Windowed Formulation of Phase--Spectral-Shift--DOS--Cosmological-Constant

Windowed Formulation of PhaseSpectral-ShiftDOSCosmological-Constant Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract We establish equivalence chain centered on generalized scattering phase , connecting KontsevichVishik (KV) determinant phase, generalized Kren spectral shift, density-of-states dierence (DOS dierence), and WignerSmith (WS) trace identity, providing rigorous variable and factor accounting under even-dimensional asymptotically hyperbolic/conformally compact (AH/CCM) and static-patch de Sitter (dS) geometry. Under strict LifshitsKren (LK) trace formula and relative trace-class assumptions, we prove substitution relation between heat kernel dierence Laplace representation and DOS slope under frequency variable ∆K(s) = Z∞ 0 e−sω2Θ′(ω)dω, where Θ(ω) = 1 2πarg detKV S(ω) , Θ′= ∆ρω=−∂ωξω . We propose family of logarithmic frequency windows W with Mellin-nullication conditions, establishing windowed Tauberian theorem : under non-trapping, no zero-energy resonance, analytic Fredholm and operatorLipschitz assumptions, smalls heat kernel nite part equivalent to logarithmic window average of Θ′ at scale µ∼s−1/2 , with error upper bound. This denes windowed integration law ∂ln µΛϕ,W (µ) = κΛΞW(µ),ΞW(µ) = ZR ωΘ′′(ω)Wln(ω/µ)dln ω=1 2πZω ∂ωTr Q(ω)W d ln ω, providing dimensionally consistent constant separation: κHK (heat kernelwindowing ratio) and κΛ (dimensional factor mapping ⟨Θ′⟩W to cosmological constant). For open channels (horizon/absorption), respectively establish provable conditions for ∂ωarg detKV b S=−iTr b S†∂ωb S via extended unitarization and relative determinant , preserving WS-trace equality. Demonstrate κ extraction procedure with one-dimensional δ potential as solvable template and staticpatch dS scalar as computable template. Propose minimal reproducible observation pipeline based on FRB baseband complex phase, providing closed-form formulas for second-order phase kernel variance scaling, dispersion/multipath leakage kernel, and injectionrecovery e- cacy analysis. Above theorems and engineering schemes jointly constitute veriable windowed formulation. Keywords : Generalized scattering phase; KV determinant; generalized Kren spectral shift; LifshitsKren trace formula; WignerSmith trace; heat kernel nite part; Tauberian; logarithmic window; relative determinant; static-patch de Sitter; FRB baseband 1 Introduction and Historical Background On even-dimensional AH/CCM geometry, Guillarmou denes KV determinant via renormalized trace (KontsevichVishik trace, TR) yielding 1 arg det KV Sn 2+iω=−2π ξ(ω) (mod 2π), where ξ is generalized Kren spectral function; its logarithmic derivative couples to TR-trace of scattering operator S , providing geometrized version of phase = spectral shift. This framework compatible with FriedelLloydBirmanKren (BK) relation, rigorously formulated in evendimensional AH/CCM scenarios. Sá BarretoWang prove: on non-trapping AH, S(ω) is Fourier integral operator (FIO) quantizing scattering relation, ensuring dierentiability, symbol and kernel regularity for ω > 0 , laying foundation for dierentiable framework of WS-trace and KV-det. Peller characterizes function classes applicable to LifshitsKren trace formula: for operatorLipschitz f , Tr(f(H)−f(H0)) = Zf′(λ)ξE(λ)dλ, holds under relative trace-class (or weaker relative class-trace) conditions. This paper places f(λ) = e−sλ , through substitution λ=ω2 , connecting heat kernel dierence with frequency-domain DOS slope. In electromagnetic multiport systems, trace