Delay Quantization, Feedback Loops, and \pi-Step Parity Transitions: From Scale Identity to \mathbb Z_2 Topology of Self-Referential Scattering Networks

Delay Quantization, Feedback Loops, and π -Step Parity Transitions: From Scale Identity to Z2 Topology of Self-Referential Scattering Networks Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract Within the unied framework of frequency-domain scattering theory and feedback networks, networks with tunable closed-loop delays exhibit highly robust phase-step and group-delay-pulse phenomena across a wide range of physical platforms: as the feedback round-trip time τ varies slowly, the total scattering phase and its frequency derivative undergo jumps of amplitude approximately π near certain parameter values, accompanied by reversals in the direction of spectral ow. Under the constraint of the scale identity κ(ω;τ) = φ′(ω;τ) π=ρrel(ω;τ) = 1 2πtr Q(ω;τ) this paper provides a rigorous spectral and topological characterization of the delay quantization ⇒π -step ⇒Z2 parity transition phenomenon. Here S(ω;τ) is a lossless scattering matrix family varying with angular frequency ω and eective round-trip delay τ , φ(ω;τ) = arg det S(ω;τ) is the total scattering phase, and Q(ω;τ) = −iS(ω;τ)†∂ωS(ω;τ) is the WignerSmith group delay matrix. Under natural assumptions of analyticity, losslessness, and simple zeros/poles, we prove that when τ traverses a family of delay quantization steps τk=τ0+k∆τ, k ∈Z, the spectral ow of zeros/poles of det S(ω;τ) in the complex frequency plane undergoes a crossing event across the real axis; by the argument principle, this corresponds to a jump of size ±π in the total phase at a xed frequency slice; accordingly, the topological index constructed from the spectral ow count ν(τ)∈ {0,1}, ν(τ+ ∆τ) = ν(τ)⊕1 undergoes a Z2 parity ip at each step. To make results computable and experimentally veriable, we rst provide an explicit analytic form for a one-dimensional single-channel scalar model Stot(ω;τ) = r0(ω) + t0(ω)2eiωτ 1−rfb(ω) eiωτ , and rigorously derive the magnitude and sign of π -steps and unit group-delay pulses using the argument principle and pole trajectory analysis. We then generalize to multi-channel matrix cases, showing that upon appropriate choice of branch for det S(ω;τ) , the main conclusions depend only on the eigenvalue spectral ow of the eective feedback block R(ω) , thus having universality across implementation platforms. 1 From the perspective of the unied time scale, tunable closed-loop delays constitute a natural topological driving parameter: each traversal of a delay quantization step corresponds to spectral ow crossing the real axis once in parameter space, thereby switching between two topological sectors in the Z2 sense. This topological ip manifests as easily measurable ngerprints of total phase and group delay π -steps on any linear lossless platform, and can be embedded into higher-layer structures such as self-referential scattering networks, spin double covers, and NullModular double covers, providing a unied frequency-domain topological readout. Keywords : delay quantization; feedback loops; scattering matrix; WignerSmith group delay; scale identity; phase steps; Z2 parity; self-referential scattering networks; topological invariants 1 Introduction & Historical Context 1.1 Delay Feedback Networks and the π Phase Jump Phenomenon From ber-loop resonators, integrated micro-ring resonators to microwave closed-loop networks and acoustic ring resonators, closed-loop feedback structures with nite round-trip times repeatedly appear across dierent physical platforms. Their common feature is that a wave packet making a round trip in the loop acquires a total phase Φ(ω;τ) = ϕ0(ω) + ωτ, where ϕ0(ω) is the additional phase introduced by the core scattering and couplers, and τ is the eective round-trip time. When Φ(ω;τ) satises integer or half-integer quantization conditions, the network's resonance, interference, and transmission zero structure undergoes signicant changes, triggering abrupt changes in output amplitude and phase. In the combination structure of optical ring resonance and MachZehnder interferometers, π - scale phase jumps in the transmission spectrum around resonance frequencies are commonly observed, along with their correspondence to interference-induced transparency eects. Related experiments and modeling indicate that these phase jumps are closely related to the coherent interference between two or more paths in the loop, and to the topological structure of resonance modes in parameter space. Similar π -jump phenomena also appear in the transmission and reection phases of systems such as phase-shifted gratings and split-ring resonators, and are often used as indicators to identify node structure and mode topology. However, above these concrete structures, there is still a lack of a unied spectral theory framework that systematically links tunable delay, phase steps, and parity topological sectors. 