Weyl sequence (additive orbit)¶

A fixed-step modular orbit that distributes irrational circle iterates asymptotically or traverses finite residues once per coprime-step cycle.

Version: v1 · 2026-10-17 · History
Domain-specific #: 14357
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Uniform Distribution Modulo One, Additive Cyclic Orbits → Mathematics

Core Idea¶

A Weyl sequence, in the additive-orbit sense, repeatedly adds one fixed step on a number space with wraparound. With an irrational step θ on the circle ℝ/ℤ, its terms can be written x_n = {x_0 + nθ}. They never repeat exactly and, in the limit, the fraction falling in each interval equals that interval's length.^[1]^[2] With an integer step s modulo a finite m, the recurrence w_(n+1) = w_n + s (mod m) visits every residue once before repeating exactly when gcd(s,m)=1.^[3]

The two forms share a construction, fixed addition followed by reduction, but have different guarantees. The irrational circle orbit has an asymptotic frequency property, without a finite-prefix rate supplied by that theorem alone.^[2] The finite counter has exact coverage over a complete cycle, then repeats.^[3] Marsaglia and Widynski explicitly call their odd-step word counters Weyl sequences and combine them with separate random-number-generator machinery.^[4]^[5] The counter's full period is not itself a proof that the transformed output is random.

This entry uses a qualified name because spectral theory also calls sequences of approximate eigenvectors Weyl sequences. That separate identity has an operator-residual test and no required modular additive step.^[6]

Structural Signature¶

Quotient carrier: The circle ℝ/ℤ or a finite residue space ℤ/mℤ fixes what it means to reduce after addition. A plain unbounded arithmetic progression does not have this modular orbit's coverage question.^[1]^[3]
Fixed increment: One θ or s is used at every step. The starting value x_0 or w_0 may shift the orbit, but changing the increment from term to term changes the construction.^[1]^[4]
State-carrying addition: Each term becomes the input to the next: x_(n+1)=x_n+θ (mod 1) or w_(n+1)=w_n+s (mod m). The closed form is starting value plus n increments, reduced in the same carrier.^[1]^[5]
Step condition for full coverage: Irrational θ makes the circle orbit nonperiodic and asymptotically equidistributed.^[2] In the finite case gcd(s,m)=1 makes the counter period m.^[3] A rational circle step produces a shorter orbit; a noncoprime integer step produces an incomplete finite orbit.^[2]^[3]

What It Is Not¶

Not every equidistributed sequence. Equidistribution is a property of many possible constructions. The Weyl sequence here requires the fixed-addition rule; an evenly distributed sequence generated by a different rule fails that recognition test.
Not Weyl's equidistribution theorem. The theorem proves a conclusion about the irrational orbit. The sequence is the constructed object, which can also be inspected as a finite prefix before invoking the theorem.^[2]
Not a general random-number generator. A generator also specifies how the counter participates in state evolution and how output is produced and assessed. A period statement about the counter cannot establish output independence, statistical suitability, or security.^[4]^[5]
Not an indefinitely noncycling integer sequence. Every finite counter repeats. Coprimality gives the maximum m-step period; without it, the period is m/gcd(s,m).^[3]
Not a universal low-discrepancy guarantee. Irrationality yields long-run uniform interval frequencies. A useful bound for a particular finite point set requires more information about the step and the requested error.^[2]
Not the spectral-theory homonym. Approximate eigenvectors are assessed by a vanishing operator residual, not by fractional parts or coprime modular increments.^[6]

Scope of Application¶

In uniform-distribution theory, the literal carrier is ℝ/ℤ: choosing an irrational θ and forming {nθ} gives a tractable deterministic orbit.^[2] Maliar and Maliar illustrate this with θ=√2 in a numerical-integration methods discussion.^[1] That illustration identifies the sequence; it does not report a particular integration calculation using those points or certify an arbitrary finite-prefix error rate.

In integer computation, the literal carrier is a finite word ring. Marsaglia's 32-bit counter adds 362437 on each call.^[4] Widynski's 64-bit counter adds an odd s.^[5] In each case the odd increment is coprime to the power-of-two modulus, so the counter visits every machine word during one complete cycle.^[3] The general ℤ/mℤ statement follows from the order of an element in an additive cyclic group: its period is m/gcd(s,m).^[3] This algebraic extension need not imply that every such orbit is customarily named a Weyl sequence.

The numerical-methods discussion considers finite point sets, while Marsaglia inserts a finite counter into a larger generator.^[1]^[4] Either use must evaluate what it needs beyond orbit coverage: finite-prefix error, correlations, output quality, or a security contract. Those properties are not part of the recurrence alone.

