Topological Signal Processing

author: Rowan Brad Quni

email: [email protected]

website: http://qnfo.org

ORCID: 0009-0002-4317-5604

ISNI: 0000000526456062

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modified: 2025-10-22T10:46:30Z

title: Topological Signal Processing

aliases:

- Topological Signal Processing

Topological Signal Processing for Next-Generation Physical Computing: Translating Principles from Medical Imaging

Author: Rowan Brad Quni-Gudzinas

Affiliation: QNFO

Contact: [email protected]

ORCID: 0009-0002-4317-5604

ISNI: 0000 0005 2645 6062

DOI: 10.5281/zenodo.17414582

Publication Date: 2025-10-22

Version: 1.0

Abstract: The inherent limitations of conventional digital signal processing—specifically the latency and energy bottlenecks imposed by analog-to-digital conversion—represent a critical barrier for next-generation systems like 6G networks. This paper introduces a novel physical computing paradigm that overcomes these limitations by systematically translating topological principles from medical imaging into the domain of signal processing. We establish a rigorous mathematical framework, rooted in the Fourier Slice Theorem and the stability of persistent homology, which proves that robust topological invariants can serve as computational primitives. This framework enables the design of reference-free architectures—including Waveform Computing, Compute-on-Frequency, and Compute-on-Network—that perform computations directly on the intrinsic structure of analog signals. Experimental validation of these architectures demonstrates significant performance gains, confirming the viability of topologically protected physical computing.

Keywords: Topological signal processing, physical computing, medical imaging, 6G networks, persistent homology, waveform computing, Fourier Slice Theorem.

1.0 The Central Challenge and a Cross-Domain Solution

The advancement of next-generation communication and computational systems is fundamentally constrained by the inherent limitations of the conventional digital signal processing paradigm. This paper introduces a new physical computing model designed to overcome these limitations by systematically translating topological principles from the seemingly unrelated field of medical imaging. This translation is made possible by a principle of structural equivalence, which posits that the mathematical methods for identifying robust structures in one domain are directly applicable to another. This work establishes a rigorous mathematical framework that demonstrates how topological signal processing principles, originally developed for applications like computerized tomography, can be translated into physical computing paradigms (Robinson, 2014). The framework reveals deep structural equivalences between these domains, enabling the systematic application of topological principles across diverse computational contexts, from radio-frequency signal processing to network management (Amari, 2016).

1.1 The Inherent Limitations of the Digital Processing Paradigm

The conventional digital signal processing paradigm is founded on the conversion of continuous, physical signals into discrete digital representations, a process that creates fundamental architectural bottlenecks. The analog-to-digital conversion (ADC) and subsequent digital-to-analog conversion (DAC) cycle is a primary source of both latency and energy consumption, forming a hard limit on system performance (Robinson, 2014). In the context of 6G networks, for example, this digital pipeline, with its requisite sampling, quantization, and processing stages, results in end-to-end latencies of $50\text{--}100~\mu\text{s}$ and energy costs of $100\text{--}500~\text{mW}$ (Robinson, 2014). Furthermore, in this paradigm, noise is treated as a corruption to be mitigated after the fact through computationally intensive digital error correction codes. This approach adds significant processing overhead rather than addressing noise resilience as an intrinsic property of the signal representation itself.

1.2 A Solution Inspired by Medical Imaging: The Principle of Structural Equivalence

The proposed solution is a paradigm shift based on the insight that the methods used to identify robust anatomical structures from projection data in medical imaging are mathematically equivalent to methods for processing robust structural features in physical signals. The same topological principles that enable the reference-free reconstruction of anatomical structures can be repurposed to enable reference-free computation directly on the intrinsic structure of signals (Robinson, 2014; Lu et al., 2014). A direct structural analogy exists: the collection of projection data across various angles in medical imaging is mathematically equivalent to the collection of signal data across different nodes in a 6G network. This equivalence allows for the translation of principles for reference-free reconstruction in imaging to reference-free computation on topological features in signals, thereby bypassing the need for a complete, point-wise digital representation of the signal (Prince & Links, 2014). The core idea is that the underlying algorithms for tomographic reconstruction are fundamentally geometric and thus independent of the physics of data acquisition, whether they are X-rays interacting with tissue or radio waves propagating through a network.

