Spectral Structure of the Mixed Hessian of the Dispersionless Toda $τ$-Function

SPECTRAL STR UCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TOD A τ -FUNCTION OLEG ALEKSEEV Abstra ct. W e analyze the mixed Hessian of the dispersionless T o da τ -function for the s -fold symmetric one-harmonic p olynomial conformal map. The inv erse branc h exhibits tw o distinct thresholds: an analytic threshold ζ c , where the dominan t square-ro ot singularit y reac hes the circle of con vergence, and a later geometric threshold ζ univ > ζ c , where the map ceases to b e univ alent. W e pro v e that the ﬁrst sp ectral instability occurs already at ζ c . In each symmetry sector, the w eighted sub critical realization has exactly one logarithmically diverging eigenv alue, whereas the remaining sp ectrum stays b ounded and, after remov al of the singular direction, conv erges to that of a compact limiting remainder. W e further contin ue the corresp onding scalar Gram functions b ey ond ζ c , showing that they admit a generalized hypergeometric description, a Cauch y–Stieltjes represen tation, and, for 1 ≤ p ≤ s , a realization as W eyl functions of b ounded Jacobi op erators. In particular, these scalar quantities remain ﬁnite at ζ univ . Th is iden tiﬁes analytic criticality , rather than loss of univ alence, as the ﬁrst sp ectral threshold of the T o da Hessian. 1. Intr oduction Planar domains, conformal maps, and harmonic momen ts play a cen tral role in sev eral closely related sub jects, including Laplacian growth, disp ersionless integrable hierarc hies, Green-function and Bergman-kernel theory , and planar matrix mo dels [ 1 – 7 ]. A common feature of these theories is that the geometry of a simply connected domain may be enco ded b y its conformal map and by the harmonic moments of its complemen t. This mak es it natural to ask not only ho w the domain itself degenerates near criticality , but also how the associated second-order response ob jects b eha v e. The present pap er studies one suc h ob ject: the mixed Hessian of the disp ersionless T o da free energy , equiv alently the co eﬃcien t matrix of a mixed logarithmic kernel attac hed to the in verse conformal map. Let D ⊂ C b e a b ounded simply connected domain with rectiﬁable b oundary , and let f : {| w | > 1 } → C \ D , f ( w ) = r w + a 0 + a 1 w − 1 + · · · , b e the unique exterior conformal map normalized by f ( w ) /w → r as w → ∞ . The parameter r > 0 is the conformal radius of D , while the remaining co eﬃcients encode the shap e of the b oundary . Let w = w ( z ) denote the inv erse branc h near inﬁnit y , normalized by w ( z ) = z /r + O ( z − 1 ) . T wo bilinear kernels are naturally attached to f . The ﬁrst is the holomorphic F ab er–Grunsky k ernel log f ( w ) − f ( w ′ ) r ( w − w ′ ) whose co eﬃcien ts control univ alence through the classical Grunsky inequalities [ 8 , 9 ]. By con trast, the present pap er is concerned with the mixed holomorphic–an tiholomorphic logarithmic kernel (1.1) log 1 − 1 w ( z ) w ( z ′ ) ! = − X m,n ≥ 1 H mn ( z /r ) − m m ( ¯ z ′ /r ) − n n , v alid for | z | and | z ′ | suﬃcien tly large. It is obtained by pulling bac k the canonical mixed logarithmic k ernel of the exterior disk under the inv erse conformal map. The co eﬃcien ts H mn are the main ob ject of the present pap er. In the geometric realization of the disp ersionless 2 D T o da hierarch y , the exterior conformal map ev olv es under comm uting ﬂows whose times ( t 0 , t 1 , t 2 , . . . ; ¯ t 1 , ¯ t 2 , . . . ) are identiﬁed with the harmonic 1 2 OLEG ALEKSEEV momen ts of C \ D . In a standard normalization, t 0 = (1 /π ) R D d 2 z is the area of D divided by π , while for n ≥ 1 the v ariables t n and ¯ t n are the holomorphic and an tiholomorphic harmonic momen ts of C \ D . The hierarc h y p ossesses a disp ersionless τ -function whose logarithm F = log τ pla ys the role of a free energy , and in a standard normalization one has H mn = r m + n ∂ 2 F ∂ t m ∂ ¯ t n . Th us ( H mn ) is the scale-inv ariant mixed Hessian of the disp ersionless free energy . The same co eﬃcien ts may b e interpreted in sev eral equiv alen t languages. In Laplacian growth and Hele–Sha w dynamics, ( H mn ) records the second-order coupling b et w een moment deformations. In the disp ersionless T o da hierarc hy , it is the matrix of mixed second deriv ativ es of log τ . In Bergman-k ernel language, one has, with our normalization, (1.2) π B C \ D ( z , z ′ ) = − ∂ z ∂ ¯ z ′ log 1 − 1 w ( z ) w ( z ′ ) ! , so the same data ma y b e recov ered from the exterior Bergman kernel in conformal co ordinates [ 10 , 11 ]. In the normal matrix mo del, F is the planar free energy , or equiv alen tly the logarithmic energy of the equilibrium measure on D [ 6 , 7 ], and ( H mn ) may be view ed as a mixed susc eptibility matrix with resp ect to momen t deformations. The main contribution of the pap er is to determine the sp ectral structure of ( H mn ) near criticality in a family for which the in v erse map can b e con trolled explicitly . The question is which part of the sp ectrum b ecomes singular near criticality , and whether that transition is gov erned by analytic criticalit y of the inv erse branch or b y the later geometric loss of univ alence. This question is genuinely diﬀeren t from the classical holomorphic Grunsky theory , whose op erator b ounds are tied directly to univ alence and quasiconformal extension [ 8 , 9 ]. In the mixed sector studied here, the ﬁrst instabilit y is triggered already at the analytic threshold and concentrates in to one logarithmically div erging stiﬀ mo de per symmetry sector, that is, an eigen-direction whose eigenv alue diverges logarithmically , while the remaining sectorial sp ectrum stays bounded. The resulting separation b et w een analytic and geometric criticality is invisible in a purely holomorphic framework. R elation to prior work. The present problem should also b e distinguished from spectral and determinan tal questions for the classical Grunsky op erator and matrix, including F redholm-eigenv alue problems for quasicircles and recent asymptotics for truncated Grunsky op erators [ 12 – 14 ]. Those w orks concern the holomorphic Grunsky data, or F redholm determinants built from them. Here the basic ob ject is instead the mixed Hessian ( H mn ) , equiv alently the mixed logarithmic kernel ( 1.1 ) ; the relev an t transition is therefore not the saturation of a b ounded holomorphic op erator norm or a truncated determinant asymptotic, but a logarithmic rank-one spike in the mixed sector at the earlier analytic threshold ζ c . With this motiv ation in hand, we no w restrict to a family in whic h the analysis can b e carried out explicitly . W e study the s -fold symmetric one-harmonic family (1.3) f ( w ) = r w + aw 1 − s , s ≥ 2 , r > 0 , a ∈ R , whic h is the simplest family in which the analytic singularit y of the inv erse map and the geometric loss of univ alence separate. Maps of the form ( 1.3 ) are standard in Laplacian growth and related free-b oundary problems [ 5 ]; they already exhibit nontrivial geometric transitions, including loss of univ alence and, for s ≥ 3 , b oundary cusps at the geometric threshold. Because they dep end on a single dimensionless shap e parameter ζ := a/r > 0 , they pro vide a minimal and fully explicit mo del in which the sp ectral b eha vior of the mixed Hessian can b e analyzed in detail. All rigorous results pro v ed b elow concern the family ( 1.3 ) ; the corresp onding extension to general p olynomial conformal maps, under an isolated dominant square-ro ot orbit hypothesis, is developed in the companion pap er [ 15 ]. SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 3 W riting x := r /z and w ( z ) − 1 = x U ( x ; ζ ) , one ﬁnds that the in v erse branch is gov erned by the algebraic equation (1.4) U = 1 + ζ x s U s . This elementary relation already contains the tw o thresholds that con trol the problem. The ﬁrst is the analytic thr eshold ζ c = ( s − 1) s − 1 s s , at which the dominan t square-ro ot singularit y of the inv erse branc h reaches the unit circle in the x -plane. The second is the ge ometric thr eshold ζ univ = 1 s − 1 , at which the conformal map ceases to b e univ alent and the b oundary dev elops its ﬁrst geometric singularit y: for s ≥ 3 , semicubical cusps, and for s = 2 , the degenerate Jouko wski segment. The problem is to determine whic h of these thresholds go v erns the ﬁrst sp ectral instability of the mixed Hessian. Our answer is that the relev an t threshold is ζ c . The s -fold symmetry of ( 1.3 ) implies H mn = 0 unless m ≡ n ( mo d s ) , so the Hessian decomposes into s indep enden t blo c ks. These blo c ks admit an exact p ositiv e Gram factorization in terms of Raney co eﬃcien ts generated b y ( 1.4 ) . After passing to a suitable w eigh ted realization, each symmetry blo c k dev elops one logarithmically div erging eigen v alue, while the remaining sectorial sp ectrum sta ys b ounded. Equiv alen tly , there is one stiﬀ mo de p er symmetry sector, so at most s logarithmically diverging eigenv alues globally . The ﬁrst sp ectral transition is therefore an analytic phenomenon of the in v erse branch rather than a manifestation of geometric non-univ alence. A second theme of the pap er concerns the intermediate regime ζ c < ζ < ζ univ , where the conformal map remains univ alent although the w eighted compact op erator realization used in the sub critical regime is no longer av ailable. In this region the blo c k-op erator picture breaks do wn, but its scalar Gram building blo c ks remain well deﬁned and contin ue analytically . More precisely , the blo c k Gram weigh ts in the sub critical op erator theory are obtained from generating functions built from the squared Raney co eﬃcien ts, and these scalar functions still admit analytic con tin uation b ey ond ζ c . W e iden tify these con tin ued functions as generalized hypergeometric ob jects, derive their resonant branc h-p oin t expansion, prov e a Cauch y–Stieltjes representation, and, in the p ositiv e range 1 ≤ p ≤ s , obtain a Jacobi-matrix realization. Th us the con tinuation of these scalar Gram data captures the same analytic branc h-p oin t mec hanism that pro duces the sub critical logarithmic instabilit y , even after the weigh ted compact realization has ceased to exist. In particular, the singularit y at ζ c is not the onset of geometric breakdo wn: the t wo lateral b oundary v alues of the con tin ued scalar data remain ﬁnite at the univ alence threshold. 1.1. Main results. The results of the pap er fall into three groups. Theorem A gives the op erator- theoretic description of the analytic threshold ζ c : for a ﬁxed weigh t parameter β > 0 , eac h symmetry blo c k splits into one logarithmically stiﬀ direction and a b ounded soft remainder. Theorem B iden tiﬁes the scalar Gram data that survive b ey ond the w eigh ted compact regime and con tinues them across the cut as generalized h yp ergeometric/Stieltjes ob jects. Finally , Prop osition C shows that this analytic transition o ccurs strictly b efore the geometric loss of univ alence at ζ univ . Fix a symmetry sector q ∈ { 1 , . . . , s } and a w eigh t parameter β > 0 . Let H ( q ) denote the in trinsic co eﬃcien t matrix of the q -th Hessian blo c k, and let e G ( q ) ( ζ ) b e the corresp onding weigh ted blo c k Gram operator. The sp ectral analysis in this pap er is carried out for these weigh ted realizations on the ﬁxed Hilb ert space ℓ 2 ( N 0 ) , where N 0 := { 0 , 1 , 2 , . . . } . The w eigh ting is only an auxiliary 4 OLEG ALEKSEEV Hilb ert-space device: it does not alter the co eﬃcien t data and do es not remov e the critical instability . The weigh ted Gram and weigh ted Hessian blo cks ha v e the same nonzero sp ectrum, so the sp ectral conclusions may be stated equiv alen tly for e G ( q ) ( ζ ) or e H ( q ) ( ζ ) . Theorem (A: Rank-one logarithmic instability) . Fix q ∈ { 1 , . . . , s } and β > 0 . The weighte d blo ck Gr am op er ator admits, as ζ ↑ ζ c , a r ank-one lo garithmic de c omp osition e G ( q ) ( ζ ) = L ( ζ ) e d ( q ) ⊗ e d ( q ) ∗ + e C ( q ) ( ζ ) , L ( ζ ) := log  1 1 − ζ 2 /ζ 2 c  → + ∞ , wher e e d ( q )  = 0 is indep endent of ζ , while e C ( q ) ( ζ ) r emains uniformly b ounde d and c onver ges in op er ator norm to a c omp act limit. In p articular, exactly one eigenvalue in the q -th se ctor diver ges lo garithmic al ly, µ ( q ) 1 ( ζ ) = ∥ e d ( q ) ∥ 2 ℓ 2 L ( ζ ) + O (1) . A l l r emaining se ctorial eigenvalues stay b ounde d; mor e pr e cisely, after r emoving the spike dir e ction, e ach ﬁxe d soft eigenvalue c onver ges to the c orr esp onding eigenvalue of the c ompr esse d c omp act limit describ e d in Pr op osition 4.8 . Theorem A is prov ed b y combining the square-ro ot singularity analysis of the in v erse branch with the blo c kwise Gram representation of the Hessian. The logarithmic term is ﬁrst isolated at the lev el of matrix entries, and the weigh ted realization for ﬁxed β > 0 then yields a compact op erator framew ork in whic h the rank-one singular part separates from the b ounded soft sector. R emark 1.1 (Contrast with the Grunsky op erator) . The classical Grunsky op erator b elongs to the holomorphic sector and is gov erned by the sharp b ound ∥ B ∥ ≤ 1 , which is equiv alen t to univ alence of the underlying map [ 9 ]. Accordingly , in the present one-parameter family , the holomorphic-sector transition is go v erned by the geometric threshold ζ univ . By con trast, the mixed Hessian studied here transitions earlier, at the analytic threshold ζ c , and the nature of the transition is diﬀerent: instead of the saturation of a b ounded op erator norm, one eigenv alue div erges logarithmically while the remaining sp ectrum stays bounded. This distinction b et w een the holomorphic and mixed sectors is gen uinely op erator-theoretic and is not visible at the level of individual matrix entries. The second principal result concerns the scalar quan tities that underlie the blo c k Gram construction itself and therefore remain meaningful b ey ond the weigh ted op erator regime. F or each p ≥ 1 , consider the generating function (1.5) G p ( u ) := X m ≥ 0 R s,p ( m ) 2 u m , u = ζ 2 . Belo w ζ c , the scalar Gram weigh ts en tering the blo c k op erators are obtained from these functions b y explicit Euler-t yp e diﬀerential op erators. Th e imp ortance of G p is therefore that they enco de, in scalar form, the same branc h-point mec hanism that driv es the logarithmic sp ectral instabilit y . These functions contin ue analytically b ey ond the radius of conv ergence u = ζ 2 c , even though the weigh ted compact op erator picture no longer p ersists there. Theorem (B: Analytic contin uation of the scalar Gram data) . F or e ach inte ger p ≥ 1 , the function G p extends fr om its disk of c onver genc e to a single-value d holomorphic function on C \ [ ζ 2 c , ∞ ) . It is a gener alize d hyp er ge ometric function, and ne ar the br anch p oint u = ζ 2 c it admits a r esonant exp ansion of the form (1.6) G p ( u ) = A ( u ) + B ( u )  1 − u ζ 2 c  2 log  1 − u ζ 2 c  , with A and B analytic ne ar ζ 2 c . The sc alar Gr am weights ar e r e c over e d fr om G p by explicit Euler-typ e diﬀer ential op er ators; these r emove the quadr atic pr efactor and pr o duc e the lo garithmic diver genc e at u = ζ 2 c . SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 5 Theorem B is obtained from the explicit co eﬃcien t ratio of R s,p ( m ) 2 , which identiﬁes G p with a generalized hy p ergeometric function. The lo cal expansion ( 1.6 ) is then derived by F rob enius analysis at the regular singular p oin t u = ζ 2 c . Moreov er, the b oundary v alues across the cut deﬁne a discontin uit y density , and G p admits a Cauc hy–Stieltjes represen tation. In the p ositiv e range 1 ≤ p ≤ s , the corresp onding Stieltjes measure is p ositiv e, which yields a Jacobi-matrix realization of G p as a W eyl m -function. The third principal result clariﬁes the relation b etw een analytic and geometric criticality . Prop osition (C: Separation of thresholds) . F or every s ≥ 2 , (1.7) ζ c < ζ univ . Henc e the lo garithmic sp e ctr al instability of The or em 1.1 o c curs while the c onformal map is stil l univalent and the b oundary is stil l smo oth. Mor e over, the analytic al ly c ontinue d sc alar data r emain ﬁnite at ζ = ζ univ (Prop ositions 6.3 and 6.5 ) . Th us the singularity of the Hessian at ζ c is not a consequence of cusp formation. It is already enco ded in the branch-point structure of the in verse map and therefore precedes geometric breakdown. 