Self-Referential Scattering and the Birth of Fermions: Riccati Square Roots, Spinor Double Cover, and a $\mathbb{Z

Self-Referential Scattering and the Birth of Fermions: Riccati Square Roots, Spinor Double Cover, and a Z2 Exchange Phase Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract Standard quantum eld theory explains the relation between spin and statistics through the spinstatistics theorem, derived from Lorentz covariance and microcausality on a continuous spacetime background. In topological approaches, the antisymmetry of fermionic wavefunctions can also be understood via the nontrivial topology of conguration spaces and associated line bundles, as in the FinkelsteinRubinstein construction for solitons in nonlinear eld theories. However, these frameworks typically assume relativistic quantum elds as the ontological starting point. Within a discrete, causal quantum cellular automaton (QCA) ontology, the universe is described as a lattice of local Hilbert spaces updated by a global unitary step. In previous work, massive excitations were interpreted as localized, self-sustained interference structures whose internal dynamics consume part of a global information-update budget, giving mass an interpretation as topological impedance in an underlying scattering network. Building on this picture, the present work proposes a dynamical and geometric origin of fermionic statistics in terms of self-referential scattering . We consider localized excitations realized as feedback loops in an eective one-dimensional scattering problem obtained from coarse-graining the QCA. The boundary response of such a loop is encoded in an impedance function or reection coecient solving a nonlinear Riccati equation, a structure well known in wave propagation and scattering theory. We show that the xed-point condition for a stable, self-referential loop forces the physical state of the excitation to live on a square-root branch of the underlying scattering data. This branch structure induces a canonical double cover of the conguration space of N identical excitations. We prove that the generator corresponding to exchanging two excitations lifts to a nontrivial loop on this double cover with holonomy (−1) , so that the N -body wavefunction transforms in the sign representation of the permutation group and satises fermionic exchange statistics. In this construction, spinor behavior and the Spin(3) double cover of SO(3) are not postulated but emerge from the necessity of taking square roots of self-referential scattering data. The internal Riccati square-root variable plays the role of a spinor amplitude whose squared modulus reproduces observable scattering characteristics. In this sense, spin 1/2  is reinterpreted as the topological ngerprint of information self-reference in a QCA-based universe. We outline an explicit realization in Dirac-type QCA models and propose engineered scattering networks in photonic and superconducting platforms to test the predicted Z2 exchange phase. Keywords: Quantum cellular automata; self-referential scattering; Riccati equation; impedance; spinstatistics theorem; spinor double cover; FinkelsteinRubinstein constraint; fermionic exchange phase 1 1 Introduction & Historical Context 1.1 SpinStatistics Theorem in Continuous Field Theory The spinstatistics theorem states that in three-plus-one-dimensional relativistic quantum eld theory, half-integer spin particles must obey FermiDirac statistics, while integer spin particles must obey BoseEinstein statistics. Modern textbook derivations rely on the following ingredients: Lorentz covariance, vacuum stability, positive-denite probabilities of local observables, and (anti)commutation relations of operators at spacelike separations (microcausality). In this framework, the distinction between fermions and bosons is one of the input conditions of eld operator commutation relations, and consistency with spin is subsequently guaranteed by structural theorems. This proof structure is highly rigorous, yet brings a widely recognized puzzle: why is there such a profound connection between spin and many-body exchange statistics, rather than being mutually independent structures? Feynman once lamented that an intuitive explanation of this simple statement remains elusive. 