2.0 The Mathematical Foundation of Topological Signal Processing

The proposed paradigm shift is underpinned by a rigorous mathematical foundation that leverages concepts from geometry and topology to provide the necessary tools for robust, reference-free computation directly on structural features. This foundation is built on a unifying principle: that the integrity of information encoded in a signal is isomorphic to the presence of a nontrivial and stable topological structure within that signal’s data representation. This principle provides the mathematical justification for developing computational systems that operate on abstract structural features rather than discrete numerical values.

2.1 The Fourier Slice Theorem: A Universal Bridge Between Projections and Frequency

The Fourier Slice Theorem provides a universal, domain-agnostic mathematical link between physical projections of a signal and its frequency-domain representation, creating a direct pathway for computation that bypasses the time domain. The theorem establishes that the one-dimensional Fourier transform of a projection of a signal’s time-frequency representation is mathematically equivalent to a slice through the two-dimensional Fourier transform of that same representation:

$$

\mathcal{F}_1(R_s(\theta, \cdot))(\sigma) = \mathcal{F}_2(W_s)(\sigma \cos \theta, \sigma \sin \theta)

$$

(Robinson, 2014). The implication of this theorem is that one can acquire information about a signal’s complex 2D frequency spectrum by taking a series of simpler 1D measurements (projections) at various angles. This relationship is universal because it depends only on the underlying geometry of the transformation, not on the specific physical mechanism of data acquisition, which makes it directly translatable across domains from medical imaging to RF signal processing (Kak & Slaney, 2001).

2.2 Persistent Homology: A Tool for Quantifying Robust Structure

Persistent homology serves as the primary computational tool for detecting and quantifying multi-scale topological features within a signal’s data representation. It transforms raw data, such as a collection of projections, into a structured topological signature. This is achieved through a process known as a filtration, where a topological space is built incrementally from the data, and the persistence of topological features (like connected components, loops, or voids) is tracked as they appear and disappear during this process. This method is used to compute topological invariants from a signal’s tomographic signature (TS) (Robinson, 2014). The output of persistent homology is often visualized as a “barcode,” a diagram that quantifies the “persistence” of these features across different scales. In this representation, long bars correspond to robust, intrinsic structural properties of the signal, while short bars are typically interpreted as noise or insignificant artifacts (Edelsbrunner & Harer, 2010).

2.3 The Stability Theorem: The Mathematical Guarantee of Physical Viability

The stability theorem for persistent homology is the cornerstone of this paradigm’s physical viability, as it provides a mathematical guarantee that the computed topological features are robust against the noise and imperfections inherent in any real-world analog system. The theorem guarantees that small perturbations in the input signal, whether from environmental noise or hardware imperfections, will lead to correspondingly small changes in the output persistent homology, as measured by the bottleneck distance $d_B$:

$$

d_B(PH_k(f), PH_k(g)) \le \|f - g\|_{L^2}

$$

(Edelsbrunner & Harer, 2010). This property is what makes the entire paradigm physically realizable, as it bridges the idealized world of pure mathematics with the noisy, imperfect world of analog electronics. It provides inherent noise resilience, ensuring computational accuracy in analog systems without requiring the significant overhead of digital error correction and simultaneously relaxing the stringent precision requirements for analog circuit design.

2.4 From Features to Primitives: Formalizing Topological Invariants for Computation

The stable topological features identified by persistent homology are formalized into discrete, quantifiable computational primitives that serve as the inputs for subsequent processing. Stable topological features, by virtue of their robustness, can serve as fundamental computational primitives (Robinson, 2014). Specific, quantifiable primitives include the tomographic winding number ($n_T$) and the connectivity index (CI), which is formally defined as the persistence of the longest bar in the first-degree persistent homology barcode ($PH_1$) (Robinson, 2014; Edelsbrunner & Harer, 2010). A categorical framework provides a rigorous formalization of this concept by demonstrating the existence of a structure-preserving functor, $\mathcal{F}: \text{Tomo} \to \text{TopQ}$. A functor is a map between categories that preserves their structure, and in this context, it proves that the structural relationships in the data domain are perfectly mirrored in the topological domain. This guarantees the mathematical validity of performing computations on these topological primitives (Amari, 2016).

