Error Controllability Under Finite-Order Discipline: PSWF/DPSS Extremal Windows Uniqueness, Integer Leading Terms (Spectral Flow/Index of Projection Pairs), and $10^{-3

Error Controllability Under Finite-Order Discipline: PSWF/DPSS Extremal Windows Uniqueness, Integer Leading Terms (Spectral Flow/Index of Projection Pairs), and 10−3 -Level Universal Constants Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract Under unied Fourier normalization b f(ξ) = RRf(t)e−2πitξ dt (frequency in cycles), we construct error discipline around time-limitingband-limiting concatenated operators: windowing main leakage, multiplicative cross-terms, and sumintegral dierence are organized into computable chains of topological integer leading terms + analytic tail terms. On the continuous side, Kc=DTBΩDT (discrete side KN,W =TNBWTN ) yields uniqueness of extremal windows and leakage identity |(I−BΩ)g∗|2 2= 1 −λ0 ; on any fundamental domain of length 1, squared-sum aliasing energy equals out-of-domain energy; multiplicative cross-terms' Hankeltype blocks yield HilbertSchmidt (HS) exact formulas; EulerMaclaurin (EM) remainder analytic tail terms are controlled in closed form by periodic Bernoulli supremum constants and BPW inequality. Based on explicit non-asymptotic eigenvalue upper bounds in natural logarithm caliber, we obtain window-shape-independent minimal integer Shannon number thresholds (ε, N⋆ 0) = (10−3,33),(10−6,42),(10−9,50) . Under denability hypotheses (trace-class dierence and strongly continuous paths), spectral ow equals index of projection pairs, identifying integer leading terms of errors as topological invariants. Complete proofs and reproducible procedures are provided. Keywords : Timeband limiting; Prolate Spheroidal Wave Functions (PSWF); Discrete Prolate Spheroidal Sequences (DPSS); Shannon number; aliasing; Hankel block; HilbertSchmidt norm; EulerMaclaurin remainder; spectral ow; index of projection pairs; de Branges 1 Introduction & Historical Context The time-limitingband-limiting problem occupies a central position in signal processing and harmonic analysis. The SlepianLandauPollak framework reveals that under concatenated constraints of nite time window [−T, T] and nite bandwidth [−Ω,Ω] , waveforms with optimal energy concentration are given by PSWF/DPSS, whose eigenvalues cluster exponentially near 1 and 0 . Engineering practice commonly encounters three types of errorswindowing out-of-band leakage, aliasing, and sumintegral dierence (Poisson/EM remainder)historically treated separately, leading to incomparable and non-reproducible thresholds and constants. This paper proposes, under unied cycles normalization, a computableprovablereproducible error discipline chain: (1) Precisely characterize windowing main leakage via principal eigenvalue λ0 of Kc=DTBΩDT (discrete side KN,W =TNBWTN ); (2) Reduce aliasing to out-of-band energy via the identity squared-sum aliasing = out-of-domain energy; (3) Provide HankelHS exact formulas and bounds for out-of-band leakage after multiplicative action; (4) Characterize integer 1 leading terms via the framework spectral ow = index of projection pairs, giving closed-form EM analytic tail terms via VaalerLittmann extrema and periodic Bernoulli constants; (5) Generate window-shape-independent minimal integer Shannon number thresholds via explicit nonasymptotic eigenvalue upper bounds in natural logarithm caliber. Thus, three blocks of errors all reduce to three computable quantities: 1−λ0 , HankelHS, and EM tail terms; integer leading terms are carried by spectral invariants. 