1.2 Topological SpinStatistics Relations and Solitons An alternative line of thought, advocated by Finkelstein and Rubinstein, examines the homotopy properties of topological soliton conguration spaces in nonlinear eld theories. They point out that when the fundamental group of the soliton conguration space is nontrivial, one can relate 2π rotations to particle exchanges via nontrivial line bundles, thus obtaining a spinstatistics correspondence. In this perspective, the wavefunction is no longer a single-valued function on conguration space, but rather a section of a line bundle; the homotopy class of soliton winding paths determines the exchange phase through the holonomy of the bundle. Further work has applied this idea to YangMills solitons and Hopf topological invariants, demonstrating geometric relationships between linking numbers and statistics. Topological methods provide geometric intuition for the spinstatistics theorem, but still take intrinsic continuous elds as fundamental objects, and typically start from known soliton models. 1.3 Quantum Cellular Automata and Discrete Ontology In recent years, quantum cellular automata (QCA) have emerged as an attempt to reconstruct quantum eld theory, and even the entire physical universe, within a framework of discrete, local unitary evolution. In the QCA picture, spacetime consists of a discrete lattice Λ with a local Hilbert space Hx at each site, and dynamics are given by a global unitary operator U acting on nite neighborhoods. In the continuum limit, appropriate choices of local coin operators can reproduce Dirac, Weyl, or Maxwell equations, thus recovering standard quantum eld theory on a discrete ontology. In a series of previous works, mass was interpreted as a geometric impedance of information propagation rates: massless excitations correspond to feedforward propagation along the light cone, while massive excitations correspond to local structures with feedback and looping, whose internal evolution speed vint and external group velocity vext satisfy an information rate conservation constraint. The corresponding microscopic picture is: a particle is a kind of selfreferential feedback loop in the QCA network, whose stable existence depends on impedance matching between input and output. 1.4 Goals and Claims of This Work This paper attempts to take a further step in the above QCA and mass = topological impedance picture, proposing the following claims: 2 1. Any localized excitation that realizes a stable rest mass in QCA can, after appropriate coarse-graining, be viewed as a self-referential feedback loop in an eective onedimensional scattering problem. 2. The spatial evolution of the boundary response (impedance or reection coecient) of this feedback loop is governed by a nonlinear Riccati equation; its steady-state solution is a xed point of a certain Möbius transformation and has a square-root discriminant structure. 3. For a system of N identical self-referential excitations, the natural quantization of the total conguration space is no longer a single-valued wavefunction, but rather a section of a double cover line bundle induced by the above Riccati structure. Particle exchange paths have Z2 holonomy on this double cover, leading to an exchange phase of (−1) . 4. Therefore, massive self-referential excitations automatically realize FermiDirac statistics in this construction; bosons correspond to pure feedforward modes without self-reference or composites of several self-referential loops. The starting point of this research is not to attempt to re-prove the spinstatistics theorem, but to construct a microscopic mechanism in discrete ontology that establishes an explicit connection between self-referential scattering, Riccati square roots, and spinor double covers, thus providing a dynamicalgeometric explanation for the existence of fermions. 2 Model & Assumptions 2.1 Underlying QCA Structure Let Λ⊂Zd be a regular lattice; this paper primarily considers cases d= 1,3 . Each site x∈Λ is associated with a nite-dimensional Hilbert space Hx∼ =Cq , and the global Hilbert space is H=O x∈ΛHx. Time evolution is given by a local unitary operator with nite neighborhood. That is, there exists a nite range R such that the single-step evolution U can be written as a nite-depth quantum circuit of local gates, respecting causality: U†AOU⊂ AO+ , where AO is the local operator algebra of region O , and O+ is a nite-thickness neighborhood of O . To connect to the continuum limit, we consider a class of Dirac-type QCA, whose single-step evolution in momentum representation can be written as U(k) = exp (−iHeff(k)∆t), where Heff(k) in the long-wavelength limit ka ≪1 approaches the Dirac Hamiltonian Heff(k)≈αk +βm, with α, β matrices satisfying the Cliord algebra, a the lattice spacing, and m the eective mass parameter. 