Two-mode dominance and deterministic parameter bias bounds for equatorial Kerr-de Sitter ringdown

TW O-MODE DOMINANCE AND DETERMINISTIC P ARAMETER BIAS BOUNDS F OR EQUA TORIAL KERR–DE SITTER RINGDO WN R UILIANG LI Abstract. W e study scalar w av es on subextremal Kerr–de Sitter space- times in a compact slow-rotation regime and at a ﬁxed ov ertone index. W orking initially at a ﬁxed cosmological constant Λ > 0 and uniformly for ( M , a ) in a compact slow-rotation set, using the meromorphic/F redholm framew ork for quasinormal mo des and a semiclassical equatorial labeling pro ved in a companion pap er, we establish a quantitativ e tw o-mo de dominance theorem in an equatorial high-frequency pack age: after exact azim uthal reduction, microlo cal equatorial lo calization, and analytic p ole selection b y en tire lo calization weigh ts constructed from equatorial pseudop oles, the k = ± ℓ sector signals are eac h go v erned b y a single quasi- normal exp onential, up to an explicitly controlled tail and an O ( ℓ −∞ ) con tribution from all other p oles. W e then develop a fully deterministic frequency-extraction stability estimate based on time-shift inv ariance, and com bine it with the tw o-mo de dominance result and the companion pap er’s inv erse stability theorem to obtain an explicit parameter bias b ound for ringdo wn-based recov ery of ( M , a ). Finally , using the compan- ion pap er’s three-parameter inv erse theorem and a damping observ able based on the scaled imaginary part of one equatorial mo de, we propagate the same deterministic error c hain to a local bias b ound for reco very of ( M , a, Λ) on compact parameter sets with | a | b ounded a wa y from 0. As a further consequence, we obtain a lo calized pseudospectral stability statemen t for the equatorial resolv ent pack age, quantifying how large microlo calized resolv ent norms enforce proximit y to the lab eled equato- rial p oles. The resulting estimates clarify the conditioning mec hanisms (start time, window length, shift step, and detector nondegeneracy) and pro vide a rigorous PDE-to-data in terface for high-frequency black-hole sp ectroscop y . Date : F ebruary 19, 2026. 2020 Mathematics Subje ct Classiﬁc ation. 35P25, 35R30, 35B40, 58J50, 83C57. Key wor ds and phr ases. Kerr–de Sitter, quasinormal modes, resonance expansion, normally hyperb olic trapping, deterministic frequency extraction, blac k-hole sp ectroscop y . 1 2 RUILIANG LI 1. Introduction The late-time resp onse of a p erturbed blac k hole is often mo deled, on a ﬁnite observ ation window, by a sup erposition of a small num ber of exp onen- tially damp ed oscillations, (1.1) y ( t ∗ ) ≈ J X j =1 A j e − iω j t ∗ , t ∗ ≥ t 0 , ℑ ω j < 0 , where y ( t ∗ ) is a time series obtained from an observ ation of the ﬁeld and the complex frequencies ω j are quasinormal mo des (QNMs). In applications y is typically not the raw wa v eform: it ma y incorp orate symmetry reduc- tion, phase-space lo calization, or other prepro cessing designed to isolate a particular dynamical channel b efore attempting a low-rank ﬁt. F rom a deterministic PDE viewp oint, using ( 1.1 ) for inference leads to t w o in tert wined stabilit y issues. Dominance and leak age. Ev en when a meromorphic resolven t and a resonance expansion are a v ailable, the time-domain represen tation is t ypically an inﬁnite sum with a remainder. A ﬁnite-mo de mo del on a prescrib ed windo w requires quantitativ e dominance of the selected p oles together with quan titativ e control of leak age from all other p oles and from the tail term on a shifted con tour. Nonnormalit y and extraction conditioning. The stationary problem is non-selfadjoin t; residue pro jectors can b e large (often called excitation fac- tors ), and pseudosp ectral and transien t eﬀects may in terfere with naiv e trun- cations or ﬁts that do not separate the relev ant phase-space c hannels. More- o v er, extracting complex frequencies from ﬁnite-time data is ill-conditioned in general and m ust b e analyzed together with branc h selection for the complex logarithm; this is classical in the Pron y/matrix-p encil literature [ 25 , 26 , 27 , 28 , 29 ]. Recent w ork on blac k-hole ringdo wn has highligh ted these mec hanisms from several complemen tary angles, including pseudosp ectral analyses, non-mo dal dynamics, and transien ts [ 14 , 20 , 15 , 21 , 18 ]. Closely related phenomena arise near a v oided crossings and exceptional p oin ts, where resonan t excitation can amplify or reshuﬄe the observ ed mo de hierarc h y [ 16 , 24 ]. Setting and scop e; t w o-mo de dominance in the equatorial pac k age. W e w ork on sub extremal Kerr–de Sitter spacetimes with ﬁxed cosmological constan t Λ > 0 and slo w rotation | a | ≤ a 0 . In this setting the stationary resolv en t is meromorphic, and con tour deformation yields resonance expan- sions for w a v e solutions within the F redholm/microlo cal framew ork dev eloped for asymptotically hyperb olic and Kerr–de Sitter geometries [ 2 , 5 , 10 , 11 ]. W e isolate an e quatorial high-fr e quency p ackage : w e ﬁx an o v ertone index n ∈ N 0 and consider large angular momen tum ℓ in the exact azimuthal sectors k = ± ℓ , using h ℓ = ℓ − 1 as semiclassical parameter. EQUA TORIAL KERR–DE SITTER RINGDOWN 3 A key p oin t of this pap er is that, within this pack age, one can justify a t w o-mo de mo del after an explicit sequence of lo calizations: exact azimuthal pro jection to k = ± ℓ , microlo cal equatorial lo calization, and analytic p ole selection by an en tire weigh t. Our tw o-mode dominance theorem is therefore not a statement ab out a generic unﬁltered ringdo wn w a v eform. Rather, it is a quantitativ e statement ab out tw o r eﬁne d signals (one for each of k = ± ℓ ) obtained b y prepro cessing, in a regime where the relev an t QNMs admit a semiclassical lab eling and can b e isolated with uniform error b ounds. More precisely , the dominance statemen t concerns the mo de-separated, microlo cal- ized signals obtained by applying the exact azimuthal pro jectors Π k ± and the equatorial cutoﬀs A ± ,h ℓ to the solution, together with the entire w eigh ts e g ± ,ℓ inserted in the in v erse Laplace representation; see Deﬁnition 5.11 . This pap er is the second part of a series. The companion pap er [ 1 ] pro v es the semiclassical equatorial QNM lab eling in the ab o v e pac k age and establishes a quan titative lo cal inv ersion map from the pair of equatorial frequencies ( ω + ,ℓ , ω − ,ℓ ) to the parameters ( M , a ). The purp ose of the present pap er is to pro vide the time-domain mec hanism needed to turn that frequency information into a deterministic inference pro cedure with explicit error b ookkeeping on a ﬁnite windo w. Strategy: microlo cal/analytic mo de selection plus deterministic extraction. Tw o ingredients are essential. First, w e separate the relev an t phase-space channel b efor e ﬁtting a low-rank time series: w e p erform exact azim uthal pro jection to k = ± ℓ and apply equatorial microlo cal cutoﬀs. Sec- ond, we select a single o v ertone uniformly in ℓ b y multiplying the resolv en t in tegrand with an entir e weigh t e g ± ,ℓ ( ω ) constructed from equatorial pseu- dop oles provi ded b y [ 1 ]. This analytic window is designed to b e compatible with con tour deformation: it suppresses all p oles except the labeled one while having controlled growth on a shifted contour, yielding a window ed resonance expansion with an explicitly con trolled tail term. A related approach—isolating individual QNM contributions using a mo de prior—app ears in recent data-analysis work under the name QNM ﬁlters [ 13 ]. The common theme is that, in a regime where QNM expansions are meaningful, prior frequency information can be used to enhance a target mo de. The k ey distinction here is that the w eights are c hosen to b e entire (indeed, p olynomial) and to interact well with semiclassical microlo calization, so that leak age from non-target p oles can b e con trolled quantitatively and uniformly in ℓ . The dep endence on a prior is therefore explicit: our results quan tify ho w windo wing, tail suppression, and extraction errors propagate in a deterministic chain, and the framework naturally accommo dates iterative use of a coarse prior follo wed b y reﬁnement. The second ingredien t is a deterministic stability theory for extracting a complex frequency from a one-mo de mo del with a con trolled remainder. W e analyze a time-shift in v ariant estimator (a one-step matrix-p encil/Pron y-t yp e construction) in an abstract Hilb ert space setting and use the pseudop ole 4 RUILIANG LI prior to ﬁx the logarithm branch. This yields an extraction error b ound that can b e combined directly with the PDE remainder. Main results. The main conclusions of the pap er can b e summarized as follo ws. • A nalytic al ly windowe d r esonanc e exp ansion and two-mo de dominanc e. After azimuthal reduction to k = ± ℓ , equatorial microlo cal lo caliza- tion, and analytic p ole selection by the weigh ts e g ± ,ℓ , the resulting signals are each gov erned by a single QNM exp onential, up to an explicitly controlled shifted-con tour tail and an O ( ℓ −∞ ) contribution from all other p oles; see Theorem 5.23 , based on the windo w ed ex- pansion of Theorem 5.14 and the equatorial p ole/sectorization inputs of [ 1 ]. • Deterministic stability of one-mo de fr e quency extr action. F or a one- mo de Hilbert-v alued signal y ( t ∗ ) = Ae − iω t ∗ + r ( t ∗ ) on a ﬁnite windo w, the time-shift estimator together with a ﬁxed logarithm branc h reco v ers ω with an explicit error b ound in terms of the residual size; see Theorem 6.6 and its equatorial sp ecialization in Corollary 6.12 . • Deterministic p ar ameter bias b ound. Combining the preceding steps with the inv erse stability theorem of [ 1 ], we obtain an explicit deter- ministic estimate for the bias of the reco vered parameters ( M , a ) in terms of the PDE remainder, the extraction window parameters, and the semiclassical conditioning of the equatorial parameter map; see Theorem 7.15 . • Thr e e-p ar ameter deterministic bias b ound when Λ varies. On com- pact three-parameter sets with | a | b ounded aw a y from 0, the com- panion pap er prov es that adding one damping observ able f W ℓ := − ( n + 1 2 ) − 1 ℑ ω + ,ℓ yields a lo cally in vertible three-parameter data map ( U ℓ , V ℓ , f W ℓ ) 7→ ( M , a, Λ). W e show that the same time-domain extraction and tail estimates propagate through this three-parameter in v erse map, pro ducing an explicit deterministic bias b ound for re- co v ery of ( M , a, Λ); see Theorem 7.20 . Relation to in v erse and sp ectral results. A useful mathematical p oin t of comparison is the in verse theorem of Uhlmann–W ang [ 12 ], who recov er the mass of a de Sitter–Sch w arzsc hild blac k hole from a single QNM with a H¨ older-t yp e stability estimate. Our setting is genuinely non-selfadjoint and in v olves rotation: we use t wo equatorial mo des to recov er ( M , a ) and propagate time-domain PDE errors through a concrete extraction map, obtaining a deterministic bias bound on a prescrib ed windo w. On the sp ectral side, h yp erb oloidal and Keldysh-t yp e schemes make biorthogonalit y and sp ectral pro jection mechanisms explicit and pro vide new p ersp ectiv es on resonance expansions [ 19 , 22 ]. Relatedly , there has b een recent progress on con v ergent complete mo de decomp ositions which go b ey ond late-time QNM sums [ 23 ]. The present pap er do es not aim at a globally complete mo de EQUA TORIAL KERR–DE SITTER RINGDOWN 5 decomp osition. Instead, it isolates a semiclassical Kerr–de Sitter pack age in whic h a t w o-mo de mo del can b e justiﬁed with explicit constants and then used for deterministic in v erse b ounds. Organization of the pap er. Section 2 in tro duces the Kerr–de Sitter setup and the functional framew ork. Section 3 prov es uniform resolv ent b ounds on shifted con tours. Section 4 establishes the resonance expansion with remainder. Section 5 develops analytic p ole selection and pro ves tw o-mo de dominance. Section 6 studies deterministic frequency extraction from ﬁnite- time data, and Section 7 propagates the resulting frequency errors through the inv erse map of [ 1 ] to obtain parameter bias b ounds. Section 8 pro v es a microlo calized pseudosp ectral resolven t b ound in the same equatorial pac k age, quan tifying ho w far the equatorial resolven t can grow aw a y from the lab eled p oles. Section 9 discusses further directions. The app endices collect analytic lemmas for p ole selection and contour subtraction, precise companion-pap er inputs, the dual-state interpretation of amplitudes, and the t w o-exp onen tial conditioning analysis. 2. Geometric/PDE setup and functional framew ork In Sections 2 – 6 w e ﬁx a cosmological constan t Λ > 0 (suppressing it from the notation) and work on sub extremal Kerr–de Sitter spacetimes ( M M ,a , g M ,a ) with ( M , a ) ranging in a compact slow-rotation parameter set K . In Section 7.6 w e extend the ﬁnal bias bounds to compact three-parameter sets ( M , a, Λ); the required uniformity of the forw ard estimates with resp ect to Λ on compact sets is summarized in App endix E . The aim of this section is to set up a time-domain framework in which one can write forward solutions via an inv erse Laplace transform, reduce exactly to azim uthal symmetry sectors, insert microlo cal equatorial cutoﬀs, and then conv ert resolven t p oles (QNMs) into ringdown terms by con tour deformation. While most ingredients are standard in the Kerr–de Sitter F redholm framew ork [ 2 , 5 , 11 ], we record the con ven tions and normalizations w e will use later; in particular w e are careful ab out the underlying spatial manifold. Since we work uniformly for ( M , a ) in a ﬁxed compact slo w–rotation set K , all auxiliary c hoices (such as the buﬀer size δ used b elo w to extend across the futur e horizons) are made once and for all, with constants uniform on K . 2.1. Spacetime region, regular time coordinate, and a ﬁxed spatial slice. Let ( M M ,a , g M ,a ) b e a sub extremal Kerr–de Sitter spacetime with parameters ( M , a ) and ﬁxed Λ > 0. Denote b y r e ( M , a ) and r c ( M , a ) the (future) even t and cosmological horizon radii (0 < r e < r c ), and let M ◦ M ,a := R t × ( r e , r c ) r × S 2 θ,φ b e the domain of outer communications in Bo y er–Lindquist co ordinates. 6 RUILIANG LI Ph ysical region v ersus analytic extension. The ph ysically relev ant spatial region is (2.1) X phys M ,a := ( r e , r c ) × S 2 θ,φ ∗ , M phys M ,a := R t ∗ × X phys M ,a . All initial data and all observ ation/measuremen t cutoﬀs considered in this pap er are assumed to b e supp orted in a ﬁxed compact subset of X phys M ,a whic h is uniformly separated from the horizons. F or the stationary F redholm theory and the redshift estimates it is, how ev er, con v enient to work on a slightly extended region across the futur e horizons; this extension is a technical device and will never en ter the cutoﬀ quantities χR ( ω ) χ used in the contour argumen t. Regular time coordinate and extension across future horizons. As is standard, w e pass to a future-regular stationary time co ordinate t ∗ and a corresp onding azim uthal co ordinate φ ∗ (Eddington–Fink elstein–t yp e “star” co ordinates) so that the metric is smo oth across the future even t and cosmological horizons; see, for instance, [ 5 , § 2] and [ 11 , § 2]. Concretely , we assume: • g M ,a is smo oth across the h ypersurfaces r = r e and r = r c in the ( t ∗ , r , θ , φ ∗ ) chart; • T := ∂ t ∗ and Φ := ∂ φ ∗ are Killing ﬁelds; • t ∗ is a global time function on a slightly extended region across the futur e horizons. W e ﬁx a small buﬀer δ > 0 and deﬁne the extended spatial region (2.2) X ◦ M ,a := ( r e − δ, r c + δ ) × S 2 θ,φ ∗ , X M ,a := [ r e − δ, r c + δ ] × S 2 θ,φ ∗ , M M ,a := R t ∗ × X ◦ M ,a . W e write Σ M ,a := { t ∗ = 0 } ⊂ M M ,a for the reference Cauc h y h yp ersurface. Compactiﬁed spatial slice and cutoﬀ insensitivity . Sev eral parts of the stationary F redholm theory are naturally formulated on a compactiﬁed spatial manifold with b oundary obtained b y closing the r –in terv al in ( 2.2 ) . W e therefore regard X ◦ M ,a as the interior of the compact manifold (2.3) X M ,a := X M ,a = [ r e − δ, r c + δ ] × S 2 θ,φ ∗ . All Cauc h y problems are posed on M M ,a = R t ∗ × X ◦ M ,a , but all Sob olev spaces are tak en on X M ,a (for instance b y extending across ∂ X M ,a in lo cal c harts). The b oundary ∂ X M ,a = { r = r e − δ } ∪ { r = r c + δ } is artiﬁcial and lies outside the physical region X phys M ,a . Consequently , as long as a cutoﬀ χ ∈ C ∞ c ( X ◦ M ,a ) is supp orted strictly inside X phys M ,a , the op erator χR ( ω ) χ dep ends only on the geometry in the ph ysical domain and is insensitiv e to the chosen completion near ∂ X M ,a . Uniformizing the underlying manifold (optional, but conv enien t). When v arying parameters ( M , a ) in a compact set K inside a slo w-rotation sub extremal range, one ma y identify the family of compactiﬁed manifolds EQUA TORIAL KERR–DE SITTER RINGDOWN 7 X M ,a with a single ﬁxed manifold X = [ r − , r + ] × S 2 b y an aﬃne reparame- terization of the radial v ariable that sends r e ( M , a ) − δ and r c ( M , a ) + δ to r − and r + , resp ectiv ely . This is a routine device (we use it in the companion pap er), and it allo ws all Sob olev norms to be tak en on a ﬁxed X while k eeping constan ts uniform on K . T o k eep notation ligh t, we will not distinguish notationally b et ween X M ,a and such an iden tiﬁed copy . 2.2. P arameter regime and geometric constan ts. F or a ﬁxed cosmo- logical constant Λ > 0, Kerr–de Sitter metrics are parametrized b y the mass M > 0 and the rotation parameter a ∈ R . In Boy er–Lindquist form, the radial function is (2.4) ∆ r ( r ; M , a ) := ( r 2 + a 2 )  1 − Λ r 2 3  − 2 M r . W e write P Λ for the sub extr emal parameter set, i.e. those ( M , a ) for whic h ∆ r has (at least) tw o distinct p ositiv e simple roots 0 < r e ( M , a ) < r c ( M , a ) corresp onding to the ev en t and cosmological horizons. W e will w ork uniformly on compact subsets of a slow-rotation sub extremal range: ﬁx a 0 > 0 and a compact set (2.5) K ⋐ P Λ ∩ {| a | ≤ a 0 } . All implicit constants in the sequel are allo w ed to dep end on Λ and on K (as w ell as on the ﬁxed cutoﬀ χ used to lo calize aw a y from the horizons), but are uniform in the high-frequency parameter ℓ → ∞ . Photon-sphere frequency scale (Sc h w arzsc hild–de Sitter). When a = 0, the trapp ed null geo desics concen trate on the photon sphere r ph = 3 M , and the corresp onding co ordinate-time orbital frequency is (2.6) Ω ph ( M ) := √ 1 − 9Λ M 2 3 √ 3 M . This quan tit y app ears as the leading real-frequency co eﬃcien t in high- ℓ equatorial QNM asymptotics in the Sc h warzsc hild–de Sitter limit; see [ 1 , § 2]. W e include ( 2.6 ) mainly as a conv enien t frequency scale when discussing uniformit y on parameter sets (Remark 5.16 ); none of the main pro ofs b elo w require the explicit closed form. Ov ertone index. Throughout, w e ﬁx an ov ertone index n ∈ N 0 . All results in the tw o-mode regime are uniform in ℓ → ∞ for this ﬁxed n . 2.3. The wa ve op erator and a conv enien t splitting. Let (2.7) P := □ g M ,a + V where V ∈ C ∞ ( M M ,a ) is stationary and axisymmetric: T ( V ) = 0 , Φ( V ) = 0 . (F or most of the pap er one may tak e V ≡ 0; allowing V costs nothing and mak es the formalism stable under low er-order p erturbations.) 8 RUILIANG LI Stationary splitting. Since co eﬃcients of P are indep enden t of t ∗ , one can write P in the form (2.8) P = ∂ 2 t ∗ + Q ∂ t ∗ + L, where Q ∈ Diﬀ 1 ( X M ,a ) and L ∈ Diﬀ 2 ( X M ,a ) are diﬀeren tial operators with co eﬃcients indep enden t of t ∗ . Concretely , Q enco des the mixed t ∗ – x deriv atives and p ossible ﬁrst-order t ∗ –terms, while L is a purely spatial second-order op erator (elliptic on Σ M ,a ). Lemma 2.1 (Existence of the splitting) . L et P = □ g + V on M M ,a with g stationary in t ∗ . Then ther e exist t ∗ –indep endent op er ators Q ∈ Diﬀ 1 ( X M ,a ) and L ∈ Diﬀ 2 ( X M ,a ) such that ( 2.8 ) holds. Mor e over, Q and L c ommute with Φ = ∂ φ ∗ . Pr o of. W rite P in lo cal co ordinates ( t ∗ , x ) on M M ,a : P = X | α |≤ 2 a α ( x ) ∂ α 0 t ∗ ∂ α ′ x , a α indep enden t of t ∗ . Collect the terms with α 0 = 2. Since P is a wa v e op erator, the co eﬃcien t of ∂ 2 t ∗ is a smo oth function that do es not v anish on X ◦ M ,a ; up to an everywhere nonzero smo oth factor coming from the choice of density in □ g , this co eﬃcien t is g t ∗ t ∗ . Because t ∗ is a global time function on the future-extended region (so dt ∗ is timelike), the slices { t ∗ = const } are spacelike and therefore g t ∗ t ∗ < 0. Multiplying P b y a smooth no where-v anishing factor dep ending only on x (see the remark b elo w), we may normalize the ∂ 2 t ∗ co eﬃcien t to equal 1. Collect the remaining ∂ t ∗ terms (including mixed deriv atives ∂ t ∗ ∂ x j ) into Q∂ t ∗ , and the rest into L . Stationarity giv es t ∗ –indep endence, and axisymmetry gives comm utation with Φ. □ R emark 2.2 (Normalization of the ∂ 2 t ∗ co eﬃcien t) . The harmless division b y a smo oth nowhere v anishing function in the pro of ab ov e do es not aﬀect the quasinormal sp ectrum. Indeed, if P is replaced by e P = µP with µ ∈ C ∞ ( X M ,a ) and µ  = 0, then the stationary family satisﬁes e P ( ω ) = µP ( ω ), and hence e P ( ω ) − 1 = P ( ω ) − 1 µ − 1 wherev er either in verse exists; in particular, the p oles (and their orders) of the meromorphic resolven t are unc hanged. 2.4. Cauc h y problem and energy spaces. Let Σ M ,a = { t ∗ = 0 } and denote initial data b y (2.9) f 0 := u | t ∗ =0 , f 1 := ( ∂ t ∗ u ) | t ∗ =0 . F or s ≥ 0 deﬁne the standard energy Sob olev space (2.10) H s := H s +1 (Σ M ,a ) × H s (Σ M ,a ) (with resp ect to any ﬁxed smo oth Riemannian metric on Σ M ,a ; diﬀerent c hoices yield equiv alen t norms on compact sets). EQUA TORIAL KERR–DE SITTER RINGDOWN 9 Theorem 2.3 (W ell-p osedness and an a priori b ound) . F or every s ≥ 0 and initial data ( f 0 , f 1 ) ∈ H s , ther e exists a unique solution u ∈ C 0 ( R t ∗ ; H s +1 (Σ M ,a )) ∩ C 1 ( R t ∗ ; H s (Σ M ,a )) to P u = 0 on M M ,a with initial data ( 2.9 ) . Mor e over, ther e exist c onstants C s , κ s ≥ 0 such that (2.11) ∥ u ( t ∗ ) ∥ H s +1 + ∥ ∂ t ∗ u ( t ∗ ) ∥ H s ≤ C s e κ s | t ∗ |  ∥ f 0 ∥ H s +1 + ∥ f 1 ∥ H s  , t ∗ ∈ R . Pr o of. W e give a self-con tained energy estimate argument; for Kerr–de Sitter the required geometric h yp otheses (existence of a smo oth global time function t ∗ on the future-extended region and smo othness across horizons) are standard and can b e found in [ 2 , 5 , 11 ]. Since dt ∗ is timelike, the op erator P = □ g M ,a + V is strictly hyperb olic with resp ect to t ∗ . After normalizing the ∂ 2 t ∗ co eﬃcien t as in Lemma 2.1 , we ma y write P = ∂ 2 t ∗ + Q∂ t ∗ + L with Q ∈ Diﬀ 1 ( X M ,a ) and L ∈ Diﬀ 2 ( X M ,a ) ha ving t ∗ –indep enden t co eﬃcients. Fix a smo oth Riemannian metric on Σ M ,a and use it to deﬁne the Sob olev norms in ( 2.10 ) . Let Λ s = (1 − ∆ Σ ) s/ 2 b e deﬁned using an elliptic Laplace-t yp e op erator ∆ Σ on Σ M ,a . F or smo oth solutions, apply Λ s to the equation and take the L 2 inner pro duct of ∂ t ∗ (Λ s u ) with ∂ 2 t ∗ (Λ s u ). Standard comm utator estimates for Λ s and Q, L (whose co eﬃcien ts are smo oth and t ∗ –indep enden t) give a diﬀerential inequality of the form d dt ∗  ∥ Λ s ∂ t ∗ u ( t ∗ ) ∥ 2 L 2 + ∥ Λ s +1 u ( t ∗ ) ∥ 2 L 2  ≤ C s  ∥ Λ s ∂ t ∗ u ( t ∗ ) ∥ 2 L 2 + ∥ Λ s +1 u ( t ∗ ) ∥ 2 L 2  , where C s dep ends on ﬁnitely many deriv atives of the co eﬃcien ts of Q, L, V . Gron wall’ s inequalit y yields ( 2.11 ) for smo oth solutions. Existence and uniqueness in the stated spaces follow by a standard densit y argumen t, using the energy estimate to pass to limits. □ R emark 2.4 . On Kerr–de Sitter, one can obtain sharp er b ounds (and decay) using redshift multipliers near horizons and normally h yp erbolic trapping estimates. F or the functional calculus b elo w, the crude exp onential b ound ( 2.11 ) is enough to justify Laplace transforms; later sections will in vok e reﬁned resolven t estimates for decay . The Cauch y propagator. F or s ≥ 0 we denote b y (2.12) U ( t ∗ ) : H s → H s , U ( t ∗ )( f 0 , f 1 ) :=  u ( t ∗ ) , ∂ t ∗ u ( t ∗ )  , the solution op erator provided by Theorem 2.3 . By stationarity , U ( t ∗ ) forms a strongly contin uous group. 2.5. First-order form ulation and the stationary family. Introduce the state vector U ( t ∗ ) :=  u ( t ∗ ) ∂ t ∗ u ( t ∗ )  . 10 RUILIANG LI Using the splitting ( 2.8 ) , the equation P u = 0 is equiv alent to the ﬁrst-order system (2.13) ∂ t ∗ U = A U, A :=  0 I − L − Q  , acting on H 0 with natural domain Dom ( A ) = H 2 (Σ M ,a ) × H 1 (Σ M ,a ). The solution op erator is thus the strongly contin uous group e t ∗ A on H s : U ( t ∗ ) = e t ∗ A U (0) . Stationary family . F or ω ∈ C deﬁne the stationary op erator on X M ,a b y the mo de substitution u ( t ∗ , x ) = e − iω t ∗ v ( x ): (2.14) P ( ω ) v := e iω t ∗ P  e − iω t ∗ v     t ∗ =const =  L − iω Q − ω 2  v . Th us e − iω t ∗ v solv es P u = 0 if and only if P ( ω ) v = 0. 2.6. Meromorphic resolv en t and quasinormal mo des. W e use a mi- crolo cal F redholm framew ork (V asy-type v ariable order Sob olev spaces) to deﬁne P ( ω ) − 1 as a meromorphic family . W e summarize the needed conclu- sions. V ariable order spaces (brieﬂy). Near the horizons, the stationary op era- tor P ( ω ) is non-elliptic and has radial p oin ts in phase space. T o form ulate a F redholm problem, one uses v ariable order Sob olev spaces H s ( X M ,a ) whose order s is chosen to enforce outgoing regularity at the future radial sets and allo w incoming singularities at the past radial sets. This choice of v ariable order is the microlo cal implementation of the standard outgoing (radiation) condition at the future ev en t and cosmological horizons on the V asy/Kerr– de Sitter extension; see [ 2 , 8 , 10 , 11 ]. W e denote the corresp onding graph space by (2.15) X s ( ω ) := { v ∈ H s ( X M ,a ) : P ( ω ) v ∈ H s − 1 ( X M ,a ) } , Y s − 1 := H s − 1 ( X M ,a ) . with norm ∥ v ∥ X s ( ω ) = ∥ v ∥ H s + ∥ P ( ω ) v ∥ H s − 1 . Theorem 2.5 (Meromorphic resolv en t and QNMs) . Fix ( M , a ) sub extr emal and cho ose an outgoing or der function s . Then ther e exists ω 0 ∈ R such that for ℑ ω > ω 0 , P ( ω ) : X s ( ω ) → Y s − 1 is invertible, and the inverse (2.16) R ( ω ) := P ( ω ) − 1 : Y s − 1 → X s ( ω ) extends mer omorphic al ly to ω ∈ C , with ﬁnite-r ank r esidues at p oles. The p oles of R ( ω ) ar e c al le d quasinormal mode frequencies (QNMs), and elements of ker P ( ω ) at a p ole ar e c al le d resonant states . EQUA TORIAL KERR–DE SITTER RINGDOWN 11 R emark 2.6 . In the high-frequency slo w-rotation regime relev ant later, the QNMs we isolate are simple and admit stable lab eling (by semiclassical quan tization). W e nevertheless allow higher-order p oles in the abstract setup b ecause they produce distinct time-domain signatures (p olynomial prefactors) and are relev an t to exceptional-p oin t phenomena. 2.7. F unctional-analytic con v en tions and mapping prop erties. W e brieﬂy collect the functional-analytic conv en tions used throughout the pap er, with emphasis on the mapping prop erties of the stationary resolven t that are needed for con tour deformation. (1) Standard and semiclassical Sobolev norms. By the conv ention ( 2.3 ) , X M ,a is a compact manifold with b oundary . W e write H s := H s ( X M ,a ) for standard Sob olev spaces deﬁned using any ﬁxed smo oth Riemannian metric on X M ,a (equiv alently , b y extending X M ,a across its b oundary and using lo cal charts). F or a temp or al-fr e quency semiclassical parameter h ω ∈ (0 , 1] w e denote by H s h ω the semiclassical Sob olev space with norm ∥ v ∥ H s h ω := ∥⟨ h ω D ⟩ s v ∥ L 2 , ⟨ h ω D ⟩ = (1 + h 2 ω ∆) 1 / 2 , where ∆ is an y ﬁxed elliptic Laplace-type op erator on X M ,a . W e will use these norms in Section 3 on dyadic blo c ks where |ℜ ω | ∼ h − 1 ω . R emark 2.7 (Two semiclassical parameters) . Two unrelated small parameters app ear in this pap er: • h ω ∼ |ℜ ω | − 1 is the temp or al-fr e quency semiclassical parameter used in the resolven t estimates of Section 3 . • h ℓ = ℓ − 1 is the angular semiclassical parameter used for equatorial microlo calization via A ± ,h ℓ in Section 2.11 and in the t wo-mode analysis (Sections 5 – 7 ). W e keep the notation distinct to av oid confusion: there is no reason to iden tify h ω with h ℓ in general. In the t w o-mo de regime we will only apply the angular microlo calization at frequencies |ℜ ω | ≍ ℓ (equiv alently h ω ≍ h ℓ ), so the co existence of the tw o parameters causes no ambiguit y . W e will rep eatedly use the elementary comparison inequalities (v alid for 0 < h ω ≤ 1 and s ≥ 0) (2.17) ∥ v ∥ H s h ω ≤ ∥ v ∥ H s , ∥ v ∥ H s ≤ C s h − s ω ∥ v ∥ H s h ω , whic h follo w from the p oin twise b ounds ⟨ h ω ξ ⟩ s ≤ ⟨ ξ ⟩ s ≤ C s h − s ω ⟨ h ω ξ ⟩ s in lo cal co ordinates. (2) V ariable order spaces and the outgoing condition at radial sets. T o formulate a F redholm problem for P ( ω ) across the horizons one uses v ariable order Sob olev spaces H s (V asy-type spaces), where the order function s = s ( x, ξ ) is c hosen so that s is larger than 1 / 2 at the future radial sets (enforcing outgoing regularity) and smaller than 1 / 2 at the past radial sets (allowing incoming singularities). W e will not repro duce the full 12 RUILIANG LI construction; see [ 2 , 10 , 11 ]. F or any such outgoing order function s we w ork with the graph space X s ( ω ) and target space Y s − 1 deﬁned in ( 2.15 ). (3) Cutoﬀ resolv en t as a map b et w een standard Sob olev spaces. All time-domain contour in tegrals in this pap er inv olv e the cutoﬀ resolv en t χR ( ω ) χ , where χ ∈ C ∞ c ( X M ,a ) is supp orted strictly inside the ph ysical region aw a y from the horizons. On the supp ort of χ the v ariable order spaces coincide with standard Sob olev spaces, and the stationary family is elliptic a w ay from the characteristic set. In particular, for each ﬁxed ω a w ay from p oles and each s ∈ R , (2.18) χR ( ω ) χ : H s − 1 ( X M ,a ) − → H s +1 ( X M ,a ) is a b ounded op erator, with b ounds depending contin uously on ω on p ole-free compact subsets of C . (4) P arameter dep endence. Whenever ( M , a ) v aries in a compact param- eter set K , we will alwa ys choose the cutoﬀs ( χ , the microlo cal equatorial cutoﬀs A ± ,h ℓ ) and the outgoing order function s so that all norms and op era- tor b ounds are uniform on K . In particular, the resolv en t b ounds on shifted con tours in Section 3 and all subsequent constants are uniform on K once a p ole-free con tour is ﬁxed (Prop osition 3.13 ). 