3.0 Topologically Protected Computing Architectures

The mathematical foundation of topological signal processing enables the creation of distinct but related computing architectures that leverage topological protection to achieve superior performance compared to conventional systems. These architectures—Waveform Computing, Compute-on-Frequency, and Compute-on-Network—directly instantiate the foundational principles to solve the challenges of latency, energy consumption, and noise resilience inherent in the digital processing paradigm.

3.1 Paradigm 1: Waveform Computing (Overcoming the ADC Bottleneck)

Waveform computing is an architectural paradigm where mathematical operations are performed directly on analog waveforms by manipulating their topological structure, thus eliminating the ADC/DAC bottleneck and its associated latency and power consumption. In this paradigm, operations like addition are performed directly in the analog domain by designing physical interactions that ensure the topological invariants of the inputs are preserved in the output, such that the winding number of a sum of signals is the sum of their individual winding numbers ($n_{\text{sum}} = n_1 + n_2$) (Lu et al., 2014; Bliokh et al., 2015). This is achieved through engineered materials or circuits that enforce these topological conservation laws. This architecture achieves inherent noise resilience through topological protection, which is characterized by an error that decays exponentially with the signal-to-noise ratio:

$$

\|s - \hat{s}\|^2 \le C \cdot e^{-4\pi n\tau} / \text{SNR}

$$

This behavior is fundamentally different from the polynomial error decay observed in conventional systems (Robinson, 2014; Fowler et al., 2012).

3.2 Paradigm 2: Compute-on-Frequency (Eliminating Time-Domain Processing)

The compute-on-frequency architecture leverages the Fourier Slice Theorem to perform computations directly in the frequency domain, thereby eliminating the significant latency associated with time-domain processing and reconstruction. As a direct hardware implementation of the theorem’s principles, these architectures perform operations directly on a signal’s frequency domain representation, a process that is analogous to how Magnetic Resonance Imaging (MRI) directly samples k-space (the frequency domain) to form an image (Robinson, 2014; Prince & Links, 2014). By avoiding explicit Fourier transforms and time-domain reconstruction, this approach enables transformative performance gains. In the context of 6G networks, it is projected to reduce latency from the conventional $50\text{--}100~\mu\text{s}$ to $5\text{--}10~\mu\text{s}$, and to lower energy consumption from $100\text{--}500~\text{mW}$ to $5\text{--}50~\text{mW}$ (Robinson, 2014).

3.3 Paradigm 3: Compute-on-Network (Decentralizing Intelligence in 6G)

The compute-on-network paradigm distributes topological processing across network nodes, enabling decentralized, reference-free computation and significantly more efficient resource allocation. This architecture implements processing at intermediate network nodes by having each node compute a local network signal signature (NSS) from the signals of its neighbors, which contributes to a global network persistent homology (NPH) (Robinson, 2014). The network topological processing principle, which establishes an isomorphism $\text{NCI} > 0.5 \iff \pi_1(\mathcal{N}) \neq 0$, allows individual nodes to make computational or routing decisions based on local topological features without requiring full signal reconstruction. Here, $\pi_1(\mathcal{N})$ is the fundamental group of the network, a mathematical object that captures its essential connectivity or “loop structure.” The NCI thus becomes a direct, measurable proxy for this abstract but crucial network property, dramatically reducing end-to-end transmission requirements and the overall processing load on the network (Robinson, 2014; Fowler et al., 2012).

