Functorial Framework for Morphological Computing

author: Rowan Brad Quni-Gudzinas

ORCID: 0009-0002-4317-5604

ISNI: 0000000526456062

modified: 2025-10-31T07:12:48Z

title: Functorial Framework for Morphological Computing

aliases:

- Functorial Framework for Morphological Computing

A Functorial Framework for Morphological Computing: A Vectorial, Topologically-Protected Solution to the Residue Number System Reconstruction Paradox

Author: Rowan Brad Quni-Gudzinas

Contact: [email protected]

ORCID: 0009-0002-4317-5604

ISNI: 0000 0005 2645 6062

DOI: 10.5281/zenodo.17491258

Publication Date: 2025-10-31

Version: 1.0

Abstract: This work introduces a formal framework for morphological computing that resolves the longstanding reconstruction paradox in Residue Number System (RNS) arithmetic by leveraging the physics of topological quantum matter. Conventional RNS-based accelerators are bottlenecked by the serial, computationally expensive conversion of a parallel residue vector into a scalar integer via the Chinese Remainder Theorem. We dissolve this bottleneck by proposing a vectorial topological encoding scheme where the computational result is the residue vector itself, physically embodied as a vector of Chern numbers in a moiré superlattice. We address the critical challenge of formal verification in physical computing by constructing a structure-preserving functor from the algebraic category of RNS computations to the physical category of adiabatic evolutions in topological phases. This functor provides a formal guarantee of computational correctness. The final result is read out directly via a multi-terminal quantum Hall conductance measurement, which yields the Chern vector without any algorithmic post-processing. This approach establishes a new paradigm of computation-by-relaxation that is formally verifiable, physically robust, and achieves end-to-end energy efficiency by eliminating the reconstruction step entirely.

Keywords: morphological computing, Residue Number Systems, topological quantum matter, functorial framework, category theory, Chern insulator, moiré materials, quantum Hall effect, physical computation, hardware-software co-design, energy-efficient computing

**I. State of the Art and Gap Identification**

A. Summary of key prior work

The scholarly context for this work is situated at the confluence of three distinct but complementary fields: morphological computing, Residue Number Systems (RNS), and topological quantum matter. The concept of morphological computing, where a system’s physical structure contributes to computation, was advanced by Pfeifer & Bongard (2007), but has been critically assessed as often lacking a rigorous information-theoretic foundation that would distinguish it from trivial physical dynamics (Müller & Hoffmann, 2017). This concern is reinforced by the argument that without a formal framework to define the relationship between a physical process and an abstract computation, claims of physical computation risk becoming unverifiable (Horsman et al., 2014). In parallel, RNS, as formalized by Szabo & Tanaka (1967), offers an architecture for carry-free parallel arithmetic. However, its practical application is persistently hindered by the computational overhead of the Chinese Remainder Theorem (CRT) reconstruction step, a bottleneck that remains a challenge even in modern applications such as homomorphic encryption (Cheon et al., 2017). Concurrently, the study of topological phases of matter has progressed from theoretical proposals (Kitaev, 2003) to experimental realizations of robust, topologically protected states in moiré materials (Sharpe et al., 2019). While these systems provide stable, quantized invariants, the literature lacks a clear protocol for encoding arbitrary computational results into these invariants and reading them out directly. Existing work on topological quantum computing has largely focused on developing fault-tolerant quantum gates using non-Abelian anyons, rather than addressing general-purpose integer arithmetic (Nayak et al., 2008). The broader field of physical computation has long explored analogues to Turing-completeness in continuous physical systems but has not yet produced scalable, noise-resilient, and formally verifiable architectures for integer arithmetic (MacLennan, 2004; Siegelmann, 1999). Finally, the persistent demand for energy-efficient computing in the post-Moore era highlights the limitations of existing RNS accelerators, which have yet to overcome the systems-level overhead imposed by the reconstruction step (Horowitz, 2014).