of WS time-delay matrix Q=−i S†∂ωS equivalent to total phase derivative: ∂ωarg det S= Tr Q . This equality measurable at both experimental and algorithmic levels, constituting observation-end interface. In black hole/static-patch dS scenarios,  relative DOSpartition functionphase shift  provides computable paradigm removing continuous spectrum divergence, combining scattering phase with one-loop free energy/partition function. This paper uses this as template for acyclic calibration of absolute constants. This paper's goal is complete alignment of above chain on frequency variable , establishing Tauberian theorem for logarithmic frequency windowing , rigorously stating dierentiability of KV-det under open channels and trace formula applicability domain, proposing implementable FRB baseband observation test. 2 Model and Assumptions 2.1 Geometry and Operators We work on two types of backgrounds:  Even-dimensional AH/CCM : (Xn+1, g) even-dimensional conformally compact scattering geometry, satisfying non-trapping and no zero-energy resonance/embedded eigenvalue . Laplace-type operator H=−∆g+V (with regular potential) and reference operator H0 form relative pair.  Static-patch de Sitter (dS) : Take static-patch region with horizon as boundary, adopt physical in/out channels, construct extended channels S or relative operator b S:= SS−1 ref . For AH/CCM, S(ω) is FIO with kernel smooth in ω ; for dS, extends to unitary after extended channels. 2 2.2 KV Determinant and Generalized Kren Denote Φ(ω) := arg detKV S(ω) , Θ(ω) := Φ(ω)/(2π) . Guillarmou establishes Φ(ω) = −2π ξω(ω) (mod 2π),Θ′(ω) = −∂ωξω(ω). where ξω(ω) := ξE(λ)λ=ω2 . 2.3 DOS and Variable Transformation In energy variable λ , ∆ρE(λ) = −ξ′ E(λ) . In frequency variable ω , ∆ρω(ω)=2ω∆ρE(ω2) = −∂ωξω(ω). Dene Θ′(ω)=∆ρω(ω),Tr Q(ω) = ∂ωΦ(ω) = 2π∆ρω(ω). Last equality is WS-trace. 2.4 Window Family and Mellin-Nullication Take compactly supported window W∈C∞ 0(R) , dene logarithmic window average ⟨f⟩W(µ) = ZR f(µeu)W(u)du =Z∞ 0 f(ω)Wln(ω/µ)dln ω. Denote Mellin transform c W(z) = RRezuW(u)du . If f(ω)∼Pkckωβk+Pm,j dm,j ω˜ βm(ln ω)j has high-frequency power-logarithmic asymptotics, require window to satisfy c W(βk) = 0,dj dzjc W(z)z=˜ βm= 0 (0 ≤j≤Jm), nullifying power and logarithmic power terms. 3 Main Results Theorem 1 (1: PhaseSpectral-ShiftDOSWS Unication) . Under assumptions of 2.1, for ω > 0 , Θ′(ω) = ∆ρω(ω) = −∂ωξω(ω),Tr Q(ω) = ∂ωΦ(ω)=2π∆ρω(ω). Moreover, Φ(ω) = −2π ξω(ω) (mod 2π) . Proof in 4.1. Theorem 2 (2: Heat KernelFrequency Domain Substitution, No Extra 2ω ) . Assume f(λ) = e−sλ satisfying LK conditions, then heat kernel dierence ∆K(s) := Tre−sH −e−sH0=Z∞ 0 e−sω2Θ′(ω)dω, holds when ξE(0+) = 0 or interpreting smallλ endpoint via Hadamard nite part. Proof in 4.2. Note : If writing in energy variable as R∞ 0e−sλ ∆ρE(λ)dλ , through substitution λ=ω2 and ∆ρω= 2ω∆ρE canceling Jacobian, right side contains no extra 2ω , dimensionally self-consistent. 3 Theorem 3 (3: Windowed Tauberian Theorem: Smalls Finite Part ↔ Logarithmic Window Average) . Assume Θ′(ω) possesses nite number of power-logarithmic asymptotic terms and controlled remainder as ω→ ∞ , with Mellin transform analytic bounded in strip ℜz > −α . Take window family W making c W nullify at above powers and logarithmic power indices. Then there exist constants CW>0 and α′>0 such that fps→0+∆K(s) = κHK CW·Θ′Wµ=1 √s+Osα′, where κHK only depends on dimension, eld content and chosen regularization scheme. Proof in 4.3. Corollary 4 (3.1: Windowed Integration Law) . Dene Λϕ,W (µ) := κΛ⟨Θ′⟩W(µ) ( κΛ has M3 dimension, in