2.2 Self-Referential Scattering Unit and Eective One-Dimensional Model Consider introducing a local structure in a nite region D⊂Λ such that there exists a feedback channel within the region: part of the incident amplitude, after undergoing local scattering, is re-injected into the same region. For modes with wavelength much larger than the size of D , an eective one-dimensional description can be adopted, compressing the entire region D into an 3 equivalent transmission line or scattering center, with incident and outgoing signals propagating along a one-dimensional coordinate x . In the frequency domain representation, suppose that for a xed frequency ω , the mode satises an eective wave equation on the one-dimensional coordinate. Its propagation in the half-space x > 0 can be characterized by a position-dependent impedance Z(x;ω) . According to electromagnetic wave and acoustic wave propagation theory, under appropriate one-dimensional approximations, the spatial evolution of Z(x;ω) satises a nonlinear Riccati equation: dZ dx=A(x;ω) + B(x;ω)Z+C(x;ω)Z2, where A, B, C are determined by medium parameters. For layered media or discrete lattice models, Z jumps between layers according to Möbius transformations: Zn+1 =anZn+bn cnZn+dn ,anbn cndn∈SL(2,C). The core requirement of the self-referential feedback structure is: at an eective boundary point x= 0 , the input impedance Zin(ω) externally exhibited by the local structure must equal the load impedance Zloop(ω) seen by the internal loop. This self-consistency condition can be written in discrete representation as Zin = Φ(Zin), where Φ is a composite mapping consisting of Möbius transformations and feedback phases from multiple layers. Denition 2.1 (Self-Referential Scattering Unit) . At a given frequency ω , a local structure is called a self-referential scattering unit if its external equivalent input impedance Zin(ω) is a xed point of some complex Möbius transformation Φ : Zin(ω) = ΦZin(ω),Φ(z) = Az +B Cz +D, AD −BC = 1. The stability of this xed point is determined by the modulus of Φ′Zin . 2.3 Riccati Fixed Point and Square-Root Discriminant In general, the Möbius transformation xed-point equation Z=AZ +B CZ +D can be reduced to a quadratic equation CZ2+ (D−A)Z−B= 0, whose solution is Z±=A−D±p(A−D)2+ 4BC 2C, provided C= 0 . Thus, the equivalent impedance of any self-referential scattering unit naturally carries a square-root branch structure, whose discriminant ∆=(A−D)2+ 4BC determines the physical properties of the two branch solutions through its phase and modulus. For lossless systems, (A, B, C, D) belong to an appropriate representation of SU(1,1) or SL(2,R) , 4 ∆ lies on a certain curve in the complex plane, and the choice of square-root function √∆ corresponds to two types of boundary conditions. We view this square-root structure as the embryonic form of a spinor: the observable impedance Z corresponds to the square of some amplitude variable ζ , i.e., Z=F(ζ2), and the multivaluedness of ζ under closed paths in parameter space will determine the exchange statistics. 2.4 Conguration Space of Identical Self-Referential Excitations Consider N well-separated self-referential scattering units in three-dimensional space, with center positions x1,...,xN∈R3 . Ignoring internal structural details, the geometric conguration space is QN=R3N\∆ SN , where ∆ is the diagonal subset where particle positions coincide, and SN is the permutation group. For d≥3 dimensions, the fundamental group of QN is SN , whose elements can be generated by exchanges of adjacent particles. In standard quantum mechanics, the many-body wavefunction Ψ is viewed as a complexvalued single-valued function on QN . In topological spinstatistics schemes, Ψ is viewed as a section of a line bundle or vector bundle, with dierent bundle structures corresponding to dierent exchange statistics. In our scheme, we will use the Riccati square-root structure internal to each self-referential scattering unit to construct a natural double cover space e QN over QN , and show that exchange paths have Z2 holonomy on e QN . 