ej,0(k)} respectively, dene single-band relative DOS ∆ρj(k) = δ(E−Ej(k)) −δ(E−Ej,0(k)). (35) Let Wµ(E) be energy weight function mapped from continuous window kernel W(ln(ω/µ)) , then there exists normalization constant CW such that Λeff(µ) = CWX jZBZ Wµ(Ej(k)) ∆ρj(k) ddk, (36) consistent with continuous spectrum expression of Theorem 3.2 in continuous limit ∆t→ 0 . Theorem 3.5 (QCA High-Energy Suppression Mechanism) . Under conditions of Theorem 3.4, if high-energy bands satisfy following two types of conditions: 1. Band Structure Symmetric Pairing: Exists band pair (j, j′) , for |E| ≥ Ec Ej′(k)≈ −Ej(k), (37) ∆ρj′(E)≈∆ρj(−E), (38) and window weight approximately even symmetric in high-energy region Wµ(E)≈ Wµ(−E) . 2. Inter-Band Harmonious Sum Rule: Exists UV scale EUV such that ZEUV 0 E2∆ρ(E) dE= 0, (39) 9 6.1 Numerical Reconstruction of Cosmological Scattering Spectrum In Matrix Universe perspective, elements of cosmological scattering matrix Scos(ω) essentially correspond to scattering amplitudes of dierent modes on FRW or de Sitter background. Can be numerically reconstructed via following steps: 1. Select a family of mode functions for scalar or tensor perturbation equations, numerically solve scattering amplitudes on given background (M, g) and reference background (M0, g0) . 2. Construct frequency-dependent scattering matrices Scos(ω) and Scos,0(ω) via numerical linear algebra, calculate relative matrix Srel(ω) . 3. Estimate scattering phase Θrel(ω) and derivative Θ′ rel(ω) on discrete frequency grid, obtaining relative DOS ∆ρcos(ω) . 4. Choose logarithmic window kernel W satisfying Tauberian conditions, calculate numerical estimates of spectral window kernel Ξcos(µ) and Λeff(µ) . This numerical workow provides direct path to verify validity of windowed Tauberian theorem and specic shape of spectral window kernel Ξ(ω) . 6.2 Design and Simulation of QCA Spectral Structure In QCA universe scheme, need to specically construct QCA models satisfying band pairing and sum rule conditions. Engineering steps: 1. On 1D or 2D lattice, starting from known constructions of Dirac-type and gauge eld QCA, add extra internal degrees of freedom and coupling parameters, making energy spectrum exhibit E↔ −E symmetric pairing in high-energy region. 2. Calculate DOS via numerical diagonalization and Brillouin zone integration, verify sum rule conditions between high-energy bands. 3. Evolve QCA on quantum simulation or classical HPC platforms, measure quasi-energy spectrum distribution and behavior of band structure with parameter variation, explore numerical magnitude of Λeff under dierent IR and UV scales. 4. Compare obtained windowed vacuum energy with ν values tted from RVM, test if QCA spectral structure can naturally produce parameter region |ν| ≪ 1 . 6.3 Spectral Windowing Method in Cosmological Data Analysis In actual cosmological data processing, can attempt introducing logarithmic frequency windowing method: 1. In data analysis of CMB, BAO, and Type Ia supernovae, replace parameterization from directly assuming form of ρde(z) to simple parameterization of Ξ(ω) , e.g., piecewise constant or low-order polynomial. 2. Encode relation between Ξ(ω) , Λeff(µ) , and wde(z) into cosmological parameter inference chain, making tted parameters directly describe spectral window kernel rather than equation of state. 3. Compare Bayesian evidence and parameter correlation of direct t w(z)  vs t Ξ(ω) then derive w(z) , test if spectral windowing framework captures data preference more naturally. This scheme can be viewed engineering-wise as a spectral variable coordinate transformation for existing cosmological MCMC frameworks, potentially oering advantages in parameter correlation and interpretability. 16 7 Discussion (risks, boundaries, past work) 7.1 Risks and Limitations The spectral windowing unied framework proposed in this paper relies on several nontrivial assumptions: 1. Geometric and Scattering Assumptions: Require (M, g) and (M0, g0) fall into asymptotically hyperbolic or conformally compact geometric categories, and satisfy even metric condition and good scattering theory, which has not been rigorously proven at full spacetime level of real universe. 2. Window Kernel and Tauberian Conditions: Windowed Tauberian theorem relies on high-energy estimates of resolvent and growth bounds of scattering phase. Current mathematical literature results on spacetimes like Kerrde Sitter are incomplete, generalizing to universe with complex matter distribution and non-stationary background requires more work. 3. Realization of QCA Sum Rule: Constructing spectral structures satisfying band pairing and sum rule in specic QCA models is a non-trivial engineering problem, requiring careful design under constraints of locality, causality, and symmetry. These risks imply many conclusions of this paper should currently be viewed as structural inferences under explicit assumptions, rather than theorems rigorously veried in all physical scenarios. 