B. Identified open problems or tensions

The confluence of these fields reveals a set of interconnected and unresolved problems. The primary issue is the reconstruction paradox, where the inherent parallelism of RNS is nullified by the serial, computationally intensive CRT reconstruction required to produce a usable scalar output. This is directly linked to the verification challenge: many proposals for physics-based computation remain descriptive, lacking the formal mathematical structure, such as that provided by category theory, needed to prove that a physical system correctly implements a specified computation (Müller & Hoffmann, 2017; Horsman et al., 2014). Even if a computational state were encoded in a topological invariant, a measurement problem persists, as no established protocol allows for the direct measurement of such an invariant to yield a final computational result without requiring further algorithmic processing. This has led to a scalar encoding failure, where attempts to map the CRT reconstruction onto a single, weighted physical observable have proven unworkable due to non-physical requirements and a misunderstanding of measurement principles. These specific issues point to a deeper ontological mismatch: prior work typically treats a physical system as a passive substrate on which an abstract algorithm is executed, whereas this work proposes an inversion where the computation is the natural dynamics of the physics itself. Furthermore, existing topological computing frameworks exhibit a lack of arithmetic universality, focusing on specialized quantum gate sets rather than the general-purpose integer arithmetic needed for many classical computing tasks. This disconnect is formalized by the absence of categorical grounding, as no prior work has constructed a functorial bridge to formally link an algebraic model of computation with a category of topological quantum phases. Finally, these issues culminate in a practical energy-latency tradeoff, where the benefits of RNS arithmetic are offset by the costs of reconstruction, yielding no net system-level advantage.

C. Positioned contribution of this work

This work presents a framework that directly addresses the identified gaps by integrating these fields through a novel, formally grounded approach. It resolves the reconstruction paradox by dissolving the problem: a vectorial topological encoding is proposed where the computational result is the vector of residues itself, rendered directly as a physical observable, eliminating the need for a scalar reconstruction. The verification challenge is met through a categorically formalized framework that constructs an explicit, structure-preserving functor from the category of RNS computations to the category of topological physical evolutions, providing a formal proof of correctness that responds to the critiques of Müller & Hoffmann (2017). The measurement problem is solved by specifying an experimentally feasible, direct multi-terminal measurement protocol where the encoded Chern vector is read out as a set of quantized Hall conductances, yielding the final result without post-processing. This approach overcomes the scalar encoding failure by embracing a parallel vector output that is native to both RNS and the physics of topological matter. By grounding the framework in a computation-by-relaxation paradigm, this work offers a concrete realization of morphological computing that is non-trivial and formally verifiable, as called for by Horsman et al. (2014). It establishes arithmetic universality for a class of topological systems by demonstrating a mapping from any integer arithmetic operation to a realizable physical protocol. Through an ontological inversion, computation is redefined not as an abstract process imposed on matter, but as the natural, deterministic relaxation of a constrained physical system. By eliminating the reconstruction bottleneck, this architecture achieves end-to-end energy efficiency, breaking the longstanding energy-latency tradeoff that has limited the utility of RNS accelerators.

**II. Theoretical Foundations of Morphological Computing**

A. From symbolic to physical computation

This framework redefines computation not as the execution of a logical sequence, but as the deterministic evolution of a constrained physical system. The process begins with the preparation of a high-energy initial state, which then naturally relaxes to a minimum-energy ground state that physically encodes the computational solution. This “computation-by-relaxation” paradigm leverages the system’s natural physics to perform computational work, thereby avoiding the von Neumann bottleneck and the energy costs associated with transistor switching. The process can be understood as a physical instantiation of coarse-graining, where microscopic degrees of freedom self-organize into a stable, macroscopic state that represents the computational result. This view is consistent with Landauer’s principle, as the energy dissipated is fundamentally linked to the physical relaxation process rather than the logical irreversibility of an abstract gate model. The computational complexity of a problem is encoded in the landscape of the system’s free energy functional, with the solution path corresponding to a geodesic in the system’s state space, a concept that aligns with the notion of thermodynamic depth (Lloyd & Pagels, 1988). The input is encoded in the preparation of the initial state, while the function to be computed is encoded in the topology of the energy landscape, thus unifying program and data within the physical substrate. The system functions as a dissipative structure, where an external energy flow is used to prepare the system far from equilibrium, and its subsequent relaxation toward equilibrium performs the computation (Prigogine, 1967).

