Unified Role of Relative Scattering Determinant in Quantum Gravity: Two-Domain Framework, Fixed-Energy BK ($p\in\{1,2\

Unied Role of Relative Scattering Determinant in Quantum Gravity: Two-Domain Framework, Fixed-Energy BK ( p∈ {1,2} Unied Version), Closed-Domain Λ -Slope, and Black Hole Pole Spectroscopy Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract Taking relative determinant as unied object, we establish rigorous and veriable theory in two types of geometric-physical scenarios: (C) relative ζ /heat kernel determinant for Euclideanized second variation operator family in compact closed domain and its volume density response to cosmological constant term; (S) xed-frequency scattering matrix on stationary exterior geometry (Schwarzschildde Sitter/Kerrde Sitter) with relative (modied) determinant, spectral shift object, and quasinormal mode (QNM) spectroscopy. This paper provides four main theorems with complete proofs: (i) Under control of weighted limiting absorption principle (LAP) and double operator integral (DOI), prove p∈ {1,2} unied version of xed-energy BirmanKren equality: for Lebesgue almost everywhere frequency ω , detpSΛ(ω) = exp−2πi Ξ(p) Λ(ω) , where p= 1 gives Ξ(1) =ξ as LifshitsKren spectral shift function, p= 2 yields Ξ(2) as cumulative antiderivative of Koplienko second-order spectral shift; (ii) In closed-domain Müller relative determinant framework, prove volume slope theorem: limµ→0+Vol4(M)−1∂Λℜlog detζ,rel(KΛ+µ2,K0+µ2) = 1 8πG (per signature convention xed in text); (iii) On physical strip ℑω > −γ0 , pole set of relative scattering determinant τp(ω) = detpS(ω)S0(ω)−1 equivalent to QNM (counting algebraic multiplicity), independent of reference S0 choice; (iv) For real frequency only phase admits equality, arg detpS=−2πΞ(p) ; while for p= 2 Carleman determinant |det2S|= expPj(1 −cos θj)≥1 , generally cannot claim |det2S|= 1 . Accordingly introduce phase-normalized determinant c detpS:= detpS/|detpS| as constrained object for frequency-domain globally meromorphic tting, providing principal angle upper bound for Fisher information. Paper concludes with parameters and acceptance standards for three reproducible experimental pipelines: closed-domain rel-zeta, exterior-domain meromorph-t, channelpseudo-unitary verication. 1 Introduction Closed-domain relative ζ /heat kernel determinant and exterior-domain relative scattering determinant share essential structurerelative phase. On closed-domain side, this phase recovers on-shell action's volume density response to cosmological constant Λ via logarithmic derivative; on exteriordomain side, it's controlled by BK/LK (and second-order Koplienko version) spectral shift function to scattering matrix phase on energy bers. This paper unies two domains into closed loop of veriable hypothesis ⇒ theorem ⇒ detailed proof: 1. Achieve xed-energy implementation under operator-Lipschitz and DOI techniques, with weighted LAP dominated convergence; 1 2. Implement regularization independence and item-wise cancellation of corner/boundary/ghost under Müller relative determinant; 3. Unify relative scattering determinant poles as QNM under analytic Fredholm framework, proving reference independence; 4. Separate block-level modulus conservation from global Carleman modulus non-constant identity under pseudo-unitary (J-unitary) framework, imposing real-axis modulus constraint via phase-normalized determinant. All mathematical expressions in text presented inline with · form, avoiding ambiguity from display/environment switching. 