3 Main Results (Theorems and Alignments) This section presents the core results of this paper, focused on three levels: 1. The square-root structure of self-referential scattering units and the Riccati equation; 2. The spinor double cover induced by the square-root structure; 3. The Z2 phase from particle exchange and its correspondence with Fermi statistics. 3.1 Self-Referential Scattering and Riccati Square Root Theorem 3.1 (Square-Root Discriminant of Self-Referential Scattering Unit) . Suppose the equivalent transfer matrix of a self-referential scattering unit is M=A B C D∈SL(2,C), and its external input impedance Zin is a xed point of the Möbius transformation, i.e., Zin = Φ(Zin) , with Φ(z)=(Az +B)/(Cz +D) . Assume C= 0 and the system is lossless, so that the spectrum of M lies on the unit circle. Then: 1. Zin satises the quadratic equation CZ2+ (D−A)Z−B= 0. 5 2. There exists a discriminant ∆=(A−D)2+ 4BC such that Zin =Z±=A−D±√∆ 2C, where √∆ is the two-valued square-root function on the complex plane. 3. If M belongs to SU(1,1) or an equivalent representation, then exactly one of the two branch solutions Z± corresponds to a stable xed point ( |Φ′(Z)|<1 ), and the other to an unstable xed point ( |Φ′(Z)|>1 ). Therefore, the external response of any stable self-referential scattering unit can be equivalently characterized by a choice of one of a pair of square-root variables ±√∆ . Proof Sketch. Writing the xed-point condition as a quadratic equation immediately yields the discriminant and two-valued solution; the stability condition is determined by the modulus of the derivative of the Möbius transformation. The lossless condition constrains the spectrum of M , thereby constraining the phase and modulus of ∆ , so that only one branch is stable. See Appendix A for details. 3.2 Emergence of Spinor Double Cover Denition 3.2 (Spinor Internal Variable) . For each self-referential scattering unit, we introduce an internal variable ζ such that √∆=Λζ2, where Λ∈C× is a nonzero constant related to the specic implementation. Dene a normalized spinor variable χ=ζ ∥ζ∥, whose overall phase redundancy is regarded as a gauge freedom. Thus, a stable self-referential scattering unit can be described by either the equivalent impedance Zin or the spinor variable χ , which satisfy Zin =F(χ2), where F is an explicit rational function. Note that χ and −χ correspond to the same Zin . Theorem 3.3 (Internal Spinor Double Cover) . Under the above assumptions, the internal state space of a single self-referential scattering unit can be viewed as a quotient space Sint ∼ =C2\{0}/{χ∼ −χ}, and there exists a natural mapping πint :C2\{0}→Sint, πint(χ)=[χ], whose kernel is {±1} . Under appropriate gauge choices and coarse-graining, the SU(2) rotation representation on C2 projects via πint to the SO(3) rotation representation on Sint , and thus the internal degrees of freedom are naturally organized into a double cover structure of Spin(3) . This structure is completely isomorphic to standard spinor theory: χ plays the role of a spin1/2 spinor, while Zin and observable scattering phases correspond to quadratic invariants. 6 3.3 Particle Exchange and Z2 Exchange Phase Theorem 3.4 (Fermionic Exchange Statistics of Self-Referential Excitations) . Consider N identical self-referential scattering units in three-dimensional space, with conguration space QN as dened in Section 2.4. Let e QN be the double cover induced by the internal spinor variables: e QN=n(x1,...,xN;χ1, . . . , χN)o∼, where the equivalence relation identies χj∼ −χj for each particle, while requiring invariance of the external Zin . Then: 1. e QN is a double cover space of QN , whose covering transformation group is generated by simultaneously changing the sign of all χj , forming a Z2 . 2. For any pair of particles i, j , their exchange operation corresponds to a closed path γij on QN . On e QN , γij lifts to two paths eγ± ij , whose endpoints dier by a global sign change: eγ+ ij (1) = −eγ− ij (1). 