Information Entropy--Geometric Unification and Windowed Generation of Cosmological Terms: From Relative Entropy Hessian to Effective Action, Poisson--Euler--Maclaurin Finite-Order Discipline, and Geometric Entropy Decomposition of Friedmann Equations

Information EntropyGeometric Unication and Windowed Generation of Cosmological Terms: From Relative Entropy Hessian to Eective Action, PoissonEulerMaclaurin Finite-Order Discipline, and Geometric Entropy Decomposition of Friedmann Equations Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Version: 1.5 Abstract Within a unied operatormeasurefunction framework, we establish an organic assembly connecting multi-order responses of relative entropy , master-scale calibrations of scattering spectra , windowed readout Toeplitz/Berezin compressions , and Nyquist PoissonEulerMaclaurin (NPE) nite-order discipline to closed derivations of geometric eective action and cosmological terms . First, under Eguchi regularized divergence and Amari α -geometry, we prove construction of the FisherRao metric and dual connections; second, under trace-class/relative trace-class perturbation and energy-dierentiable scattering theory assumptions, we present a theorem-level statement of the master scale trinity φ′(ω) π=−ξ′(ω) = 1 2πtr Q(ω), Q =−iS†∂ωS pointing out distributional sense corrections at thresholds/long-range potentials. Next, selecting PaleyWiener / de Branges / Hardy environments, employing symmetric smooth allocation ( bg=pb h , h=g∗˜g ), we place Kw,h =P Mw1/2Cg·C˜gMw1/2P into Schatten trace class and provide explicit upper bounds . Subsequently, unifying Fourier conventions and distinguishing Poisson zero-aliasing criterion ( ∆<2π/B ) from Shannon no-aliasing reconstruction ( ∆< π/B ) in their multiplicative constant dierences; under double-layer tail control of near band-limited, we provide EM remainder with ζ(2m) explicit constants. Using ToeplitzFIO diagonal-type wave-front relation, we prove windowingcompression convolution singularity non-increase (holds on T∗X\0 away from zero cut, with bandlimited/near band-limited windowing as global inclusion). In a minimal computable model of linearized gravity, we provide explicit coecients from fourth-order response to curvature quadratic invariants , thereby obtaining the scale integral law for volume terms Λeff (µ)−Λeff (µ0) = Zµ µ0 Ξ(ω)dln ω, [Ξ] = L−2, and provide sucient conditions for positivity/monotonicity of Ξ plus local non-monotonicity boundaries at resonances/thresholds. The action unies as Seff [g] = Zd4x√−ghR−2Λeff (µ) 16πG +αR2+βRµν Rµν +···i, 1 where α, β are dimensionless. Using three-dimensional S3 heat kernelcounting function curvature docking example, we close the spectralgeometric interpretation of FRW curvature term, demonstrating via one-dimensional δ potential and AB scattering the windowing mechanism of single-peak saturation/peak-family quasi-logarithmic accumulation. Appendices provide complete proofs of all theorems, constant estimates and dimensional tables, plus reproducible experimental/numerical script essentials. MSC : 53Bxx; 83C05; 58J35; 46E22; 47B35; 42A38; 94A17; 81U40 Keywords : Information geometry; Eguchi regularized divergence; FisherRao metric; Amari α -connection; Bregman/Hessian; spectral shift function; BirmanKrein; WignerSmith group delay; Toeplitz/Berezin compression; Schatten trace class criteria; Poisson summation; Euler Maclaurin remainder constants; wave-front set and ToeplitzFIO; heat kernel/SeeleyDeWitt; spectral action; running vacuum; FRW geometric entropy