1.2. Organization of the pap er. Section 2 analyzes the inv erse conformal map, derives the Raney- n um b er form ulas, and identiﬁes the square-ro ot singularity at the analytic threshold. Section 3 in tro duces the scale-inv ariant T o da Hessian, prov es the blo c kwise Gram factorization, and records the scalar Gram quantities that will later b e contin ued across the threshold. Section 4 develops the weigh ted sp ectral theory in the sub critical regime 0 < ζ < ζ c and prov es Theorem A. Section 5 studies the scalar contin uation problem b ey ond ζ c , including the h yp ergeometric represen tation, the resonan t expansion, the Cauch y–Stieltjes formula, and the Jacobi realization. Finally , Section 6 treats the geometric threshold ζ univ , prov es the strict separation ( 1.7 ) , and shows that the contin ued scalar data remain ﬁnite at the univ alence b oundary . The app endices con tain the uniform Raney asymptotics, the en trywise rank-one extraction, the h yp ergeometric con tinuation argumen t, and the computation of the branc h-p oin t co eﬃcien t. 2. Inversion of the conformal map This section analyzes the in v erse conformal map and extracts the information needed for the sp ectral analysis of the Hessian. It provides three inputs for later use: the functional equation for the in v erse branc h (Prop osition 2.1 ), the explicit T aylor co eﬃcien ts in terms of Raney num bers (Prop osition 2.4 ), and the lo cation and square-ro ot nature of the dominan t singularity (Lemma 2.2 ). The last point generates the universal m − 3 / 2 co eﬃcien t deca y resp onsible for the logarithmic sp ectral div ergence established in Section 4 . Throughout this manuscript we in troduce the scale-free v ariable (2.1) x := r z . 2.1. The generating function and its functional equation. W e enco de the in verse branc h w ( z ) ∼ z /r as z → ∞ b y factoring out the leading linear growth: (2.2) w ( z ) = z r U ( x ; ζ ) − 1 , where U is a scalar function to b e determined. The normalization w ( z ) ∼ z /r corresp onds to U (0; ζ ) = 1 . Prop osition 2.1 (F unctional equation for the in v erse map) . L et f ( w ) = r w + aw 1 − s with ζ = a/r , and let w ( z ) b e the inverse br anch normalize d by w ( z ) ∼ z /r at inﬁnity. Deﬁne U ( x ; ζ ) by ( 2.2 ) . Then U satisﬁes the algebr aic functional e quation (2.3) U ( x ; ζ ) = 1 + ζ x s U ( x ; ζ ) s , U (0; ζ ) = 1 . 6 OLEG ALEKSEEV Pr o of. Substituting ( 2.2 ) into the deﬁning relation z = r w + aw 1 − s and using x = r /z giv es 1 = U − 1 + ζ x s U s − 1 . Multiplying by U yields ( 2.3 ). □ Throughout w e restrict to the one-parameter family with ζ := a/r > 0 . F or the present one- parameter family , c hanging the sign of a amoun ts to a rigid rotation of the image domain by π /s . 2.2. Square-root criticality. W e now determine the lo cal b eha vior of U ( x ; ζ ) at its dominant singularit y . This is the k ey analytic input for the sp ectral analysis: the square-ro ot nature of the singularit y pro duces the universal m − 3 / 2 co eﬃcien t asymptotics that drive the logarithmic div ergence of the Hessian. Lemma 2.2 (Critical p oin t and lo cal expansion) . The function U ( x ; ζ ) deﬁne d by U = 1 + ζ x s U s is a p ower series in the c ombination ζ x s , with r adius of c onver genc e ζ c = ( s − 1) s − 1 /s s in that variable. Equivalently, for ﬁxe d ζ > 0 its r adius of c onver genc e in x is ( ζ c /ζ ) 1 /s . Its only singularity on | ζ x s | = ζ c is at ζ x s = ζ c . At this p oint the function has a squar e-r o ot br anch p oint with the lo c al exp ansion (2.4) U ( x ; ζ ) = s s − 1 − κ s 1 − ζ x s ζ c + O  1 − ζ x s ζ c  , ζ x s → ζ c , wher e κ = p 2 s/ ( s − 1) 3 > 0 . Pr o of. Consider the implicit function F ( y , x ; ζ ) := y − 1 − ζ x s y s . The T a ylor branch U ( x ; ζ ) is deﬁned by F ( U, x ; ζ ) = 0 with U (0; ζ ) = 1 . A singularity of this branch occurs where the implicit function theorem fails, i.e., where ∂ y F = 0 : F ( U, x ; ζ ) = 0 , ∂ y F ( U, x ; ζ ) = 1 − sζ x s U s − 1 = 0 . F rom the second equation, ζ x s U s − 1 = 1 /s . Substituting into the ﬁrst equation: U = 1 + ζ x s U s = 1 + U s , whic h gives U c := s/ ( s − 1) . Substituting back: ζ c = 1 s U s − 1 c = 1 s  s − 1 s  s − 1 = ( s − 1) s − 1 s s . Since U is algebraic, its singularities o ccur only at discriminan t p oin ts of the p olynomial F ( y , x ; ζ ) = y − 1 − ζ x s y s in the v ariable y . Equiv alen tly , they o ccur exactly when the system F ( y , x ; ζ ) = 0 , ∂ y F ( y , x ; ζ ) = 0 has a common ro ot. The computation ab o v e sho ws that the only nonzero critical v alue of the pro duct ζ x s is ζ c . T o determine the lo cal b eha vior, we verify that the singularit y is simple (i.e., ∂ y y F  = 0 at the critical p oin t): ∂ y y F = − s ( s − 1) ζ x s U s − 2 = − ( s − 1) U c = − ( s − 1) 2 s  = 0 . Moreo v er, near any critical p oint with ζ x s = ζ c and y = U c , the T aylor expansion of F has the form F ( y , x ; ζ ) = 1 2 ∂ y y F ( U c , x ; ζ ) ( y − U c ) 2 − U s c ( ζ x s − ζ c ) + O  | y − U c | 3 + | ζ x s − ζ c | | y − U c |  , SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 7 b ecause ∂ y F ( U c , x ; ζ ) = 0 at criticalit y and the co eﬃcient of ( ζ x s − ζ c ) is − U s c . Since ∂ y y F ( U c , x ; ζ )  = 0 , the W eierstrass preparation theorem gives a square-root branch point, hence ( 2.4 ) . Matching the leading co eﬃcien ts yields 1 2 ∂ y y F ( U c , x ; ζ ) κ 2 + U s c ζ c = 0 . Using ∂ y y F ( U c , x ; ζ ) = − ( s − 1) 2 /s and U s c ζ c = 1 / ( s − 1) , we obtain κ 2 = 2 s ( s − 1) 3 , as claimed. □ R emark 2.3 (Geometry in the z -plane) . Returning to the original v ariable z via x = r /z , the condition | ζ x s | = ζ c b ecomes | z | = r ( ζ /ζ c ) 1 /s . Thus the s branc h p oin ts of the inv erse map w ( z ) lie on a circle whose radius shrinks tow ard r as ζ ↑ ζ c . When ζ = ζ c , the branc h points reach the circle | z | = r , which is the image of the unit circle under the leading term z = r w of the conformal map. This is the geometric manifestation of the analytic threshold. 2.3. Raney num b ers and co eﬃcien t formulas. The T aylor coeﬃcients of p o w ers of U admit a classical closed form. W e deﬁne these co eﬃcien ts directly and then identify them with the Raney n um b ers. F or any p ∈ Z , write (2.5) U ( x ; ζ ) p = ∞ X n =0 R s,p ( n ) ( ζ x s ) n . Prop osition 2.4 (Raney num b ers) . F or al l inte gers s ≥ 2 , p ≥ 1 , and n ≥ 0 , (2.6) R s,p ( n ) = p sn + p  sn + p n  . F or p ≥ 1 , these ar e the classic al R aney numb ers; the c ase p = 1 gives the F uss–Catalan numb ers [ 16 , 17 ]. Pr o of. Set U ( t ) = 1 + V ( t ) with V (0) = 0 . Then V = t (1 + V ) s , which is the Lagrange form V = t Φ( V ) with Φ( v ) = (1 + v ) s . W e apply the Lagrange–Bürmann inv ersion formula to the comp osite function F ( V ) = (1 + V ) p = U p . The standard co eﬃcien t extraction [ 18 , 19 ] gives, for n ≥ 1 , [ t n ] F ( V ( t )) = 1 n [ v n − 1 ] F ′ ( v )Φ( v ) n . Since F ′ ( v ) = p (1 + v ) p − 1 and Φ( v ) n = (1 + v ) sn , we obtain [ t n ] U ( t ) p = p n [ v n − 1 ](1 + v ) sn + p − 1 = p n  sn + p − 1 n − 1  . The iden tit y p n  sn + p − 1 n − 1  = p sn + p  sn + p n  yields ( 2.6 ) for n ≥ 1 . F or n = 0 , w e ha v e R s,p (0) = U (0) p = 1 (when p  = 0 ), which agrees with ( 2.6 ). □ R emark 2.5 (Extension to p ≤ 0 ) . The generating function U ( x ; ζ ) p is well-deﬁned for all p ∈ Z . F or p = 0 , we ha ve U 0 ≡ 1 , so R s, 0 ( n ) = δ n, 0 . F or p < 0 , the form ula ( 2.6 ) can b e extended using the generalized binomial coeﬃcient, but requires separate treatment at v alues of n where sn + p = 0 . Since all applications in this pap er inv olv e p ≥ 1 (the index p in the Hessian expansion), w e restrict to p ositiv e p in the main statemen t. 8 OLEG ALEKSEEV Corollary 2.6 (Con volution property) . F or any inte gers k ≥ 1 and p 1 , . . . , p k ∈ Z , and any m ≥ 0 , (2.7) X n 1 + ··· + n k = m n i ≥ 0 k Y i =1 R s,p i ( n i ) = R s,p 1 + ··· + p k ( m ) . Pr o of. Multiply the generating series ( 2.5 ) and extract the co eﬃcien t of ( ζ x s ) m . □ W e no w record the expansions needed for the Hessian k ernel in Section 3 . The k ernel inv olv es the logarithm log (1 − ( w ¯ w ′ ) − 1 ) , which w e will expand in p o w ers of w − 1 ; hence we need the follo wing. Lemma 2.7 (Po w ers of the in v erse map) . F or e ach inte ger p ≥ 1 , (2.8) w ( z ) − p =  r z  p ∞ X m =0 R s,p ( m ) ( ζ x s ) m = ∞ X m =0 R s,p ( m ) ζ m x p + ms . In p articular, the exp onents of x app e aring in the exp ansion of w ( z ) − p ar e exactly { p + ms : m ≥ 0 } . Pr o of. F rom ( 2.2 ), w − 1 = ( r /z ) U = xU , so w − p = x p U p . Expanding U p via ( 2.5 ) gives ( 2.8 ). □ R emark 2.8 (Co eﬃcien t asymptotics) . The square-ro ot singularity ( 2.4 ) implies, via standard transfer theorems [ 20 ], the univ ersal large- n asymptotics (2.9) R s,p ( n ) = A s,p n − 3 / 2 ζ − n c  1 + O ( n − 1 )  , n → ∞ , with an explicit amplitude A s,p > 0 dep ending on s and p . The exp onen t − 3 / 2 is universal for simple algebraic singularities of square-ro ot t yp e. This is the only asymptotic information ab out the inv erse map that enters the Hessian estimates in Section 4 ; the detailed deriv ation is giv en in App endix A . 3. The Hessian of the τ -function This section constructs the main ob ject of the pap er: the mixed Hessian of log τ with resp ect to the disp ersionless T o da times. W e w ork throughout in the scale-free v ariables in tro duced in Section 2 , which remov es the trivial dep endence on the conformal radius and yields a Hessian that dep ends only on the shap e parameter ζ = a/r . The section is organized as follo ws. W e ﬁrst recall the kernel represen tation due to Wiegmann– Zabro din and use it to deﬁne the scale-inv ariant Hessian (Subsection 3.1 ). W e then expand the k ernel using the Raney mac hinery from Section 2 , obtaining an exact Gram representation as a sum of rank-one op erators (Subsection 3.3 ). Finally , w e analyze the Gram weigh ts and identify the analytic threshold ζ c at which the naive unw eighted Gram realization breaks do wn (Subsection 3.4 ). 3.1. Kernel representation and scale-inv ariant form ulation. The disp ersionless T o da hierarch y asso ciates to a simply connected domain D ⊂ C a τ -function whose logarithm F = log τ serv es as a generating function for the harmonic moments. In the presen t paper, ho w ever, w e use this framework only for the p olynomial family ( 1.3 ) . The general discussion from Section 1 is included only to place the problem in its geometric and integrable con text. F or the maps ( 1.3 ) with 0 < ζ < ζ univ , the b oundary is real-analytic and the Wiegmann–Zabro din k ernel identit y applies. The mixed second deriv atives of F with resp ect to the T o da times ( t m , ¯ t n ) are then expressed via the in v erse conformal map w ( z ) as follows [ 4 ]: (3.1) ∂ 2 F ∂ t m ∂ ¯ t n = − mn [ z − m ][ ¯ z ′− n ] log 1 − 1 w ( z ) w ( z ′ ) ! , where z and z ′ are treated as indep enden t complex v ariables and the co eﬃcien t extraction [ z − m ] refers to the Laurent expansion at inﬁnity . SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 9 The right-hand side of ( 3.1 ) dep ends on the conformal radius r through the norm alization w ( z ) ∼ z /r at inﬁnity . This dep endence is inessen tial for sp ectral questions: it can b e absorb ed b y passing to the scale-free v ariable x = r /z in tro duced in Section 2 . In terms of x , we ha ve w ( z ) − 1 = x U ( x ; ζ ) by ( 2.2 ), and the kernel becomes (3.2) K ( x, ¯ x ′ ) := − log  1 − x ¯ x ′ U ( x ; ζ ) U ( x ′ ; ζ )  . This expression dep ends on ( x, ¯ x ′ , ζ ) alone, with no residual r -dependence. Deﬁnition 3.1 (Scale-in v ariant Hessian) . The sc ale-invariant Hessian is the inﬁnite matrix H = ( H mn ) m,n ≥ 1 deﬁned by (3.3) H mn := mn [ x m ][ ¯ x ′ n ] K ( x, ¯ x ′ ) . The prefactor mn arises from the coeﬃcient extraction in ( 3.1 ) and ensures that H has the correct homogeneit y to act on mo de sequences. Concretely , H is the natural scale-in v ariant v ersion of the T o da Hessian. R emark 3.2 (Op erator viewp oin t and role of renormalization) . The scale-inv ariant Hessian H = ( H mn ) m,n ≥ 1 is naturally a p ositiv e semideﬁnite quadratic form on ﬁnitely supp orted sequences. Sp ectral statemen ts require a compatible Hilb ert realization, and the standard ℓ 2 ( N ) top ology is not stable at analytic criticality . The appropriate framew ork is the weigh ted Gram realization introduced later in Deﬁnition 4.1 ; all sp ectral statemen ts in Sections 4 – 5 refer to these renormalized operators. 