1.2 Scattering Phase, Density of States, and Time Delay In quantum scattering and wave scattering theory, the profound relationship between the phase of the scattering matrix S(ω) and time delay and density of states has been systematically established by work from multiple directions. The group delay and WignerSmith time delay matrix introduced by Wigner and Smith Q(ω) = −iS(ω)†∂ωS(ω) characterizes the average dwell time of wave packets in the scattering potential eld, having direct physical meaning for quantum, acoustic, and electromagnetic wave scattering. On the other hand, Friedel and Levinson-type theorems show that under appropriate conditions, there is a linear relationship between the scattering phase derivative and the dierence in density 2 of states with and without interaction. For one-dimensional or partial-wave scattering, one obtains the form d dEδl(E)∝ρl(E)−ρ(0) l(E), where δl is the partial wave phase shift, and ρl and ρ(0) l are the densities of states with and without interaction, respectively. Such results have been reformulated in recent mathematical physics work as topological index pairings between spectral shift functions and time delays, introducing a clear K-theory and spectral ow perspective into scattering theory. On the experimental side, Wigner delay has been directly measured in atomic scattering, waveguides, and optical structures, and linked to resonance lifetimes and local density of states. Thus, unifying scattering phase, group delay, and density of states under a single scale identity is a natural theoretical development direction. 1.3 Spectral Flow, Topological Invariants, and Z2 Structure Spectral ow characterizes the continuous evolution of operator spectra in parameter space, and its relationship with topological invariants, especially integer and Z2 indices, has been systematically studied in various situations. For unitary scattering matrices, zero/pole trajectories induced by parameter changes can be characterized through the argument principle and index pairings, leading to conclusions similar to topological Levinson theorems: the total change in phase equals the spectral ow count. In many systems, each time spectral ow crosses the real axis due to parameter changes, the total phase only undergoes half a circle of winding, corresponding to a jump of π rather than 2π . This suggests the existence of a natural double-cover structure: each crossing event in the base parameter space corresponds to two sectors in the lifted space, distinguished by Z2 parity. This structure is formally isomorphic to spin double covers, page-change phenomena in Fermi statistics, and double-cover sectors in NullModular geometry. 1.4 Goals and Structure of This Paper This paper focuses on scattering networks with tunable closed-loop delays, formalizing them as a parameter family S(ω;τ)∈U(N), where ω∈R is the frequency and τ∈R is the controllable eective round-trip time. Under the constraint of the scale identity κ(ω;τ) = 1 π∂ωφ(ω;τ) = ρrel(ω;τ) = 1 2πtr Q(ω;τ) this paper establishes the following three main conclusions: 1. Under natural assumptions of analyticity and non-degeneracy, the zero/pole spectral ow varying with τ forms a series of isolated crossing events in the complex frequency plane, each corresponding to a pole or zero crossing the real axis once. 2. Each crossing event induces a jump of size ±π in the total phase φ(ω;τ) at a xed frequency slice; the corresponding jump in the frequency integral of scale density or group delay is one unit. 3 3. The Z2 index ν(τ) dened by N(τ) mod 2 , where N(τ) is the topological count constructed from spectral ow, ips once at each delay quantization step, forming the unied structure delay quantization ⇒π -step ⇒Z2 parity transition. Theoretically, this paper provides spectral and topological proofs of the above structure and illustrates its universality through one-dimensional scalar and multi-channel matrix models; in applications, this paper proposes a series of experimental schemes based on optical, microwave, and acoustic platforms to measure π -steps and reconstruct Z2 indices, providing frequency-domain readouts for self-referential scattering networks and double-cover structures. 