Functorial Framework for Morphological Computing

In the paper's words

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.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. 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).

   from I. State of

2. 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).

   from I. State of

How it is done

1. 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.

   from A

2. 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).

   from I. State of

3. 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.

   from II.

4. 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 $M = \prod_i p_i$. | Category RNS : The category RNS is defined.

   from Appendix A: Formal Verification of the Functorial Framework

What would falsify it

1. 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.

   from I. State of

2. 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.

   from I. State of

3. 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.

   from Appendix

Where the claim stops

1. 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).

   from I. State of

2. 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).

   from I. State of

3. 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.

   from Appendix

Where to read

1. I. State of898 words · 4 min
2. II.737 words · 3 min
3. Appendix A: Formal Verification of the Functorial Framework518 words · 2 min
4. Appendix418 words · 2 min