3.2. Expansion of the k ernel and blo c k structure. W e no w expand the kernel ( 3.2 ) using the Raney formalism from Section 2 . The logarithm expands as (3.4) K ( x, ¯ x ′ ) = ∞ X p =1 1 p  x ¯ x ′ U ( x ; ζ ) U ( x ′ ; ζ )  p = ∞ X p =1 1 p x p ¯ x ′ p U ( x ; ζ ) p U ( x ′ ; ζ ) p . By Lemma 2.7 , each pow er U ( x ; ζ ) p has the expansion U ( x ; ζ ) p = ∞ X m =0 R s,p ( m ) ( ζ x s ) m = ∞ X m =0 R s,p ( m ) ζ m x sm , and similarly for the conjugate factor. Substituting in to ( 3.4 ) and collecting p o w ers of x m ¯ x ′ n yields: Prop osition 3.3 (Kernel co eﬃcien ts) . The c o eﬃcient [ x m ][ ¯ x ′ n ] K vanishes unless m ≡ n ( mo d s ) . When this c ongruenc e holds, (3.5) [ x m ][ ¯ x ′ n ] K = X p ≥ 1 p ≡ m (mo d s ) 1 p R s,p ( k ) R s,p ( l ) ζ k + l wher e k = ( m − p ) /s and l = ( n − p ) /s . Pr o of. The monomial x m ¯ x ′ n arises from the p -th term in ( 3.4 ) when m = p + sk and n = p + sl for some k , l ≥ 0 . This requires m ≡ p ≡ n ( mo d s ) . Summing ov er all contributing v alues of p giv es ( 3.5 ). □ The selection rule has an immediate structural consequence for the Hessian. Corollary 3.4 (Block decomp osition) . The sc ale-invariant Hessian de c omp oses as (3.6) H = s M q =1 H ( q ) , 10 OLEG ALEKSEEV wher e H ( q ) is the r estriction of H to indic es m, n ≡ q ( mo d s ) . Explicitly, identifying the q -th blo ck with an op er ator on ℓ 2 ( N 0 ) via the c orr esp ondenc e m = q + j s ↔ j , the matrix elements ar e (3.7) H ( q ) j 1 j 2 = H q + j 1 s, q + j 2 s , j 1 , j 2 ≥ 0 . All sp ectral statements in this pap er are formulated and prov ed blo c kwise. The s -fold rotational symmetry of the conformal map th us reduces the sp ectral problem to s indep enden t comp onen ts. 3.3. Gram representation. The factorized structure of ( 3.5 ) , namely , a pro duct of Raney co ef- ﬁcien ts, one dep ending on m and one on n , suggests rewriting the Hessian as a sum of rank-one op erators. This Gr am r epr esentation is the key algebraic structure underlying the spectral analysis. Prop osition 3.5 (Gram representation) . Deﬁne ve ctors v ( p ) = ( v ( p ) m ) m ≥ 1 by (3.8) v ( p ) m :=    m √ p R s,p ( k ) ζ k , if m = p + k s for some k ≥ 0 , 0 , otherwise . Then the sc ale-invariant Hessian admits the exact r epr esentation (3.9) H = ∞ X p =1 v ( p ) ⊗ v ( p ) ∗ , wher e v ⊗ v ∗ denotes the r ank-one op er ator ( v ⊗ v ∗ ) mn = v m v n . Pr o of. Com bining Deﬁnition 3.1 with Prop osition 3.3 , w e hav e for m = p + k s and n = p + l s : H mn = mn X p ′ ≡ m 1 p ′ R s,p ′ ( k ′ ) R s,p ′ ( l ′ ) ζ k ′ + l ′ , where k ′ = ( m − p ′ ) /s and l ′ = ( n − p ′ ) /s . On the other hand, ∞ X p =1 v ( p ) m v ( p ) n = X p ≡ m ≡ n mn p R s,p ( k ) R s,p ( l ) ζ k + l , with k = ( m − p ) /s and l = ( n − p ) /s . These expressions coincide. Finally , for each ﬁxed ( m, n ) only ﬁnitely many p con tribute (b ecause v ( p ) m = 0 unless p ≤ m and p ≡ m ( mo d s ) ), so ( 3.9 ) holds as an en trywise identit y , equiv alen tly as a quadratic-form identit y on the vector space of ﬁnitely supp orted sequences on N . □ R emark 3.6 (Structure of the Gram representation) . The Gram form ( 3.9 ) mak es p ositivit y of H manifest: it is a sum of p ositive semideﬁnite rank-one op erators. More imp ortan tly , it separates t wo sources of complexity: (i) The individual contribution of eac h logarithmic mo de p , enco ded in the v ector v ( p ) , and (ii) the c ol le ctive mixing of mo des within eac h symmetry sector, which determines the eigenv alue distribution. The sp ectral b eha vior of H is controlled b y the in terpla y b et w een these t wo eﬀects. As we sho w next, the critical phenomenon at ζ = ζ c arises from the b orderline summabilit y of the Gram data, and not from any individual term b ecoming singular. 3.4. Gram w eigh ts and the analytic threshold. The Gram represen tation expresses H as a sup erposition of rank-one contributions. Whether this sum deﬁnes a b ounded op erator on the naive un w eighted ℓ 2 -space dep ends on the size of the individual terms, measured b y their squared norms. Deﬁnition 3.7 (Gram weigh ts) . F or p ≥ 1 , the p -th Gr am weight is (3.10) σ p ( ζ ) := ∥ v ( p ) ∥ 2 ℓ 2 = ∞ X m =0 ( p + ms ) 2 p R s,p ( m ) 2 ζ 2 m . SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 11 The Gram w eigh ts are the natural quan tities controlling op erator b ounds for the Hessian. Their b eha vior as ζ v aries reve als the analytic threshold. Prop osition 3.8 (Threshold for Gram w eigh ts) . F or e ach ﬁxe d p ≥ 1 , one has σ p ( ζ ) < ∞ if and only if ζ < ζ c . In p articular, the analytic thr eshold for the sc alar Gr am weights is exactly ζ c . Pr o of. By Prop osition A.1 , the Raney co eﬃcien ts satisfy R s,p ( m ) ∼ A s,p m − 3 / 2 ζ − m c as m → ∞ . Hence the summand in ( 3.10 ) has the b orderline form ( p + ms ) 2 p R s,p ( m ) 2 ζ 2 m ∼ s 2 A 2 s,p p m − 1  ζ ζ c  2 m . The factor m − 1 is summable exactly when ζ < ζ c , and not summable at ζ = ζ c . This prov es the claim. □ R emark 3.9 (Nature of the threshold) . Prop osition 3.8 shows that ζ c is exactly the threshold at which the v ectors v ( p ) cease to b elong to ℓ 2 . Hence the elementary Gram realization of this subsection is v alid only for ζ < ζ c . The co eﬃcien t matrix H , how ever, remains w ell deﬁned throughout the geometric regime ζ < ζ univ , and the weigh ted framew ork of Section 4 is introduced to study the singular b eha vior near ζ c on a ﬁxed Hilb ert scale. The strict inequality ζ c < ζ univ , prov ed later in Prop osition 6.3 , pro duces the intermediate regime ζ c < ζ < ζ univ in which spectral criticalit y precedes geometric breakdown. The detailed analysis of these tw o regimes is carried out in Sections 4 and 5 . 4. Subcritical phase ( 0 < ζ < ζ c ) This section develops the op erator-theoretic picture in the analytic sub critical regime. Prop o- sition 3.5 gives a canonical Gram factorization of each symmetry block of the Hessian. Although the unrenormalized Gram weigh ts σ p ( ζ ) diverge as ζ ↑ ζ c (Prop osition 3.8 ), one can renormalize the column mo des b y an explicit diagonal w eigh t. The resulting compact p ositiv e op erators exhibit a single logarithmically diverging eigen v alue, while the remaining soft sp ectrum sta ys b ounded and admits a well-deﬁned limiting description after compression. 4.1. Op erator setting, blo c k factorization, and w eigh ted renormalization. Fix a symmetry sector q ∈ { 1 , . . . , s } , and write p j := q + j s, j ∈ N 0 , for the indices b elonging to that blo c k; see Corollary 3.4 . Restricting the Gram vectors v ( p ) from ( 3.8 ) to the q -th blo c k, we obtain a syn thesis map V ( ζ ) , initially deﬁned on ﬁnitely supp orted sequences, whose j -th column is the restriction of v ( p j ) . The blo c k Hessian and blo c k Gram op erators are then (4.1) H ( q ) ( ζ ) = V ( ζ ) V ( ζ ) ∗ , G ( q ) ( ζ ) = V ( ζ ) ∗ V ( ζ ) . A t this stage, V ( ζ ) is used only on ﬁnitely supp orted sequences, and ( 4.1 ) is an identit y of matrix co eﬃcien ts (equiv alently , of quadratic forms on ﬁnitely supp orted v ectors). The diﬃculty is that the unw eigh ted realization is not stable as ζ ↑ ζ c . Indeed, Prop osition A.2 sho ws that the Raney co eﬃcien ts satisfy R s,p ( m ) ≤ C s p M p ζ − m c m − 3 / 2 , M := s s − 1 . A ccordingly , the j -th column of V ( ζ ) carries the noncritical bac kground growth p j M p j in the blo c k index j , together with the b orderline transfer tail m − 3 / 2 in the summation v ariable m . T o obtain a compact op erator on a ﬁxed Hilb ert space, one therefore rescales the columns by an explicit diagonal weigh t that remo v es this background gro wth but leav es the gen uine logarithmic singularity un touc hed. 12 OLEG ALEKSEEV Deﬁnition 4.1 (W eighted realization) . Fix β > 0 , set M := s s − 1 , and deﬁne (4.2) w j := p 3 2 + β j M p j , j ∈ N 0 . Let (4.3) H β := n x = ( x j ) j ≥ 0 : ∥ x ∥ 2 H β = X j ≥ 0 | x j | 2 w 2 j < ∞ o , with inner pro duct ⟨ x, y ⟩ H β := X j ≥ 0 x j y j w 2 j . Let W : H β → ℓ 2 ( N 0 ) b e the unitary map ( W x ) j = w j x j . The renormalized synthesis op erator is e V ( ζ ) := V ( ζ ) W − 1 : ℓ 2 ( N 0 ) → ℓ 2 ( N 0 ) , initially deﬁned on ﬁnitely supp orted sequences and extended by con tin uit y . Its asso ciated w eigh ted Gram and Hessian op erators are (4.4) e G ( q ) ( ζ ) := e V ( ζ ) ∗ e V ( ζ ) , e H ( q ) ( ζ ) := e V ( ζ ) e V ( ζ ) ∗ . The structure of the w eigh ts in ( 4.2 ) is dictated by the Raney asymptotics. The factor M p j comp ensates the exp onen tial dep endence on the blo c k index. The exp onen t 3 / 2 matc hes the b orderline tail m − 3 / 2 , and the additional parameter β > 0 provides a small margin that turns the residual p − 2 j b eha vior in to an absolutely summable sequence. In particular, the w eigh ted realization is ζ -indep enden t. R emark 4.2 (In trinsic ob ject versus weigh ted realization) . The co eﬃcien t matrix of the Hessian blo c k is the intrinsic ob ject. The w eigh ts ( 4.2 ) do not mo dify these coeﬃcients and do not remov e the critical instability; they only realize the same co eﬃcien t data in a ﬁxed, ζ -indep enden t Hilb ert scale in which the singular part b ecomes visible as a compact-plus-rank-one op erator. The singular direction is already present at the co eﬃcien t lev el and is enco ded b y the intrinsic amplitudes d ( q ) j ; see ( B.4 ). The vector e d ( q ) =  d ( q ) j /w j  j ≥ 0 ∈ ℓ 2 ( N 0 ) is simply its realization in the weigh ted scale. V arying β > 0 changes this realization, but not the underlying one-dimensional singular direction. Th us Theorems 4.6 and 4.7 are statements ab out the weigh ted realization for ﬁxed β > 0 , whereas their in trinsic conten t is the existence of a single dominan t singular direction in each symmetry sector. The ﬁrst consequence of this construction is compactness for every ﬁxed sub critical parameter. Prop osition 4.3 (Sub critical compactness) . Fix q ∈ { 1 , . . . , s } and β > 0 . F or every 0 < ζ < ζ c , the op er ator e V ( ζ ) : ℓ 2 ( N 0 ) → ℓ 2 ( N 0 ) is Hilb ert–Schmidt. Conse quently, e G ( q ) ( ζ ) and e H ( q ) ( ζ ) ar e tr ac e class, henc e c omp act. Pr o of. By construction, the j -th column of e V ( ζ ) is e V ( ζ ) e j = v ( p j ) w j , SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 13 restricted to the q -blo c k. Therefore (4.5) ∥ e V ( ζ ) ∥ 2 HS = X j ≥ 0 ∥ e V ( ζ ) e j ∥ 2 ℓ 2 = X j ≥ 0 ∥ v ( p j ) ∥ 2 ℓ 2 w 2 j = X j ≥ 0 σ p j ( ζ ) w 2 j . It is thus enough to b ound σ p ( ζ ) uniformly in p . Using the closed form for R s,p ( m ) from Prop osition 2.4 , the Gram w eigh ts may b e written as (4.6) σ p ( ζ ) = p X m ≥ 0  sm + p m  2 ζ 2 m , p ≥ 1 . W e claim that there exists C s > 0 suc h that, for all p ≥ 1 and m ≥ 1 , (4.7)  sm + p m  ≤ C s M p ζ − m c m − 1 / 2 . Indeed, set N := sm + p , and K := N − m = ( s − 1) m + p . A t wo-sided Stirling b ound giv es  N m  ≤ C N N + 1 2 m m + 1 2 K K + 1 2 = C 1 √ m r N K exp  N log N − m log m − K log K  , with an absolute constant C > 0 . W riting t := p/m ≥ 0 , so that N = m ( s + t ) and K = m ( s − 1 + t ) , the exp onen t b ecomes m (( s + t ) log ( s + t ) − ( s − 1 + t ) log ( s − 1 + t )) . By the conv exit y estimate used in Prop osition A.2 , ( s + t ) log ( s + t ) − ( s − 1 + t ) log ( s − 1 + t ) ≤ log ( ζ − 1 c ) + t log M , hence exp  N log N − m log m − K log K  ≤ ζ − m c M p . Moreo v er, r N K = r s + t s − 1 + t ≤ r s s − 1 , uniformly in t ≥ 0 . This prov es ( 4.7 ). No w ﬁx 0 < ζ < ζ c and set η := ζ /ζ c ∈ (0 , 1) . Combining ( 4.6 ) with ( 4.7 ) , and separating the term m = 0 , we obtain σ p ( ζ ) = p + p X m ≥ 1  sm + p m  2 ζ 2 m ≤ p + p C 2 s M 2 p X m ≥ 1 m − 1 η 2 m . Since P m ≥ 1 m − 1 η 2 m = − log (1 − η 2 ) < ∞ , this yields (4.8) σ p ( ζ ) ≤ C ′ s ( ζ ) p M 2 p , for some ﬁnite constan t C ′ s ( ζ ) dep ending on ζ , but not on p . Finally , using w 2 j = p 3+2 β j M 2 p j and ( 4.5 ), σ p j ( ζ ) w 2 j ≤ C ′ s ( ζ ) p j M 2 p j p 3+2 β j M 2 p j = C ′ s ( ζ ) p − 2 − 2 β j . Because p j = q + j s ∼ sj and β > 0 , the series P j ≥ 0 p − 2 − 2 β j con v erges. Hence e V ( ζ ) is Hilb ert– Sc hmidt. Therefore e G ( q ) ( ζ ) = e V ( ζ ) ∗ e V ( ζ ) , e H ( q ) ( ζ ) = e V ( ζ ) e V ( ζ ) ∗ , are trace class, and in particular compact. □ Lemma 4.4 (Isosp ectralit y of the weigh ted blo c k realizations) . F or e ach 0 < ζ < ζ c , the c omp act p ositive op er ators e H ( q ) ( ζ ) and e G ( q ) ( ζ ) have the same nonzer o eigenvalues, c ounte d with multiplicities. 14 OLEG ALEKSEEV Pr o of. By Prop osition 4.3 , the op erator e V ( ζ ) : ℓ 2 ( N 0 ) → ℓ 2 ( N 0 ) is b ounded. Therefore the standard relation b et w een e V e V ∗ and e V ∗ e V applies. If e G ( q ) ( ζ ) x = µ x with µ > 0 , then e V ( ζ ) x  = 0 and e H ( q ) ( ζ )  e V ( ζ ) x  = e V ( ζ ) e V ( ζ ) ∗ e V ( ζ ) x = e V ( ζ ) e G ( q ) ( ζ ) x = µ e V ( ζ ) x. The conv erse implication is identic al, with e V ( ζ ) ∗ in place of e V ( ζ ) . Multiplicities are preserved b y the same corresp ondence. □ Corollary 4.5 (T ransfer from the w eighted Gram blo c k to the weigh ted Hessian blo c k) . Fix q ∈ { 1 , . . . , s } , β > 0 , and 0 < ζ < ζ c . Then e G ( q ) ( ζ ) and e H ( q ) ( ζ ) have the same nonzer o eigenvalues, c ounte d with multiplicities. In p articular, every nonzer o-eigenvalue statement pr ove d b elow for e G ( q ) ( ζ ) holds verb atim for e H ( q ) ( ζ ) . Pr o of. The ﬁrst statement is exactly Lemma 4.4 . The remaining assertion follo ws b ecause all sp ectral conclusions in Sections 4.2 – 4.3 are form ulated only in terms of the nonzero eigen v alues and their m ultiplicities. □ 4.2. Rank-one logarithmic spik e at ζ c . W e now isolate the singular part of the weigh ted blo c k Gram op erator as ζ ↑ ζ c . The p oin t is that, after the weigh ted realization of Section 4.1 , the b orderline div ergence is carried by a single vector e d ( q ) , whereas the remaining part stays compact and conv erges in norm. This is the op erator-theoretic core of Theorem A. Theorem 4.6 (Rank-one decomp osition) . Fix q ∈ { 1 , . . . , s } and β > 0 . Ther e exist a nonzer o ve ctor e d ( q ) ∈ ℓ 2 ( N 0 ) , indep endent of ζ , and a family of c omp act op er ators e C ( q ) ( ζ ) on ℓ 2 ( N 0 ) such that, for every 0 < ζ < ζ c , (4.9) e G ( q ) ( ζ ) = L ( ζ ) e d ( q ) ⊗ e d ( q ) ∗ + e C ( q ) ( ζ ) , L ( ζ ) := log 1 1 − ζ 2 /ζ 2 c . Mor e over, sup 0 <ζ <ζ c ∥ e C ( q ) ( ζ ) ∥ < ∞ , e C ( q ) ( ζ ) → e C ( q ) ∗ in op er ator norm as ζ ↑ ζ c , for some c omp act limit op er ator e C ( q ) ∗ . The ve ctor e d ( q ) is describ e d explicitly in App endix B . Its intrinsic, weight-indep endent amplitudes d ( q ) ar e deﬁne d in ( B.4 ) , and e d ( q ) j = d ( q ) j w j is their r e alization in the weighte d sc ale. Pr o of. The detailed co eﬃcien t analysis is contained in App endix B ; here we only record the reduction that pro duces ( 4.9 ). Entrywise extr action of the lo garithmic term. By Lemma B.1 , eac h matrix element of e G ( q ) ( ζ ) is given by a single series in the summation index m . Lemma B.3 applies the uniform Raney expansion of Lemma A.3 to the tail of this series and con trols the initial range b y the global b ound of Prop osition A.2 . As a result, if η := ζ ζ c ∈ (0 , 1) , and ( K η ) j 1 j 2 := η | j 1 − j 2 | e d ( q ) j 1 e d ( q ) j 2 , j 1 , j 2 ∈ N 0 , then (4.10) e G ( q ) ( ζ ) = L ( ζ ) K η + R ( q ) ( ζ ) , SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 15 where R ( q ) ( ζ ) is Hilb ert–Sc hmidt, sup 0 <ζ <ζ c ∥ R ( q ) ( ζ ) ∥ HS < ∞ , and R ( q ) ( ζ ) → e C ( q ) ∗ in Hilb ert–Sc hmidt norm as ζ ↑ ζ c . In particular, R ( q ) ( ζ ) is compact for eac h ζ , and the con v ergence also holds in op erator norm. W e denote this op erator-norm limit by e C ( q ) ∗ ; thus the R ( q ) ∗ of App endix B and the e C ( q ) ∗ used in the main text are the same op erator. R emoval of the r esidual T o eplitz factor. Set K 1 := e d ( q ) ⊗ e d ( q ) ∗ . The kernel K η diﬀers from K 1 only by the T o eplitz damping factor η | j 1 − j 2 | . By Lemma B.4 , (4.11) L ( ζ ) ∥ K η − K 1 ∥ − → 0 , ζ ↑ ζ c . Therefore L ( ζ ) K η = L ( ζ ) K 1 + L ( ζ ) ( K η − K 1 ) , and the second term is negligible at the op erator lev el. Final de c omp osition. Combining ( 4.10 ) with ( 4.11 ), we obtain e G ( q ) ( ζ ) = L ( ζ ) K 1 +  R ( q ) ( ζ ) + L ( ζ )( K η − K 1 )  . Deﬁne e C ( q ) ( ζ ) := R ( q ) ( ζ ) + L ( ζ )( K η − K 1 ) . Eac h e C ( q ) ( ζ ) is compact, since R ( q ) ( ζ ) is Hilb ert–Sc hmidt and K η − K 1 is Hilb ert–Sc hmidt by Lemma B.4 . Moreov er, sup 0 <ζ <ζ c ∥ e C ( q ) ( ζ ) ∥ ≤ sup 0 <ζ <ζ c ∥ R ( q ) ( ζ ) ∥ + sup 0 <ζ <ζ c L ( ζ ) ∥ K η − K 1 ∥ < ∞ , and e C ( q ) ( ζ ) → e C ( q ) ∗ in op erator norm as ζ ↑ ζ c , b ecause R ( q ) ( ζ ) → e C ( q ) ∗ in Hilb ert–Sc hmidt norm and L ( ζ )( K η − K 1 ) → 0 in op erator norm. Since K 1 = e d ( q ) ⊗ e d ( q ) ∗ , this is exactly ( 4.9 ). □ 4.3. Spectral asymptotics. W e now conv ert the rank-one decomp osition of Theorem 4.6 into sp ectral information. The conclusion is that the singular part pro duces one stiﬀ eigen v alue in each symmetry blo c k, while the remaining sp ectrum sta ys b ounded and con v erges to the compressed limit op erator. Theorem 4.7 (Sp ectral asymptotics at the threshold) . Fix q ∈ { 1 , . . . , s } and β > 0 . L et µ ( q ) 1 ( ζ ) ≥ µ ( q ) 2 ( ζ ) ≥ · · · ≥ 0 denote the eigenvalues of e G ( q ) ( ζ ) , c ounte d with multiplicity, and let e d ( q )  = 0 b e the spike ve ctor fr om The or em 4.6 . Set Γ ( q ) := ∥ e d ( q ) ∥ 2 ℓ 2 . Then, as ζ ↑ ζ c , (4.12) µ ( q ) 1 ( ζ ) = Γ ( q ) L ( ζ ) + O (1) , sup 0 <ζ <ζ c µ ( q ) k ( ζ ) < ∞ ( k ≥ 2) . In p articular, in the q -th symmetry blo ck exactly one eigenvalue diver ges lo garithmic al ly, and this eigenvalue is simple for ζ suﬃciently close to ζ c . 