7.2 Relation to Existing Work Regarding cosmological constant problem, vast literature explores weak running behavior of vacuum energy density with H or curvature via renormalization and running vacuum models. In comparison, contribution of this paper lies in: 1. Explicitly rewriting cosmological constant problem as relative spectral problem on scattering phasespectral shiftDOS chain, controlling all scale dependence with uni- ed time scale density κ(ω) . 2. Introducing logarithmic frequency window kernel and nite-order Tauberian theorem, unifying nite part of heat kernel dierence with scattering spectral information, giving Λeff(µ) expression fully accounted for in dimension and variable. 3. Revealing structural mechanism of natural suppression of vacuum energy via high-energy spectral pairing and inter-band sum rule in QCA universe framework, attributing magnitude of Λeff to power-law ratio between IR and UV scales, rather than ne-tuning of individual degrees of freedom. Regarding unication of QCA and QFT, this paper follows and extends systematic analysis of QCA continuous limits by Farrelly, BisioD'Ariano, Sellapillay, and Brun, introducing discrete version of cosmological constant windowing on this basis. 7.3 Future Work Directions Future work can proceed along following directions: 1. On specic cosmological backgrounds (e.g., structured de Sitter, non-at FRW), utilizing Vasy's microlocal analysis methods and Guillarmou's scattering operator theory, rigorously establish Tauberian theorems satisfying conditions required by this paper. 2. Construct high-dimensional QCA models with clear QFT continuous limits, systematically explore their band structure and DOS, nding natural parameter regions satisfying sum rule conditions. 3. Interface spectral windowing framework with other unied time 17 scale related results like black hole entropy, NullModular double cover, generalized entropy conditions, forming a Universe Terminal Object structure on larger scale. 8 Conclusion Based on unied time scale, scattering phasespectral shiftdensity of states chain, and QCA universe axioms, this paper provides a systematic spectral windowing restatement of cosmological constant and dark energy problem. By establishing windowed Tauberian theorem in even-dimensional asymptotically hyperbolic and de Sitter geometries, equivalence is made between nite part of smalls heat kernel dierence and logarithmic frequency window average of scattering phase derivative, thereby obtaining a cosmological constant mother formula fully determined by unied scale density κ(ω) and window kernel W . In Matrix Universe representation, cosmological scattering matrix of Physical Universe relative to Reference Universe gives relative DOS, windowed integral rewrites cosmological constant as accumulation of relative spectral quantities; in QCA Universe, symmetric pairing of band structure and inter-band harmonious sum rule naturally bring cancellation of high-energy parts and power-law suppression of vacuum energy, making magnitude of Λeff mainly controlled by ratio of IR scale to UV scale. Finally, connecting spectral windowing results with running vacuum models and dark energy observational constraints shows unied time scale framework can naturally produce equivalent equation of state approximating wde(z)≈ −1 and possibly slowly evolving, compatible with current data. Overall, cosmological constant problem transforms from dicult problem of summing absolute zero-point energies to windowed sum rule problem of relative DOS under unied time scale, laying foundation for realizing more complete unication in Matrix Universe and QCA Universe. Acknowledgements The authors thank colleagues and anonymous reviewers in related elds for discussions and suggestions on scattering theory, QCA continuous limits, and cosmological data analysis, which signicantly inuenced structure and presentation of this paper. Code Availability Main results of this paper are based on analytical derivations. Numerical verication involving heat kernel, scattering phase, and QCA DOS can be implemented on open source platforms using general numerical linear algebra libraries and spectral methods. Relevant example codes and numerical scripts can be organized and made public in subsequent work, or provided upon reasonable request. References [1] Particle Data Group, Dark Energy, Review of Particle Physics , 2024 update. 18 [2] J. Solà Peracaula, The Cosmological Constant Problem and Running Vacuum in the Expanding Universe, Philosophical Transactions of the Royal Society A 380, 2022. [3] J. Solà Peracaula et al., Cosmological Constant vis-à-vis Dynamical Vacuum: Bold Challenging the Λ CDM, International Journal of Modern Physics A 31, 2016. [4] J. de Cruz Pérez, J. Solà Peracaula, Running Vacuum in BransDicke Theory: A Possible Cure for the H0 and σ8 Tensions, Astroparticle Physics , 2024. [5] L. A. Escamilla et al., The State of the Dark Energy Equation of State circa 2023, Journal of Cosmology and Astroparticle Physics , 2024. [6] C. Guillarmou, Generalized Krein Formula, Determinants and Selberg Zeta Function for Convex Co-Compact Manifolds, Communications in Mathematical Physics 277, 2008. [7] C. Guillarmou, Spectral Characterization of PoincaréEinstein Manifolds, Journal of Dierential Geometry 