2 Setting, Notation, and Veriable Hypotheses 2.1 Spectrum, Ideals, and Modied Determinants Take separable Hilbert space H . Denote self-adjoint operator pair (HΛ, H0) , dierence V=HΛ−H0 . Schatten ideal Sp standard denition. For K∈S1 take Fredholm determinant det(I+K) ; for K∈ S2 take Carleman determinant det2(I+K) = det(I+K) exp(−K) . If U unitary with U−I∈S2 , spectral angles {θj} ∈ ℓ2 satisfy |det2(U)|= expPj(1−cos θj)≥1 , arg det2(U) = Pj(θj−sin θj) . 2.2 Spectral Shift Objects and DOI First-order spectral shift ξ and second-order spectral shift measure η respectively satisfy Tr(f(HΛ)− f(H0)) = Rf′(E)ξ(E)dE , Tr(f(HΛ)−f(H0)−f′(H0)V) = Rf′′(E)dη(E) , function class taking operator-Lipschitz/appropriate Besov intersection. Cumulative antiderivative Ξ(2)(E) = η((−∞, E)) , normalized Ξ(2)(−∞) = 0 . Double operator integral representation f(HΛ)−f(H0) = RR Φf(λ, µ)dEΛ(λ)V dE0(µ) , where Φf(λ, µ) = (f(λ)−f(µ))/(λ−µ) has Schur/Haagerup bound. 2.3 Weighted LAP and Energy Fiberization There exist s > 1 2 , energy window I , constant CI such that |⟨x⟩−s(H#−λ∓i0)−1⟨x⟩−s| ≤ CI holds for λ∈I , #∈ {Λ,0} . Stationary exterior region (SdS/KdS) stationary under time Killing eld with frequency ω , partial wave decomposition yields channel matrix Sℓm(ω) . 2.4 Closed-Domain Relative ζ -Determinant and Volume Slope Euclideanized second variation operator family KΛ with reference K0 matching principal symbol, boundary conditions and FaddeevPopov ghost pairing consistent, zero modes/threshold resonances removed via deprojection. Dierence heat kernel Krel(t) = Tre−t(KΛ+µ2)−e−t(K0+µ2) has shorttime expansion, dene log detζ,rel(KΛ+µ2,K0+µ2) = −R∞ 0t−1Krel(t)dt . Metric signature and action convention xed as ∂ΛSon - shell = (8πG)−1Vol4(M) . 2.5 Exterior-Domain Reference and Pseudo-Unitary On strip ℑω∈(−γ0,0] choose reference scattering matrix S0(ω) , require analyticity on same sheet without zero/poles. Each channel constructs energy ux quadratic form η via JostWronskian normalization making S† ℓmηSℓm =η . 2 2.6 Veriable Hypotheses (Assumption Box) (H - AC) : Wave operators exist and complete, AC part admits energy berization; (H - LAP) : Weighted LAP (parameter s > 1/2 , constant CI ); (H - LK/DOI) : Poisson smoothing fε∈OL , |fε|OL ≤C/ε , DOI kernel has uniform Schur/Haagerup bound; (H - Sp) : For a.e. ω∈I , χ(−∞,ω](HΛ)−χ(−∞,ω](H0)∈Sp and SΛ(ω)S0(ω)−1−I∈Sp (typical p= 2 ); (H - relDet) : Principal symbol consistent, boundary/ghost matching, no zero modes or dereso'd, dierence heat kernel has short-time expansion; (H - Ref) : Reference S0 analytic on strip without zero/poles; (H - Can) : Channel energy ux gauge xed, block-level pseudo-unitary holds. 