2.8. F orw ard Laplace transform and an explicit in v ersion form ula. Let u b e the unique solution to P u = 0 with initial data ( f 0 , f 1 ) ∈ H s . Deﬁne the forward (Heaviside-truncated) solution u + ( t ∗ , x ) := 1 t ∗ ≥ 0 u ( t ∗ , x ) . As a distribution on R t ∗ × X M ,a , u + satisﬁes an inhomogeneous equation whose source is supp orted at t ∗ = 0 and enco des the initial data. Lemma 2.8 (Distributional source at t ∗ = 0) . L et P = ∂ 2 t ∗ + Q∂ t ∗ + L as in ( 2.8 ) . If u solves P u = 0 for t ∗ > 0 and has initial data ( f 0 , f 1 ) at t ∗ = 0 , then (2.19) P u + = δ ′ ( t ∗ ) f 0 + δ ( t ∗ )  f 1 + Qf 0  in D ′ ( R t ∗ × X M ,a ) . Pr o of. W rite u + = H u with H = 1 t ∗ ≥ 0 . Then ∂ t ∗ ( H u ) = δ ( t ∗ ) u (0 , · ) + H ∂ t ∗ u and ∂ 2 t ∗ ( H u ) = δ ′ ( t ∗ ) u (0 , · ) + δ ( t ∗ ) ∂ t ∗ u (0 , · ) + H ∂ 2 t ∗ u. Since Q and L are t ∗ –indep enden t and act only in x , w e ha v e Q ( H u ) = H ( Qu ) and L ( H u ) = H ( Lu ). Substituting into P ( H u ) and using P u = 0 for t ∗ > 0 giv es P u + = δ ′ ( t ∗ ) f 0 + δ ( t ∗ ) f 1 + Q  δ ( t ∗ ) f 0  = δ ′ ( t ∗ ) f 0 + δ ( t ∗ ) ( f 1 + Qf 0 ) , as claimed. □ EQUA TORIAL KERR–DE SITTER RINGDOWN 13 F ourier–Laplace transform. F or ℑ ω suﬃcien tly large (dep ending on the gro wth b ound ( 2.11 )), deﬁne the forward F ourier–Laplace transform (2.20) b u + ( ω , x ) := Z ∞ 0 e iω t ∗ u ( t ∗ , x ) dt ∗ . Since u + is supp orted in t ∗ ≥ 0, b u + extends holomorphically to a half-plane ℑ ω > ω 1 . T aking the F ourier transform in t ∗ of ( 2.19 ) and using ( 2.14 ) yields: Lemma 2.9 (Resolven t iden tity for the Laplace transform) . F or ℑ ω suﬃ- ciently lar ge, (2.21) P ( ω ) b u + ( ω ) =  f 1 + Qf 0  − iω f 0 . Equivalently, (2.22) b u + ( ω ) = R ( ω )  f 1 + ( Q − iω ) f 0  , R ( ω ) = P ( ω ) − 1 . Pr o of. T aking the F ourier transform of ( 2.19 ) in t ∗ with kernel e iω t ∗ giv es [ P u + ( ω ) = b δ ′ ( ω ) f 0 + b δ ( ω ) ( f 1 + Qf 0 ) = ( − iω ) f 0 + ( f 1 + Qf 0 ) . On the other hand, stationarit y of co eﬃcien ts implies [ P u + ( ω ) = P ( ω ) b u + ( ω ), with P ( ω ) given b y ( 2.14 ) . This pro ves ( 2.21 ) , and ( 2.22 ) follo ws by applying R ( ω ). □ In v erse Laplace representation. By the standard inv ersion form ula for F ourier–Laplace transforms of forward-supported distributions, for any C > ω 1 and all t ∗ > 0, (2.23) u ( t ∗ ) = 1 2 π Z ℑ ω = C e − iω t ∗ R ( ω )  f 1 + ( Q − iω ) f 0  dω , where the in tegral is taken along the horizon tal line ℑ ω = C and interpreted as an oscillatory integral in appropriate Sob olev spaces. F orm ula ( 2.23 ) is the starting p oin t for contour deformation: shifting the con tour do wn w ard across QNM p oles pro duces ringdown terms. 2.9. Residues, higher-order p oles, and time-domain con tributions. Theorem 2.5 implies that R ( ω ) has at most ﬁnite-order p oles. W e record the standard residue-to-time-domain dictionary , since it will b e used later when discussing p ossible exceptional-p oint (Jordan) eﬀects. Let ω 0 b e a pole of R ( ω ) of order m ≥ 1. Then there exist ﬁnite-rank op erators A − j suc h that (2.24) R ( ω ) = m X j =1 A − j ( ω − ω 0 ) j + R hol ( ω ) , with R hol holomorphic near ω 0 . 14 RUILIANG LI Lemma 2.10 (Time-domain con tribution of a p ole) . L et F ( ω ) b e holomorphic ne ar ω 0 with values in a Banach sp ac e, and c onsider the c ontour inte gr al I ( t ∗ ) := 1 2 π Z Γ e − iω t ∗ R ( ω ) F ( ω ) dω , wher e Γ is a smal l p ositively oriente d cir cle ar ound ω 0 . Then for t ∗ > 0 , (2.25) I ( t ∗ ) = i e − iω 0 t ∗ m X j =1 ( − it ∗ ) j − 1 ( j − 1)! A − j F ( ω 0 ) + i e − iω 0 t ∗ m X j =1 j − 1 X q =1 ( − it ∗ ) j − 1 − q ( j − 1 − q )! A − j F ( q ) ( ω 0 ) q ! . In p articular, if the p ole is simple ( m = 1 ), then I ( t ∗ ) = i e − iω 0 t ∗ A − 1 F ( ω 0 ) . Pr o of. Insert the Lauren t expansion ( 2.24 ) and the T aylor expansion F ( ω ) = P q ≥ 0 F ( q ) ( ω 0 ) q ! ( ω − ω 0 ) q in to the in tegral. The residue theorem gives con tri- butions only from terms with ( ω − ω 0 ) − 1 . Since I ( t ∗ ) carries the prefactor 1 / (2 π ), the residue theorem contributes an additional factor of i . W rit- ing e − iω t ∗ = e − iω 0 t ∗ P p ≥ 0 ( − it ∗ ) p p ! ( ω − ω 0 ) p and collecting the ( ω − ω 0 ) − 1 co eﬃcien ts yields ( 2.25 ). □ In our main high-frequency equatorial regime, the relev ant QNMs are simple, and the ﬁrst (simple p ole) form ula suﬃces. W e keep Lemma 2.10 to make clear how higher-order p oles w ould manifest in the time domain by p olynomial factors in t ∗ . 2.10. Exact symmetry reduction in the azim uthal n um b er. Since P is axisymmetric, [ P , Φ] = 0, hence P and P ( ω ) preserve the F ourier mo des in φ ∗ . The commutation relations [ P , Φ] = 0 and [ L, Φ] = [ Q, Φ] = 0 are exact on the extended manifold, since the extension across the horizons is p erformed in an axisymmetric fashion; moreov er, our radial cutoﬀs dep end only on r and the sym b ols a ± deﬁning A ± ,h ℓ are taken indep enden t of φ ∗ . F or k ∈ Z set D k := { v ∈ C ∞ ( X M ,a ) : Φ v = ik v } , L 2 ( X M ,a ) = M k ∈ Z L 2 k , L 2 k := D k L 2 . Then Q and L restrict to Q k and L k on each k -subspace, and (2.26) P k ( ω ) = L k − iω Q k − ω 2 , R k ( ω ) = P k ( ω ) − 1 . The inv ersion formula ( 2.23 ) holds in eac h k sector by applying the orthogonal pro jector Π k on to L 2 k : (2.27) Π k u ( t ∗ ) = 1 2 π Z ℑ ω = C e − iω t ∗ R k ( ω ) Π k  f 1 + ( Q − iω ) f 0  dω . EQUA TORIAL KERR–DE SITTER RINGDOWN 15 Th us the entire problem decouples exactly in k . Later, w e will fo cus on the e quatorial high-fr e quency regime in whic h | k | and the total angular frequency are b oth large and comparable. 2.11. Microlo cal equatorial cutoﬀs (angular semiclassical parameter h ℓ ). T o isolate the equatorial high-frequency pac k age relev an t for tw o-mode dominance, we introduce a microlo cal cutoﬀ on the sphere whic h lo calizes near the conic set where the azim uthal momen tum saturates the total angular momen tum. This construction is indep endent of the temp oral-frequency semiclassical parameter used later for resolv ent estimates (see Remark 2.7 ). Semiclassical notation on S 2 . Let ( θ , φ ∗ ) b e coordinates on S 2 and ( ξ θ , ξ φ ) the dual v ariables. The (p ositiv e) Laplacian has principal symbol p S 2 ( θ , φ ∗ ; ξ θ , ξ φ ) = ξ 2 θ + ξ 2 φ sin 2 θ . Fix an angular semiclassical parameter h ℓ ∈ (0 , h 0 ] (later w e will take h ℓ = ℓ − 1 ) and write the rescaled cov ariables η θ = h ℓ ξ θ , η φ = h ℓ ξ φ . The unit cosphere bundle corresp onds to p S 2 = 1, i.e. η 2 θ + η 2 φ / sin 2 θ = 1. Equatorial conic sets. Deﬁne the outgoing/incoming equatorial sets (2.28) E ± := n ( θ , φ ∗ ; η θ , η φ ) ∈ S ∗ S 2 : θ = π 2 , η θ = 0 , η φ = ± 1 o . These corresp ond to geo desics conﬁned to the equatorial plane with max- imal azim uthal momen tum. Cho ose semiclassical symbols a ± ∈ S 0 ( T ∗ S 2 ) supp orted in a small conic neighborho o d of E ± ⊂ S ∗ S 2 and in an ann u- lar neigh b orhoo d of the unit cosphere. Concretely , we may assume a ± is supp orted where ρ ( θ , φ, η ) :=  η 2 θ + η 2 φ sin 2 θ  1 / 2 ∈ [1 / 2 , 2] , and where ( θ , φ ; η /ρ ) lies in a suﬃciently small neigh b orhoo d of E ± . W e also imp ose a + a − = 0, and take a ± ≡ 1 on a smaller conic neighborho od of E ± . F or later commutation with the exact azimuthal pro jectors Π k , we choose a ± indep endent of φ ∗ . Equatorial microlocal cutoﬀs. Let Op h ℓ b e any semiclassical quantization on S 2 . Deﬁne b ounded semiclassical pseudo diﬀeren tial op erators (2.29) A ± ,h ℓ := Op h ℓ ( a ± ) : L 2 ( S 2 ) → L 2 ( S 2 ) , and extend them to X M ,a = ( r e − δ, r c + δ ) × S 2 b y acting trivially in r : A ± ,h ℓ := I r ⊗ Op h ℓ ( a ± ) . In particular, A ± ,h ℓ comm utes with multiplication b y any cutoﬀ dep ending only on r . Since a ± is independent of φ ∗ , A ± ,h ℓ comm utes with Φ and therefore with the exact F ourier pro jectors Π k on L 2 ( X M ,a ). W e will apply 16 RUILIANG LI these cutoﬀs either directly to time-domain solutions u ( t ∗ , · ) (for ﬁxed t ∗ ), or to the stationary resolven t R ( ω ) b y sandwiching: (2.30) R ± ,h ℓ ( ω ) := A ± ,h ℓ R ( ω ) A ∗ ± ,h ℓ . R emark 2.11 (Uniform b oundedness) . Since A ± ,h ℓ ∈ Ψ 0 h ℓ ( S 2 ) and is extended trivially in r , it is uniformly b ounded on standard Sob olev spaces: for each s ∈ R there exists C s suc h that (2.31) ∥ A ± ,h ℓ ∥ H s → H s ≤ C s , 0 < h ℓ < h 0 . W e will use ( 2.31 ) rep eatedly to insert equatorial lo calization in to time- domain expansions and remainder b ounds. R emark 2.12 (Preference for microlocal cutoﬀs) . In Kerr–de Sitter, separation of v ariables ties the angular eigenfunctions to the frequency ω (spheroidal har- monics). F or time-domain analysis it is therefore preferable to use micr olo c al sectorial cutoﬀs suc h as A ± ,h ℓ , which do not dep end on ω and capture the equatorial geometry directly in phase space. In the s lo w-rotation regime, the equatorial pack age is stable under the stationary dynamics, and one can pro v e a microlo cal sectorization result excluding non-equatorial p oles up to O ( ℓ −∞ ) errors; see App endix D . Equatorial sector of initial data. F or s ≥ 0 deﬁne the equatorial data subspaces H s ± ,h ℓ := n ( f 0 , f 1 ) ∈ H s : ( A ± ,h ℓ f 0 , A ± ,h ℓ f 1 ) = ( f 0 , f 1 ) + O ( h ∞ ℓ ) o , in terpreting O ( h ∞ ℓ ) in H s . Later, we will c ho ose h ℓ = ℓ − 1 and combine this loc alization with the azim uthal mo de reduction to isolate the k = ± ℓ equatorial pack age. Output of Section 2 . The k ey output of this section is the explicit resolv en t represen tation ( 2.23 ) (and its k -reduced v ersion ( 2.27 ) ), whic h con verts time- domain analysis into a contour problem for the stationary resolv ent R ( ω ). The microlo cal cutoﬀs A ± ,h ℓ set up the sectorial framework needed later to pro v e a tw o-mode dominance statemen t b y excluding all but a single QNM in an appropriate strip. 3. Resol vent bounds on a shifted contour The inv erse Laplace representation ( 2.23 ) suggests the following strategy: shift the con tour from ℑ ω = C ≫ 1 down to ℑ ω = − ν < 0, picking up the residues of the quasinormal p oles (ringdown terms) and leaving a remainder in tegral on ℑ ω = − ν . T o make this rigorous, we need a priori b ounds for the stationary resolven t R ( ω ) = P ( ω ) − 1 on the shifted con tour Γ − ν := { ω ∈ C : ℑ ω = − ν } . The purp ose of this section is to give suc h b ounds in a form adapted to later contour deformation, EQUA TORIAL KERR–DE SITTER RINGDOWN 17 including (i) uniformit y on compact parameter sets, (ii) semiclassical high- frequency rescalings, and (iii) b ounds for ω –deriv atives needed for rep eated in tegration by parts. Standing geometric input. All estimates b elo w ultimately rely on t w o microlo cal dynamical facts for Kerr–de Sitter: (1) Stable r adial p oint structur e at the (futur e) horizons (redshift), which giv es uniform propagation/regularity estimates at the radial sets. (2) Normal ly hyp erb olic tr apping of the null bic haracteristic ﬂo w on the trapp ed set in the domain of outer communication (photon sphere dynamics), which gives semiclassical high-energy resolv en t estimates. F or the scalar wa v e equation on Kerr–de Sitter, (1) is part of the general F redholm framework for QNMs, and (2) is established in the Kerr–de Sitter setting (in the full sub extremal range) and yields the semiclassical estimates w e use; see e.g. [ 2 , 3 , 11 ]. W e will treat these as standard microlo cal inputs, and derive from them the precise op erator b ounds needed for the con tour argumen t. 3.1. Choice of shifted contour and cutoﬀ resolv en ts. Fix ν > 0. W e denote the horizontal line (3.1) Γ − ν := { ω ∈ C : ℑ ω = − ν } . In later sections we will c ho ose ν so that Γ − ν lies strictly b etw een t w o resonance lay ers in the equatorial high-frequency sector; in particular, Γ − ν will av oid QNM p oles in that sector. F or the presen t section, we only assume: Assumption 3.1 (No p oles on the c on tour) . F or the p ar ameter r ange under c onsider ation, R ( ω ) has no p ole on Γ − ν . Equivalently, P ( ω ) is invertible (as a F r e dholm op er ator) for every ω ∈ Γ − ν . Since the p ole set is discrete for eac h ﬁxed parameter v alue, this assump- tion can alwa ys b e arranged b y a small p erturbation of ν . When parameters v ary , p ole-freeness of a ﬁxed contour is an op en condition by analytic F red- holm theory , hence one ma y w ork on any compact set K con tained in the corresp onding op en p ole-free region; see § 3.7 and Prop osition 3.13 . Cutoﬀ resolven t. Let χ ∈ C ∞ c ( X M ,a ) b e a smo oth cutoﬀ supp orted in the ph ysical region supp χ ⊂ ( r e + δ 2 , r c − δ 2 ) × S 2 . W e also assume that χ ≡ 1 on a neighborho o d of the supp orts of all initial data and observ ation regions. Deﬁne the cutoﬀ resolv ent (3.2) R χ ( ω ) := χ R ( ω ) χ. All con tour estimates in the time domain ma y b e formulated in terms of R χ ( ω ) once we choose χ so that χ ≡ 1 on a neigh b orhoo d of the spacetime supp ort of the compactly supp orted forcing F ϑ in tro duced in Section 4.1 ; see Lemma 4.2 b elo w. In particular b F ϑ ( ω ) = χ b F ϑ ( ω ), and w e may insert χ on the right of the resolven t without c hanging the con tour integrals we consider. 18 RUILIANG LI 3.2. Semiclassical rescaling for high frequencies. The high-frequency regime relev an t for ringdo wn and QNM asymptotics corresp onds to |ℜ ω | ≫ 1 with ℑ ω b ounded b elo w. W e use a dyadic semiclassical rescaling to access uniform estimates. 3.2.1. Dyadic p artition in ℜ ω . Let ψ ∈ C ∞ c ((1 / 2 , 2)) satisfy P j ∈ Z ψ (2 − j | σ | ) = 1 for all | σ | ≥ 1. F or σ ∈ R write 1 = ψ 0 ( σ ) + X j ≥ 1 ψ j ( σ ) , ψ j ( σ ) := ψ (2 − j | σ | ) , j ≥ 1 , with ψ 0 supp orted where | σ | ≤ 2. On the shifted contour ω = σ − iν , w e th us decomp ose the integral into low and high frequency parts. F or eac h dy adic blo c k j ≥ 1, set (3.3) h ω := h j := 2 − j , σ ∼ h − 1 ω , z := h ω ω . R emark 3.2 (T emp oral v ersus angular semiclassics) . In this section h ω is the dyadic temp or al–fr e quency semiclassical parameter asso ciated with the magnitude of ℜ ω ; it is unrelated to the angular parameter h ℓ = ℓ − 1 used for equatorial microlo calization in Section 2.11 . W e keep the subscripts to av oid am biguit y; see Remark 2.7 . Then on Γ − ν w e hav e (3.4) z = h ω σ − ih ω ν, ℜ z ∈ [1 / 2 , 2] · sgn( σ ) , ℑ z = − ν h ω . Th us z sta ys in a compact set, and its imaginary part is O ( h ω ). 3.2.2. Semiclassic al stationary op er ator. Recall from ( 2.14 ) that P ( ω ) = L − iω Q − ω 2 . Deﬁne the semiclassical op erator (dep ending on z and h ω ) by (3.5) P h ω ( z ) := h 2 ω P ( h − 1 ω z ) = h 2 ω L − ih ω z Q − z 2 . W e view P h ω ( z ) as a semiclassical diﬀeren tial operator of order 2 on the compact manifold X M ,a . In the semiclassical quantization where hD x is the basic diﬀerential, h 2 ω L ∈ Diﬀ 2 h , hQ ∈ Diﬀ 1 h , z 2 ∈ Diﬀ 0 h , and ( 3.5 ) is a standard semiclassical Helmholtz-type family . Denote by R h ω ( z ) the semiclassical resolv en t (3.6) R h ω ( z ) := P h ω ( z ) − 1 , whenev er the inv erse exists (as a F redholm inv erse). Note the exact identit y (3.7) R ( ω ) = P ( ω ) − 1 = h 2 ω R h ω ( z ) , z = h ω ω . EQUA TORIAL KERR–DE SITTER RINGDOWN 19 3.2.3. Semiclassic al Sob olev norms. Let H s h ω ( X ) b e the semiclassical Sob olev space on X deﬁned b y (3.8) ∥ u ∥ H s h ω := ∥ Op h ( ⟨ ξ ⟩ s ) u ∥ L 2 , for an y ﬁxed semiclassical quan tization Op h and ⟨ ξ ⟩ = (1 + | ξ | 2 ) 1 / 2 . On compact manifolds, these norms are equiv alen t for diﬀerent quantizations and are uniformly comparable for h ω ∈ (0 , h 0 ]. 3.3. Microlo cal resolv en t estimate in a strip. W e no w state the semi- classical estimate in an O ( h ω ) neighborho o d of the real axis; in particular it applies on the line ℑ z = − ν h ω whic h is the rescaled v ersion of the shifted con tour. The pro of is microlo cal and rests on (i) radial p oin t estimates at the horizons and (ii) a normally h yp erbolic trapping estimate at the trapp ed set. 3.3.1. Ge ometric assumptions on the semiclassic al ﬂow. Let p h ( z ) denote the semiclassical principal symbol of P h ω ( z ). Since the O ( h ω )-imaginary parts enter only in low er order terms (see ( 3.4 )), the semiclassical principal sym b ol is real and giv en b y (3.9) p 0 ( x, ξ ; E ) := σ 2 ( h 2 ω L )( x, ξ ) − E 2 , E := ℜ z ∈ R , where σ 2 ( h 2 ω L ) is the degree-2 homogeneous principal symbol of h 2 ω L . The Hamilton vector ﬁeld H p 0 on T ∗ X \ 0 generates the bicharacteristic ﬂow. In the Kerr–de Sitter case, the c haracteristic set Char ( p 0 ) = { p 0 = 0 } has a compact trapp ed set K E inside the domain of outer comm unication, and the ﬂow on K E is normally hyperb olic. Moreo v er, near the horizons (where r = r e and r = r c ) the ﬂow has radial p oin ts with a stable sour c e/sink structure (redshift). Both structures p ersist under small p erturbations and hold uniformly on compact parameter sets; see [ 2 , 3 , 11 ]. 3.3.2. A semiclassic al estimate with normal ly hyp erb olic tr apping. W e no w state the estimate w e will use; it is a direct sp ecialization of the microlo cal estimate for op erators with normally h yp erbolic trapping (W unsch–Zw orski) com bined with the Kerr–de Sitter F redholm setup (V asy-type v ariable order spaces, or the reﬁnemen ts in [ 10 , 11 ]). Theorem 3.3 (Semiclassical resolv en t b ound near the real axis) . Fix ν ∗ > 0 , a c omp act set I ⋐ R \ { 0 } , and a c omp act p ar ameter set K in a sub extr emal Kerr–de Sitter r ange. L et χ ∈ C ∞ c ( X ) b e supp orte d away fr om the artiﬁcial b oundaries r = r e − δ and r = r c + δ . Then ther e exist h 0 > 0 , C > 0 and N ≥ 0 such that for al l ( M , a ) ∈ K , al l 0 < h ω < h 0 , and al l (3.10) z ∈ C , ℜ z ∈ I , |ℑ z | ≤ ν ∗ h ω , satisfying a p ole-separation condition (3.11) dist  z , Res( P h ω )  ≥ c 0 h ω , 20 RUILIANG LI for some ﬁxe d c 0 > 0 , the cutoﬀ r esolvent ob eys the b ound (3.12) ∥ χ R h ω ( z ) χ ∥ H s − 1 h ω → H s h ω ≤ C h − 1 ω log(1 /h ω ) , ∀ s ∈ R . The c onstants C, h 0 may b e chosen uniformly for ( M , a ) ∈ K , ℜ z ∈ I , and al l s in b ounde d intervals. R emark 3.4 (On the p ole-separation condition) . The condition ( 3.11 ) is automatic if the strip {ℑ z ≥ − ν h ω } is p ole-free. In our application we delib erately choose ν so that ﬁnitely many p oles lie ab ov e ℑ z = − ν h ω (these will pro duce the ringdown terms), but w e also choose ν b etwe en r esonanc e layers so that the line ℑ z = − ν h ω sta ys at a distance ≳ h ω from all p oles; see § 5 where this is v eriﬁed in the equatorial high-frequency sector. R emark 3.5 (On the h − 1 ω log (1 /h ω ) loss) . The h − 1 ω log (1 /h ω ) loss is character- istic of normally hyperb olic trapping. In some settings it can b e improv ed to h − 1 ω or h − 1+ ε ω using reﬁned anisotropic sym b ol classes and resonance pro jector tec hnology (e.g. [ 9 , 8 ]), but h − 1 ω log (1 /h ω ) suﬃces for the con tour shift and for quantitativ e tw o-mode dominance. 3.3.3. Outline and r efer enc es for the r esolvent estimate. W e do not repro- duce the full microlo cal argumen t leading to the semiclassical b ound ( 3.12 ) ; instead we record a precise prov enance of the ingredien ts. The estimate is a standard consequence of com bining: (i) the V asy-type F redholm framework for stationary op erators on Kerr–de Sitter (including radial p oin t estimates at the horizons), and (ii) the semiclassical resolven t b ound for normally h yp erbolic trapping. Radial sets (redshift) and F redholm setup. The construction of v ariable order spaces and the outgoing F redholm problem for P ( ω ) on Kerr–de Sitter, together with the corresp onding radial p oin t estimates at the horizons, is dev elop ed in [ 2 ] and reﬁned in later works; see also [ 11 ] for global estimates in the full sub extremal range. These results pro vide a microlo cal a priori estimate aw a y from the trapp ed set, with loss h − 1 ω at most. Normally h yp erb olic trapping. Near the trapp ed set in the domain of outer communication, the n ull bicharacteristic ﬂo w is normally hyperb olic. The corresp onding semiclassical resolven t estimate with the characteristic h − 1 ω log (1 /h ω ) loss is prov ed for normally hyperb olic trapp ed sets in [ 3 ]; see also [ 9 , 8 ] for further reﬁnements in related settings. Applied to the semiclassical family P h ω ( z ) with ℑ z = O ( h ω ), this yields the b ound ∥ χR h ω ( z ) χ ∥ H s − 1 h ω → H s h ω ≤ C h − 1 ω log(1 /h ω ) , uniformly for z in the compact region ( 3.4 ) and for parameters in compact subsets. Gluing and in v ertibilit y on the con tour. A standard partition of unity in phase space glues the elliptic region, the radial regions, and the trapp ed region estimates in to a global b ound. Under the dyadic p ole-separation EQUA TORIAL KERR–DE SITTER RINGDOWN 21 h yp othesis ( 3.11 ) (equiv alently , p ole-freeness on the relev an t dyadic blo c k of Γ − ν ), the ﬁnite-dimensional obstruction in the F redholm problem v anishes and P h ω ( z ) is inv ertible, so the a priori estimate yields the op erator norm b ound ( 3.12 ). ω –deriv atives. Bounds for ∂ m ω R ( ω ) on Γ − ν follo w from the resolven t diﬀeren tiation identit y ∂ ω R ( ω ) = − R ( ω )( ∂ ω P ( ω )) R ( ω ), together with the p olynomial structure of P ( ω ) in ω and the dyadic resolven t b ounds on Γ − ν established b elo w. 3.4. Bac k to the ω –plane: b ounds on R ( ω ) on Γ − ν . W e now translate Theorem 3.3 in to b ounds directly on R ( ω ) on the shifted contour. Since the contour is the union of a compact piece (lo w frequency) and dy adic high-frequency blo c ks, w e treat them separately . 3.4.1. L ow fr e quency b ound. Let Ω low := { ω ∈ Γ − ν : |ℜ ω | ≤ 2 } . By Assumption 3.1 , R ( ω ) is holomorphic on Ω low and hence b ounded on it as an op erator b et w een any ﬁxed Sob olev spaces (or the graph spaces in ( 2.15 ) ). Th us: Lemma 3.6 (Low frequency b ound) . Fix s ∈ R . Under Assumption 3.1 ther e exists C low > 0 such that (3.13) sup ω ∈ Ω low ∥ χR ( ω ) χ ∥ H s − 1 → H s ≤ C low . 3.4.2. High fr e quency b ound on dyadic blo cks. Let Ω high := { ω ∈ Γ − ν : |ℜ ω | ≥ 1 } . F or ω ∈ Ω high , c ho ose j ≥ 1 such that ψ j ( ℜ ω )  = 0 and set h ω = 2 − j , z = h ω ω . Then z satisﬁes ( 3.10 ) . Using ( 3.7 ) and b oundedness of χ on H s h ω w e obtain: Prop osition 3.7 (High frequency b ound on Γ − ν ) . Fix ν > 0 , a c omp act p ar ameter set K , and χ as ab ove. Assume the p ole-sep ar ation c ondition ( 3.11 ) holds on the dyadic blo cks of Γ − ν . Then for every s ∈ R ther e exist h 0 > 0 and C > 0 such that for al l ω ∈ Γ − ν with |ℜ ω | ≥ h − 1 0 , (3.14) ∥ χR ( ω ) χ ∥ H s − 1 h ω → H s h ω ≤ C h ω log(1 /h ω ) , h ω := |ℜ ω | − 1 . Equivalently, in dyadic form: if ψ j ( ℜ ω )  = 0 and h ω = 2 − j , then ∥ χR ( ω ) χ ∥ H s − 1 h ω → H s h ω ≤ C h ω log(1 /h ω ) . Pr o of. Set z = h ω ω so that R ( ω ) = h 2 ω R h ω ( z ) by ( 3.7 ) . Theorem 3.3 giv es ∥ χR h ω ( z ) χ ∥ H s − 1 h ω → H s h ω ≤ C h − 1 ω log (1 /h ω ). Multiplying b y h 2 ω yields ( 3.14 ). □ 22 RUILIANG LI 3.5. ω –deriv ativ e bounds. F or the quantitativ e tail estimates in Section 4 w e will integrate b y parts in the inv erse Laplace represen tation; this requires b ounds for ∂ m ω ( χR ( ω ) χ ) on horizon tal lines. On dyadic high-frequency blocks w e w ork with the rescaled parameter z = h ω ω and the semiclassical resolven t R h ω ( z ) = P h ω ( z ) − 1 . The following lemma upgrades the basic semiclassical b ound of Theorem 3.3 to b ounds for z –deriv atives, using only holomorph y and the p ole-separation condition ( 3.11 ). Lemma 3.8 (Cauch y estimates for z –deriv atives) . Assume the hyp otheses of The or em 3.3 for some ﬁxe d ν ∗ > 0 , and let c 0 > 0 b e the p ole-sep ar ation c onstant in ( 3.11 ) . Fix s ∈ R and m ∈ N 0 . Then ther e exist h 0 > 0 and C s,m > 0 such that for al l 0 < h ω < h 0 and al l z satisfying ℜ z ∈ I , |ℑ z | ≤  ν ∗ − c 0 2  h ω , dist  z , Res( P h ω )  ≥ c 0 h ω , one has the op er ator b ound (3.15) ∥ χ ∂ m z R h ω ( z ) χ ∥ H s − 1 h ω → H s h ω ≤ C s,m h − m − 1 ω log(1 /h ω ) . Pr o of. Fix z as ab o ve and set ρ := c 0 2 h ω . By ( 3.11 ) , the closed disk D ( z , ρ ) con tains no p ole of R h ω ( · ). Hence R h ω ( ζ ) is holomorphic in ζ with v alues in b ounded op erators H s − 1 h ω → H s h ω on D ( z , ρ ). F or ζ ∈ ∂ D ( z , ρ ) w e hav e dist  ζ , Res( P h ω )  ≥ dist  z , Res( P h ω )  − | ζ − z | ≥ c 0 2 h ω , and also |ℑ ζ | ≤ |ℑ z | + | ζ − z | ≤  ν ∗ − c 0 2  h ω + c 0 2 h ω = ν ∗ h ω . Th us Theorem 3.3 applies on ∂ D ( z , ρ ) (with the p ole-separation constant reduced to c 0 / 2), and yields sup ζ ∈ ∂ D ( z,ρ ) ∥ χR h ω ( ζ ) χ ∥ H s − 1 h ω → H s h ω ≤ C h − 1 ω log(1 /h ω ) . The Cauch y integral form ula gives, as an identit y of b ounded operators H s − 1 h ω → H s h ω , ∂ m z R h ω ( z ) = m ! 2 π i Z ∂ D ( z,ρ ) R h ω ( ζ ) ( ζ − z ) m +1 dζ , so taking norms and using | ζ − z | = ρ and | ∂ D ( z , ρ ) | = 2 π ρ we obtain ∥ χ ∂ m z R h ω ( z ) χ ∥ ≤ m ! ρ − m sup ζ ∈ ∂ D ( z,ρ ) ∥ χR h ω ( ζ ) χ ∥ . Since ρ ≍ h ω , this yields ( 3.15 ) after absorbing m ! and c − m 0 in to C s,m . □ Prop osition 3.9 (Dyadic ω -deriv ative b ounds) . Assume the hyp otheses of Pr op osition 3.7 . Fix s ∈ R and m ∈ N 0 . Then ther e exist c onstants C s,m > 0 and h 0 > 0 such that for every ω = σ − iν ∈ Γ − ν with | σ | ∼ h − 1 ω and every 0 < h ω < h 0 , (3.16) ∥ χ ∂ m ω R ( ω ) χ ∥ H s h ω → H s +1 h ω ≤ C s,m h ω log(1 /h ω ) . The same b ound holds with ∂ m σ in plac e of ∂ m ω on Γ − ν . EQUA TORIAL KERR–DE SITTER RINGDOWN 23 Pr o of. On a dyadic blo ck, set z = h ω ω so that R ( ω ) = h 2 ω R h ω ( z ) by ( 3.7 ) . Since z dep ends linearly on ω , we hav e ∂ ω = h ω ∂ z , and hence ∂ m ω R ( ω ) = h 2+ m ω ∂ m z R h ω ( z ) . Along Γ − ν w e hav e |ℑ z | = ν h ω . Applying Lemma 3.8 with ν ∗ = 2 ν and with the p ole-separation constan t c 0 replaced b y min ( c 0 , ν ) (whic h preserv es ( 3.11 )) gives, with s replaced b y s + 1, ∥ χ ∂ m z R h ω ( z ) χ ∥ H s h ω → H s +1 h ω ≤ C s,m h − m − 1 ω log(1 /h ω ) . Multiplying by h 2+ m ω yields ( 3.16 ) . Finally , along Γ − ν one has ∂ σ = ∂ ω , so the ∂ m σ statemen t follows. □ Lemma 3.10 (Standard Sob olev b ounds on the shifted contour) . Assume L emma 3.6 , Pr op osition 3.7 , and Pr op osition 3.9 . Fix s ∈ R and m ∈ N 0 . Then ther e exist c onstants C s,m > 0 , A s ≥ 0 and B m ≥ 0 (dep ending on ν , the c omp act p ar ameter set K , and χ ) such that for al l ω = σ − iν ∈ Γ − ν , (3.17) ∥ χ ∂ m ω R ( ω ) χ ∥ H s → H s +1 ≤ C s,m ⟨ σ ⟩ A s  log(2 + ⟨ σ ⟩ )  B m . The same b ound holds with ∂ m σ in plac e of ∂ m ω . Pr o of. F or | σ | ≤ 2, the b ound follows from holomorphy of χR ( ω ) χ on the compact set Ω low (cf. Lemma 3.6 ), after p ossibly increasing the constant. F or | σ | ≥ 1, set h ω := ⟨ σ ⟩ − 1 . Applying Prop osition 3.9 with s replaced by s + 1 yields, on the corresp onding dy adic blo ck, ∥ χ ∂ m ω R ( ω ) χ ∥ H s h ω → H s +1 h ω ≤ C s,m h ω log(1 /h ω ) . Using the comparison inequalities ( 2.17 ) (with s + 1) we obtain ∥ χ ∂ m ω R ( ω ) χf ∥ H s +1 ≤ C s h − ( s +1) ω ∥ χ ∂ m ω R ( ω ) χf ∥ H s +1 h ω ≤ C s,m h − s ω log(1 /h ω ) ∥ f ∥ H s . Since h − 1 ω ∼ ⟨ σ ⟩ on the blo c k and log (1 /h ω ) ≲ log (2 + ⟨ σ ⟩ ), this gives ( 3.17 ) with (for instance) A s = s and B m = 1. The ∂ m σ b ound follo ws from ∂ m σ R ( σ − iν ) = ∂ m ω R ( ω ) | ω = σ − iν . □ 3.6. Equatorial microlo cal cutoﬀs and sectorial b ounds. Recall the microlo cal equatorial cutoﬀs A ± ,h ℓ from ( 2.29 ) and the sandwiched resolven t R ± ,h ℓ ( ω ) = A ± ,h ℓ R ( ω ) A ∗ ± ,h ℓ from ( 2.30 ) . By Remark 2.11 the op erators A ± ,h ℓ are uniformly b ounded on H s ( X ) for each ﬁxed s , with constan ts indep enden t of h ℓ . Corollary 3.11 (Sectorial resolv ent b ounds in standard Sob olev spaces) . Under the hyp otheses of L emma 3.10 , for every s ∈ R and m ∈ N 0 ther e exist c onstants C s,m > 0 and exp onents A s , B m ≥ 0 such that for al l ω = σ − iν ∈ Γ − ν and al l 0 < h ℓ < h 0 , (3.18) ∥ χ A ± ,h ℓ ∂ m ω R ( ω ) A ∗ ± ,h ℓ χ ∥ H s → H s +1 ≤ C s,m ⟨ σ ⟩ A s  log(2 + ⟨ σ ⟩ )  B m . The same b ound holds with ∂ m σ in plac e of ∂ m ω . 