B. Residue number systems as the optimal physical logic

Residue Number Systems provide the ideal mathematical structure for this physical computing paradigm. The core advantage of RNS is its inherent parallelism; arithmetic operations on each residue channel are performed independently of all others, which eliminates the carry propagation that fundamentally limits conventional binary arithmetic. This mathematical independence maps directly and naturally onto a physical architecture of decoupled subsystems, such as distinct regions of a mesoscopic material, where each prime modulus corresponds to a unique, topologically protected computational channel. The Chinese Remainder Theorem guarantees a bijective mapping between a global integer and its corresponding vector of local residues, ensuring that the vector representation is both complete and unambiguous. The product ring structure, $\prod_{i=1}^k \mathbb{Z}/p_i\mathbb{Z}$, is not merely a mathematical convenience but is the formal expression of a physically decomposable system, making RNS the natural logic for a modular hardware implementation. The isomorphism of algebras guaranteed by the CRT ensures that all essential ring-theoretic properties, such as distributivity and associativity, are preserved in the vectorial representation, providing a faithful encoding of the computation. Furthermore, the use of small prime moduli provides a balance between dynamic range, physical realizability, and intrinsic error detection, as any error affecting a single channel produces an out-of-range vector that is immediately detectable. RNS thus functions as a non-linear error-detecting code, a property that is maintained and physically grounded in the topological embodiment.

C. Topological protection for robust information encoding

To ensure that the physical embodiment of RNS is robust, information is encoded in global topological invariants that are intrinsically resilient to local noise, defects, and thermal perturbations. The Chern number, a global property of a system’s electronic band structure, serves as the physical carrier of a residue value. Fractional Chern Insulators (FCIs), which have been realized in moiré superlattices, offer a suitable physical platform where the ground-state degeneracy and associated Chern number can be engineered to be prime-bounded, thereby creating a direct physical realization of an RNS residue channel. In this architecture, the moiré pattern is not a passive substrate but an active computational landscape whose electronic properties can be programmed by tuning physical parameters like twist angle and external electric fields. The topological degeneracy of the ground state provides a natural Hilbert space for storing a residue, with each distinct Chern sector functioning as a stable, noise-immune memory element. The robustness of this encoding is physically guaranteed by the spectral gap of the system, which exponentially suppresses fluctuations that could cause unintended transitions between different topological sectors. Information can be read out by leveraging the chiral edge states characteristic of such systems, which carry a quantized current directly proportional to the bulk Chern number, enabling a non-invasive, contact-based measurement. This system exhibits topological quantum order, where long-range quantum entanglement provides a deep, intrinsic fault tolerance that protects the encoded information beyond the simple energy barrier of the spectral gap (Wen, 2002).

**III. The Functorial Framework: A Formal Bridge from Computation to Physics**

A. Defining the category of RNS computations

To formalize the link between computation and physics, we first define the category of RNS computations. The Chinese Remainder Theorem establishes a ring isomorphism $\mathbb{Z}/M\mathbb{Z} \cong \prod_{i=1}^k \mathbb{Z}/p_i\mathbb{Z}$ for a set of pairwise coprime moduli $\{p_i\}$, which provides the foundational product structure of RNS. Based on this, the category RNS is defined with objects as the residue rings $\mathbb{Z}/p_i\mathbb{Z}$ and morphisms as the ring homomorphisms induced by modular arithmetic operations, such as addition and multiplication. This categorical structure formally captures the parallel and independent nature of RNS, where each residue channel corresponds to an operation on a distinct object in the product category. For instance, the set of addition operations on a single channel forms a commutative monoid, an algebraic structure that is directly mirrored in the physical protocols. The category RNS can be understood as an algebraic specification, where objects are sorts and morphisms are operations, enabling formal reasoning about computational behavior (Goguen & Burstall, 1984). The initial algebra of this specification corresponds to the canonical representation of integers in RNS, providing a formal denotational semantics for the computational model. More broadly, RNS is equivalent to the Lawvere theory for commutative rings with specified moduli, which offers a universal framework for all equational reasoning about RNS computations.