83, 2009. [8] A. Vasy, Microlocal Analysis of Asymptotically Hyperbolic Spaces and High-Energy Resolvent Estimates, MSRI Publications 60, 2012. [9] S. Dyatlov, Scattering Phase Asymptotics with Fractal Remainders, Communications in Mathematical Physics 339, 2015. [10] T. Farrelly, A Review of Quantum Cellular Automata, Quantum 4, 368 (2020). [11] G. M. D'Ariano, P. Perinotti, A. Bisio, From Quantum Cellular Automata to Quantum Field Theory, various works, 20142017. [12] K. Sellapillay et al., A Discrete Relativistic Spacetime Formalism for 1+1 QED from QCA, Scientic Reports 12, 2022. [13] T. A. Brun, G. Chiribella, C. M. Scandolo, Quantum Electrodynamics from Quantum Cellular Automata, Entropy 27, 2025. A Spectral Data, Scattering Matrix, and Krein Trace Formula This appendix provides detailed statement of spectral and scattering theory background used in this paper. A.1 Self-Adjoint Pairs and Spectral Shift Function Let H and H0 be self-adjoint operators satisfying (H−i)−1−(H0−i)−1 is trace class. Then for any Schwartz function f , tr(f(H)−f(H0)) = ZR f′(λ)ξE(λ) dλ, (79) 19 where ξE(λ) is spectral shift function. Derivative of spectral shift function at almost all λ gives relative DOS: ∆ρE(λ) = −ξ′ E(λ) . In presence of xed-energy scattering theory, scattering matrix S(λ) and spectral shift function are related by BirmanKrein formula: det S(λ) = exp−2πiξE(λ). (80) On asymptotically hyperbolic and conformally compact manifolds, Guillarmou extended this relation to more general KV determinant background through in-depth analysis of scattering operator and relative determinant, unifying scattering determinant phase with generalized spectral shift function. A.2 WignerSmith Group Delay and Unied Scale Density Frequency derivative of scattering matrix S(ω) gives WignerSmith group delay matrix Q(ω) = −iS(ω)†∂ωS(ω). (81) Under trace class condition, trace tr Q(ω) = ∂ωarg det S(ω) = 2πΘ′(ω), (82) thus Θ′(ω) = (2π)−1tr Q(ω). (83) Unied time scale density is dened as κ(ω)=Θ′(ω)=∆ρω(ω). (84) In Matrix Universe and QCA Universe, denition of Q(ω) can be generalized to cosmological scattering matrix and QCA scattering map, making unied time scale span continuous and discrete frameworks. B Technical Details of Windowed Tauberian Theorem B.1 Mellin Transform and Logarithmic Window Kernel Mellin transform of function f(ω) dened as M[f](s) = Z∞ 0 ωs−1f(ω) dω. (85) Introduction of logarithmic window kernel W(ln(ω/µ)) makes ⟨f⟩W(µ) = ZR f(ω)W(ln(ω/µ)) d ln ω (86) correspond in Mellin space to ⟨f⟩W(µ) = 1 2πiZΓM[f](s)c W(s)µ−sds, (87) where c W(s) = ZR esuW(u) du. (88) Mellin vanishing conditions ensure c W(s) has zeros at several low-order points, cancelling dominant singular terms of heat kernel expansion in residue calculation, retaining only nite geometric part. 20 B.2 From Scattering Phase to Heat Kernel Finite Part Substituting f(ω)=Θ′(ω) and f(ω)=∆ρω(ω) , simple relation exists between corresponding M[f](s) and LaplaceMellin transform of heat kernel ∆K(s) . By choosing positions of Γ and Γ′ , can simultaneously analyze smalls behavior of ∆K(s) and largeω behavior of Θ′(ω) on complex plane, obtaining via HardyLittlewood Karamata type Tauberian theorem FPs→0∆K(s)∼κΛΞW(µ), (89) and giving explicit error bounds. In cosmological applications, nite-order Tauberian version is sucient: only need to control error up to certain nite order sγ . C QCA Band Pairing, Sum Rule, and Suppression Exponent C.1 Construction of Band Pairing Structure In specic QCA models, band pairing can be achieved via following strategies: 1. Require local update rules to possess generalized particleantiparticle symmetry and time reversal symmetry in appropriate sense, producing E↔ −E symmetry in energy spectrum. 2. Introduce extra Z2 or Z4 symmetry in internal degrees of freedom, making high-energy bands appear in pairs, with coupling structure automatically satisfying symmetric pairing condition. Such structures partially appear in Dirac-type and QED-type QCA, requiring further parameter tuning and constraints to obtain spectral structures satisfying assumptions of Theorem 3.5. C.2 Spectral Condition of Sum Rule Inter-band harmonious sum rule ZEUV 0 E2∆ρ(E) dE= 0 (90) can be understood as a kind of relative energy squared conservation: Physical QCA and Reference QCA have same energy squared weighted density of states in UV region. In Dirac-type models, if Reference QCA and Physical QCA dier in UV region only by IR mass and nite topological modes, this sum rule can be naturally satised or achieved by adjusting nite high-energy coupling parameters. By numerical tting of DOS, can verify approximate validity of sum rule, and estimate impact of its deviation on Λeff , determining magnitude of suppression exponent γ . C.3 Suppression Exponent and Parameter Space In actual calculation, multi-scale analysis of QCA spectrum yields Λeff(µ)≤C0E4 IR +C1E4 IREIR/EUVγ1+··· , (91) 21 where C0 relates to low-energy DOS structure, γ1 to precision of sum rule. By searching for regions in parameter space where C0 is also signicantly reduced, suppression eect can be further enhanced, explaining observationally tiny Λeff . 22