24 RUILIANG LI Pr o of. Com bine Lemma 3.10 with the uniform b oundedness ( 2.31 ) : Set T := χA ± ,h ℓ ( ∂ m ω R ( ω )) A ∗ ± ,h ℓ χ . Then ∥ T ∥ H s → H s +1 ≤ ∥ χA ± ,h ℓ ∥ H s +1 → H s +1 ∥ χ ( ∂ m ω R ( ω )) χ ∥ H s → H s +1 × ∥ A ∗ ± ,h ℓ χ ∥ H s → H s . Absorb the A ± ,h ℓ op erator norms into the constan t. □ 3.7. Uniformity in parameters and admissible shifted contours. F or the later inv erse/quan titativ e statements w e v ary ( M , a ) in a compact slow- rotation set K . Since the con tour shift is meaningful only when the shifted line do es not cross p oles, we isolate the precise uniform hypothesis needed for parameter-uniform remainder estimates. Deﬁnition 3.12 (Admissible shifted contour) . Fix ν > 0 and a compact parameter set K . W e sa y that Γ − ν is uniformly admissible on K if: (1) Γ − ν con tains no p ole of R ( ω ) for any ( M , a ) ∈ K (equiv alently , P ( ω ) is inv ertible for all ω ∈ Γ − ν and all ( M , a ) ∈ K ). (2) There exist constants c 0 > 0 and σ 0 ≥ 1 such that for every ( M , a ) ∈ K and every ω = σ − iν with | σ | ≥ σ 0 , setting h ω := | σ | − 1 and z = h ω ω , one has the uniform dy adic separation (3.19) dist  z , Res( P h ω )  ≥ c 0 h ω . Item (2) is the genuinely semiclassical condition: it rules out p oles that approac h the shifted line at a rate faster than O ( h ω ) on high-frequency blo c ks, which is exactly what is needed to apply Theorem 3.3 and Lemma 3.8 . W e now explain why item (2) can b e ensured by choosing ν b et w een damping la y ers. Prop osition 3.13 (Uniform high-frequency p ole separation b et w een damp- ing la yers) . Fix a c omp act slow–r otation p ar ameter set K 0 and a c omp act set I ⋐ R \ { 0 } as in ( 3.10 ) . L et n ∈ N 0 b e an overtone index. Then ther e exist c onstants ν = ν n > 0 , c 0 > 0 and σ 0 ≥ 1 such that item (2) in Deﬁnition 3.12 holds for this choic e of ν for every ( M , a ) ∈ K 0 . L et U ν := n ( M , a ) ∈ P Λ ∩ {| a | ≤ a 0 } : Γ − ν c ontains no p ole of R ( · ; M , a ) o . Then U ν is op en in ( M , a ) (analytic F r e dholm the ory), henc e for any c omp act set K ⋐ U ν ∩ K 0 the c ontour Γ − ν is uniformly admissible on K in the sense of Deﬁnition 3.12 . This p ar ameter lo c alization is harmless for the pr esent p ap er, sinc e al l later quantitative statements (including the inverse step imp orte d fr om [ 1 ] ) ar e lo c al in ( M , a ) ; if desir e d, one may c over a lar ger c omp act set by ﬁnitely many such c omp acts and p atch the r esulting c onstants. Pr o of. High-fr e quency p ole sep ar ation. In the slow-rotation regime, Dy atlo v’s semiclassical Bohr–Sommerfeld description of Kerr–de Sitter resonances ([ 6 ], building on [ 5 ]) gives a high-frequency asymptotic distribution of p oles in rescaled coordinates z = h ω ω with ℜ z in a ﬁxed compact interv al I and EQUA TORIAL KERR–DE SITTER RINGDOWN 25 |ℑ z | = O ( h ω ). F or each ﬁxed ov ertone index n there is a damping layer in whic h the p oles lie, and adjacent lay ers are separated b y a p ositiv e gap whic h is uniform for ℜ z ∈ I and for ( M , a ) in compact subsets of the slow-rotation range. Cho osing ν = ν n strictly b et w een the n th and ( n + 1)st la yers, we obtain that for all suﬃciently small h ω (and hence for all | σ | suﬃcien tly large), the line ℑ z = − ν h ω sta ys at distance ≳ h ω from every p ole with ℜ z ∈ I . This is exactly ( 3.19 ) (after ﬁxing σ 0 and c 0 ) and prov es item (2). Op enness of p ole-fr e eness. By Theorem 2.5 , for eac h ﬁxed ( M , a ) the p oles of R ( ω ) are zeros of a F redholm determinan t associated with P ( ω ). Moreov er, b y [ 10 ] (see also [ 11 ]) these p oles dep end real–analytically on parameters in the slow–rotation range. In particular, the set U ν of parameters for whic h P ( ω ) is inv ertible for all ω ∈ Γ − ν is op en in ( M , a ) by analytic F redholm theory . Th us item (1) in Deﬁnition 3.12 holds on any compact K ⋐ U ν . Com bining this with the high–frequency separation established ab o v e yields the stated uniform admissibilit y on K ⋐ U ν ∩ K 0 . □ R emark 3.14 . Throughout the remainder of the pap er w e ﬁx ν = ν n as in Prop osition 3.13 (for the ov ertone n under consideration) and assume that Γ − ν is uniformly admissible on our c hosen compact parameter set K in the sense of Deﬁnition 3.12 . The equatorial tw o-mo de mec hanism will later use a ﬁner se ctorial p ole separation, but the global admissibilit y recorded here is the only input needed for the contour deformation itself. 3.8. Summary of usable b ounds. F or later reference, we collect the resolv ent b ounds on the shifted contour Γ − ν = { ω = σ − iν : σ ∈ R } , with ν > 0 c hosen so that Γ − ν con tains no p oles (Assumption 3.1 ). In the low- frequency region Ω low the cutoﬀ resolv ent is uniformly b ounded (Lemma 3.6 ). In the high-frequency regime | σ | ≫ 1, set h ω = ⟨ σ ⟩ − 1 on each dyadic blo ck. • Semiclassic al dyadic b ounds (use d in the stationary analysis). Prop o- sition 3.9 yields, for every s ∈ R and m ∈ N 0 , (3.20) ∥ χ ∂ m ω R ( ω ) χ ∥ H s h ω → H s +1 h ω ≤ C s,m h ω log(1 /h ω ) . Here ω = σ − iν with | σ | ∼ h − 1 ω . • Standar d Sob olev b ounds (use d for c ontour inte gr ation). Lemma 3.10 implies the p olynomial control (3.21) ∥ χ ∂ m ω R ( ω ) χ ∥ H s → H s +1 ≤ C s,m ⟨ σ ⟩ A s  log(2 + ⟨ σ ⟩ )  B m , ω = σ − iν. • Se ctorial b ounds with e quatorial cutoﬀs. Corollary 3.11 yields the analogous estimate for the sandwiched resolv ent, uniformly in the angular parameter h ℓ : (3.22) ∥ χ A ± ,h ℓ ∂ m ω R ( ω ) A ∗ ± ,h ℓ χ ∥ H s → H s +1 ≤ C s,m ⟨ σ ⟩ A s  log(2 + ⟨ σ ⟩ )  B m . Here ω = σ − iν and 0 < h ℓ < h 0 . All of the ab o v e b ounds also hold with ∂ m σ in place of ∂ m ω on Γ − ν . 26 RUILIANG LI 4. Resonant exp ansion with remainder In this section w e deform the inv erse Laplace con tour in ( 2.23 ) to obtain a time-domain r esonant exp ansion (ringdown) plus a c ontr ol le d r emainder (tail). The analytic input is the meromorphic con tin uation of the stationary resolv en t R ( ω ) = P ( ω ) − 1 (Theorem 2.5 ), while the quantitativ e input is the shifted-con tour resolven t b ounds from Section 3 . The contour deformation argument in Kerr–de Sitter is b y now stan- dard in microlo cal scattering theory . The meromorphic contin uation and F redholm framework for the stationary family P ( ω ) are due to V asy [ 2 ], with global uniformity in the full sub extremal parameter range reﬁned b y P etersen–V asy [ 11 ]. Our purp ose in this section is therefore not to reprov e those foundational results, but to record the deformation in a form that is completely explicit and quantitativ ely usable later: we insert a smo oth time cutoﬀ pro ducing a Sch w artz forcing term, and we k eep track of the remainder as an op erator with p olynomial-in-time decay on the shifted con tour. The gen uinely new inputs of the present pap er b egin in Section 5 , where the general expansion is com bined with equatorial microlo cal sectorization and analytic windows to isolate a ﬁnite set of leading QNMs. A purely formal contour shift is straigh tforward; the main tec hnical p oin t is to make the argument har d and usable by (i) remo ving the singularity at t ∗ = 0 and pro ducing rapid decay in the frequency domain, and (ii) obtaining explicit remainder b ounds suitable for subsequen t in v erse steps. W e achiev e (i) by a smo oth time cutoﬀ. 4.1. A smooth time cutoﬀ and compactly supported forcing. Let ϑ ∈ C ∞ ( R ) satisfy (4.1) ϑ ( t ∗ ) = 0 for t ∗ ≤ 0 , ϑ ( t ∗ ) = 1 for t ∗ ≥ 1 . Let u solv e P u = 0 with initial data ( f 0 , f 1 ) at t ∗ = 0 as in Section 2 , and set (4.2) u ϑ ( t ∗ , x ) := ϑ ( t ∗ ) u ( t ∗ , x ) . Then u ϑ = u for t ∗ ≥ 1 and u ϑ ≡ 0 for t ∗ ≤ 0, so u ϑ has v anishing initial data at t ∗ = 0. Lemma 4.1 (Compactly supp orted forcing) . L et P = ∂ 2 t ∗ + Q∂ t ∗ + L b e the stationary splitting ( 2.8 ) . Then u ϑ satisﬁes (4.3) P u ϑ = F ϑ with F ϑ = ϑ ′′ u + 2 ϑ ′ ∂ t ∗ u + ϑ ′ Qu, wher e F ϑ is supp orte d in the slab { 0 < t ∗ < 1 } and is smo oth in t ∗ with values in C ∞ ( X M ,a ) if ( f 0 , f 1 ) ar e smo oth. Mor e over, for every s ≥ 0 and m ∈ N ther e exists C = C s,m such that (4.4) sup t ∗ ∈ R m X j =0 ∥ ∂ j t ∗ F ϑ ( t ∗ ) ∥ H s ≤ C ∥ ( f 0 , f 1 ) ∥ H s + m , wher e H s + m = H s + m +1 × H s + m as in ( 2.10 ) . EQUA TORIAL KERR–DE SITTER RINGDOWN 27 Pr o of. Using P u = 0 for all t ∗ > 0 and the pro duct rule, P ( ϑu ) = ϑP u + [ ∂ 2 t ∗ , ϑ ] u + [ Q∂ t ∗ , ϑ ] u = 0 + ( ϑ ′′ u + 2 ϑ ′ ∂ t ∗ u ) + ϑ ′ Qu, whic h is ( 4.3 ) . Since ϑ ′ , ϑ ′′ are supported in (0 , 1), so is F ϑ . The b ound ( 4.4 ) follo ws from the well-posedness estimate ( 2.11 ) and the fact that ϑ has compactly supp orted deriv ativ es: eac h ∂ j t ∗ F ϑ is a ﬁnite linear com bination of terms ϑ ( ℓ ) ∂ k t ∗ u and ϑ ( ℓ ) Q∂ k t ∗ u with k ≤ j + 1, hence is controlled by ﬁnitely man y energy norms of ( u, ∂ t ∗ u ) on 0 ≤ t ∗ ≤ 1. □ Lemma 4.2 (Choice of the spatial cutoﬀ for the forcing) . L et ϑ b e as in Se ction 4.1 , and let F ϑ b e deﬁne d by ( 4.3 ) . Assume that the initial data ( f 0 , f 1 ) ar e supp orte d in a c omp act set K 0 ⋐ X phys M ,a . Then ther e exists a c omp act set K prop ⋐ X phys M ,a such that supp u ( t ∗ , · ) ⊂ K prop for al l t ∗ ∈ [0 , 1] , supp F ϑ ⊂ [0 , 1] × K prop . In p articular, after cho osing the cutoﬀ χ ∈ C ∞ c ( X phys M ,a ) so that χ ≡ 1 on a neighb orho o d of K prop , we have F ϑ = χF ϑ , b F ϑ ( ω ) = χ b F ϑ ( ω ) , ω ∈ C . Pr o of. Since the wa v e op erator is second order hyperb olic with resp ect to the stationary time t ∗ , it enjo ys ﬁnite propagation sp eed. Thus, if ( f 0 , f 1 ) are supp orted in K 0 ⋐ X M ,a , then for each t ∗ ∈ [0 , 1] the supp ort of u ( t ∗ , · ) is contained in the domain of dep endence of K 0 at time t ∗ . As t ∗ ∈ [0 , 1] ranges ov er a compact interv al and K 0 is contained in the interior of X M ,a , the union of these supp orts is con tained in some compact set K prop ⋐ X phys M ,a . By deﬁnition ( 4.3 ) , F ϑ is a linear combination of u and ∂ t ∗ u with co eﬃcien ts supp orted where ϑ ′ and ϑ ′′ are nonzero, hence where t ∗ ∈ [0 , 1]. Therefore supp F ϑ ⊂ [0 , 1] × K prop . Cho osing χ ≡ 1 near K prop giv es F ϑ = χF ϑ , and the identit y for b F ϑ follo ws since the F ourier–Laplace transform acts only in t ∗ . □ Assumption 4.3 (Interior supp ort and cutoﬀ conv en tion) . Fix a c omp act set K 0 ⋐ X phys M ,a . In al l c ontour deformation ar guments in Se ctions 4 – 5 , we r estrict to initial data ( f 0 , f 1 ) supp orte d in K 0 . L et K prop ⋐ X phys M ,a b e the c omp act pr op agation set asso ciate d with K 0 given by L emma 4.2 , so that supp F ϑ ⊂ [0 , 1] × K prop for every such solution. We ﬁx a cutoﬀ χ ∈ C ∞ c ( X phys M ,a ) with χ ≡ 1 on a neighb orho o d of K prop . Then the for cing satisﬁes F ϑ = χF ϑ and henc e (4.5) b F ϑ ( ω ) = χ b F ϑ ( ω ) for al l ω ∈ C . R emark 4.4 (Compact supp ort is not restrictiv e) . Assumption 4.3 is imp osed only to justify the insertion of the spatial cutoﬀ χ in the resolv ent identit y . Since all statemen ts in this paper concern the localized ﬁeld χu in the ph ysical region, one can alwa ys reduce to this setting b y ﬁnite propagation sp eed: for a ﬁxed observ ation cutoﬀ χ and time slab t ∗ ∈ [0 , 1], the v alues of χu ( t ∗ ) 28 RUILIANG LI dep end only on the restriction of the initial data to a compact subset of X M ,a determined b y the bac kw ard domain of dependence of supp χ o v er [0 , 1]. Replacing ( f 0 , f 1 ) b y a compactly supp orted cutoﬀ of the data in this subset leav es χu ( t ∗ ) unchanged for t ∗ ∈ [0 , 1], and therefore leav es F ϑ and the subsequent expansion for t ∗ ≥ 1 unc hanged. 4.2. Laplace transform of the forced problem and rapid decay in frequency. Deﬁne the (forward) F ourier–Laplace transform of the compactly supp orted forcing: (4.6) b F ϑ ( ω , x ) := Z ∞ 0 e iω t ∗ F ϑ ( t ∗ , x ) dt ∗ = Z 1 0 e iω t ∗ F ϑ ( t ∗ , x ) dt ∗ . Since F ϑ is smo oth and compactly supp orted in t ∗ , b F ϑ is en tire in ω and rapidly decaying on horizon tal lines. Lemma 4.5 (Sch w artz b ounds in ω ) . Fix s ≥ 0 . F or every N ∈ N ther e exists C s,N > 0 such that for al l ω ∈ C , (4.7) ∥ b F ϑ ( ω ) ∥ H s ≤ C s,N (1 + | ω | ) − N ∥ ( f 0 , f 1 ) ∥ H s + N . Mor e over, for every m ∈ N , (4.8) ∥ ∂ m ω b F ϑ ( ω ) ∥ H s ≤ C s,N ,m (1 + | ω | ) − N ∥ ( f 0 , f 1 ) ∥ H s + N + m . Pr o of. Fix N ∈ N and s ≥ 0. Step 1: the b ound for | ω | ≤ 1 . Since F ϑ is supp orted in [0 , 1], w e ha ve the crude estimate ∥ b F ϑ ( ω ) ∥ H s ≤ Z 1 0 ∥ F ϑ ( t ∗ ) ∥ H s dt ∗ ≤ sup t ∗ ∈ [0 , 1] ∥ F ϑ ( t ∗ ) ∥ H s , | ω | ≤ 1 . Using ( 4.4 ) with m = 0 giv es sup t ∗ ∥ F ϑ ( t ∗ ) ∥ H s ≤ C s ∥ ( f 0 , f 1 ) ∥ H s +1 . Since (1 + | ω | ) − N ≥ 2 − N when | ω | ≤ 1, this yields ( 4.7 ) on | ω | ≤ 1 (after p ossibly enlarging the constant and the Sob olev index on the righ t). Step 2: the b ound for | ω | ≥ 1 . Integrating b y parts N times in ( 4.6 ) gives b F ϑ ( ω ) = 1 ( iω ) N Z 1 0 e iω t ∗ ∂ N t ∗ F ϑ ( t ∗ ) dt ∗ , | ω | ≥ 1 , where b oundary terms v anish b ecause F ϑ is smo oth and supp orted in (0 , 1). T aking H s norms and using ( 4.4 ) with m = N yields ∥ b F ϑ ( ω ) ∥ H s ≤ C s,N | ω | − N ∥ ( f 0 , f 1 ) ∥ H s + N ≤ C s,N (1 + | ω | ) − N ∥ ( f 0 , f 1 ) ∥ H s + N . This holds for | ω | ≥ 1. whic h together with Step 1 prov es ( 4.7 ). Step 3: b ounds for ∂ m ω b F ϑ . Diﬀerentiatin g under the integral gives ∂ m ω b F ϑ ( ω ) = Z 1 0 ( it ∗ ) m e iω t ∗ F ϑ ( t ∗ ) dt ∗ . EQUA TORIAL KERR–DE SITTER RINGDOWN 29 F or | ω | ≤ 1, estimate as in Step 1 and use ( 4.4 ) with m replaced by m . F or | ω | ≥ 1, integrate by parts N times and b ound ∂ N t ∗  ( it ∗ ) m F ϑ ( t ∗ )  using ( 4.4 ) with m replaced b y N + m . This yields ( 4.8 ). □ Resolv en t represen tation. Since P u ϑ = F ϑ and u ϑ v anishes for t ∗ ≤ 0, the same Laplace transform argument as in Lemma 2.9 gives, for ℑ ω suﬃcien tly large, (4.9) P ( ω ) b u ϑ ( ω ) = b F ϑ ( ω ) , so b u ϑ ( ω ) = R ( ω ) b F ϑ ( ω ) . In v erting the Laplace transform then yields, for any C large enough and all t ∗ > 0, (4.10) u ϑ ( t ∗ ) = 1 2 π Z ℑ ω = C e − iω t ∗ R ( ω ) b F ϑ ( ω ) dω . Since u = u ϑ for t ∗ ≥ 1, it suﬃces to analyze ( 4.10 ). 4.3. Con tour deformation and extraction of residues. Fix ν > 0 and assume the shifted con tour Γ − ν is p ole-free in the sense of Prop osition 3.13 (for the parameter set under consideration). Let χ ∈ C ∞ c ( X M ,a ) b e a spatial cutoﬀ supp orted in the physical region and equal to 1 on the region where w e observe u (as in ( 3.2 )). T runcated contour argument. Let C b e the inv ersion height in ( 4.10 ) . F or R > 1, consider the rectangle R R := { ω ∈ C : |ℜ ω | ≤ R, − ν ≤ ℑ ω ≤ C } . Let ∂ R R b e its p ositiv ely orien ted b oundary . Cho ose R > 1 so that ∂ R R do es not pass through p oles of R ( ω ) (p ossible since p oles are discrete). T o mak e the v anishing of the v ertical side integrals completely robust, w e will later let R → ∞ along a subsequence for whic h the vertical sides sta y a quan titativ ely con trolled distance from p oles in the strip − ν ≤ ℑ ω ≤ C . Lemma 4.6 (Pole-a v oiding truncation radii) . Fix ν > 0 and C > 0 . Ther e exist an exp onent A > 0 and a se quenc e R n → ∞ such that, for every n , (4.11) dist  ± R n + iη , Res( P )  ≥ (1 + R n ) − A ∀ η ∈ [ − ν, C ] , and ∂ R R n is p ole-fr e e. Pr o of. The resonance set in a ﬁxed horizontal strip is discrete and satisﬁes a p olynomial coun ting b ound: there exist p ≥ 0 and C p > 0 suc h that (4.12) #  Res( P ) ∩ R R  ≤ C p (1 + R ) p , R ≥ 1 , coun ting p oles with m ultiplicit y . F or Kerr–de Sitter this follows from the analytic F redholm setup for P ( ω ) together with semiclassical parametrix estimates; see, for example, [ 2 , § 2] and the global reﬁnements in [ 11 ]. 30 RUILIANG LI Fix A > p + 2. F or R ≥ 1 consider the excluded subset of [ R, R + 1], F R := [ ω j ∈ Res( P ) − ν ≤ℑ ω j ≤ C |ℜ ω j |≤ R +1  ℜ ω j − (1 + R ) − A , ℜ ω j + (1 + R ) − A  . By ( 4.12 ), its Leb esgue measure satisﬁes |F R | ≤ 2(1 + R ) − A #  Res( P ) ∩ R R +1  ≤ 2 C p (1 + R ) p − A , whic h is < 1 for all suﬃcien tly large R . Th us for large R w e can c ho ose R ′ ∈ [ R, R + 1] \ F R . By construction, R ′ sta ys at distance at least (1 + R ) − A from the real parts of all p oles in the strip − ν ≤ ℑ ω ≤ C , hence ( 4.11 ) holds (with R n = R ′ for a sequence R → ∞ ). Finally , since p oles are discrete w e ma y p erturb R n b y at most (1 + R n ) − A / 2 to ensure that ∂ R R n itself a v oids p oles, without sp oiling ( 4.11 ). □ In what follo ws we tak e R from this sequence and, for notational simplicity , k eep writing R instead of R n . All limits as R → ∞ are understo od along this p ole-a voiding sequence. By ( 4.5 ) w e hav e b F ϑ ( ω ) = χ b F ϑ ( ω ). Therefore, after applying χ to the in v ersion form ula ( 4.10 ) we ma y write the in tegrand as I ( ω ) := e − iω t ∗ χR ( ω ) χ b F ϑ ( ω ) . Then I ( ω ) is meromorphic in ω , with p oles exactly at the QNMs. By the residue theorem, (4.13) Z ∂ R R I ( ω ) dω = 2 π i X ω j ∈ Res( P ) ∩R R Res ω = ω j  I ( ω )  , where the s um runs o ver p oles in the interior of R R (coun ted with m ultiplic- it y). W riting the b oundary in tegral as the sum of four sides yields (4.14) Z ℑ ω = C, |ℜ ω |≤ R I ( ω ) dω − Z ℑ ω = − ν, |ℜ ω |≤ R I ( ω ) dω + Z ℜ ω = R I ( ω ) dω + Z ℜ ω = − R I ( ω ) dω = 2 π i X ω j ∈ Res( P ) ∩R R Res ω = ω j I ( ω ) . The next lemma shows that the tw o vertical side in tegrals v anish as R → ∞ . Lemma 4.7 (V ertical side integrals v anish) . L et ν > 0 and C > 0 b e as ab ove, and let R → ∞ along the p ole-avoiding se quenc e fr om L emma 4.6 . Then for e ach ﬁxe d t ∗ > 0 , (4.15) lim R →∞  Z ℜ ω = R I ( ω ) dω + Z ℜ ω = − R I ( ω ) dω  = 0 in H s +1 ( X M ,a ) , pr ovide d ( f 0 , f 1 ) ∈ H s + N with N suﬃciently lar ge (dep ending on s ). EQUA TORIAL KERR–DE SITTER RINGDOWN 31 Pr o of. W e estimate the right v ertical side; the left side is iden tical. W rite ω = R + iη with η ∈ [ − ν, C ] and R → ∞ as ab o v e. Step 1: de c ay of the for cing tr ansform. By Lemma 4.5 , for every M ∈ N there exists C s,M > 0 suc h that (4.16) ∥ b F ϑ ( R + iη ) ∥ H s ≤ C s,M (1 + R ) − M ∥ ( f 0 , f 1 ) ∥ H s + M , η ∈ [ − ν, C ] , uniformly in η on the compact in terv al [ − ν, C ]. Step 2: p olynomial c ontr ol of the r esolvent on the vertic al sides. By Lemma 4.6 w e hav e the quan titative p ole separation dist ( R + iη , Res ( P )) ≥ (1 + R ) − A uniformly for η ∈ [ − ν, C ]. In any ﬁxed strip − ν ≤ ℑ ω ≤ C , the cutoﬀ resolv ent χR ( ω ) χ is p olynomially b ounded in |ℜ ω | a wa y from p oles; this is a standard consequence of the Kerr–de Sitter F redholm framework and semiclassical dyadic estimates (see, e.g., [ 2 , § 2] and [ 11 ]). Consequen tly , for eac h s there exist K = K ( s, A ) and C s,K > 0 suc h that (4.17) sup η ∈ [ − ν,C ] ∥ χR ( R + iη ) χ ∥ H s → H s +1 ≤ C s,K (1 + R ) K . Step 3: c oncluding the vanishing. Using ( 4.16 ) – ( 4.17 ) and | e − i ( R + iη ) t ∗ | = e η t ∗ ≤ e C t ∗ ,     Z ℜ ω = R I ( ω ) dω     H s +1 ≤ Z C − ν e η t ∗ ∥ χR ( R + iη ) χ ∥ H s → H s +1 ∥ b F ϑ ( R + iη ) ∥ H s dη ≤ e C t ∗ ( C + ν ) C s,K C s,M (1 + R ) K − M ∥ ( f 0 , f 1 ) ∥ H s + M . Cho osing M > K + 2 yields decay as R → ∞ , and the same estimate holds on the left side ℜ ω = − R . This prov es ( 4.15 ). □ Letting R → ∞ in ( 4.14 ) and using Lemma 4.7 yields the basic contour deformation. T o k eep the argument completely explicit (and to av oid any unnecessary global assumptions on residue pro jectors), w e form ulate the residue contribution as a symmetric trunc ation in ℜ ω . F or R > 1 suc h that ∂ R R a v oids p oles, set (4.18) S ν,R ( t ∗ ) := X ω j ∈ Res( P ) ∩R R Res ω = ω j  I ( ω )  , where p oles are counted with m ultiplicit y . (If ω j is a p ole of order m j , then Res ω = ω j I ( ω ) denotes the co eﬃcient of ( ω − ω j ) − 1 in the Laurent expansion of I at ω j .) Deﬁnition 4.8 (Symmetric truncation sums) . Let ν > 0 and let ( R n ) b e the p ole-a voiding sequence of Lemma 4.6 . Given a family of vectors ( v j ) ω j ∈ Res( P ) , ℑ ω j > − ν in a Banach space B , we write (4.19) X sym ℑ ω j > − ν v j := lim n →∞ X ω j ∈ Res( P ) ℑ ω j > − ν |ℜ ω j |≤ R n v j , 32 RUILIANG LI whenev er the limit exists in B . In all o ccurrences b elo w the limit exists and is indep endent of the particular choice of p ole-a v oiding sequence, b ecause it is uniquely determined b y contour deformation (see Theorem 4.9 ). Then ( 4.14 ) and Lemma 4.7 giv e, for each such R , Z ℑ ω = C, |ℜ ω |≤ R I ( ω ) dω = Z ℑ ω = − ν, |ℜ ω |≤ R I ( ω ) dω + 2 π i S ν,R ( t ∗ ) + o R →∞ (1) in H s +1 . Since the integrand on each horizon tal line is absolutely in tegrable in σ = ℜ ω (b y the p olynomial b ounds for χR ( ω ) χ and the rapid decay of b F ϑ ), the truncated horizontal integrals con v erge to the full in tegrals as R → ∞ . Consequen tly , (4.20) Z ℑ ω = C I ( ω ) dω = Z ℑ ω = − ν I ( ω ) dω + 2 π i lim R →∞ S ν,R ( t ∗ ) , and the limit exists in H s +1 ( X M ,a ). The pro of b elo w sho ws that the limit is unique (indep enden t of the particular sequence of R → ∞ a v oiding p oles), b ecause it is equal to the diﬀerence of tw o absolutely conv ergen t contour in tegrals. 4.4. Resonan t expansion and remainder op erator. W e now translate the con tour deformation ( 4.20 ) in to a time-domain expansion. The p ole con tributions are naturally ordered by the truncation parameter R in the rectangular argument, and w e record conv ergence in this intrinsic sense. Theorem 4.9 (Resonant expansion with remainder) . Fix ν > 0 and assume Γ − ν is p ole-fr e e (Pr op osition 3.13 ). Assume the interior supp ort/cutoﬀ c onvention of Assumption 4.3 . L et u solve P u = 0 with initial data ( f 0 , f 1 ) ∈ H s + N for N suﬃciently lar ge, supp orte d in K 0 . L et ϑ b e the time cutoﬀ ( 4.1 ) and let χ b e the sp atial cutoﬀ ﬁxe d in Assumption 4.3 . Then for al l t ∗ ≥ 1 we have the identity in H s +1 ( X M ,a ) : (4.21) χu ( t ∗ ) = X sym ℑ ω j > − ν U ω j ( t ∗ ) ( f 0 , f 1 ) + R ν ( t ∗ ) ( f 0 , f 1 ) , Her e P sym denotes the symmetric trunc ation sum of Deﬁnition 4.8 , which is the natur al or dering induc e d by c ontour deformation; in p articular, no a priori absolute c onver genc e of the p ole series is assume d. The terms ar e as fol lows: (1) F or e ach QNM p ole ω j of or der m j ≥ 1 , the c orr esp onding r esonant c ontribution is a ﬁnite sum (4.22) U ω j ( t ∗ ) ( f 0 , f 1 ) = i e − iω j t ∗ m j − 1 X p =0 t p ∗ C j,p ( f 0 , f 1 ) , EQUA TORIAL KERR–DE SITTER RINGDOWN 33 with C j,p ﬁnite-r ank op er ators H s + N → H s +1 ( X M ,a ) dep ending on χ, ϑ . If the p ole is simple ( m j = 1 ), then (4.23) U ω j ( t ∗ ) ( f 0 , f 1 ) = i e − iω j t ∗ χ Π ω j χ b F ϑ ( ω j ) , Π ω j := Res ω = ω j R ( ω ) . (2) The r emainder op er ator is given explicitly by the shifte d-c ontour inte gr al (4.24) R ν ( t ∗ ) ( f 0 , f 1 ) = 1 2 π Z ℑ ω = − ν e − iω t ∗ χR ( ω ) χ b F ϑ ( ω ) dω . The limit in ( 4.21 ) exists and is indep endent of the p articular se quenc e of trunc ation r adii R → ∞ avoiding p oles, sinc e it is uniquely determine d by the diﬀer enc e χu − R ν . Mor e over, the r emainder on the shifte d c ontour enjoys arbitr arily high p olynomial de c ay in t ∗ . F or every m ∈ N ther e exists C s,m > 0 such that (4.25) ∥ R ν ( t ∗ ) ( f 0 , f 1 ) ∥ H s +1 ≤ C s,m e − ν t ∗ (1 + t ∗ ) − m ∥ ( f 0 , f 1 ) ∥ H s + m + N , t ∗ ≥ 1 . If the p ar ameters r ange in a c omp act set K c ontaine d in the p ole-fr e e r e gion for Γ − ν (as in Pr op osition 3.13 ), then the c onstants C s,m c an b e chosen uniformly for ( M , a ) ∈ K . Pr o of. W e b egin with the in version formula ( 4.10 ) for u ϑ and apply χ . By ( 4.5 ) (equiv alen tly , Assumption 4.3 ), we ha v e χu ϑ ( t ∗ ) = 1 2 π Z ℑ ω = C I ( ω ) dω , I ( ω ) = e − iω t ∗ χR ( ω ) χ b F ϑ ( ω ) , with I as in § 4.3 . Step 1: c ontour deformation and existenc e of the p ole-sum limit. By ( 4.20 ) , χu ϑ ( t ∗ ) = 1 2 π Z ℑ ω = − ν I ( ω ) dω + i lim R →∞ S ν,R ( t ∗ ) . The ﬁrst term is exactly the remainder ( 4.24 ), hence (4.26) i lim R →∞ S ν,R ( t ∗ ) = χu ϑ ( t ∗ ) − R ν ( t ∗ ) . In particular, the limit exists in H s +1 and is unique: it is the diﬀerence of t w o w ell-deﬁned con tour integrals. Since u = u ϑ for t ∗ ≥ 1, this yields ( 4.21 ) once we identify the residue con tributions. Step 2: structur e of e ach r esidue c ontribution. F or each ﬁxed R , the sum S ν,R ( t ∗ ) in ( 4.18 ) is ﬁnite. Expanding R ( ω ) in a Lauren t series at a p ole ω j and using Lemma 2.10 gives that the residue of I ( ω ) at ω j is a ﬁnite linear com bination of terms t p ∗ e − iω j t ∗ times ﬁnite-rank op erators applied to b F ϑ ( ω j ) and its ω –deriv atives, which yields the general form ( 4.22 ) . In the simple p ole case, the residue is exactly ( 4.23 ). 34 RUILIANG LI Th us, for each R , i S ν,R ( t ∗ ) = X ω j ∈ Res( P ) ℑ ω j > − ν |ℜ ω j |≤ R U ω j ( t ∗ ) ( f 0 , f 1 ) , and taking R → ∞ gives the p ole contribution in ( 4.21 ). Step 3: r emainder b ound. W rite ω = σ − iν on Γ − ν : R ν ( t ∗ ) = e − ν t ∗ 1 2 π Z R e − iσ t ∗ χR ( σ − iν ) χ b F ϑ ( σ − iν ) dσ . Set B ( σ ) := χR ( σ − iν ) χ b F ϑ ( σ − iν ) . Using the ω –deriv ative b ounds from Section 3 (Prop osition 3.9 in dy adic form, com bined with lo w-frequency boundedness), w e ha ve p olynomial bounds for ∂ ℓ σ ( χR ( σ − iν ) χ ) in op erator norms H s → H s +1 , uniformly in σ ∈ R . Com bining these p olynomial b ounds with the Sc hw artz decay of ∂ q ω b F ϑ (Lemma 4.5 ) shows that for eac h m , (4.27) Z R ∥ ∂ m σ B ( σ ) ∥ H s +1 dσ ≤ C s,m ∥ ( f 0 , f 1 ) ∥ H s + m + N . In tegrating by parts m times in σ giv es Z R e − iσ t ∗ B ( σ ) dσ = 1 ( it ∗ ) m Z R e − iσ t ∗ ∂ m σ B ( σ ) dσ, hence by ( 4.27 ), ∥ R ν ( t ∗ ) ∥ H s +1 ≤ e − ν t ∗ 1 2 π t − m ∗ Z R ∥ ∂ m σ B ( σ ) ∥ H s +1 dσ ≤ C s,m e − ν t ∗ t − m ∗ ∥ ( f 0 , f 1 ) ∥ H s + m + N . Replacing t − m ∗ b y (1 + t ∗ ) − m giv es ( 4.25 ). □ R emark 4.10 (On conv ergence and on residue pro jectors) . The expansion ( 4.21 ) is stated as a limit of symmetric truncations in ℜ ω . This is the natural notion of con vergence coming from the contour deformation argument: the partial sums are precisely the ﬁnite residue sums pro duced by expanding rectangles R R . W e emphasize that w e do not claim any global b ound such as ∥ χ Π ω j χ ∥ ≲ (1 + | ω j | ) K for general QNM p oles. F or non-selfadjoint problems suc h b ounds can fail dramatically , and the size of residue pro jectors is tied to pseudosp ectral eﬀects and transien t growth; see for instance [ 14 , 17 ] for discussions in the blac k hole context. F or the goals of this pap er, a global residue-growth theory is unnecessary . In Sections 5 – 7 we alw ays apply exact azim uthal pro jection and microlo- cal ﬁltering to the equatorial high-frequency sector. In that setting, the residue pro jectors ar e quantitativ ely con trolled: the microlocalized pro jec- tors asso ciated with the lab eled equatorial p oles satisfy a p olynomial b ound EQUA TORIAL KERR–DE SITTER RINGDOWN 35 (Prop osition 5.18 ), while every other p ole ab o v e the con tour is suppressed b y O ( ℓ −∞ ) after equatorial cutoﬀ (Theorem 5.4 ). As a consequence, the windo w ed expansions used later are not only conv ergen t but come with explicit uniform tail b ounds, and the dynamics in the equatorial window reduces to ﬁnitely man y leading mo des plus a sup erp olynomial remainder. R emark 4.11 (Time-shifted ringdo wn windo w) . F or ringd own one often begins observing at a late time T 0 . Let ϑ T 0 ( t ∗ ) := ϑ ( t ∗ − T 0 ) and deﬁne u ϑ T 0 = ϑ T 0 u . Then u ϑ T 0 = u for t ∗ ≥ T 0 + 1, and the entire argument ab ov e applies with b F ϑ T 0 ( ω ) = e iω T 0 b F ϑ ( ω ). All estimates remain unchanged up to constants dep ending on T 0 through the trivial factor e ν T 0 . 