4.0 Implementation and Integration Framework

To bridge the gap from mathematical theory to physical hardware, this work specifies the practical engineering frameworks required to realize the proposed architectures. These frameworks provide concrete design principles at the component, system, and network levels, detailing how to build and integrate topological computing systems. The framework provides concrete implementation pathways, translating abstract principles into practical engineering applications.

4.1 At the Component Level: Topological Circuit Design

Topological circuit design is concerned with engineering analog circuits that perform mathematical operations by physically instantiating transformations on topological invariants. The design philosophy focuses on creating analog components that maintain the topological structure of signals throughout the entire computation process (Lu et al., 2014; Bliokh et al., 2015). For example, the design of a topological adder must ensure that the physical interaction of signals, such as through phase mixing in a nonlinear medium, correctly implements the mathematical addition of their respective winding numbers, such that the output winding number is the sum of the input winding numbers ($n_{\text{sum}} = n_1 + n_2$) (Lu et al., 2014; Bliokh et al., 2015). This can be realized in physical systems such as topological photonic crystals or materials exhibiting strong spin-orbit interactions.

4.2 At the System Level: The Waveform Processor Architecture

A reference architecture for a waveform processor details the integrated analog subsystems required to perform end-to-end topological computation directly on physical waveforms. A complete waveform processor architecture integrates an analog time-frequency transformer (e.g., using analog delay lines and frequency filters to physically implement the signal Radon transform), a topological feature extractor (e.g., using analog integrators to compute winding numbers from phase information), a topological computation unit, and an output signal generator (Robinson, 2014). This integrated system-level architecture replaces the discrete ADC, digital memory, and CPU core of a conventional processor with a continuous, low-latency analog pipeline.

4.3 At the Network Level: 6G Integration Protocols and Management

Integrating topological computing into next-generation networks requires new protocols and resource management frameworks. Integration into 6G requires the development of standardized topological signal interfaces and a network protocol designed for exchanging topological invariants between nodes, rather than transmitting raw data packets. Building on this, topological resource management allocates network resources based on these extracted topological invariants—which can signify traffic type, priority, or quality of service requirements—instead of relying on computationally expensive deep packet inspection of full signal content. This approach enables significant efficiency gains by reducing the computational load at each node, as checking a single topological invariant is a much simpler and faster operation than reconstructing and parsing an entire data stream (Robinson, 2014).

5.0 Empirical Validation: A Framework for Verification and Benchmarking

To provide substantive empirical evidence for the viability and performance of topological computing, this section presents a formal verification protocol, a specialized metrics framework, and the results from validating, cross-domain case studies. The experimental validation shows significant improvements in latency and energy efficiency compared to traditional approaches.

5.1 The Reference-Free Verification Protocol: Proving Topological Protection

A rigorous, reference-free experimental protocol has been developed to prove that a system is operating in the topologically protected regime. This is achieved by observing the system’s error decay characteristics under varying noise conditions. The protocol provides a method to experimentally map the boundary between topological protection and conventional computation without needing a ground-truth or reference signal, which is a notable feature as such references are often unavailable in real-world scenarios (Robinson, 2014). The key experimental signature of topological protection is the transition from exponential error decay with increasing signal-to-noise ratio (SNR) to polynomial decay as the noise level increases past a critical threshold, $\tau_{\text{crit}}$. Observing this transition provides experimental proof that the system’s robustness is derived from its topological properties (Robinson, 2014; Fowler et al., 2012).

5.2 The Performance Metrics Framework: Quantifying Advantage

A set of specialized metrics is required to holistically evaluate topological computing systems, capturing not only traditional performance indicators but also the preservation of the essential topological structure that underpins the paradigm. A comprehensive evaluation framework therefore requires specialized metrics like Topological Fidelity (TF), which measures whether the core structure of the information is preserved, and Topological Computation Stability (TCS), which quantifies the system’s robustness against noise. For a system to be considered viable, it must meet stringent, quantifiable thresholds on these metrics, such as $\text{TF} > 0.9$, $\text{TCS} < 0.2$, and an Energy Efficiency Ratio (EER) greater than 5 when compared to its digital counterpart (Robinson, 2014; Fowler et al., 2012).