3. If we regard the many-body state as a section of a line bundle on e QN and require that this section changes sign under the covering transformation, then parallel transport along γij gives the wavefunction a phase of (−1) : Ψ(γij(1)) = −Ψ(γij(0)). Therefore, in this quantization scheme, the geometric realization of particle exchange necessarily corresponds to fermionic antisymmetric statistics. Proof Sketch. This is a concrete realization of the FinkelsteinRubinstein scheme on self-referential spinor internal degrees of freedom. The key is: viewing particle winding paths as closed curves on e QN , their lift in the covering space has nontrivial Z2 holonomy. Choosing a line bundle where the covering transformation corresponds to wavefunction sign change yields Fermi statistics. See Appendix B for details. 4 Proofs This section provides proof outlines for the above theorems, with technical details expanded in the appendices. 4.1 Proof of Theorem 3.1 From the xed-point condition Z=AZ +B CZ +D we obtain CZ2+ (D−A)Z−B= 0. This is a quadratic equation in Z ; as long as C= 0 we can write the explicit solution Z±=A−D±√∆ 2C,∆=(A−D)2+ 4BC. 7 Assuming the system is lossless, i.e., M∈SU(1,1) or a similar group, means that the eigenvalues of M lie on the unit circle, and M is closely related to the intrinsic phase of the corresponding scattering matrix S . The Möbius transformation Φ(z) = Az +B Cz +D has derivative Φ′(z) = 1 (Cz +D)2. Substituting Z± , we can evaluate |Φ′(Z±)| . Under the lossless condition, the moduli of (CZ±+D) are reciprocals, so exactly one of the two branch solutions satises |Φ′(Z)|<1 , corresponding to a stable xed point and a stable self-referential scattering structure; the other branch is unstable, corresponding to a nonphysical solution or excited state. A detailed analysis comparing with the variable phase method and Levinson theorem is given in Appendix A. 4.2 Proof of Theorem 3.3 The discriminant ∆ is an invariant of the trace and determinant of M ; in the lossless case, ∆ typically lies on a complex plane curve passing through the origin. For each frequency ω and momentum k , we can write ∆(ω, k)=Λ2(ω, k)ζ4(ω, k), where Λ= 0 is a gauge choice. Thus √∆=Λζ2, and Zin can be rewritten as a rational function F(ζ2) . Viewing ζ as coordinates of a twodimensional complex vector, we introduce normalization χ=ζ ∥ζ∥∈C2\{0}, and identify χ∼ −χ to obtain the quotient structure of the internal state space. Standard group theory results show that the natural representation of SU(2) on C2 projects via the quotient map to the representation of SO(3) on S2 , and SU(2) is the double cover of SO(3) . Thus, the internal state has a double cover structure completely equivalent to a spin1/2 spinor. 4.3 Proof of Theorem 3.4 The topological properties of QN have mature conclusions: in three-dimensional space, the fundamental group of QN is the permutation group SN , whose generators can be viewed as adjacent particle exchange paths. The FinkelsteinRubinstein proof shows: if there exists a double cover e QN→QN with covering transformation group Z2 , and 2π spatial rotations and particle exchange paths are related to nontrivial closed loops in e QN , then one can construct a line bundle such that the many-body state, as a section of this line bundle, changes sign under the covering transformation, thus realizing Fermi statistics. In our construction, each particle carries an internal spinor variable χj , with χj∼ −χj corresponding to the same physical impedance. Combining the internal variables of all particles naturally yields e QN . Evolution along the exchange path γij not only winds in position space, but also winds around the square-root branch cut in internal parameter space, causing the overall phase of (χi, χj) to undergo a 2π winding. This corresponds to nontrivial Z2 holonomy on e QN . Choosing a line bundle where the covering transformation corresponds to wavefunction sign change yields Ψ(. . . , xi,xj, . . . ) = −Ψ(. . . , xj,xi, . . . ). Therefore, self-referential scattering excitations naturally realize fermionic exchange statistics. 8 5 Model Apply This section shows how to implement the above self-referential scattering structure in concrete QCA models, and relate it to Dirac mass and topological impedance. 