4.5. Azim uthal and equatorial sectorial expansions. Since [ P , Φ] = 0, the expansion decouples exactly in the azimuthal n umber k (Section 2.10 ). Moreo ver, applying b ounded semiclassical microlo cal cutoﬀs A ± ,h ℓ (Sec- tion 2.11 ) yields sectorial expansions with the same remainder b ounds. Corollary 4.12 (Sectorial resonant expansion) . Fix k ∈ Z and let Π k b e the ortho gonal pr oje ctor onto the k –th azimuthal subsp ac e. L et A ± ,h ℓ b e the e quatorial micr olo c al cutoﬀ ( 2.29 ) (extende d trivial ly in r ). Then for t ∗ ≥ 1 , (4.28) χA ± ,h ℓ Π k u ( t ∗ ) = X sym ℑ ω j > − ν χA ± ,h ℓ Π k U ω j ( t ∗ ) ( f 0 , f 1 ) + χA ± ,h ℓ Π k R ν ( t ∗ ) ( f 0 , f 1 ) . Mor e over, the r emainder satisﬁes the same b ound as ( 4.25 ) (with c onstants indep endent of h ). Pr o of. Apply χA ± ,h ℓ Π k to both sides of ( 4.21 ) . Since A ± ,h ℓ is uniformly b ounded on H s +1 and commutes with e − iω t ∗ , the decomp osition remains v alid, and the remainder estimate follows from ( 4.25 ). □ Bridge to tw o-mo de dominance. Corollary 4.12 is the precise form of the resonant expansion needed for the next step: after sp ecializing to the equatorial high-frequency sector and choosing ν b et w een resonance lay ers, the sum ov er ℑ ω j > − ν collapses to a single QNM in each of the equatorial k = ± ℓ sectors, yielding a t w o-mo de ringdown mo del plus an exp onen tially deca ying tail. 5. Tw o-mode dominance in the equa torial high-frequency sector This section is the microlo cal–spectral core of the paper. Starting from the forced evolution u ϑ constructed in Section 2 , we build t w o reﬁned, mo de- separated, equatorially lo calized signals u (+ ,ℓ ) ( t ∗ ) and u ( − ,ℓ ) ( t ∗ ) which, for large angular momentum ℓ , are gov erned by a single lab eled quasinormal exp onen tial in eac h azimuthal sector k = ± ℓ . The output is a deterministic t w o-exp onen tial mo del with an explicit tail term, suitable for the stability analysis of Section 6 . 36 RUILIANG LI Throughout this section w e adopt the in terior supp ort/cutoﬀ conv en tion of Assumption 4.3 from Section 4 : the initial data are supp orted in a ﬁxed compact set K 0 ⋐ X phys M ,a , and the spatial cutoﬀ χ ∈ C ∞ c ( X phys M ,a ) is ﬁxed so that b F ϑ ( ω ) = χ b F ϑ ( ω ) for all ω ∈ C , cf. ( 4.5 ) . This is a tec hnical device allo wing us to mo ve χ freely inside resolv en t/Laplace expressions without altering the observed signal. Tw o issues require particular care in a uniform (in ℓ ) error analysis. First, quasinormal contributions are identiﬁed b y contour deformation in the complex ω –plane, so any frequency ﬁlter must be holomorphic across the deformation region in order not to in tro duce spurious singularities. Second, the generator is non-selfadjoint: residue pro jectors can ha v e large norms (often called excitation factors ), and a scalar suppression factor alone do es not exclude leak age from unw anted p oles unless it is paired with quan titativ e con trol of the corresp onding microlo calized pro jectors. Our strategy combines three mechanisms. (a) Companion-pap er inputs in the equatorial pac k age. F or each k = ± ℓ and ﬁxed ov ertone index n , the companion pap er [ 1 ] constructs equatorial pseudop oles ω ♯ j,ℓ,k and identiﬁes nearby true QNMs ω j,ℓ,k , uniformly in ( M , a ) on compact slow–rotation sets. Moreov er, after equatorial microlo calization, p oles outside the lab eled equatorial disks are suppressed b y O ( ℓ −∞ ). W e record these inputs in Subsection 5.1 in a form tailored to the residue calculus used b elo w. (b) Entire analytic windo ws built from equatorial pseudopoles. W e build an entire w eight e g ± ,ℓ ( ω ) which equals 1 at the target pseudop ole ω ♯ ± ,ℓ and v anishes at the lo w er pseudop oles ω ♯ j, ± ,ℓ , j ≤ n − 1, while remaining uniformly controlled on the shifted contour Γ − ν ; see Lemma 5.6 . Because the weigh t is entire, it is compatible with contour deformation and do es not in tro duce additional residues. (c) La y ering. A vertical gap b et w een consecutiv e equatorial ov ertone lay ers allo ws us to choose a shifted con tour heigh t ν separating the n th lay er from the ( n + 1)st (Lemma 5.15 ). This yields exp onen tial decay for the tail term and keeps the residue b o okk eeping uniform on the ﬁxed parameter set K . 5.1. Companion-pap er inputs: equatorial lab eling and microlo cal sectorization. Fix a cosmological constan t Λ > 0 and a compact slo w– rotation parameter set K ⋐ P Λ ∩ {| a | ≤ a 0 } , with a 0 > 0 suﬃciently small. W e also ﬁx an o v ertone index n ∈ N 0 . The companion pap er [ 1 ] pro vides a high–frequency equatorial pack age in the exact azimuthal sectors k = ± ℓ : pseudop oles ω ♯ j, ± ,ℓ and nearby true QNMs ω j, ± ,ℓ , stable lab eling, and microlo cal control of the corresp onding resolv en t singularities after equatorial localization. W e b egin by recording the isolation and simplicit y of the lab eled p oles. F or the presen t paper it is crucial that the lab eled disks are disjoint and EQUA TORIAL KERR–DE SITTER RINGDOWN 37 that the p oles inside them are simple, b ecause this eliminates Jordan–blo c k eﬀects in the regime where w e pro v e t w o-mo de dominance. Theorem 5.1 (Input from Part I: equatorial p ole labeling and isolation) . Fix n ∈ N 0 . Ther e exist c onstants c sep > 0 and ℓ 0 ∈ N such that for every ℓ ≥ ℓ 0 , every ( M , a ) ∈ K , e ach sign ± , and e ach overtone index j ∈ { 0 , 1 , . . . , n + 1 } , the disks (5.1) D j, ± ,ℓ :=  ω ∈ C : | ω − ω ♯ j, ± ,ℓ ( M , a ) | < c sep  ar e p airwise disjoint and e ach c ontains exactly one simple p ole of R ( ω ) , de- note d ω j, ± ,ℓ ( M , a ) . Mor e over, for every N ∈ N one has the sup er–p olynomial pseudop ole pr oximity estimate (5.2) | ω j, ± ,ℓ ( M , a ) − ω ♯ j, ± ,ℓ ( M , a ) | = O ( ℓ − N ) , 0 ≤ j ≤ n + 1 , with c onstants uniform on K . R emark 5.2 (Origin in the companion pap er) . Theorem 5.1 is a streamlined reform ulation of the stable lab eling and sup er–polynomial appro ximation results in [ 1 ]; see in particular [ 1 , Theorem 15 and Prop osition 20]. Simplicit y for ℓ ≫ 1 follows from the same barrier–top Grushin reduction and the quan titativ e nondegeneracy of the asso ciated scalar quantization function; compare [ 1 , Remark 21] and the analyticity framework of [ 10 , 11 ]. W e collect further p oin ters and a sectorization pro of outline in App endix D . W e also need a microlo cal selection statemen t: after equatorial microlo- calization and azimuthal pro jection to k = ± ℓ , all p oles outside the lab eled equatorial disks are in visible up to O ( ℓ −∞ ). Since w e must control p ossible higher-order p oles aw a y from the equatorial disks, it is conv enien t to w ork with the generalized Lauren t co eﬃcients. Deﬁnition 5.3 (Generalized Lauren t coeﬃcients at a p ole) . Let ω 0 ∈ Res ( P ) b e a p ole of R ( ω ) of order m ω 0 ≥ 1. W e write the Laurent expansion near ω 0 as (5.3) R ( ω ) = m ω 0 X q =1 ( ω − ω 0 ) − q Π [ q ] ω 0 + R hol ω 0 ( ω ) , where R hol ω 0 is holomorphic near ω 0 and the op erators Π [ q ] ω 0 are ﬁnite rank. In particular, Π [1] ω 0 = Π ω 0 = Res ω = ω 0 R ( ω ). Theorem 5.4 (Input from P art I: equatorial microlo cal sectorization) . Assume The or em 5.1 and ﬁx ν > 0 such that Γ − ν is admissible (Pr op osi- tion 3.13 ). F or every N ∈ N , every Sob olev index s ≥ 0 , and every inte ger q ≥ 1 , ther e exist c onstants C s,N ,q > 0 and ℓ 1 ∈ N such that for al l ℓ ≥ ℓ 1 , al l ( M , a ) ∈ K , e ach sign ± , and every p ole ω j ∈ Res ( P ) with ℑ ω j > − ν satisfying ω j / ∈ [ 0 ≤ m ≤ n D m, ± ,ℓ , 38 RUILIANG LI one has the op er ator b ound (5.4)   χ A ± ,h ℓ Π k ± Π [ q ] ω j χ   H s − 1 → H s +1 ≤ C s,N ,q ℓ − N . R emark 5.5 (In terpretation) . Estimate ( 5.4 ) is a microlo cal selection state- men t: the equatorial cutoﬀ A ± ,h ℓ restricts to a small phase–space neighbor- ho od of the equatorial trapp ed set, and the Grushin reduction in [ 1 ] shows that the only p oles which pro duce non-negligible singularities in that c hannel are the lab eled equatorial p oles inside the disks D m, ± ,ℓ . Poles outside these disks are negligible to the equatorial channel up to O ( ℓ −∞ ), ev en though they are gen uine p oles of the full resolv en t. See Appendix D for a pro of outline. 5.2. Analytic frequency ﬁlters compatible with con tour deformation. Fix an ov ertone index n ∈ N 0 . Throughout this section we w ork in the equatorial high-frequency regime ℓ ≫ 1 and we set (5.5) h ℓ := ℓ − 1 ∈ (0 , h 0 ] , k ± := ± ℓ ∈ Z . Recall from the companion pap er [ 1 ] that, for each k = ± ℓ , the equatorial barrier-top pack age pro duces pseudop oles ω ♯ j,ℓ,k (one for eac h ov ertone j ) and nearb y true QNMs ω j,ℓ,k . W e denote (5.6) ω ♯ ± ,ℓ := ω ♯ n,ℓ,k ± ( M , a ) , ω ± ,ℓ := ω n,ℓ,k ± ( M , a ) , and write ω ♯ j, ± ,ℓ := ω ♯ j,ℓ,k ± , ω j, ± ,ℓ := ω j,ℓ,k ± for the low er o vertone lab els. 5.2.1. Interp olation weights and a uniform mo diﬁc ation. F or each sign ± and each ℓ , consider the degree- n interpolation p olynomial (5.7) g ± ,ℓ ( ω ) := n − 1 Y j =0 ω − ω ♯ j, ± ,ℓ ω ♯ ± ,ℓ − ω ♯ j, ± ,ℓ , ω ∈ C . By construction, (5.8) g ± ,ℓ ( ω ♯ ± ,ℓ ) = 1 , g ± ,ℓ ( ω ♯ j, ± ,ℓ ) = 0 (0 ≤ j ≤ n − 1) . Th us g ± ,ℓ cancels the low er pseudop oles while keeping the target pseudop ole unc hanged. A uniformity issue arises b ecause ω ♯ ± ,ℓ ∼ ℓ and g ± ,ℓ has ﬁxed degree: on b ounded frequency sets, g ± ,ℓ ma y amplify by a factor ∼ ℓ n , which would en ter the constants in the shifted-con tour remainder estimate. W e eliminate this by inserting a high-order zero at ω = 0. Fix once and for all (5.9) m 0 := n + 2 , and deﬁne the mo diﬁe d analytic window (5.10) e g ± ,ℓ ( ω ) :=  ω ω ♯ ± ,ℓ  m 0 g ± ,ℓ ( ω ) . EQUA TORIAL KERR–DE SITTER RINGDOWN 39 Then e g ± ,ℓ is entire (in fact a p olynomial of degree n + m 0 ), it retains ( 5.8 ) , and it is uniformly b ounded on compact subsets of C as ℓ → ∞ since m 0 ≥ n . Lemma 5.6 (Uniform b ounds for the mo diﬁed analytic weigh ts) . Fix ν > 0 . Assume ℓ is suﬃciently lar ge so that the denominators in ( 5.7 ) ar e uniformly b ounde d away fr om 0 (which fol lows fr om The or em 5.1 and the disjointness of the disks D j, ± ,ℓ ). Then for every inte ger r ≥ 0 ther e exist c onstants C r > 0 and ℓ r ∈ N such that for al l ℓ ≥ ℓ r , al l ( M , a ) ∈ K , e ach sign ± , and al l ω ∈ Γ − ν , (5.11)   ∂ r ω e g ± ,ℓ ( ω )   ≤ C r (1 + | ω | ) n + m 0 . In p articular, for every ﬁxe d R > 0 , sup ℓ ≥ ℓ r sup ω ∈ Γ − ν | ω |≤ R | ∂ r ω e g ± ,ℓ ( ω ) | < ∞ . Pr o of. W rite e g ± ,ℓ ( ω ) = ω m 0 p ± ,ℓ ( ω ) with p ± ,ℓ ( ω ) := ( ω ♯ ± ,ℓ ) − m 0 n − 1 Y j =0  ω − ω ♯ j, ± ,ℓ  n − 1 Y j =0  ω ♯ ± ,ℓ − ω ♯ j, ± ,ℓ  . By the disk disjointness in Theorem 5.1 , the denominators are b ounded aw a y from 0 uniformly for ℓ ≥ ℓ r and ( M , a ) ∈ K . Since | ω ♯ j, ± ,ℓ | ≲ ℓ for ﬁxed j and large ℓ (see [ 1 ]), w e obtain on Γ − ν the b ound | e g ± ,ℓ ( ω ) | ≤ C ℓ n − m 0 (1 + | ω | ) n + m 0 ≤ C (1 + | ω | ) n + m 0 , using | ω ♯ ± ,ℓ | ∼ ℓ and m 0 ≥ n . F or deriv ativ es, note that e g ± ,ℓ is a p olynomial of degree n + m 0 . Fix r ≥ 0 and apply Cauch y’s estimate with radius 1: | ∂ r ω e g ± ,ℓ ( ω ) | ≤ r ! sup | z − ω | =1 | e g ± ,ℓ ( z ) | . Using the previous b ound and | z | ≤ | ω | + 1 on the circle yields ( 5.11 ). □ 5.2.2. Prior-dep endent ﬁltering and non-cir cularity. The analytic windo ws e g ± ,ℓ are built from frequency priors, namely the pseudop oles ω ♯ j, ± ,ℓ ( M , a ). This dep endence is explicit: in ringdown inference one t ypically has an external prior (for example from the inspiral phase) and uses it to design ﬁlters which suppress non-target con tributions; see [ 13 ] for a data-analysis p erspective. Our contribution is to form ulate such a prior-dep enden t ﬁltering step in a w a y that is compatible with con tour deformation and admits uniform semiclassical leak age b ounds. 40 RUILIANG LI R emark 5.7 (Prior dep endence in the deterministic pipeline) . In the determin- istic bias chain of Sections 6 – 7 the prior enters in t w o places: (i) it ﬁxes the analytic windo w e g ± ,ℓ and thereb y the mo de-separated signal; (ii) it ﬁxes the branc h of the logarithm used when conv erting a complex ratio of time shifts in to a frequency (Lemma 6.5 ). Neither step is circular: one may take the prior from a diﬀerent data segmen t (e.g. inspiral), or from an initial coarse estimate, and then iterate (prior → ﬁltered signal → extracted frequency → up dated prior). Lemma 5.8 (Robustness of pseudop ole-based w eigh ts under small prior mismatc h) . Fix n ∈ N 0 and a c omp act p ar ameter set K . F or ˜ p ∈ K and e ach ℓ , deﬁne the mo diﬁe d analytic window e g ˜ p ± ,ℓ by ( 5.10 ) , using the pseudop oles ω ♯ j, ± ,ℓ ( ˜ p ) in plac e of ω ♯ j, ± ,ℓ ( M , a ) . Set (5.12) δ ± ,ℓ ( p, ˜ p ) := max 0 ≤ j ≤ n   ω ♯ j, ± ,ℓ ( p ) − ω ♯ j, ± ,ℓ ( ˜ p )   , d ♯, ± ,ℓ ( ˜ p ) := min 0 ≤ m 0 and ℓ 0 ∈ N dep ending only on K and n such that for al l ℓ ≥ ℓ 0 , al l p, ˜ p ∈ K with δ ± ,ℓ ( p, ˜ p ) ≤ c 0 d ♯, ± ,ℓ ( ˜ p ) , and e ach sign ± , one has (5.13) e g ˜ p ± ,ℓ  ω j, ± ,ℓ ( p )  = δ j n + O ( ℓ −∞ ) + O  δ ± ,ℓ ( p, ˜ p ) d ♯, ± ,ℓ ( ˜ p )  , 0 ≤ j ≤ n, wher e the implicit c onstants ar e uniform for p, ˜ p ∈ K . Pr o of. Fix a sign ± and ℓ ≫ 1. Let Ω ♯ j := ω ♯ j, ± ,ℓ ( ˜ p ) and Ω j := ω ♯ j, ± ,ℓ ( p ) for 0 ≤ j ≤ n , so that d ♯ = d ♯, ± ,ℓ ( ˜ p ) and δ = δ ± ,ℓ ( p, ˜ p ). Apply Prop osition C.1 with m = n to the Lagrange weigh t G n built from the no des Ω ♯ j . Recalling that G n coincides with the p olynomial g ˜ p ± ,ℓ in ( 5.7 ) deﬁned using the pseudop oles at ˜ p , we obtain g ˜ p ± ,ℓ (Ω n ) = 1 + O ( δ /d ♯ ) , g ˜ p ± ,ℓ (Ω j ) = O ( δ /d ♯ ) (0 ≤ j ≤ n − 1) . Next, ev aluate the extra factor in ( 5.10 ) at Ω j . Since Ω j = O ( ℓ ) and Ω ♯ n = O ( ℓ ) uniformly on K , the ratio (Ω j / Ω ♯ n ) m 0 is uniformly b ounded for 0 ≤ j ≤ n . Hence the same b ounds hold with g ˜ p ± ,ℓ replaced by e g ˜ p ± ,ℓ and with Ω j in place of Ω ♯ j . Finally , the true p oles satisfy ω j, ± ,ℓ ( p ) = ω ♯ j, ± ,ℓ ( p ) + O ( ℓ −∞ ) by ( 5.2 ) . Since e g ˜ p ± ,ℓ is a p olynomial of ﬁxed degree and is uniformly b ounded (together with its deriv ativ es) on the union of the disks D j, ± ,ℓ ( p ), a T aylor expansion at Ω j giv es the additional O ( ℓ −∞ ) term in ( 5.13 ). □ Lemma 5.9 (Scaling of pseudop ole mismatc h with high frequency) . Fix n ∈ N 0 and a c omp act p ar ameter set K . Ther e exist c onstants C > 0 and ℓ 0 ∈ N such that for al l ℓ ≥ ℓ 0 , al l p, ˜ p ∈ K , and e ach sign ± , (5.14) δ ± ,ℓ ( p, ˜ p ) ≤ C ℓ | p − ˜ p | , EQUA TORIAL KERR–DE SITTER RINGDOWN 41 wher e δ ± ,ℓ is deﬁne d in ( 5.12 ) and | · | is any ﬁxe d norm on the ( M , a ) p ar ameter sp ac e. Pr o of. F or eac h ﬁxed ov ertone index 0 ≤ j ≤ n and each sign ± , write ω ♯ j, ± ,ℓ ( p ) = ℓ b ω ♯ j, ± ,ℓ ( p ) with b ω ♯ j, ± ,ℓ ( p ) := ℓ − 1 ω ♯ j, ± ,ℓ ( p ). The barrier–top quanti- zation map in the companion pap er provides b ω ♯ j, ± ,ℓ as a smo oth function of p , with uniform C 1 b ounds on K for ℓ ≫ 1; see [ 1 , § 3– § 5 and App endix B]. In particular, after enlarging C if necessary , sup ℓ ≥ ℓ 0 sup p ∈K   D p b ω ♯ j, ± ,ℓ ( p )   ≤ C, 0 ≤ j ≤ n. By the mean v alue theorem, | b ω ♯ j, ± ,ℓ ( p ) − b ω ♯ j, ± ,ℓ ( ˜ p ) | ≤ C | p − ˜ p | for all p, ˜ p ∈ K . Multiplying b y ℓ giv es | ω ♯ j, ± ,ℓ ( p ) − ω ♯ j, ± ,ℓ ( ˜ p ) | ≤ C ℓ | p − ˜ p | . T aking the maximum o v er 0 ≤ j ≤ n yields ( 5.14 ). □ R emark 5.10 (Interpreting the prior-robustness hypothesis) . Lemma 5.8 quan tiﬁes ho w muc h pseudop ole mismatc h one can tolerate while k eeping the analytic window selective. The hypothesis is δ ± ,ℓ ( p, ˜ p ) ≪ d ♯, ± ,ℓ ( ˜ p ), where d ♯, ± ,ℓ is the minimal separation among the ﬁrst n + 1 pseudop oles at ˜ p . F or ﬁxed n and compact K , the separation d ♯, ± ,ℓ is bounded b elo w uniformly for ℓ ≫ 1: consecutiv e ov ertones hav e a uniform vertical gap (Lemma 5.15 ; cf. App endix D ), and the pseudop ole pro ximit y ( 5.2 ) prev ents accidental coalescence. Com bining this with Lemma 5.9 shows that, when one uses analytic cancellation of lo wer ov ertones ( n ≥ 1), a suﬃcient robustness condition is a prior accuracy of order | p − ˜ p | ≲ ℓ − 1 . Tw o simpliﬁcations are worth stressing. First, for the fundamental mo de ( n = 0) one may take e g ± ,ℓ ≡ 1, so no frequency prior is needed for the dominance mechanism. Second, for n ≥ 1 one can av oid ﬁne priors altogether b y using the band-isolation identit y in Prop osition 5.21 (and Corollary 5.22 ), whic h dep ends only on the existence of a uniform la yer gap and not on the precise lo cations of the pseudop oles. In practice, one ma y use the preceding discussion in an iterative manner. One ﬁrst applies the n = 0 pipeline (for w hic h one may tak e e g ± ,ℓ ≡ 1) to obtain a coarse estimate of the relev an t parameters from the leading equatorial pair. This coarse estimate can then b e used as ˜ p when constructing the windo ws e g ˜ p ± ,ℓ for n ≥ 1, either to deﬂate low er o v ertones or to incorp orate additional observ ables in the three-parameter map. The deterministic error b ounds apply at each reﬁnemen t step as long as the up dated prior remains in the compact neighborho od where the companion inv erse theorem is v alid. 5.2.3. Analytic-windowe d evolution and r eﬁne d signals. W e now combine the analytic weigh ts with the forced resonance expansion of Section 4 . Deﬁnition 5.11 (Analytic-window ed ev olution and reﬁned equatorial sig- nals) . Let C > 0 b e large as in Lemma 4.5 , so that b F ϑ ( ω ) is holomorphic 42 RUILIANG LI for ℑ ω > − C . F or each ℓ and eac h sign ± , deﬁne the analytic-window ed ev olution op erator (5.15) W ( ± ,ℓ ) u ( t ∗ ) := 1 2 π Z ℑ ω = C e − iω t ∗ e g ± ,ℓ ( ω ) R ( ω ) b F ϑ ( ω ) dω , t ∗ > 0 , where e g ± ,ℓ is giv en by ( 5.10 ) . The corresp onding reﬁned e quatorial mo de- separated signals are (5.16) u (+ ,ℓ ) ( t ∗ ) := χ A + ,h ℓ Π k + W (+ ,ℓ ) u ( t ∗ ) , u ( − ,ℓ ) ( t ∗ ) := χ A − ,h ℓ Π k − W ( − ,ℓ ) u ( t ∗ ) , view ed as elements of H s +1 ( X M ,a ) for t ∗ > 0. R emark 5.12 (Time-domain realization of the analytic window) . Since e g ± ,ℓ is a p olynomial, e g ± ,ℓ ( i∂ t ∗ ) is a ﬁnite-order diﬀeren tial op erator in t ∗ . F ormally one may write W ( ± ,ℓ ) u = e g ± ,ℓ ( i∂ t ∗ ) u ϑ microlo cally on the supp ort of χ . W e use this only as intuition; all arguments b elo w pro ceed through the frequency-domain contour integrals. R emark 5.13 (Role of holomorphy) . The resonan t con tributions in Section 4 arise by deforming a horizon tal contour in the complex ω -plane. An y window that is not holomorphic across the deformation region w ould introduce additional singularities and sp oil the residue b o okk eeping. This is the reason w e insist on the entire (or at least holomorphic) c haracter of e g ± ,ℓ . 5.3. Analytic-window ed sectorial resonant expansion. W e ﬁrst record the con tour deformation expansion satisﬁed by the reﬁned signals ( 5.16 ) . Compared to Theorem 4.9 , the only additional input is the presence of the en tire weigh t e g ± ,ℓ in the integrand. In particular, the p ole contributions m ust b e stated in a form that remains v alid for higher-order p oles: by Lemma 2.10 , deriv atives of the holomorphic factor e g ± ,ℓ ( ω ) b F ϑ ( ω ) app ear when the p ole order exceeds 1. Theorem 5.14 (Analytic-windo w ed sectorial resonant expansion) . Fix Λ > 0 , a c omp act p ar ameter set K ⋐ P Λ ∩ {| a | ≤ a 0 } with a 0 smal l, and ℓ ∈ N . L et ν > 0 b e such that the shifte d c ontour Γ − ν satisﬁes the uniform p ole- sep ar ation hyp othesis Pr op osition 3.13 of Se ction 3 . Assume the interior supp ort/cutoﬀ c onvention of Assumption 4.3 . Then for every t ∗ > 0 the r eﬁne d signals ( 5.16 ) admit the exp ansion (5.17) u ( ± ,ℓ ) ( t ∗ ) = X sym ω j ∈ Res( P ): ℑ ω j > − ν U ( ± ,ℓ ) ω j ( t ∗ ) + R ( ν ) ± ,ℓ ( t ∗ ) , wher e: (1) F or e ach p ole ω j of or der m j ≥ 1 with gener alize d L aur ent c o eﬃcients Π [ q ] ω j , the windowe d p ole c ontribution is the ﬁnite sum (5.18) U ( ± ,ℓ ) ω j ( t ∗ ) = i e − iω j t ∗ m j X q =1 q − 1 X r =0 ( − it ∗ ) q − 1 − r ( q − 1 − r )! r ! χA ± ,h ℓ Π k ± Π [ q ] ω j χ ∂ r ω  e g ± ,ℓ b F ϑ  ( ω j ) . EQUA TORIAL KERR–DE SITTER RINGDOWN 43 In p articular, if ω j is a simple p ole, then (5.19) U ( ± ,ℓ ) ω j ( t ∗ ) = i e − iω j t ∗ χA ± ,h ℓ Π k ± Π ω j χ  e g ± ,ℓ ( ω j ) b F ϑ ( ω j )  , Π ω j := Res ω = ω j R ( ω ) . (2) The r emainder is the shifte d-c ontour inte gr al (5.20) R ( ν ) ± ,ℓ ( t ∗ ) := 1 2 π Z ℑ ω = − ν e − iω t ∗ e g ± ,ℓ ( ω ) χA ± ,h ℓ Π k ± R ( ω ) b F ϑ ( ω ) dω . Mor e over, for every m ∈ N and every s ≥ 0 ther e exists N m ∈ N and a c onstant C s,m > 0 such that for al l ( f 0 , f 1 ) ∈ H s + m + N m supp orte d in K 0 and al l t ∗ ≥ 1 , (5.21) ∥ R ( ν ) ± ,ℓ ( t ∗ ) ∥ H s +1 ≤ C s,m e − ν t ∗ (1 + t ∗ ) − m ∥ ( f 0 , f 1 ) ∥ H s + m + N m , with c onstants uniform for ( M , a ) ∈ K and for al l suﬃciently lar ge ℓ . Pr o of. Start from the deﬁnition ( 5.15 ) and apply the op erators χA ± ,h ℓ Π k ± . Since e g ± ,ℓ is entire and b F ϑ is holomorphic in {ℑ ω > − C } , the map ω 7− → e g ± ,ℓ ( ω ) χA ± ,h ℓ Π k ± R ( ω ) b F ϑ ( ω ) is meromorphic in {ℑ ω > − C } with p oles precisely at the quasinormal p oles of P . W e deform the contour from ℑ ω = C do wn to ℑ ω = − ν . The vertical sides v anish as in Lemma 4.7 , using Sch w artz decay of b F ϑ (Lemma 4.5 ) and p olynomial gro wth of e g ± ,ℓ (Lemma 5.6 with r = 0). The residue theorem yields ( 5.17 ) with remainder ( 5.20 ). T o obtain the explicit p ole con tribution ( 5.18 ) , ﬁx a p ole ω j and integrate the meromorphic Banach-v alued function ω 7− → e − iω t ∗ e g ± ,ℓ ( ω ) χA ± ,h ℓ Π k ± R ( ω ) χ b F ϑ ( ω ) o ver a small p ositiv ely orien ted circle around ω j con tained in {ℑ ω > − ν } and disjoint from other p oles. Expanding R ( ω ) in a Laurent series ( 5.3 ) at ω j and applying Lemma 2.10 with F ( ω ) = e g ± ,ℓ ( ω ) b F ϑ ( ω ) giv es ( 5.18 ) (and ( 5.19 ) when m j = 1). It remains to pro v e the tail b ound ( 5.21 ) . W rite ω = σ − iν with σ ∈ R and set I ± ,ℓ ( σ ) := e g ± ,ℓ ( σ − iν ) χA ± ,h ℓ Π k ± R ( σ − iν ) b F ϑ ( σ − iν ) , view ed as an op erator H s + m + N m → H s +1 . Then R ( ν ) ± ,ℓ ( t ∗ ) = e − ν t ∗ 2 π Z R e − iσ t ∗ I ± ,ℓ ( σ ) dσ. In tegrating by parts m times in σ and arguing as in the pro of of Theorem 4.9 giv es (5.22) ∥ R ( ν ) ± ,ℓ ( t ∗ ) ∥ H s +1 ≤ e − ν t ∗ 2 π t − m ∗ Z R   ∂ m σ I ± ,ℓ ( σ )   H s + m + N m → H s +1 dσ. W e estimate the integrand b y the pro duct rule: eac h ∂ σ diﬀeren tiates e g ± ,ℓ ( σ − iν ), the resolv ent factor R ( σ − iν ), or b F ϑ ( σ − iν ). 44 RUILIANG LI The deriv ativ es of the windo w satisfy p olynomial b ounds in σ uniform in ℓ by Lemma 5.6 . The deriv ativ es of the forced transform are Sc h wartz in σ with v alues in H s + m b y Lemma 4.5 . Finally , by Prop osition 3.13 and the resolv en t estimates of Section 3 (see Prop osition 3.9 ), the op erator norms of ∂ q σ R ( σ − iν ) b et ween the relev an t Sob olev spaces grow at most p olynomially in | σ | , uniformly for ( M , a ) ∈ K . Com bining these b ounds yields   ∂ m σ I ± ,ℓ ( σ )   H s + m + N m → H s +1 ≤ C s,m (1 + | σ | ) C m (1 + | σ | ) − M for some C m ≥ 0 and for arbitrary M , provided N m is chosen large enough to absorb deriv ative losses in the resolven t b ounds. Cho osing M > C m + 2 mak es the right-hand side in tegrable in σ , and inserting this into ( 5.22 ) yields ( 5.21 ) for t ∗ ≥ 1 (after replacing t − m ∗ b y (1 + t ∗ ) − m ). □ 5.4. La y ering and equatorial microlo cal selection. Theorem 5.14 re- duces the reﬁned signals to a weigh ted sum of resonant contributions plus an exp onen tially decaying tail. T o obtain a one-mo de mo del in eac h k = ± ℓ sector, we isolate the equatorial p oles of a ﬁxed o vertone and show that all other p oles contribute only O ( ℓ −∞ ) after equatorial microlo calization. 5.4.1. Uniform sep ar ation of e quatorial overtone layers. Lemma 5.15 (Uniform equatorial la y er separation) . Fix n ∈ N 0 . Ther e exist ν n > 0 , ℓ 0 ∈ N , and (after p ossibly shrinking the c omp act p ar ameter set K ) a c onstant c 0 > 0 such that for al l ℓ ≥ ℓ 0 , al l ( M , a ) ∈ K , and e ach sign ± , (5.23) ℑ ω ♯ n, ± ,ℓ > − ν n > ℑ ω ♯ n +1 , ± ,ℓ , and the shifte d c ontour Γ − ν n is admissible in the sense of Pr op osition 3.13 : it is p ole-fr e e, and on e ach dyadic blo ck of fr e quencies one has the semiclassic al p ole sep ar ation (5.24) dist  z , Res( P h ω )  ≥ c 0 h ω , z := h ω ω , h ω := ⟨ ω ⟩ − 1 , for al l ω ∈ Γ − ν n with |ℜ ω | suﬃciently lar ge. Pr o of. The strict separation ( 5.23 ) for pseudop oles is part of the barrier-top quan tization pac k age in [ 1 ]. Since the pseudop oles depend contin uously (indeed smo othly) on parameters, after shrinking K if necessary w e ma y c ho ose ν n uniformly for ( M , a ) ∈ K . F or the admissibilit y of Γ − ν n , we use the uniform contour construction of Section 3 . At high frequency , the band/lay er structure for Kerr–de Sitter resonances in ﬁxed-width strips ([ 6 ], together with the equatorial lo calization and lab eling in [ 1 ]) shows that there is a vertical gap b etw een the n th and ( n + 1)st equatorial la yers, and choosing ν n in that gap yields the dy adic semiclassical separation ( 5.24 ) . At bounded frequencies there are only ﬁnitely man y p oles in an y strip ℑ ω ≥ − ν n − 1; b y slightly adjusting ν n if nec essary , w e ma y also arrange that Γ − ν n con tains no p ole. This giv es the desired admissibilit y uniformly on the compact set K . □ EQUA TORIAL KERR–DE SITTER RINGDOWN 45 R emark 5.16 (On the size of the parameter set) . The choice of a single con tour heigh t ν used throughout the subsequen t error b ookkeeping is a uniform statement in parameters. If the relev an t high-frequency in v ariants v ary to o widely on a large parameter set, a single ν ma y not exist. F or the lo cal inv erse problem pursued in [ 1 ] (and for the present deterministic extraction chain), it is natural to w ork on a suﬃciently small compact K where the high-frequency structure is uniform. 