decomposition 1 Introduction & Historical Context Information geometry characterizes statistical manifolds via Hessian metrics and α -connections generated by divergences; second-order response of relative entropy yields FisherRao metric, thirdorder response corresponds to AmariChentsov tensor and α -connection. Bregman divergence induces dual at (Hessian) structure and Legendre dual coordinates in exponential families. In spectralscattering theory, the LifshitzKrein trace formula and BirmanKrein identity relate spectral shift function ξ with scattering determinant; FriedelLloyd and WignerSmith unify phase derivative, group delay, and density-of-states dierence under the same calibration. Heat kernel/Seeley DeWitt expansion and spectral action principle provide standard tools for bridging geometric invariantswindowed spectra. This paper closes these elements under theorem-level assumptions into a logical chain from relative entropymaster scalewindowingNPEheat kernelFRW. 2 Model & Assumptions 2.1 Fourier Convention, Variables, and Window Kernel General Declaration Fix b f(ω) = ZR e−iωxf(x)dx, f(x) = 1 2πZR eiωx b f(ω)dω. Throughout, we uniformly use frequency ω to record energy variables (readers may view E≡ ω ). Windows wµ take smoothed logarithmic windows, satisfying wµ∈C∞ 0∩L∞ with supp wµ⊂ [µ0, µ] , µ>µ0>0 ; specically one may take wµ(ω) = ψ(ω) ω, ψ ∈C∞ 0, ψ ≡1 on [µ0, µ]◦, smoothly cut o at ω=µ0, µ (if covering interval around ω≈0 , rst take µ0>0 then take limit). General declaration : Readout kernel h defaults to Bochner positive denite ( b h≥0 , b h∈L1 ), thus admitting bg=pb h∈L2 such that h=g∗˜g . 2 2.2 Information Divergence and Dual Flatness Regularized divergence D(θ∥θ0) with second/third/fourth-order responses gij =∂i∂jD|θ0, Tijk =∂i∂j∂kD|θ0,Kijkl =∂i∂j∂k∂lD|θ0. Denote K:= Kijij as full contraction of fourth-order response tensor (unrelated to Kw,h ). Induce FisherRao and α -connection: Γ(α)ijk = Γ(0)ijk +α 2Tijk . Bregman divergence Dψ makes g=∇2ψ , ∇(±1) at. 2.3 Master Scale, ScatteringSpectral Shift, and Threshold Clauses Self-adjoint pair (H0, H) satises trace-class or relative trace-class perturbation, wave operators complete; S(ω) unitary and weakly dierentiable. Spectral shift ξ(ω) satises det S(ω) = e−2πiξ(ω) . Denition 1 (Total Scattering Phase) . Let φ(ω) := 1 2iLog det S(ω), taking the branch consistent with threshold phase renormalization and continuous as ω→+∞ . Then φ′(ω) = 1 2itr S−1∂ωS=1 2tr Q(ω), Q =−iS†∂ωS, hence φ′(ω) π=−ξ′(ω) = 1 2πtr Q(ω), holding in distributional sense on discrete threshold set Σ . 2.4 Toeplitz/Berezin Compression and Readout Take PaleyWiener / de Branges / Hardy space H , orthogonal projection P ( norm |P|= 1 ). Let w∈C∞ 0∩L∞ , h=g∗˜g as above. Denition 2 (Relative Spectral Projection Dierence) . Denote Π as the distributional kernel of relative spectral projection dierence for self-adjoint pair (H0, H) in energy representation (equivalent to relative spectral measure), such that tr(Kw,hΠ) = Zw(ω) [h∗!ρrel](ω)dω, where ρrel(ω) = φ′(ω) π=1 2πtr Q(ω) , Q=−iS†∂ωS . Dene Kw,h := P Mw1/2Cg·C˜gMw1/2P, Obs(w, h) = tr(Kw,hΠ) = Zw(ω) [h∗!ρrel](ω)dω. 3 2.5 NPE Discipline and Near Band-Limited Strictly band-limited : supp b f⊂[−B, B] . Near band-limited : R|ω|>B |b f|dω ≤ε and R|ω|>B |b f|2dω ≤ε2 . Criterion distinction (detailed in Theorem 3): Poisson zero-aliasing term ∆<2π/B ; Shannon no-aliasing reconstruction ∆< π/B . 