Weyl sequence (additive orbit)¶
A fixed-step modular orbit that distributes irrational circle iterates asymptotically or traverses finite residues once per coprime-step cycle.
Core Idea¶
A Weyl sequence, in the additive-orbit sense, repeatedly adds one fixed step with wraparound. On the circle ℝ/ℤ, an irrational step gives a nonrepeating orbit whose interval-visit frequencies approach interval lengths.[ref-3f01af0f1f79][ref-88599a267bba] In finite ℤ/mℤ, a coprime step visits every residue before repeating.[^ref-1ac2e27a2790] These are different coverage tests: an asymptotic limit and a complete finite cycle. Neither proves that a short prefix is balanced or a generator's output is random.
Scope of Application¶
The scope is fixed-addition orbits on quotient additive carriers; the carrier and step determine which coverage test applies.
- Mathematical circle orbit. The fractional parts
{x_0+nθ}have limiting interval frequencies whenθis irrational.[^ref-88599a267bba] - Numerical-methods illustration. Maliar and Maliar give
{n√2}as an example; they do not report a specific integration run with those points.[^ref-3f01af0f1f79] - 32-bit generator counter. Marsaglia calls
d += 362437a Weyl sequence insidexorwow.[^ref-9edaa466cbf5] - 64-bit generator counter. Widynski uses an odd-step 64-bit counter in a different generator.[^ref-82539000449b]
The counter periods describe state coverage, not output quality or security. Spectral theory also calls approximate eigenvectors Weyl sequences; that homonym has a different recognition test.[^ref-50ec42f63927]
Clarity¶
Write down three things before calling a sequence a Weyl additive orbit: the quotient carrier, the fixed step, and the coverage condition. For a circle orbit, test irrationality and state an asymptotic interval-frequency conclusion.[^ref-88599a267bba] For a finite counter, test coprimality and state a full-cycle period conclusion.[^ref-1ac2e27a2790] A plain unbounded arithmetic progression, a sequence generated by varying steps, and a theorem about the orbit are different objects. A general equidistributed sequence need not use this fixed-addition rule.
Manages Complexity¶
The start, step, and carrier describe a long sequence without listing its terms. Additive group supplies the internal addition on either quotient carrier. Recurrence relation supplies the order-one update rule plus initial value. Recurrence captures dependence on the preceding state and return: exact after a finite cycle, or arbitrarily close to an earlier circle state after enough irrational steps. The latter follows from all-interval equidistribution.[^ref-88599a267bba] These three relationships explain different parts of the construction, and none supplies a finite-prefix error bound by itself.
Abstract Reasoning¶
For m=12, step 5 is coprime to 12, so it visits all twelve residues before returning. Step 4 has period 12/gcd(4,12)=3, so it visits only three residues from a given start.[^ref-1ac2e27a2790] With θ=√2 on the circle, the step is irrational, so the orbit has no exact return; over a growing prefix, interval frequencies approach interval lengths.[ref-3f01af0f1f79][ref-88599a267bba] In either case, decide what a later application actually needs: full-cycle coverage, a finite-prefix error estimate, or output-quality evidence. Those are separate questions.
Knowledge Transfer¶
The same fixed-addition rule can generate mathematical fractional parts and word-sized modular counters. Marsaglia and Widynski place such counters inside different generator designs.[ref-9edaa466cbf5][ref-82539000449b] The transferable part is the orbit rule and its typed step test. The circle's asymptotic conclusion does not become a finite counter's period claim, and a counter's period does not become a randomness guarantee. Calling an unrelated repeated process a Weyl sequence would lose the quotient-arithmetic recognition test.
Relationships to Other Abstractions¶
Current abstraction Weyl sequence (additive orbit) Domain-specific
Parents (3) — more general patterns this builds on
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Weyl sequence (additive orbit) is part of Additive group Domain-specific
The quotient additive group is an identity-bearing carrier inside the Weyl orbit.