5.4.2. Suppr ession of lower lab ele d p oles by the analytic window. Lemma 5.17 (Oﬀ-target suppression of lab eled equatorial p oles) . Assume The or em 5.1 . Fix n ∈ N 0 and deﬁne e g ± ,ℓ by ( 5.10 ) . Then for e ach sign ± and every N ∈ N ther e exists ℓ 0 ( N ) such that for al l ℓ ≥ ℓ 0 ( N ) and ( M , a ) ∈ K , (5.25) e g ± ,ℓ ( ω n, ± ,ℓ ) = 1 + O ( ℓ − N ) , e g ± ,ℓ ( ω j, ± ,ℓ ) = O ( ℓ − N ) (0 ≤ j ≤ n − 1) . Pr o of. W rite ω j, ± ,ℓ = ω ♯ j, ± ,ℓ + ε j, ± ,ℓ with ε j, ± ,ℓ = O ( ℓ −∞ ) by ( 5.2 ) . The in terp olation iden tities ( 5.8 ) giv e e g ± ,ℓ ( ω ♯ ± ,ℓ ) = 1 and e g ± ,ℓ ( ω ♯ j, ± ,ℓ ) = 0 for 0 ≤ j ≤ n − 1. Since the disks are disjoint, the denominators ω ♯ ± ,ℓ − ω ♯ j, ± ,ℓ are b ounded aw ay from 0 uniformly , hence e g ± ,ℓ and its deriv ativ es are uniformly b ounded on S j ≤ n D j, ± ,ℓ . The estimates in ( 5.25 ) then follow b y T a ylor’s theorem. □ 5.4.3. Polynomial c ontr ol of lab ele d e quatorial r esidue pr oje ctors. In non- selfadjoin t sp ectral problems, residue pro jectors ma y hav e large norms (some- times called excitation factors ). A scalar suppression such as ( 5.25 ) m ust therefore b e paired with quantitativ e bounds on the microlo calized pro jectors. Prop osition 5.18 (Polynomial b ound for microlo calized lab eled residue pro jectors) . Assume The or em 5.1 . Fix s ≥ 0 and n ∈ N 0 . Ther e exist c onstants K = K ( s, n ) ≥ 0 , ℓ 0 ∈ N , and C s,n > 0 such that for al l ( M , a ) ∈ K , al l ℓ ≥ ℓ 0 , e ach sign ± , and e ach j ∈ { 0 , 1 , . . . , n } , (5.26)   χ A ± ,h ℓ Π k ± Π ω j, ± ,ℓ χ   H s − 1 → H s +1 ≤ C s,n ℓ K , Π ω j, ± ,ℓ := Res ω = ω j, ± ,ℓ R ( ω ) . Pr o of. This is a direct consequence of the equatorial Grushin reduction in [ 1 , App endix B]. W e recall the main steps. Fix ( M , a ) ∈ K , ℓ large, a sign ± , and j ∈ { 0 , 1 , . . . , n } , and set R ± ,ℓ ( ω ) := χ A ± ,h ℓ Π k ± R ( ω ) χ, ω ∈ D j, ± ,ℓ . By Theorem 5.1 , R ± ,ℓ has a unique simple p ole in D j, ± ,ℓ , lo cated at ω = ω j, ± ,ℓ . The Grushin reduction pro vides a decomp osition, v alid on D j, ± ,ℓ , (5.27) R ± ,ℓ ( ω ) = R hol ± ,ℓ ( ω ) + E ( ± ,ℓ ) + ( ω ) q ± ,ℓ ( ω ) − 1 E ( ± ,ℓ ) − ( ω ) , where R hol ± ,ℓ is holomorphic, the op erators E ( ± ,ℓ ) ± ( ω ) satisfy polynomial b ounds in ℓ as maps H s − 1 → H s +1 , and q ± ,ℓ is a scalar holomorphic function with 46 RUILIANG LI a simple zero at ω j, ± ,ℓ and (5.28)   ∂ ω q ± ,ℓ ( ω j, ± ,ℓ )   ≥ c ∗ > 0 uniformly in ( M , a ) ∈ K and ℓ large. T aking residues in ( 5.27 ) yields Res ω = ω j, ± ,ℓ R ± ,ℓ ( ω ) = E ( ± ,ℓ ) + ( ω j, ± ,ℓ )  ∂ ω q ± ,ℓ ( ω j, ± ,ℓ )  − 1 E ( ± ,ℓ ) − ( ω j, ± ,ℓ ) , and ( 5.28 ) together with the p olynomial b ounds on E ( ± ,ℓ ) ± giv es ( 5.26 ). □ R emark 5.19 (Quan titativ e con trol of nonnormalit y in the equatorial pack- age) . In non-selfadjoin t scattering problems, large residue norms (“excitation factors”) and the asso ciated pseudosp ectral b eha vior can in principle inv ali- date naive truncations of resonance expansions on ﬁnite time windows; see, for instance, [ 20 , 22 , 21 , 18 ]. Prop osition 5.18 shows that, after azim uthal re- duction and microlo calization to the equatorial trapp ed channel, the residue pro jectors asso ciated with the lab eled family ha v e at most polynomial growth in ℓ on the spatial Sob olev scale. Combined with the sup er–p olynomial sec- torization estimate of Theorem 5.4 , this implies that non-equatorial p oles are quantitatively invisible to the equatorial pack age up to O ( ℓ −∞ ) errors. This is the mechanism that allo ws us to pass from an abstract resonance expansion to a uniform tw o-mode mo del in Section 5.6 . Corollary 5.20 (Oﬀ-equatorial leak age remains sup er–p olynomially small after prepro cessing) . Assume The or em 5.1 and The or em 5.4 , and ﬁx s ≥ 0 and n ∈ N 0 . L et ν > 0 b e such that Γ − ν is uniformly admissible on K (Deﬁnition 3.12 ). Then for every N ∈ N ther e exist ℓ 0 ( N ) ∈ N and C N > 0 such that for al l ℓ ≥ ℓ 0 ( N ) , al l ( M , a ) ∈ K , e ach sign ± , and al l t ∗ ≥ 1 , (5.29)      X sym ω j ∈ Res( P ): ℑ ω j > − ν, ω j / ∈ S 0 ≤ m ≤ n D m, ± ,ℓ U ( ± ,ℓ ) ω j ( t ∗ )      H s +1 ≤ C N ℓ − N ∥ ( f 0 , f 1 ) ∥ H s + N , and the same b ound holds after applying any b ounde d dete ctor O : H s +1 → C . Pr o of. F or eac h p ole ω j in the summation, the windo wed con tribution U ( ± ,ℓ ) ω j ( t ∗ ) is given by ( 5.18 ) , hence it is a ﬁnite sum of terms of the form e − iω j t ∗ t q − 1 − r ∗ χA ± ,h ℓ Π k ± Π [ q ] ω j χ ∂ r ω  e g ± ,ℓ b F ϑ  ( ω j ). The sectorization estimate in Theorem 5.4 gives ∥ χA ± ,h ℓ Π k ± Π [ q ] ω j χ ∥ H s − 1 → H s +1 = O ( ℓ − N ) for arbitrary N , uniformly for p oles outside the lab eled disks. Moreo ver, b y Lemma 4.5 and Lemma 5.6 , the deriv ativ es ∂ r ω  e g ± ,ℓ b F ϑ  ( ω j ) decay rapidly in ℜ ω j on the strip ℑ ω j ≥ − ν , up to p olynomial factors. Finally , exp onen tial energy deca y yields a parameter–uniform sp ectral gap aw ay from ω = 0 (see the discussion in the pro of of Theorem 5.23 ), whic h controls the p olynomial time factors uniformly on t ∗ ≥ 1 for all p oles in the strip. Using p olynomial resonance counting in ﬁxed-width strips [ 6 , § 3], the symmetric sum con v erges absolutely and the total contribution is O ( ℓ − N ) in H s +1 for every N . Applying a b ounded detector O preserves the same b ound. □ EQUA TORIAL KERR–DE SITTER RINGDOWN 47 5.5. Isolation of a ﬁxed ov ertone band by subtracting tw o con tour shifts. The interpolation window e g ± ,ℓ pro vides an algebraic wa y to suppress the low er ov ertones j ≤ n − 1. There is also a contour-theoretic mechanism: subtracting tw o resonance expansions corresp onding to tw o diﬀeren t con tour heigh ts isolates a vertical band of p oles. W e record this alternativ e since it clariﬁes the role of lay ering and is useful in extensions. Let n ≥ 1. Apply Lemma 5.15 with the index n to obtain a pole-free heigh t ν n > 0 separating the n th equatorial lay er from the ( n + 1)st. Apply Lemma 5.15 again with the index n − 1; after p ossibly shrinking K and increasing ℓ 0 , we obtain a second p ole-free height ν n − 1 > 0 separating the ( n − 1)st la y er from the n th, and we ma y assume (5.30) 0 < ν n − 1 < ν n . Then, for ℓ ≥ ℓ 0 , the n th equatorial pseudop oles satisfy (5.31) − ν n < ℑ ω ♯ n, ± ,ℓ ≤ − ν n − 1 , while all equatorial pseudop oles from o vertones j ≤ n − 1 lie strictly ab o ve ℑ ω = − ν n − 1 and all from o v ertones j ≥ n + 1 lie strictly b elo w ℑ ω = − ν n . Recall that in Theorem 5.14 the ν –dep enden t remainder is given by the line integral along ℑ ω = − ν : (5.32) R ( ν ) ± ,ℓ ( t ∗ ) := 1 2 π Z ℑ ω = − ν e − iω t ∗ e g ± ,ℓ ( ω ) χA ± ,h ℓ Π k ± R ( ω ) b F ϑ ( ω ) dω . Prop osition 5.21 (Band isolation b y con tour subtraction) . Assume the hyp otheses of The or em 5.14 . L et n ≥ 1 and let ν n − 1 < ν n b e as ab ove. Then for every t ∗ > 0 one has the exact identity (5.33) R ( ν n − 1 ) ± ,ℓ ( t ∗ ) − R ( ν n ) ± ,ℓ ( t ∗ ) = X sym ω j ∈ Res( P ): − ν n < ℑ ω j ≤− ν n − 1 U ( ± ,ℓ ) ω j ( t ∗ ) , wher e U ( ± ,ℓ ) ω j ( t ∗ ) ar e the windowe d p ole c ontributions deﬁne d in ( 5.18 ) . Pr o of. W e in tegrate the meromorphic H s − 1 → H s +1 –v alued function ω 7− → e − iω t ∗ e g ± ,ℓ ( ω ) χA ± ,h ℓ Π k ± R ( ω ) b F ϑ ( ω ) o v er the b oundary of the rectangle with horizon tal edges ℑ ω = − ν n − 1 and ℑ ω = − ν n and vertical edges ℜ ω = ± R . The vertical contributions v anish along a p ole-av oiding sequence R → ∞ b y Lemma 4.7 , using Sc hw artz deca y of b F ϑ (Lemma 4.5 ), polynomial growth of ∥ χR ( ω ) χ ∥ on compact ℑ ω –in terv als a wa y from p oles, and the p olynomial b ounds for e g ± ,ℓ from Lemma 5.6 . The residue theorem yields ( 5.33 ) ; the residue at each p ole is precisely the window ed con tribution ( 5.18 ) by Lemma 2.10 . □ Corollary 5.22 (Band-isolated one-mo de mo del in the equatorial sector) . Assume in addition The or ems 5.1 and 5.4 . L et n ≥ 1 and cho ose ν n − 1 < ν n as ab ove. Then for every N ∈ N ther e exists ℓ 0 ( N ) such that for al l ℓ ≥ ℓ 0 ( N ) , (5.34) O  R ( ν n − 1 ) ± ,ℓ ( t ∗ ) − R ( ν n ) ± ,ℓ ( t ∗ )  = a ± ,ℓ e − iω n, ± ,ℓ t ∗ + O ( ℓ − N ) , 48 RUILIANG LI uniformly for t ∗ ∈ [1 , ∞ ) and ( M , a ) ∈ K , wher e a ± ,ℓ ar e the c orr esp onding dete ctor amplitudes. Pr o of. By Prop osition 5.21 , the left-hand side is a symmetric sum ov er p ole con tributions from the strip − ν n < ℑ ω ≤ − ν n − 1 . By the construction of ν n − 1 and ν n , this strip con tains no equatorial p oles from ov ertones j ≤ n − 1 and no equatorial p oles from ov ertones j ≥ n + 1. The sectorization estimate ( 5.4 ) implies that, after applying the equatorial cutoﬀ and the detector O , all p oles outside the equatorial disks D n, ± ,ℓ con tribute O ( ℓ − N ). Finally , Theorem 5.1 yields that eac h D n, ± ,ℓ con tains exactly one p ole, namely ω n, ± ,ℓ , whic h is simple; th us the contribution is a single exp onen tial. □ 5.6. Tw o-mo de dominance theorem. W e no w combine the analytic- windo w ed resonance expansion (Theorem 5.14 ) with the p ole suppression Lemma 5.17 and the microlo cal b ounds from Prop osition 5.18 and Theo- rem 5.4 . Theorem 5.23 (Two-mode dominance in the equatorial high-frequency pac k age) . Fix Λ > 0 , a c omp act p ar ameter set K ⋐ P Λ ∩ {| a | ≤ a 0 } with a 0 smal l, and an overtone index n ∈ N 0 . L et ν > 0 b e chosen as in L emma 5.15 , and assume The or ems 5.1 and 5.4 as wel l as Assumption 4.3 . Then ther e exist ℓ 1 ∈ N and N 0 ∈ N such that for al l ℓ ≥ ℓ 1 , al l ( M , a ) ∈ K , and al l initial data ( f 0 , f 1 ) ∈ H s + N 0 supp orte d in K 0 , the r eﬁne d e quatorial signals ( 5.16 ) satisfy, for t ∗ ≥ 1 , (5.35) u (+ ,ℓ ) ( t ∗ ) = e − iω + ,ℓ t ∗ A + ,ℓ ( f 0 , f 1 ) + E + ,ℓ ( t ∗ ) , u ( − ,ℓ ) ( t ∗ ) = e − iω − ,ℓ t ∗ A − ,ℓ ( f 0 , f 1 ) + E − ,ℓ ( t ∗ ) , wher e: (1) ω ± ,ℓ = ω n,ℓ, ± ℓ ( M , a ) ar e the lab ele d QNMs in the e quatorial k = ± ℓ se ctors; (2) the amplitudes A ± ,ℓ ar e ﬁnite-r ank op er ators H s + N 0 → H s +1 ( X M ,a ) given by the simple-p ole r esidue formula (5.36) A ± ,ℓ ( f 0 , f 1 ) := i χA ± ,h ℓ Π k ± Π ω ± ,ℓ χ  e g ± ,ℓ ( ω ± ,ℓ ) b F ϑ ( ω ± ,ℓ )  , Π ω ± ,ℓ := Res ω = ω ± ,ℓ R ( ω ); (3) the err ors satisfy the fol lowing quantitative tail–plus–le akage b ound: for every m ∈ N and every N ′ ∈ N ther e exist an inte ger N m,N ′ ∈ N and a c onstant C s,m,N ′ > 0 such that for al l ( f 0 , f 1 ) ∈ H s + m + N m,N ′ supp orte d in K 0 , (5.37) ∥E ± ,ℓ ( t ∗ ) ∥ H s +1 ≤ C s,m,N ′  e − ν t ∗ (1+ t ∗ ) − m + ℓ − N ′  ∥ ( f 0 , f 1 ) ∥ H s + m + N m,N ′ , t ∗ ≥ 1 , with c onstants uniform in ( M , a ) ∈ K and ℓ ≥ ℓ 1 . In p articular, the c ombine d e quatorial high-fr e quency signal u eq ,ℓ ( t ∗ ) := u (+ ,ℓ ) ( t ∗ ) + u ( − ,ℓ ) ( t ∗ ) ob eys (5.38) u eq ,ℓ ( t ∗ ) = e − iω + ,ℓ t ∗ A + ,ℓ ( f 0 , f 1 ) + e − iω − ,ℓ t ∗ A − ,ℓ ( f 0 , f 1 ) + E ℓ ( t ∗ ) , EQUA TORIAL KERR–DE SITTER RINGDOWN 49 with ∥E ℓ ( t ∗ ) ∥ H s +1 b ounde d by the right-hand side of ( 5.37 ) after summing the ± estimates. Pr o of. Apply Theorem 5.14 . The resonan t sum in ( 5.17 ) is a sum of windo wed p ole con tributions U ( ± ,ℓ ) ω j ( t ∗ ) plus the tail R ( ν ) ± ,ℓ ( t ∗ ). W e split the p oles ω j with ℑ ω j > − ν in to three classes. (i) Poles in the lab ele d e quatorial disks with 0 ≤ j ≤ n − 1 . These are the lo w er ov ertones ω j, ± ,ℓ , j ∈ { 0 , 1 , . . . , n − 1 } . By Theorem 5.1 , these p oles are simple, hence ( 5.19 ) applies. Lemma 5.17 gives | e g ± ,ℓ ( ω j, ± ,ℓ ) | = O ( ℓ − N 1 ) for ev ery N 1 . Moreov er, by Lemma 4.5 and | ω j, ± ,ℓ | ∼ ℓ , for every N 2 one has ∥ b F ϑ ( ω j, ± ,ℓ ) ∥ H s − 1 ≤ C s,N 2 ℓ − N 2 ∥ ( f 0 , f 1 ) ∥ H s + m + N 2 . Finally , Prop osition 5.18 gives a p olynomial b ound on the microlo calized residue pro jectors. Com bining these b ounds and summing ov er 0 ≤ j ≤ n − 1 yields a total con tribution O ( ℓ − N ′ ) in H s +1 for any N ′ . (ii) Poles outside the lab ele d e quatorial disks. Let ω j ∈ Res ( P ) satisfy ℑ ω j > − ν and ω j / ∈ S 0 ≤ m ≤ n D m, ± ,ℓ . F or suc h p oles w e use the general windo w ed formula ( 5.18 ) . Each term in volv es a generalized Laurent coeﬃcient Π [ q ] ω j m ultiplied b y a deriv ativ e ∂ r ω ( e g ± ,ℓ b F ϑ )( ω j ) with 0 ≤ r ≤ q − 1. By the pro duct rule, Lemma 4.5 , and Lemma 5.6 , these deriv ativ es hav e rapid decay in ℜ ω j (uniformly on ℑ ω j ≥ − ν ) up to p olynomial factors, while Theorem 5.4 giv es ∥ χA ± ,h ℓ Π k ± Π [ q ] ω j χ ∥ H s − 1 → H s +1 = O ( ℓ − N ) for arbitrary N . Since w e w ork on a compact sub extremal slo w–rotation set K , exp onen tial lo cal energy deca y implies a uniform sp ectral gap: there exists ν gap > 0 (dep ending only on K ) such that the cutoﬀ resolven t has no p oles in {ℑ ω > − ν gap } except for the simple stationary p ole at ω = 0; see [ 5 , 2 , 11 ]. After azimuthal pro jection Π k ± with k ± = ± ℓ  = 0, the stationary p ole is annihilated. Consequen tly , ev ery p ole ω j in class (ii) satisﬁes ℑ ω j ≤ − ν gap , and p olynomial-in-time factors from p ossible higher-order p oles are uniformly con trolled on t ∗ ≥ 1 b y the elementary estimate (5.39) sup t ∗ ≥ 1 t p ∗ | e − iω j t ∗ | ≤ sup t ≥ 0 t p e − ν gap t ≤ C p ν − p gap , p ∈ N 0 , v alid uniformly for all p oles with ℑ ω j ≤ − ν gap . Using p olynomial resonance coun ting in strips [ 6 , § 3], we ma y sum ov er all p oles in class (ii) and obtain a total contribution O ( ℓ − N ′ ) in H s +1 for every N ′ . (iii) The tar get p ole ω ± ,ℓ = ω n, ± ,ℓ . By Theorem 5.1 the target p ole is simple, hence ( 5.19 ) applies. Lemma 5.17 gives e g ± ,ℓ ( ω ± ,ℓ ) = 1 + O ( ℓ − N ′ ), and the leading contribution is precisely the exp onential term in ( 5.35 ) with amplitude ( 5.36 ) . The multiplicativ e O ( ℓ − N ′ ) error is absorb ed in to the leak age term in ( 5.37 ). Finally , the tail estimate e − ν t ∗ (1 + t ∗ ) − m follo ws from ( 5.21 ) in Theo- rem 5.14 . Summing the ± statements gives ( 5.38 ). □ 50 RUILIANG LI R emark 5.24 (Bandpass alternative for o vertone selection) . The o vertone iso- lation in Theorem 5.23 is achiev ed by the en tire factor e g ± ,ℓ , whic h suppresses the low er ov ertones 0 , . . . , n − 1 while k eeping uniform control on Γ − ν . Corol- lary 5.22 provides an indep enden t mechanism: subtracting t w o contour shifts at heights ν n − 1 < ν n isolates the n th ov ertone lay er directly . The resulting band-isolated signal satisﬁes a one-mo de mo del with an O ( ℓ − N ) error and no exp onen tially decaying tail. All deterministic extraction statements in Section 6 apply v erbatim to either c hoice of ﬁltered time series. R emark 5.25 (Nontrivialit y and genericity of the amplitudes) . Theorem 5.23 giv es an explicit residue form ula for the mo de amplitudes. F or ﬁxed ℓ and ﬁxed mo de lab el, the set of detectors that annihilate the leading mo de is a prop er closed hyperplane in the detector space; see Lemma 5.26 . Thus for generic equatorially lo calized initial data and generic detectors the leading scalar amplitudes are nonzero. A quan titativ e lo w er b ound (a “detectability” condition) is discussed in Section 7.2 , and App endix B records the dual-state in terpretation in the simple-p ole regime. Lemma 5.26 (Generic detectors do not annihilate a ﬁxed mo de) . L et X b e a Banach sp ac e and 0  = u ∈ X . Then { Λ ∈ X ∗ : Λ( u ) = 0 } is a pr op er close d hyp erplane of the dual sp ac e X ∗ . In p articular, its c omple- ment is op en and dense in X ∗ . Pr o of. The map X ∗ ∋ Λ 7→ Λ( u ) ∈ C is contin uous and nonzero, hence its kernel is a closed co dimension-one subspace. Prop erness follows from Hahn–Banac h: there exists Λ 0 ∈ X ∗ with Λ 0 ( u )  = 0. □ 5.7. Implications for the in v erse problem. Theorem 5.23 provides the PDE-to-data input needed in Section 7 : after equatorial microlo calization, azim uthal selection k = ± ℓ , and analytic frequency lo calization at ω ♯ ± ,ℓ , the time-domain signal is a single exp onen tial e − iω ± ,ℓ t ∗ plus an explicitly con trolled tail. Combining the + ℓ and − ℓ sectors yields a tw o-exponential mo del whose frequencies are precisely the equatorial QNMs used in the in v erse theorem of the companion pap er [ 1 ]. 6. Deterministic frequency extraction st ability In this section we record a deterministic, quan titative stabilit y statemen t for extracting a complex frequency from a scalar ringdown signal. The p oin t here is not to introduce a new signal-pro cessing algorithm, but to isolate the p erturbation mec hanism used later in the PDE-to-parameter pip eline: after the microlo cal and analytic lo calizations of Section 5 , eac h equatorial pack age reduces to a single exp onen tially damp ed sinusoid plus a deterministic tail, and the remaining frequency recov ery step is an elementary consequence of time-shift in v ariance. Closely related ideas app ear in classical Pron y-type EQUA TORIAL KERR–DE SITTER RINGDOWN 51 metho ds and in the matrix-pencil/ESPRIT literature; see [ 25 , 28 , 29 ] for represen tativ e analyses from diﬀerent viewp oints. 6.1. F rom a ﬁeld to a scalar time series. T o av oid any clash with the energy spaces H s ﬁxed in Section 2 , we denote b y H obs the Hilb ert space in whic h our time-dep enden t ﬁeld is observed. In our application, H obs = H s +1 ( X M ,a ). Let O : H obs → C b e a b ounded linear “detector”. Giv en a time-dep enden t ﬁeld v ( t ∗ ) ∈ H obs , deﬁne the observ ed scalar signal y ( t ∗ ) := O ( v ( t ∗ )) . If v ( t ∗ ) = A e − iω t ∗ + r ( t ∗ ) with A ∈ H obs , ω ∈ C , then (6.1) y ( t ∗ ) = a e − iω t ∗ + ρ ( t ∗ ) , a := O ( A ) ∈ C , ρ ( t ∗ ) := O ( r ( t ∗ )) . All b ounds b elow will b e expressed in terms of ρ and a ; in particular, (6.2) | ρ ( t ∗ ) | ≤ ∥ O ∥ ∥ r ( t ∗ ) ∥ H obs . Th us, applying the results of this section with y = O ( u ( ± ,ℓ ) ) yields determin- istic frequency b ounds for ω ± ,ℓ as so on as the detector do es not annihilate the leading amplitude ( a  = 0). 6.2. A robust one-mo de estimator based on time-shift in v ariance. W e b egin with the one-mo de mo del (6.3) y ( t ) = a e − iω t + ρ ( t ) , t ∈ [ T 0 , T 0 + T ] , where ω ∈ C is unknown (with ℑ ω < 0 in ringdown applications), a ∈ C is an unkno wn nonzero amplitude, and ρ is an unkno wn deterministic error (tail + measurement noise). W e write t (instead of t ∗ ) for notational simplicit y . 6.2.1. Weighte d shift R ayleigh quotient. Fix a shift step ∆ > 0 with ∆ < T . Let w ∈ L ∞ ([ T 0 , T 0 + T − ∆]) b e a nonnegative w eigh t suc h that (6.4) 0 ≤ w ( t ) ≤ 1 , w ( t ) ≡ 1 on [ T 0 + ∆ , T 0 + T − 2∆] , so that b oth t and t + ∆ lie in the observ ation windo w on the essential supp ort of w . Deﬁne the weigh ted inner pro duct and norm ⟨ f , g ⟩ w := Z T 0 + T − ∆ T 0 w ( t ) f ( t ) g ( t ) dt, ∥ f ∥ w := ⟨ f , f ⟩ 1 / 2 w . Let ( S ∆ f )( t ) := f ( t + ∆) (deﬁned a.e. on [ T 0 , T 0 + T − ∆]). If ρ ≡ 0, then S ∆ y = z y with z := e − iω ∆ , so z is an eigenv alue of the shift. Motiv ated by this, deﬁne the shift R ayleigh quotient estimator (6.5) b z := ⟨ S ∆ y , y ⟩ w ⟨ y , y ⟩ w whenev er ⟨ y , y ⟩ w > 0 . The estimator b z is determined en tirely from the observ ed scalar time series y . W e next quantify its stabilit y under deterministic p erturbations. 52 RUILIANG LI R emark 6.1 (Relation to Pron y , matrix p encils, and ESPRIT) . In the noiseless mo del y ( t ) = Ae ω t , the shift op erator ( S ∆ y )( t ) := y ( t + ∆) acts on the one- dimensional Krylo v space span { y } b y m ultiplication with z = e ω ∆ . The estimator ( 6.5 ) is precisely the Rayleigh quotien t of S ∆ restricted to this subspace, and may b e viewed as the 1 × 1 case of the classical matrix- p encil and ESPRIT constructions for sums of exp onen tials; see [ 25 , 26 ] and the discussion in [ 29 ]. The no velt y here is not the signal-processing mec hanism, but the deterministic propagation of PDE remainder b ounds and measurement p erturbations into explicit frequency error estimates that are uniform in the high-frequency parameter ℓ . 6.2.2. A quantitative p erturb ation lemma. Let y 0 ( t ) := a e − iω t denote the pure exp onential comp onen t, so y = y 0 + ρ and S ∆ y 0 = z y 0 . The k ey p oin t is that the shift S ∆ ev aluates ρ on a shifte d part of the observ ation window, so the stability estimate m ust con trol b oth ρ and S ∆ ρ in the same weigh ted norm. Lemma 6.2 (Stability of the shift Rayleigh quotien t) . L et y = y 0 + ρ with y 0 ( t ) = ae − iω t , a  = 0 , and z = e − iω ∆ . Assume ℑ ω ≤ 0 so that | z | ≤ 1 . Deﬁne the r elative err or sizes (6.6) ε 0 := ∥ ρ ∥ w ∥ y 0 ∥ w , ε 1 := ∥ S ∆ ρ ∥ w ∥ y 0 ∥ w , ε := max { ε 0 , ε 1 } . If ε 0 ≤ 1 / 4 , then ⟨ y , y ⟩ w ≥ (1 − 2 ε 0 ) ∥ y 0 ∥ 2 w ≥ 1 2 ∥ y 0 ∥ 2 w and (6.7) | b z − z | ≤ ( ε 0 + ε 1 )(1 + ε 0 ) 1 − 2 ε 0 . In p articular, if ε ≤ 1 / 8 , then (6.8) | b z − z | ≤ 3 ε, | b z | ≤ | z | + 3 ε. Pr o of. W rite D := ⟨ y , y ⟩ w and N := ⟨ S ∆ y , y ⟩ w . W e estimate the denominator ﬁrst: D = ⟨ y 0 + ρ, y 0 + ρ ⟩ w = ∥ y 0 ∥ 2 w + 2 ℜ⟨ y 0 , ρ ⟩ w + ∥ ρ ∥ 2 w . By Cauch y–Sc h w arz, |⟨ y 0 , ρ ⟩ w | ≤ ∥ y 0 ∥ w ∥ ρ ∥ w = ε 0 ∥ y 0 ∥ 2 w , ∥ ρ ∥ 2 w = ε 2 0 ∥ y 0 ∥ 2 w , hence D ≥ ∥ y 0 ∥ 2 w − 2 ε 0 ∥ y 0 ∥ 2 w = (1 − 2 ε 0 ) ∥ y 0 ∥ 2 w . If ε 0 ≤ 1 / 4 this gives D ≥ 1 2 ∥ y 0 ∥ 2 w > 0, so b z is w ell-deﬁned. F or the numerator we use S ∆ y 0 = z y 0 : N = ⟨ S ∆ ( y 0 + ρ ) , y 0 + ρ ⟩ w = ⟨ z y 0 , y 0 ⟩ w + ⟨ z y 0 , ρ ⟩ w + ⟨ S ∆ ρ, y 0 ⟩ w + ⟨ S ∆ ρ, ρ ⟩ w . Subtracting z D and regrouping, N − z D = ⟨ S ∆ ρ, y 0 ⟩ w − z ⟨ ρ, y 0 ⟩ w + ⟨ S ∆ ρ, ρ ⟩ w − z ⟨ ρ, ρ ⟩ w = ⟨ S ∆ ρ − z ρ, y 0 ⟩ w + ⟨ S ∆ ρ − z ρ, ρ ⟩ w = ⟨ S ∆ ρ − z ρ, y 0 + ρ ⟩ w . EQUA TORIAL KERR–DE SITTER RINGDOWN 53 W e b ound the p erturbation term in the w eigh ted norm: ∥ S ∆ ρ − z ρ ∥ w ≤ ∥ S ∆ ρ ∥ w + | z | ∥ ρ ∥ w ≤ ∥ S ∆ ρ ∥ w + ∥ ρ ∥ w = ( ε 1 + ε 0 ) ∥ y 0 ∥ w , using | z | ≤ 1. Therefore, | N − z D | ≤ ∥ S ∆ ρ − z ρ ∥ w ∥ y 0 + ρ ∥ w ≤ ( ε 0 + ε 1 ) ∥ y 0 ∥ w  ∥ y 0 ∥ w + ∥ ρ ∥ w  = ( ε 0 + ε 1 )(1 + ε 0 ) ∥ y 0 ∥ 2 w . Dividing by the denominator low er b ound yields | b z − z | =     N D − z     = | N − z D | D ≤ ( ε 0 + ε 1 )(1 + ε 0 ) ∥ y 0 ∥ 2 w (1 − 2 ε 0 ) ∥ y 0 ∥ 2 w = ( ε 0 + ε 1 )(1 + ε 0 ) 1 − 2 ε 0 , whic h is ( 6.7 ). F or the crude b ound, assume ε ≤ 1 / 8. Then ε 0 ≤ 1 / 8 and ε 1 ≤ 1 / 8, so | b z − z | ≤ (2 ε )(1 + ε ) 1 − 2 ε ≤ 2 ε · (9 / 8) 3 / 4 = 3 ε, pro ving ( 6.8 ). The b ound on | b z | follows from | b z | ≤ | z | + | b z − z | . □ R emark 6.3 (A con v enien t suﬃcien t condition for ε 1 ≲ ε 0 ) . If the weigh t satisﬁes a shift-compatibilit y condition of the form w ( t − ∆) ≤ C w w ( t ) a.e. on [ T 0 + ∆ , T 0 + T − ∆] (after extending w b y 0 outside its domain), then a change of v ariables gives ∥ S ∆ f ∥ w ≤ C 1 / 2 w ∥ f ∥ w and hence ε 1 ≤ C 1 / 2 w ε 0 . In particular, for the un weigh ted c hoice w ≡ 1 one has C w = 1. W e do not imp ose such a condition in general; instead, w e keep ε 1 explicit since it is directly controlled by the same tail b ounds as ε 0 in our PDE application. 6.3. F rom shift eigen v alue to frequency: branch con trol. The estima- tor ( 6.5 ) returns b z ≈ z = e − iω ∆ . T o con vert this into an estimate for ω , we m ust choose a branc h of the complex logarithm. 6.3.1. L o garithm on a c ontr ol le d neighb orho o d. W e denote by Log the princi- pal branch on C \ ( −∞ , 0]. Lemma 6.4 (Lo cal Lipschitz con trol for the logarithm) . L et z ∈ C \ { 0 } and assume b z satisﬁes | b z − z | ≤ 1 2 | z | . Then b z is nonzer o and (6.9) | Log b z − Log z | ≤ 2 | z | | b z − z | , wher e Log is any holomorphic br anch of the lo garithm on a simply c onne cte d domain c ontaining the close d se gment b etwe en z and b z and avoiding 0 . Pr o of. Consider the straight-line path γ ( s ) = z + s ( b z − z ), s ∈ [0 , 1]. The assumption implies | γ ( s ) | ≥ | z | − | b z − z | ≥ | z | / 2 for all s , so γ a v oids 0. F or an y holomorphic branch of Log on a domain con taining γ , the fundamental theorem of calculus yields Log b z − Log z = Z 1 0 γ ′ ( s ) γ ( s ) ds = Z 1 0 b z − z γ ( s ) ds. T aking absolute v alues and using | γ ( s ) | ≥ | z | / 2 giv es ( 6.9 ). □ 54 RUILIANG LI 6.3.2. Br anch sele ction using a pseudop ole prior. The map ω 7→ z = e − iω ∆ is 2 π / ∆ p eriodic in ℜ ω . Thus, ev en if one determines z exactly , the frequency is only determined mo dulo 2 π / ∆ in real part. In our application this ambiguit y is remo v ed b y the semiclassical prior ω ♯ ± ,ℓ (the pseudop ole) and the analytic lo calization window of Section 5 . The next lemma mak es this branch selection explicit and reduces it to the principal logarithm of a ratio close to 1. Lemma 6.5 (Branch selection from a prior) . Fix ∆ > 0 and let ω ♯ ∈ C b e a prior. Set z ♯ := e − iω ♯ ∆ . Assume that z = e − iω ∆ and b z satisfy (6.10) | z − z ♯ | ≤ 1 4 | z ♯ | , | b z − z | ≤ 1 2 | z | . Then b z /z ♯ b elongs to the disk { ζ : | ζ − 1 | ≤ 5 / 8 } ⊂ C \ ( −∞ , 0] , so the princip al lo garithm Log is holomorphic at b z /z ♯ . Deﬁne (6.11) Log ω ♯ ( b z ) := Log  b z z ♯  − iω ♯ ∆ , b ω := i ∆ Log ω ♯ ( b z ) . Then Log ω ♯ is a holomorphic lo garithm br anch on a neighb orho o d of b z , satisﬁes exp(Log ω ♯ ( b z )) = b z and Log ω ♯ ( z ♯ ) = − iω ♯ ∆ , and mor e over (6.12) | b ω − ω | ≤ 2 ∆ | z | | b z − z | . Pr o of. F rom | z − z ♯ | ≤ 1 4 | z ♯ | we obtain | z | ≥ | z ♯ | − | z − z ♯ | ≥ 3 4 | z ♯ | . Hence | b z − z ♯ | ≤ | b z − z | + | z − z ♯ | ≤ 1 2 | z | + 1 4 | z ♯ | ≤ 1 2 · 3 4 | z ♯ | + 1 4 | z ♯ | = 5 8 | z ♯ | . Dividing by | z ♯ | giv es | b z /z ♯ − 1 | ≤ 5 / 8, proving that b z /z ♯ lies in the stated disk. This disk is con tained in the half-plane {ℜ ζ ≥ 3 / 8 } and therefore av oids ( −∞ , 0]; thus the principal Log is holomorphic there, and ( 6.11 ) deﬁnes a holomorphic function of b z . The identit y exp ( Log ω ♯ ( b z )) = b z follo ws from exp(Log( b z /z ♯ )) = b z /z ♯ and exp( − iω ♯ ∆) = z ♯ . F or the Lipschitz b ound, note that b ω − ω = i ∆  Log( b z /z ♯ ) − Log( z /z ♯ )  , b z z ♯ − z z ♯ = b z − z z ♯ . Moreo ver, | z /z ♯ | = | z | / | z ♯ | ≥ 3 / 4 and the hypothesis | b z − z | ≤ 1 2 | z | implies   b z z ♯ − z z ♯   ≤ 1 2 | z /z ♯ | . Applying Lemma 6.4 to z /z ♯ and b z /z ♯ giv es | Log( b z /z ♯ ) − Log( z /z ♯ ) | ≤ 2 | z /z ♯ |    b z − z z ♯    = 2 | z | | b z − z | . Multiplying by 1 / ∆ yields ( 6.12 ). □ 6.3.3. A deterministic one-mo de fr e quency stability the or em. W e now com- bine Lemma 6.2 with the logarithm con trol ab o ve. Deﬁne the frequency estimator (6.13) b ω := i ∆ Log( b z ) , b z as in ( 6.5 ) , EQUA TORIAL KERR–DE SITTER RINGDOWN 55 where Log is a chosen logarithm branc h. In applications w e will take Log = Log ω ♯ from Lemma 6.5 with ω ♯ = ω ♯ ± ,ℓ . Theorem 6.6 (Deterministic one-mo de frequency extraction) . L et y ( t ) = ae − iω t + ρ ( t ) on [ T 0 , T 0 + T ] with a  = 0 and ℑ ω ≤ 0 . Fix ∆ ∈ (0 , T ) and a weight w satisfying ( 6.4 ) , and deﬁne b z by ( 6.5 ) . L et ε 0 , ε 1 , ε b e as in ( 6.6 ) and set z = e − iω ∆ . Assume ε ≤ min { 1 / 8 , | z | / 20 } . Then b z  = 0 , | b z − z | ≤ 1 2 | z | , and for any holomorphic lo garithm br anch Log on a simply c onne cte d domain c ontaining the se gment b etwe en z and b z and avoiding 0 , the estimator ( 6.13 ) satisﬁes (6.14) | b ω − ω | ≤ 10 ∆ | z | ε = 10 ∆ e −ℑ ω ∆ max {∥ ρ ∥ w , ∥ S ∆ ρ ∥ w } ∥ ae − iω t ∥ w . Pr o of. By Lemma 6.2 , if ε ≤ 1 / 8 then | b z − z | ≤ 3 ε . The additional assumption ε ≤ | z | / 20 implies | b z − z | ≤ 3 | z | / 20 < | z | / 2. Lemma 6.4 then yields | Log b z − Log z | ≤ 2 | z | | b z − z | ≤ 6 | z | ε ≤ 10 | z | ε. Finally , | b ω − ω | = 1 ∆ | Log b z − Log z | ≤ 10 ∆ | z | ε, whic h is ( 6.14 ). Since | z | = | e − iω ∆ | = e ℑ ω ∆ , the last iden tit y follo ws. □ R emark 6.7 (Cho osing the shift step ∆) . The b ound ( 6.14 ) mak es the dep en- dence on the shift step explicit. T aking ∆ to o small ampliﬁes the prefactor ∆ − 1 and provides little phase separation, so that the ratio b z b ecomes sensi- tiv e to the residual term. T aking ∆ to o large reduces the a v ailable ﬁtting in terv al [ T 0 , T 0 + T − ∆] and introduces the factor | z | − 1 = e −ℑ ω ∆ , which can signiﬁcan tly magnify errors when the damping is strong. A practical c hoice is to take ∆ as a ﬁxed fraction of the windo w length T while keeping T − ∆ comfortably larger than the exp ected damping time, s o that b oth the signal energy and the phase separation remain visible. No optimal choice is claimed here; the p oin t is that the stabilit y constan t is explicit and can b e balanced against the observ ation constrain ts. R emark 6.8 (Rayleigh-quotien t viewp oin t and relation to Prony-t yp e meth- o ds) . The estimator ( 6.5 ) can b e view ed as a weigh ted shift R ayleigh quotient . Indeed, if we equip L 2 ([ T 0 , T 0 + T − ∆]) with the w eigh ted inner pro duct ⟨· , ·⟩ w from ( 6.5 ), then b z = ⟨ S ∆ y , y ⟩ w ⟨ y , y ⟩ w is the Ra yleigh quotien t of the shift op erator S ∆ ev aluated at the test v e ctor y . In the exact one-mo de case ρ ≡ 0, y is an eigenfunction of S ∆ and the quotien t returns the corresp onding eigenv alue z = e − iω ∆ . F or m ulti-exp onen tial signals y ( t ) = P J j =1 a j e − iω j t , higher-dimensional v ariants arise by restricting S ∆ to span { y , S ∆ y , . . . , S J − 1 ∆ y } , leading to generalized eigenv alue problems 56 RUILIANG LI that underlie Prony/matrix-pencil/ESPRIT constructions; see, for instance, [ 25 , 28 , 29 ]. The tw o-exp onen tial conditioning b ounds in Prop osition F.1 quan tify the exp ected loss of stability as frequencies approach each other (a regime relev an t near coalescence and exceptional p oin ts). R emark 6.9 (Interpretation of the branch condition) . Lemma 6.5 sho ws that once a prior ω ♯ lo calizes z = e − iω ∆ to a small neighborho o d of z ♯ = e − iω ♯ ∆ , the correct logarithm branch is selected b y taking the principal logarithm of b z /z ♯ . In our Kerr–de Sitter application, the pseudop ole ω ♯ ± ,ℓ and the analytic windo w of Section 5 provide such lo calization, with an error O ( ℓ −∞ ). 6.4. Applying the one-mo de theorem to the equatorial ringdown pac k age. W e now sp ecialize to the PDE output of Theorem 5.23 . Fix ℓ ≥ ℓ 1 and parameters ( M , a ) ∈ K . F or deﬁniteness, consider the + sector (the − sector is identical). Let v ( t ) := u (+ ,ℓ ) ( t ) b e the H obs -v alued signal (here H obs = H s +1 ( X M ,a )), so that b y ( 5.35 ): v ( t ) = e − iω + ,ℓ t A + ,ℓ + E + ,ℓ ( t ) , t ≥ 1 , with A + ,ℓ := A + ,ℓ ( f 0 , f 1 ) ∈ H s +1 ( X M ,a ) and ω + ,ℓ = ω n,ℓ,ℓ ( M , a ). Applying a detector O gives a scalar signal y ( t ) = O ( v ( t )) of the form (6.15) y ( t ) = a + ,ℓ e − iω + ,ℓ t + ρ + ,ℓ ( t ) , a + ,ℓ := O ( A + ,ℓ ) , ρ + ,ℓ ( t ) := O ( E + ,ℓ ( t )) . By ( 6.2 ) and ( 5.37 ), for every m ∈ N and every N ′ ∈ N , (6.16) | ρ + ,ℓ ( t ) | ≤ C s,m,N ′ ∥ O ∥  e − ν t (1 + t ) − m + ℓ − N ′  ∥ ( f 0 , f 1 ) ∥ H s + m + N m,N ′ , t ≥ 1 . In particular, on an y window [ T 0 , T 0 + T ] with T 0 ≥ 1, w e ha v e (6.17) ∥ ρ + ,ℓ ∥ L 2 ([ T 0 ,T 0 + T ]) ≤ C s,m,N ′ ∥ O ∥  e − ν T 0 (1 + T 0 ) − m + ℓ − N ′  √ T × ∥ ( f 0 , f 1 ) ∥ H s + m + N m,N ′ . T o apply Theorem 6.6 , w e compare the residual energies ∥ ρ + ,ℓ ∥ w and ∥ S ∆ ρ + ,ℓ ∥ w to the dominant-mode energy ∥ a + ,ℓ e − iω + ,ℓ t ∥ w . A conv enien t lo w er b ound for ∥ ae − iω t ∥ w is obtained by integrating | e − iω t | 2 = e 2 ℑ ω t . Lemma 6.10 (Lo w er b ound for the dominant-mode energy) . L et y 0 ( t ) = ae − iω t with a  = 0 and ℑ ω < 0 . L et w satisfy ( 6.4 ) . Then (6.18) ∥ y 0 ∥ 2 w ≥ | a | 2 Z T 0 + T − 2∆ T 0 +∆ e 2 ℑ ω t dt = | a | 2 e 2 ℑ ω ( T 0 +∆) − e 2 ℑ ω ( T 0 + T − 2∆) − 2 ℑ ω . In p articular, (6.19) ∥ y 0 ∥ w ≥ | a | e ℑ ω ( T 0 + T − 2∆) √ T − 3∆ , ( T > 3∆) . EQUA TORIAL KERR–DE SITTER RINGDOWN 57 Pr o of. Since w ≡ 1 on [ T 0 + ∆ , T 0 + T − 2∆], we hav e ∥ y 0 ∥ 2 w = Z T 0 + T − ∆ T 0 w ( t ) | a | 2 e 2 ℑ ω t dt ≥ | a | 2 Z T 0 + T − 2∆ T 0 +∆ e 2 ℑ ω t dt. whic h gives ( 6.18 ) . The crude b ound ( 6.19 ) follo ws from the inequalit y R I e 2 ℑ ω t dt ≥ | I | · min I e 2 ℑ ω t . □ Lemma 6.11 (Con trolling the shifted residual by the L 2 tail) . L et ρ ∈ L 2 ([ T 0 , T 0 + T ]) and let w satisfy ( 6.4 ) . Then (6.20) max {∥ ρ ∥ w , ∥ S ∆ ρ ∥ w } ≤ ∥ ρ ∥ L 2 ([ T 0 ,T 0 + T ]) . Pr o of. Since 0 ≤ w ≤ 1 and the w -in tegration domain is contained in [ T 0 , T 0 + T ], ∥ ρ ∥ 2 w = Z T 0 + T − ∆ T 0 w ( t ) | ρ ( t ) | 2 dt ≤ Z T 0 + T T 0 | ρ ( t ) | 2 dt. Similarly , ∥ S ∆ ρ ∥ 2 w = Z T 0 + T − ∆ T 0 w ( t ) | ρ ( t + ∆) | 2 dt ≤ Z T 0 + T − ∆ T 0 | ρ ( t + ∆) | 2 dt = Z T 0 + T T 0 +∆ | ρ ( s ) | 2 ds ≤ Z T 0 + T T 0 | ρ ( s ) | 2 ds. T aking square ro ots yields ( 6.20 ). □ Com bining Lemmas 6.10 – 6.11 with ( 6.17 ) yields an explicit relative error b ound. W e summarize the result in a corollary . Corollary 6.12 (Equatorial QNM extraction error b ound) . Fix ℓ ≥ ℓ 1 and let y ( t ) b e the observe d + –e quatorial signal ( 6.15 ) on [ T 0 , T 0 + T ] with T 0 ≥ 1 and T > 3∆ . Assume a + ,ℓ = O ( A + ,ℓ )  = 0 an d ℑ ω + ,ℓ ≤ − c < 0 on K . Deﬁne (6.21) ε + ,ℓ := max {∥ ρ + ,ℓ ∥ w , ∥ S ∆ ρ + ,ℓ ∥ w } ∥ a + ,ℓ e − iω + ,ℓ t ∥ w . Then for every m ∈ N ther e exist a c onstant C s,m > 0 and an inte ger N = N ( s, m ) such that (6.22) ε + ,ℓ ≤ C s,m ∥ O ∥ | a + ,ℓ | e − ( ν + ℑ ω + ,ℓ ) T 0 (1 + T 0 ) − m √ T √ T − 3∆ ∥ ( f 0 , f 1 ) ∥ H s + m + N , 58 RUILIANG LI and if ε + ,ℓ ≤ min { 1 / 8 , | z + ,ℓ | / 20 } (with z + ,ℓ = e − iω + ,ℓ ∆ ), the estimator ( 6.13 ) satisﬁes (6.23) | b ω + ,ℓ − ω + ,ℓ | ≤ 10 ∆ | z + ,ℓ | ε + ,ℓ . A n identic al statement holds for the − se ctor. Pr o of. Lemma 6.11 giv es max {∥ ρ + ,ℓ ∥ w , ∥ S ∆ ρ + ,ℓ ∥ w } ≤ ∥ ρ + ,ℓ ∥ L 2 ([ T 0 ,T 0 + T ]) . Using ( 6.17 ) w e obtain an explicit upp er b ound for the numerator in ( 6.21 ) . F or the denominator w e use Lemma 6.10 and the uniform bound ℑ ω + ,ℓ ≤ − c < 0 on K , which yields the low er b ound ( 6.19 ) with constants absorb ed into C s,m . This gives ( 6.22 ) . Finally , apply Theorem 6.6 to obtain ( 6.23 ). □ R emark 6.13 (Interpretation) . The factor e − ( ν + ℑ ω ) T 0 in ( 6.22 ) reﬂects a basic signal-to-tail comp etition: the dominan t mo de deca ys lik e e ℑ ω t , while the tail deca ys like e − ν t . Since ν is chosen b elow the dominan t mo de (i.e. ν > |ℑ ω | b y construction in Section 5 ), the relativ e tail size improv es exp onen tially as T 0 increases, up to the p oin t where the absolute signal b ecomes to o small for measurement. Our b ounds make this tradeoﬀ explicit and deterministic. 