5.1 Self-Referential Defects in One-Dimensional DiracQCA Consider a one-dimensional Dirac-type QCA, whose single-step evolution can be written as a quantum walk: U=S+⊗|↑⟩⟨↑|+S−⊗|↓⟩⟨↓|◦(I⊗C), where S± shift the state left or right by one step, and C is a 2×2 coin matrix, e.g., C(θ) = cos θsin θ −sin θcos θ. In momentum representation, the eigenvalues of U(k) are e∓iω(k) , satisfying the dispersion relation cos ω(k) = cos θcos(ka), which in the long-wavelength limit yields an eective Dirac equation, with mass m related to θ . Against this background, modifying the coin matrix or introducing a local loop at a single site or nite sub-chain can realize an eective scattering center. For modes with wavelength much larger than the defect region size, their scattering is characterized by a 2×2 single-channel scattering matrix S(k) = r(k)t′(k) t(k)r′(k). By explicitly modeling the internal loop of the defect region as additional boundary and feedback channels, its equivalent impedance can be written as a function Z(x;k) satisfying a Riccati equation, whose value at the defect region periphery gives r(k) . Related techniques are closely related to the variable phase method. 5.2 Mass, Bound States, and Self-Referential Feedback For defects with self-referential feedback, there exist certain frequencies ω and momenta k such that the pole condition 1−r(ω)eiθloop = 0 holds, where θloop is the additional loop phase. These poles correspond to bound or quasi-bound states, which in the continuum limit of QCA manifest as localized particles whose frequency deviates from the massless mode dispersion relation, dening an eective mass m . On the other hand, the reection coecient can be written as r(ω) = eiδ(ω), where δ(ω) is the scattering phase. Through the Levinson theorem and variable phase method, δ(ω) can be related to the number of bound states and Riccati phase functions. The pole condition can be rewritten as a self-consistent equation for impedance, whose solution has a two-valued square-root structure, thus introducing a natural internal spinor variable for the bound state. 9 B.3 FR Data of Self-Referential Scattering Excitations In this paper's construction, the Riccati discriminant square root internal to each self-referential scattering unit provides a natural two-valued degree of freedom χ∼ −χ . Combining the positions and internal spinors of all particles yields e QN=n(x1,...,xN;χ1, . . . , χN)o∼, whose projection onto QN forgets all χj . The lift of exchange paths on e QN has the following property:  Performing spatial exchange alone without changing internal spinors corresponds to a closed path;  However, due to the existence of the Riccati square-root branch cut, adiabatic evolution along this path causes the overall phase of (χi, χj) to wind around the origin once, thus corresponding to a nontrivial element on e QN . Therefore, the self-referential scattering structure automatically provides the double cover and holonomy data required by the FR scheme. Choosing a quantization rule where the covering transformation corresponds to wavefunction sign change yields Fermi statistics; choosing the covering transformation as trivial action yields Bose statistics. This paper emphasizes: under the natural dynamics of massive self-referential excitations, stability and unitarity require choosing the former, thus locking massive and Fermi statistics together in this construction. C Remarks on 2+1 Dimensions and Anyonic Generalizations Although this paper primarily focuses on three-plus-one dimensions, in two-plus-one dimensions, self-referential scattering and Riccati square-root structure may produce richer statistical behavior. In two-plus-one dimensions, the fundamental group of the N -particle conguration space is the braid group BN , whose representations can yield anyonic statistics, allowing continuous interpolation of phase or matrix representations during particle exchange between bosons and fermions. In self-referential scattering networks, the combination of the multivaluedness of internal spinor variables χ and braid group representations promises to give a concrete realization of a class of self-referential anyons: their statistical phase is no longer limited to ±π , but is related to the winding number of the Riccati discriminant in parameter space. Systematic analysis of this case requires extending this paper's QCAscattering construction to two-dimensional lattices and performing unied modeling of the topology of multi-particle braid paths and the geometric structure of internal self-referential feedback, left for future work. 16