16 OLEG ALEKSEEV Mor e over, if ψ 1 ( ζ ) is a unit eigenve ctor asso ciate d with µ ( q ) 1 ( ζ ) , then after ﬁxing its phase one has (4.13) ψ 1 ( ζ ) − → e d ( q ) ∥ e d ( q ) ∥ ℓ 2 in ℓ 2 as ζ ↑ ζ c . Final ly, (4.14) µ ( q ) 1 ( ζ ) = Γ ( q ) L ( ζ ) + Λ ( q ) ﬁn + o (1) , ζ ↑ ζ c , wher e Λ ( q ) ﬁn := ⟨ e d ( q ) , e C ( q ) ∗ e d ( q ) ⟩ ∥ e d ( q ) ∥ 2 ℓ 2 . Theorem 4.7 has a clear numerical signature: after weigh ted renormalization, exactly one eigenv alue in each symmetry blo c k gro ws on the logarithmic scale L ( ζ ) , while the remaining eigenv alues sta y b ounded. Figure 1 illustrates this separation for the blo ck operator e G ( q ) ( ζ ) in the sector q = 1 for s = 3 and s = 5 . The top ro w shows that the leading branc h µ ( q ) 1 ( ζ ) is asymptotically aﬃne in L ( ζ ) , whereas the b ottom row conﬁrms that after division by L ( ζ ) only the stiﬀ branc h survives and all soft branches tend to zero. Pr o of. W rite e d := e d ( q ) , b d := e d / ∥ e d ∥ ℓ 2 , P := b d ⊗ b d ∗ . By Theorem 4.6 , (4.15) e G ( q ) ( ζ ) = Γ ( q ) L ( ζ ) P + e C ( q ) ( ζ ) , where each e C ( q ) ( ζ ) is compact self-adjoin t, K q := sup 0 <ζ <ζ c ∥ e C ( q ) ( ζ ) ∥ < ∞ , and e C ( q ) ( ζ ) → e C ( q ) ∗ in op erator norm as ζ ↑ ζ c . Only one eigenvalue c an diver ge. If x ⊥ e d and ∥ x ∥ = 1 , then P x = 0 , hence ⟨ x, e G ( q ) ( ζ ) x ⟩ = ⟨ x, e C ( q ) ( ζ ) x ⟩ . Using the one-dimensional trial space Span { e d } in the min–max characterization of µ ( q ) 2 ( ζ ) , w e obtain (4.16) µ ( q ) 2 ( ζ ) ≤ sup ∥ x ∥ =1 x ⊥ e d ⟨ x, e G ( q ) ( ζ ) x ⟩ ≤ ∥ e C ( q ) ( ζ ) ∥ ≤ K q . Therefore every eigen v alue with index k ≥ 2 remains uniformly b ounded. On the other hand, the Ra yleigh quotient of b d gives µ ( q ) 1 ( ζ ) ≥ ⟨ b d , e G ( q ) ( ζ ) b d ⟩ = Γ ( q ) L ( ζ ) + ⟨ b d , e C ( q ) ( ζ ) b d ⟩ ≥ Γ ( q ) L ( ζ ) − K q . Since L ( ζ ) → + ∞ , it follo ws that µ ( q ) 1 ( ζ ) → + ∞ while µ ( q ) k ( ζ ) , k ≥ 2 , stay b ounded. This pro v es ( 4.12 ), and the simplicit y of µ ( q ) 1 ( ζ ) for ζ close to ζ c follo ws from the sp ectral gap µ ( q ) 1 ( ζ ) − µ ( q ) 2 ( ζ ) → + ∞ . The top eigenve ctor aligns with the spike dir e ction. Let ψ 1 ( ζ ) b e a unit eigen v ector corresp onding to µ ( q ) 1 ( ζ ) . Applying I − P to the eigenv alue equation e G ( q ) ( ζ ) ψ 1 ( ζ ) = µ ( q ) 1 ( ζ ) ψ 1 ( ζ ) and using ( I − P ) P = 0 , w e obtain ( I − P ) e C ( q ) ( ζ ) ψ 1 ( ζ ) = µ ( q ) 1 ( ζ )( I − P ) ψ 1 ( ζ ) . SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 17 2 4 6 8 10 L ( ζ ) = l o g ( 1 / δ ) 0.5 1.0 1.5 2.0 2.5 l e a d i n g e i g e n v a l u e µ 1 ( ζ ) s = 3 , q = 1 , β = 1 µ 1 t a i l f i t : 0 . 2 4 0 L + 0 . 2 3 8 2 4 6 8 10 L ( ζ ) = l o g ( 1 / δ ) 1.0 1.5 2.0 2.5 s = 5 , q = 1 , β = 1 µ 1 t a i l f i t : 0 . 1 9 9 L + 0 . 5 1 9 1 0 4 1 0 3 1 0 2 1 0 1 δ = 1 − ζ 2 / ζ 2 c 1 0 9 1 0 7 1 0 5 1 0 3 1 0 1 n o r m a l i z e d e i g e n v a l u e s µ k / L ( ζ ) (logarithmic y-scale) µ 1 / L µ 2 / L µ 3 / L µ 4 / L 1 0 4 1 0 3 1 0 2 1 0 1 δ = 1 − ζ 2 / ζ 2 c 1 0 9 1 0 7 1 0 5 1 0 3 1 0 1 µ 1 / L µ 2 / L µ 3 / L µ 4 / L E x p l i c i t l o g a r i t h m i c d i v e r g e n c e o f t h e s t i f f m o d e : µ 1 ∼ c L ( ζ ) a n d µ k ≥ 2 / L ( ζ ) → 0 Figure 1. Logarithmic sp ectral asymptotics of the weigh ted Gram block e G ( q ) ( ζ ) for s = 3 , 5 in sector q = 1 ( β = 1 , N = 30 ). T op row: the leading eigenv alue µ ( q ) 1 ( ζ ) plot- ted against L ( ζ ) , sho wing the asymptotically aﬃne law µ ( q ) 1 ( ζ ) = Γ ( q ) L ( ζ ) + O (1) ; the dashed line is a tail linear ﬁt. Bottom ro w: the normalized eigenv alues µ ( q ) k ( ζ ) /L ( ζ ) . The ratio µ ( q ) 1 ( ζ ) /L ( ζ ) approaches a positive constant, whereas µ ( q ) k ( ζ ) /L ( ζ ) → 0 for k ≥ 2 . Hence (4.17) ∥ ( I − P ) ψ 1 ( ζ ) ∥ ≤ ∥ e C ( q ) ( ζ ) ∥ µ ( q ) 1 ( ζ ) ≤ K q µ ( q ) 1 ( ζ ) = O  L ( ζ ) − 1  . If a ( ζ ) := ⟨ ψ 1 ( ζ ) , b d ⟩ , then | a ( ζ ) | 2 = 1 − ∥ ( I − P ) ψ 1 ( ζ ) ∥ 2 = 1 + O  L ( ζ ) − 2  . After ﬁxing the phase so that a ( ζ ) ≥ 0 , we obtain ( 4.13 ). Asymptotics of the diver ging eigenvalue. Since ψ 1 ( ζ ) is normalized, µ ( q ) 1 ( ζ ) = ⟨ ψ 1 ( ζ ) , e G ( q ) ( ζ ) ψ 1 ( ζ ) ⟩ = Γ ( q ) L ( ζ ) | a ( ζ ) | 2 + ⟨ ψ 1 ( ζ ) , e C ( q ) ( ζ ) ψ 1 ( ζ ) ⟩ . 18 OLEG ALEKSEEV By ( 4.17 ) , Γ ( q ) L ( ζ ) | a ( ζ ) | 2 = Γ ( q ) L ( ζ ) + o (1) , while ( 4.13 ) and the op erator-norm conv ergence of e C ( q ) ( ζ ) imply ⟨ ψ 1 ( ζ ) , e C ( q ) ( ζ ) ψ 1 ( ζ ) ⟩ → ⟨ b d , e C ( q ) ∗ b d ⟩ = ⟨ e d , e C ( q ) ∗ e d ⟩ ∥ e d ∥ 2 ℓ 2 . This is exactly ( 4.14 ). □ The remaining sp ectral data are gov erned by the compression of the limiting remainder to the orthogonal complement of the spike direction. Prop osition 4.8 (Con v ergence of the soft sp ectrum) . Fix q ∈ { 1 , . . . , s } and β > 0 . Deﬁne b d ( q ) := e d ( q ) ∥ e d ( q ) ∥ ℓ 2 , P := b d ( q ) ⊗ b d ( q ) ∗ , Q := I − P , and let e C ( q ) ∗ , ⊥ := Q e C ( q ) ∗ Q   ( e d ( q ) ) ⊥ . L et µ ( q ) 2 , ∗ ≥ µ ( q ) 3 , ∗ ≥ · · · ≥ 0 b e the eigenvalues of e C ( q ) ∗ , ⊥ , liste d in nonincr e asing or der and c ounte d with multiplicity. Then, for e ach ﬁxe d n ≥ 2 , (4.18) µ ( q ) n ( ζ ) − → µ ( q ) n, ∗ as ζ ↑ ζ c . Prop osition 4.8 identiﬁes the limiting b ounded sp ectrum that remains after the stiﬀ direction is remo v ed. This b ounded part is naturally seen at the level of the compressed op erator, not in the full diverging spectrum. Figure 2 illustrates this soft regime for s = 3 and s = 5 : the upp er panels sho w the con vergence of the ﬁrst soft branc hes as ζ ↑ ζ c , while the low er panels display near-critical ﬁnite- ζ soft sp ectral proﬁles across the symmetry sectors. T ogether with Figure 1 , this mak es the stiﬀ/soft decomp osition visually explicit. Pr o of. W rite e d := e d ( q ) , b d := e d / ∥ e d ∥ ℓ 2 , P := b d ⊗ b d ∗ , and Q := I − P . Let ψ 1 ( ζ ) b e a normalized eigen v ector corresp onding to the simple eigenv alue µ ( q ) 1 ( ζ ) , and deﬁne P ζ := ψ 1 ( ζ ) ⊗ ψ 1 ( ζ ) ∗ , Q ζ := I − P ζ . By ( 4.13 ), ∥ P ζ − P ∥ ≤ 2 ∥ ψ 1 ( ζ ) − b d ∥ ℓ 2 → 0 , and therefore (4.19) ∥ P ζ − P ∥ → 0 , ∥ Q ζ − Q ∥ → 0 as ζ ↑ ζ c . Since P ζ is the sp ectral pro jection of the simple top eigenv alue, the restriction of e G ( q ) ( ζ ) to Ran Q ζ has eigenv alues precisely µ ( q ) 2 ( ζ ) ≥ µ ( q ) 3 ( ζ ) ≥ · · · . It is therefore enough to prov e norm conv ergence of the compressed op erators: (4.20)   Q ζ e G ( q ) ( ζ ) Q ζ − Q e C ( q ) ∗ Q   − → 0 . Using ( 4.15 ), we ha v e Q ζ e G ( q ) ( ζ ) Q ζ = Γ ( q ) L ( ζ ) Q ζ P Q ζ + Q ζ e C ( q ) ( ζ ) Q ζ . Because P = b d ⊗ b d ∗ is rank one and Q ζ = I − P ζ , one has the exact iden tit y Q ζ P Q ζ = ( Q ζ b d ) ⊗ ( Q ζ b d ) ∗ , hence ∥ Q ζ P Q ζ ∥ = ∥ Q ζ b d ∥ 2 = 1 − |⟨ ψ 1 ( ζ ) , b d ⟩| 2 = ∥ ( I − P ) ψ 1 ( ζ ) ∥ 2 . Therefore ( 4.17 ) implies (4.21) L ( ζ ) ∥ Q ζ P Q ζ ∥ = L ( ζ ) ∥ ( I − P ) ψ 1 ( ζ ) ∥ 2 = O  L ( ζ ) − 1  − → 0 . SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 19 0.15 0.20 0.25 0.30 0.35 0.40 1 / L ( ζ ) , w h e r e L ( ζ ) = l o g ( 1 / ( 1 − ζ 2 / ζ 2 c ) ) 1 0 8 1 0 7 1 0 6 1 0 5 1 0 4 1 0 3 s o f t e i g e n v a l u e s µ ( 1 ) k ( ζ ) s = 3 , q = 1 , β = 1 µ 2 µ 3 µ 4 µ 5 µ 6 0.15 0.20 0.25 0.30 0.35 0.40 1 / L ( ζ ) , w h e r e L ( ζ ) = l o g ( 1 / ( 1 − ζ 2 / ζ 2 c ) ) 1 0 8 1 0 7 1 0 6 1 0 5 1 0 4 1 0 3 s = 5 , q = 1 , β = 1 µ 2 µ 3 µ 4 µ 5 µ 6 2 4 6 8 10 g l o b a l e i g e n v a l u e i n d e x k 1 0 1 0 1 0 8 1 0 6 1 0 4 µ ( q ) k a t ζ / ζ c = 0 . 9 9 9 9 s = 3 , ζ / ζ c = 0 . 9 9 9 9 q = 1 q = 2 q = 3 2 4 6 8 10 g l o b a l e i g e n v a l u e i n d e x k 1 0 1 1 1 0 9 1 0 7 1 0 5 1 0 3 s = 5 , ζ / ζ c = 0 . 9 9 9 9 q = 1 q = 2 q = 3 q = 4 q = 5 Near-critical soft-spectrum trajectories and sector-wise decay of the compressed proxy Figure 2. Soft sp ectrum after remo v al of the stiﬀ direction for s = 3 , 5 ( β = 1 , N = 40 ). T op row: the soft branches µ ( q ) 2 ( ζ ) , . . . , µ ( q ) 6 ( ζ ) in sector q = 1 , plotted against 1 /L ( ζ ) . Their ﬂattening as 1 /L ( ζ ) → 0 shows that the soft branches remain b ounded as ζ ↑ ζ c . Bottom ro w: ﬁnite- ζ snapshots of the soft branc hes for all sectors q = 1 , . . . , s at ζ /ζ c = 0 . 9999 . Next, ∥ Q ζ e C ( q ) ( ζ ) Q ζ − Q e C ( q ) ( ζ ) Q ∥ ≤ 2 ∥ e C ( q ) ( ζ ) ∥ ∥ Q ζ − Q ∥ → 0 by ( 4.19 ), while ∥ Q e C ( q ) ( ζ ) Q − Q e C ( q ) ∗ Q ∥ ≤ ∥ e C ( q ) ( ζ ) − e C ( q ) ∗ ∥ → 0 b y Theorem 4.6 . T ogether with ( 4.21 ), this yields ( 4.20 ). Both Q ζ e G ( q ) ( ζ ) Q ζ and e C ( q ) ∗ , ⊥ are compact self-adjoint op erators. Hence ( 4.20 ) and the min–max principle imply conv ergence of every ﬁxed ordered eigen v alue, whic h is exactly ( 4.18 ). □ R emark 4.9 (Soft mo des) . Prop osition 4.8 shows that, after the unique stiﬀ direction e d ( q ) is remo v ed, the remaining sp ectrum has a ﬁnite limit as ζ ↑ ζ c . Thus the b ounded eigen v alues µ ( q ) 2 ( ζ ) , µ ( q ) 3 ( ζ ) , . . . are gov erned at leading order by the compressed limit op erator e C ( q ) ∗ , ⊥ . In particular, the logarithmic div ergence of Theorem 4.7 is en tirely carried b y the rank-one spike. 20 OLEG ALEKSEEV 5. Continua tion of scalar Gram da t a beyond the anal ytic threshold 5.1. In trinsic Hessian, w eigh ted realization, and con tinued scalar data. Three related ob jects m ust b e kept distinct b eyond the analytic threshold. First, the intrinsic Hessian co eﬃcien ts H mn and the corresp onding unw eigh ted blo c k co eﬃcien ts are attached to the conformal map itself and therefore remain meaningful as long as the geometric map exists. Second, the weigh ted blo c k op erators e G ( q ) ( ζ ) from Section 4 , constructed for a ﬁxed choice of β > 0 , are a ﬁxed-space Hilb ert realization of the sub critical theory . Their role is to separate the logarithmically singular rank-one part from the b ounded soft remainder. Third, the scalar quantities G p , σ p , and ρ p enco de the same branc h-p oin t information at the co eﬃcien t lev el and are the ob jects that can b e contin ued b ey ond the weigh ted op erator regime. The weigh ted compact-op erator framew ork of Section 4 is inheren tly sub critical. As ζ ↑ ζ c , the scalar Gram weigh ts σ p ( ζ ) acquire the b orderline logarithmic div ergence isolated in Theorem 4.6 . A ccordingly , the ﬁxed-space w eigh ted blo c k realization used in Section 4 is not con tin ued past ζ c : no b ounded con tin uation of the compact op erators e G ( q ) ( ζ ) on the same weigh ted Hilb ert space is asserted here. What do es contin ue is the scalar data built from the squared Raney co eﬃcien ts. The purp ose of this section is to isolate those scalar quan tities, contin ue them analytically b ey ond the radius of conv ergence, and sho w that they still enco de the branch-point mechanism resp onsible for the sub critical sp ectral instability . Th us this is a scalar contin uation theory , not an extension of the w eigh ted op erator picture b ey ond ζ c . Since the geometric map remains univ alen t up to the larger threshold ζ univ ; see Prop osition 6.1 , the interv al ζ c < ζ < ζ univ should b e understoo d as a regime in which the ﬁxed-space w eigh ted sp ectral realization breaks do wn b efore geometric univ alence is lost. The argument has three parts. First, w e introduce the scalar functions G p ( u ) , where u = ζ 2 , and sho w that they are generalized hypergeometric functions 2 s F 2 s − 1 with parametric excess γ = 2 ; see Prop osition 5.4 and Corollary 5.5 . This yields a canonical analytic contin uation to the slit plane C \ [ ζ 2 c , ∞ ) . Second, we analyze the singular p oin t u = ζ 2 c and prov e the resonant local expansion G p ( u ) = A ( u ) + B ( u )  1 − u ζ 2 c  2 log  1 − u ζ 2 c  , with A and B analytic and B ( ζ 2 c ) < 0 ; see Theorem 5.9 . The Euler op erator that reconstructs the scalar Gram w eights from G p remo v es the quadratic prefactor and thereb y conv erts this mild resonan t singularity in to the logarithmic divergence found in Section 4 ; see Corollary 5.10 . Third, the branch-cut discon tin uity of G p giv es a Cauc h y–Stieltjes representation; see Prop osition 5.16 . In the range 1 ≤ p ≤ s , the representing measure is p ositiv e, and G p b ecomes the W eyl m -function of a b ounded Jacobi op erator J p ; see Prop osition 5.19 and Theorem 5.21 . Throughout this section w e work with the con tin uation v ariable u = ζ 2 . Later, in the Stielt- jes/Jacobi interpretation, we shall also use the recipro cal sp ectral v ariable t = u − 1 . T o av oid confusion, we reserv e ρ p ( u ) for the jump densit y of the contin ued scalar Gram w eigh t and ϱ p ( t ) for the density of the representing measure of G p . F or a function f deﬁned on the slit plane, w e write Disc f ( u ) := f ( u + i 0) − f ( u − i 0) , u ∈ ( ζ 2 c , ∞ ) , for the discontin uit y across the branch cut. SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 21 5.2. F rom Gram weigh ts to scalar generating functions. W e b egin b y separating the genuinely analytic part of the Gram weigh ts from the p olynomial prefactor coming from the blo c k index. Recall from Deﬁnition 3.7 that (5.1) σ p ( ζ ) = ∥ v ( p ) ∥ 2 = ∞ X m =0 ( p + ms ) 2 p R s,p ( m ) 2 ζ 2 m . F or ζ < ζ c this series con v erges, while for ζ ≥ ζ c it div erges. The divergen t b ehavior is entirely enco ded in the squared Raney co eﬃcients, and it is therefore natural to strip a w ay the explicit p olynomial factor ( p + ms ) 2 /p . Deﬁnition 5.1 (Squared Raney generating function) . F or p ≥ 1 , deﬁne (5.2) G p ( u ) := ∞ X m =0 R s,p ( m ) 2 u m , u ∈ C , initially in the disk | u | < ζ 2 c . The point of Deﬁnition 5.1 is that the original Gram w eigh t is reco v ered from G p b y a ﬁxed second-order Euler op erator. Thus the contin uation problem for σ p reduces to the con tin uation problem for G p . Lemma 5.2 (Euler op erator identit y) . F or | u | < ζ 2 c , (5.3) σ p ( ζ ) = 1 p  p + s u d du  2 G p ( u )    u = ζ 2 . Pr o of. F or each monomial u m one has  p + s u d du  u m = ( p + sm ) u m , hence 1 p  p + s u d du  2 u m = ( p + sm ) 2 p u m . Applying this term wise to the absolutely con v ergen t series ( 5.2 ) in the disk | u | < ζ 2 c , and then setting u = ζ 2 , gives ( 5.3 ). □ F ormula ( 5.3 ) is the basic bridge b et ween the subcritical op erator theory and the con tin uation theory developed b elo w. The op erator picture detects the singularit y through the logarithmic gro wth of σ p ( ζ ) , while the scalar contin uation problem is gov erned by the analytic structure of G p ( u ) at the branc h p oin t u = ζ 2 c . 