D= 4 making Λ∼M2 self-consistent), then ∂ln µΛϕ,W (µ) = κΛΞW(µ),ΞW(µ) = ZωΘ′′(ω)Wln(ω/µ)dln ω=1 2πZω ∂ωTr Q W d ln ω . Proof in 4.4. Theorem 5 (4: Open Channels: Extended Unitarization and Relative Determinant) . Assume static-patch dS/black hole scattering operator through channel extension becomes S(ω) unitary and dierentiable, or there exists reference propagator Sref making b S=SS−1 ref satisfy b S−1∈S1 , ∂ωb S∈S1 . Then ∂ωarg det KV b S(ω) = −iTrb S†∂ωb S, diering from phase given by S route only by constant, thus equivalent at ΞW level. Proof in 4.5. Theorem 6 (5: Threshold Finite Part and Even-Dimensional Logarithmic Terms) . Under assumptions of 2.1 and no zero-energy resonance condition, Θ′(ω) possesses Hadamard nite part at ω→0+ ; even-dimensional ωmlog ω type terms can be nullied by above window family, with nite part and windowed limit commuting. Proof in 4.6. 4 Proofs 4.1 Proof of Theorem 1 (i) KVKren equivalence : Guillarmou proves on even-dimensional AH/CCM −2πi ∂zξ(z) = TR∂zSn 2+izS−1 n 2+iz, thus arg detKV S(n 2+iω) = −2π ξ(ω) (mod 2π) . This yields Θ′=−∂ωξω . (ii) DOSspectral shift : LifshitsKren denes ∆ρE=−ξ′ E , variable substitution yields ∆ρω= 2ω∆ρE(ω2) = −∂ωξω . (iii) WS-trace : For unitary S , Q=−iS†∂ωS , Tr Q=∂ωarg det S . Substituting Φ=2πΘ yields Tr Q= 2πΘ′ . Electromagnetic multiport version holds in measurable framework. Three steps combined yield proposition. 4 4.2 Proof of Theorem 2 By LK trace formula (Peller) for relative pair (H, H0) with f(λ) = e−sλ , ∆K(s) = Trf(H)−f(H0)=Z∞ 0 f′(λ)ξE(λ)dλ. Integration by parts yields ∆K(s) = h−e−sλξE(λ)i∞ 0++Z∞ 0 e−sλ ∆ρE(λ)dλ. High-energy end e−sλ →0 vanishes; low-energy end requires ξE(0+)=0 or Hadamard nite part interpretation (Theorem 5). Substituting λ=ω2 , dλ = 2ω dω with ∆ρω= 2ω∆ρE(ω2) canceling Jacobian yields ∆K(s) = Z∞ 0 e−sω2∆ρω(ω)dω =Z∞ 0 e−sω2Θ′(ω)dω. Proof complete. 4.3 Proof of Theorem 3 (Windowed Tauberian) Three steps. (a) Frequency domain decomposition and window Mellin-nullication . Assume Θ′(ω) = K X k=1 ckωβk+X m Jm X j=0 dm,j ω˜ βm(ln ω)j+r(ω), r 's Mellin transform analytic bounded in ℜz > −α . Take W∈C∞ 0 making c W(βk)=0 , dj dzjc W(z)|z=˜ βm= 0 . Then ⟨Θ′⟩W(µ) = ⟨r⟩W(µ). (b) Laplace saddle and scale matching . Denote u= ln(ω/µ) , then ∆K(s) = ZR e−sµ2e2uΘ′(µeu)du ≈Θ′(µ)ZR e−sµ2e2udu +··· Take µ=s−1/2 placing saddle at u= 0 . By stationary phase/saddle approximation, there exist CW and α′>0 such that fps→0∆K(s) = κHK CW⟨Θ′⟩W(µ=s−1/2) + O(sα′). ( CW can write as constant multiple of Re−e2uW(u)du ; exact coecient determined by regularization and window normalization.) (c) OL constant and relative class-trace stability . By Peller's OL estimate, OL constant of f(λ) = e−sλ bounded as s↓0 (specically depends on Besov norm), combined with relative class-trace assumption yielding error bound consistency. Combined above proves theorem. 5 4.4 Proof of Corollary 3.1 (Windowed Integration Law) Dierentiating ⟨Θ′⟩W(µ) = RΘ′(ω)W(ln(ω/µ)) dln ω : ∂ln µ⟨Θ′⟩W=−ZΘ′(ω)∂ln ωW d ln ω=ZωΘ′′(ω)W d ln ω. Using ∂ωTr Q= 2πΘ′′ yields statement. 4.5 Proof of Theorem 4 (Open Channels) (i) Extended channels : In static-patch dS, view horizon as in/out scattering channels extending to unitary S . Unitarity and dierentiability ensure ∂ωarg det S= Tr Qext . (ii) Relative determinant : Choose reference Sref (Rindler/outer region) making b S−1 , ∂ωb S∈ S1 . KV determinant dierentiable with ∂ωlog det KV b S= TRb S−1∂ωb S=−iTrb S†∂ωb S, equality holding relies on b S 's quasi-unitarity and ideal class conditions. Diers from phase given by S only by constant, equivalent at ΞW level. 