2.6 Eective Action and Dimensions Take c=ℏ= 1 . Action written as Seff =Zd4x√−ghR−2Λeff(µ) 16πG +αR2+βRµνRµν +···i, in four dimensions α, β dimensionless; [Λeff] = L−2 . Dimensional table in Appendix J. 3 Main Results (Theorems and Alignments) Theorem 3 (Relative Entropy Hessian and α -Connection) . Second-order response of relative entropy yields FisherRao metric, third-order response via Γ(α)ijk = Γ(0)ijk +α 2Tijk generates α - connection; Bregman divergence induces dual at (Hessian) structure. Theorem 4 (Master Scale Trinity: Sucient Conditions and Threshold Corrections) . Under assumptions of 3, det S(ω) = e−2πiξ(ω) and ξ′(ω) = −1 2πtr Q(ω) holds almost everywhere; at ω∈Σ or long-range potentials holds in distributional sense with renormalized phase. Theorem 5 (Poisson Zero-Aliasing Criterion and Shannon Reconstruction Criterion) . Under the present Fourier convention, if supp b f⊂[−B, B] , then X n∈Z f(n∆) = 1 ∆X k∈Zb f2πk ∆ with k= 0 aliasing terms strictly zero if and only if ∆<2π/B ; Shannon no-aliasing reconstruction requires ∆< π/B . Theorem 6 (NPE: EulerMaclaurin Explicit Constants and Near Band-Limited Tails) . If f∈ C2m[a, b] and is (B, ε) -near band-limited, |Rm| ≤ 2ζ(2m) (2π)2m(b−a) sup [a,b]|f(2m)|+O(ε),sup |f(2m)| ≤ C B2m|f|∞. Theorem 7 (ToeplitzFIO Pseudolocality and Singularity Non-Increase) . Let w∈C∞ , h∈ S , P be ToeplitzFIO with diagonal-type wave-front relation. For any distribution u and any open cone domain U⋐T∗X\0 away from zero cut, WFP MwChP u∩U⊆WF(u)∩U. Remark 8 . When energy-shell windowing (band-limited/near band-limited) excludes low-frequency neighborhood of |ξ| ≈ 0 , we obtain global inclusion WF(PMwChPu)⊆WF(u), and when WF(u)=∅ , the above inclusion is strict. 4 Theorem 9 (Fourth-Order Response → Curvature Quadratic Terms: Minimal Computable Model and Coecients) . Under linearization gµν =ηµν+hµν in harmonic gauge, decompose by scalar/transverse traceless (TT) and dene windowed spectral weights Ns=Zd4k k4W(k)|As(k)σ(k)|2,Nt=Zd4k k4W(k)|At(k)hTT(k)|2, where W is determined by ρrel, w, h . Then Z√−gK=c1Z√−g R2+c2Z√−g RµνRµν + (total derivative) , with c1=Ns 36 , c2=Ns 12 +Nt 4. Normalization declaration : The denition of Ns,t has absorbed all (2π) factors and measure constants in the unied Fourier convention of this section; under dierent conventions, rescaling is required accordingly. Theorem 10 (Volume Term Scale Integral Law and Positivity/Monotonicity of Ξ ) . After information free energy windowing, Λeff(µ)−Λeff(µ0) = Zµ µ0 Ξ(ω)dln ω, Ξ(ω) = ⟨K, ρrel⟩wω,h,[Ξ] = L−2. If ρrel(ω)≥0 and induced two-point kernel with wω, h are non-negative/Bochner positive denite, then Ξ(ω)≥0 and monotonically non-decreasing in ln µ ; if ρrel sign-variable, only window-averaged sense quasi-monotonicity is obtained, or change Ξ to quadratic form to obtain strict non-negativity. Thresholds/resonance clusters can cause local non-monotonicity, but when peak families are nearuniformly dense in ln ω with slowly varying weights, Ξ is nearly constant over wide intervals, exhibiting quasi-logarithmic accumulation. Theorem 11 (FRW Curvature Term SpectralGeometric Docking) . Three-dimensional constantcurvature manifold heat kernel asymptotic Tr e−t∆∼(4πt)−3/2hVol + t 6ZR+O(t2)i, t ↓0, for S3(L) has R= 6/L2 , Vol = 2π2L3 . Windowed counting function sub-leading term ∝RR∝ κVol consistent with FRW's −κ/a2 term; window shape only alters coecients, does not break homogeneity and isotropy. 4 Proofs 4.1 Theorem 1 (Relative Entropy Hessian and α -Connection) Follows from Eguchi's contrast functional and AmariChentsov tensor denition. Realized via Bregman potential in exponential families for dual atness. 