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Weyl sequence (additive orbit) is part of Recurrence relation Domain-specific
An order-one recurrence equation with an initial state is an internal defining rule of each orbit.
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Weyl sequence (additive orbit) is a decomposition of Recurrence Prime
The arithmetic orbit frames a portable prior-state dependence and exact or near-return pattern.
Hierarchy paths (7) — routes to 7 parentless roots
- Weyl sequence (additive orbit) → Additive group → Group → Monoid → Semigroup → Set and Membership
- Weyl sequence (additive orbit) → Recurrence
- Weyl sequence (additive orbit) → Recurrence relation → Recursion
- Weyl sequence (additive orbit) → Additive group → Group → Monoid → Identity Element
- Weyl sequence (additive orbit) → Additive group → Group → Monoid → Semigroup → Closure
- Weyl sequence (additive orbit) → Additive group → Group → Monoid → Semigroup → Associativity → Invariance
- Weyl sequence (additive orbit) → Additive group → Group → Monoid → Semigroup → Associativity → Symmetry
Neighborhood in Abstraction Space¶
Weyl sequence (additive orbit) sits in a sparse region of the domain-specific corpus (76th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Integer Classifications & Arithmetic Functions (45 abstractions)
Nearest neighbors
- Quotient stack — 0.83
- Ramanujan's sum — 0.83
- Completion of a ring — 0.83
- Commutative ring — 0.82
- Brandt matrix — 0.82
Computed from structural-signature embeddings · 2026-10-17
References¶
[^ref-3f01af0f1f79]: Lilia Maliar and Serguei Maliar, “Numerical Methods for Large-Scale Dynamic Economic Models,” in Handbook of Computational Economics, vol. 3 (2014), chapter 7, printed p. 386, one-based PDF p. 62, https://web.stanford.edu/~maliars/Files/Chapter7HandbookCE2014.pdf. Authored methods chapter; the √2 sequence is an illustration, not a reported integration experiment.
[^ref-9edaa466cbf5]: George Marsaglia, “Xorshift RNGs,” Journal of Statistical Software 8, no. 14 (2003), printed p. 4, DOI 10.18637/jss.v008.i14, https://www.jstatsoft.org/article/view/v008i14. Original published paper; full text checked in a mirrored PDF at https://www.dkl9.net/archive/marsaglia_xorshift_rng.pdf, one-based PDF p. 5, because journal PDF text extraction failed.
[^ref-82539000449b]: Bernard Widynski, “Middle-Square Weyl Sequence RNG,” arXiv:1704.00358v7 (2022), §§2, 4, https://arxiv.org/html/1704.00358v7. Original preprint; Theorem A supports the counter-period claim. Later output-quality or cryptographic assertions are not relied upon here.
[^ref-88599a267bba]: Steve Pincus and Rudolf E. Kalman, “Not All (Possibly) ‘Random’ Sequences Are Created Equal,” Proceedings of the National Academy of Sciences 94, no. 8 (1997): 3513–3518, “Multiples of Irrationals,” DOI 10.1073/pnas.94.8.3513, https://www.pnas.org/doi/10.1073/pnas.94.8.3513. Original research article; supports fixed-difference correlation and the qualitative theorem's lack of a rate.
[^ref-1ac2e27a2790]: Louisiana State University, “Cyclic Group Supplement,” mathematics course notes, Theorems 6–7 and the ℤ/12ℤ example, one-based PDF pp. 2–3, https://www.math.lsu.edu/~adkins/m7200/cyclicgroup.pdf. University proof of order m/gcd(s,m) and the coprime full-generation condition; it does not assign the Weyl name to every cyclic orbit.
[^ref-50ec42f63927]: Martin Peev, Functional Analysis II, unofficial lecture notes for Prof. Phan Thành Nam, LMU Munich, Theorem 4.15, printed p. 50, one-based PDF p. 50, https://www.math.lmu.de/~nam/LectureNotesFA1718.pdf. University source for the separate approximate-eigenvector homonym.
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