6.5. Optional: extracting tw o frequencies from a t w o-exp onen tial scalar signal. The preceding analysis is the most eﬃcien t route for our Kerr–de Sitter application b ecause the PDE lo calizations isolate each mo de separately . F or completeness, we record a deterministic stability statement for the two-exp onential mo del, whic h ma y b e useful if one only observ es the com bined signal. This is a v ery small instance of Prony-t ype reconstruction; see [ 28 , 25 , 29 ] for broader frameworks and sharp discussions of conditioning. Consider (6.24) y ( t ) = a 1 e − iω 1 t + a 2 e − iω 2 t + ρ ( t ) , t ∈ [ T 0 , T 0 + T ] , with ω 1  = ω 2 and a 1 a 2  = 0. Fix ∆ > 0 with 2∆ < T and deﬁne discrete samples y j := y ( T 0 + j ∆) , j = 0 , 1 , 2 , 3 . In the noiseless case ρ ≡ 0, the sequence ( y j ) satisﬁes the order-2 recurrence (6.25) y j +2 = s 1 y j +1 − s 2 y j , s 1 = z 1 + z 2 , s 2 = z 1 z 2 , z k := e − iω k ∆ . Solving for ( s 1 , s 2 ) from j = 0 , 1 yields z 1 , z 2 as the ro ots of λ 2 − s 1 λ + s 2 = 0. The next lemma quantiﬁ es stabilit y under deterministic p erturbations of the four samples. Lemma 6.14 (F our-sample Pron y stability (lo cal)) . Assume ρ ≡ 0 in ( 6.24 ) and let y j b e the noiseless samples y j = a 1 z j 1 + a 2 z j 2 with a 1 a 2  = 0 and z 1  = z 2 . Set ∆ 0 := y 0 y 2 − y 2 1 = a 1 a 2 ( z 1 − z 2 ) 2  = 0 , R := max {| z 1 | , | z 2 |} . EQUA TORIAL KERR–DE SITTER RINGDOWN 59 Supp ose p erturb e d samples e y j = y j + e j ar e given, with | e j | ≤ η . L et ( e s 1 , e s 2 ) b e obtaine d by solving the p erturb e d line ar system  e y 1 − e y 0 e y 2 − e y 1   e s 1 e s 2  =  e y 2 e y 3  , and let e z 1 , e z 2 b e the r o ots of λ 2 − e s 1 λ + e s 2 = 0 . Assume in addition that the noise level is smal l in the sense (6.26) η ≤ c 0 | a 1 a 2 | | z 1 − z 2 | 4 , for a suﬃciently smal l absolute c onstant c 0 > 0 . Then, after lab eling ( e z 1 , e z 2 ) to match ( z 1 , z 2 ) , one has (6.27) | e z k − z k | ≤ C ( R, a 1 , a 2 ) η | a 1 a 2 | | z 1 − z 2 | 3 , k = 1 , 2 . Conse quently, cho osing a lo garithm br anch as in L emma 6.5 , one obtains (6.28) | e ω k − ω k | ≤ C ( R, a 1 , a 2 ) ∆ | z k | η | a 1 a 2 | | z 1 − z 2 | 3 . Pr o of. Step 1: stability of the r e curr enc e c o eﬃcients. F or the noiseless data, the linear system for ( s 1 , s 2 ) has determinan t ∆ 0 = y 0 y 2 − y 2 1  = 0, and Cramer’s rule gives (6.29) s 1 = y 0 y 3 − y 1 y 2 ∆ 0 , s 2 = y 1 y 3 − y 2 2 ∆ 0 . F or the p erturb ed data, the matrix determinant is e ∆ 0 := e y 0 e y 2 − e y 2 1 and e s 1 = e y 0 e y 3 − e y 1 e y 2 e ∆ 0 , e s 2 = e y 1 e y 3 − e y 2 2 e ∆ 0 . W rite δ s ℓ := e s ℓ − s ℓ and δ ∆ 0 := e ∆ 0 − ∆ 0 . A direct expansion yields (6.30) | δ ∆ 0 | ≤ | e 0 | | y 2 | + | y 0 | | e 2 | +2 | y 1 | | e 1 | + | e 0 | | e 2 | + | e 1 | 2 ≤ η  | y 0 | +2 | y 1 | + | y 2 |  +2 η 2 . Similarly , expanding the numerators sho ws (6.31)   ( e y 0 e y 3 − e y 1 e y 2 ) − ( y 0 y 3 − y 1 y 2 )   +   ( e y 1 e y 3 − e y 2 2 ) − ( y 1 y 3 − y 2 2 )   ≤ C y η , where C y dep ends only on | y 0 | , . . . , | y 3 | . Since y j = a 1 z j 1 + a 2 z j 2 , we hav e the crude b ound | y j | ≤ | a 1 | | z 1 | j + | a 2 | | z 2 | j ≤ ( | a 1 | + | a 2 | ) R j ≤ ( | a 1 | + | a 2 | ) max { 1 , R 3 } , j = 0 , 1 , 2 , 3 . (In the ringdown regime ℑ ω k ≤ 0, one has | z k | ≤ 1 and hence R ≤ 1, so the righ t-hand side reduces to | a 1 | + | a 2 | .) Thus C y can b e b ounded in terms of max { 1 , R 3 } and | a 1 | + | a 2 | . Under the smallness assumption ( 6.26 ) (with c 0 c hosen suﬃciently small dep ending on R , a 1 , a 2 through C y ), ( 6.30 ) implies | δ ∆ 0 | ≤ 1 2 | ∆ 0 | , hence (6.32) | e ∆ 0 | ≥ 1 2 | ∆ 0 | . 60 RUILIANG LI Com bining ( 6.32 ) with ( 6.31 ) yields (6.33) | δ s 1 | + | δ s 2 | ≤ C 1 ( R, a 1 , a 2 ) η | ∆ 0 | = C 1 ( R, a 1 , a 2 ) η | a 1 a 2 | | z 1 − z 2 | 2 . Step 2: stability of the r o ots. Let p ( λ ) = λ 2 − s 1 λ + s 2 = ( λ − z 1 )( λ − z 2 ) and e p ( λ ) = λ 2 − e s 1 λ + e s 2 . Set δ p ( λ ) := e p ( λ ) − p ( λ ) = − δ s 1 λ + δ s 2 . Cho ose r := | z 1 − z 2 | / 4. By ( 6.33 ) and the smallness assumption ( 6.26 ) , we ma y ensure (by shrinking c 0 further if needed) that | δ s 1 | + | δ s 2 | ≤ c | z 1 − z 2 | 2 with c > 0 small enough so that (6.34) | δ p ( λ ) | < | p ( λ ) | for all λ with | λ − z k | = r, k = 1 , 2 . Indeed, on | λ − z 1 | = r w e hav e | λ − z 2 | ≥ | z 1 − z 2 | − r = 3 | z 1 − z 2 | / 4, hence | p ( λ ) | = | λ − z 1 | | λ − z 2 | ≥ r · 3 4 | z 1 − z 2 | = 3 16 | z 1 − z 2 | 2 , while | δ p ( λ ) | ≤ | δ s 1 | | λ | + | δ s 2 | and | λ | ≤ | z 1 | + r ≤ | z 1 | + | z 1 − z 2 | / 4. The corresp onding estimate for the circle around z 2 is identical. By Rouch ´ e’s theorem, ( 6.34 ) implies that e p has exactly one zero in each disk D k := {| λ − z k | < r } . Label these zeros as e z k ∈ D k . No w use the iden tit y e p ( e z k ) = 0, i.e. p ( e z k ) = δ s 1 e z k − δ s 2 . Since p ( e z k ) = ( e z k − z k )( e z k − z 3 − k ) and e z k ∈ D k , we hav e | e z k − z 3 − k | ≥ | z 1 − z 2 | − r = 3 | z 1 − z 2 | / 4. Therefore, | e z k − z k | ≤ | δ s 1 | | e z k | + | δ s 2 | | e z k − z 3 − k | ≤ 4 3 | z 1 − z 2 |  | δ s 1 | ( | z k | + r ) + | δ s 2 |  ≤ C 2 ( R ) | δ s 1 | + | δ s 2 | | z 1 − z 2 | . Com bining this with ( 6.33 ) yields ( 6.27 ). Finally , ( 6.28 ) follo ws from Lemma 6.4 (or Lemma 6.5 if a prior is a v ailable) applied to z k and e z k . □ R emark 6.15 (Tw o-exp onen tial extraction and the main pip eline) . Lemma 6.14 sho ws that recov ering b oth no des from only four consecutive samples suf- fers a sev ere separation-dep enden t loss: even in the lo cal regime where the reconstruction map is Lipsc hitz, the error constan t scales at least like | a 1 a 2 | − 1 | z 1 − z 2 | − 3 . The factor | z 1 − z 2 | − 2 comes from in v erting the 2 × 2 Han- k el matrix (whose determinant is a 1 a 2 ( z 1 − z 2 ) 2 ), and an additional | z 1 − z 2 | − 1 comes from conv erting p erturb ed symmetric functions into p erturb ed ro ots. In Kerr–de Sitter, the equatorial splitting satisﬁes |ℜ ω + ,ℓ − ℜ ω − ,ℓ | ∼ | a | , so | z 1 − z 2 | ∼ ∆ | a | for small ∆ and the constant can blo w up like (∆ | a | ) − 3 when a is small. The PDE lo calization of Section 5 a voids this loss by separating the tw o mo des b efor e extraction, reducing the problem to tw o stable one-mo de extractions. EQUA TORIAL KERR–DE SITTER RINGDOWN 61 6.6. Output for the in v erse problem. Combining Section 5 with the one-mo de stabilit y Theorem 6.6 yields: • a deterministic pro cedure to estimate ω n,ℓ, ± ℓ ( M , a ) from a time- domain signal after equatorial microlo calization and azimuthal selec- tion; • an explicit b ound for the frequency error in terms of the tail size (con trolled b y the shifted-con tour resolv en t) and an y additional mea- suremen t p erturbation. In the next section we feed these frequency error b ounds into the parameter reco v ery stabilit y estimate from the ﬁrst paper to obtain an explicit parameter- bias b ound for ringdown-based in v ersion. 7. Applica tion: p arameter bias bound f or ringdown-based inversion This section closes the deterministic pip eline time-domain ringdown data = ⇒ QNM frequencies = ⇒ ( M , a ) , and pro duces an explicit p ar ameter bias b ound in terms of: (i) the PDE tail remainder on a shifted contour (Sections 3 – 5 ), (ii) deterministic mea- suremen t p erturbations in the time series, and (iii) the conditioning of the frequency-extraction functional (Section 6 ). The estimates b elo w are entirely deterministic and k eep track of the parameters that go vern conditioning: start time T 0 , windo w length T , shift step ∆, and (crucially) nondegeneracy of the chosen detector for the extracted mo des. 7.1. The in v ersion map from equatorial QNMs (input from the companion pap er). Fix Λ > 0, a compact parameter set K ⋐ P Λ ∩ {| a | ≤ a 0 } , with a 0 > 0 suﬃcien tly small, and ﬁx an ov ertone index (7.1) n ∈ N 0 . F or ℓ ≫ 1 let ω ± ,ℓ ( M , a ) := ω n,ℓ, ± ℓ ( M , a ) denote the lab eled simple equatorial QNMs (Section 5 ). Deﬁne the normalized real observ ables (7.2) U ℓ ( M , a ) := ℜ ω + ,ℓ ( M , a ) + ω − ,ℓ ( M , a ) 2 ℓ , V ℓ ( M , a ) := ℜ ω + ,ℓ ( M , a ) − ω − ,ℓ ( M , a ) 2 ℓ . and the e quatorial QNM data map (7.3) G ℓ ( M , a ) := ( U ℓ ( M , a ) , V ℓ ( M , a )) ∈ R 2 . The follo wing theorem is the frequency-to-parameter stability input pro ved in the companion pap er (w e restate it for conv enience). 62 RUILIANG LI Theorem 7.1 (Lo cal in v ersion from equatorial QNMs; stabilit y) . Ther e exist ℓ 0 ∈ N and c onstants c ∗ , C ∗ > 0 such that for every ℓ ≥ ℓ 0 : (1) G ℓ : K → R 2 is r e al-analytic and lo c al ly invertible at every p oint of K , with Jac obian uniformly b ounde d away fr om 0 : (7.4) inf ( M ,a ) ∈K | det D G ℓ ( M , a ) | ≥ c ∗ . (2) (Quantitative lo c al stability.) F or al l ( M , a ) , ( M ′ , a ′ ) ∈ K suﬃciently close, (7.5) | ( M , a ) − ( M ′ , a ′ ) | ≤ C ∗ |G ℓ ( M , a ) − G ℓ ( M ′ , a ′ ) | . Mor e over, in terms of the fr e quencies, (7.6) | ( M , a ) − ( M ′ , a ′ ) | ≤ C ∗ ℓ  |ℜ ( ω av − ω ′ av ) | + |ℜ ( ω dif − ω ′ dif ) |  , wher e ω av = 1 2 ( ω + ,ℓ + ω − ,ℓ ) and ω dif = 1 2 ( ω + ,ℓ − ω − ,ℓ ) . R emark 7.2 (A condition num b er viewp oin t) . It is useful to interpret ( 7.5 ) – ( 7.6 ) as an inv erse-problem condition n um b er statement. Deﬁne the r aw- fr e quency map F ℓ ( M , a ) :=  ℜ ω av ( M , a ) , ℜ ω dif ( M , a )  ∈ R 2 . Then G ℓ = F ℓ /ℓ b y ( 7.2 ) . Diﬀerentiating giv es D F ℓ = ℓ D G ℓ and consequently   ( D F ℓ ) − 1   = 1 ℓ   ( D G ℓ ) − 1   . Th us the lo cal in v erse from unsc ale d frequencies carries an additional factor 1 /ℓ compared to the inv erse of G ℓ . In particular, the constant C ∗ /ℓ in ( 7.6 ) is precisely the (uniform) Lipsc hitz constan t of the lo cal inv erse when the data are taken to b e ( ℜ ω av , ℜ ω dif ). This is the source of the high-frequency conditioning gain in Theorem 7.15 b elo w. 7.2. Time-domain observ ation mo del and detectability of the sep- arated mo des. Fix ℓ ≥ ℓ 1 (as in Theorem 5.23 ) and let ( M , a ) ∈ K b e the true parameters. Let u b e the (scalar) w av e solution with initial data ( f 0 , f 1 ) ∈ H s + N . Equatorial mo de-separated signals. W e work with the reﬁned equatorial windo w ed signals u ( ± ,ℓ ) ( t ) constructed in ( 5.16 ): u (+ ,ℓ ) ( t ) = χA + ,h ℓ Π + ℓ W (+ ,ℓ ) u ( t ) , u ( − ,ℓ ) ( t ) = χA − ,h ℓ Π − ℓ W ( − ,ℓ ) u ( t ) , where W ( ± ,ℓ ) is the in v erse Laplace integral with the modiﬁed analytic weigh t e g ± ,ℓ . By Theorem 5.23 , for t ≥ 1, (7.7) u ( ± ,ℓ ) ( t ) = e − iω ± ,ℓ t A ± ,ℓ + E ± ,ℓ ( t ) , ∥ E ± ,ℓ ( t ) ∥ H s +1 ≤ C s,m,N ′  e − ν t (1 + t ) − m + ℓ − N ′  ∥ ( f 0 , f 1 ) ∥ H s + m + N m,N ′ . EQUA TORIAL KERR–DE SITTER RINGDOWN 63 R emark 7.3 (Choice of the leak age exp onent) . The term ℓ − N ′ in ( 7.7 ) enco des semiclassically small leak age from other p oles in the same equatorial pack age; it is uniform on ﬁxed compact parameter sets. F or the later stabilit y estimates w e ﬁx a large N ′ once and for all, and tak e m as large as needed. Scalar detector and ideal signals. Let O : H s +1 ( X M ,a ) → C b e a b ounded linear functional (Section 6.1 ). Deﬁne the ide al mo de-separated scalar signals y ± ( t ) := O  u ( ± ,ℓ ) ( t )  . Then by ( 7.7 ), (7.8) y ± ( t ) = a ± ,ℓ e − iω ± ,ℓ t + ρ ± ,ℓ ( t ) , a ± ,ℓ := O ( A ± ,ℓ ) , ρ ± ,ℓ ( t ) := O ( E ± ,ℓ ( t )) . The tail terms satisfy , for every m , (7.9) | ρ ± ,ℓ ( t ) | ≤ ∥ O ∥ ∥ E ± ,ℓ ( t ) ∥ H s +1 ≤ C s,m,N ′ ∥ O ∥  e − ν t (1 + t ) − m + ℓ − N ′  ∥ ( f 0 , f 1 ) ∥ H s + m + N m,N ′ , t ≥ 1 . Assumption 7.4 (Mode detectabilit y on the c hosen window) . On the observation window under c onsider ation, we assume that the dete ctor do es not annihilate the extr acte d dominant c omp onents: (7.10) a + ,ℓ  = 0 , a − ,ℓ  = 0 . Equivalently, for any admissible weight w that is not a.e. zer o on [ T 0 , T 0 + T − ∆] , the r efer enc e signals y 0 , ± ( t ) := a ± ,ℓ e − iω ± ,ℓ t satisfy ∥ y 0 , ± ∥ w > 0 . A l l fr e quency and p ar ameter estimates b elow ar e c onditional on ( 7.10 ) and on a quantitative signal-to-r esidual smal lness c ondition, expr esse d thr ough the r elative r esidual sizes ε ± in ( 7.14 ) . R emark 7.5 (Dual-state interpretation of detectability) . Assumption 7.4 admits a clean op erator-theoretic in terpretation in terms of left and right resonan t states. F or a simple p ole ω = ω 0 of the meromorphic family R ( ω ) = P ( ω ) − 1 , App endix B (recall also the general framework of Keldysh expansions [ 31 , 30 ]) shows that the residue pro jector has rank one and can b e written, after a c hoice of normalization, as (7.11) Π ω 0 f = ⟨ f , v ω 0 ⟩ ⟨ ∂ ω P ( ω 0 ) u ω 0 , v ω 0 ⟩ u ω 0 , where u ω 0 is a (right) resonant state and v ω 0 is the corresp onding dual (left) resonan t state. Consequently , for the equatorial poles ω ± ,ℓ app earing in Theorem 5.23 , the residue contributions tak e the form A ± ,ℓ = χ A ± ,h ℓ Π k ± Π ω ± ,ℓ b F ϑ ( ω ± ,ℓ ) =  ⟨ b F ϑ ( ω ± ,ℓ ) , v ± ,ℓ ⟩ ⟨ ∂ ω P ( ω ± ,ℓ ) u ± ,ℓ , v ± ,ℓ ⟩  χ A ± ,h ℓ Π k ± u ± ,ℓ . 64 RUILIANG LI Applying the observ ation functional O therefore yields a ± ,ℓ = ⟨ b F ϑ ( ω ± ,ℓ ) , v ± ,ℓ ⟩ ⟨ ∂ ω P ( ω ± ,ℓ ) u ± ,ℓ , v ± ,ℓ ⟩ × O  χ A ± ,h ℓ Π k ± u ± ,ℓ  . (7.12) The dete ctability condition a ± ,ℓ  = 0 th us fails precisely when either the c hosen observ able annihilates the pro jected right resonan t state, or when the data are orthogonal (in the dual pairing) to the left resonan t state. In particular, for ﬁxed ( M , a ) and ℓ , the set of non-detectable data is a co dimension-one subspace, and nonnormality enters through the normalization denominator in ( 7.12 ), which is the standard excitation-factor mechanism. Lemma 7.6 (Generic initial data do not annihilate a ﬁxed extracted mo de) . Fix ( M , a ) ∈ K , ℓ ≥ ℓ 1 , and a b ounde d dete ctor O : H s +1 → C . F or e ach sign ± , the map H s + N ∋ ( f 0 , f 1 ) 7− → a ± ,ℓ = O  A ± ,ℓ ( f 0 , f 1 )  ∈ C is a c ontinuous line ar functional. If it is not identic al ly zer o, then its kernel is a pr op er close d hyp erplane, and its c omplement is op en and dense. Pr o of. By ( 7.7 ) , ( f 0 , f 1 ) 7→ A ± ,ℓ ( f 0 , f 1 ) ∈ H s +1 is linear. Its contin uit y in the H s + N top ology follo ws from the construction in Theorem 5.23 : A ± ,ℓ is obtained by comp osing b ounded op erators on the data with the ﬁnite-rank residue pro jector at the simple pole ω ± ,ℓ . Comp osing with the bounded functional O giv es a con tin uous linear functional on H s + N . If it is not iden tically zero, its k ernel is a prop er closed h yp erplane, hence its complemen t is op en and dense. □ Prop osition 7.7 (Generic detectability for a ﬁxed mo de and a ﬁxed detector) . Fix ( M , a ) ∈ K and ℓ ≥ ℓ 1 , and let O : H s +1 → C b e a b ounde d dete ctor. F or e ach sign ± , exactly one of the fol lowing alternatives holds: (a) The dete ctor is blind to the extr acte d ± mo de in the sense that a ± ,ℓ ( f 0 , f 1 ) = 0 for al l data ( f 0 , f 1 ) ∈ H s + N . (b) The set of data ( f 0 , f 1 ) ∈ H s + N for which a ± ,ℓ ( f 0 , f 1 )  = 0 is op en and dense. If b oth signs fal l under alternative (b) , then the joint dete ctability set  ( f 0 , f 1 ) ∈ H s + N : a + ,ℓ ( f 0 , f 1 )  = 0 and a − ,ℓ ( f 0 , f 1 )  = 0  is op en and dense. Pr o of. F or each sign ± , Lemma 7.6 shows that the amplitude map ( f 0 , f 1 ) 7→ a ± ,ℓ ( f 0 , f 1 ) is a con tin uous linear functional on H s + N . If it is identically zero, w e are in alternative (a). Otherwise its kernel is a prop er closed h yp erplane and its complement is op en and dense, whic h is alternativ e (b). If b oth amplitude functionals are nontrivial, then the in tersection of the tw o op en dense complements is again op en and dense. □ EQUA TORIAL KERR–DE SITTER RINGDOWN 65 R emark 7.8 (Characterization of blind detectors) . In the rank-one setting ( 7.11 ) – ( 7.12 ) , the range of the residue contribution A ± ,ℓ is contained in the one-dimensional space spanned by χA ± ,h ℓ Π k ± u ± ,ℓ . Consequen tly , if χA ± ,h ℓ Π k ± u ± ,ℓ  = 0, then alternativ e (a) in Prop osition 7.7 holds if and only if O  χA ± ,h ℓ Π k ± u ± ,ℓ  = 0 , a co dimension-one condition on O in the dual space of H s +1 . In particular, blindness is nongeneric among b ounded detectors, and can b e mitigated in practice by using m ultiple indep enden t c hannels, as describ ed b elo w. R emark 7.9 (Avoiding acciden tal annihilation) . Assumption 7.4 is intrinsic: if a detector annihilates a mo de, no deterministic metho d can recov er its frequency from that channel. If one has access to multiple b ounded detectors O 1 , . . . , O J , one ma y run the entire extraction pro cedure in each channel and select the channel(s) with the largest observ ed dominant energy ∥ e y j, ± ∥ w on the chosen window. This reduces the practical risk of acciden tal annihilation and is compatible with the deterministic framework developed here. Measured signals and measurement noise. W e allo w an additional deterministic measurement p erturbation η ± ( t ) on the observ ation window [ T 0 , T 0 + T ]: (7.13) e y ± ( t ) = y ± ( t ) + η ± ( t ) = a ± ,ℓ e − iω ± ,ℓ t +  ρ ± ,ℓ ( t ) + η ± ( t )  | {z } =: r ± ( t ) , t ∈ [ T 0 , T 0 + T ] , where T 0 ≥ 1 and T > 0. W e measure errors in the w eigh ted norm ∥ · ∥ w of Section 6.2 , w ith a ﬁxed shift ∆ ∈ (0 , T ) and a weigh t w satisfying ( 6.4 ) . When we use the explicit lo wer b ounds from Lemma 6.10 , we additionally assume T > 3∆. 7.3. F requency estimation from eac h separated mo de. Apply the shift Ra yleigh quotient estimator ( 6.5 ) (Section 6.2 ) to each e y ± on the window [ T 0 , T 0 + T ] with step ∆ and w eigh t w : b z ± := ⟨ S ∆ e y ± , e y ± ⟩ w ⟨ e y ± , e y ± ⟩ w , b ω ± := i ∆ Log( b z ± ) , where Log is chosen consistently with the prior ω ♯ ± ,ℓ as in Lemma 6.5 (or equiv alently Remark 6.9 ). Deﬁne the relative residual sizes (7.14) ε ± := max {∥ r ± ∥ w , ∥ S ∆ r ± ∥ w } ∥ a ± ,ℓ e − iω ± ,ℓ t ∥ w = max {∥ ρ ± ,ℓ + η ± ∥ w , ∥ S ∆ ( ρ ± ,ℓ + η ± ) ∥ w } ∥ a ± ,ℓ e − iω ± ,ℓ t ∥ w . Prop osition 7.10 (Deterministic frequency error from tail+measurement) . Assume ( 7.10 ) and ℑ ω ± ,ℓ ≤ 0 . If ε ± ≤ min { 1 / 8 , | z ± | / 20 } wher e z ± = 66 RUILIANG LI e − iω ± ,ℓ ∆ , then (7.15) | b ω ± − ω ± ,ℓ | ≤ 10 ∆ | z ± | ε ± . In p articular, (7.16) |ℜ ( b ω ± − ω ± ,ℓ ) | ≤ | b ω ± − ω ± ,ℓ | . Pr o of. This is Theorem 6.6 applied to e y ± with residual r ± . The deﬁnition ( 7.14 ) matc hes the relativ e error quantit y ε in Theorem 6.6 (with ρ replaced b y r ± ). □ Corollary 7.11 (A suﬃcient condition for staying in the lab eled frequency disk) . Assume The or em 5.1 and ﬁx the overtone index n use d thr oughout Se ction 5 . L et c sep > 0 b e the disk r adius in ( 5.1 ) and write ω ± ,ℓ := ω n, ± ,ℓ and ω ♯ ± ,ℓ := ω ♯ n, ± ,ℓ . Cho ose ℓ suﬃciently lar ge so that | ω ± ,ℓ − ω ♯ ± ,ℓ | ≤ c sep / 4 (which holds by ( 5.2 ) ). If, in addition to the hyp otheses of Pr op osition 7.10 , the r esidual size satisﬁes (7.17) ε ± ≤ c sep 40 ∆ | z ± | , z ± = e − iω ± ,ℓ ∆ , then the extr acte d fr e quency ob eys b ω ± ∈ D n, ± ,ℓ . Pr o of. By Prop osition 7.10 and ( 7.17 ), | b ω ± − ω ± ,ℓ | ≤ 10 ∆ | z ± | ε ± ≤ c sep 4 . T ogether with | ω ± ,ℓ − ω ♯ ± ,ℓ | ≤ c sep / 4, this giv es | b ω ± − ω ♯ ± ,ℓ | ≤ c sep / 2, hence b ω ± ∈ D n, ± ,ℓ b y ( 5.1 ). □ Corollary 7.12 (Explicit con trol of ε ± and the frequency error) . Assume Assumption 7.4 and ﬁx window p ar ameters T 0 ≥ 1 , T > 0 , ∆ ∈ (0 , T ) and a weight w as in ( 6.4 ) . L et y 0 , ± ( t ) := a ± ,ℓ e − iω ± ,ℓ t and write the me asur e d signals as in ( 7.13 ) , e y ± = y 0 , ± + r ± with r ± = ρ ± ,ℓ + η ± . Then (7.18) ε ± ≤ ∥ r ± ∥ L 2 ([ T 0 ,T 0 + T ]) ∥ y 0 , ± ∥ w ≤ ∥ ρ ± ,ℓ ∥ L 2 ([ T 0 ,T 0 + T ]) + ∥ η ± ∥ L 2 ([ T 0 ,T 0 + T ]) ∥ y 0 , ± ∥ w . Mor e over, inte gr ating the p ointwise tail b ound ( 7.9 ) yields, for every m ∈ N and N ′ ∈ N , (7.19) ∥ ρ ± ,ℓ ∥ L 2 ([ T 0 ,T 0 + T ]) ≤ C s,m,N ′ ∥ O ∥  e − ν T 0 (1+ T 0 ) − m + ℓ − N ′  √ T ∥ ( f 0 , f 1 ) ∥ H s + m + N m,N ′ . In p articular, if ε ± ≤ min { 1 / 8 , | z ± | / 20 } (with z ± = e − iω ± ,ℓ ∆ ), then (7.20) | b ω ± − ω ± ,ℓ | ≤ 10 ∆ | z ± | ∥ ρ ± ,ℓ ∥ L 2 ([ T 0 ,T 0 + T ]) + ∥ η ± ∥ L 2 ([ T 0 ,T 0 + T ]) ∥ y 0 , ± ∥ w . If, in addition, T > 3∆ , then L emma 6.10 pr ovides the lower b ound ∥ y 0 , ± ∥ w ≥ | a ± ,ℓ |  Z T 0 + T − 2∆ T 0 +∆ e 2 ℑ ω ± ,ℓ t dt  1 / 2 , EQUA TORIAL KERR–DE SITTER RINGDOWN 67 which makes ( 7.18 ) – ( 7.20 ) ful ly explicit in terms of the amplitude | a ± ,ℓ | . Pr o of. The ﬁrst inequalit y in ( 7.18 ) is Lemma 6.11 applied to r ± . The second inequalit y is the triangle inequality ∥ r ± ∥ L 2 ≤ ∥ ρ ± ,ℓ ∥ L 2 + ∥ η ± ∥ L 2 . The tail estimate ( 7.19 ) follo ws b y integrating ( 7.9 ) o v er the window. Finally , ( 7.20 ) is the frequency b ound ( 7.15 ) with ε ± con trolled by ( 7.18 ). □ Lemma 7.13 (Separating tail and measuremen t con tributions) . L et y 0 , ± ( t ) := a ± ,ℓ e − iω ± ,ℓ t and r ± = ρ ± ,ℓ + η ± . Then (7.21) ε ± ≤ ε tail ± + ε meas ± , wher e (7.22) ε tail ± := max {∥ ρ ± ,ℓ ∥ w , ∥ S ∆ ρ ± ,ℓ ∥ w } ∥ y 0 , ± ∥ w , ε meas ± := max {∥ η ± ∥ w , ∥ S ∆ η ± ∥ w } ∥ y 0 , ± ∥ w . Mor e over, sinc e 0 ≤ w ≤ 1 , (7.23) max {∥ f ∥ w , ∥ S ∆ f ∥ w } ≤ ∥ f ∥ L 2 ([ T 0 ,T 0 + T ]) for al l f ∈ L 2 ([ T 0 , T 0 + T ]) , and ther efor e ε tail ± ≤ ∥ ρ ± ,ℓ ∥ L 2 ([ T 0 ,T 0 + T ]) ∥ y 0 , ± ∥ w , ε meas ± ≤ ∥ η ± ∥ L 2 ([ T 0 ,T 0 + T ]) ∥ y 0 , ± ∥ w . Pr o of. The ﬁrst inequality ( 7.21 ) follows from the triangle inequalit y: max {∥ r ± ∥ w , ∥ S ∆ r ± ∥ w } ≤ max {∥ ρ ± ,ℓ ∥ w , ∥ S ∆ ρ ± ,ℓ ∥ w } + max {∥ η ± ∥ w , ∥ S ∆ η ± ∥ w } . and dividing by ∥ y 0 , ± ∥ w giv es ( 7.21 ) . The b ound ( 7.23 ) is Lemma 6.11 applied to f . □ 7.4. F rom frequency errors to parameter errors. Deﬁne the estimated QNM observ ables from the extracted frequencies: (7.24) b U ℓ := ℜ b ω + + b ω − 2 ℓ , b V ℓ := ℜ b ω + − b ω − 2 ℓ , b G ℓ := ( b U ℓ , b V ℓ ) . Let ( c M , b a ) b e deﬁned by lo cal in version: (7.25) ( c M , b a ) := G − 1 ℓ  b G ℓ  , where G − 1 ℓ is the lo cal inv erse guaranteed by Theorem 7.1 on a neighborho o d of the true p oin t ( M , a ). Lemma 7.14 (Observ able error in terms of frequency errors) . L et δ ω ± := b ω ± − ω ± ,ℓ . Then (7.26) | b U ℓ − U ℓ | ≤ | δ ω + | + | δ ω − | 2 ℓ , | b V ℓ − V ℓ | ≤ | δ ω + | + | δ ω − | 2 ℓ , and henc e (7.27) | b G ℓ − G ℓ ( M , a ) | ≤ √ 2 2 ℓ  | δ ω + | + | δ ω − |  . 68 RUILIANG LI Pr o of. By deﬁnition, b U ℓ − U ℓ = ℜ δ ω + + δ ω − 2 ℓ , b V ℓ − V ℓ = ℜ δ ω + − δ ω − 2 ℓ . Th us | b U ℓ − U ℓ | ≤ | δ ω + | + | δ ω − | 2 ℓ and similarly for V . The b ound ( 7.27 ) follo ws from | ( x, y ) | ≤ √ 2 max {| x | , | y |} and ( 7.26 ). □ W e can now state the main parameter-bias b ound. 7.5. Assumptions and quantitativ e inputs for the bias estimate. F or the reader’s con v enience, we record here the precise ingredients that enter the deterministic parameter-bias b ound. Throughout this subsection w e ﬁx ( M , a ) ∈ K and an integer ℓ ≥ max { ℓ 0 , ℓ 1 } . (A1) Tw o-mo de ringdo wn mo del and tail con trol. By the t wo-mode dominance result (Theorem 5.23 ), the mo de-separated signals pro duced b y the analytic windows satisfy , for t ≥ 1, O  u ( ± ,ℓ ) ( t )  = a ± ,ℓ e − iω ± ,ℓ t + ρ ± ,ℓ ( t ) , with ρ ± ,ℓ ob eying the explicit tail b ounds ( 7.9 ) . In particular, the residual on a windo w [ T 0 , T 0 + T ] admits an a priori estimate in terms of ( T 0 , T ), the shifted-con tour decay rate ν , and the high-frequency leak age order ℓ − N ′ . (A2) Detectability of the extracted mo des. W e assume that the chosen detector do es not annihilate the dominant extracted comp onen ts, a + ,ℓ  = 0 , a − ,ℓ  = 0 , that is, Assumption 7.4 holds on the window. (A3) F requency extraction with a small residual. W e measure the observ ed signals e y ± ( t ) = a ± ,ℓ e − iω ± ,ℓ t + r ± ( t ) on [ T 0 , T 0 + T ] in the weigh ted norm ∥ · ∥ w of Section 6.2 , with shift ∆ ∈ (0 , T ), and deﬁne the relative residual sizes ε ± b y ( 7.14 ) . When ε ± satisfy the quan titative signal-to- residual smallness conditions app earing in Proposition 7.10 , the shift Ra yleigh quotien t estimator yields deterministic frequency errors b ounded b y ( 7.15 ). (A4) Conditioning of the parameter in v erse map. The companion pap er provides a lo cal in v erse of the equatorial frequency map G ℓ = ( U ℓ , V ℓ ) with a uniform stability constant C ∗ , as stated in Theorem 7.1 ; equiv alently , the Jacobian of G ℓ is uniformly nondegenerate on K . With these inputs ﬁxed, the next theorem giv es a deterministic parameter bias b ound in terms of the residual sizes ε ± . Theorem 7.15 (Parameter bias b ound from ringdown-based in v ersion) . Fix Λ > 0 , K , and n as in ( 7.1 ) . Cho ose ℓ ≥ max { ℓ 0 , ℓ 1 } so that: (i) the two-mo de dominanc e the or em (The or em 5.23 ) holds, and (ii) the inverse map the or em (The or em 7.1 ) holds. Fix an observation window [ T 0 , T 0 + T ] with T 0 ≥ 1 , a shift ∆ ∈ (0 , T ) , and a weight w as in ( 6.4 ) . EQUA TORIAL KERR–DE SITTER RINGDOWN 69 Assume the inputs (A1)–(A4) fr om Se ction 7.5 , and deﬁne ε ± by ( 7.14 ) . If ε ± ≤ min { 1 / 8 , | z ± | / 20 } with z ± = e − iω ± ,ℓ ∆ , then the r e c onstructe d p ar ameters ( 7.25 ) satisfy the deterministic err or b ound (7.28) | ( c M , b a ) − ( M , a ) | ≤ 5 √ 2 C ∗ ∆ ℓ  ε + | z + | + ε − | z − |  , wher e C ∗ is the inverse stability c onstant fr om The or em 7.1 . Mor e over, splitting ε ± as in L emma 7.13 , one obtains a de c omp osition into a PDE tail bias term and a measuremen t bias term: (7.29) | ( c M , b a ) − ( M , a ) | ≤ B tail + B meas , wher e (7.30) B tail := 5 √ 2 C ∗ ∆ ℓ ε tail + | z + | + ε tail − | z − | ! , B meas := 5 √ 2 C ∗ ∆ ℓ  ε meas + | z + | + ε meas − | z − |  . In p articular, using ( 7.9 ) to gether with L emma 6.11 and L emma 6.10 , one obtains explicit upp er b ounds for ε tail ± (and henc e for B tail ) in terms of ( T 0 , T , ∆ , w ) , the tail de c ay r ate ν , and the amplitudes a ± ,ℓ . Pr o of. By Prop osition 7.10 , | δ ω ± | = | b ω ± − ω ± ,ℓ | ≤ 10 ∆ | z ± | ε ± . Insert this into Lemma 7.14 : | b G ℓ − G ℓ ( M , a ) | ≤ √ 2 2 ℓ  10 ∆ | z + | ε + + 10 ∆ | z − | ε −  = 5 √ 2 ∆ ℓ  ε + | z + | + ε − | z − |  . No w apply the in verse stability estimate ( 7.5 ) from Theorem 7.1 : | ( c M , b a ) − ( M , a ) | ≤ C ∗ | b G ℓ − G ℓ ( M , a ) | , whic h yields ( 7.28 ) . The split ( 7.29 ) – ( 7.30 ) follo ws by combining ( 7.28 ) with Lemma 7.13 . □ Corollary 7.16 (Explicit bias b ound in terms of the PDE tail and measure- men t noise) . Assume the hyp otheses of The or em 7.15 and, in addition, that T > 3∆ . L et y 0 , ± ( t ) := a ± ,ℓ e − iω ± ,ℓ t and r ± = ρ ± ,ℓ + η ± as in ( 7.13 ) . Then (7.31) | ( c M , b a ) − ( M , a ) | ≤ 5 √ 2 C ∗ ∆ ℓ X ς ∈{ + , −} 1 | z ς | ∥ r ς ∥ L 2 ([ T 0 ,T 0 + T ]) ∥ y 0 ,ς ∥ w . Mor e over, by L emma 6.10 , ∥ y 0 ,ς ∥ w ≥ | a ς ,ℓ |  Z T 0 + T − 2∆ T 0 +∆ e 2 ℑ ω ς ,ℓ t dt  1 / 2 , 70 RUILIANG LI and ther efor e (7.32) | ( c M , b a ) − ( M , a ) | ≤ 5 √ 2 C ∗ ∆ ℓ X ς ∈{ + , −} 1 | z ς | ∥ ρ ς ,ℓ ∥ L 2 ([ T 0 ,T 0 + T ]) + ∥ η ς ∥ L 2 ([ T 0 ,T 0 + T ]) | a ς ,ℓ | ×  Z T 0 + T − 2∆ T 0 +∆ e 2 ℑ ω ς ,ℓ t dt  − 1 / 2 . In p articular, inte gr ating the p ointwise b ound ( 7.9 ) gives ∥ ρ ς ,ℓ ∥ L 2 ([ T 0 ,T 0 + T ]) ≤ C s,m,N ′ ∥ O ∥  e − ν T 0 (1 + T 0 ) − m + ℓ − N ′  √ T × ∥ ( f 0 , f 1 ) ∥ H s + m + N m,N ′ . Pr o of. By Lemma 6.11 