2 Model & Assumptions 2.1 Frequency-Domain Scattering Matrix, Total Phase, and Group Delay Consider a linear lossless network with N external channels, whose frequency-domain scattering matrix is denoted S(ω;τ)∈CN×N, ω ∈R, τ ∈I⊂R, where I is a parameter interval. Losslessness means that for each real frequency ω and τ∈I , we have S(ω;τ)†S(ω;τ) = IN. For xed τ , assume S(·;τ) admits analytic continuation into the upper half-plane, with poles corresponding to resonances or quasi-bound states; for xed ω , assume S(ω;·) is analytic on I . Dene the total scattering phase φ(ω;τ) = arg det S(ω;τ)∈R/2πZ, and x a continuous branch in a neighborhood of a chosen reference point (ω∗, τ∗) such that φ(ω∗, τ∗)=0 . The WignerSmith group delay matrix is dened as Q(ω;τ) = −iS(ω;τ)†∂ωS(ω;τ). For unitary matrix families, we obtain ∂ωφ(ω;τ) = ℑ∂ωlog det S(ω;τ) = 1 2tr Q(ω;τ), yielding the scale density κ(ω;τ) := 1 π∂ωφ(ω;τ) = 1 2πtr Q(ω;τ). In the standard scattering setting, κ(ω;τ) can be identied with the relative density of states ρrel(ω;τ) , the dierence in density of states with and without the scattering potential. This identi- cation makes the scale identity κ(ω;τ) = φ′(ω;τ) π=ρrel(ω;τ) = 1 2πtr Q(ω;τ) the unifying mother formula connecting phase, time, and density of states. 4 2.2 Tunable-Delay Feedback Loop Model Given a delay-free core network S0(ω) , introduce a closed-loop branch between some of its ports with round-trip delay τ , whose frequency-domain description is D(ω;τ)=eiωτ IM, M ≤N. Using the Redheer star product or Schur complement, the core network and delay block can be combined into an eective scattering matrix S(ω;τ) = S0(ω) + S1(ω)IM−R(ω)eiωτ −1S2(ω), where R(ω) is an eective feedback block, and S1, S2 are coupling matrices. When the core is lossless and the delay block is pure phase, S(ω;τ) remains a unitary matrix for each real frequency ω . Poles and some zeros are controlled by detIM−R(ω)eiωτ = 0. Let λj(ω) be the eigenvalues of R(ω) ; the corresponding poles satisfy 1−λj(ω)eiωτ = 0 ⇐⇒ eiωτ =λj(ω)−1. In a one-dimensional scalar minimal model, the core network is described by complex reection coecient r0(ω) and transmission coecient t0(ω) , with feedback branch reection coecient rfb(ω) . The total scattering coecient is Stot(ω;τ) = r0(ω) + t0(ω)2eiωτ 1−rfb(ω) eiωτ , whose denominator 1−rfb(ω)eiωτ zero points give pole trajectories. 2.3 Analyticity and Non-Degeneracy Assumptions Proofs of subsequent theorems rely on the following assumption. Assumption 1 (Assumption A (Analyticity and Non-Degeneracy)) . 1. For each τ∈I , S(·;τ) admits analytic continuation into the upper half-plane, with all zeros/poles of nite order and only nitely many in compact regions. 2. For each ω∈R , S(ω;·) is analytic on I . 3. There exists a sequence {τk} ⊂ I with corresponding frequencies {ωk} ⊂ R such that in a neighborhood of each (ωk, τk) , det S(ω;τ) has exactly one zero or pole zk(τ) crossing the real axis, satisfying zk(τk) = ωk, ∂τℑzk(τk)= 0, and no other zeros/poles simultaneously cross the real axis in the same neighborhood. When Assumption A is satised, (ωk, τk) is called a crossing event, and {τk} is called a family of delay quantization steps. We will see that when the eigenvalues of R(ω) move along the unit circle with approximately equal spacing, τk approximately forms an arithmetic sequence τk≃τ0+k∆τ, k ∈Z, where ∆τ is given by the average round-trip phase quantization condition. 5 3 Main Results (Theorems and Alignments) This section presents the main theorems on delay-driven spectral ow, π -steps, and Z2 indices under Assumption A, and aligns them with the scale identity. 3.1 Delay-Driven Spectral Flow and the Argument Principle For xed τ∈I , suppose det S(·;τ) has zeros {zj(τ)} and poles {pk(τ)} (counted with multiplicity) in the upper half-plane, satisfying appropriate growth conditions. Taking a closed contour Γ surrounding the real axis interval [ω1, ω2] , the argument principle gives 1 2π∆Γarg det S(·;τ) = Nzero(τ)−Npole(τ), where Nzero(τ), Npole(τ) are the numbers of zeros and poles inside Γ , respectively. Choosing the standard keyhole path, we obtain the real-axis integral form 1 πφ(ω2;τ)−φ(ω1;τ)=Nzero(τ)−Npole(τ). For a xed frequency window [ω1, ω2] , as τ varies continuously, the zero/pole trajectories {zj(τ), pk(τ)} evolve continuously in the complex frequency plane. Whenever a zero or pole crosses the real axis, the count on the right changes by ±1 , inducing a step in the total phase within that frequency window. 