5.3. Hypergeometric structure and analytic con tin uation. W e next identify the analytic structure of the scalar generating function G p from Deﬁnition 5.1 . The ﬁrst p oin t is that its radius of con vergence is exactly the analytic threshold u = ζ 2 c , which is already enco ded in the large- m asymptotics of the Raney co eﬃcien ts. Prop osition 5.3 (Radius of conv ergence) . The series ( 5.2 ) has r adius of c onver genc e ζ 2 c . Mor e pr e cisely, as m → ∞ , (5.4) R s,p ( m ) 2 = C s,p m 3 ζ − 2 m c  1 + o (1)  , wher e C s,p > 0 dep ends only on ( s, p ) . Pr o of. By Prop osition A.1 , equiv alen tly by the squared asymptotic form ula ( A.2 ) from App endix A , one has R s,p ( m ) 2 ∼ C s,p m − 3 ζ − 2 m c . This is exactly ( 5.4 ), and the radius of conv ergence is therefore ζ 2 c . □ 22 OLEG ALEKSEEV The represen tation ( 5.2 ) is lo cal in u . T o contin ue G p b ey ond | u | < ζ 2 c , one needs a closed analytic mo del. This is pro vided by a generalized hypergeometric expression, whose existence ultimately reﬂects the fact that the co eﬃcien t ratio R s,p ( m + 1) 2 /R s,p ( m ) 2 is rational in m . Prop osition 5.4 (Hypergeometric represen tation) . F or | u | < ζ 2 c , one has (5.5) G p ( u ) = 2 s F 2 s − 1  α 1 , . . . , α 2 s β 1 , . . . , β 2 s − 1    ζ − 2 c u  , wher e the p ar ameter multisets ar e (5.6) { α i } = 2 ×  p + k s : k = 0 , . . . , s − 1  , { β j } = { 1 } ∪ 2 ×  p + l s − 1 : l = 1 , . . . , s − 1  . Pr o of. See App endix C . □ Corollary 5.5 (P arametric excess) . F or the hyp er ge ometric data ( 5.5 ) – ( 5.6 ) , γ := X j β j − X i α i = 2 for every s ≥ 2 and p ≥ 1 . Pr o of. A direct computation gives X i α i = 2 s − 1 X k =0 p + k s = 2 p + ( s − 1) , while X j β j = 1 + 2 s − 1 X l =1 p + l s − 1 = 1 + 2 p + s. Hence γ = 2 . □ The p ositivit y of the parametric excess has t w o immediate consequences. First, the branch p oin t of G p o ccurs at ξ := ζ − 2 c u = 1 , that is, at u = ζ 2 c . Second, the h yp ergeometric diﬀeren tial equation is of resonant type there. In particular, γ = 2 forces a logarithmic term with prefactor (1 − ξ ) 2 , rather than a pure algebraic singularity . Lemma 5.6 (Reduced hypergeometric data after cancellation) . L et c p denote the numb er of c anc el le d c ommon upp er and lower p ar ameters in ( 5.5 ) , c ounte d with multiplicity, and set q p := 2 s − 1 − c p . After those c anc el lations, the germ of G p at u = 0 is r epr esente d by a r e duc e d gener alize d hyp er- ge ometric function of typ e q p +1 F q p with the same p ar ametric exc ess 2 . In p articular, the r e duc e d e quation is nontrivial and has ﬁnite singular p oints only at ξ = 0 and ξ = 1 , wher e ξ = ζ − 2 c u . Pr o of. Cancelling a common upp er and low er parameter remo v es the same quan tit y from P i α i and P j β j , so the parametric excess is unchanged. Since Corollary 5.5 gives excess 2 b efore cancellation, the reduced equation also has excess 2 . If the reduction w ere of type 1 F 0 , its unique upp er parameter would hav e to equal − 2 in order to ha v e excess 2 , whic h is imp ossible b ecause every surviving upp er parameter comes from the p ositive list ( 5.6 ) . Th us q p ≥ 1 , so the reduced form is genuinely of t yp e q p +1 F q p . F or such equations the only ﬁnite singular p oin ts are 0 and 1 in the v ariable ξ ; see [ 21 , §16.8, §16.11]. □ Corollary 5.7 (Analytic con tin uation) . The function G p extends to a single-value d holomorphic function on the slit plane C \ [ ζ 2 c , ∞ ) . SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 23 Pr o of. By Lemma 5.6 , after cancellation of common upp er and low er parameters the germ at u = 0 is represented by a reduced generalized hypergeometric function of type q p +1 F q p with the same parametric excess 2 . F or that reduced hypergeometric diﬀeren tial equation, the only ﬁnite singular p oin ts in the v ariable ξ = ζ − 2 c u are ξ = 0 and ξ = 1 ; equiv alen tly , the only ﬁnite singular points in the v ariable u are u = 0 and u = ζ 2 c ; see [ 21 , §16.8, §16.11]. Hence the germ at the origin contin ues uniquely along every path in C \ [ ζ 2 c , ∞ ) , which is simply connected. This yields a single-v alued holomorphic contin uation of G p to the slit plane. □ 5.4. Resonan t expansion at the branch p oin t. W e no w analyze the singularit y of G p at the endp oin t u = ζ 2 c of the disk of conv ergence. Because the parametric excess is the integer γ = 2 , the local F rob enius basis at the branc h p oin t is resonant, and the singular term is of the form (1 − u/ζ 2 c ) 2 log(1 − u/ζ 2 c ) . Lemma 5.8 (Reduced lo cal mo del at the branc h p oin t) . L et ξ = ζ − 2 c u , and write the germ of G p at ξ = 0 in its r e duc e d hyp er ge ometric form q p +1 F q p , obtaine d by c anc el ling any c ommon upp er and lower p ar ameters in ( 5.5 ) . Then ξ = 1 is a r e gular singular p oint of the r e duc e d diﬀer ential e quation. Its lo c al exp onents ar e { 0 , 2 } if q p = 1 , 0 , 1 , . . . , q p − 1 , 2 if q p ≥ 2 . In p articular, the exp onent 0 is simple, while the exp onent 1 is either absent or simple. Henc e no terms of the form log (1 − ξ ) or (1 − ξ ) log (1 − ξ ) c an o c cur in the lo c al c ontinuation of the germ. Pr o of. By Lemma 5.6 , after cancellation G p satisﬁes a reduced generalized h yp ergeometric equation of type q p +1 F q p with parametric excess γ = 2 . F or suc h an equation, the lo cal exponents at ξ = 1 are 0 , 1 , . . . , q p − 1 , γ ; see [ 21 , §16.8]. Substituting γ = 2 giv es the stated list. The exp onen t 0 is therefore simple, and the exp onen t 1 is either absent or simple. Consequently , the lo cal contin uation of the germ selected b y the T aylor series at ξ = 0 cannot con tain logarithmic terms of order (1 − ξ ) 0 or (1 − ξ ) 1 . The ﬁrst order at whic h a logarithmic sector may o ccur is thus (1 − ξ ) 2 . □ Theorem 5.9 (Resonant expansion) . Ther e exist functions A ( u ) and B ( u ) , analytic in a neighb orho o d of u = ζ 2 c , such that (5.7) G p ( u ) = A ( u ) + B ( u )  1 − u ζ 2 c  2 log  1 − u ζ 2 c  , as u → ζ 2 c in C \ [ ζ 2 c , ∞ ) . Mor e over, B ( ζ 2 c ) < 0 for al l s ≥ 2 and p ≥ 1 . Pr o of. Set ξ := u/ζ 2 c . By Corollary 5.7 , the germ deﬁned b y the T aylor series at ξ = 0 extends holomorphically to the slit plane C \ [1 , ∞ ) . By Lemma 5.6 , after cancellation of any common upper and low er parameters in ( 5.5 ) , this germ is a reduced generalized h yp ergeometric function of type q p +1 F q p with parametric excess γ = 2 . By Lemma 5.8 , the lo cal exp onen ts at ξ = 1 are 0 and 1 with no m ultiplicity , so logarithmic terms of order (1 − ξ ) 0 and (1 − ξ ) 1 are excluded. F or a reduced generalized hypergeometric equation of t yp e q p +1 F q p with p ositiv e integer parametric excess m , the connection formulas at ξ = 1 on the slit plane imply that the contin ued germ has a lo cal represen tation consisting of an analytic part plus one resonant term of the form (1 − ξ ) m log (1 − ξ ) with analytic co eﬃcien t; see [ 21 , §16.8(ii), §16.11]. Since here m = γ = 2 , it follo ws that G p ( ζ 2 c ξ ) = A 0 ( ξ ) + B 0 ( ξ )(1 − ξ ) 2 log(1 − ξ ) , where A 0 and B 0 are analytic near ξ = 1 . The exp onen t analysis from Lemma 5.8 shows that no lo w er-order logarithmic terms o ccur. 24 OLEG ALEKSEEV Returning to the v ariable u = ζ 2 c ξ , we obtain ( 5.7 ) with A ( u ) := A 0 ( u/ζ 2 c ) , B ( u ) := B 0 ( u/ζ 2 c ) . The co eﬃcient B ( ζ 2 c ) is computed explicitly in Appendix D . More precisely , Prop osition D.1 yields B ( ζ 2 c ) < 0 for all s ≥ 2 and p ≥ 1 . Hence the logarithmic sector is nontrivial and starts precisely at order (1 − ξ ) 2 log(1 − ξ ) . □ The singularity in ( 5.7 ) is mild: the function G p ( u ) itself stays ﬁnite at u = ζ 2 c . The logarithmic div ergence app ears only after one returns to the Gram w eigh ts via the Euler op erator from Lemma 5.2 . In this sense, the scalar con tin uation problem is strictly softer than the op erator-theoretic one. Corollary 5.10 (Mechanism of Gram-w eight div ergence) . As ζ ↑ ζ c , e quivalently u = ζ 2 ↑ ζ 2 c , the Gr am weights satisfy σ p ( ζ ) = e A p + 2 s 2 p B ( ζ 2 c ) log  1 − ζ 2 ζ 2 c  + o (1) , wher e e A p ∈ R , and B ( ζ 2 c ) < 0 is the c o eﬃcient fr om The or em 5.9 . In p articular, σ p ( ζ ) → + ∞ lo garithmic al ly, in agr e ement with Pr op osition 3.8 . Pr o of. Apply Lemma 5.2 to the expansion ( 5.7 ) , with u = ζ 2 , and write w := 1 − u/ζ 2 c . Since A ( u ) and B ( u ) are analytic at u = ζ 2 c , one has G p ( u ) = A ( u ) + B ( ζ 2 c ) w 2 log w + O ( w 3 log w ) . The Euler op erator  p + s u d du  2 preserv es analyticity , so the con tribution of A ( u ) is a constant plus o (1) , while O ( w 3 log w ) con tributes only o (1) . F or the singular term, (5.8)  p + s u d du  2  w 2 log w  = 2 s 2 log w + 3 s 2 + O  w log w  , w ↓ 0 . Hence σ p ( ζ ) = e A p + 2 s 2 p B ( ζ 2 c ) log w + o (1) , for some e A p ∈ R . Since w = 1 − ζ 2 /ζ 2 c , this is the claimed expansion. Finally , B ( ζ 2 c ) < 0 and log(1 − ζ 2 /ζ 2 c ) → −∞ , hence σ p ( ζ ) → + ∞ . □ 5.5. Contin ued Gram weigh ts and edge densit y. The Euler identit y from Lemma 5.2 contin ues to make sense on the slit plane once G p has b een analytically contin ued. This giv es a canonical con tin uation of the scalar Gram weigh ts. Deﬁnition 5.11 (Contin ued scalar Gram weigh ts) . F or u ∈ C \ [ ζ 2 c , ∞ ) , deﬁne (5.9) σ cont p ( u ) := 1 p  p + s u d du  2 G p ( u ) . F or | u | < ζ 2 c , this agrees with the original Gram weigh t σ p ( ζ ) under the substitution u = ζ 2 , b y Lemma 5.2 . The function σ cont p inherits the branch cut [ ζ 2 c , ∞ ) from G p . Its jump across the cut is the natural sup ercritical analogue of the logarithmic divergence on the sub critical side. Deﬁnition 5.12 (Discontin uit y densit y) . F or u > ζ 2 c , deﬁne (5.10) ρ p ( u ) := − 1 2 π i Disc σ cont p ( u ) = − 1 2 π i  σ cont p ( u + i 0) − σ cont p ( u − i 0)  . SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 25 The sign conv en tion in ( 5.10 ) is chosen so that the edge v alue is p ositiv e when B ( ζ 2 c ) < 0 , in agreemen t with the standard resolven t orientation. Theorem 5.13 (Nonv anishing edge density) . The limit ρ p ( ζ 2 c ) := lim u ↓ ζ 2 c ρ p ( u ) exists and is nonzer o for al l s ≥ 2 and p ≥ 1 . Mor e pr e cisely, (5.11) ρ p ( ζ 2 c ) = − 2 s 2 p B ( ζ 2 c ) . Pr o of. Let u > ζ 2 c and set w := 1 − u/ζ 2 c ∈ ( −∞ , 0) . By Theorem 5.9 , in a punctured neigh b orhoo d of u = ζ 2 c one has G p ( u ) = A ( u ) + B ( u ) w 2 log w , with A and B analytic near ζ 2 c . Since A and B are single-v alued across the cut, only the logarithm con tributes to the jump. On the principal branch, Disc log w = 2 π i , u > ζ 2 c , hence (5.12) Disc G p ( u ) = 2 π i B ( u ) w 2 , u > ζ 2 c close to ζ 2 c . Because the Euler operator has analytic co eﬃcien ts aw a y from the endp oin t, it may b e applied separately to the upp er and low er holomorphic b oundary v alues on the cut. Subtracting the t w o resulting expressions shows that it comm utes with the discontin uity op eration. Therefore, Disc σ cont p ( u ) = 1 p  p + s u d du  2 Disc G p ( u ) = 2 π i p  p + s u d du  2  B ( u ) w 2  . No w u = ζ 2 c (1 − w ) , so u ( d/du ) = − (1 − w )( d/dw ) . Since B ( u ) = B ( ζ 2 c ) + O ( w ) , one ﬁnds  p + s u d du   B ( u ) w 2  = − 2 s B ( ζ 2 c ) w + O ( w 2 ) , and applying the same op erator once more yields (5.13)  p + s u d du  2  B ( u ) w 2  = 2 s 2 B ( ζ 2 c ) + O ( w ) , u → ζ 2 c . Substituting into ( 5.10 ) gives ρ p ( u ) = − 1 p  2 s 2 B ( ζ 2 c ) + O ( w )  , and letting u ↓ ζ 2 c pro v es ( 5.11 ). □ R emark 5.14 (Matching across the analytic threshold) . Corollary 5.10 and Theorem 5.13 describ e the same resonan t singularity from opp osite sides of the threshold. On the sub critical side ( u ↑ ζ 2 c ) , the Euler op erator conv erts the term B ( u )  1 − u ζ 2 c  2 log  1 − u ζ 2 c  in to a bare logarithm, pro ducing the div ergence σ p ( ζ ) → + ∞ . On the sup ercritical side ( u > ζ 2 c ) , the same logarithm acquires the jump Disc log  1 − u ζ 2 c  = 2 π i, and this yields the nonzero edge v alue ρ p ( ζ 2 c ) . Thus the subcritical blo w-up and the sup ercritical edge density are tw o b oundary manifestations of one and the same branch-point coeﬃcient B ( ζ 2 c ) . 26 OLEG ALEKSEEV 0.0 0.2 0.4 0.6 0.8 1.0 u / ζ 2 c 0 50 100 150 200 f σ p ( u ) ( a ) s = 3 : f σ p ( u ) p = 1 p = 2 p = 3 p = 4 p = 5 0.0 0.2 0.4 0.6 0.8 1.0 u / ζ 2 c 0 20 40 60 80 100 120 ( b ) s = 5 : f σ p ( u ) p = 1 p = 2 p = 3 p = 4 p = 7 1 2 3 4 5 6 u / ζ 2 c 0 20 40 60 ρ p ( u ) ( c ) s = 3 : ρ p ( u ) p = 1 p = 2 p = 3 p = 4 p = 5 1 2 3 4 5 6 u / ζ 2 c 0 10 20 30 ( d ) s = 5 : ρ p ( u ) p = 1 p = 2 p = 3 p = 4 p = 7 C o n t i n u e d G r a m w e i g h t f σ p ( u ) a n d d i s c o n t i n u i t y d e n s i t y ρ p ( u ) Figure 3. Con tin ued Gram weigh t σ cont p ( u ) and discontin uit y density ρ p ( u ) for s = 3 , 5 . T op ro w: the analytically contin ued Gram w eigh t σ cont p ( u ) on the subcritical side 0 < u < ζ 2 c . Bottom row: the discontin uit y density ρ p ( u ) on the sup ercritical side u > ζ 2 c . The op en circles at u = ζ 2 c mark the edge v alues ρ p ( ζ 2 c ) > 0 . F or larger v alues of p , the densit y becomes negative aw a y from the edge, sho wing that positivity at u = ζ 2 c do es not p ersist on the en tire sup ercritical branch. T o illustrate the contin uation across the critical point, w e plot in Figure 3 the con tin ued scalar Gram weigh t σ cont p ( u ) , which agrees with the sub critical Gram w eigh t for u < ζ 2 c , together with the sup ercritical discon tin uit y density ρ p ( u ) for sev eral v alues of p . The lo w er panels show that the edge v alue ρ p ( ζ 2 c ) = − 2 s 2 p B ( ζ 2 c ) is p ositiv e, in agreement with the lo cal expansion at the branc h p oin t. At the same time, the additional curves with larger p sho w that ρ p ( u ) need not remain p ositiv e for all u > ζ 2 c : after starting from a p ositiv e edge v alue, the density ma y cross zero and b ecome negativ e further along the sup ercritical branc h. Thus the sign of ρ p is controlled locally near u = ζ 2 c , but is not globally ﬁxed on the whole contin uation domain. This n umerical b eha vior is consisten t with the edge asymptotics prov ed abov e, while showin g that the p ositivit y statement is lo cal in u and do es not extend to the full sup ercritical branc h. SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 27 5.6. Cauc h y–Stieltjes represen tation. The hypergeometric con tinuation of G p on the slit plane pro duces canonical b oundary v alues across the cut and therefore a Cauc h y–Stieltjes representation. This representation does not require an y p ositivit y assumption and is v alid for every p ≥ 1 . Lemma 5.15 (Growth at inﬁnity) . Fix p ≥ 1 . As | u | → ∞ in C \ [ ζ 2 c , ∞ ) , (5.14) G p ( u ) = O  | u | − p/s (1 + log | u | )  . Conse quently, for F p ( η ) := η − 1 G p (1 /η ) , one has (5.15) F p ( η ) = O  | η | p/s − 1 (1 + log(1 / | η | ))  , η → 0 , η / ∈ [0 , 1 /ζ 2 c ] . Pr o of. By Prop osition 5.4 , G p ( u ) is a generalized hypergeometric function in the v ariable ξ = u/ζ 2 c , with numerator parameters α k = p + k s , k = 0 , . . . , s − 1 , eac h o ccurring with multiplicit y 2 . The generalized h yp ergeometric diﬀeren tial equation has regular singular p oin ts at ξ = 0 , 1 , ∞ , and the lo cal exp onen ts at ξ = ∞ are precisely the numerator parameters; see [ 21 , §16.11]. Because each exp onen t app ears with multiplicit y 2 , the asymptotic expansion contains at most one logarithm p er exp onen t: G p ( u ) = s − 1 X k =0 u − α k  A k log u + B k  + O  | u | − ( p +1) /s log | u |  , uniformly on every