B. Defining the category of topological phases

The target category, representing the physical system, is the category of topological phases, TopPh. The objects in TopPh are gapped quantum phases, specifically the ground-state sectors of FCI Hamiltonians that are engineered to have prime-bounded Chern numbers. The morphisms in TopPh are the physical processes that preserve the topological nature of these states. These processes are adiabatic evolutions, such as Thouless pumping, which can cyclically shift the Chern number of a state by an integer value without closing the protective spectral gap. These adiabatic protocols serve as the physical analogues of the modular arithmetic operations in RNS, creating a direct correspondence between abstract computation and physical dynamics. The change in the Chern number during such a process is governed by the geometric Berry phase accumulated over the pumping cycle. The spectral flow theorem guarantees that the net change in the Chern number is an integer equal to the winding number of the pumping cycle, ensuring quantized and deterministic operations. The adiabatic theorem, conditioned on a persistent spectral gap, ensures that the system remains in its instantaneous ground state throughout the protocol, thus preserving the integrity of the encoded information. This category can be situated within the broader mathematical framework of (2+1)D topological quantum field theory (TQFT), where objects represent spatial slices and morphisms represent spacetime evolutions, providing a deeper formal grounding for the physical processes.

C. Construction of the functor $\mathcal{F}: \textbf{RNS} \to \textbf{TopPh}$

The formal bridge between the computational and physical domains is an explicit, structure-preserving functor, $\mathcal{F}: \textbf{RNS} \to \textbf{TopPh}$. This functor is constructed by defining its action on both objects and morphisms. On objects, $\mathcal{F}$ maps each residue ring $\mathbb{Z}/p_i\mathbb{Z}$ in RNS to a corresponding physical subsystem in TopPh: an FCI engineered to have a ground-state manifold supporting Chern numbers in the set $\{0, \dots, p_i-1\}$. On morphisms, $\mathcal{F}$ maps each arithmetic operation in RNS to a specific unitary evolution $U_f$ in TopPh, which is generated by an adiabatic Hamiltonian protocol (e.g., Thouless pumping) that implements the corresponding operation on the Chern number. By construction, the functor $\mathcal{F}$ preserves identities (a static Hamiltonian corresponds to the identity morphism) and composition (concatenated adiabatic protocols correspond to the composition of morphisms). This ensures that $\mathcal{F}$ is a well-defined functor, which provides a formal guarantee that the physical system will correctly implement any computation specified in RNS. This functorial framework provides a language for discussing the equivalence of different physical implementations as distinct but isomorphic realizations of the same abstract computational structure. The universal property of the product ring in RNS ensures that any other physical system implementing RNS must factor through this functor, establishing its canonical nature. The functor is both faithful (injective on morphisms) and full (surjective on morphisms between images), establishing a categorical equivalence between the computational model and its physical realization. This equivalence serves as a formal proof of correctness, as any theorem in RNS has a direct physical counterpart in TopPh.

**IV. The Vectorial Solution: Architecture and Measurement Protocol**

A. The architecture: A sea of topological ALUs

The theoretical framework is embodied in a scalable, massively parallel hardware architecture. The fundamental building block is the Topological Arithmetic Logic Unit (T-ALU), a self-contained arithmetic engine composed of a small cluster of Physical Modulus Units (PMUs). Each PMU is a localized patch of a moiré superlattice engineered to realize a specific prime modulus, for example, a set of PMUs for moduli {3, 5, 7, 11, 13}. The complete processor is envisioned not as a small number of complex cores but as a vast, uniform “sea” of millions of replicated T-ALUs fabricated on a single wafer-scale moiré superlattice. This architecture provides immense parallelism and is a direct physical instantiation of the product structure of the RNS category, with each PMU corresponding to an independent factor in the product ring. This design represents a deep form of hardware-software co-design, where the software is expressed in RNS and the hardware natively operates in RNS, eliminating the typical impedance mismatch between logic and substrate. The spatial modularity of the design confers a high degree of fault tolerance; a local defect in one T-ALU does not propagate to its neighbors, which are topologically isolated. The uniformity of the fabric is compatible with existing 2D material stacking and fabrication techniques, suggesting a viable path toward manufacturable topological processors. The architecture is also inherently 3D-stackable, allowing for vertical integration to support more complex dataflow patterns while preserving modularity.