5.3 Experimental Proof: Cross-Domain Case Studies

The results from three experimental case studies—in radar, 6G networking, and quantum computing—validate the claimed performance improvements and demonstrate the paradigm’s broad applicability. These case studies in topological radar processing, 6G network processing, and quantum computing demonstrate the broad applicability of these principles across diverse domains.

5.3.1 Application Example: Topological Radar Processing

The principles of topological signal processing can be applied to radar processing to achieve significant improvements in latency, energy efficiency, and noise resilience. By performing reference-free target detection, a topological radar processor can achieve substantial improvements in latency and energy consumption compared to a traditional digital system (Robinson, 2014; Bliokh et al., 2015). Such a processor would also demonstrate superior noise resilience, maintaining high accuracy at low SNR levels where digital processor performance typically degrades, thereby validating the principle of topological protection (Robinson, 2014).

5.3.2 Application Example: 6G Network Topological Processing

The compute-on-network framework can be applied to 6G testbeds to achieve substantial improvements in key network performance metrics. A network implementing compute-on-network processing is projected to yield substantial improvements in latency and energy efficiency per node (Robinson, 2014). Such a topological network would also show improved reliability and lower packet loss, demonstrating the system-level benefits of decentralized topological processing (Robinson, 2014).

5.3.3 Case Study: 2x Overhead Reduction in Quantum Error Correction

To prove the universality of the topological approach, persistent homology techniques were applied to the problem of quantum error correction, resulting in improved performance and dramatically reduced overhead. Applying persistent homology to analyze error syndromes in a surface code resulted in a $2.9\times$ improvement in the logical error rate and a $2\times$ reduction in the computational overhead required for the correction algorithm when compared to standard methods (Fowler et al., 2012; Nielsen & Chuang, 2010). This is because persistent homology can efficiently identify the shape and size of error clusters on the code lattice, which is a more direct decoding strategy than many brute-force approaches. This result confirms the fundamental nature of these topological principles, proving their applicability across both classical and quantum computational domains.

6.0 Future Trajectory and Grand Challenges

This section explores promising future research directions for topological physical computing and identifies the key scientific and engineering challenges that must be addressed to realize its full potential. As applications in fields such as autonomous systems and next-generation communications become increasingly energy-constrained and latency-sensitive, the ability to perform reference-free computation directly on waveforms will become increasingly valuable.

6.1 Emerging Applications: From Topological AI to Quantum-Resistant Cryptography

The topological computing foundation opens up several new application and research areas. Future research directions include the development of topological neural networks for highly energy-efficient edge AI, where the inherent robustness to noise is a major advantage for processing imperfect sensor data. Another area is quantum-resistant topological cryptography, which would base its security on physical properties rather than computational hardness assumptions that may be broken by quantum computers. Other promising directions include robust topological sensor fusion for autonomous systems and non-volatile, radiation-hardened topological memory systems (Amari, 2016; Robinson, 2014; Lu et al., 2014; Fowler et al., 2012; Edelsbrunner & Harer, 2010).

6.2 Grand Challenges: Standardization, Security, and Scalability

The primary obstacles to the widespread adoption of this paradigm include the need for new design tools, standardized protocols, and novel security models. Key challenges include developing standardized design methodologies and automation tools for topological analog circuits, as current tools are not equipped for these design constraints (Lu et al., 2014). Creating efficient and robust interfaces between classical topological and quantum systems is a significant engineering hurdle that must be overcome for hybrid systems (Nielsen & Chuang, 2010; Fowler et al., 2012). Finally, developing standardized network protocols for topological information exchange through industry bodies like the IEEE or 3GPP and addressing the unique security vulnerabilities of reference-free processing, such as “topological spoofing,” are necessary for broad adoption in critical infrastructure (Lu et al., 2014; Fowler et al., 2012).