Clarity¶

The phrase uniformly distributed hides two different tests here. For {nθ}, the statement concerns the limit of interval visit frequencies as the number of terms grows.^[2] For w_n mod m, the statement concerns exact enumeration of a finite set during one full cycle.^[3] An engineer cannot replace one test with the other merely because both are called uniform.

The step condition also clarifies what failed when a sequence repeats early. If θ is rational, a circle orbit returns after finitely many steps.^[2] If gcd(s,m)>1, an integer orbit skips whole residue classes.^[3] These are failures of full coverage, not evidence about the quality of some separate output transformation.

Manages Complexity¶

The recurrence compresses a potentially long point list into a carrier, starting phase, and one step. Its orbit structure can then be analyzed without enumerating every term: check irrationality for the continuous circle,^[2] or compute one greatest common divisor for the finite counter.^[3] Changing √2 to another irrational changes the terms while preserving the fractional-part construction and asymptotic result.^[1]^[2] Changing 362437 to another odd 32-bit step preserves the finite counter form and full period.^[4]^[3]

That compression has a limit. The same few parameters do not summarize finite-prefix discrepancy in every irrational case, and they do not summarize a complete generator's output behavior. The exact orbit test is a starting point for those evaluations, not their substitute.^[2]^[4]

Abstract Reasoning¶

To classify a proposed sequence, first write its state space and update rule. If successive terms differ by the same amount modulo the declared carrier, test the increment. Irrationality on ℝ/ℤ yields an all-interval limiting frequency; a rational step yields a finite orbit.^[2] Coprimality in ℤ/mℤ yields a complete finite cycle; otherwise m/gcd(s,m) gives the shorter period.^[3]

For example, with m=12, step 5 generates all twelve residues, whereas step 4 reaches only three residues before returning. The relevant proof is that the additive order of step s is m/gcd(s,m); the starting phase merely shifts which residue comes first.^[3] A favorable orbit result licenses a coverage conclusion only. To judge a sampling or generator use, separately ask about finite prefixes and dependence between adjacent or transformed terms.^[2]

Knowledge Transfer¶

The rule transfers literally between mathematical fractional-part constructions^[1] and finite modular counters:^[4]^[5] choose a quotient carrier, iterate a fixed additive step, and test whether the step closes the orbit early. Marsaglia and Widynski use different word sizes and generator surroundings while retaining that finite-counter role.^[4]^[5]

The consequences do not transfer unchanged. An irrational infinite orbit and a finite coprime counter have different notions of uniformity and different questions of rate, period, and output. Outside quotient arithmetic, calling any regular walk or rotation a Weyl sequence would retain a verbal resemblance while losing the mathematical tests. A more general repeated-state pattern may deserve separate Prime consideration, but the present name is tied to this arithmetic mechanism.

Examples¶

Canonical — irrational fractional parts¶

Maliar and Maliar give x_n={n√2} as a Weyl-sequence example.^[1] The first two fractional parts are about 0.4142 and 0.8284; subsequent terms repeatedly add √2 and discard the integer part. Irrationality means no exact finite return, and Weyl's theorem gives limiting interval frequencies equal to interval lengths.^[2] The finite prefix may nevertheless show structure, and adjacent points have a fixed modular difference.^[2]

Mapped back: quotient carrier → ℝ/ℤ; fixed increment → √2; state-carrying addition → {x_n+√2}; step condition → irrationality. The resulting coverage claim is asymptotic interval frequency, without an asserted universal finite-prefix rate.

Applied / In Practice — a 32-bit counter inside xorwow¶

Marsaglia's xorwow example includes d += 362437 on a 32-bit unsigned word and calls that counter a Weyl sequence.^[4] Since 362437 is odd, the counter alone has period 2^32, traversing every 32-bit value before it repeats.^[3] The generator also has an xorshift state and combines its value with the counter.^[4] The behavior of that combined output is a separate question.

Mapped back: quotient carrier → ℤ/(2^32)ℤ; fixed increment → 362437; state-carrying addition → update d once per call; step condition → odd is coprime to 2^32. The resulting coverage claim is one visit to every counter state in a full 2^32-step cycle.

Widynski's separate middle-square implementation uses the same role pattern on a 64-bit counter with an odd step and proves the corresponding 2^64 counter cycle.^[5] Its different surrounding algorithm shows the Weyl counter as a component of another generator design.

Structural Tensions¶

T1: Coverage versus dependence. A fixed step gives a short proof of long-run circle coverage or a complete finite cycle. The same fixed relation makes adjacent raw terms predictable: Pincus and Kalman demonstrate two-dimensional correlation in {nθ}.^[2] Favoring simplicity and proof transparency retains that dependence; demanding random-like output adds transformation and new evaluation obligations. Marsaglia's generator uses a separate xorshift state alongside its Weyl counter.^[4] Diagnostic: Is the claim about where the orbit goes, or about the statistical behavior of values emitted by a different algorithm?