closed subsector of the slit plane. The smallest exp onen t is α 0 = p/s , which yields ( 5.14 ). Substituting u = 1 /η gives ( 5.15 ). □ Prop osition 5.16 (Stieltjes represen tation) . Ther e exists a ﬁnite r e al signe d Bor el me asur e ν p of b ounde d total variation, supp orte d on [0 , 1 /ζ 2 c ] , such that (5.16) G p ( u ) = Z 1 /ζ 2 c 0 dν p ( t ) 1 − ut , u ∈ C \ [ ζ 2 c , ∞ ) . Mor e over, ν p  [0 , 1 /ζ 2 c ]  = G p (0) = 1 , and ν p is absolutely c ontinuous on the op en interval (0 , 1 /ζ 2 c ) , with density (5.17) dν p dt ( t ) = 1 2 π i t Disc G p  1 t  , 0 < t < 1 /ζ 2 c . Pr o of. Set F p ( η ) := η − 1 G p (1 /η ) . Since G p is holomorphic on C \ [ ζ 2 c , ∞ ) , the function F p is holomorphic on C \ [0 , 1 /ζ 2 c ] . F rom the T aylor expansion G p ( u ) = 1 + O ( u ) at u = 0 , one obtains F p ( η ) = η − 1 + O ( η − 2 ) , | η | → ∞ . On the other hand, Lemma 5.15 gives F p ( η ) = o ( | η | − 1 ) , η → 0 , a w ay from the cut. Let η ∈ C \ [0 , 1 /ζ 2 c ] , and apply Cauch y’s theorem to a k eyhole contour around the interv al [0 , 1 /ζ 2 c ] , with outer radius R and inner radius ε . The contribution of the outer circle v anishes as R → ∞ 28 OLEG ALEKSEEV b ecause F p ( ξ ) = O ( ξ − 1 ) , while the inner-circle contribution v anishes as ε → 0 by the b ound ( 5.15 ) . P assing to the limit yields (5.18) F p ( η ) = 1 2 π i Z 1 /ζ 2 c 0 Disc F p ( t ) t − η dt. Deﬁne a signed Borel measure on [0 , 1 /ζ 2 c ] by dν p ( t ) := 1 2 π i Disc F p ( t ) dt. Its total v ariation is ﬁnite, and this is exactly where the endp oin t estimates enter. If 0 < t < 1 /ζ 2 c is close to the right endpoint and u = 1 /t , then u > ζ 2 c and Theorem 5.9 give s Disc G p ( u ) = 2 π i B ( u )  1 − u ζ 2 c  2 . Hence Disc F p ( t ) = t − 1 Disc G p (1 /t ) = O  1 ζ 2 c − t  2  , t ↑ 1 ζ 2 c , so the density is integrable at the righ t endp oin t. Near t = 0 , ( 5.15 ) gives Disc F p ( t ) = O  t p/s − 1 (1 + | log t | )  , whic h is integrable b ecause p/s > 0 . Therefore | Disc F p ( t ) | is integrable on [0 , 1 /ζ 2 c ] , and the resulting measure ν p has b ounded total v ariation. Th us ( 5.18 ) b ecomes F p ( η ) = Z 1 /ζ 2 c 0 dν p ( t ) t − η . Substituting η = 1 /u and m ultiplying by u gives ( 5.16 ). Next, Disc F p ( t ) = t − 1 Disc G p (1 /t ) , whic h yields the jump formula ( 5.17 ). Ev aluating ( 5.16 ) at u = 0 gives ν p  [0 , 1 /ζ 2 c ]  = G p (0) = 1 . Finally , ν p is real b ecause G p has real T aylor co eﬃcien ts and therefore satisﬁes Sch warz reﬂection on the slit plane. □ R emark 5.17 . F or general p , the represen tation ( 5.16 ) – ( 5.17 ) is used only as an analytic description of the contin ued scalar Gram data. Positivit y is not required at this stage. The op erator-theoretic in terpretation enters only later, in the range 1 ≤ p ≤ s , where ν p b ecomes a p ositiv e measure and one can pass to the Jacobi/OPRL framework. 5.7. Jacobi realization in the p ositive range. F or general p , Prop osition 5.16 yields only a signed representing measure for G p . F rom this p oin t on ward w e restrict to the p ositiv e-measure range 1 ≤ p ≤ s . In that range, p ositivit y is supplied by an external Hausdorﬀ-momen t theorem for Raney num b ers. W e record exactly the part of that input used here, in the normalization relev an t for the present pap er, b efore passing to the Jacobi/OPRL framework. Lemma 5.18 (External Hausdorﬀ-momen t input in the presen t normalization) . Assume s ≥ 2 and 0 ≤ p ≤ s . Then ther e exists a pr ob ability me asur e ν s,p supp orte d on [0 , τ s ] , τ s := s s ( s − 1) s − 1 = ζ − 1 c , such that R s,p ( n ) = Z τ s 0 x n dν s,p ( x ) , n ≥ 0 . SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 29 Equivalently, the r esc ale d se quenc e { R s,p ( n ) ζ n c } n ≥ 0 is a Hausdorﬀ moment se quenc e on [0 , 1] . This lemma is the only plac e wher e the external R aney-moment p ositivity input is use d. In what fol lows, we use only the existenc e of the p ositive me asur e and the supp ort endp oint τ s = ζ − 1 c ; no explicit formula for the density is ne e de d. These facts ar e supplie d by [ 22 , Thm. 5] to gether with [ 23 ]. F or b ackgr ound on the c orr esp onding R aney distributions, se e also [ 24 ]. Pr o of. By Prop osition 2.4 , R s,p ( n ) = p sn + p  sn + p n  . In the notation of [ 22 ], this is exactly the Raney sequence A n ( s, p ) . Therefore [ 22 , Thm. 5], as corrected in [ 23 ], applies with the parameter pair ( p, r ) = ( s, p ) : the hypotheses are satisﬁed b ecause s ≥ 1 and 0 ≤ p ≤ s . It follows that { R s,p ( n ) } n ≥ 0 is represen ted b y a probability measure supp orted on [0 , τ s ] , with τ s = s s ( s − 1) s − 1 . In the normalization of the presen t pap er, τ s = ζ − 1 c , whic h gives exactly the stated momen t represen tation. After the rescaling x = τ s y , this is equiv alen t to the Hausdorﬀ momen t statement for { R s,p ( n ) ζ n c } n ≥ 0 on [0 , 1] . □ Prop osition 5.19 (P ositivit y for 1 ≤ p ≤ s ) . Assume 1 ≤ p ≤ s . Then the me asur e ν p in Pr op osition 5.16 c an b e chosen to b e a p ositive pr ob ability me asur e supp orte d on [0 , 1 /ζ 2 c ] . Equivalently, after the r esc aling t 7→ t ζ 2 c , the se quenc e { R s,p ( n ) 2 ζ 2 n c } n ≥ 0 is a Hausdorﬀ moment se quenc e. Pr o of. By Lemma 5.18 , for 1 ≤ p ≤ s there exists a probability measure ν s,p supp orted on [0 , τ s ] = [0 , ζ − 1 c ] such that R s,p ( n ) = Z τ s 0 x n dν s,p ( x ) , n ≥ 0 . Let e ν p b e the pushforward of ν s,p ⊗ ν s,p under the multiplication map ( x, y ) 7→ xy . Then e ν p is a probability measure supp orted on [0 , τ 2 s ] = [0 , ζ − 2 c ] , and for every n ≥ 0 , Z 1 /ζ 2 c 0 t n d e ν p ( t ) = Z Z ( xy ) n dν s,p ( x ) dν s,p ( y ) =  Z τ s 0 x n dν s,p ( x )  2 = R s,p ( n ) 2 . Th us the rescaled sequence { R s,p ( n ) 2 ζ 2 n c } n ≥ 0 is a Hausdorﬀ moment sequence on [0 , 1] . Consequen tly , Z 1 /ζ 2 c 0 d e ν p ( t ) 1 − ut = X n ≥ 0 R s,p ( n ) 2 u n = G p ( u ) , | u | < ζ 2 c . Since b oth sides are holomorphic on C \ [ ζ 2 c , ∞ ) , the identit y theorem extends this representation to the whole slit plane. On the other hand, Prop osition 5.16 already provides a ﬁnite signed measure ν p on [0 , 1 /ζ 2 c ] with the same Stieltjes transform: G p ( u ) = Z 1 /ζ 2 c 0 dν p ( t ) 1 − ut , u ∈ C \ [ ζ 2 c , ∞ ) . Hence Z 1 /ζ 2 c 0 d ( ν p − e ν p )( t ) 1 − ut = 0 30 OLEG ALEKSEEV on the slit plane. By uniqueness of the Cauch y–Stieltjes transform of a ﬁnite compactly supp orted measure, one has ν p = e ν p . Therefore the measure in Prop osition 5.16 is in fact p ositiv e, and b eing a probabilit y measure it has total mass 1 . □ Once p ositivity is av ailable, the standard orthogonal-p olynomial machinery pro duces a Jacobi op erator with sp ectral measure ν p . Deﬁnition 5.20 (Jacobi op erator asso ciated with G p ) . Assume 1 ≤ p ≤ s , and let { P ( p ) n } n ≥ 0 b e the monic orthogonal p olynomials with resp ect to the p ositiv e measure ν p from Prop osition 5.19 . Then there exist co eﬃcients a ( p ) n > 0 , b ( p ) n ∈ R , suc h that (5.19) tP ( p ) n ( t ) = P ( p ) n +1 ( t ) + b ( p ) n P ( p ) n ( t ) + ( a ( p ) n ) 2 P ( p ) n − 1 ( t ) , with P ( p ) − 1 ≡ 0 . The asso ciated Jacobi op erator J p is the b ounded self-adjoin t op erator on ℓ 2 ( N 0 ) with tridiagonal matrix J p =        b ( p ) 0 a ( p ) 1 0 · · · a ( p ) 1 b ( p ) 1 a ( p ) 2 . . . 0 a ( p ) 2 b ( p ) 2 . . . . . . . . . . . . . . .        . Since supp ( ν p ) ⊂ [0 , 1 /ζ 2 c ] , one has σ ( J p ) ⊂ [0 , 1 /ζ 2 c ] . Theorem 5.21 (W eyl function identit y) . Assume 1 ≤ p ≤ s . Then G p is the W eyl m -function of J p in the r e cipr o c al sp e ctr al p ar ameter: (5.20) G p ( u ) = ⟨ e 0 , ( I − uJ p ) − 1 e 0 ⟩ , u ∈ C \ [ ζ 2 c , ∞ ) , wher e e 0 = (1 , 0 , 0 , . . . ) T . Equivalently, for u  = 0 , the formula holds whenever u − 1 ∈ ρ ( J p ) . Pr o of. By the sp ectral theorem, ⟨ e 0 , ( I − uJ p ) − 1 e 0 ⟩ = Z 1 /ζ 2 c 0 dν p ( t ) 1 − ut , u ∈ C \ [ ζ 2 c , ∞ ) , where ν p is the sp ectral measure of J p at e 0 . By Prop osition 5.19 , this is precisely the p ositiv e represen ting measure of G p , and the right-hand side equals G p ( u ) by ( 5.16 ) . This pro v es ( 5.20 ) ; see, for example, [ 25 , Ch. I I I]. □ Theorem 5.22 (Stieltjes–Perron inv ersion) . Assume 1 ≤ p ≤ s . Then the p ositive me asur e ν p fr om Pr op osition 5.19 is absolutely c ontinuous on (0 , 1 /ζ 2 c ) , and its density is (5.21) ϱ p ( t ) = 1 π t ℑ G p  1 t + i 0  = 1 2 π i t Disc G p  1 t  , 0 < t < 1 /ζ 2 c . Pr o of. Prop osition 5.16 constructs a representing measure ν p that is absolutely contin uous on the op en interv al (0 , 1 /ζ 2 c ) , with density given b y the jump formula ( 5.17 ) . In the range 1 ≤ p ≤ s , Prop osition 5.19 shows that the same function G p also admits a p ositiv e represen ting measure. By uniqueness of the Stieltjes transform of a ﬁnite compactly supp orted measure, this p ositiv e measure coincides with the ν p from Prop osition 5.16 . Hence ν p is p ositiv e and its density in the sp ectral v ariable t can therefore b e written in the Stieltjes–Perron form displa y ed in ( 5.21 ) ; see, for example, [ 25 , Ch. I I I]. □ SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 31 20 40 60 80 100 m o d e i n d e x p j = q + j s 0.25 0.00 0.25 0.50 0.75 s i g n e d c o m p o n e n t s φ k ( j ) ( a ) s = 3 , q = 1 , β = 1 : s i g n e d s o f t m o d e s φ 2 φ 3 φ 4 φ 5 25 50 75 100 125 150 175 m o d e i n d e x p j = q + j s 0.25 0.00 0.25 0.50 0.75 ( b ) s = 5 , q = 1 , β = 1 : s i g n e d s o f t m o d e s φ 2 φ 3 φ 4 φ 5 S t r u c t u r e o f t h e s o f t m o d e s o f Q d f G ( q ) ( ζ ) Q d | d Figure 4. Structure of the ﬁrst soft mo des of e C ( q ) ∗ , ⊥ for s = 3 , 5 , with q = 1 , β = 1 , and N = 40 . The curves sho w the signed comp onen ts of the eigenv ectors ϕ ( q ) 2 , ϕ ( q ) 3 , ϕ ( q ) 4 , ϕ ( q ) 5 , corresp onding to the soft eigenv alues µ ( q ) 2 , ∗ , µ ( q ) 3 , ∗ , µ ( q ) 4 , ∗ , µ ( q ) 5 , ∗ . As the mo de index increases, the eigenv ectors b ecome more oscillatory while remaining concen trated near lo w v alues of the lattice index p j = q + j s . R emark 5.23 (No dal structure of the soft eigenv ectors) . The Jacobi realization suggests a further structural prop ert y of the compact remainder e C ( q ) ∗ in tro duced in Theorem 4.6 . Let ϕ ( q ) 2 , ϕ ( q ) 3 , . . . denote the eigenv ectors of e C ( q ) ∗ , ⊥ , ordered by decreasing eigenv alue µ ( q ) 2 , ∗ ≥ µ ( q ) 3 , ∗ ≥ · · · , and view eac h ϕ ( q ) k as a function on the lattice { p j = q + j s } j ≥ 0 . Numerically , ϕ ( q ) k exhibits exactly k − 1 sign c hanges in the index j (Figure 4 ), a pattern stable across all v alues of s , q , and β tested. This oscillation coun t is the hallmark of classical Sturm oscillation for self-adjoint Jacobi matrices [ 25 , Ch. I II]. F or the range 1 ≤ p ≤ s , each scalar factor G p j admits such a Jacobi realization b y Theorem 5.21 . The observ ed no dal prop ert y of the full matrix e C ( q ) ∗ is therefore consisten t with, though not a direct corollary of, the scalar Jacobi framew ork, since the Gram en tries mix contributions from diﬀeren t Jacobi op erators J p j . W e lea ve a rigorous no dal theorem for e C ( q ) ∗ as an op en problem. The results of this section pro vide the analytic framew ork for the intermediate regime ζ c < ζ < ζ univ . The generating functions G p ( u ) extend to the slit plane via their hypergeometric structure, hav e a resonant singularit y of parametric excess γ = 2 at the branc h p oin t, and admit a nonv anishing edge density; for 1 ≤ p ≤ s , G p also has a Jacobi–W eyl realization. Thus the scalar sp ectral data p ersist b ey ond the loss of the weigh ted compact op erator picture. Geometric singularity on the domain b oundary o ccurs only at ζ = ζ univ : for s ≥ 3 this is cusp formation, whereas for s = 2 the critical map degenerates to the classical Jouk owski segmen t. W e treat this geometric threshold next in Section 6 . 6. The geometric threshold ζ univ W e now return to the geometric side of the problem. F or the p olynomial map f ( w ) = r w + aw 1 − s , ζ := a r > 0 , the parameter ζ univ = 1 s − 1 32 OLEG ALEKSEEV marks the loss of univ alence of the exterior conformal map. This threshold is indep endent of the w eigh ted sp ectral theory developed in Section 4 and of the scalar contin uation theory developed in Section 5 . The purp ose of the present section is to compare these t wo notions of criticality . W e ﬁrst c haracterize the geometric transition itself. Below ζ univ the b oundary is a smo oth Jordan curv e. At ζ univ , the critical geometry dep ends on s : for s ≥ 3 the b oundary develops semicubical cusps, whereas for s = 2 the Jouko wski trace degenerates to a line segment. Ab o v e ζ univ , the map is no longer injectiv e; for s ≥ 3 the b oundary self-intersects, while for s = 2 the unit-circle trace remains an ellipse although the exterior map ceases to b e one-to-one. W e then sho w that ζ c < ζ univ , so the logarithmic sp ectral instability from Theorems 4.6 and 4.7 o ccurs while the conformal map is still univ alen t. Finally , we pro ve that the analytically contin ued scalar Gram data remain ﬁnite at u = ζ 2 univ . Thus the sp ectral singularit y at ζ c and the geometric singularit y at ζ univ are genuinely distinct. 