3 Main Theorems and Conclusions Theorem 1 (3.1: Fixed-Energy BK: p∈ {1,2} Unied Version) . Under (H - AC) , (H - LAP) , (H - LK/DOI) , (H - Sp) , for Lebesgue almost everywhere ω∈I : when p= 1 , det SΛ(ω) = exp − 2πi ξΛ(ω) ; when p= 2 , det2SΛ(ω) = exp −2πi Ξ(2) Λ(ω) . Thus arg detpSΛ(ω) = −2πΞ(p) Λ(ω) . Theorem 2 (3.2: Closed-Domain Volume Slope) . Under (H - relDet) and deresonance projection, limµ→0+Vol4(M)−1∂Λℜlog detζ,rel(KΛ+µ2,K0+µ2) = 1 8πG (per text signature convention). Theorem 3 (3.3: τp Poles = QNM, Reference Independent) . Let τp(ω) = detpS(ω)S0(ω)−1 . Under (H - Ref) , on strip ℑω∈(−γ0,0] , pole set of τp coincides with S poles (QNM) counting algebraic multiplicity. If changing reference to e S0 still satisfying (H - Ref) , then τp/eτp is analytic outer function without zeros/poles on strip, leaving pole set unchanged. Theorem 4 (3.4: Real-Frequency Phase and Modulus; Phase-Normalized Determinant) . Blocklevel: If S† ℓm(ω)ηSℓm(ω) = η , then |det Sℓm(ω)|= 1 . Global: generally only arg detpS(ω) = −2πΞ(p)(ω) holds. When S(ω) unitary with S(ω)−I∈S2 , |det2S(ω)|= exp Pj(1−cos θj(ω))≥ 1 . Dene c detpS(ω) = detpS(ω)/|detpS(ω)| , bτp(ω) = τp(ω)/|τp(ω)| as real-axis modulus equals 1 constrained objects. 4 Proof of Theorem 3.1 (DOILAP Dominated Convergence to Fixed Energy) Proof strategy overview : Approximate step function via Poisson smoothing fε , apply DOI expression with weighted LAP establishing uniform Sp domination inequality, then exchange limit ε↓0 at Lebesgue points of spectral shift object, nally identify scattering phase via AC berization and exponentiate to determinant equality. Dierence between p= 1 and p= 2 carried by rst/second-order trace formulas. Step 1 (Poisson smoothing and DOI kernel bound) : Take fε(λ) = 1 2+1 πarctan((ω−λ)/ε) . Then fε∈OL with |fε|OL ≤C/ε . DOI expression fε(HΛ)−fε(H0) = RR Φfε(λ, µ)dEΛ(λ)V dE0(µ) , where |Φfε|Schur ≤C/ε . Step 2 (Weighted LAP and Schatten domination) : Write weighted projection boundary value resolvent form via Stone formula, apply (H - LAP) yielding |⟨x⟩−sR#(ω±i0)⟨x⟩−s| ≤ CI . By BirmanSolomyak type estimate obtain |fε(HΛ)−fε(H0)|Sp≤CI(C/ε)Mp(I) , where Mp(I) = supλ∈I|⟨x⟩−s(RΛ(λ±i0) −R0(λ±i0))⟨x⟩−s|Sp bounded. 3 Step 3 ( p= 1 : spectral shift and BK) : First-order trace formula yields Tr(fε(HΛ)− fε(H0)) = Rf′ ε(E)ξ(E)dE . Taking ε↓0 with dominated convergence yields Tr(χ(−∞,ω](HΛ)− χ(−∞,ω](H0)) = ξ(ω) at Lebesgue points of ω . AC berization with stationary scattering shows det S(ω) = exp(−2πi ξ(ω)) . Step 4 ( p= 2 : Koplienko phase) : Second-order trace formula yields Trfε(HΛ)−fε(H0)− f′ ε(H0)V=Rf′′ ε(E)dη(E) . Integrate right side twice by parts, taking ε↓0 yields Ξ(2)(ω) = η((−∞, ω)) . Fixed-energy implementation same as above, thus det2S(ω) = exp(−2πi Ξ(2)(ω)) . QED. 5 Proof of Theorem 3.2 (Relative Heat Kernel Item-Wise Cancellation, Tauberian Exchange, and Signature Convention) Step 1 (Logarithmic derivative heat kernel representation) : ∂Λlog detζ,rel =−R∞ 0t−1∂ΛKrel(t)dt , where Krel(t) = Tr(e−t(KΛ+µ2)−e−t(K0+µ2)) . Step 2 (Short-time expansion and item-wise cancellation) : Under principal symbol consistency, boundary/ghost pairing consistency, multiplicative anomaly vanishing, Krel(t)∼Pk≥0arel kt(k−d)/2 ( d= 4 ), local coecients (including GHY, corners, ghost pairing) cancel item-wise except volume term arel 0 , i.e., arel k>0= 0 . Step 3 (Tauberian exchange and volume slope) : Introduce small mass µ > 0 controlling large t part, split R∞ 