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| Format: | Recurso digital |
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Zenodo
2026
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| Online-Zugang: | https://doi.org/10.5281/zenodo.20141579 |
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Inhaltsangabe:
- <p>Every integer encodes a braid. And the vast majority encode two. Zeckendorf's theorem — a standard result in combinatorial number theory — guarantees that every positive integer has a unique decomposition into non-consecutive Fibonacci numbers [1]. Herein we observe that the indices of this decomposition directly specify braid generators over the Fibonacci anyon model.</p> <p>Applying Zeckendorf's theorem independently to both the Fibonacci and Lucas number sets, we construct the Fibonacci-Lucas Coordinate Tube (FLCT), an anyonic coordinate system that maps every integer from 0 to F(445) ≈10^92 — the verified kernel range, exhaustively cross-checked against TQSim and Qrack at every structurally foundational address — to structurally addressable braid words, from which gate classification and phase structure extend naturally via integer arithmetic over the bounding algebra of ℚ(√5). The dual-chart construction itself is unbounded; its theoretical terminus, an attractor-driven singularity derived from the same ℚ(√5) algebra, is the subject of a further companion paper.</p> <p>Each decomposition yields a canonical generator set which then admits further algebraic ordering flexibility under the principles that compose this coordinate system. That reordering methodology, however, will be explored at length in a subsequent paper. This paper develops the Zeckendorf-reachable specialization; the structural ruleset itself applies to the entirety of braidspace, treated in Paper 2 (The Prism), which decomposes the 10^2118-word landscape into 8,010 structurally labeled cells via a mathematical construct called the MDA, or Multiphasic Data Adapter.</p> <p>For any address, the braid generators, phase class, cyclotomic field, superselection sector, gap signature, and other quantum classifications are computable in O(log n) bit-operations from integer arithmetic alone — no simulation, no search. Current compilation paradigms brute-force braid space as featureless terrain, iterating Solovay-Kitaev or Monte Carlo against target unitaries; FLCT sidesteps this approach for the Zeckendorf-reachable subset by exposing the rule-based structure already present. Every classification reduces to a single modular operation.</p> <p>FLCT provides a direct coordinate mapping to the same F- and R-matrices confirmed by recent quantum hardware [2, 3], reproducing these unitaries on conventional classical hardware (a laptop). This is an addressing and classification scheme, not a universal gate-approximation compiler; approximation guarantees for arbitrary targets are outside the scope of this paper.</p>