2 Model & Assumptions Fourier and units : b f(ξ) = RRf(t)e−2πitξ dt , ξ in cycles; Plancherel: |f|2=|b f|2 . Projections : Time-limiting DTf=1[−T,T ]f ; band-limiting BΩf=F−1(1[−Ω,Ω] b f) . Concatenated operators : Continuous side Kc=DTBΩDT ; discrete side KN,W =TNBWTN ( TN lengthN restriction, BW band-limiting projection for [−W, W ]⊂[−1 2,1 2] ). Shannon number : c=πTΩ , N0= 2TΩ = 2c/π (continuous); N0= 2NW (discrete). Both sides aligned via N0 . BPW : If supp bg⊂[−Ω,Ω] , then |g(m)|2≤(2πΩ)m|g|2 . Norms : |·|2,|·|∞ ; for any bounded operator |A|op ≤ |A|HS . Fundamental domain : Any length-1 interval I= [a, a + 1) , for aliasing energy normalization. 3 Main Results (Theorems and Alignments) Theorem 1 (Theorem 1: Extremal Window Uniqueness and Leakage Identity) . Let Kc=DTBΩDT be a compact self-adjoint positive operator acting on L2(R) . Its largest eigenvalue λ0∈(0,1) is simple, with corresponding eigenfunction g∗ ( |g∗|2= 1 ) unique (up to phase), satisfying |BΩg∗|2 2=λ0,|(I−BΩ)g∗|2 2= 1 −λ0. The discrete side KN,W =TNBWTN is completely parallel, with principal vector being the rst DPSS, also simple. Alignment 2 (Alignment 1: Squared-Sum Aliasing = Out-of-Domain Energy) . For any w∈L2(R) and any fundamental domain I= [a, a + 1) of length 1, ZIX k=0 bw(ξ+k)2dξ =ZR\Ibw(η)2dη. When I aligns with physical passband [−Ω,Ω] (via translation/scaling), the right side is out-ofdomain energy relative to that passband; taking w=g∗ yields Aliasing(g∗;I)=1−λ0 . Theorem 3 (Theorem 2: HS Exact Formula and Bounds for Multiplicative Cross-Terms) . For x∈L∞∩L2 and W∈(0,1 2] , |(I−BW)MxBW|2 HS =ZR|bx(δ)|2σW(δ)dδ, σW(δ) = min(2W, |δ|). Furthermore, for any w∈L2 , |(I−BW)Mxw|2≤ |(I−BW)MxBW|op|w|2+|x|∞|(I−BW)w|2, 2 thus for extremal window g∗ (Theorem 1): |(I−BW)Mxg∗|2≤ |(I−BW)MxBW|HS +|x|∞p1−λ0. Moreover, (I−BW)MxBW≡0 if and only if x is a.e. constant. Theorem 4 (Theorem 3: Analytic Tail Bounds for EM Remainder) . For g∈W2p,1(R)∩L2(R) , |R2p(g)| ≤ 2ζ(2p) (2π)2p|g(2p)|L1. If additionally supp bg⊂[−Ω,Ω] and local evaluation on time-domain length L , |R2p(g)| |g|2≤2ζ(2p)√LΩ2p. Sucient thresholds achieving |R2p(g)|/|g|2≤10−3 are √LΩ4≤4.6197 ×10−4,√LΩ6≤4.9148 ×10−4,√LΩ8≤4.9797 ×10−4. Theorem 5 (Theorem 4: Spectral Flow = Index of Projection Pairs: Topologizing Integer Leading Terms) . Take smooth frequency multiplier ϕ∈C∞ c(R) , let Π = F−1MϕF and orthogonal projection P=1[1/2,∞)(Π) . Let modulation group Uθf(t) = e2πiθtf(t) , Pθ=UθPU∗ θ . If (i) P−Pθ∈ S1 ; (ii) θ7→ Uθ strongly continuous, then self-adjoint path A(θ) = 2Pθ−I admits spectral ow, with SfA(θ)θ∈[θ0,θ1]= indP, Pθ1−indP, Pθ0∈Z. Thus integer leading terms of sumintegral dierence can be identied as relative indices along paths; analytic tail terms controlled by Theorem 3. Theorem 6 (Theorem 5: KRD Non-Asymptotic Principal Value Bound and Minimal Integer Thresholds) . Let N0= 2TΩ (continuous) or N0= 2NW (discrete), then the principal value satises 1−λ0≤10 exp −(⌊N0⌋−7)2 π2log(50N0+ 25)!, and dene the minimal integer threshold achieving leakage bound ε : N⋆ 0(ε) := min nn∈N: 10 exp−(n−7)2 π2log(50n+25) ≤εo. Numerical values (natural logarithm, oor in exponent): (ε, N⋆ 0, c⋆, NW⋆) = (10−3,33,π 2·33,16.5),(10−6,42,π 2·42,21.0),(10−9,50,π 2·50,25.0). 