0=Rt0 0+R∞ t0 . Former dominated by arel 0 , latter under deresonance projection has uniform bound. Exchanging µ↓0 with volume density limit yields Vol−1 4∂Λℜlog detζ,rel =∂Λc0 . By text convention ∂ΛSon - shell = (8πG)−1Vol4 , alignment yields ∂Λc0=1 8πG . QED. 6 Proof of Theorem 3.3 (Analytic Fredholm and Reference Independence) Step 1 (Analytic Fredholm) : On strip ℑω∈(−γ0,0] , write S(ω) = I+K(ω) where K(ω) is Sp -valued meromorphic family. Determinant Dp(ω) = detp(I+K(ω)) meromorphic, its zero order equals kernel dimension (algebraic multiplicity) of I+K(ω) . Step 2 (Relativization and pole counting) : Dene τp(ω) = detp(S(ω)S0(ω)−1) . If S0 analytic nonzero on strip, τp shares poles and orders with S , poles being QNM. Step 3 (Reference independence) : If choosing another e S0 also satisfying condition, then τp/eτp= detp(S0e S−1 0) is analytic outer function without zeros/poles, leaving pole set unchanged. QED. 7 Proof of Theorem 3.4 (Block-Level Pseudo-Unitary and Global Carleman Modulus) Block level : By S† ℓmηSℓm =η and det(η−1S† ℓmηSℓm)=1 obtain |det Sℓm|= 1 . Global phase : By Theorem 3.1 obtain arg detpS(ω) = −2πΞ(p)(ω) . Global modulus ( p= 2 ) : If S(ω) unitary with S(ω)−I∈S2 , spectral angles {θj(ω)} ∈ ℓ2 yield |det2S(ω)|= exp Pj(1 −cos θj(ω))≥1 . In general J-unitary case modulus non-constant, thus phase-normalization c detp natural object for real-axis modulus constraint. QED. 4 8 Globally Meromorphic Fitting and Fisher Projection Geometry (For Data-Side Implementation) On strip ℑω∈[−γ0,0] parametrize log bτp(ω) = PJ j=1 log ω−ωj ω−ωj+iQ(ω) , where ωj are lower half-plane poles, Q low-order entire function taking purely imaginary values on real axis. Enforce conjugate pairing and phase-normalized modulus equals 1, suppress false poles via strip cross-validation. Proposition 5 (8.1: Fisher Principal Angle Upper Bound) . For whitened observation yk=ℑlog bτp(ωk)+ ϵk , Jacobian J with constraint submanifold tangent space projection PM yields restricted Fisher FM= (PMJ)⊤(PMJ) . If ϑ is maximum principal angle between range(J) and range(PM) , then variance reduction factor R ≤ 1/|sin ϑ| . Proof in Appendix F. 9 Reproducible Experimental Protocols (P1P3) P1 | rel-zeta (closed domain) : Grid step h (three levels), heat kernel window t∈[tmin, tmax] ( tmin ∼c h2 ), extrapolation order N∈ {2,3} , small mass µ (three to ve logarithmic points). Target quantity Vol−1 4∂Λℜlog detζ,rel . Acceptance: slope error <1% ; drift under dierent corner triangulations/gauges <0.5% . P2 | meromorph-t (exterior domain) : Fit log bτp recovering {ωj} . Priors: pairwise symmetric, strip analytic, real-axis modulus constraint (on bτp ), and ℜlog det2(ω)≥0 (if using p= 2 ). Acceptance: CRLB improvement over mode-by-mode ≥1.3× ; false alarm rate ≤5% ; cross-strip consistent. P3 | bh-channels (pseudo-unitary and BK phase) : JostWronskian normalization constructs η , compute |S† ℓmηSℓm −η| and phase closure arg c detpS+2πΞ(p) . Acceptance: pseudo-unitary residual <10−12 , phase closure <10−3 radians; converges as a+b/ℓmax with ℓmax . 