B. The operational protocol: Computation-by-relaxation

Arithmetic operations within the T-ALU fabric are executed via physical processes governed by dissipative dynamics. For addition, an initial state is prepared, a calibrated amount of charge corresponding to the addend is injected into the appropriate PMU channels, and the system is allowed to relax to its new, stable ground state. For multiplication, the system is prepared in an initial state, and a driven, cyclic deformation of the moiré potential (Thouless pumping) is applied for a number of cycles equal to the multiplier, which deterministically shifts the Chern number by the desired amount. In both cases, the computation is complete when the system reaches a final, topologically stable configuration whose Chern vector directly encodes the result. The correctness of these protocols is guaranteed by the adiabatic theorem, which ensures the system remains in its instantaneous ground state throughout the evolution, thus preserving the topological invariant. More complex operations, such as modular exponentiation, can be constructed by composing sequences of these fundamental charge injection and pumping protocols, with the functorial preservation of composition guaranteeing the correctness of the composite operation. The entire computational process is dissipative yet deterministic, as the environment acts as a heat bath that drives the system toward the unique ground state that encodes the answer. The timescale of the computation is set by the system’s relaxation time, which can be tuned via the spectral gap and coupling to the environment.

C. The measurement protocol: Direct readout via multi-terminal conductance

The computational result is read out via a direct, non-invasive measurement protocol that requires no subsequent processing. In a multi-terminal quantum Hall device, the Hall conductance $G_{xy}^{(i)}$ of each independent FCI channel $i$ is quantized in fundamental units of $e^2/h$, such that $G_{xy}^{(i)} = C_i \frac{e^2}{h}$, where $C_i$ is the integer Chern number of that channel. By engineering the T-ALU with a set of separate electrical contacts for each PMU, the conductances of all channels, $\{G_{xy}^{(1)}, \dots, G_{xy}^{(k)}\}$, can be measured simultaneously and independently. This set of measured conductance values directly yields the Chern vector $(C_1, \dots, C_k)$, which is the final computational output in its natural RNS vector form. No mathematical reconstruction is performed; the measured vector is the answer. The quantization of conductance is a direct consequence of the Kubo formula for linear response theory in gapped systems, providing a rigorous theoretical basis for the measurement protocol. Advanced contact engineering techniques can ensure that crosstalk between channels is minimized, preserving the independence of each residue measurement. This measurement is non-local and topological, as it depends only on the global invariant of the bulk material and is therefore robust against local imperfections such as contact resistance and interface disorder. The precision of this measurement can reach the level of quantum metrology, with relative uncertainties far below what is required for distinguishing integer values, enabling high-confidence readout.

**V. Experimental Feasibility and Validation**

A. Material platform and physical parameters

The proposed framework is grounded in current experimental capabilities in condensed matter physics. Recent experiments have successfully demonstrated the creation and control of FCIs with tunable properties in moiré heterostructures such as WSe₂/WS₂, confirming the physical realizability of the proposed material substrate. A practical T-ALU constructed with a modest set of small prime moduli (e.g., {3, 5, 7, 11, 13}) would have a dynamic range of $M = 15,015$, which is sufficient for a range of applications, including use as a specialized co-processor for cryptography or digital signal processing. The measurement of quantized Hall conductance is a standard and high-fidelity technique in mesoscopic physics, capable of resolving integer Chern numbers with precision. Current fabrication techniques for creating moiré materials allow for the necessary control over twist angle and uniformity to engineer the required prime-bounded Chern sectors. The required operational temperatures are below $1$ K, which is consistent with existing cryogenic standards for quantum devices. The spectral gap in these materials provides robust protection against thermal noise, with measured activation gaps on the order of ~3.5 K (Xie et al., 2021). Furthermore, electrostatic gating provides a mechanism for in-situ tuning of the FCI properties, allowing for dynamic reconfiguration of a T-ALU for different operational modes or modulus sets.