4 Proofs 4.1 Proof of Theorem 1 Compactness and self-adjointness . BΩ and DT are orthogonal projections, (DTBΩ)(t, s) = 1[−T,T ](t)sin(2πΩ(t−s)) π(t−s) is square-integrable on [−T, T]2 , so DTBΩ is HilbertSchmidt, thus Kc= DTBΩDT is compact self-adjoint. 3 Commutativity and simplicity . Let x=t/T ∈[−1,1] , c=πTΩ . Classical PSWF satises (1 −x2)y′′(x)−2xy′(x)+(χ−c2x2)y(x)=0. Write Lcy:= −d dx(1 −x2)dy dx+c2x2y , reformulating as Lcy=χy . Lc is a self-adjoint Sturm Liouville operator on [−1,1] , with endpoints being regular singular points, spectrum purely discrete with each eigenvalue simple; eigenfunctions' zero counts match their indices (oscillation theorem). Slepian commutativity shows Kc and Lc can be simultaneously diagonalized, so geometric multiplicity of λ0 equals multiplicity of corresponding χ0 , hence 1, with principal eigenfunction unique (up to phase). Leakage identity . By orthogonal projection property of BΩ and |g∗|2= 1 , λ0=⟨Kcg∗, g∗⟩=⟨BΩg∗, g∗⟩=|BΩg∗|2 2,|(I−BΩ)g∗|2 2= 1 −λ0. Discrete side KN,W with commuting second-order dierence operator forms discrete Sturm theory, principal value also simple. 4.2 Proof of Alignment 1 R\I=Fk=0(I+k) is a countable disjoint decomposition. Since bw∈L2 , Pk=0 RI|bw(ξ+k)|2dξ ≤ |bw|2 2<∞ , Tonelli applies. Variable substitution η=ξ+k yields ZIX k=0 |bw(ξ+k)|2dξ =X k=0 ZI+k|bw(η)|2dη =ZR\I|bw(η)|2dη. 4.3 Proof of Theorem 2 HS exact formula . Frequency-domain kernel K(ξ, η) = 1|ξ|>W 1|η|≤Wbx(ξ−η). HS norm squared ZZ |K|2=Z|η|≤WZ|ξ|>W |bx(ξ−η)|2dξdη =ZR|bx(δ)|2mW(δ)dδ, where mW(δ) = meas{η∈[−W, W ] : |η+δ|> W } . Geometrically the measure of complementintersection between length2W interval and its translation: mW(δ) = min(2W, |δ|) . This yields the stated formula. Bounds . Decompose (I−BW)Mx= (I−BW)MxBW+ (I−BW)Mx(I−BW), apply |A|op ≤ |A|HS , |(I−BW)| ≤ 1 and triangle inequality to obtain general bound; taking w=g∗ and using Theorem 1's |(I−BW)g∗|2≤√1−λ0 yields stated formula. If (I−BW)MxBW≡0 , then for any band-limited input BWw , bx∗(bw·1[−W,W ]) support remains in [−W, W ] , forcing supp bx⊂ {0} ; combined with x∈L∞ leaves only constant functions. 4 4.4 Proof of Theorem 3 Periodic Bernoulli's Fourier expansion gives  B2p(·) (2p)! ∞ =2ζ(2p) (2π)2p. EM remainder formula R2p(g) = ZR g(2p)(t)B2p({t}) (2p)! dt, thus |R2p(g)| ≤ 2ζ(2p) (2π)2p|g(2p)|L1. If supp bg⊂[−Ω,Ω] , then |g(2p)|L1≤√L|g(2p)|L2≤√L(2πΩ)2p|g|2 . The (2π)2p in denominator and BPW's (2π)2p completely cancel, yielding stated formula and threshold values. 4.5 Proof of Theorem 4 Dene A(θ)=2Pθ−I . By hypotheses (i)(ii) and regularity of spectral projections, A(θ) is a strongly continuous path of self-adjoint Fredholm operators. Spectral ow is dened as signed count of zero crossings; on the other hand, relative index ind(P, Q) when P−Q∈ S1 can be dened via relative dimension, satisfying additivity and homotopy invariance. Subdivide [θ0, θ1] into small intervals making 0 a regular value on each segment; locally, spectral ow equals jumps in rank(Pθ|ranP) ; relative index also records the same jumps. Concatenate and use homotopy invariance to obtain Sf(A(θ))θ1 θ0= ind(P, Pθ1)−ind(P, Pθ0). Frequency-domain smoothing ϕ∈C∞ c and (when necessary) time-domain localization ensure P− Pθ∈ S1 ; in strong operator topology let ϕ→1[−W,W ] , integer invariant, thus establishing topological origin of integer leading terms. 