4.6 Proof of Theorem 5 (Threshold Finite Part) AH/CCM spectrum at ω→0 contains ωmlog ω form terms. Non-trapping and no zero-energy resonance ensure resolvent threshold control, FIO structure yields kernel singularity order; accordingly Θ′(ω) possesses Hadamard nite part. After window family satises logarithmic nullication, nite part and windowed limit commute, proof complete. 5 Modeled Examples 5.1 One-Dimensional δ Potential (Solvable Verication) Let V(x) = α δ(x) . Partial wave degenerates, scattering phase δ(ω) = arctan(−α/2ω) . Thus Φ(ω)=2δ(ω),Θ′(ω) = 1 2π∂ωΦ = 1 π α 4ω2+α2. Substituting into Theorem 2 veries closed-form computability consistency of ∆K(s) = R∞ 0e−sω2Θ′(ω)dω ; take window with c W(0) = 1 , c W(1) = 0 , numerically verify windowed Tauberian error order O(sα′) . 5.2 Static-Patch dS Scalar Template ( κ Extraction) Following AlbrychiewiczNeiman, for massless scalar greybody factor/transmission phase δℓ(ω) and relative DOS written as partial wave sum, Θ′(ω) = Pℓ(2ℓ+ 1)δ′ ℓ(ω)/π . LawParmentier's relative DOSpartition function yields consistency with one-loop free energy. Numerically compute ⟨Θ′⟩W(µ) on low-frequency cuto ω≤µ , matching with smalls heat kernel nite part (Seeley DeWitt bulk term), extract κHK =fps→0∆K(s) CW⟨Θ′⟩W(s−1/2), κΛ determined by dimensional matching , as numerical demonstration of acyclic calibration. 6 6 Engineering Scheme (FRB Baseband) 6.1 Observable Kernel Reconstruct system transfer b S(ω) = Hsys(ω)Href(ω)−1 from cross-spectrum/multiport network, take Φ(ω) = arg detKV b S , dene b Θ(ω) = Φ(ω) 2π,b ΞW(µ) = Zω ∂2 ωb Θ(ω)W(ln(ω/µ)) dln ω=1 2πZω ∂ωTr b Q W d ln ω. WS-trace's electromagnetic measurability provides direct estimator for this construction. 6.2 Leakage Kernel and Variance Phase-level residual dispersion ϕDM =KDM ω−1 and thin-screen broadening ϕsca =Ksca ω−3 induce ΞDM =ω ∂2 ωϕDM = + 2KDM ω−2,Ξsca = + 12Ksca ω−3. Second-order derivative noise amplication: if phase noise spectrum near-white with channel width ∆ω , discrete second-order dierence operator yields Varb Ξ(ω)≃Cω2 ∆ω4σ2 ϕ(ω), C determined by discrete kernel spectral norm. After windowing provides closed-form upper bound for Var[b ΞW] by W 's L2 norm. 6.3 Data, Pipeline, and Ecacy CHIME/FRB published approximately 140 baseband events containing coherent dedispersion and polarization information, satisfying phase-level access; we provide minimal reproducible pipeline: read and calibrate  relative determinant  phase unwrapping  regularized dierentiation (Tikhonov/TV on ln ω axis)  windowing  shape consistency test/upper limit. Injectionrecovery experiment: inject Ξinj(ω) = A ω−1ψ(ln ω) and ω−2 , ω−3 templates, compare recovery biasvariance with Fisher-CR lower bound, assess sample stacking ecacy curve. 7 Discussion: Risks, Boundaries, Related Work Mathematical core of this framework is even-dimensional KVKren equivalence and FIO structure of AH/CCM; odd dimensions require alternative introduction. Threshold log terms and open channel dierentiability require strict relative class-trace and branch continuity. In black hole/staticpatch dS, relative DOSpartition function provides computable anchor. Observation-wise, Ξ 's second-order derivative noise amplication and dispersion/multipath leakage need windowing and regularization control. This paper's windowing law is structural equality , absolute numerical mapping depends on κHK , κΛ calibration. 