6.1. Geometric threshold. F or the comparison with the analytic threshold ζ c , we only need the lo cation of the geometric threshold at which the exterior map ceases to b e univ alen t. The ﬁner lo cal description of the b oundary singularit y at the threshold is classical and will not b e used b elo w. Prop osition 6.1 (Geometric threshold) . L et f ( w ) = r w + aw 1 − s = r  w + ζ w 1 − s  , s ≥ 2 , ζ := a/r > 0 . Then the univalenc e thr eshold on the exterior disk is ζ univ = 1 s − 1 . Mor e pr e cisely: (1) if ζ < ζ univ , then f is univalent on {| w | > 1 } and extends inje ctively to | w | = 1 ; (2) if ζ > ζ univ , then f is not univalent on {| w | > 1 } ; (3) if ζ = ζ univ , then f ′ ( w ) = r  1 − ( s − 1) ζ w − s  vanishes exactly at the s p oints w s = 1 on the unit cir cle. Pr o of. F or w = e iθ , write z ( θ ) := f ( e iθ ) = r  e iθ + ζ e − i ( s − 1) θ  . Supp ose z ( θ ) = z ( ϕ ) . Then e iθ − e iϕ = ζ ( e − i ( s − 1) ϕ − e − i ( s − 1) θ ) . T aking absolute v alues and setting δ = ( θ − ϕ ) / 2 , we obtain | sin δ | = ζ | sin(( s − 1) δ ) | ≤ ( s − 1) ζ | sin δ | . Hence, if ( s − 1) ζ < 1 , then sin δ = 0 , so θ ≡ ϕ ( mo d 2 π ) . Th us the b oundary trace is injective. Moreo v er, f ′ ( w ) = r  1 − ( s − 1) ζ w − s  , so for ( s − 1) ζ < 1 all critical p oin ts satisfy | w | s = ( s − 1) ζ < 1 . Therefore f ′ ( w )  = 0 on | w | ≥ 1 , so f is a holomorphic lo cal biholomorphism on {| w | > 1 } . Since f ( w ) = r w + O (1) as w → ∞ , the map is prop er as a map from the exterior disk to its image. Hence f is a cov ering map of ﬁnite degree onto its image. The normalization at inﬁnit y shows that this degree is 1 : for | z | suﬃcien tly large, the equation f ( w ) = z has exactly one solution in | w | > 1 , namely the branc h with w = z /r + O (1) . Therefore the cov ering is trivial, so f is univ alent on {| w | > 1 } and extends injectiv ely to | w | = 1 . If ( s − 1) ζ > 1 , then the equation f ′ ( w ) = 0 has solutions with | w | s = ( s − 1) ζ > 1 , SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 33 so f ′ v anishes inside the exterior domain. Since a holomorphic injectiv e map cannot hav e a critical p oin t, f is not univ alent on {| w | > 1 } . Finally , if ( s − 1) ζ = 1 , then f ′ ( w ) = 0 is equiv alent to w s = 1 , so the critical p oin ts lie exactly on the unit circle. □ R emark 6.2 . A t the critical v alue ζ = ζ univ , the b oundary singularity is standard. F or s ≥ 3 , the image of | w | = 1 develops s semicubical cusps; for s = 2 , the Jouk o wski trace degenerates to a line segmen t. These local descriptions are classical and are not needed for the sp ectral analysis b elo w. 6.2. Separation of the analytic and geometric thresholds. With the geometric threshold iden tiﬁed, we no w compare it with the analytic threshold ζ c = ( s − 1) s − 1 /s s . Prop osition 6.3 (Separation of thresholds) . F or every s ≥ 2 , ζ c < ζ univ . Equivalently, (6.1) ζ c ζ univ =  s − 1 s  s < 1 . In p articular, the lo garithmic sp e ctr al tr ansition describ e d by The or ems 4.6 and 4.7 takes plac e while the c onformal map is stil l univalent and the b oundary is stil l smo oth. Pr o of. The iden tit y ( 6.1 ) is immediate: ζ c ζ univ = ( s − 1) ( s − 1) s − 1 s s =  s − 1 s  s . Since ( s − 1) /s < 1 , the righ t-hand side is strictly less than 1 . □ 6.3. Regularit y of the contin ued scalar data at ζ univ . W e ﬁnally sho w that the contin ued scalar data do not develop an y new singularit y at the geometric threshold. This is the ﬁnal step in separating analytic and geometric criticality . Since ζ univ > ζ c , the p oint u univ := ζ 2 univ lies on the branch cut [ ζ 2 c , ∞ ) of the analytically con tinued functions G p ( u ) . How ever, u univ is an in terior p oin t of the cut, not its endp oin t. The only singular endp oin t pro duced by the h yp ergeometric con tin uation is u = ζ 2 c . Thus one exp ects ﬁnite lateral v alues at u = u univ , and the next prop osition conﬁrms this. The key p oin t is that every in terior p oin t of the cut is an ordinary p oin t of the h yp ergeometric diﬀeren tial equation. Lemma 6.4 (Interior points of the cut are ordinary p oin ts) . Fix p ≥ 1 and u 0 > ζ 2 c . L et G p, + and G p, − denote the analytic c ontinuations of the germ of G p fr om u = 0 to a neighb orho o d of u 0 thr ough the upp er and lower half-planes, r esp e ctively. Then e ach br anch extends holomorphic al ly to a ful l disk c enter e d at u 0 . In p articular, G p, ± ( u 0 ) , G ′ p, ± ( u 0 ) , G ′′ p, ± ( u 0 ) ar e ﬁnite. Pr o of. By Prop osition 5.4 , G p satisﬁes the generalized hypergeometric diﬀeren tial equation in the v ariable ξ = ζ − 2 c u . This equation is F uchsian, and its only ﬁnite singular p oin ts are ξ = 0 and ξ = 1 ; equiv alen tly , in the v ariable u the only ﬁnite singular p oin ts are u = 0 and u = ζ 2 c ; see [ 21 , §16.8,§16.11]. Hence every u 0 > ζ 2 c with u 0  = ζ 2 c is an ordinary p oin t of the diﬀeren tial equation. Cho ose r > 0 so small that the closed disk D ( u 0 , r ) a v oids { 0 , ζ 2 c } . On D ( u 0 , r ) , the diﬀerential equation has analytic co eﬃcien ts. Standard lo cal ODE theory therefore implies that any solution deﬁned on a connected op en subset of D ( u 0 , r ) extends uniquely to a holomorphic solution on the whole disk. Applying this to the branc hes G p, + and G p, − , initially deﬁned on the upp er and lo w er 34 OLEG ALEKSEEV half-disks, gives the claimed holomorphic extensions. Their v alues and ﬁrst t w o deriv ativ es at u 0 are therefore ﬁnite. □ Prop osition 6.5 (Regularit y at the geometric threshold) . F or every p ≥ 1 , the later al values G p ( u ± i 0) , σ cont p ( u ± i 0) , ar e ﬁnite for every u > ζ 2 c . In p articular, they ar e ﬁnite at u = ζ 2 univ , and the disc ontinuity density satisﬁes ρ p ( ζ 2 univ ) < ∞ . Pr o of. Fix u 0 > ζ 2 c . By Lemma 6.4 , eac h lateral branc h of G p extends holomorphically to a neigh b orhoo d of u 0 . Hence the limits G p ( u 0 ± i 0) exist and are ﬁnite, and so do the ﬁrst tw o lateral deriv ativ es. No w σ cont p is obtained from G p b y the Euler op erator σ cont p ( u ) = 1 p  p + s u d du  2 G p ( u ) , whose co eﬃcien ts are analytic at every ﬁnite u . Therefore σ cont p ( u 0 ± i 0) is ﬁnite for every u 0 > ζ 2 c , and in particular at u 0 = ζ 2 univ . Finally , Deﬁnition 5.12 gives ρ p ( u 0 ) = − 1 2 π i  σ cont p ( u 0 + i 0) − σ cont p ( u 0 − i 0)  , whic h is therefore ﬁnite as w ell. □ The three regimes of the problem ma y now be summarized as follows: (i) F or 0 < ζ < ζ c , each symmetry sector is describ ed by a compact weigh ted Gram op erator with one stiﬀ eigenv alue diverging logarithmically as ζ ↑ ζ c , while the remaining sectorial sp ectrum stays bounded. Equiv alen tly , there are at most s logarithmically diverging eigenv alues globally . (ii) F or ζ c < ζ < ζ univ , the weigh ted op erator realization breaks down, but the scalar data G p , σ cont p , and ρ p admit a canonical analytic con tin uation, with Jacobi–W eyl interpretation for 1 ≤ p ≤ s . (iii) At ζ = ζ univ , the conformal map loses univ alence; for s ≥ 3 the b oundary develops cusp singularities, whereas for s = 2 the critical map degenerates to the Jouko wski segmen t. In b oth cases, the contin ued scalar data remain ﬁnite. This completes the pro of that analytic criticalit y and geometric criticality are distinct in the symmetric one-harmonic family . Conclusion W e hav e shown that for the s -fold symmetric one-harmonic p olynomial family the ﬁrst sp ectral instabilit y of the mixed Hessian of the disp ersionless T o da τ -function is g ov erned b y analytic criticalit y of the in verse map, not by the later geometric loss of univ alence of the conformal map. The square-ro ot singularit y of the in v erse branch pro duces the borderline co eﬃcien t decay that driv es the transition. On the sub critical side, for any ﬁxed β > 0 , the weigh ted Hilb ert realization of Deﬁnition 4.1 separates this mec hanism into one stiﬀ direction and a b ounded soft sector in each symmetry block. A ccordingly , in eac h such weigh ted realization eac h sector contains exactly one logarithmically div erging eigenv alue, while the remaining sectorial spectrum sta ys b ounded; after removing the spik e direction, each ﬁxed soft eigenv alue conv erges to the corresp onding eigenv alue of a compact limiting remainder. The in trinsic co eﬃcien t-lev el conten t is the existence of a single dominant singular direction in each sector from whic h these w eigh ted spikes are realized. Beyond the analytic SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 35 threshold, the weigh ted compact realization ceases to apply , but the scalar Gram data remain well deﬁned after analytic contin uation. They are described by generalized h yp ergeometric functions, admit a Cauc h y–Stieltjes represen tation, and in the p ositiv e range ﬁt naturally in to the Jacobi/W eyl framew ork. The geometric threshold lies strictly b ey ond the analytic one. Consequen tly , the T o da Hessian b ecomes sp ectrally singular while the conformal map is still univ alen t and the b oundary is still smo oth, and the con tinued scalar data ha ve ﬁnite upper and lo wer lateral b oundary v alues at the univ alence threshold. In this sense, analytic and geometric criticalit y are genuinely distinct in the presen t family . The results also admit a natural heuristic in terpretation in adjacent frameworks. In the Laplacian- gro wth language, the rank-one logarithmic spike singles out a distinguished stiﬀ deformation mo de b efore an y geometric singularit y app ears. In the matrix-mo del or p oten tial-theoretic language, the pro v ed statement is that the mixed second-order resp onse concen trates on to a single dominant sectorial direction at criticality . These interpretiv e remarks are not used in the pro ofs and are included only to indicate the structural scop e of the result. The argument isolates a structural mec hanism that should persist beyond the one-harmonic leaf: a dominan t branc h p oin t, borderline co eﬃcient decay , and p ositiv e Gram structure. Several extensions therefore suggest themselves. F or more general p olynomial conformal maps one exp ects several comp eting dominan t orbits and hence a ﬁnite-rank version of the present instabilit y mechanism. It w ould also b e natural to understand the b ounded soft sector more conceptually , p ossibly through hidden Jacobi or total-p ositivit y structure, and to sharp en the leading-order theory by deriving asymptotics for sp ectral gaps, sector dep endence, and soft eigen v alues. A ckno wledgments The study w as implemented in the framew ork of the Basic Research Program at HSE Universit y in 2026. Appendix A. Asymptotics of Raney numbers This app endix provides explicit Stirling–type asymptotics for the Raney n um b ers in ( 2.6 ) . These form ulas are used in the Gram estimates and in App endix D . Prop osition A.1 (Explicit m − 3 / 2 asymptotics) . Fix inte gers s ≥ 2 , p ≥ 1 , and set ζ c := ( s − 1) s − 1 s s , M := s s − 1 . As m → ∞ , (A.1) R s,p ( m ) = A s,p ζ − m c m − 3 / 2  1 + O ( m − 1 )  , A s,p = p √ 2 π s p − 1 2 ( s − 1) p + 1 2 = p M p p 2 π s ( s − 1) . Conse quently, (A.2) R s,p ( m ) 2 = p 2 2 π s 2 p − 1 ( s − 1) 2 p +1 ζ − 2 m c m − 3  1 + O ( m − 1 )  . Pr o of. Starting from ( 2.6 ) and writing factorials via Gamma functions, R s,p ( m ) = p sm + p Γ( sm + p + 1) Γ( m + 1) Γ(( s − 1) m + p + 1) = p Γ( sm + p ) Γ( m + 1) Γ(( s − 1) m + p + 1) . F or ﬁxed s, p w e apply Stirling’s formula Γ( z ) = √ 2 π z z − 1 2 e − z  1 + O ( z − 1 )  , z → + ∞ , 36 OLEG ALEKSEEV to the three Gamma factors with arguments sm + p , m + 1 , and ( s − 1) m + p + 1 . The exp onen tial terms e − sm e m e ( s − 1) m cancel, while the p o w er terms pro duce ( sm ) sm + p − 1 2 m m + 1 2 (( s − 1) m ) ( s − 1) m + p + 1 2 =  s s ( s − 1) s − 1  m m − 3 / 2 s p − 1 2 ( s − 1) p + 1 2  1 + O ( m − 1 )  . Since ζ − 1 c = s s / ( s − 1) s − 1 , this gives ( A.1 ); squaring yields ( A.2 ). □ Prop osition A.2 (Uniform upp er b ound) . Ther e exists a c onstant C s > 0 such that for al l inte gers p ≥ 1 and m ≥ 1 , (A.3) R s,p ( m ) ≤ C s p M p ζ − m c m − 3 / 2 . Mor e over, the same b ound holds with m − 3 / 2 r eplac e d by (1 + m ) − 3 / 2 for al l m ≥ 0 . Pr o of. W e start from the closed form ( 2.6 ): R s,p ( m ) = p sm + p  sm + p m  , m ≥ 1 . Set N := sm + p and K := N − m = ( s − 1) m + p . Using a standard t wo-sided Stirling b ound (e.g. Robbins’ b ound), there is an absolute constan t C > 0 such that  N m  ≤ C N N + 1 2 m m + 1 2 K K + 1 2 = C 1 √ m r N K exp  N log N − m log m − K log K  . W rite t := p/m ≥ 0 , so that N = m ( s + t ) and K = m ( s − 1 + t ) . The exp onen t simpliﬁes to N log N − m log m − K log K = m h ( s + t ) log ( s + t ) − ( s − 1 + t ) log ( s − 1 + t ) i . Deﬁne Φ( t ) := log  s s ( s − 1) s − 1  + t log  s s − 1  − h ( s + t ) log ( s + t ) − ( s − 1 + t ) log ( s − 1 + t ) i . A direct computation giv es Φ(0) = Φ ′ (0) = 0 and Φ ′′ ( t ) = 1 ( s + t )( s − 1 + t ) > 0 , so Φ is conv ex with a global minimum at t = 0 . Hence Φ( t ) ≥ 0 for all t ≥ 0 , i.e. ( s + t ) log ( s + t ) − ( s − 1 + t ) log ( s − 1 + t ) ≤ log ( ζ − 1 c ) + t log ( M ) . Substituting back yields exp  N log N − m log m − K log K  ≤ ζ − m c M p . Also p N /K ≤ p s/ ( s − 1) = √ M and p sm + p ≤ p sm . Combining these b ounds giv es R s,p ( m ) ≤ p sm · C 1 √ m · √ M · ζ − m c M p ≤ C s p M p ζ − m c m − 3 / 2 , for a constan t C s dep ending only on s . The extension to m = 0 follows by enlarging C s , since R s,p (0) = 1 and (1 + m ) − 3 / 2 = 1 at m = 0 . □ Lemma A.3 (Uniform one-term expansion in the tail region) . Fix s ≥ 2 and 0 < θ < 1 . Ther e exists a c onstant C s,θ > 0 such that for al l inte gers p ≥ 1 and m ≥ 1 satisfying p ≤ θ m , (A.4) R s,p ( m ) = A s,p ζ − m c m − 3 / 2  1 + ε s,p ( m )  , | ε s,p ( m ) | ≤ C s,θ m , wher e A s,p is as in ( A.1 ) . SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 37 Pr o of. W rite t := p/m , so 0 ≤ t ≤ θ , and write p = tm . Using the Gamma-form expression from the pro of of Prop osition A.1 , consider log  R s,p ( m ) A s,p ζ − m c m − 3 / 2  . Applying the logarithmic Stirling expansion log Γ( z ) =  z − 1 2  log z − z + 1 2 log(2 π ) + O ( z − 1 ) to the three Gamma factors with arguments m ( s + t ) , m + 1 , and m ( s − 1 + t ) + 1 , one obtains log R s,p ( m ) = − m log ζ c + log A s,p − 3 2 log m + O ( m − 1 ) , uniformly for t ∈ [0 , θ ] . Equiv alen tly , log  R s,p ( m ) A s,p ζ − m c m − 3 / 2  = O ( m − 1 ) , uniformly for t ∈ [0 , θ ] . Here the terms of order m , the log m con tribution, and the explicit t - dep enden t prefactor enco ded in A s,p cancel identically; the remainder is uniform b ecause all auxiliary functions of t are smo oth on the compact interv al [0 , θ ] . Exp onentiating yields R s,p ( m ) A s,p ζ − m c m − 3 / 2 = 1 + O ( m − 1 ) , uniformly for p ≤ θ m , which is exactly ( A.4 ). □ Appendix B. Rank–one t ail extra ction in the Gram blocks This app endix records the co eﬃcien t–level structure b ehind the rank–one decomp osition in Theorem 4.6 . Throughout w e ﬁx a symmetry sector q ∈ { 1 , . . . , s } , write p j := q + j s, j ∈ N 0 , and use the Gram v ectors v ( p ) from ( 3.8 ) . Recall the block matrix G ( q ) ( ζ ) determined co eﬃcien t wise b y the Gram representation in ( 4.1 ), and its weigh ted conjugate e G ( q ) ( ζ ) = W − 1 G ( q ) ( ζ ) W − 1 , where W = diag ( w j ) and w j is given b y ( 4.2 ). B.1. En try form ula. Lemma B.1 (Entries of the Gram blo c k) . F or j 1 ≤ j 2 (and ∆ := j 2 − j 1 ≥ 0 ) one has (B.1) G ( q ) j 1 j 2 ( ζ ) = ∞ X m =0 ( p j 2 + sm ) 2 √ p j 1 p j 2 R s,p j 1 ( m + ∆) R s,p j 2 ( m ) ζ 2 m +∆ . Conse quently, (B.2) e G ( q ) j 1 j 2 ( ζ ) = 1 w j 1 w j 2 G ( q ) j 1 j 2 ( ζ ) , j 1 , j 2 ∈ N 0 . Pr o of. By deﬁnition, G ( q ) j 1 j 2 ( ζ ) = ⟨ v ( p j 1 ) , v ( p j 2 ) ⟩ in ℓ 2 ( N ) , and v ( p ) is supp orted on indices p + sm . F or j 1 ≤ j 2 , the supp orts o verlap precisely at p j 2 + sm = p j 1 + s ( m + ∆) , m ≥ 0 . Inserting ( 3.8 ) at these indices yields ( B.1 ). The conjugated entry relation ( B.2 ) is immediate from ( 4.4 ). □ 38 OLEG ALEKSEEV B.2. The logarithmic tail and the singular v ector. W rite η := ζ ζ c ∈ (0 , 1) , δ := 1 − η 2 = 1 − ζ 2 ζ 2 c , L ( ζ ) = X m ≥ 1 η 2 m m = log 1 1 − ζ 2 /ζ 2 c . By Remark 2.8 (or App endix A ), for each ﬁxed p ≥ 1 , (B.3) R s,p ( m ) = A s,p ζ − m c m − 3 / 2  1 + O ( m − 1 )  , m → ∞ , with an explicit constan t A s,p > 0 . Deﬁne the intrinsic amplitudes and their w eighted realizations by (B.4) d ( q ) j := s A s,p j √ p j , e d ( q ) j := d ( q ) j w j , j ∈ N 0 . Then e d ( q ) := ( e d ( q ) j ) j ≥ 0 ∈ ℓ 2 ( N 0 ) , and it is the v ector app earing in Theorem 4.6 . Lemma B.2 (Uniform exp onen tial gap a w ay from the critical slop e) . Fix s ≥ 2 and θ > 0 . Ther e exist c onstants C s,θ , c s,θ > 0 such that for al l inte gers p, m ≥ 1 satisfying p ≥ θ m , R s,p ( m ) ≤ C s,θ p M p ζ − m c m − 3 / 2 e − c s,θ p . Pr o of. W rite t := p/m ∈ [ θ , ∞ ) . By the Stirling estimate used in the pro of of Proposition A.2 , one has R s,p ( m ) ≤ C s p m − 3 / 2 exp  m Ψ s ( t )  , where Ψ s ( t ) := ( s + t ) log ( s + t ) − ( s − 1 + t ) log ( s − 1 + t ) . Equiv alently , R s,p ( m ) ≤ C s p M p ζ − m c m − 3 / 2 exp  p Ξ s ( t )  , with Ξ s ( t ) := Ψ s ( t ) − log( ζ − 1 c ) t − log M . The function Ξ s is contin uous on [ θ , ∞ ) . The con v exity argumen t used in Prop osition A.2 yields Ψ s ( t ) − t log M ≤ log ( ζ − 1 c ) , with equality only at t = 0 , so Ξ s ( t ) < 