4.6 Proof of Theorem 5 Discrete-side original formula with NW as parameter: 1−λ0≤10 exp−(⌊2NW⌋−7)2 π2log(100NW + 25). Setting N0= 2NW yields 1−λ0≤10 exp−(⌊N0⌋−7)2 π2log(50N0+ 25). Continuous-side bound expressed in c via N0= 2c/π gives same unied form. For given ε , scan minimal integer n such that right side ≤ε to obtain N⋆ 0(ε) . Numerical values as stated. 5 Model Applications Continuousdiscrete mapping : (T, Ω) ↔(N, W ) aligned via N0= 2TΩ=2NW ; typically take N≈2T , W≈Ω (when units consistent). 5 Consistency of two calibers : DTBΩDT and BΩDTBΩ have identical non-zero spectra (proposition: AB and BA have identical non-zero spectra). Frequency-domain leakage: |(I−BΩ)g∗|2 2= 1−λ0 ; time-domain leakage: |(I−DT)w∗|2 2= 1 −λ0 . Fundamental domain consistency : Choose fundamental domain consistent with [−Ω,Ω] (translation/scaling aligned), Alignment 1's right side is out-of-domain energy relative to that passband, thus for w=g∗ , aliasing energy equals 1−λ0 . 6 Engineering Proposals (1) Threshold-driven parameter selection Given leakage tolerance ε∈ {10−3,10−6,10−9} , consult Theorem 5 for minimal integer N⋆ 0 , accordingly set (T, Ω) or (N, W ) . (2) Computable bounds for cross-terms One FFT obtains bx , compute ΞW(x) := ZR|bx(δ)|2min(2W, |δ|)dδ1/2 . Budget formula |(I−BW)Mxg∗|2≤ΞW(x) + |x|∞p1−λ0. If bx pre-ltered and narrowband, ΞW(x) signicantly reduced. (3) EM order selection Given (L, Ω) , choose smallest p∈ {2,3,4} such that √LΩ2p≤ 10−3/(2ζ(2p)) . (4) Multi-taper/multi-passband Take rst K≈ ⌊N0⌋ DPSS for multi-taper; aliasing budget along Alignment 1 accumulates per taper, cross-terms estimated blockwise per Theorem 2 by bx 's energy distribution. 7 Discussion (Risks, Boundaries, Past Work) Denability boundaries : Spectral ow = index of projection pairs relies on P−Pθ∈ S1 . Sharp band-limiting projection and pure modulation dierence generally non-trace-class, requiring frequency-domain smoothing rst and (when necessary) time-domain localization, then taking spectral projection, nally approaching limit in strong operator topology; integer leading terms insensitive to regularization details. Scales and constants : Under cycles normalization, BPW's (2π)m and EM constant denominator (2π)2p completely cancel; KRD threshold expressed in natural logarithm with log(50N0+25) , oor in exponent yields minimal integer. Conservative vs tight : Can generate thresholds per Theorem 5's tight version ( 33,42,50 ), or under extreme risk aversion choose larger integers, forming conservative redundancy. Historical threads : Timeband extrema, Toeplitz index/winding number, spectral ow/relative index, and one-sided extrema constitute theoretical backbone; non-asymptotic thresholds connect classical asymptotics with engineering parameterization. 8 Conclusion Under unied normalization and parameter mapping, three error typesmain leakage, multiplicative cross-terms, and sumintegral dierenceare incorporated into operator-theoretic discipline of integer leading terms + analytic tail terms: 6  Main leakage precisely characterized by λ0 , with explicit non-asymptotic bounds generating window-shape-independent minimal integer thresholds;  Cross-terms quantied via HankelHS exact formulas, yielding computable bounds without heuristic constants;  EM remainder under cycles normalization exhibits  (2π) complete cancellation, with closedform constants directly interfacing timeband parameters;  Integer leading terms (spectral ow/index of projection pairs) provide topological origin. The resulting nite-order discipline achieves both engineering implementation and complete mathematical anchoring. A Notation, Units, and Basic Tools  Normalization: b f(ξ) = RRf(t)e−2πitξ dt , ξ in cycles.  Projections: DTf=1[−T,T ]f , BΩ=F−11[−Ω,Ω]F .  Norms: |A|op ≤ |A|HS , Plancherel: |f|2=|b f|2 .  Shannon number: N0= 2TΩ=2NW . B AB and BA Have Identical Non-Zero Spectra If ABx =λx with λ= 0 , then Bx = 0 and BA(Bx) = λ(Bx) ; reverse direction similar. Thus DTBΩDT and BΩDTBΩ have identical non-zero spectra; leakage identities in both calibers are equivalent. C PSWF SturmLiouville Structure and Principal Value Simplicity Variable x=t/T ∈[−1,1] , c=πTΩ . PSWF satises (1 −x2)y′′(x)−2xy′(x)+(χ−c2x2)y(x)=0. Write Lcy:= −d dx(1 −x2)dy dx+c2x2y, then Lc is self-adjoint second-order dierential operator on [−1,1] . Endpoints x=±1 are regular singular points, spectrum purely discrete with each eigenvalue simple; eigenfunctions' zero counts match indices. Slepian commutativity shows Kc and Lc share orthogonal eigensystem, so Kc 's principal eigenvalue has geometric multiplicity 1. Discrete side : Toeplitz-type prolate matrix commutes with tridiagonal dierence operator; discrete Sturm oscillation theorem guarantees principal value simplicity. 7 D Squared-Sum Aliasing = Out-of-Domain Energy Details For I= [a, a + 1) , we have R\I=Fk=0(I+k) (disjoint). For any w∈L2 , X k=0 ZI|bw(ξ+k)|2dξ =X k=0 ZI+k|bw(η)|2dη =ZR\I|bw(η)|2dη. Squared-sum means taking modulus squared of each translated channel before summing prior to integration; distinguished from engineering measures of periodize rst then take modulus squared (which contain cross-terms). E HankelHS Geometric Measure Piecewise Calculation For xed δ , the set S(δ) = {η∈[−W, W ] : |η+δ|> W}. If |δ| ≤ 2W , then S(δ) is union of two endpoint pieces each of length |δ|/2 , measure |δ| ; if |δ|>2W , then S(δ)=[−W, W] entirely, measure 2W . Thus mW(δ) = meas S(δ) = min(2W, |δ|), yielding Theorem 2's HS exact formula. F EM Remainder and  (2π) Cancellation Details Periodic Bernoulli sup constant  B2p(·) (2p)! ∞ =2ζ(2p) (2π)2p. If supp bg⊂[−Ω,Ω] , then |g(2p)|L1≤√L|g(2p)|L2≤√L(2πΩ)2p|g|2, substituting into EM remainder bound, denominator (2π)2p and BPW's (2π)2p completely cancel, obtaining |R2p(g)| |g|2≤2ζ(2p)√LΩ2p. G Reproducible Checklist (Pseudocode) KRD threshold (natural logarithm) def N0_star(eps): n = 1 while True: U = 10*exp(- (n-7)**2 / (pi**2*log(50*n + 25)) ) if U <= eps: return n, (pi*n/2), (n/2) # (N0*, c*, NW*) n += 1 HankelHS cross-term 8 # xhat: Fourier samples on grid delta with spacing ddelta XiW_sq = sum( abs(xhat)**2 * minimum(2*W, abs(delta)) ) * ddelta XiW = sqrt(XiW_sq) EM remainder threshold # choose smallest p in {2,3,4} with # sqrt(L) * Omega**(2*p) <= 1e-3/(2*zeta(2*p)) 9