5 4.2 Theorem 2 (Master Scale Trinity: Sucient Conditions and Threshold Corrections) Proof in three steps: 1. Spectral shift function denition : Dened by LifshitzKrein trace formula. 2. Scattering determinant relation : BirmanKrein identity yields det S=e−2πiξ . 3. Derivative relation : Dierentiability of stationary scattering derives ξ′=−(2π)−1tr Q . Threshold/long-range potential cases hold in distributional sense with phase renormalization correction. 4.3 Theorem 3 (Poisson Zero-Aliasing Criterion and Shannon Reconstruction Criterion) By Poisson formula and present Fourier convention, b f(2πk/∆) = 0 ( k= 0 ) if and only if ∆<2π/B . Shannon no-aliasing reconstruction requires stricter condition: ∆< π/B . 4.4 Theorem 4 (NPE: EulerMaclaurin Explicit Constants and Near BandLimited Tails) Employ DLMF's EM remainder constants and Bernstein-type derivative bounds. Near band-limited tails enter O(ε) . 4.5 Theorem 5 (ToeplitzFIO Pseudolocality and Singularity Non-Increase) Hörmander pseudolocality yields WF(Mwu)⊆WF(u) , while Ch is smoothing. ToeplitzFIO diagonal-type wave-front relation implies: for any U⋐T∗X\0 , WF(PMwChPu)∩U⊆WF(u)∩U. Under band-limited/near band-limited windowing, can take U covering entire T∗X\0 , thus WF(PMwChPu)⊆WF(u) . 4.6 Theorem 6 (Fourth-Order Response → Curvature Quadratic Terms: Minimal Computable Model and Coecients) From linearized decomposition obtain Kijkl contribution; its full contraction K matches R2, RµνRµν with coecients c1=Ns/36, c2=Ns/12 + Nt/4 . Scalar mode : Linearized curvature R(1) =−6□σ , thus R2= 36 k4σ2 , R(1) µν Rµν(1) = 12 k4σ2 . TT mode : R(1) = 0 , RµνRµν =1 4k4(hTT)2 . Matching windowed fourth-order kernel weights yields coecients c1,2 . 4.7 Theorem 7 (Volume Term Scale Integral Law and Positivity/Monotonicity of Ξ ) Low-frequency cluster (Poisson's k= 0 ) dominates volume term. When ρrel(ω)≥0 and kernel/window non-negative/Bochner positive denite, Ξ≥0 ; if ρrel sign-variable, requires window averaging or change to quadratic form. 6 Tauberian control when peak families near-uniformly dense in ln ω ensures quasi-logarithmic intervals. 4.8 Theorem 8 (FRW Curvature Term SpectralGeometric Docking) Use S3 spectrum λn=n(n+ 2)/L2 , multiplicity (n+ 1)2 and Tauberian theorem to recover heat kernel sub-leading term and dock with FRW curvature term. 5 Model Applications 5.1 One-Dimensional δ Potential: Single-Peak Saturation and Quasi-Logarithmic Accumulation Take V(x) = λδ(x) . Under the present unit convention expressing in energy variable E≡ω , phase shift written as δ(E) = δk(E)=−arctan λ 2k(E), k(E) = √E( may take 2m= 1), hence relative density of states ρrel(E) = 1 π dδ dE =1 π dδ dk dk dE  below identify E with ω. Subsequently employ analytic integration of logarithmic window with Lorentzian peak to demonstrate single-peak saturation/peak-family quasi-logarithmic accumulation, compatible with above formula. For smooth logarithmic window I(µ;µ0) = Zµ µ0 Γ (ω−ω0)2+ Γ2 dω ω has closed form I(µ;µ0) = Γ ω2 0+ Γ2ln µ µ0−Γ 2(ω2 0+ Γ2)ln (µ−ω0)2+ Γ2 (µ0−ω0)2+ Γ2 +ω0 ω2 0+ Γ2harctan µ−ω0 Γ−arctan µ0−ω0 Γi, two classes of ln µ exactly cancel, single-peak saturation ; when peak families near-uniformly dense in ln ω with slowly varying weights, quasi-logarithmic accumulation emerges. Reproducible experimental essentials (example parameters) : λ= 1 ; µ0= 10−3 , scan µ to 103 ; window width smoothing parameter σ= 0.05 ; kernel h(ω) = e−ω2/2σ2 h take σh= 0.1 . 5.2 AB Scattering: Windowing TopologySpectral Density Dierence Ideal AB model phase shift energy-independent, tr Q= 0 ; nite-radius/screened models introduce energy dependence, windowed dierence forms eective contribution to curvature/topological terms, non-analytic points correspond to steps/cusps in Ξ . 