T2: Infinite guarantee versus finite budget. Irrationality ensures asymptotic circle equidistribution.^[2] An integration task consumes a finite prefix; choosing a practical prefix needs a bound or direct assessment for the chosen step, dimension, and test sets. A finite counter has exact full-cycle coverage.^[3] A short prefix can still be unbalanced; that follows from the difference between full-cycle and prefix counts. Diagnostic: What has been proved or measured about the number of terms actually used?

Structural–Framed Character¶

Vocabulary travel: Step, orbit, and coverage travel widely, but Weyl sequence here requires quotient arithmetic and a particular fixed-addition rule. Evaluative weight: Whether a step is irrational or coprime is an exact mathematical fact; whether a finite prefix is useful or an output is good depends on a chosen task and separate criteria. Human-practice dependence: The sequence can be defined without a practitioner, though numerical analysts and generator designers choose steps and evaluation targets. Institutional origin: No institution confers membership; Maliar and Maliar and Marsaglia use the name in mathematical and computing work, respectively.^[1]^[4] Import versus recognition: One can recognize the rule from the recurrence and prove its coverage on the declared carrier; calling an unrelated ordered list a Weyl sequence by analogy imports a label without its test.

The portable skeleton is repeated fixed-step state update, a possible subject for a broader Prime question. That skeleton alone cannot produce the irrationality or coprimality diagnostics. Its character: predominantly structural inside a tightly specified arithmetic frame, with use-dependent evaluation kept separate from the named sequence.

Structural Core vs. Domain Accent¶

The skeletal relation is state_(n+1)=T(state_n) with one fixed transformation and state handoff. A broader Prime devoted to that fixed-transform handoff remains a future question: one could test whether successive applications of an image filter to its prior result and repeated treatment of a material by the same chemical process preserve this exact state-handoff relation and yield a useful shared inference. Those are proposed tests, not attested Weyl instances. If each filter run starts from the original image, or changing chemical conditions make the treatment rule vary on each pass, the proposed fixed-transform relation fails. The live Recurrence instead captures state dependence with return or near return; Iteration adds progress and stopping commitments that an intentionally infinite Weyl orbit does not automatically meet.

The domain-bound mechanism is addition in ℝ/ℤ or ℤ/mℤ, followed by the irrationality or gcd test that makes the respective coverage proof work.^[1]^[4]^[3] Remove that mechanism and one is left with a generic repeated update, not a Weyl sequence. The name applies to unlike mathematical and computational instances while the necessary roles remain within quotient arithmetic. Its reach across those settings supports a domain-specific node, not a substrate-independent Prime.

This entry is part of Additive group, is part of Recurrence relation and is a decomposition of Recurrence.

Additive group supplies the quotient carrier and addition internal to both forms. Recurrence relation supplies the indexed update equation and initial state; each requested term reduces to the base value at index zero. Recurrence captures the one-step dependence and return pattern: exact after a finite counter's cycle, or arbitrarily close to earlier circle states in the irrational case. The latter is a mathematical inference from all-interval equidistribution.^[2] These three parents carry different parts of the identity.

Iteration describes state-carrying repeated application together with progress and stopping commitments; an infinite Weyl orbit supplies the former without automatically supplying the latter. Cycle captures a closed return and can illuminate a finite counter, but the irrational circle orbit never closes. Algorithm can describe a finite procedure that computes a requested prefix, whereas the mathematical infinite sequence is not itself a terminating procedure.

Relationships to Other Abstractions¶

Current abstraction Weyl sequence (additive orbit) Domain-specific

Parents (3) — more general patterns this builds on

Weyl sequence (additive orbit) is part of Additive group Domain-specific

The quotient additive group is an identity-bearing carrier inside the Weyl orbit.
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.
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.

Neighborhood in Abstraction Space¶

Weyl sequence (additive orbit) sits in a sparse region of the domain-specific corpus (76^th 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

Not to Be Confused With¶

Equidistributed sequence: the all-interval limiting-frequency property, which other constructions can also satisfy. This entry specifies how a particular orbit is generated.
Equidistribution theorem: the implication from irrational circle step to that limiting property; it is a proposition, not the ordered orbit.^[2]
Cyclic group: an algebraic structure whose element order explains the finite counter's period, but does not specify this cross-form named sequence or the circle's asymptotic statement.^[3]
Pseudorandom number generator: a complete mechanism with output and a quality contract. A Weyl counter can be one component without supplying that contract.^[4]^[5]
Spectral-theory Weyl sequence: unit approximate eigenvectors with an operator residual tending to zero. It shares a name, not the modular-addition identity.^[6]

References¶

[1] 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. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k

[2] 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. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t

[3] 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. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r

[4] 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. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o

[5] 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. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h

[6] 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. registry ↩a ↩b ↩c