and 0 ≤ w ≤ 1, one has max {∥ r ς ∥ w , ∥ S ∆ r ς ∥ w } ≤ ∥ r ς ∥ L 2 ([ T 0 ,T 0 + T ]) . Th us ε ς ≤ ∥ r ς ∥ L 2 ([ T 0 ,T 0 + T ]) / ∥ y 0 ,ς ∥ w . Insert this in to ( 7.28 ) to obtain ( 7.31 ) . The inequality ( 7.32 ) follo ws from Lemma 6.10 and the triangle inequalit y ∥ r ς ∥ L 2 ≤ ∥ ρ ς ,ℓ ∥ L 2 + ∥ η ς ∥ L 2 . Finally , integrating ( 7.9 ) o v er [ T 0 , T 0 + T ] yields the display ed L 2 b ound for ρ ς ,ℓ . □ 7.6. Three-parameter ringdown in v ersion: deterministic bias b ound for ( M , a, Λ) . In many applications the cosmological constant is treated as kno wn. F rom a mathematical viewp oin t, ho w ever, it is natural to ask whether the same high-frequency equatorial pack age can supp ort lo cal recov ery of the thr e e Kerr–de Sitter parameters ( M , a, Λ). At the sp ectral lev el this requires a third indep enden t real observ able. The companion pap er [ 1 ] sho ws that, aw a y from a = 0, adding one damping observ able extracted from the imaginary part of a single equatorial mo de yields suc h a three- parameter inv erse theorem. In this subsection we propagate the deterministic PDE-to-data error chain through that three-parameter inv erse map. Thr e e-p ar ameter inverse map fr om e quatorial QNMs (input fr om the c om- p anion p ap er). Let (7.33) K (3) ⋐  ( M , a, Λ) : Λ > 0 , ( M , a ) ∈ P Λ , 0 < a 1 ≤ | a | ≤ a 0  b e a compact three-parameter slow-rotation set with a 1 > 0. Fix an ov ertone index n as in ( 7.1 ), and for ℓ ≫ 1 let ω ± ,ℓ ( M , a, Λ) := ω n,ℓ, ± ℓ ( M , a, Λ) denote the lab eled simple equatorial QNMs. Deﬁne U ℓ and V ℓ as in ( 7.2 ) (no w viewed as functions of ( M , a, Λ)), and deﬁne the additional damping observ able (7.34) f W ℓ ( M , a, Λ) := − ℑ ω + ,ℓ ( M , a, Λ) n + 1 2 . Set (7.35) H ℓ ( M , a, Λ) :=  U ℓ ( M , a, Λ) , V ℓ ( M , a, Λ) , f W ℓ ( M , a, Λ)  ∈ R 3 . EQUA TORIAL KERR–DE SITTER RINGDOWN 71 Theorem 7.17 (Lo cal three-parameter inv ersion from equatorial QNMs; stabilit y) . Fix n and K (3) as ab ove. Then ther e exist ℓ 0 ∈ N and C (3) ∗ > 0 such that for every ℓ ≥ ℓ 0 and every p oint µ := ( M , a, Λ) ∈ K (3) ther e exists a neighb orho o d N ⊂ K (3) of µ on which the map H ℓ is inje ctive and satisﬁes the quantitative stability estimate (7.36) | µ − µ ′ | ≤ C (3) ∗   H ℓ ( µ ) − H ℓ ( µ ′ )   , µ, µ ′ ∈ N . Mor e over, H ℓ is r e al-analytic on K (3) . Pr o of. This is pro v ed in the companion pap er [ 1 , Theorem 51] (see also [ 1 , § 5.8] for the construction of the damping observ able and the underlying Jacobian mechanism). □ R emark 7.18 (Uniformity in Λ) . The forward analysis in Sections 2 – 6 w as presen ted with Λ > 0 ﬁxed. All inputs used there (F redholm/meromorphic resolv en t theory , normally hyperb olic trapping estimates, sectorization in the equatorial pack age, and the Grushin reduction underlying Prop osition 5.18 ) dep end smo othly on the bac kground parameters, hence remain v alid uni- formly on compact parameter sets; a more formal uniformity statement is recorded in App endix E . In particular, after restricting to K (3) in ( 7.33 ) and shrinking auxiliary constants if needed, all constants in Theorem 5.23 and Prop osition 7.10 may b e taken uniform on K (3) . Pr op agation of time-domain err ors to the thr e e-p ar ameter data map. Let µ = ( M , a, Λ) ∈ K (3) b e the true parameters and let ℓ b e suﬃciently large so that Theorem 5.23 , Prop osition 7.10 , and Theorem 7.17 apply . As in Section 7.2 , w e observe the mo de-separated scalar signals e y ± ( t ) = a ± ,ℓ e − iω ± ,ℓ t + r ± ( t ) on [ T 0 , T 0 + T ], deﬁne ε ± b y ( 7.14 ) , and construct the estimated frequencies b ω ± b y ( 6.13 ). Deﬁne the estimated three-comp onen t data (7.37) b H ℓ :=  b U ℓ , b V ℓ , c f W ℓ  , where (7.38) b U ℓ := ℜ b ω + + b ω − 2 ℓ , b V ℓ := ℜ b ω + − b ω − 2 ℓ , c f W ℓ := − ℑ b ω + n + 1 2 . Lemma 7.19 (Data-map p erturbation b ound in terms of frequency errors) . L et δ ω ± := b ω ± − ω ± ,ℓ . Then (7.39)   b H ℓ − H ℓ ( µ )   ≤ √ 2 2 ℓ  | δ ω + | + | δ ω − |  + | δ ω + | n + 1 2 . Pr o of. By ( 7.38 ) and ( 7.2 ) (with Λ treated as a v ariable), | b U ℓ − U ℓ | ≤ 1 2 ℓ  |ℜ δ ω + | + |ℜ δ ω − |  ≤ 1 2 ℓ  | δ ω + | + | δ ω − |  , 72 RUILIANG LI and similarly | b V ℓ − V ℓ | ≤ 1 2 ℓ  | δ ω + | + | δ ω − |  . Moreo v er,   c f W ℓ − f W ℓ   = 1 n + 1 2 |ℑ δ ω + | ≤ | δ ω + | n + 1 2 . Com bining these estimates and using p | x | 2 + | y | 2 ≤ √ 2 max {| x | , | y |} ≤ √ 2 2 ( | x | + | y | ) gives ( 7.39 ). □ Thr e e-p ar ameter deterministic bias b ound. Theorem 7.20 (Three-parameter parameter bias b ound from ringdown-based in v ersion) . Fix n and K (3) as in ( 7.33 ) . Cho ose ℓ lar ge so that The or em 5.23 , Pr op osition 7.10 , and The or em 7.17 hold on K (3) . Fix an observation window [ T 0 , T 0 + T ] with T 0 ≥ 1 , a shift ∆ ∈ (0 , T ) , and a weight w as in ( 6.4 ) . Assume the mo de dete ctability c ondition ( 7.10 ) and deﬁne ε ± by ( 7.14 ) . If ε ± ≤ min { 1 / 8 , | z ± | / 20 } with z ± = e − iω ± ,ℓ ∆ , then the r e c onstructe d p ar ameters b µ := ( c M , b a, b Λ) deﬁne d by inverting b H ℓ satisfy (7.40) | b µ − µ | ≤ 5 √ 2 C (3) ∗ ∆ ℓ  ε + | z + | + ε − | z − |  + 10 C (3) ∗ ∆ ( n + 1 2 ) ε + | z + | . Pr o of. By Prop osition 7.10 , (7.41) | δ ω ± | ≤ 10 ∆ | z ± | ε ± . Insert ( 7.41 ) into Lemma 7.19 to obtain   b H ℓ − H ℓ ( µ )   ≤ √ 2 2 ℓ  10 ∆ | z + | ε + + 10 ∆ | z − | ε −  + 1 n + 1 2 10 ∆ | z + | ε + . Using √ 2 2 · 10 = 5 √ 2 gives (7.42)   b H ℓ − H ℓ ( µ )   ≤ 5 √ 2 ∆ ℓ  ε + | z + | + ε − | z − |  + 10 ∆ ( n + 1 2 ) ε + | z + | . Finally , apply the three-parameter in v erse stability estimate ( 7.36 ) from Theorem 7.17 : | b µ − µ | ≤ C (3) ∗   b H ℓ − H ℓ ( µ )   , and combine with ( 7.42 ) to obtain ( 7.40 ). □ 7.7. Practical remarks on conditioning and c hoice of windo w param- eters. Theorem 7.15 is deterministic and mak es explicit several imp ortant conditioning mechanisms. (i) The high-frequency gain and condition num ber. The factor 1 /ℓ in ( 7.28 ) is inherited from the normalization in ( 7.2 ) and from the companion- pap er stabilit y b ound ( 7.6 ) . Equiv alently , it is the 1 /ℓ impro v ement in the condition num b er of the raw-frequency inv erse map discussed in Remark 7.2 . EQUA TORIAL KERR–DE SITTER RINGDOWN 73 (ii) T ail v ersus signal. The dominant mo de decays lik e e ℑ ω t while the remainder decays like e − ν t . Since ν is chosen strictly ab ov e the mo de damping (Section 5 ), the ratio improv es like e − ( ν −|ℑ ω | ) T 0 = e − ( ν + ℑ ω ) T 0 as T 0 increases, cf. Remark 6.13 . This gives a deterministic justiﬁcation for c ho osing a ringdo wn start time after prompt resp onse has decay ed, sub ject to the obvious limitation that the absolute signal | a ± ,ℓ e − iω ± ,ℓ t | also decreases with T 0 . (iii) Shift step ∆ and branch con trol. The frequency extraction error scales like 1 / ∆ in ( 7.15 ) , while the branch am biguity scales lik e 2 π / ∆ in ℜ ω . In the present setting, the pseudop ole prior ω ♯ ± ,ℓ and the analytic window e g ± ,ℓ lo calize eac h frequency to a neighborho o d where Lemma 6.5 selects a consisten t logarithm branch. (iv) Detectabilit y and amplitude dep endence. All b ounds depend on the amplitudes through ∥ y 0 , ± ∥ − 1 w , hence on | a ± ,ℓ | − 1 . This is intrinsic: if a detector nearly annihilates a mo de, then the corresp onding frequency is p oorly observ ed on that channel. Assumption 7.4 makes the nonv anishing requiremen t explicit, and Lemma 7.6 sho ws that the requirement is generic for ﬁxed ( M , a ) and ﬁxed ℓ . Quan titatively , the smallness conditions ε ± ≤ 1 / 8 in Prop osition 7.10 are exactly signal-to-residual inequalities on the chosen windo w; they enco de ho w large the dominan t-mo de energy must b e relativ e to the tail and the measurement p erturbation for deterministic extraction to b e stable. 8. Pseudospectral st ability in the equa torial p a cka ge The stationary wa v e op erator is non-selfadjoint, and QNM p oles alone do not control the size of the resolven t a wa y from the sp ectrum: pseudosp ectral eﬀects and transient growth can occur ev en when all p oles lie strictly in the lo wer half-plane [ 14 , 17 , 20 , 18 ]. In our setting, ho w ever, w e do not w ork with the full resolven t but rather with an e quatorial ly micr olo c alize d resolv en t in a ﬁxed semiclassical pack age. The purp ose of this section is to record a quantitativ e b ound sho wing that, within this pack age, the lo calized resolv en t cannot develop large pseudomo de growth far aw a y from the lab eled p oles: its blo w-up is conﬁned to a controlled neighborho o d of the p oles, with p olynomial dep endence on ℓ . 8.1. A microlo calized resolven t b ound near lab eled equatorial p oles. Fix an o v ertone index n ∈ N 0 and a Sob olev order s ≥ 0. F or eac h ℓ and sign ± , deﬁne the lo calized resolven t family (8.1) R ± ,ℓ ( ω ) := χ A ± ,h ℓ Π k ± R ( ω ) χ, k ± = ± ℓ, h ℓ = ℓ − 1 . Deﬁnition 8.1 (Lo calized ϵ -pseudosp ectrum in the equatorial pac k age) . Fix s ≥ 0 and n ∈ N 0 . F or eac h ℓ and sign ± , and for ϵ > 0, we deﬁne the 74 RUILIANG LI lo calized ϵ -pseudosp ectrum (8.2) Σ ( ± ,ℓ ) ϵ := n ω ∈ [ 0 ≤ j ≤ n D j, ± ,ℓ :   R ± ,ℓ ( ω )   H s − 1 → H s +1 > ϵ − 1 o . This set measures the growth of the microlo calized resolven t within the lab eled equatorial neighborho o d. On each disk D j, ± ,ℓ around a lab eled equatorial p ole ω j, ± ,ℓ (with 0 ≤ j ≤ n ), the Grushin reduction from [ 1 , App endix B] yields the decomp osition ( 5.27 ) , with a scalar quan tization function q ± ,ℓ ha ving a simple zero at ω j, ± ,ℓ and satisfying the uniform normalization ( 5.28 ). Lemma 8.2 (Quantization function controls distance to a simple p ole) . Possibly after shrinking the disk r adius in the deﬁnition of D j, ± ,ℓ (uniformly in ( M , a ) and ℓ ), ther e exist c onstants c q , r q > 0 such that for al l ( M , a ) ∈ K , al l ℓ suﬃciently lar ge, e ach sign ± , e ach j ∈ { 0 , 1 , . . . , n } , and al l ω ∈ D j, ± ,ℓ with | ω − ω j, ± ,ℓ | ≤ r q , (8.3) | q ± ,ℓ ( ω ) | ≥ c q | ω − ω j, ± ,ℓ | . Pr o of. Fix ( M , a ), ℓ , ± , and j . Since q ± ,ℓ is holomorphic on D j, ± ,ℓ and has a simple zero at ω j, ± ,ℓ , T aylor’s theorem gives q ± ,ℓ ( ω ) = ( ω − ω j, ± ,ℓ ) ∂ ω q ± ,ℓ ( ω j, ± ,ℓ ) + 1 2 ( ω − ω j, ± ,ℓ ) 2 ∂ 2 ω q ± ,ℓ ( ξ ) , for some ξ on the line segmen t b et w een ω and ω j, ± ,ℓ . The Grushin reduction pro vides uniform bounds on ∂ 2 ω q ± ,ℓ on D j, ± ,ℓ (see [ 1 , Appendix B]); thus there is C q > 0 with sup D j, ± ,ℓ | ∂ 2 ω q ± ,ℓ | ≤ C q uniformly in ( M , a ) ∈ K and ℓ large. Cho ose r q > 0 so that 1 2 r q C q ≤ 1 2 c ∗ , where c ∗ > 0 is the low er b ound in ( 5.28 ). Then for | ω − ω j, ± ,ℓ | ≤ r q , | q ± ,ℓ ( ω ) | ≥ | ω − ω j, ± ,ℓ | | ∂ ω q ± ,ℓ ( ω j, ± ,ℓ ) |− 1 2 | ω − ω j, ± ,ℓ | 2 sup D j, ± ,ℓ | ∂ 2 ω q ± ,ℓ | ≥ 1 2 c ∗ | ω − ω j, ± ,ℓ | . This prov es ( 8.3 ) with c q = 1 2 c ∗ . □ Prop osition 8.3 (Localized resolven t b ound near lab eled equatorial p oles) . Assume The or em 5.1 . Fix s ≥ 0 and n ∈ N 0 . Ther e exist c onstants K ≥ 0 , ℓ 0 ∈ N , and C > 0 such that for al l ( M , a ) ∈ K , al l ℓ ≥ ℓ 0 , e ach sign ± , e ach j ∈ { 0 , 1 , . . . , n } , and al l ω ∈ D j, ± ,ℓ \ { ω j, ± ,ℓ } , (8.4)   R ± ,ℓ ( ω )   H s − 1 → H s +1 ≤ C ℓ K  1 + 1 | ω − ω j, ± ,ℓ |  . Pr o of. Fix ( M , a ), ℓ , ± , and j . On D j, ± ,ℓ , the decomp osition ( 5.27 ) giv es R ± ,ℓ ( ω ) = R hol ± ,ℓ ( ω ) + E ( ± ,ℓ ) + ( ω ) q ± ,ℓ ( ω ) − 1 E ( ± ,ℓ ) − ( ω ) . By [ 1 , App endix B], the holomorphic term and the Grushin op erators s atisfy p olynomial b ounds in ℓ as maps H s − 1 → H s +1 , uniformly on D j, ± ,ℓ . Th us EQUA TORIAL KERR–DE SITTER RINGDOWN 75 there exist K ≥ 0 and C 0 > 0 suc h that sup ω ∈ D j, ± ,ℓ   R hol ± ,ℓ ( ω )   H s − 1 → H s +1 ≤ C 0 ℓ K , sup ω ∈ D j, ± ,ℓ   E ( ± ,ℓ ) ± ( ω )   H s − 1 → H s +1 ≤ C 0 ℓ K . F or ω with | ω − ω j, ± ,ℓ | ≤ r q , Lemma 8.2 yields | q ± ,ℓ ( ω ) | − 1 ≤ c − 1 q | ω − ω j, ± ,ℓ | − 1 . Com bining these b ounds giv es ( 8.4 ) for suc h ω . If | ω − ω j, ± ,ℓ | ≥ r q , then | ω − ω j, ± ,ℓ | − 1 ≤ r − 1 q and the same inequalit y follows (with a larger constant) from the uniform p olynomial b ounds on D j, ± ,ℓ . □ Corollary 8.4 (Lo calized pseudosp ectral disks near lab eled p oles) . Assume The or em 5.1 and ﬁx s ≥ 0 and n ∈ N 0 . L et Σ ( ± ,ℓ ) ϵ b e the lo c alize d ϵ - pseudosp e ctrum fr om Deﬁnition 8.1 , and let Z ± ,ℓ := { ω j, ± ,ℓ : 0 ≤ j ≤ n } . Then ther e exist K ≥ 0 , ℓ 0 ∈ N , and C > 0 such that for every ℓ ≥ ℓ 0 and every ϵ > 0 , (8.5) Σ ( ± ,ℓ ) ϵ ⊂ [ ω j ∈Z ± ,ℓ B  ω j , C ℓ K ϵ  . Pr o of. Let ω ∈ Σ ( ± ,ℓ ) ϵ . Then ω ∈ D j, ± ,ℓ for some 0 ≤ j ≤ n and ∥R ± ,ℓ ( ω ) ∥ H s − 1 → H s +1 > ϵ − 1 . By Prop osition 8.3 , ϵ − 1 < C ℓ K  1 + 1 | ω − ω j, ± ,ℓ |  . If ϵ ≥ (2 C ℓ K ) − 1 then the conclusion ( 8.5 ) is trivial after enlarging the constan t. Otherwise ϵ − 1 ≥ 2 C ℓ K , hence the righ t-hand side forces 1 | ω − ω j, ± ,ℓ | ≥ 1 2 C − 1 ℓ − K ϵ − 1 , i.e. | ω − ω j, ± ,ℓ | ≤ 2 C ℓ K ϵ . This yields ( 8.5 ). □ R emark 8.5 (Scop e of the pseudosp ectral statement) . The inclusion ( 8.5 ) is a lo c alize d pseudosp ectral statement: it con trols the growth of the resolven t only after restricting to a ﬁxed equatorial high-frequency pack age and inserting microlo cal cutoﬀs. It do es not claim that the glob al Kerr (or Kerr–de Sitter) pseudosp ectrum is small; rather it iden tiﬁes a regime in which the relev an t c hannel is quantitativ ely stable, whic h is precisely what is needed for the deterministic bias b ounds in Section 7 . 8.2. A remark on extensions to higher spin. The analysis in this pap er is presented for the scalar wa v e equation, since the barrier-top equatorial quan tization and Grushin reduction used in Prop osition 5.18 are currently established in that setting. At a structural level, ho wev er, sev eral steps are not tied to spin 0: the meromorphic/F redholm framew ork for stationary families, the contour-shift resonance expansion with remainder, the analytic- windo w calculus (en tire w eigh ts), and the deterministic frequency-extraction stabilit y theory apply to any stationary op erator family for which (i) one has a resonance expansion in a strip and (ii) one can control the relev ant microlo calized sp ectral pro jectors. F or Kerr–de Sitter, a robust linear and nonlinear stability framework for the Einstein v acuum equations with Λ > 0 w as dev elop ed b y Hintz–V asy [ 7 ], suggesting that analogous microlo cal 76 RUILIANG LI resolv ent technology should b e a v ailable for tensorial op erators once an appropriate high-frequency lab eling and sectorization pack age is established. Carrying out the equatorial semiclassical quan tization and the asso ciated pro jector b ounds for the T eukolsky system (or for gauge-ﬁxed linearized Einstein op erators) remains an in teresting op en direction. 9. Discussion and outlook This pap er is the second part of a series whose goal is to place a concrete v ersion of high-frequency black-hole sp ectroscop y on a fully deterministic mathematical fo oting. Building on the semiclassical quan tization and labeling theory developed in the companion pap er [ 1 ], we w ork in the slo w-rotation compact regime and fo cus on an equatorial high-frequency pack age with a ﬁxed ov ertone index. Starting from the resolven t-based resonant expansion with remainder established in Section 4 , w e show that a suitable analytic prepro cessing of the time signal isolates the target ov ertone with quantitativ e con trol (Section 5 ). W e then establish deterministic stability of one- and t w o-frequency extraction from ﬁnite-length, p erturb ed data (Section 6 ), and propagate the resulting frequency errors through the lo cal inv erse map from equatorial quasinormal frequencies to parameters, obtaining a quantitativ e parameter bias b ound (Section 7 ). P ositioning relativ e to curren t ringdo wn debates. Sev eral recen t w orks empha- size that nonnormality can manifest itself through pseudosp ectral sensitivity , transien t gro wth, and non-mo dal dynamics even when all QNMs are damp ed [ 14 , 20 , 15 , 21 , 18 ]. Our results are compatible with these p henomena b ecause the main theorems concern r eﬁne d signals obtained by an explicit sequence of reductions and lo calizations: azimuthal pro jection, equatorial microlo cal ﬁltering, and an en tire (p olynomial) analytic windo w. The deterministic remainder b ounds quan tify what is discarded b y these steps on a prescrib ed late-time window. In particular, the t w o-mo de dominance theorem should not b e read as a claim that a generic unﬁltered ringdo wn wa v eform is uni- v ersally tw o-mode; rather it identiﬁes a semiclassical regime and a con trolled prepro cessing map in which a tw o-mode mo del is justiﬁed. Comparison with single-frequency inv erse results. In the mathematical in- v erse literature on black-hole sp ectroscop y , a useful reference p oin t is the result of Uhlmann–W ang [ 12 ], who recov er the black-hole mass from a single quasinormal mo de. Our setting is diﬀerent in tw o complemen tary wa ys. First, w e w ork in a high-frequency equatorial pack age and use a p air of frequencies ω ± ,ℓ to recov er a p air of parameters ( M , a ), with a conditioning gain of order 1 /ℓ in the slo w-rotation regime. Second, rather than proving iden tiﬁabilit y from exact sp ectral data, w e quantify how ﬁnite-windo w ring- do wn truncation and deterministic measuremen t p erturbations propagate through the extraction step and the inv erse map; this leads to the explicit bias b ound ( 7.28 ). EQUA TORIAL KERR–DE SITTER RINGDOWN 77 Analytic p ole selection and implementable ﬁltering. A key constrain t in time-domain ringdown analysis is that “mo de selection” m ust b e compatible with the analytic structure of the Laplace transform: a non-holomorphic frequency cutoﬀ destro ys the con tour-shift argument and generally has no clean interpretation in the resolv ent calculus. Our approach therefore uses an entir e (indeed, p olynomial) w eigh t e g ± ,ℓ ( ω ) which v anishes at all p oles in the relev an t o vertone strip except for the target p ole ω ± ,ℓ , and equals 1 at ω ± ,ℓ . The construction pro ceeds b y Lagrange in terpolation on the pseudop ole lattice and a low-degree correction factor whic h suppresses the growth of the interpolation p olynomial on the shifted con tour; see App endix C and Prop osition 5.18 . This is precisely the p oin t where we av oid making any global claim ab out the size of residue pro jectors: the window is designed so that the weighte d p ole sum is absolutely conv ergen t for the windo wed signal, while the unw eigh ted full resonan t expansion is used only in the sense of con v ergence of truncated sums (Section 4 ). Time-domain meaning. Because e g ± ,ℓ is a p olynomial in ω , multiplica- tion by e g ± ,ℓ ( ω ) on the Laplace side corresp onds to applying a ﬁnite-order diﬀeren tial op erator e g ± ,ℓ ( i∂ t ∗ ) in the time domain. In discrete data, this can b e implemented by stable ﬁnite-diﬀerence combinations of a ﬁxed num b er of time shifts (dep ending on the c hosen ov ertone index n but indep endent of ℓ ). This provides a direct bridge b et w een analytic p ole calculus and an algorithmically realizable prepro cessing step. Relation to QNM ﬁlters in data analysis. There is a natural connection b e- t w een our analytic window construction and recent work in the gravitational- w av e literature on QNM ﬁlters, whic h aim to remov e or isolate selected QNMs by applying a frequency-domain ﬁlter and then transforming bac k to the time domain; see, for instance, Ma et al. [ 13 ]. The common theme is that (in a regime where QNM expansions are meaningful) one seeks a prepro cessing map whose action on an ideal sum of damp ed exp onen tials is transparen t. Our analytic windows diﬀer in tw o resp ects that are imp ortant for the deterministic resolven t analysis pursued here. First, the weigh t e g ± ,ℓ is en tire (indeed, p olynomial), so it is automatically compatible with contour deformation on the Laplace side and yields a remainder term that can b e b ounded directly by shifted-contour resolven t estimates. Second, the window is constructed to lo calize a single o v ertone in a sp eciﬁc semiclassical pack age and to in teract cleanly with the microlo cal sectorization estimate from the companion pap er, thereb y pro ducing quantitativ e tw o-mode dominance in a Sob olev top ology . Nonnormalit y , excitation factors, and pseudosp ectra. Quasinormal-mo de problems are fundamentally non-selfadjoint, and sp ectral data can b e highly sensitiv e to p erturbations. This sensitivit y is naturally quantiﬁed by pseu- dosp ectra and has b een explored in the blac k-hole context in [ 14 ]; see also the discussion of the structural role of the underlying scalar pro duct in [ 17 ]. Recent n umerical studies compute pseudosp ectra for Kerr in the scalar 78 RUILIANG LI case b y casting the problem into a non-selfadjoin t eigen v alue formulation on h yp erboloidal slices and examining the norm dep endence of the resolv ent; see for instance [ 18 ] and the broader hyperb oloidal p ersp ectiv e in [ 19 ]. F rom the viewp oin t of deterministic ringdown inv ersion, the main analytic diﬃcult y is not only that the spectrum may be sensitive, but also that residue pro jectors (often referred to as excitation factors in the physics literature) ma y ha v e large op erator norms and can in principle mask small scalar weigh ts. In the equatorial barrier-top pack age the Grushin reduction from [ 1 ] yields an explicit p olynomial b ound for the relev ant microlo calized residue pro jectors (Prop osition 5.18 ). Com bined with the O ( ℓ −∞ ) suppression pro duced by the analytic weigh ts e g ± ,ℓ , this giv es a tw o-mode dominance statemen t that is robust against hidden large excitation factors in the regime considered here. Understanding how far this p olynomial-con trol regime extends (and how it deteriorates outside it) is an imp ortan t direction for further work. T ransient dynamics and sup erradian t ampliﬁcation. The analysis in this series is in ten tionally late-time and deterministic: w e extract frequencies from a window [ T 0 , T 0 + T ] chosen b ey ond the initial prompt resp onse, and we treat everything outside the selected ﬁnite-dimensional mo del as a con trolled remainder. Recen t w ork emphasizes, how ev er, that nonnormality can manifest itself through large tr ansient eﬀects ev en when all QNMs are damp ed. In particular, the b eha vior of truncated QNM sums and the possibility of transient growth/superradiant ampliﬁcation ha v e b een analyzed in [ 20 , 15 ]; see also the ov erview [ 21 ]. F rom an inv erse-problem p erspective, suc h transients may create an additional bias source if the observ ation window ov erlaps the growth phase. The deterministic framework dev elop ed here suggests tw o mathematically clean wa ys to incorp orate these phenomena: either enlarge T 0 so that the transient comp onen t is absorb ed in to the exp onen tially damp ed tail, or extend the signal mo del to include a ﬁnite num ber of non-mo dal con tributions, leading naturally to higher-rank p encil/Pron y estimators and to condition num bers go v erned b y V andermonde- t yp e matrices. The tw o-frequency stability results in Section 6.5 provide a ﬁrst step tow ard suc h m ulti-comp onen t deterministic mo dels. Scop e of the t wo-mode regime and failure mechanisms. The tw o-mode domi- nance result (Theorem 5.23 ) rests on three structural inputs: (1) Pole isolation and layer gaps : the target p ole ω ± ,ℓ is isolated in an o v ertone strip, and the shifted contour can b e placed in a p ole-free region with uniform resolv en t bounds (Section 3.7 ). (2) Micr olo c al se ctorization : after applying the equatorial/azimuthal lo calization A ± ,h ℓ Π k ± , all p oles outside the equatorial pack age con- tribute only O ( ℓ −∞ ) to the windo w ed signal (App endix D ). (3) L ate-time suppr ession of tr ansients : the start time T 0 is chosen so that the remainder term pro duced by con tour shifting is small compared to the selected p ole con tribution (Sections 4 and 5 ). EQUA TORIAL KERR–DE SITTER RINGDOWN 79 In addition, detectabilit y (Assumption 7.4 ) excludes the degenerate case where the selected mo de is annihilated by the observ able or b y dual orthogo- nalit y . These hypotheses highlight concrete mec hanisms by which the tw o-mo de regime can fail. Near the b oundary of the sub extremal parameter set P Λ (e.g. in near-extremal limits), sp ectral gaps ma y shrink and diﬀeren t QNM families can approac h each other, making p ole isolation delicate and p oten tially enhancing nonnormal eﬀects. Ev en aw a y from such limits, p o or choices of observ able or initial data can yield v ery small amplitudes, amplifying the impact of an y residual tail. Finally , for early start times the prompt resp onse and transien t growth dominate the signal and in v alidate an y asymptotic t w o-mo de mo del. Exceptional p oin ts and near-coalescence. Exceptional points and av oided crossings pro vide another mec hanism b y whic h in v erse problems become ill-conditioned: as t w o mo des approac h coalescence, sp ectral pro jectors and in v erse maps can lose regularity , and small data errors can translate in to large parameter bias. This phenomenon has b een highligh ted in black- hole ringdown studies from the viewp oin t of non-Hermitian physics; see for instance [ 16 ] and the analysis of resonant excitation in [ 24 ]. In the presen t series the same mechanism appears transparently through the condition- n um b er interpretation of the parameter map (Section 7.7 ): the constants in the bias b ounds reﬂect the lo cal in v erse Lipschitz constant, which necessarily deteriorates in near-coalescence regimes. A systematic extension of the deterministic extraction and bias analysis to neigh b orhoo ds of exceptional p oin ts would require a careful treatment of generalized resonant states and p ossibly higher-order p oles, together with sharp stabilit y b ounds for conﬂuen t Pron y systems, cf. [ 29 ]. Completeness, Keldysh-type expansions, and global mo de decomp ositions. The present w ork is delib erately lo cal in the sp ectral plane: it relies on isolating a ﬁxed o v ertone band and a pair of equatorial branches, rather than on global completeness of quasinormal mo des. A t the same time, there has b een notable recent progress on global expansion frameworks. A h yp erboloidal Keldysh-type approach to quasinormal mo de expansions has b een dev elop ed in [ 22 ], and complementary p ersp ectiv es on complete mo de decomp ositions b ey ond pure QNM sums hav e b een prop osed, for instance, in [ 23 ]. Connecting such global expansion theories with the deterministic in v erse analysis pursued here would require uniform quantitativ e control of remainders (tails, branch-cut contributions, and high o vertones) in norms compatible with the inv erse map, together with robust b ounds on the growth of the relev an t sp ectral pro jectors. F urther directions. The present series fo cuses on the scalar wa v e equation on Kerr–de Sitter, where the meromorphic resolv ent theory and the high- frequency semiclassical mac hinery are particularly clean. Extending the in v erse framew ork to other ﬁeld spins and to more general observ ation mo dels (for instance, m ultiple channels or non-equatorial pro jections) would b e 80 RUILIANG LI v aluable. Another natural direction is to relax the small-rotation restriction b y combining the equatorial semiclassical pac k age of [ 1 ] with global estimates for the full sub extremal Kerr–de Sitter family , suc h as [ 11 ]. Finally , while the de Sitter setting provides exp onen tial decay and a voids asymptotically ﬂat branc h-cut tails, it w ould b e of substantial in terest to develop analogues of the present deterministic inv erse b ounds for asymptotically ﬂat Kerr, where p olynomial late-time tails and contin uum contributions are una v oidable and m ust b e incorp orated in to the error b ookkeeping. Appendix A. Fourier–Lapla ce conventions and the distributional source term This app endix ﬁxes sign conv en tions and records the distributional iden ti- ties used in Section 2 . A.1. F orw ard F ourier–Laplace transform. F or a distribution v sup- p orted in t ≥ 0 we use the forward F ourier–Laplace transform b v ( ω ) := Z ∞ 0 e iω t v ( t ) dt, ℑ ω > C 0 , and the inv erse form ula (for suitable C > C 0 ) v ( t ) = 1 2 π Z ℑ ω = C e − iω t b v ( ω ) dω . With this conv en tion one has, for test functions ϕ , ⟨ δ, ϕ ⟩ = ϕ (0) , ⟨ δ ′ , ϕ ⟩ = − ϕ ′ (0) , and therefore (A.1) b δ ( ω ) = 1 , b δ ′ ( ω ) = − iω . A.2. Resolv en t identit y for the Cauc h y problem. Let u solv e the w av e equation P u = 0 with initial data ( u, ∂ t ∗ u ) | t ∗ =0 = ( f 0 , f 1 ), and let u + := 1 t ∗ ≥ 0 u . A direct computation in distributions gives P u + = δ ′ ( t ∗ ) f 0 + δ ( t ∗ ) ( f 1 + Qf 0 ) . T aking the F ourier–Laplace transform and using ( A.1 ) yields P ( ω ) b u + ( ω ) = ( f 1 + Qf 0 ) − iω f 0 , so that, whenever R ( ω ) = P ( ω ) − 1 exists, b u + ( ω ) = R ( ω )  f 1 + ( Q − iω ) f 0  . This is precisely Lemma 2.9 . EQUA TORIAL KERR–DE SITTER RINGDOWN 81 A.3. Motiv ation for the forced formulation. In the main text we av oid distributional source terms by passing to the forced problem with a smo oth cutoﬀ in time: we set u ϑ ( t ∗ ) := ϑ ( t ∗ ) u ( t ∗ ) with a ﬁxed ϑ ∈ C ∞ supp orted in (0 , ∞ ) and equal to 1 for t ∗ ≥ 1. Then u ϑ satisﬁes P u ϑ = F ϑ with F ϑ smo oth and compactly supp orted in time, and u ϑ = u for t ∗ ≥ 1. This is the starting p oin t for the con tour deformation argumen ts in Section 4 and for the analytically window ed ev olution in Section 5 . Appendix B. Dual resonant st a tes, bior thogonality, and excit a tion amplitudes This app endix records a standard rank-one formula for the residue pro- jector at a simple p ole of a meromorphic F redholm resolven t and explains ho w the sc alar amplitude seen by a detector factors through a pairing with a dual (left) resonant state. While the contour deformation argumen ts of Section 4 do not rely on these iden tities, they provide an op erator-theoretic in terpretation of the detectabilit y requirement in Section 7.2 and clarify ho w nonnormality may en ter through the size of the residue pro jector. F or bac kground on meromorphic F redholm families and Keldysh-t yp e expansions w e refer to [ 31 , 30 , 22 ]. B.1. Rank-one structure of the residue at a simple p ole. Let X , Y b e complex Banach spaces and let P ( ω ) : X → Y b e a holomorphic family of F redholm op erators of index 0 in a neighbourho o d of ω 0 ∈ C . Assume that P ( ω ) is in vertible for some ω in this neigh b ourhoo d, and denote the meromorphic inv erse by R ( ω ) = P ( ω ) − 1 . Supp ose that ω 0 is a simple p ole of R and that dim k er P ( ω 0 ) = dim k er P ( ω 0 ) ∗ = 1 , where P ( ω 0 ) ∗ : Y ∗ → X ∗ denotes the Banac h space adjoin t. Cho ose nonzero v ectors u 0 ∈ ker P ( ω 0 ) ⊂ X , v 0 ∈ ker P ( ω 0 ) ∗ ⊂ Y ∗ , and use the duality pairing ⟨· , ·⟩ : Y × Y ∗ → C . Since v 0 annihilates Ran P ( ω 0 ), we hav e ⟨ P ( ω 0 ) w , v 0 ⟩ = 0 for all w ∈ X . Lemma B.1 (Residue pro jector at a simple p ole) . Under the ab ove assump- tions, (B.1)  ∂ ω P ( ω 0 ) u 0 , v 0   = 0 , and the r esidue op er ator Π ω 0 := Res ω = ω 0 R ( ω ) : Y → X is the r ank-one op er ator (B.2) Π ω 0 f = ⟨ f , v 0 ⟩ ⟨ ∂ ω P ( ω 0 ) u 0 , v 0 ⟩ u 0 , f ∈ Y . In p articular, Ran Π ω 0 = k er P ( ω 0 ) and k er Π ω 0 ⊃ Ran P ( ω 0 ) . 