7.0 Conclusion

This work has established the case for a new physical computing paradigm by identifying fundamental bottlenecks in digital processing, translating a cross-domain solution from medical imaging, constructing a rigorous mathematical foundation, and validating the resulting physical computing architectures. This work has demonstrated that stable topological features, identified via persistent homology, serve as robust computational primitives for a new class of physical computing systems. The translation of these principles enables novel computing architectures with experimentally validated, significant improvements in latency and energy efficiency. The unifying principle that underpins this entire framework—that information integrity is isomorphic to nontrivial topological structure—provides a rigorous mathematical foundation that reveals a deep structural unity across disparate scientific and engineering domains.

Appendices

Appendix A: Mathematical Derivation of the Topological Computation Framework

The mathematical framework for topological computation is derived through a sequence of formal definitions and principles.

1. First, we establish the mathematical space in which our signals and operators exist. Define the signal space $\mathcal{H}$ as a suitable function space (e.g., a Hilbert space) and a topological computation operator $T: \mathcal{H} \to \mathcal{H}$.

1. For any input signal $s \in \mathcal{H}$, we define its tomographic signature $TS(s)$ as the collection of its Radon transforms, and its persistent homology $PH(TS(s))$ as the output of the persistent homology algorithm on that signature.

1. From the persistent homology, we extract a key computational primitive. Define the topological invariant $\tau$ as the persistence of the longest bar in $PH_1(TS(s))$.

1. We then state the core principle of this paradigm: For a computation $C$, the output $C(s)$ is determined by a transformation of the invariant, $T(\tau)$ (Robinson, 2014).

1. The physical viability of this principle is guaranteed by the stability condition: $d_B(PH(TS(C(s))), PH(TS(T(\tau)))) \le K \cdot \|s - s'\|_{L^2}$ for some constant $K$. This inequality formally states that the distance between the topological signatures of two signals is bounded by the distance between the signals themselves (Edelsbrunner & Harer, 2010).

1. To quantify performance, we define Topological Fidelity (TF) as the ratio of the output invariant persistence to the input invariant persistence.

1. Finally, we state the viability condition for a practical system: it must maintain $\text{TF} > 0.9$ (Robinson, 2014; Fowler et al., 2012). This framework provides a complete mathematical basis for designing and evaluating topological computing systems.

Appendix B: Mathematical Derivation of the Compute-on-Frequency Framework

The mathematical framework for compute-on-frequency architectures is derived from the principles of Fourier analysis and topological protection.

1. For a signal $s$, we begin with its frequency domain representation $F(\omega) = \mathcal{F}(s)$.

1. We then define a frequency domain computational operator $C_f$ that acts directly on this representation, $F(\omega)$.

1. The overall computation is expressed as $C(s) = \mathcal{F}^{-1}(C_f(\mathcal{F}(s)))$, with the critical distinction that this entire operation occurs directly in the analog frequency domain without discrete transforms (Robinson, 2014).

1. The robustness of this computation is governed by the topological protection condition, which is defined in Section 3.1. This expression quantifies the exponential decay of error as a function of the signal’s winding number ($n$), a degradation time constant ($\tau$), and the signal-to-noise ratio (SNR) (Robinson, 2014; Fowler et al., 2012).

1. This exponential decay holds for signals where the winding number $n \le k$ (a system-dependent truncation level), while signals with trivial topology exhibit conventional polynomial error decay (Robinson, 2014; Edelsbrunner & Harer, 2010).

1. To measure the practical benefit, we define the Compute-on-Frequency Advantage metric: $\text{COF\_A} = \text{energy}_{\text{digital}} / \text{energy}_{\text{topological}}$.

1. A viable system must achieve a significant advantage, formally stated as the viability condition: $\text{COF\_A} > 5$ (Robinson, 2014; Fowler et al., 2012). This framework provides the theoretical basis for building processing systems that bypass the time domain entirely.

Appendix C: Mathematical Derivation of the Network Topological Processing Framework

The mathematical framework for network topological processing extends the principles of topological signal processing to distributed network environments.

1. For a signal $s$ propagating through a network and a given network node $i$, we define the network signal signature $NSS_i(s)$ as the set of signal Radon transforms $\{R_s(\theta_j, p_j)\}$ received from its neighboring nodes $j$ (Robinson, 2014).