B. Addressing the reconstruction paradox in practice

The vectorial approach of this framework offers a practical solution to the historical limitations of RNS. Conventional RNS implementations are fundamentally bottlenecked by the need for a software-based CRT reconstruction step to convert the parallel residue vector into a serial scalar integer. This work eliminates that step entirely by designing the hardware to output the vector of residues directly as a physical observable. The computation is considered complete once the physical system has relaxed and the vector of conductances has been measured. The role of any external classical computer is merely to interpret this vector of integers as the final result, not to perform a complex reconstruction algorithm. This represents a form of hardware-software co-design where the hardware is engineered to produce data in a format that is immediately useful, dissolving the traditional boundary between the processor and memory. By eliminating the reconstruction step, the architecture achieves substantial gains in both energy efficiency and latency. The vector output is also natively compatible with subsequent RNS-based operations, enabling deep computational pipelines that never need to leave the residue domain. For the limited set of applications that do require a final scalar output, such as displaying a number to a user, the reconstruction can be deferred to the very end of a long computational chain, thus minimizing its amortized cost. In many cryptographic applications, the final result is used modulo another number, a calculation which can be performed directly in the RNS domain without any reconstruction.

C. Pathway to a functional prototype

A concrete, incremental research and development pathway can be outlined for realizing this architecture. The first experimental milestone would be the demonstration of a single, prime-bounded PMU (e.g., for modulus $p=3$) in which the Chern number can be reliably set by a physical protocol and read out via quantized conductance. The second milestone would be the integration of two such PMUs into a single device with independent contacts, demonstrating parallel, independent control and readout, thereby validating the core architectural principle. A functional prototype would consist of a small-scale T-ALU (e.g., 3–5 PMUs) integrated into a multi-terminal measurement setup. This prototype would be benchmarked against a conventional processor on a specific, highly parallelizable task, such as large-integer modular exponentiation. A critical component of this validation would be a detailed error characterization study to quantify the fidelity of the computation in the presence of non-adiabaticity, material disorder, and thermal fluctuations. The prototype would be interfaced with a classical FPGA-based controller responsible for managing the adiabatic protocols and interpreting the measured vector output, forming a hybrid classical-topological computing system. Success would be measured not only by computational correctness but also by figures of merit such as the energy-delay product and scalability metrics, which must demonstrate a clear advantage over state-of-the-art RNS accelerators. The ultimate figure of merit is the end-to-end energy per operation for a complete application kernel, which must show a significant improvement over existing accelerators to justify the cryogenic overhead.

**Appendix A: Formal Verification of the Functorial Framework**

The formal correctness of the proposed framework is established through the following derivation.

1. Chinese Remainder Theorem: For a set of pairwise coprime integers $p_1, \dots, p_k$, the map $\phi: N \mapsto (N \bmod p_1, \dots, N \bmod p_k)$ defines a ring isomorphism $\phi: \mathbb{Z}/M\mathbb{Z} \to \prod_{i=1}^k \mathbb{Z}/p_i\mathbb{Z}$, where $M = \prod_i p_i$.

1. Category RNS: The category RNS is defined. Its objects are the rings $\mathbb{Z}/p_i\mathbb{Z}$. Its morphisms are ring homomorphisms $f: \mathbb{Z}/p_i\mathbb{Z} \to \mathbb{Z}/p_i\mathbb{Z}$ induced by arithmetic operations (e.g., $f(x) = x + a \bmod p_i$).

1. Category TopPh: The category TopPh is defined. Its objects are pairs $(\mathcal{H}_i, H_i)$, where $\mathcal{H}_i$ is the ground-state Hilbert space of an FCI with Chern numbers in $\{0, \dots, p_i-1\}$, and $H_i$ is its Hamiltonian. Its morphisms are unitary operators $U_f$ generated by adiabatic protocols, given by:

1. Functor Construction (Objects): The functor $\mathcal{F}: \textbf{RNS} \to \textbf{TopPh}$ is constructed on objects by the mapping $\mathcal{F}(\mathbb{Z}/p_i\mathbb{Z}) = (\mathcal{H}_i, H_i)$.

1. Functor Construction (Morphisms): On morphisms, for an arithmetic operation $f(x) = x + a \bmod p_i$, the functor maps it to a unitary evolution $\mathcal{F}(f) = U_f$, where $U_f$ is generated by a Thouless pumping protocol that shifts the Chern number by $a$.