7 6 Engineering Proposals 1. Group delay measurement chain : Measure multi-port S(ω) and dierentiate phase to obtain Q(ω) , construct Ξ(ω) and Λeff(µ) curves, NPE constants provide error bands. 2. Toeplitz/Berezin numerical spectrology : Implement Kw,h and monitor |Kw,h|1 ; semiclassical regime approximate trace by symbol integration and assess remainder by EM constants. 3. FRW curvature windowing verication : On S3/H3/ three-torus compare heat kernel subleading term with windowed counting function, verify spectralgeometric docking of −κ/a2 . 7 Discussion Master scale trinity holds under trace-class/relative trace-class and dierentiability assumptions; long-range potentials and thresholds corrected in distributional sense. Symmetric smooth allocation of Toeplitz/Berezin provides checkable trace-class upper bounds; NPE discipline forms niteorder error budget by ζ(2m) constants and frequency-domain tail control; windowingcompression convolution non-increases singularity. Fourth-order response to R2, RµνRµν coecients veriable in minimal model; positivitymonotonicity conditions for Ξ explicit, peak-family statistics support quasi-logarithmic intervals. Windowed interpretation of FRW curvature term closed via S3 example. Extensions to open systems or non-unitary S require dissipative scattering framework, where tr Q loses positivity-preservation. 8 Conclusion Completing theorem-level closure from information divergencemaster scalewindowing NPEheat kernelFRW : (i) Master scale trinity holds under theorem-level assumptions; (ii) Toeplitz/Berezin compression enters trace class via symmetric smooth allocation with explicit upper bounds ( |P|= 1 ); (iii) Poisson zero-aliasing and Shannon reconstruction criteria separated with consistent constants; (iv) EM remainder has ζ(2m) constants, near band-limited tails controllable; (v) Windowingcompressionconvolution non-increases singularity (strictly non-increasing under energy-shell windowing); (vi) Fourth-order response to curvature quadratic term coecients explicitly veriable ; (vii) Volume term obeys scale integral law, positivity and quasi-logarithmic mechanism of Ξ clear; (viii) FRW curvature term spectralgeometric docking complete. These results provide veriable technical foundation for unied scheme of information geometry × spectralscattering × cosmology. 8 Acknowledgements, Code Availability No proprietary code used; appendices contain reproducible experimental/numerical script essentials and parameter tables. References  Amari, S.-i.; Nagaoka, H. Methods of Information Geometry . AMSOUP, 2000 (Chs. 24: FisherRao and α -connection; Ch. 8: dual atness).  Eguchi, S. A dierential geometric approach to statistical inference on the basis of contrast functionals. Hiroshima Math. J. 15 (1985) 341391.  Birman, M. S.; Krein, M. G. On the theory of wave and scattering operators. 1962 (see Yafaev, Chs. 68).  Yafaev, D. R. Mathematical Scattering Theory . AMS, 1992/2005 (Ch. 8: scattering matrix and spectral shift; Ch. 10: thresholds).  Simon, B. Trace Ideals and Their Applications . 2nd ed., AMS, 2005 (Ch. 2: Schatten ideals; HS × HS ⊂ S1).  Boutet de Monvel, L.; Guillemin, V. 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A Fourier Convention and Variable Unication Provide the paper's xed Fourier pair and all (2π) factor absorption rules, declare equivalent use of ω≡E , list dimensional consistency under transformations. 9