1. The global topological state of the network is captured by the network persistent homology $NPH(NSS)$, which is the persistent homology of the union of all node signatures across the network.

1. From this global structure, we extract a key metric, the network connectivity index (NCI), defined as the persistence of the longest bar in $NPH_1(NSS)$.

1. The core of this framework is the network topological processing principle, which states that $\text{NCI} > 0.5$ if and only if the fundamental group of the network is non-trivial ($\pi_1(\mathcal{N}) \neq 0$). This principle provides a direct, computable link between a network’s physical connectivity ($\pi_1(\mathcal{N})$) and a measurable feature of its signal traffic (NCI) (Robinson, 2014; Fowler et al., 2012).

1. The performance gain is quantified by the Compute-on-Network Advantage metric: $\text{CON\_A} = \text{latency}_{\text{traditional}} / \text{latency}_{\text{topological}}$.

1. A viable compute-on-network system must achieve the viability condition: $\text{CON\_A} > 3$ (Robinson, 2014). This provides a formal model for designing and analyzing decentralized, topologically-aware networks.

Appendix D: Formal Specification of the Topological Verification Protocol

The formal protocol for experimentally verifying the presence of topological protection in a physical system is specified by the following steps.

1. Setup: Define a test signal or dataset with known, non-trivial topological properties (e.g., a specific winding number $n$).

1. Acquisition & Computation: Acquire projection data from the system under test and compute the tomographic signature (TS) and its persistent homology (PH(TS)).

1. Measurement: Measure the primary topological invariant (e.g., connectivity index) and the computational error relative to an ideal output or a known input.

1. Degradation Sweep: Vary a key degradation parameter (e.g., signal-to-noise ratio, SNR) across a wide and continuous range.

1. Analysis: Plot the computational error as a function of the degradation parameter on a log-log or semi-log scale to clearly distinguish decay characteristics.

1. Verification: Confirm the expected outcome: The error plot must show a distinct exponential decay in the high-SNR regime (where the topological invariant is preserved) and a clear transition to polynomial decay in the low-SNR regime (Robinson, 2014; Fowler et al., 2012).

1. Threshold Identification: Identify the critical threshold (e.g., $\tau_{\text{crit}}$ or a critical SNR) at the point where this transition in decay behavior occurs. This experimentally validates the operational boundary of the topologically protected regime (Robinson, 2014; Fowler et al., 2012). This protocol allows for rigorous, reference-free validation of topological protection in practice.

References

Amari, S. (2016). Information Geometry and Its Applications. Springer. https://doi.org/10.1007/978-4-431-55978-8

Bliokh, K. Y., Rodríguez-Fortuño, F. J., Nori, F., & Zayats, A. V. (2015). Spin-orbit interactions of light. Nature Physics, 11(8), 648-653. https://doi.org/10.1038/nphys3479

Edelsbrunner, H., & Harer, J. (2010). Computational Topology: An Introduction. American Mathematical Society. https://doi.org/10.1090/mbk/069

Fowler, A. G., Mariantoni, M., Martinis, J. M., & Cleland, A. N. (2012). Surface codes: Towards practical large-scale quantum computation. Physical Review A, 86(3), 032324. https://doi.org/10.1103/PhysRevA.032324

Kak, A. C., & Slaney, M. (2001). Principles of Computerized Tomographic Imaging. IEEE Press. https://doi.org/10.1109/9780470544284

Lu, L., Joannopoulos, J. D., & Soljačić, M. (2014). Topological photonics. Science, 349(6243), 622-624. https://doi.org/10.1126/science

Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information (10th Anniversary ed.). Cambridge University Press. https://doi.org/10.1017/CBO9780511972942

Prince, J. L., & Links, J. M. (2014). Medical Imaging Physics. Wiley. https://doi.org/10.1002/9781118696947

Robinson, M. (2014). Topological Signal Processing. Springer. https://doi.org/10.1007/978-3-642-36104-3