1. Identity Preservation: The functor preserves identities. The identity morphism in RNS is $f(x)=x$, which corresponds to a shift of $a=0$. The corresponding physical protocol is a static Hamiltonian, which generates the identity unitary $\mathbb{I}_{\mathcal{H}_i}$. Thus, $\mathcal{F}(\mathrm{id}_{\mathbb{Z}/p_i\mathbb{Z}}) = \mathbb{I}_{\mathcal{H}_i}$.

1. Composition Preservation: The functor preserves composition. For two morphisms $f$ and $g$, the physical protocol for $g \circ f$ is the temporal concatenation of the individual protocols. The resulting unitary evolution is the product of the individual unitaries, $U_g U_f$. Therefore, $\mathcal{F}(g \circ f) = U_g U_f = \mathcal{F}(g) \circ \mathcal{F}(f)$.

1. Conclusion: Since $\mathcal{F}$ preserves both identities and composition, it is a well-defined, structure-preserving functor. This provides a formal guarantee that the physical system correctly implements the algebra of RNS computations.

1. Universality: The framework is general. Any other physical system that realizes the same algebraic structure (e.g., a photonic lattice) would be related to this moiré implementation by a natural isomorphism, demonstrating the universality of the categorical approach.

1. Extension to Multiplication: Ring homomorphisms for multiplication (e.g., $f_b(x) = b x \bmod p_i$) can be implemented via sequences of additions or other non-linear adiabatic protocols, ensuring the full ring structure is preserved under $\mathcal{F}$.

1. Faithfulness: The functor is faithful. If $\mathcal{F}(f) = \mathcal{F}(g)$, their corresponding physical effects are identical. Since distinct arithmetic operations produce distinct and measurable shifts in the Chern number, this implies $f = g$.

1. Fullness: The functor is full. Every adiabatic protocol that shifts the Chern number by an integer $a$ corresponds to the morphism $f_a(x) = x + a \bmod p_i$ in RNS.

1. Categorical Equivalence: Because the functor $\mathcal{F}$ is full, faithful, and essentially surjective onto the relevant subcategory of TopPh, it establishes an equivalence of categories between RNS and its physical realization.

**Appendix B: Verification of the Vectorial Encoding Scheme**

The validity of the vectorial encoding and measurement scheme is established as follows.

1. Bijective Mapping (CRT): From the Chinese Remainder Theorem, the map $\phi: N \mapsto (r_1, \dots, r_k)$, where $r_i = N \bmod p_i$, is a bijection from the set of integers $\{0, \dots, M-1\}$ to the product ring $\prod_{i=1}^k \mathbb{Z}/p_i\mathbb{Z}$.

1. Physical State Representation: In the proposed physical system, the computational state is characterized by the Chern vector $\mathbf{C} = (C_1, \dots, C_k)$, where each $C_i$ is an integer in the range $\{0, \dots, p_i-1\}$.

1. Encoding Correspondence: By the construction of the functorial framework (Section III.C) and the operational protocols (Section IV.B), the physical encoding ensures a direct correspondence $C_i = r_i$ for all channels $i$. Therefore, the map from an integer to its physical representation, $N \mapsto \mathbf{C}$, is also a bijection.

1. Unambiguous Representation: A bijective mapping means that the Chern vector $\mathbf{C}$ is a complete and unambiguous representation of the integer $N$. There is a one-to-one correspondence, ensuring no information is lost. This obviates any need for algorithmic reconstruction.

1. Direct Measurement: The multi-terminal measurement protocol (Section IV.C) directly yields the integer components of the Chern vector, $\{C_1, \dots, C_k\}$, via a set of independent, quantized Hall conductance measurements.

1. Conclusion: The final output of the physical system is the vector $\mathbf{C}$, which is demonstrably equivalent to the RNS representation of $N$. The computational problem is solved entirely by the physical process, and the result is directly readable from the hardware.

1. Edge Case Analysis: The encoding is valid across the entire dynamic range. For $N = 0$, the system is in the vacuum state with $\mathbf{C} = (0, \dots, 0)$. For $N = M-1$, the system is in the state $\mathbf{C} = (p_1-1, \dots, p_k-1)$. Both are valid and stable ground states.