7 8 Conclusion This paper rigorously aligns phasespectral-shiftDOSWSheat-kernel chain on frequency variable, providing substitution theorem without extra 2ω and logarithmic frequency windowing Tauberian theorem, establishing windowed integration law maintaining measurable WS-trace equality under open channels. Demonstrate κ extraction route with δ potential and static-patch dS templates, providing minimal reproducible scheme for FRB baseband. This framework connects spectral geometry with observable phase analysis as veriable methodology. A Notation and Factor Accounting  Frequency/energy: λ=ω2 , dλ = 2ω dω .  Spectral shift/DOS: ∆ρE=−ξ′ E , ∆ρω(ω)=2ω∆ρE(ω2) .  Scattering phase: Φ = arg detKV S , Θ=Φ/2π .  Core identities: Θ′= ∆ρω=−∂ωξω,Tr Q=∂ωΦ=2π∆ρω,∆K(s) = Z∞ 0 e−sω2Θ′(ω)dω.  Logarithmic window average: ⟨f⟩W(µ) = Rf(ω)W(ln(ω/µ)) dln ω , dln ω=dω/ω .  Observable kernel: ΞW(µ) = ∂ln µ⟨Θ′⟩W=RωΘ′′ W d ln ω=1 2πRω ∂ωTr Q W d ln ω . B LK Trace Formula and Operator-Lipschitz Proposition 7 (B.1 (LK)) . For self-adjoint pair (H, H0) satisfying relative class-trace assumption making f(H)−f(H0)∈S1 , with f operator-Lipschitz, then Trf(H)−f(H0)=Rf′(λ)ξE(λ)dλ . Proof essentials : Peller's OL criterion ( f∈B1 ∞1 ) combined with HelerSjöstrand representation; HilbertSchmidt estimate of resolvent dierence ensures class-trace. For f(λ) = e−sλ , its OL constant bounded as s↓0 . C Window Family Construction and Mellin-Nullication Take smooth compact window W with RW= 1 . To nullify power laws ωβk and ω˜ βm(ln ω)j , require c W(βk)=0,dj dzjc W(z)z=˜ βm= 0. Construction method: start with mother window W0 , form nite linear combination W= PℓaℓW0(·−uℓ) , coecients determined by nullication linear equations. Mellin-wavelet (logarithmic axis partition of unity) framework ensures numerical stability. 8 D KV-det Dierentiability for Open Channels Proposition 8 (D.1) . Assume reference Sref makes b S=SS−1 ref satisfy b S−1∈S1 , ∂ωb S∈S1 , with b S quasi-unitary. Then KV determinant exists with ∂ωlog det KV b S= TRb S−1∂ωb S=−iTrb S†∂ωb S. Proof essentials : TR multiplicative property and logarithmic derivative denition; quasiunitarity reduces TR to trace. Static-patch dS with Rindler/outer region as reference satises ideal class conditions. E Threshold ω→0 Finite Part Proposition 9 (E.1) . Non-trapping and no zero-energy resonance imply resolvent threshold controllability, Θ′ 's logarithmic singularity at most nite order. For even-dimensional ωmlog ω terms, choosing c W nullifying at corresponding indices and their derivatives yields fpω→0Θ′= limµ↓0⟨Θ′⟩W(µ) . Proof essentials : FIO structure and analytic Fredholm theory yield kernel threshold form; windowed limit commutes with nite part by dominated convergence and nullication conditions. F FRB Pipeline Discrete Implementation and Error Propagation F.1 Phase unwrapping and relative determinant : Multi-beam dierence, cross-polarization and injection noise give reference Href , ensuring continuous Φ(ω) via principal value phase and branch patching. F.2 Second-derivative regularization : On ln ω axis use Tikhonov/TV, regularization parameter take L-curve or GCV. Second-order dierence kernel D(2) spectral norm |D(2)| ∼ ∆ω−2 , thus Varb Ξ(ω)≃C ω2∆ω−4σ2 ϕ(ω). F.3 Leakage kernel : Dispersion ϕDM =KDMω−1⇒ΞDM = + 2KDMω−2 (positive sign); Thin-screen broadening ϕsca =Kscaω−3⇒Ξsca = + 12Kscaω−3 . After windowing b ΞW= ΞW+⟨ΞDM⟩W+⟨Ξsca⟩W+ noise, shape separability ensured by power index dierence and windowed frequency band decomposition. F.4 Injectionrecovery : Inject Ξinj(ω) ( ω−1 , ω−2 , ω−3 ) into public baseband, recover b ΞW through full pipeline, report |b A/A −1| versus window width relationship and Fisher-CR lower bound. End of Main Text and Appendices 9