Quantum Arithmetic: A New Mathematical Framework from Discrete Modular Arithmetic to Cross-Domain Signal Processing

This article introduces the open-source project Quantum Arithmetic, a discrete modular arithmetic framework based on Fibonacci-style shift operations that can discover hidden orbital structures in high-dimensional noisy data. Its core contribution, the QA Coherence Index (QCI), quantifies the degree of data structuring and has been validated effective in 6 domains including EEG signal detection and financial volatility prediction, outperforming traditional methods in predictive capabilities.

In the fields of signal processing, neural networks, and financial modeling, traditional continuous mathematical methods struggle with high-dimensional noisy data. Quantum Arithmetic, developed by GitHub user 1r0nw1ll, is not only a theoretical framework but also a practical tool validated in 6 domains, aiming to discover structured patterns from seemingly random data.

Quantum Arithmetic is core based on Fibonacci-style shifts of (b,e) in modulo N space: d = b+e(mod N), a = b+2e(mod N), generating three types of orbits with periods 1 (singularity), 8 (satellite), and 24 (universe) (when N=24, 576 state pairs self-organize into three categories). The 16 derived invariants correspond to rational trigonometric chromogeometric quantities, and the core identity C²+F²=G² is a special case of Wildberger's theorem. The most practical QA Coherence Index (QCI) measures the extent to which a time series follows predictions from the T operator: a high QCI indicates structured dynamics (predictable patterns), while a low QCI is close to random noise. QCI does not replace traditional methods but provides additional predictive power after controlling for traditional factors.

QCI has been validated effective in 6 domains:

1. EEG Epilepsy Detection: After controlling for delta wave power, QCI provides an additional 0.21 R² explanatory power (p<1e-33 for 10 patients), capturing topological orbital information not identified by traditional spectral analysis;
2. Electromyography (EMG) Pathological Analysis: Increases R² by 0.61, reflecting motor unit recruitment structure and aiding early diagnosis of neuromuscular diseases;
3. Climate Teleconnection: La Niña events have a 97% probability of being satellite orbits (p<1e-6);
4. Financial Volatility Prediction: After controlling for realized volatility, the partial correlation coefficient with future volatility is -0.22 (weak but reference-worthy);
5. Audio Classification: The partial correlation coefficient between orbit transition rate and target variable is +0.75 (p=0.020), exceeding first-order lag autocorrelation;
6. Atmospheric Reanalysis: The partial correlation coefficient with future variability is -0.20 (p<1e-5).

The project emphasizes verifiability: Each empirical claim is accompanied by a machine-verifiable certificate (including schema, verifier, and witness data), with 186 certificate families passing CI meta-validation; At the code level, 6 axioms ensure discrete integrity (prohibiting zero/floating-point states, etc.), and the axiom checker serves as a pre-commit hook and CI process—zero errors are a hard requirement. The tool ecosystem is layered:

• qa_observer: Installable via pip, provides TopographicObserver, QCI calculation, etc., suitable for quick start;
• qa_lab: Research infrastructure layer, including proxy routing, PIM kernel, etc.;
• qa_alphageometry_ptolemy: A collection of verifiers for 186 certificate families.

The value of Quantum Arithmetic lies in its unified perspective on structured data across different domains, revealing discrete orbital structures (which traditional methods treat as continuous or random). For data scientists and ML engineers, its significance includes:

1. New dimension in feature engineering: QCI complements existing feature sets and provides incremental predictive power;
2. New idea for anomaly detection: Low QCI indicates abnormal/noisy regions;
3. Cross-domain transfer: Effective patterns from EEG may apply to financial sequences. The project demonstrates the irreplaceability of first-principles-based mathematical insights in the era of deep learning, and is worth exploring by researchers in signal processing and time series analysis.