3.2 Delay Quantization Steps and Crossing Events In networks with delay branches, the zero/pole equation can often be written as detIM−R(ω)eiωτ = 0. Let λj(ω) be eigenvalues of R(ω) ; the pole condition is 1−λj(ω)eiωτ = 0. If λj(ω) = |λj(ω)|eiϕj(ω) , taking logarithms gives the approximate pole location ωj,n(τ) = 1 τϕj(ωj,n)+2πn −i ln |λj(ωj,n)|−1. When |λj(ω)|≲1 and τ varies on macroscopic scales, the real part is approximately ℜωj,n(τ)≃ϕj+ 2πn τ. Imagining n xed and τ increasing, poles move along trajectories contracting from the far end toward the origin, approaching the real axis under appropriate conditions. Through small loss or coupling adjustments, one can construct situations where poles cross the real axis, realizing crossing events in Assumption A. Since ωτ is dimensionless, crossing events typically correspond to the condition ωkτk+ϕj(ωk)≃(2mk+ 1)π, mk∈Z, i.e., round-trip phase satises half-integer quantization, naturally dening a family of approximately equally-spaced delay steps {τk} . 6 3.3 Main Theorem: π -Steps and Z2 Parity Transitions Near a crossing event, det S(ω;τ) can be written in local factorization det S(ω;τ)=(ω−zk(τ))mkgk(ω;τ), where mk= +1 corresponds to a zero, mk=−1 to a pole, and gk is analytic and nonzero in a neighborhood. Dene the local phase jump at τk ∆φk= lim ϵ→0+φ(ωk;τk+ϵ)−φ(ωk;τk−ϵ), and the normalized jump number ∆nk=1 π∆φk. Theorem 2 (Theorem 3.1: π -Step and Unit Jump) . Under Assumption A, for each crossing event (ωk, τk) , the local phase change satises ∆φk=±π, ∆nk=±1. See Section 4 and Appendix A for the proof. The core is that when zk(τ) crosses the real axis, ωk−zk(τ) wraps around the origin by half a circle in the complex plane, so arg(ωk−zk(τ)) jumps by ±π , while the analytic factor gk contributes continuous phase not aecting the jump count. By accumulating all crossing events with delay less than τ , dene the global spectral ow count N(τ) = X τk<τ ∆nk∈Z, ν(τ) = N(τ) mod 2 ∈ {0,1}. Theorem 3 (Theorem 3.2: Z2 Parity Transition) . Under Assumption A, the topological index ν(τ) = N(τ) mod 2 undergoes a parity ip at each delay step τk , i.e., ν(τk+ 0) = ν(τk−0) ⊕1. In particular, if τk forms an approximately arithmetic sequence τk≃τ0+k∆τ , then as τ increases monotonically along I , ν(τ) executes an approximately ideal Z2 square wave. The above results make precise the relationship between delay quantization and π -steps, Z2 topological sectors: each pole or zero crossing the real axis corresponds to a unit spectral ow event, driving a ip in the topological index. 3.4 Unied Time Readout Under the Scale Identity By the scale identity κ(ω;τ) = 1 π∂ωφ(ω;τ) = 1 2πtr Q(ω;τ), taking the frequency window [ωk−δω, ωk+δω] , dene I(τ) = Zωk+δω ωk−δω κ(ω;τ) dω=1 πφ(ωk+δω;τ)−φ(ωk−δω;τ). 7 Proposition 4 (Proposition 3.3: Unit Jump in Scale Density Integral) . Under Assumption A, for each crossing event (ωk, τk) , there exists suciently small δω > 0 such that I(τk+ 0) −I(τk−0) = ∆nk=±1. That is, the integral of the group delay trace tr Q(ω;τ) in a small frequency window jumps by one unit at each delay quantization step. Thus, the topological index ν(τ) can be dened not only through the jumps in total phase in parameter space, but also equivalently described through jumps in the frequency integral of scale density or relative density of states. 4 Proofs This section provides proof outlines for the main theorems, deferring technical details to Appendices A and B. 4.1 Local Argument Analysis and Proof of Theorem 3.1 In a neighborhood of the crossing event (ωk, τk) , write det S(ω;τ) as det S(ω;τ)=(ω−z(τ))mg(ω;τ), where z(τk) = ωk , ∂τℑz(τk)= 0 , m=±1 , and g is analytic with g(ωk;τk)= 0 . For xed ω=ωk , consider the function h(τ) = ωk−z(τ) = a(τ)−ib(τ), where a(τ) = ωk− ℜz(τ) , b(τ) = ℑz(τ) . Near τk , a(τk)= 0 , b(τk)=0 , and ∂τb(τk)= 0 , so when τ traverses τk , the vector h(τ) crosses the real axis in the complex plane. Standard complex analysis geometry shows: ∆ arg h:= lim ϵ→0+arg h(τk+ϵ)−arg h(τk−ϵ)=±π, with sign determined by the signs of a(τk) and ∂τb(τk) . Since φ(ωk;τ) = marg h(τ) + arg g(ωk;τ), and g is nonzero in a neighborhood, arg g(ωk;τ) can be chosen as a continuous branch, so in the local limit ∆φk=m∆ arg h=±π. Thus ∆nk= ∆φk/π =±1 , proving Theorem 3.1. See Appendix A for detailed proof. 