10 Discussion and Outlook Under explicitly veriable analytic hypotheses, this paper completes four main conclusions: p∈ {1,2} unied version of xed-energy BK, closed-domain volume slope, reference independence of relative scattering determinant poles = QNM, and real-frequency phasemodulus decomposition, providing reproducible experimental pipelines. Limitations: LAP constant may deteriorate under strong trapping or extreme spin; non-local boundaries and singular geometry require separate verication of multiplicative anomaly; statistical side needs robust regularization against model bias. Future work includes: extending modulusphase formula for detp under Kren spaces; seamlessly incorporating BK version for dierential forms/electromagnetic elds; testing stability of reference independent poles using multi-station strip data. A Complete Derivation of DOILAP Dominated Convergence A.1 Kernel Bound and Weight Insertion Take fε(λ) = 1 2+1 πarctanω−λ ε . fε∈OL , |fε|OL ≤C/ε . DOI expression yields fε(HΛ)−fε(H0) = RR Φfε(λ, µ)dEΛ(λ)V dE0(µ) , where Φfε satises supλR|Φfε(λ, µ)|dµ ≤C/ε , supµR|Φfε(λ, µ)|dλ ≤ C/ε . Insert ⟨x⟩±s obtaining fε(HΛ)−fε(H0) = RR(⟨x⟩−sdEΛ(λ)) (⟨x⟩sV⟨x⟩s) (dE0(µ)⟨x⟩−s) Φfε(λ, µ) . 5 A.2 Schatten Domination Inequality By Haagerup/Schur bound with Hölder inequality (on Sp ), |fε(HΛ)−fε(H0)|Sp≤ |Φfε|Schur · supλ∈I|⟨x⟩−sE′ Λ(λ)⟨x⟩−s| · |⟨x⟩sV⟨x⟩s|Sp·supµ∈I|⟨x⟩−sE′ 0(µ)⟨x⟩−s| . Use Stone formula E′ #(λ) = π−1ℑR#(λ+i0) with (H - LAP) obtaining |fε(HΛ)−fε(H0)|Sp≤ CIε−1|⟨x⟩sV⟨x⟩s|Sp . In scattering setting better use dierence resolvent form |⟨x⟩−s(RΛ(λ±i0)−R0(λ±i0))⟨x⟩−s|Sp controlling |⟨x⟩sV⟨x⟩s|Sp , thus obtaining unied domination |fε(HΛ)−fε(H0)|Sp≤CIε−1Mp(I) . A.3 Lebesgue Points and Limit Exchange Let ω be Lebesgue point of spectral shift object. By above obtain family of ε -uniformly integrable dominating function M(ω) = supε<ε0|fε(HΛ)−fε(H0)|Sp∈L1 loc(I) . Thus limit of DOI-trace as ε↓0 commutes with local integration over ω , obtaining xed-energy version of rst/second-order trace formula, completing Theorem 3.1 proof. B Relative Heat Kernel Item-Wise Cancellation and Λ -Slope Re- nement B.1 Short-Time Expansion and Local Coecients For Laplace-type operator K# (including gauge-ghost pairing) have Tr(e−tK#)∼Pj≥0aj(K#)t(j−d)/2 . Under principal symbol consistency with boundary/ghost matching, relative dierence arel j>0= aj(KΛ)−aj(K0)=0 ; corner and boundary term coecients also cancel in relative dierence (Wodzicki residue zero ensures multiplicative anomaly absent). B.2 Logarithmic Derivative and Volume Term Relative ζ written as ζrel(s;µ) = 1 Γ(s)R∞ 0ts−1e−tµ2Krel(t)dt . Dierentiating with respect to Λ , only arel 0= Vol4(M)c0 contributes, yielding ∂Λζ′ rel(0; µ) = −∂Λarel 0R∞ 0t−1e−tµ2dt + nite terms. Volume density with µ↓0 exchange, by text convention ∂Λc0=1 8πG , obtaining Theorem 3.2. B.3 Tauberian Exchange Error Estimate Take t0=µ−2α ( α∈(0,1) ), R∞ t0t−1e−tµ2Krel(t)dt controlled by spectral gap and deresonance projection as O(µ2(1−α)) , while (0, t0) segment error after higher-order coecient cancellation becomes O(t1/2 0) = O(µ−α) coecient nullication term, overall can take α making total error o(1) . C Reference Independence of Relative Scattering Determinant and Pole Counting C.1 Analytic Fredholm Tools Write S(ω) = I+K(ω) , K(ω) being Sp -valued meromorphic family. Then Dp(ω) = detp(I+K(ω)) meromorphic with zero order equal to dim ker(I+K(ω)) . 