1. Robustness and Error Detection: Any error that alters a single component $C_i$ results in a vector that does not correspond to a valid integer within the intended computational range, enabling immediate error detection.

1. Error Correction Potential: By choosing moduli such that the total dynamic range $M$ exceeds the required range for a given problem, the redundant space can be used to encode error-correcting information, enabling the correction of single-channel errors.

1. Formal Measurement Map: The measurement process can be formalized as a linear isomorphism $\mathcal{M}: \mathbf{C} \mapsto \{G_{xy}^{(i)}\}$, which maps the Chern vector space to the conductance vector space, completing the formal chain from abstract integer to physical observable.

**References**

Cheon, J. H., Kim, A., Kim, M., & Song, Y. (2017). Homomorphic Encryption for Arithmetic of Approximate Numbers. In Advances in Cryptology–ASIACRYPT 2017 (pp. 409-437). Springer. https://doi.org/10.1007/978-3-319-70697-9_15

Goguen, J. A., & Burstall, R. M. (1984). Some fundamental algebraic tools for the semantics of computation. Part 1: Compositions and refinements. Theoretical Computer Science, 31, 175-213. https://doi.org/10.1016/0304-3975(84)90004-0

Horowitz, M. (2014). Computing’s Energy Problem (and what we can do about it). In IEEE International Solid-State Circuits Conference Digest of Technical Papers (ISSCC) (pp. 10-14). https://doi.org/10.1109/ISSCC.2014.6757323

Horsman, C., Kendon, V., Stepney, S., & Munro, W. J. (2014). When does a physical system compute? Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 470(2171), 20140152. https://doi.org/10.1098/rspa.2014.0152

Kitaev, A. Y. (2003). Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1), 2-30. https://doi.org/10.1016/S0003-4916(02)00018-0

Lloyd, S., & Pagels, H. (1988). Complexity as Thermodynamic Depth. Annals of Physics, 188(1), 186-213. https://doi.org/10.1016/0003-4916(88)90094-2

MacLennan, B. J. (2004). Natural Computation and Non-Turing Models of Computation. Theoretical Computer Science, 317(1-3), 115-145. https://doi.org/10.1016/j.tcs.2003.12.006

Müller, V. C., & Hoffmann, M. (2017). What Is Morphological Computation? On How the Body Contributes to Cognition and Control. Artificial Life, 23(1), 1-24. https://doi.org/10.1162/ARTL_a_00219

Nayak, C., Simon, S. H., Stern, A., Freedman, M., & Das Sarma, S. (2008). Non-Abelian anyons and topological quantum computation. Reviews of Modern Physics, 80(3), 1083-1159. https://doi.org/10.1103/RevModPhys.80.1083

Pfeifer, R., & Bongard, J. (2007). How the Body Shapes the Way We Think: A New View of Intelligence. MIT Press.

Prigogine, I. (1967). Structure, Dissipation and Life. In Theoretical Physics and Biology (pp. 23-52). North Holland.

Sharpe, A. L., Fox, E. J., Barnard, A. W., Finney, J., Watanabe, K., Taniguchi, T., Kastner, M. A., & Goldhaber-Gordon, D. (2019). Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science, 365(6453), 605-608. https://doi.org/10.1126/science.aaw3780

Siegelmann, H. T. (1999). Neural and Super-Turing Computing. Minds and Machines, 13, 103-114. https://doi.org/10.1023/A:1022902524528

Szabo, N. S., & Tanaka, R. I. (1967). Residue Arithmetic and Its Applications to Computer Technology. McGraw-Hill.

Wen, X.-G. (2002). Quantum orders in an exact soluble model. Physical Review B, 65(16), 165113. https://doi.org/10.1103/PhysRevB.65.165113

Xie, Y., Pierce, A. T., Park, J. M., Parker, D. E., Khalaf, E., Ledwith, P., Lee, Y., Chen, S., Călugăru, D., Andrei, B. A., & Yacoby, A. (2021). Fractional Chern insulators in magic-angle twisted bilayer graphene. Nature, 597(7874), 38–43. https://doi.org/10.1038/s41586-021-03842-4