4.2 Scale Density Integral and Proof of Proposition 3.3 By I(τ) = 1 πφ(ωk+δω;τ)−φ(ωk−δω;τ), we can write I(τk+ 0) −I(τk−0) = 1 π∆φ(ωk+δω)−∆φ(ωk−δω). 8 Choose δω suciently small such that in the rectangular region [ωk−δω, ωk+δω]×[τk−δτ, τk+δτ] there is only one crossing zero or pole, and its trajectory crosses the midline of the frequency window. Using the analysis of the local factor (ω−z(τ))m in Appendix A, we know that in this region ∆φ(ω;·) is a piecewise constant function of ω , with the dierence in values on either side of ωk being ±π . Thus I(τk+ 0) −I(τk−0) = ±1. This proves Proposition 3.3. By the denition of N(τ) and integer addition structure, clearly ν(τ) = N(τ) mod 2 ips once at each step, hence Theorem 3.2 holds. 4.3 Finite-Order EulerMaclaurin and Numerical Error Control In actual numerical and experimental data processing, scale density integrals are often approximated by nite sampling as discrete sums. Let Ih(τ) = h N X n=0 κ(ωn;τ), ωn=ωk−δω +nh, where h= (2δω)/N . The EulerMaclaurin formula gives Ih(τ) = Zωk+δω ωk−δω κ(ω;τ) dω+O(h2), provided κ(ω;τ) has bounded second derivatives in the frequency window. As long as the sampling step h is suciently small, the phase step height of ±1 is only smoothed, not erased or ipped. Appendix B provides standard estimates for the EulerMaclaurin remainder, showing that the singularity near poles is only smoothed at nite resolution, without changing the spectral ow count. Thus the topological index ν(τ) is robust to nite resolution and noise. 5 Model Applications This section returns to concrete models, demonstrating the implementation of the main theorems in single-channel scalar and multi-channel matrix models, and discussing local linearization in selfreferential scattering networks. 5.1 Single-Channel Reection-Type Feedback Model Consider the single-channel model Stot(ω;τ) = r0(ω) + t0(ω)2eiωτ 1−rfb(ω) eiωτ . In a small frequency window, adopt the slow-variation approximation, treating r0, t0, rfb as constants r0, t0, rfb ∈C , satisfying |r0|2+|t0|2= 1,|rfb| ≤ 1. 9 C Relationship with Self-Referential Scattering Networks and DoubleCover Geometry This appendix discusses the position of this paper's results in the broader context of self-referential scattering networks and double-cover structures. C.1 Self-Referential Scattering Networks and Nonlinear Feedback In self-referential scattering networks, the response in feedback loops can depend on the network's own output states or history, making the scattering matrix a nonlinear or time-varying object. Typical examples include feedback structures with gain saturation, nonlinear phase modulation, or adaptive control. Near a working point, the nonlinear network can be linearized to obtain an eective scattering matrix Seff(ω;τ) and corresponding feedback block Reff(ω) . As long as the linearized system satises Assumption A, all conclusions about spectral ow, π -steps, and Z2 indices in this paper remain valid. This shows that in the parameter space of self-referential networks, a family of local regions can be identied where the network's topological behavior is assembled from several π -step units. C.2 Z2 Double Cover and Half-Phase Winding The π -steps discovered in this paper are essentially half-circle windings in phase space. Considering the phase map of det S(ω;τ) in the complex plane, its natural value space is R/2πZ . If lifted to the double-cover space R/πZ , then each π jump corresponds to one full winding in the double-cover space, and the Z2 index ν(τ) characterizes the number of page-turns of the lifted path between two pages. This structure has formal parallelism with the spin double cover Spin(n)→SO(n) and the fermion statistics phenomenon of two-winding identity: in scattering phase space, there are only two types of sectors, distinguished by odd or even numbers of π -jumps. C.3 Role in Unied Time Scale and Boundary Geometry In the framework of unied time scale and boundary time geometry, the scale identity unies scattering phase derivative, relative density of states, and WignerSmith group delay trace into a single time scale density. 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