6 C.2 Relativization and Pole Transfer Assume S0 analytic without zeros/poles on strip, dene τp(ω) = detp(S(ω)S0(ω)−1) = detp(I+ K(ω)) ·detp(S0(ω)−1) . Latter factor analytic nonzero, thus τp poles synchronize with Dp , order being QNM algebraic multiplicity. C.3 Reference Independent Outer Function Factor If changing reference to e S0 , then τp/eτp= detp(S0e S−1 0) is analytic without zeros/poles (outer function), leaving pole set and multiplicity unchanged. D Pseudo-Unitary: Channel Construction and Global Carleman Modulus D.1 Energy Flux Quadratic Form and J-Unitary Take Jost solutions uin/out at both ends of radial equation with Wronskian normalization, making energy ux F=ℑ(u ∂ru) consistent at both ends. Accordingly dene channel quadratic form η= diag(1,−1) making S† ℓmηSℓm =η . D.2 Block-Level Determinant Unit Modulus and Global Phase Finite-dimensional block directly yields |det Sℓm|= 1 . Global direct sum under S2 setting only preserves phase equality; if globally unitary with S−I∈S2 , spectral angle expansion yields |det2S|= exp(P(1 −cos θj)) ≥1 . E Koplienko Phase Fixed-Energy Construction ( p= 2 ) E.1 Second-Order Trace Formula and DOI For fε have Trfε(HΛ)−fε(H0)−f′ ε(H0)V=Rf′′ ε(E)dη(E) . Integrate right side twice by parts yielding −Rf′ ε(E)dΞ(2)(E) . E.2 Dominated Convergence and Lebesgue Points By Appendix A's S2 domination with |f′ ε|L1≤C obtain integrable domination, as ε↓0 , f′ ε converges to δω (weak sense) recovering Ξ(2)(ω) at Lebesgue points. E.3 Scattering Phase and Determinant Fixed-energy implementation with AC berization identies Ξ(2)(ω) as scattering phase secondorder spectral shift antiderivative, exponentiating yields det2S(ω) = exp(−2πi Ξ(2)(ω)) . F Proof of Fisher Projection Geometry Upper Bound F.1 Model and Projection Whitened observation y=Jθ +ϵ , hard constraint C(θ) = 0 dening dierentiable submanifold M with tangent space projection PM satisfying P2 M=PM . 7 F.2 Principal Angle and Spectral Bound Let ϑ be maximum principal angle between range(J) and range(PM) , have |PMv|≥|sin ϑ| |v| for all v∈range(J) . Thus v⊤FMv=|PMJv|2≥sin2ϑ|Jv|2=v⊤(sin2ϑ F)v , yielding FM⪰sin2ϑ F . Taking maximum eigenvalue yields Tr(F−1 M)≤Tr(F−1)/sin2ϑ , i.e., variance reduction factor R ≤ 1/|sin ϑ| . QED. G Reproducible Experiment Parameters and Error Budget (Brief Table) G.1 P1 (closed domain) : h∈ {h0, h0/2, h0/4} ; tmin ∼c h2 , tmax satisfying semiclassical window; µ taking {µ0, µ0/3, µ0/9, µ0/27} . Extrapolation using bilinear (for (µ, h) ) and Richardson (for t - window) hybrid. Tolerance: slope <1% ; drift under dierent corner triangulations/gauges <0.5% . G.2 P2 (exterior domain) : Strip ℑω∈[−γ0,0] uniform sampling; pole number J jointly determined by AIC/BIC and strip cross-validation; penalty term constrains Q(ω) degree and realaxis purely imaginary condition; prior ℜlog det2≥0 only as soft regularization. Tolerance: CRLB improvement ≥1.3 , false alarm ≤5% . G.3 P3 (channels) : Extrapolation radius, matching radius, integration step calibrated via grid search; Wronskian normalization dierence <10−12 ; |S† ℓmηSℓm −η|∞<10−12 ; phase closure <10−3 radians; converges as a+b/ℓmax with ℓmax . End of Main Text and Appendices 8