82 RUILIANG LI Pr o of. Since R has a simple p ole at ω 0 , there exists a holomorphic family R 0 ( ω ) near ω 0 and a ﬁnite-rank op erator Π ω 0 suc h that (B.3) R ( ω ) = 1 ω − ω 0 Π ω 0 + R 0 ( ω ) . Expand P ( ω ) = P ( ω 0 ) + ( ω − ω 0 ) ∂ ω P ( ω 0 ) + O  ( ω − ω 0 ) 2  and use P ( ω ) R ( ω ) = Id Y for ω  = ω 0 . Substituting ( B.3 ) giv es ( ω − ω 0 ) − 1 P ( ω 0 )Π ω 0 +  ∂ ω P ( ω 0 )Π ω 0 + P ( ω 0 ) R 0 ( ω 0 )  + O ( ω − ω 0 ) = Id Y . The co eﬃcien t of ( ω − ω 0 ) − 1 m ust v anish, so P ( ω 0 )Π ω 0 = 0 and hence Ran Π ω 0 ⊂ ker P ( ω 0 ) = span { u 0 } . Thus there exists ℓ ∈ Y ∗ suc h that (B.4) Π ω 0 f = ⟨ f , ℓ ⟩ u 0 , f ∈ Y . Similarly , the identit y R ( ω ) P ( ω ) = Id X implies Π ω 0 P ( ω 0 ) = 0, so ℓ annihi- lates Ran P ( ω 0 ). Since dim k er P ( ω 0 ) ∗ = 1, the cok ernel is one-dimensional and therefore ℓ is a nonzero multiple of v 0 , say ℓ = c v 0 . Applying the functional v 0 to the constan t term identit y and using v 0 ◦ P ( ω 0 ) = 0 yields  ∂ ω P ( ω 0 )Π ω 0 f , v 0  = ⟨ f , v 0 ⟩ , f ∈ Y . With ( B.4 ) and ℓ = c v 0 , the left-hand side equals c ⟨ f , v 0 ⟩ ⟨ ∂ ω P ( ω 0 ) u 0 , v 0 ⟩ . Since v 0  = 0, this forces ( B.1 ) and c = ⟨ ∂ ω P ( ω 0 ) u 0 , v 0 ⟩ − 1 , whic h gives ( B.2 ). □ R emark B.2 (Normalisation) . The represen tation ( B.2 ) is inv ariant under rescaling u 0 7→ αu 0 and v 0 7→ β v 0 . One may therefore imp ose the normalisa- tion (B.5)  ∂ ω P ( ω 0 ) u 0 , v 0  = 1 , in which case Π ω 0 f = ⟨ f , v 0 ⟩ u 0 . In Hilb ert settings where X = Y (or after iden tifying X with a graph space on Y ), this corresp onds to the familiar righ t/left eigenv ector normalisation for non-selfadjoint eigen v alue problems. B.2. Detector amplitudes as dual pairings. W e no w connect Lemma B.1 to the resonance expansion of Section 4 . Recall that the stationary family for the wa v e op erator is deﬁned by ( 2.14 ) , and that the resonant term asso ciated with a p ole ω j is obtained by taking the residue of the integrand e − iω t ∗ χR ( ω ) χ b F ϑ ( ω ) (cf. Theorem 4.9 ). Assume that ω 0 is a p ole of R ( ω ) whic h is simple and has one-dimensional resonan t and co-resonan t spaces, so that Lemma B.1 applies. Let u 0 b e a (righ t) resonant state and v 0 a dual (left) resonant state, normalised by ( B.5 ) . Then the residue con tribution can b e written as (B.6) χ Π ω 0 χ b F ϑ ( ω 0 ) =  χ b F ϑ ( ω 0 ) , v 0  χu 0 . EQUA TORIAL KERR–DE SITTER RINGDOWN 83 Th us, for any b ounded detector O : H s +1 → C supp orted where χ ≡ 1, the corresp onding scalar amplitude satisﬁes (B.7) O  χ Π ω 0 χ b F ϑ ( ω 0 )  =  χ b F ϑ ( ω 0 ) , v 0  O ( χu 0 ) . In particular, the nonv anishing of an observ ed mo de amplitude factors in to t wo indep enden t requirements: (i) the detector must not annihilate the spatial proﬁle χu 0 , and (ii) the initial data (equiv alently , the forcing b F ϑ ( ω 0 )) m ust hav e nontrivial pairing with the dual state v 0 . R emark B.3 (Excitation factors and nonnormalit y) . If one uses an unnor- malised pair ( u 0 , v 0 ), the amplitude inv olv es the denominator ⟨ ∂ ω P ( ω 0 ) u 0 , v 0 ⟩ in ( B.2 ). Smallness of this quantit y corresp onds to a large residue op erator norm and to large “excitation factors” in the ph ysics literature; it also fea- tures in recent w ork on av oided crossings and exceptional p oin ts in black-hole ringdo wn [ 24 , 16 ]. This is a precise sense in which nonnormalit y ma y magnify observ ed ringdown amplitudes, ev en when the deca y rates are ﬁxed. B.3. Relation to Keldysh expansions. The rank-one formula ( B.2 ) is the simplest instance of a more general principle: near a p ole of a meromorphic F redholm resolv ent, the principal part can b e written explicitly in terms of righ t/left ro ot vectors (Jordan c hains) and their biorthogonality relations. F or higher-order p oles this pro duces the p olynomial prefactors in time that already app ear in the general resonance expansion of Theorem 4.9 ; in matrix and op erator-p encil language this is precisely the con tent of Keldysh-t yp e theorems. In the black-hole setting, a hyperb oloidal formulation makes these biorthogonal constructions particularly natural and has recently b een exploited to obtain sp ectral v ersions of resonance expansions; see [ 22 ] and the references therein, as w ell as the c omplemen tary con v ergent mo de-sum p erspective dev elop ed in [ 23 ]. In the present paper we do not assume any global completeness of resonant states. Instead, we isolate a ﬁnite set of p oles in the equatorial pack age (Sections 5 – 7 ) and control all remaining p oles b y a contour remainder and by microlo cal sectorization. App endix B is included only to provide a structural in terpretation of the amplitudes and of Assumption 7.4 . Appendix C. Anal ytic pole selection and band isola tion This app endix isolates the complex-analytic mechanism b ehind the “ana- lytic p ole selection” used in Section 5 . The guiding principle is: R esidue c alculus is only available for mer omorphic inte gr ands. Consequen tly , frequency lo calization in a con tour representation cannot b e ac hiev ed by inserting non-holomorphic cutoﬀs (for instance dep ending on ℜ ω ), but it c an b e ac hieved by multiplying by carefully c hosen entir e w eigh ts of controlled growth. 84 RUILIANG LI C.1. Why non-holomorphic cutoﬀs cannot be used in residue calcu- lus. Consider a contour in tegral of the form Z ℑ ω = C e − iω t R ( ω ) b F ( ω ) dω , where R ( ω ) is meromorphic. If one inserts a factor dep ending on ℜ ω —for instance ψ ( h ℜ ω ) for a smo oth cutoﬀ ψ —then the integrand is no longer holomorphic (or meromorphic) as a function of the complex v ariable ω . In that case one cannot justify deforming the contour and collecting residues” b y Cauch y’s theorem. F or this reason, Section 5.2 uses the entir e weigh ts e g ± ,ℓ deﬁned in ( 5.7 ) ; m ultiplying by an en tire function preserves meromorphicit y . C.2. Lagrange weigh ts at pseudop oles: an abstract lemma. W e now record a general interpolation mechanism which is not sp eciﬁc to Kerr– de Sitter, and whic h may b e useful whenev er one has a ﬁnite collection of (semi)classical pseudop oles approximating true p oles. Fix n ∈ N 0 and let { Ω ♯ j } n j =0 ⊂ C b e pairwise distinct. F or each m ∈ { 0 , 1 , . . . , n } deﬁne the Lagrange p olynomial (C.1) G m ( ω ) := Y 0 ≤ j ≤ n j  = m ω − Ω ♯ j Ω ♯ m − Ω ♯ j , ω ∈ C . Then G m is en tire of degree n and satisﬁes G m (Ω ♯ m ) = 1 and G m (Ω ♯ j ) = 0 for j  = m . Prop osition C.1 (Robust p ole selection b y pseudop ole in terp olation) . L et { Ω ♯ j } n j =0 and G m b e as ab ove, and set d ♯ := min j  = k | Ω ♯ j − Ω ♯ k | > 0 . Assume that { Ω j } n j =0 ⊂ C satisﬁes | Ω j − Ω ♯ j | ≤ δ, j = 0 , 1 , . . . , n, for some δ ∈ (0 , d ♯ / 8] . Then for every m ∈ { 0 , . . . , n } one has | G m (Ω m ) − 1 | ≤ C Lag n δ d ♯ , (C.2) | G m (Ω j ) | ≤ C Lag n δ d ♯ , j ∈ { 0 , . . . , n } \ { m } , (C.3) wher e C Lag n > 0 dep ends only on n . Mor e over, for every ﬁxe d η ∈ R ther e is a c onstant C η ,n such that (C.4) | G m ( σ + iη ) | ≤ C η ,n (1 + | σ | ) n , σ ∈ R , uniformly in m . EQUA TORIAL KERR–DE SITTER RINGDOWN 85 Pr o of. Fix m . Step 1: uniform b ounds ne ar the interp olation no des. Let r := d ♯ / 4 and consider the closed disks B j := {| ω − Ω ♯ j | ≤ r } , j = 0 , . . . , n . They are pairwise disjoint by deﬁnition of d ♯ . F or ω ∈ B j and k  = j , w e ha v e | ω − Ω ♯ k | ≤ | Ω ♯ j − Ω ♯ k | + r ≤  1 + 1 4  | Ω ♯ j − Ω ♯ k | , and also | ω − Ω ♯ k | ≥ | Ω ♯ j − Ω ♯ k | − r ≥  1 − 1 4  | Ω ♯ j − Ω ♯ k | . Using ( C.1 ) and the fact that eac h denominator satisﬁes | Ω ♯ m − Ω ♯ k | ≥ d ♯ , we obtain a crude b ound (C.5) sup ω ∈ S n j =0 B j | G m ( ω ) | ≤  5 3  n . (The constan t is immaterial; any uniform b ound dep ending only on n is suﬃcien t.) Step 2: Lipschitz b ound on e ach disk. Since G m is holomorphic, Cauch y’s estimate on each disk B j giv es sup ω ∈ B j | G ′ m ( ω ) | ≤ 1 r sup ω ∈ B j | G m ( ω ) | ≤ 1 r  5 3  n = C Lag n 1 d ♯ , with C Lag n := 4(5 / 3) n . Step 3: evaluate at p erturb e d no des. Since δ ≤ d ♯ / 8 = r / 2, each p erturb ed no de Ω j lies in B j . F or j = m we write G m (Ω m ) − 1 = G m (Ω m ) − G m (Ω ♯ m ) , and apply the mean v alue theorem along the segment from Ω ♯ m to Ω m inside B m to obtain | G m (Ω m ) − 1 | ≤ sup ω ∈ B m | G ′ m ( ω ) | | Ω m − Ω ♯ m | ≤ C Lag n δ d ♯ , whic h is ( C.2 ). Similarly , for j  = m we use G m (Ω ♯ j ) = 0 and obtain ( C.3 ). Step 4: p olynomial gr owth on horizontal lines. The growth b ound ( C.4 ) is immediate from ( C.1 ) since G m is a degree- n p olynomial: for ω = σ + iη , | G m ( ω ) | ≤ Y j  = m | ω | + | Ω ♯ j | | Ω ♯ m − Ω ♯ j | ≤  1 d ♯  n Y j  = m  | σ | + | η | + | Ω ♯ j |  ≤ C η ,n (1 + | σ | ) n , absorbing max j | Ω ♯ j | into C η ,n (for ﬁxed n ). □ R emark C.2 (Sp ecialization to the weigh ts g ± ,ℓ and e g ± ,ℓ ) . In Section 5 , for eac h sign ± w e ﬁx the pseudopoles Ω ♯ j = ω ♯ j, ± ,ℓ (0 ≤ j ≤ n ) from Theorem 5.1 , and w e tak e m = n in ( C.1 ) . The resulting Lagrange polynomial G n is exactly the interpolation weigh t g ± ,ℓ in ( 5.7 ) . The mo diﬁed window e g ± ,ℓ in ( 5.10 ) 86 RUILIANG LI is obtained b y multiplying g ± ,ℓ b y the additional factor ( ω /ω ♯ ± ,ℓ ) m 0 ; this preserv es the interpolation identities ( 5.8 ) and do es not aﬀect the residue calculus since it is entire. Since the true p oles satisfy ω j, ± ,ℓ = ω ♯ j, ± ,ℓ + O ( ℓ −∞ ) and the la yer gaps are uniform, Prop osition C.1 yields g ± ,ℓ ( ω ± ,ℓ ) = 1 + O ( ℓ −∞ ) , g ± ,ℓ ( ω j, ± ,ℓ ) = O ( ℓ −∞ ) (0 ≤ j ≤ n − 1) . Because ( ω /ω ♯ ± ,ℓ ) m 0 is uniformly b ounded on S 0 ≤ j ≤ n D j, ± ,ℓ , the same esti- mates hold with g ± ,ℓ replaced by e g ± ,ℓ ; this is Lemma 5.17 . C.3. P olynomial growth and contour shifts for the forced problem. The window ed evolution in Section 5.2 in v olves integrals of the form Z ℑ ω = C e − iω t g ( ω ) R ( ω ) b F ϑ ( ω ) dω , with g en tire and of p olynomial growth on horizontal lines (for e g ± ,ℓ one has degree n + m 0 ). Assume that g is entire and satisﬁes, for every ﬁxed η ∈ R , a p olynomial gro wth b ound (C.6) | g ( σ + iη ) | ≤ C η (1 + | σ | ) K , σ ∈ R , for some K ≥ 0. Because F ϑ ( t ∗ , · ) is smo oth and compactly supp orted in t ∗ , its F ourier–Laplace transform satisﬁes (for every N ∈ N and ﬁxed η ∈ R ) (C.7) ∥ b F ϑ ( σ + iη ) ∥ H s − 1 ≤ C s,N ,η (1 + | σ | ) − N , σ ∈ R , obtained by rep eated integration b y parts in t ∗ . Moreov er, for | ω | large the stationary family P ( ω ) is elliptic and R ( ω ) = P ( ω ) − 1 is p olynomially b ounded in op erator norm on eac h horizon tal line. Consequen tly , if g satisﬁes ( C.6 ) (for e g ± ,ℓ this holds with K = n + m 0 ), then the integrand decays as |ℜ ω | → ∞ along horizontal segments, and the v ertical sides in the standard rectangular contour argument v anish as the rectangle expands. This justiﬁes con tour shifts for the window ed integral. C.4. Band isolation by contour subtraction. Besides interpolation w eights, the pap er uses another analytic to ol which is often more robust: b and isolation by subtracting tw o con tour represen tations. This metho d isolates the p oles in a horizontal strip without requiring any smallness of w eigh ts at other p oles. Prop osition C.3 (Contour subtraction isolates a p ole band) . L et R ( ω ) b e a mer omorphic family of b ounde d op er ators b etwe en ﬁxe d Sob olev sp ac es, and let g b e entir e satisfying a p olynomial gr owth b ound ( C.6 ) on horizontal lines. Assume b F ( ω ) satisﬁes the r apid de c ay ( C.7 ) on horizontal lines. Fix ν 1 < ν 2 such that R ( ω ) has no p oles on the lines ℑ ω = − ν 1 and ℑ ω = − ν 2 . Deﬁne I ν ( t ) := 1 2 π Z ℑ ω = − ν e − iω t g ( ω ) R ( ω ) b F ( ω ) dω . EQUA TORIAL KERR–DE SITTER RINGDOWN 87 Then for every t ≥ 0 one has the identity (C.8) I ν 1 ( t ) − I ν 2 ( t ) = i X ω j ∈ Poles( R ) − ν 2 < ℑ ω j < − ν 1 e − iω j t Res ω = ω j  g ( ω ) R ( ω ) b F ( ω )  , wher e the sum r anges over p oles in the op en strip − ν 2 < ℑ ω < − ν 1 . Pr o of. Let Γ R b e the p ositiv ely oriented rectangle with horizontal sides on ℑ ω = − ν 1 and ℑ ω = − ν 2 and vertical sides at ℜ ω = ± R . By the residue theorem, Z Γ R e − iω t g ( ω ) R ( ω ) b F ( ω ) dω = 2 π i X ω j ∈ Poles( R ) ω j ∈ Int(Γ R ) e − iω j t Res ω = ω j  g ( ω ) R ( ω ) b F ( ω )  . The contribution of the horizon tal sides is exactly 2 π ( I ν 1 ( t ) − I ν 2 ( t )). The con tribution of the vertical sides tends to 0 as R → ∞ b y the deca y of b F in ( C.7 ) , the p olynomial gro wth of g , and the p olynomial b ound of R ( ω ) on horizon tal lines. Letting R → ∞ yields ( C.8 ). □ R emark C.4 (Relation to Prop osition 5.21 ) . Prop osition 5.21 in Section 5 is a sp ecialization of Prop osition C.3 to the forced Kerr–de Sitter ev olution, with the entire window g ( ω ) = e g ± ,ℓ ( ω ) and with ν 1 , ν 2 c hosen b et w een consecutive o v ertone lay ers. In particular, one may also take g ≡ 1 to isolate the ov ertone strip purely by contour subtraction; see Remark 5.24 . C.5. Microlo cal sectorization (input from the companion pap er). The analytic weigh ts and con tour subtraction isolate which p oles c an c on- tribute . T o show that only the equatorial p oles contribute to the microlo cal- ized signals, w e additionally use the sectorization estimate of Theorem 5.4 (see App endix D and Remark D.1 ): after applying the equatorial cutoﬀ A ± ,h ℓ and the azim uthal pro jector Π k ± , all other p oles ab o v e the c hosen contour are suppressed by O ( ℓ −∞ ). Appendix D. Comp anion-p aper inputs used in Sections 5 – 7 Sections 5 and 7 rely on three families of high–frequency inputs prov ed in the companion pap er [ 1 ]: (i) a stable lab eling of the equatorial branches together with sup er–polynomial pseudop ole appro ximation and ev en tual p ole simplicit y; (ii) a quan titative microlo cal se ctorization statem en t con trolling the generalized Laurent co eﬃcien ts of the resolven t after equatorial and azim uthal lo calization; and (iii) a quantitativ e lo cal inv ersion estimate for reco v ering ( M , a ) from the equatorial pair of QNMs. The goal of this appendix is not to repro duce the companion pro ofs in full, but to mak e the dep endence of the presen t pap er explicit: w e indicate precisely where eac h input app ears 88 RUILIANG LI in [ 1 ] and we provide a short pro of outline for sectorization, since this is the most technical black b o x used in the t wo-mode dominance argument. Throughout this app endix w e work on the compact slo w–rotation set K ﬁxed in Section 5 and suppress the dep endence of constants on K . D.1. Equatorial p ole lab eling, simplicity , and o v ertone gaps. Dy- atlo v’s semiclassical quan tization of Kerr–de Sitter QNMs in a ﬁxed-width strip (slo w rotation) pro duces an explicit pseudop ole set and sho ws that true QNMs are sup er–p olynomially close to it; see [ 6 , 5 ]. In the form used in [ 1 ], this app ears as [ 1 , Theorem 15] together with a stable lab eling reformulation [ 1 , Prop osition 20]. Sp ecializing to the equatorial pair k = ± ℓ and to the ﬁrst n + 1 o vertones yields exactly the statemen t recorded in Theorem 5.1 of the main text, including the disjoin tness of the equatorial disks and the sup er–polynomial pro ximity es timate ( 5.2 ). Simplicit y of the lab eled p oles for ℓ ≫ 1 follows from the same barrier–top Grushin reduction that pro duces the quan tization function: the eﬀectiv e scalar quantization function has a simple zero at each lab eled p ole, uniformly on compact parameter sets. This is recorded in [ 1 , Remark 21] and is com- patible with the analytic F redholm framew ork and the analytic dep endence of isolated simple p oles; compare [ 10 , 11 ]. Finally , the choice of a contour height separating the n th and ( n + 1)st equatorial la y ers (Lemma 5.15 in the main text) is a consequence of the barrier–top normal form: for ﬁxed sign ± one has an expansion of the form −ℑ ω ♯ j, ± ,ℓ ( M , a ) = ( j + 1 2 ) λ ± ( M , a ) + O ( ℓ − 1 ) , where λ ± ( M , a ) > 0 is the co ordinate-time Lyapuno v exp onen t of the cor- resp onding e quatorial trapp ed null orbit; see [ 1 , § 5] and the geometric computation in [ 1 , App endix C]. Since λ ± is contin uous and strictly p ositive on the sub extremal set, it has a p ositiv e minimum on K , which giv es a uniform vertical gap b et w een consecutive o v ertones for ℓ ≫ 1. D.2. Microlo cal sectorization and suppression of non-equatorial p oles. W e next indicate why the sectorization estimate in Theorem 5.4 holds. The k ey p oin t is that the residue op erators Π [ q ] ω j arising from the Lauren t expansion of the resolven t are op erators on the sp atial Sob olev scale H s ( X ); the relev ant mapping prop ert y is therefore H s − 1 ( X ) → H s +1 ( X ), matc hing the wa y the forced transform b F ϑ ( ω ) enters the residue calculus. R emark D.1 (Proof outline for Theorem 5.4 ) . W e sk etc h the argumen t underlying ( 5.4 ) ; full details (including parameter-uniform b ounds and the Grushin setup) are in [ 1 , § 5 and App endix B]. Step 1: k –mo de reduction and angular semiclassical scaling. After applying the azimuthal pro jector Π k ± and setting h = h ℓ = ℓ − 1 , the stationary equation b ecomes a semiclassical problem for a non-selfadjoint family P ± ( h, ω ) with real principal symbol on T ∗ X . The microlo cal cutoﬀ A ± ,h is chosen so that its semiclassical w a vefron t set lies in a small neigh b orhoo d of the EQUA TORIAL KERR–DE SITTER RINGDOWN 89 equatorial trapp ed set corresp onding to the sign ± ; the supp orts for + and − are disjoint by construction. Step 2: microlo cal inv ertibilit y outside the equatorial disks. F or ω in a ﬁxed strip {ℑ ω > − ν } , the Hamilton ﬂo w has no trapped tra jectories in the complemen t of the equatorial neigh b orhoo d selected by A ± ,h . Using elliptic regularity where P ± ( h, ω ) is elliptic and propagation of semiclassical singularities where it is of real principal type [ 2 ], one constructs a microlo cal in v erse E ± ( h, ω ) satisfying A ± ,h P ± ( h, ω ) E ± ( h, ω ) = A ± ,h + O ( h ∞ ) on H s − 1 → H s − 1 , uniformly for ω a wa y from the equatorial disks D m, ± ,ℓ . Step 3: Grushin reduction in the equatorial phase-space c hannel. Near the equatorial trapp ed set one sets up a Grushin problem for P ± ( h, ω ) with auxiliary op erators ( R + , R − ) so that the extended op erator is microlo cally in v ertible with inv erse E ± ( h, ω ). The eﬀective Hamiltonian E − + ( h, ω ) is a ﬁnite-dimensional matrix whose determinan t is (after normalization) the quan tization function; its zeros are exactly the equatorial p oles inside the disks D m, ± ,ℓ , and it is uniformly inv ertible outside these disks with in v erse O ( h ∞ ). Consequently , mo dulo O ( h ∞ ) remainders, the meromorphic part of the microlo calized resolven t is entirely captured by the equatorial p oles. Step 4: Cauch y form ula for Laurent co eﬃcien ts. Let ω j / ∈ S m ≤ n D m, ± ,ℓ b e a p ole with ℑ ω j > − ν . Cho ose a small circle γ around ω j con tained in {ℑ ω > − ν } and disjoint from the equatorial disks. Since A ± ,h Π k ± R ( ω )Π k ± A ± ,h is holomorphic on and inside γ modulo O ( h ∞ ), Cauch y’s formula giv es Π [ q ] ω j = 1 2 π i I γ ( ω − ω j ) q − 1 R ( ω ) dω , and hence χA ± ,h Π k ± Π [ q ] ω j χ = O ( h ∞ ) on H s − 1 → H s +1 after comp osing with χ and using the radial p oin t/propagation estimates needed to mov e b et w een microlo cal regions. Conv erting h ∞ to ℓ − N yields ( 5.4 ). D.3. Lo cal in v ersion from a pair of equatorial QNMs. F or complete- ness, we record where the stabilit y estimate used in Section 7 is prov ed in the companion paper. Theorem 7.1 in the main text is exactly the t wo–parameter true–QNM in v erse theorem [ 1 , Theorem 48], whose pro of combines the pseu- dop ole inv erse theorem [ 1 , Theorem 35] with the C 1 pseudop ole–QNM pro ximit y estimate [ 1 , Lemma 45]. Appendix E. Uniformity with respect to the cosmological const ant on comp act sets This app endix records a parameter-uniformity statement used when pass- ing from the ﬁxed-Λ analysis in Sections 2 – 6 to the three-parameter bias b ound in Section 7.6 . The p oin t is that the constants in the energy estimates, resolv en t b ounds, microlo cal cutoﬀs, and Grushin reductions can b e chosen uniformly on compact three-parameter sets. 90 RUILIANG LI Prop osition E.1 (Uniform forward constan ts on compact ( M , a, Λ) sets) . L et K (3) b e a c omp act subset of the slow-r otation sub extr emal Kerr–de Sitter p ar ameter r e gion. Then one may cho ose the a uxiliary ge ometric p ar ameters (buﬀer size δ , the r e gular time function t ∗ , and the sp atial slic e X M ,a ) uni- formly on K (3) , and al l c onstants in the forwar d analysis of Se ctions 2 – 6 c an b e taken uniform for ( M , a, Λ) ∈ K (3) . In p articular, onc e a p ole-fr e e c ontour is ﬁxe d uniformly on K (3) , the c onstants in The or em 5.23 and Pr op o- sition 7.10 may b e chosen uniformly for ( M , a, Λ) ∈ K (3) . Pr o of. W e indicate the three inputs where the background parameters en ter. (i) Ge ometric c onstructions and Sob olev c onventions. On the sub extremal region, the horizon radii r e ( M , a, Λ) and r c ( M , a, Λ) dep end smo othly on the parameters, and the same holds for the metric co eﬃcients in star co ordinates. Since K (3) is compact and con tained in the sub extremal set, the separation r c − r e has a p ositiv e low er b ound on K (3) . Thus one may choose a single buﬀer size δ > 0 so that the extended radial interv al ( r e − δ, r c + δ ) is w ell- deﬁned for all ( M , a, Λ) ∈ K (3) . With this c hoice, the spatial slices and the asso ciated Sob olev norms v ary smo othly with the parameters and remain uniformly equiv alen t on K (3) . (ii) F r e dholm setup and r e dshift estimates. The meromorphic/F redholm resol- v en t framew ork for Kerr–de Sitter [ 2 ] is stable under smo oth p erturbations of the metric. On compact parameter families one can choose the v ariable- order radial p oin t spaces and the redshift weigh ts with uniform constan ts; in particular, the energy estimate of Theorem 2.3 holds with constan ts uniform on K (3) . (iii) High-ener gy r esolvent estimates and the e quatorial p ackage. Normally h yp erbolic trapping p ersists throughout the sub extremal Kerr–de Sitter family , and the dynamical quan tities en tering the semiclassical estimate dep end con tin uously on the parameters; see [ 3 , 11 ]. Therefore the semi- classical cutoﬀ resolv en t bound in Theorem 3.3 can b e arranged uniformly for ( M , a, Λ) ∈ K (3) after ﬁxing a compact energy set I ⋐ R \ { 0 } . The equatorial sectorization and Grushin reductions used later are prov ed in the companion pap er with parameter-uniform estimates on compact sets [ 1 , § 5 and App endix B]. Com bining these observ ations giv es the claimed uniformit y on K (3) . □ Appendix F. Two-exponential models near coalescence and exceptional points This app endix complements Lemma 6.14 b y making explicit the algebraic condition num b ers gov erning t w o-frequency extraction. The goal is not to introduce a new algorithm, but to record in a self-contained wa y wh y an y “t w o-mo de-at-once” reconstruction b ecomes ill-conditioned as the no des approac h coalescence (the exceptional p oint limit). EQUA TORIAL KERR–DE SITTER RINGDOWN 91 F.1. Hankel determinan ts and V andermonde conditioning. Consider the noiseless tw o-exponential data (F.1) y j = a 1 z j 1 + a 2 z j 2 , j ∈ N 0 , with a 1 a 2  = 0 and z 1  = z 2 . The basic linear-algebraic ob ject is the 2 × 2 Hank el matrix H 0 :=  y 0 y 1 y 1 y 2  . A direct computation giv es the factorization (F.2) H 0 =  1 1 z 1 z 2   a 1 0 0 a 2   1 z 1 1 z 2  , so in particular (F.3) det H 0 = y 0 y 2 − y 2 1 = a 1 a 2 ( z 1 − z 2 ) 2 . Th us ∥ H − 1 0 ∥ necessarily blo ws up at least like | a 1 a 2 | − 1 | z 1 − z 2 | − 2 as z 1 → z 2 . This is the ﬁrst mechanism b ehind the loss in Lemma 6.14 . The second mec hanism is ro ot sensitivit y: ev en if one kno ws the symmetric functions s 1 = z 1 + z 2 and s 2 = z 1 z 2 accurately , con v erting them into the individual no des requires taking ro ots of λ 2 − s 1 λ + s 2 = 0, and the map ( s 1 , s 2 ) 7→ ( z 1 , z 2 ) has deriv ativ e blowing up like | z 1 − z 2 | − 1 . F.2. A deterministic conditioning estimate. Prop osition F.1 (Conditioning of the t wo-node Pron y map) . L et y j b e the noiseless samples ( F.1 ) and let e y j = y j + e j for j = 0 , 1 , 2 , 3 , wher e | e j | ≤ η . Assume a 1 a 2  = 0 and z 1  = z 2 , and set ∆ 0 := y 0 y 2 − y 2 1 = a 1 a 2 ( z 1 − z 2 ) 2 , R := max {| z 1 | , | z 2 |} . Ther e exists an absolute c onstant c 0 > 0 (dep ending only on crude b ounds on R, | a 1 | , | a 2 | ) such that if η ≤ c 0 | a 1 a 2 | | z 1 − z 2 | 4 , then the Pr ony r e c onstruction describ e d in L emma 6.14 pr o duc es no des e z 1 , e z 2 which c an b e lab ele d so that (F.4) max k =1 , 2 | e z k − z k | ≤ C ( R, a 1 , a 2 ) η | a 1 a 2 | | z 1 − z 2 | 3 . In p articular, the lo c al Lipschitz c onstant of the inverse map fr om four samples to two no des blows up at le ast like | a 1 a 2 | − 1 | z 1 − z 2 | − 3 as z 1 → z 2 . Pr o of. The statement is a reorganization of the pro of of Lemma 6.14 : ( F.3 ) sho ws the | z 1 − z 2 | − 2 loss when solving the 2 × 2 Hankel system for ( s 1 , s 2 ), and Rouc h ´ e’s theorem together with the identit y p ( e z k ) = ( e z k − z k )( e z k − z 3 − k ) yields an additional factor | z 1 − z 2 | − 1 in passing from ( s 1 , s 2 ) to the individual ro ots. All constan ts can be track ed explicitly and are collected in to C ( R, a 1 , a 2 ). □ 92 RUILIANG LI F.3. The exceptional p oin t limit and conﬂuen t mo dels. If z 1 = z 2 =: z in ( F.1 ), the mo del collapses to a c onﬂuent exp onen tial sum: one has y j = ( b 0 + b 1 j ) z j , b 0 = a 1 + a 2 , b 1 = a 2 − a 1 , whic h corresponds, in the original time v ariable, to a mo de of the form ( α + β t ) e − iω t . In this regime the “tw o distinct no des” in verse problem is ill-p osed, and stable reco very requires conﬂuent Prony-t ype metho ds and a diﬀerent parametrization. Quan titative stability b ounds for conﬂuent Pron y systems, and sharp dep endence on the separation scale in the near- conﬂuen t regime, are discussed in [ 29 ]. F or the Kerr–de Sitter application, this is the analytic reason why near exceptional p oin ts one exp ects strong parameter sensitivity when attempting to separate nearb y mo des without prior microlo cal isolation. References [1] R. 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Two-mode dominance and deterministic parameter bias bounds for equatorial Kerr-de Sitter ringdown

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