0 for ev ery t > 0 . Moreov er, Ξ s ( t ) → − log M < 0 as t → ∞ . Hence sup t ≥ θ Ξ s ( t ) = − c s,θ < 0 , whic h gives the stated b ound. □ Lemma B.3 (Logarithmic tail with Hilb ert–Sc hmidt remainder) . L et η = ζ /ζ c ∈ (0 , 1) and L ( ζ ) = log 1 1 − η 2 . Deﬁne the r ank–one T o eplitz kernel ( K η ) j 1 j 2 := η | j 1 − j 2 | e d ( q ) j 1 e d ( q ) j 2 , j 1 , j 2 ∈ N 0 . Then, for every 0 < ζ < ζ c , the op er ator R ( q ) ( ζ ) := e G ( q ) ( ζ ) − L ( ζ ) K η is Hilb ert–Schmidt on ℓ 2 ( N 0 ) , with sup ζ <ζ c ∥ R ( q ) ( ζ ) ∥ HS < ∞ . Mor e over, as ζ ↑ ζ c , R ( q ) ( ζ ) c onver ges in Hilb ert–Schmidt norm to a limit Hilb ert–Schmidt op er ator R ( q ) ∗ , henc e also in op er ator norm. In p articular, for e ach ﬁxe d ( j 1 , j 2 ) one has the entrywise exp ansion (B.5) e G ( q ) j 1 j 2 ( ζ ) = e d ( q ) j 1 e d ( q ) j 2 η | j 1 − j 2 | L ( ζ ) + ( R ( q ) ∗ ) j 1 j 2 + o (1) , ζ ↑ ζ c , and the err or is o (1) in ℓ 2 –matrix (Hilb ert–Schmidt) sense. SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 39 Pr o of. By symmetry of the Gram k ernel, it is enough to treat the case j 1 ≤ j 2 . W rite ∆ := j 2 − j 1 ≥ 0 . By Lemma B.1 , e G ( q ) j 1 j 2 ( ζ ) = X m ≥ 0 U m ( j 1 , j 2 ; ζ ) , where U m ( j 1 , j 2 ; ζ ) := 1 w j 1 w j 2 ( p j 2 + sm ) 2 √ p j 1 p j 2 R s,p j 1 ( m + ∆) R s,p j 2 ( m ) ζ 2 m +∆ . Since p j 2 ≥ p j 1 and p j 2 ≥ ∆ , the cutoﬀ M j 1 j 2 := max { 2 p j 1 , 2 p j 2 , 2∆ , 1 } reduces here to M j 1 j 2 = 2 p j 2 . W e split the sum into the initial range 0 ≤ m < 2 p j 2 and the tail m ≥ 2 p j 2 . T ail r e gion. If m ≥ 2 p j 2 , then p j 2 ≤ m/ 2 and p j 1 ≤ ( m + ∆) / 2 , so Lemma A.3 applies to both Raney factors (with θ = 1 2 ). More explicitly , R s,p j 1 ( m + ∆) = A s,p j 1 ζ − m − ∆ c ( m + ∆) − 3 / 2  1 + O (( m + 1) − 1 )  , and R s,p j 2 ( m ) = A s,p j 2 ζ − m c m − 3 / 2  1 + O (( m + 1) − 1 )  , uniformly for m ≥ 2 p j 2 . Since j 1 ≤ j 2 , the quantities p j 1 , p j 2 , ∆ are ﬁxed along the tail sum, while ( p j 2 + sm ) 2 = s 2 m 2  1 + O (( m + 1) − 1 )  and ( m + ∆) − 3 / 2 m − 3 / 2 = m − 3  1 + O (( m + 1) − 1 )  . Hence ( p j 2 + sm ) 2 √ p j 1 p j 2 R s,p j 1 ( m + ∆) R s,p j 2 ( m ) ζ 2 m +∆ = s 2 A s,p j 1 A s,p j 2 √ p j 1 p j 2 η ∆ η 2 m m  1 + O (( m + 1) − 1 )  . Using ( B.4 ) and m − 1 = ( m + 1) − 1 + O (( m + 1) − 2 ) , we obtain (B.6) U m ( j 1 , j 2 ; ζ ) = e d ( q ) j 1 e d ( q ) j 2 η ∆ η 2 m m + 1 + E tail m ( j 1 , j 2 ; ζ ) , with (B.7) | E tail m ( j 1 , j 2 ; ζ ) | ≤ C e d ( q ) j 1 e d ( q ) j 2 η ∆ η 2 m ( m + 1) 2 , m ≥ 2 p j 2 , where C dep ends only on s and β . Summing ov er m ≥ 2 p j 2 sho ws that E tail j 1 j 2 ( ζ ) := X m ≥ 2 p j 2 E tail m ( j 1 , j 2 ; ζ ) deﬁnes a uniformly Hilb ert–Sc hmidt k ernel, since | E tail j 1 j 2 ( ζ ) | ≤ C e d ( q ) j 1 e d ( q ) j 2 X m ≥ 0 1 ( m + 1) 2 ≤ C ′ e d ( q ) j 1 e d ( q ) j 2 , and e d ( q ) ∈ ℓ 2 ( N 0 ) . Initial r ange. W rite E init j 1 j 2 ( ζ ) := X 0 ≤ m< 2 p j 2 U m ( j 1 , j 2 ; ζ ) . 40 OLEG ALEKSEEV The term m = 0 is estimated b y Prop osition A.2 : | U 0 ( j 1 , j 2 ; ζ ) | ≤ C M − p j 2 p − 1 − β j 1 p 1 − β j 2 (1 + ∆) − 3 / 2 . F or 1 ≤ m < 2 p j 2 , Prop osition A.2 applies to R s,p j 1 ( m + ∆) , while Lemma B.2 with θ = 1 2 applies to R s,p j 2 ( m ) b ecause p j 2 ≥ m/ 2 . Using ζ 2 m +∆ ≤ ζ 2 m +∆ c , one gets | U m ( j 1 , j 2 ; ζ ) | ≤ C e − cp j 2 p − 1 − β j 1 p − 1 − β j 2 ( p j 2 + sm ) 2 ( m + 1) 3 / 2 ( m + ∆ + 1) 3 / 2 . Since m < 2 p j 2 , we ha v e p j 2 + sm ≤ (1 + 2 s ) p j 2 , and therefore 2 p j 2 − 1 X m =1 | U m ( j 1 , j 2 ; ζ ) | ≤ C e − cp j 2 p − 1 − β j 1 p 1 − β j 2 X m ≥ 1 1 ( m + 1) 3 ≤ C ′ e − cp j 2 p − 1 − β j 1 p 1 − β j 2 . Hence | E init j 1 j 2 ( ζ ) | ≤ C ′′ e − c ′ p j 2 p − 1 − β j 1 p 1 − β j 2 , for constants C ′′ , c ′ > 0 indep enden t of ζ . This b ound is square summable in ( j 1 , j 2 ) , so E init ( ζ ) is uniformly Hilbert–Schmidt. Moreov er, for each ﬁxed ( j 1 , j 2 ) the sum is ﬁnite and therefore conv erges term wise as ζ ↑ ζ c . Summing ( B.6 ) ov er m ≥ 2 p j 2 giv es X m ≥ 2 p j 2 U m ( j 1 , j 2 ; ζ ) = e d ( q ) j 1 e d ( q ) j 2 η ∆ X m ≥ 2 p j 2 η 2 m m + 1 + E tail j 1 j 2 ( ζ ) . Reinstating the omitted harmonic terms pro duces the correction E harm j 1 j 2 ( ζ ) := e d ( q ) j 1 e d ( q ) j 2 η ∆ 2 p j 2 − 1 X m =0 η 2 m m + 1 . Since 2 p j 2 − 1 X m =0 η 2 m m + 1 ≤ 1 + log (1 + 2 p j 2 ) , w e obtain | E harm j 1 j 2 ( ζ ) | ≤ C e d ( q ) j 1 e d ( q ) j 2  1 + log(1 + p j 2 )  . Because e d ( q ) j ≍ p − 1 − β j , the kernel on the righ t-hand side is Hilb ert–Sc hmidt: X j 1 ,j 2 ≥ 0 e d ( q ) j 1 2 e d ( q ) j 2 2  1 + log(1 + p j 2 )  2 < ∞ . Th us e G ( q ) j 1 j 2 ( ζ ) = e d ( q ) j 1 e d ( q ) j 2 η ∆ X m ≥ 0 η 2 m m + 1 + E init j 1 j 2 ( ζ ) + E tail j 1 j 2 ( ζ ) + E harm j 1 j 2 ( ζ ) . Since P m ≥ 0 η 2 m m +1 = η − 2 L ( ζ ) , we write η − 2 L ( ζ ) = L ( ζ ) +  η − 2 − 1  L ( ζ ) . The co eﬃcient  η − 2 − 1  L ( ζ ) sta ys b ounded as η ↑ 1 , and K η is uniformly Hilb ert–Sc hmidt. Therefore the extra term  η − 2 − 1  L ( ζ ) K η is uniformly Hilb ert–Sc hmidt as w ell. Collecting all Hilb ert–Sc hmidt pieces into R ( q ) ( ζ ) yields e G ( q ) ( ζ ) = L ( ζ ) K η + R ( q ) ( ζ ) , with sup ζ <ζ c ∥ R ( q ) ( ζ ) ∥ HS < ∞ . SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 41 Finally , each constituen t of R ( q ) ( ζ ) has a p oint wise limit as ζ ↑ ζ c , and the dominating Hilb ert– Sc hmidt kernels displa y ed ab o v e are indep enden t of ζ . Dominated con v ergence therefore yields R ( q ) ( ζ ) → R ( q ) ∗ in Hilb ert–Sc hmidt norm as ζ ↑ ζ c , hence also in op erator norm. □ B.3. F rom entrywise asymptotics to a rank–one op erator. The prefactor η | j 1 − j 2 | in ( B.5 ) tends to 1 as ζ ↑ ζ c . The next lemma quantiﬁes that, after weigh ting, it can b e absorb ed in to the b ounded remainder. Lemma B.4 (Remov al of the T o eplitz prefactor) . L et D = ( e d ( q ) j ) j ≥ 0 ∈ ℓ 2 ( N 0 ) and deﬁne op er ators K η : ( K η ) j 1 j 2 := η | j 1 − j 2 | D j 1 D j 2 , K 1 := D ⊗ D ∗ , 0 < η ≤ 1 . Then K η − K 1 → 0 in Hilb ert–Schmidt norm as η ↑ 1 . In p articular, ∥ K η − K 1 ∥ → 0 , L ( ζ ) ∥ K η − K 1 ∥ → 0 as ζ ↑ ζ c . Pr o of. By ( B.4 ), ( 4.2 ) and the explicit constant A s,p = p M p √ 2 π s ( s − 1) in ( A.1 ), one has D j = e d ( q ) j = c s p − 1 − β j , c s := r s 2 π ( s − 1) , hence a j := D 2 j ≤ C (1 + j ) − 2 − 2 β and P j ≥ 0 a j < ∞ . Set b d := P j ≥ 0 a j a j + d for d ≥ 0 . By symmetry , ∥ K η − K 1 ∥ 2 HS = X j 1 ,j 2 ≥ 0 (1 − η | j 1 − j 2 | ) 2 a j 1 a j 2 ≍ X d ≥ 0 (1 − η d ) 2 b d , up to an absolute factor. Moreo ver, since a j + d ≤ C (1 + d ) − 2 − 2 β for all j ≥ 0 , b d = X j ≥ 0 a j a j + d ≤ C (1 + d ) − 2 − 2 β X j ≥ 0 a j ≤ C ′ (1 + d ) − 2 − 2 β . Fix N ∈ N . Using 1 − η d ≤ d (1 − η ) for d ≤ N and 1 − η d ≤ 1 for d > N , we get X d ≥ 0 (1 − η d ) 2 b d ≤ (1 − η ) 2 N X d =0 d 2 b d + X d>N b d ≤ C ′ (1 − η ) 2 N X d =1 d − 2 β + C ′ X d>N d − 2 − 2 β . The last tail is O ( N − 1 − 2 β ) . The partial sum satisﬁes P N d =1 d − 2 β = O ( N 1 − 2 β ) if β < 1 2 , O ( log N ) if β = 1 2 , and O (1) if β > 1 2 . Cho ose N := ⌊ (1 − η ) − 1 ⌋ to balance the tw o terms. This yields      O  (1 − η ) 1+2 β  , 0 < β < 1 2 , O  (1 − η ) 2 log 1 1 − η  , β = 1 2 , O  (1 − η ) 2  , β > 1 2 , η ↑ 1 . In particular, ∥ K η − K 1 ∥ HS = O  (1 − η ) min { 1 , 1 2 + β }  up to an extra factor q log 1 1 − η when β = 1 2 . Since min { 1 , 1 2 + β } ≥ min { 1 , β } for all β > 0 , this is the required Hilb ert–Schmidt estimate. Finally , ∥ K η − K 1 ∥ ≤ ∥ K η − K 1 ∥ HS and L ( ζ ) ∼ log 1 1 − η with (1 − η ) γ log 1 1 − η → 0 for an y γ > 0 , yield L ( ζ ) ∥ K η − K 1 ∥ → 0 . This prov es the lemma. □ 42 OLEG ALEKSEEV Appendix C. Hypergeometric continua tion of G p W e give a pro of of the hypergeometric representation stated in Prop osition 5.4 . W e keep the notation { α i } 2 s i =1 and { β j } 2 s − 1 j =1 from ( 5.6 ). Pr o of of Pr op osition 5.4 . W rite G p ( u ) = X m ≥ 0 a m u m , a m := R s,p ( m ) 2 , R s,p ( m ) = p sm + p  sm + p m  . Using R s,p ( m ) = p Γ( sm + p ) Γ( m + 1)Γ(( s − 1) m + p + 1) , one computes R s,p ( m + 1) R s,p ( m ) = Q s − 1 k =0 ( sm + p + k ) ( m + 1) Q s − 1 l =1 (( s − 1) m + p + l ) . Squaring and factorizing the linear terms gives a m +1 a m = ζ − 2 c Q s − 1 k =0  m + p + k s  2 ( m + 1) 2 Q s − 1 l =1  m + p + l s − 1  2 , b ecause s 2 s ( s − 1) 2 s − 2 = ζ − 2 c . No w consider the generalized hypergeometric series H ( u ) := 2 s F 2 s − 1  α 1 , . . . , α 2 s β 1 , . . . , β 2 s − 1    ζ − 2 c u  , with parameter multisets { α i } = 2 ×  p + k s : k = 0 , . . . , s − 1  , { β j } = { 1 } ∪ 2 ×  p + l s − 1 : l = 1 , . . . , s − 1  . Its co eﬃcien ts b m satisfy b 0 = 1 and the standard ratio form ula b m +1 b m = ζ − 2 c Q 2 s i =1 ( m + α i ) ( m + 1) Q 2 s − 1 j =1 ( m + β j ) . By the chosen parameter sets, this is exactly the ratio display ed ab o ve for a m +1 /a m . Since a 0 = R s,p (0) 2 = 1 = b 0 , the sequences coincide for all m ≥ 0 , and therefore G p ( u ) = H ( u ) . This pro v es ( 5.5 ). □ Appendix D. The logarithmic coefficient a t the branch point This app endix computes the co eﬃcient B ( ζ 2 c ) in the resonan t lo cal expansion ( 5.7 ) of G p at the branc h p oin t u = ζ 2 c . Recall that the radius of conv ergence of G p equals ζ 2 c . Prop osition D.1 (Explicit v alue of B ( ζ 2 c ) ) . F or al l inte gers s ≥ 2 and p ≥ 1 , the c o eﬃcient B ( ζ 2 c ) in ( 5.7 ) satisﬁes (D.1) B ( ζ 2 c ) = − p 2 4 π s 2 p − 1 ( s − 1) 2 p +1 . In p articular, B ( ζ 2 c ) < 0 . SPECTRAL STRUCTURE OF THE MIXED HESSIAN OF THE DISPERSIONLESS TODA τ -FUNCTION 43 Pr o of. W rite G p ( u ) = P m ≥ 0 R s,p ( m ) 2 u m and introduce the scaled v ariable ξ := u/ζ 2 c , so that G p ( ζ 2 c ξ ) = X m ≥ 0 a m ξ m , a m := R s,p ( m ) 2 ζ 2 m c . By Prop osition A.1 , one has the reﬁned asymptotic (D.2) a m = C s,p m − 3 + O ( m − 4 ) , C s,p := p 2 2 π s 2 p − 1 ( s − 1) 2 p +1 . On the other hand, the p olylogarithm Li 3 ( ξ ) = P m ≥ 1 m − 3 ξ m has the classical singular expansion at ξ = 1 (see, e.g., DLMF §25.12 [ 21 ]) Li 3 ( ξ ) = (analytic at ξ = 1 ) − 1 2 (1 − ξ ) 2 log(1 − ξ ) + O  (1 − ξ ) 2  , ξ → 1 . Deﬁne the remainder series H ( ξ ) := G p ( ζ 2 c ξ ) − C s,p Li 3 ( ξ ) = X m ≥ 1  a m − C s,p m − 3  ξ m . By ( D.2 ) , the co eﬃcien ts satisfy a m − C s,p m − 3 = O ( m − 4 ) , hence P m ≥ 1 m 2   a m − C s,p m − 3   < ∞ . Therefore the series for H ( ξ ) , H ′ ( ξ ) and H ′′ ( ξ ) conv erge absolutely at ξ = 1 , so H extends to a C 2 -function on [0 , 1] and therefore H ( ξ ) = H (1) + H ′ (1)( ξ − 1) + 1 2 H ′′ (1)( ξ − 1) 2 + o  (1 − ξ ) 2  , ξ → 1 . In particular, H con tributes no (1 − ξ ) 2 log(1 − ξ ) term. Consequen tly , the (1 − ξ ) 2 log (1 − ξ ) co eﬃcien t in G p ( ζ 2 c ξ ) is exactly − 1 2 C s,p , coming from C s,p Li 3 ( ξ ) . Comparing with ( 5.7 ) yields B ( ζ 2 c ) = − 1 2 C s,p , which is ( D.1 ). □ Corollary D.2 (Rational–ov er– π structure) . F or al l inte gers s ≥ 2 and p ≥ 1 one has π B ( ζ 2 c ) ∈ Q and π B ( ζ 2 c ) < 0 . Conse quently, the e dge value in The or em 5.13 satisﬁes ρ p ( ζ 2 c ) ∈ π − 1 Q . References [1] M. Mineev-W einstein, P . B. Wiegmann, and A. Zabro din, “Integrable structure of interface dynamics,” Physic al R eview L etters , v ol. 84, no. 22, pp. 5106–5109, 2000. [2] P . B. Wiegmann and A. Zabro din, “Conformal maps and integrable hierarchies,” Communic ations in Mathematic al Physics , v ol. 213, no. 3, pp. 523–538, 2000. [3] A. Marshako v, P . Wiegmann, and A. Zabro din, “Integrable structure of the Dirichlet b oundary problem in t w o dimensions.,” Communic ations in Mathematic al Physics , vol. 227, pp. 131–153, 2002. [4] I. Kosto v, I. Krichev er, M. Mineev-W einstein, P . Wiegmann, and A. Zabro din, “tau-function for analytic curves,” in R andom Matric es and their Applic ations (P . Bleher and A. Its, eds.), v ol. 40, pp. 285–299, Cambridge, U.K.: Cambridge Univ ersit y Press, 01 2001. [5] B. Gustafsson and Y.-L. Lin, “On the dynamics of roots and p oles for solutions of the P olubarino v a–Galin equation,” Annales F ennici Mathematici , v ol. 38, no. 1, pp. 259–286, 2013. [6] P . B. Wiegmann and A. Zabrodin, “Large N expansion for the 2D Dyson gas,” Journal of Physics A: Mathematic al and Gener al , v ol. 39, no. 28, pp. 8933–8964, 2006. [7] L.-P . T eo, “Conformal mappings and the disp ersionless T o da hierarch y ,” Communic ations in Mathematic al Physics , vol. 292, pp. 391–415, 2009. [8] P . L. Duren, Univalent F unctions , vol. 259 of Grund lehr en der mathematischen Wissenschaften . New Y ork: Springer-V erlag, 1983. [9] C. Pommerenk e, Boundary Behaviour of Conformal Maps , vol. 299 of Grund lehr en der mathe- matischen Wissenschaften . Berlin: Springer-V erlag, 1992. 44 OLEG ALEKSEEV [10] S. R. Bell, The Cauchy T r ansform, Potential The ory, and Conformal Mapping . Bo ca Raton, FL: CRC Press, 1992. [11] M. Jeong, “The b ergman kernel for circular m ultiply connected domains,” Paciﬁc Journal of Mathematics , vol. 233, no. 1, pp. 71–86, 2007. [12] M. Sc hiﬀer, “F redholm eigenv alues and Grunsky matrices,” Annales Polonici Mathematici , v ol. 39, no. 1, pp. 149–164, 1981. [13] Y. Shen, “F redholm eigenv alue for a quasi-circle and grunsky functionals,” Annales F ennici Mathematici , vol. 35, no. 2, pp. 581–593, 2010. [14] K. Johansson and F. Viklund, “Coulomb gas and the Grunsky op erator on a Jordan domain with corners.” arXiv preprin t, 2023. [15] O. Alekseev, “Univ ersalit y of rank-one logarithmic instabilit y in disp ersionless to da hierarc h y ,” 2025. Preprint. [16] G. N. Raney , “F unctional comp osition patterns and p o w er series reversion,” T r ansactions of the A meric an Mathematic al So ciety , v ol. 94, no. 3, pp. 441–451, 1960. [17] R. L. Graham, D. E. Kn uth, and O. Patashnik, Concr ete Mathematics: A F oundation for Computer Scienc e . Reading, MA: Addison-W esley , 2 ed., 1994. [18] P . Henrici, Applie d and Computational Complex Analysis, V olume 1: Power Series, Inte gr ation, Conformal Mapping, L o c ation of Zer os . New Y ork: John Wiley & Sons, 1974. [19] L. Comtet, A dvanc e d Combinatorics: The Art of Finite and Inﬁnite Exp ansions . Dordrech t: D. Reidel Publishing Company , 1974. [20] P . Fla jolet and R. Sedgewick, Analytic Combinatorics . Cam bridge: Cambridge Universit y Press, 2009. [21] NIST Digital Library of Mathematical F unctions, “ NIST Digital Library of Mathematical F unctions.” https://dlmf.nist.gov/ . Release 1.2.1 (2025-06-15), accessed 2026-01-18. [22] J.-G. Liu and R. L. P ego, “On generating functions of Hausdorﬀ moment sequences,” T r ansactions of the Americ an Mathematic al So ciety , v ol. 368, pp. 8499–8518, Dec. 2016. [23] J.-G. Liu and R. L. Pego, “Corrigendum to “on generating functions of hausdorﬀ moment sequences”,” T r ansactions of the Americ an Mathematic al So ciety , vol. 379, no. 4, 2026. [24] P . J. F orrester and D.-Z. Liu, “Raney distributions and random matrix theory ,” Journal of Statistic al Physics , vol. 158, no. 5, pp. 1051–1082, 2015. [25] N. I. Akhiezer, The Classic al Moment Pr oblem and Some R elate d Questions in A nalysis . Edin burgh: Oliver & Boyd, 1965. T ranslated from the Russian b y N. Kemmer. Dep ar tment of Ma thema tics, HSE University (Na tional Research University Higher School of Economics), Moscow, Russia

Spectral Structure of the Mixed Hessian of the Dispersionless Toda $τ$-Function

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment