Self-Referential Scattering Networks: Connection Matrix Synthesis, J-Unitary Robustness, and Floquet Band-Edge Topology

Self-Referential Scattering Networks: Connection Matrix Synthesis, J -Unitary Robustness, and Floquet Band-Edge Topology Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Version 1.10 Abstract In Self-Referential Scattering Networks (SSN), this paper provides closed-loop methodology from designimplementationreadoutfalsication to theorem-level guarantees . Under trace-class calibration , establish mod-two equivalence between global half-phase ( √det covering) holonomy, spectral shift, spectral ow through −1 , and discriminant transversality. Compared to previous draft, this version completes veriable details and checkable constants at ve key junctures: (i) Section 3 adds bridging lemma for  log det regularization BirmanKren −1 spectral owmod-two intersection number with half-page self-contained proof ; (ii) Section 4 provides quantitative structural lemma for  no spurious crossing  after star product, specifying comparison principle and design line for principal block minimal gradient gmin and mutual coupling upper bound rmax ; (iii) Section 5 formulates binarized projection + majority voting as concentration inequality , providing explicit relation between misclassication rate and sample number with correlation correction ; (iv) Section 6 characterizes robust domain via J -inner-product normalized imaginary Rayleigh quotient (Kren angle) , constructs polarization homotopy , provides feasible upper bound for threshold function ε0(η, β) , uniformly ensuring Cayley denominator invertibility and discriminant non-crossing; (v) Section 7 formulates phase-type Floquet index truncation independence and gauge independence as theorems, completing regularization consistency and failure detection via det2 in HilbertSchmidt scenario . Engineering side provides simulatable Schur-closed form for two-port couplermicroringgain prototype, quantifying square-root scaling and threshold selection basis for group delay double-peak merging. Keywords : Closed-loop scattering; Redheer star product; Schur complement; Herglotz/Nevanlinna; discriminant; spectral shift and spectral ow; mod-two Levinson; J -unitary; Kren angle; Floquet phase-type index. 1 Calibration and Basic Objects Frequency domain and variables : ω as angular frequency (static ω∈R ; periodic systems ω∈[−π/T, π/T] ), upper half-plane calibration takes z=ω+ i0+ . Exchange with unit circle calibration via Cayley map when necessary, maintaining orientation consistency. Scattering and star product : Portized node scattering Sj(z) through interconnection (including feedback) yields closed-loop S⟲(z) via Redheer star product and Schur complement. Cayley duality (unied notation): S⟲(z)=(I−iK(z))(I+ iK(z))−1, K(z) = −iI+S⟲(z)−1I−S⟲(z). 1 Trace-class and regularization : All results related to spectral shift ξ and BirmanKren uniformly assume S⟲−I∈S1 using det . For innite dimensions (e.g., Floquet sidebands) rst truncate nitely, use det2 when necessary; this paper's mod-two conclusions independent of det /det2 choice (see Appendix F). J -unitary calibration : J=J†=J−1 , S♯=J−1S†J , S†JS =J . Non-Hermitian robustness stated under this calibration. 2 Discriminant, Local Model, and Transversality On parameter manifold X dene codimension-one piecewise smooth submanifold family D=FbDb , whose local models include Jost zeros, threshold opening/closing, embedded eigenvalues, and EP coalescence. After removal X◦=X\D . For closed path γ⊂X◦ , mod-two intersection number I2(γ, D) is parity of transverse points. Endpoints/thresholds excised via small semicircles merged into D boundary component. Closed-loop implementation level can use D=n(ω, ϑ) : σminI−C(ω, ϑ)Sii(ω, ϑ)= 0o, equivalent to det(I−CSii)=0 at general position. 3 Mod-Two Half-PhaseSpectral-ShiftSpectral-FlowIntersection: Bridging Lemma and Equivalence Theorem (Trace-Class) Denition 1 (3.1: Global Half-Phase) . ν√det S⟲(γ) = expiIγ 1 2d arg det S⟲∈ {±1}. Lemma 2 (3.2: log det Regularization and Endpoint Treatment) . If along closed path γ almost everywhere S⟲ unitary with S⟲−I∈S1 , then log det S⟲ admits continuous continuation along γ and is integrable; after treating endpoints/thresholds per 2 small semicircle excision rule, Hγ 1 2d arg det S⟲ well-dened ( mod 2π ). Lemma 3 (3.3: BirmanKren −1 Spectral Flow Bridge: Two-Step Veriable Details) . Denote ξ(ω) as spectral shift. Then (i) By BK formula det S(ω) = exp{−2πiξ(ω)} , arbitrary branch change makes ξ7→ ξ+n ( n∈Z ), contributing exp{iH1 2d(2πn)}= 1 to half-phase, thus mod-two invariant ; (ii) Let S(τ) = V(τ)eiΦ(τ)U(τ) be local Schur form with at τ=τc only one simple eigenphase ϕj crossing π ( −1 ), then ξ jump is ±1 while Sf−1= 1 . Multiple crossings decompose into nite simple crossings, thus exp−iπIγ dξ= (−1)Sf−1(S⟲◦γ). Proposition 4 (3.4: Base-Point and Branch Independent Mod-Two Property) . For arbitrary base point and continuous branch continuation, ν√det S⟲(γ) invariant; allowed endpoint treatment equivalent to adding boundary homotopy on D , mod-two value preserved. Theorem 5 (3.5: Four-Fold Equivalence) . If along γ almost everywhere S⟲ unitary with S⟲−I∈ S1 , then ν√det S⟲(γ) = exp−iπIγ dξ= (−1)Sf−1S⟲◦γ= (−1)I2(γ,D). 2 4 No Spurious Crossing After Star Product and Z2 Combinatorial Law (Quantitative Version) Block notation : S(k)= S(k) ee S(k) ei S(k) ie S(k) ii !, k = 1,2. Lemma 6 (4.1: Structural Lemma for No Spurious Crossing: Quantitative Conditions with Tubular Separation) . Assume in neighborhood U⊂X◦ satisfying (i) Schur invertibility lower bound: σminI−S(1) ii S(2) ii ≥δ > 0 ; (ii) Mutual coupling smallness: ∥S(1) ei ∥2,∥S(2) ie ∥2≤ρ < 1 ; (iii) Tubular separation : For zero sets D(k) of k= 1,2 exists uniform tubular radius τ∗>0 , take 0< τ ≤τ∗ such that Nτ(D(1)) , Nτ(D(2))⊂U and disjoint. Dene principal discriminant Φk(ϑ) = detI−CS(k) ii  and principal block minimal gradient gmin = inf ϑ∈Umin|∇ϑΦ1(ϑ)|,|∇ϑΦ2(ϑ)|. Then mutual coupling residual upper bound exists rmax ≤∥S(1) ei ∥2∥S(2) ie ∥2 δ2, and when rmax <(τgmin)2 network discriminant is transverse disjoint union of subnet discriminants: Dnet =D(1) ⊔D(2) , with I2(γ, Dnet) = I2(γ, D(1)) + I2(γ, D(2)) mod 2. Proof essentials : By tubular separation and mean value theorem, outside tubes have |Φ1(ϑ)| ∧ |Φ2(ϑ)| ≥ τgmin , thus |Φ1(ϑ)Φ2(ϑ)| ≥ (τgmin)2 ; when rmax <(τgmin)2 , mutual coupling remainder insucient to introduce new zeros outside tubes. In each single tube use implicit function theorem obtaining zero set as normal small deformation of original zero set, obtaining transverse disjoint union since tubes non-intersecting. Engineering design line (veriable) Take δ= infUσmin(I−S(1) ii S(2) ii ) . If ∥S(1) ei ∥2∥S(2) ie ∥2≤1 2δ2(τgmin)2, then rmax ≤1 2(τgmin)2<(τgmin)2 , satisfying lemma sucient condition. Boundary and failure mode 4.2 : When δ→0+ or ρ→1− , near-tangent crossing and resonance-induced spurious crossing may occur. Should real-time monitor σmin(I−CSii) margin and gmin numerical estimate, shrink U or recongure ports when necessary. Theorem 7 (4.3: Z2 Combinatorial Law) . Under lemma conditions, νnet =ν(1) ⊙ν(2) (componentwise multiplication, mod-two addition). 3 5 Binarized Projection, Concentration Inequality, and Dealiasing Phase increment and binarization : ∆ϕab =1 2harg det Sγa;θb+δ−arg det Sγa;θb−δicont,Π(∆ϕab) = 1{|∆ϕab|≥π/2}. Denition 8 (Eective Phase Window) . Assume experiment/simulation uses measurement grid G={(a, b)} , dene ∆ϕab and binarization rule Π(·) per above formula. Dene ∆ϕeff := ess inf (a,b)∈G |∆ϕab|. When bounded additive noise ε exists with |ε| ≤ ϵ , adopt threshold condition ∆ϕeff >π 2+ 2ϵ ; accordingly obtain Theorem 5.1's misclassication rate and sample complexity estimate. Hypothesis (independence and sub-Gaussian) Measurement samples {c ∆ϕ(n) ab }N n=1 independent, Ec ∆ϕ(n) ab = ∆ϕab , sub-Gaussian with proxy variance σ2 . Theorem 9 (5.1: Error Bound and Sample Complexity for Majority Voting) . Assume measurement samples {c ∆ϕ(n) ab }N n=1 mutually independent, Ec ∆ϕ(n) ab = ∆ϕab , sub-Gaussian with proxy variance σ2 ; with bounded additive bias |ε| ≤ ϵ . Denote m:= ∆ϕeff −π 2−2ϵ > 0, q := P|c ∆ϕ(n) ab |<π 2. Then q≤2 exp−m2 2σ2,P( majority voting misjudgment )≤exp−2N1 2−q2. Given target error δ∈(0,1) , sucient condition is N≥log(1/δ) 21 2−2e−m2/(2σ2)2, requiring m>σ√2 log 4 ensuring q < 1 2 for majority voting convergence. If oversampling correlation exists, replace N in above formula with eective sample number Neff . Proposition 10 (5.2: Dealiasing and Secondary Evidence Fusion) . For multiple crossings or nearthreshold masking: (i) Adopt multi-window sliding and anchor continuity strategy; (ii) Fuse group delay double-peak merging, conrming crossing only when both ngerprints synchronize; (iii) When crosstalk occurs, add redundant columns nearly orthogonal to existing sensitivity, recalculate Gram criterion until full rank. 6 J -Unitary Robustness: Kren Angle, Polarization Homotopy, and Threshold Function Kren angle and angle gap : Assume ⟨ψj(τ), Jψj(τ)⟩ = 0 (non-neutral eigenstate), otherwise κj undened, and that parameter point viewed as robust domain boundary and excluded. Dene 4 κj(τ) = Im ⟨ψj(τ), J S−1(∂τS)ψj(τ)⟩ ⟨ψj(τ), J ψj(τ)⟩, consistent with ∂τϕj in unitary limit J=I , used as phase slope . Angle gap dened as η:= min jinf τdistϕj(τ), π + 2πZ∈(0, π]. Near J -unitary there exist c±(ε)→1 making c−|∂τϕj| ≤ |κj| ≤ c+|∂τϕj| . Note: κj only as slope control term, not participating in η denition . Lemma 11 (6.1: Estimate from (S†JS −J) to (K−K♯) ) . Assume S(τ) pointwise invertible along considered parameter domain, with sup τ|S(τ)|<∞,sup τ|S(τ)−1|<∞,|S†JS −J| ≤ ε, β = inf τσmin(I+ iK(τ)) >0. Then constant C=Cβ, sup |S|,sup |S−1| exists making |K−K♯| ≤ C ε. Proof essentials : Use Fréchet dierential of Cayley inverse map K=−i(I+S)−1(I−S) with J -conjugation and bounded multiplier inequality. Construction 6.2 (polarization homotopy) Let Kt= (1 −t)K+t1 2(K+K♯), St= (I−iKt)(I+ iKt)−1, t ∈[0,1]. By Lemma 6.1 and Neumann lemma obtain σminI+ iKt(τ)≥β−t 2C ε, thus when ε < 2β/C , (I+iKt) invertible throughout t∈[0,1] , St well-dened. In near J -unitary calibration, constant α > 0 exists (depending on |S| , |S−1| , β ) making ε0(η, β) := min 2β C, α sin2η 2 feasible upper bound : when ∥S†JS −J∥ ≤ ε<ε0(η, β) with η > 0 , homotopy {St} doesn't intersect D , with Cayley denominator invertible throughout. Theorem 12 (6.3: Homotopy Robustness) . If homotopy {St} satisfying above formula exists without intersecting D throughout, then ν√det S⟲ equals unitary limit value. Square root asymptotics (engineering ngerprint) : Dominant branching 2×2 eective subspace yields arg det S(t) = arg det S(tc)±arctanκ1/2|t−tc|1/2+O|t−tc|3/2, group delay exhibits symmetric double-peak merging, peak separation ∆ω=Cp|t−tc|+O|t− tc|3/2 . Higher-order roots (EP order >2 ) mod-two equivalent to square root. 5 7 Floquet-SSN: Phase-Type Band-Edge Index, Truncation and Gauge Independence Denition 13 (7.1: Phase-Type Index) . Truncate sideband to |n| ≤ N obtaining nite-dimensional S(N) F(ω) , dene ν(N) F= expi 2Zπ/T −π/T ∂ωarg det S(N) F(ω) dω∈ {±1}. Equivalently, ν(N) F= expi 2Zπ/T −π/T Im ∂ωlog det S(N) F(ω) dω. Theorem 14 (7.2: Truncation Independence: Norm/HS Version) . If S(N) F→SF in operator norm or HilbertSchmidt topology, with endpoints ω=±π/T having no N -migrating branching, then N∗ exists making ν(N) F stable for N≥N∗ ; dene νF=ν(N∗) F . In HilbertSchmidt scenario , replace det dierential with Koplienko spectral shift and det2 derivative recipe, combined with Appendix F mod-two consistency, obtaining same νF . Strong convergence alone insucient to ensure det2 and second-order trace formula well-denedness, thus not included in theorem premise. Lemma 15 (7.3: Gauge Independence) . If SF(ω)7→ UL(ω)SF(ω)UR(ω) where UL,R continuous with |det UL,R|= 1 , satisfying band-edge gluing condition [arg det UL+ arg det UR]π/T −π/T ∈4πZ, then Zπ/T −π/T ∂ωarg det SF(ω) dω mod-two value invariant. Theorem 16 (7.4: Band-Edge Equivalence) . If endpoints satisfy square root local model with Theorem 7.2 holding, then νF= (−1)I2[−π/T,π/T ],DF. Failure mode 7.5 (detection and fault tolerance threshold) When truncation induces endpoint pseudo-branching, monitor min ω∈{±π/T}σmin I−CS(N) ii (ω)≥δF, requiring ν(N) F=ν(N+1) F=ν(N+2) F continuous three-order consistency to judge stable. 6 8 Prototype and SOP 8.1 Two equivalent paths for single crossing : (i) Complex parameter small loop: λ circles Jost zero once in complex plane; (ii) Real parameter traverse + frequency domain readout: Real parameter traverses D , select single branch in frequency domain using anchor continuity. Both homotopy equivalent. Along any closed loop from complex parameter small loop or real parameter traverse + return closure, have Id arg det S=±2π, expi 2Id arg det S=−1, fully consistent with 3 half-phasespectral-ow parity. 8.2 Two-port couplermicroringgain prototype (simulatable) : Coupler C(κ) = √1−κ2iκ iκ√1−κ2,C(ω, t) = ρeiϕ(ω,t). Eective scattering (Schur-closure) S⟲=See +SeiI−CSii−1CSie. In critical neighborhood of CSii →I falls back to 8.1 square root model, directly reproducing ∆φ=π and group delay double-peak merging. 8.3 SOP and criteria : Port de-embedding  sweep feedback crossing D  use ∆ϕeff > π/2+2ϵ and double-peak merging as passing condition; triple ngerprint asynchrony vetoes causal chain, returning to column controllability procedure for redundant column addition or frequency window reselection. 9 Readout and WignerSmith Unied Calibration ∂ωlog det S= trS−1∂ωS, ∂ωarg det S= Im trS−1∂ωS. In unitary case let Q(ω) = −iS†∂ωS , then ∂ωarg det S= tr Q(ω) . In J -unitary case can also write ∂ωarg det S= Im tr(S♯∂ωS) . 10 Conclusion This paper rigorizes half-phasespectral-shiftspectral-owintersection four-fold equivalence via bridging lemma ; ensures star-product interconnection discriminant transverse disjoint union and Z2 combinatorial law via quantitative structural lemma ; guarantees binarized projection reproducibility via concentration inequality ; characterizes near J -unitary robust domain and threshold via Kren angle and polarization homotopy ; closes band-edge topological readout chain via phase-type Floquet index and truncation/gauge independence theorems . Combined with minimal prototype and SOP, provides programmable implementation and counterfactual verication for closed-loop self-consistency ⇒ square-root criticality ⇒ double Riemann sheet ⇒Z2 half-phase. 7 References (Selected) [1] R. Redheer, On a Certain Linear Fractional Transformation, Pacic J. Math. , 9 (1959) 871 893. [2] J. Gough, M. R. James, The Series Product and Its Application to Quantum Feedforward and Feedback Networks, IEEE TAC , 54 (2009) 25302544. [3] B. Simon, Trace Ideals and Their Applications , 2nd ed., AMS (2005). [4] M. Sh. Birman, M. G. Kren, On the Theory of Wave and Scattering Operators, Sov. Math. Dokl. , 3 (1962) 740744. [5] L. Koplienko, Trace Formula for Perturbations of Class S2 , Sb. Math. , 122 (1983) 457486. [6] E. P. Wigner, Lower Limit for the Energy Derivative of the Scattering Phase Shift, Phys. Rev. , 98 (1955) 145147; F. T. Smith, Lifetime Matrix in Collision Theory, Phys. Rev. , 118 (1960) 349356. [7] T. Ya. Azizov, I. S. Iokhvidov, Linear Operators in Spaces with an Indenite Metric , Wiley (1989). [8] D. Z. Arov, H. Dym, J -Contractive Matrix-Valued Functions and Related Topics , CUP (2008). [9] I. C. Fulga, F. Hassler, A. R. Akhmerov, Scattering Formula for the Topological Quantum Number, Phys. Rev. B , 85 (2012) 165409. [10] M. S. Rudner, N. H. Lindner, E. Berg, M. Levin, Anomalous Edge States..., Phys. Rev. X , 3 (2013) 031005. A Notation and Regularization Notation : J (Kren metric), S♯=J−1S†J ; S1/S2 (trace-class/HS); det⋆ (nite-dimensional take det , HS take det2 ); D, DF (discriminant/band-edge discriminant); Λ , Πτ , η , Q=−iS†∂ωS , ν√det S⟲ , νF . Calibration : Main text adopts S1 . HS case stabilizes with det2 ; four-fold equivalence and νF mod-two value independent of det /det2 choice (see Appendix F). B CoveringLifting and Z2 Reduction Square covering p:z7→ z2 corresponds to multiplication by two in cohomology: s:X◦→U(1) exists making s2= det S⟲ if and only if [det S⟲]∈2H1(X◦;Z) . Its Z2 reduction is this paper's global half-phase invariant. C Square Root Puiseux Asymptotics and Error For 2×2 eective subblock M(t) with ∆(t)≈∆′(tc)(t−tc) , Cayley mapping to S(t)=(I−iM)(I+ iM)−1 has leading term ±arctanκ1/2|t−tc|1/2+O|t−tc|3/2, group delay exhibits symmetric double-peak merging, peak separation ∆ω=Cp|t−tc|+O|t− tc|3/2 . Substituting into 5's ∆ϕeff yields sample complexity estimate. 8 D Target-to-Device Execution Checklist and ℓ Open-loop calibration and de-embedding  choose W and step size  compute Sθb(ω) = ∂θbarg det S(ω) integrating into M  obtain Π(M) via π/2±τ checking rank  solve Π(M)x=ℓ selecting columns  trigger column-by-column with ∆φ=π , double-peak merging as stop  counterfactual verication and archive. ℓ= (1 −ν⋆)/2∈ {0,1}m ( ν⋆= +1 7→ 0 , ν⋆=−17→ 1 ). E Kren Angle and Homotopy Threshold under J -Unitary Assume S†JS =J , denote X=S−1˙ S . By dierentiation X♯=−X ( J -skew-Hermitian). For (λj= eiθj, vj) : ˙ θj=Im ⟨vj, JX vj⟩ v† jJvj ,κj=Im ⟨vj, JX vj⟩ v† jJvj . From above two formulas pointwise obtain κj=˙ θj , thus |˙ θ1−˙ θ2|=|κ1−κ2|. By Lemma 6.1 obtain ∥K−K♯∥ ≤ C ε . Take Kt= (1 −t)K+t(K+K♯)/2 , St= (I−iKt)(I+ iKt)−1 . By Neumann lemma obtain σmin(I+iKt)≥β−t 2C ε , thus when ε < 2β/C , St well-dened. If ∥S†JS −J∥ ≤ ε with η > 0 , constant α > 0 exists making ε0(η, β) = min{2β/C, α sin2(η/2)} feasible upper bound, uniformly ensuring Cayley denominator invertibility and discriminant noncrossing. F Regularization Independence (Unied Proposition) Proposition 17 (F.1: det /det2 Mod-Two Consistency) . Assume along closed path γ almost everywhere S⟲ unitary. If S⟲−I∈S1 then expiIγ 1 2∂log det S⟲= (−1)Sf−1= (−1)I2. If only S⟲−I∈S2 , take S1 approximation family S⟲ ϵ→S⟲ in HS topology, dene half-phase via det2 , then lim ϵ→0expiIγ 1 2∂log det S⟲ ϵ= expiIγ 1 2∂log det 2S⟲, with both sides' mod-two value consistent with Sf−1 , I2 . G Floquet Truncation, Convergence, and Failure Mode Truncate SF sideband to |m| ≤ M obtaining S(M) F . If P|m|>M ∥Km∥ → 0 ( operator norm or HS convergence; this condition implies operator norm convergence ), with endpoints having no M -migrating branching, then supω∥S(M+1) F−S(M) F∥ → 0 , M0 exists making ν(M) F stable for M≥M0 . Detection quantities: endpoint singular value threshold δF and plateau stability criterion ( ν(M) F=ν(M+1) F=ν(M+2) F ). When abnormal drift occurs increase M or reduce coupling bandwidth to avoid pseudo-branching. 9