Pith. sign in

REVIEW 4 minor 12 references

For a valid siteswap, first-throw and last-catch extensions preserve validity precisely when the old landing times stay distinct and outside one residue interval modulo the extended period; ground-state patterns always satisfy this.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 10:41 UTC pith:WX6MIKHL

load-bearing objection Small, correct paper giving exact validity criteria for two natural siteswap extensions; worth refereeing but not a big deal.

arxiv 2607.29255 v1 pith:WX6MIKHL submitted 2026-07-31 cs.DM math.CO

About two results for new valid juggling sequences

Hugo Parada (SPHINX , IECL) This is my paper
classification cs.DM math.CO MSC 05A05
keywords siteswapjuggling sequencespermutation testfirst-throw extensionlast-catch extensionlanding timesmodular arithmeticground state
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces two simple ways to extend a valid siteswap (a numeric notation for juggling patterns): inserting extra throws right after the first few throws, and appending throws to record the final catches. It proves exact necessary and sufficient conditions, in terms of where the original throws land modulo the new period, for either operation to produce another valid pattern with the same number of balls. These conditions are always satisfied when the pattern starts from the ground state, so ground-state sequences can be extended freely in both directions. For arbitrary patterns, the conditions give a concrete modular check. The work matters because it gives jugglers and combinatorists a rigorous way to build longer valid patterns from short ones.

Core claim

For a valid b-ball siteswap s of period p, the first-throw extension F_{i,q}(s) is valid if and only if the original landing times are pairwise distinct modulo p+q and avoid the residues b-q, ..., b-1; the last-catch extension B_{i,q}(s) is valid if and only if the modified landing times (the i largest bumped by q) are pairwise distinct modulo p+q and avoid the residues p+b-i, ..., p+b-i+q-1. Whenever these hold, the extension is again a b-ball siteswap of period p+q. As a direct corollary, every ground-state pattern admits both extensions for every admissible i and q, and remains ground-state.

What carries the argument

The permutation test (a sequence is valid iff the map i ↦ i+a_i mod p is a permutation of Z_p) is the engine. Both extension theorems shift the landing times of the old throws by q, add q new throws landing in a contiguous residue block, and ask that the union be a complete residue system modulo p+q. The forbidden residue intervals are exactly the landing positions of the inserted or appended throws.

Load-bearing premise

The results assume the standard siteswap model, where validity is exactly the distinctness of landing times modulo the period (the permutation test); if simultaneous catches or throws (multiplex or synchronous) were allowed, the stated conditions would not be sufficient.

What would settle it

Enumerate all valid b-ball siteswaps of small period p (say up to 6), apply F_{i,q} and B_{i,q} for every admissible i and q, and compare each result's validity (via the permutation test) with the paper's residue conditions; any mismatch would refute the theorems.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Ground-state patterns always admit both extensions: for any q and any admissible i, F_{i,q}(s) and B_{i,q}(s) are valid b-ball siteswaps of period p+q, and they are again ground-state.
  • For arbitrary valid siteswaps, the theorems give a single modular check that decides validity for all i at once: if the landing times pass the test for one i, they pass for every i.
  • The operations preserve ball count and increase the period by q, so iterating them generates infinite families of b-ball patterns from a single seed.
  • The forward and backward constructions are not symmetric: a pattern and its cyclic shift can behave oppositely (51 fails forward while its shift 15 succeeds, and the situation reverses for last catches), so the choice of starting point matters when extending excited patterns.
  • The paper shows the constructions compose with state-loop splicing and local swaps, yielding a long valid routine from 5555; this demonstrates the criteria are useful for building actual juggling sequences.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The residue conditions can be reinterpreted as a 'forbidden window' in the cyclic group Z_{p+q}; for a random valid siteswap, the probability that a random shift or state passes the test could be estimated, giving a quantitative sense of how often excited patterns can be extended.
  • The ground-state corollary suggests the set of ground-state siteswaps of a fixed ball count is closed under these two operations, so they generate an algebraic structure (a monoid) on ground-state sequences; this could connect to known results on state graphs.
  • The last-catch construction selects the i largest landing times, which correspond to the throws with the longest flight durations; one could imagine generalizations that bump a different selection of landing times, or adapt both operations to synchronous or multiplex siteswaps, which the paper explicitly leaves open.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper studies two operations on simple (non-multiplex, non-synchronous) siteswap sequences: the first-throw extension F_{i,q}(s), in which the first i throws are raised by q and q throws of height b−i are inserted after them, and the last-catch extension B_{i,q}(s), in which the i throws with largest landing times are raised by q and q throws of height b−i are appended. The main theorems (Theorems 2.3 and 2.9) give necessary and sufficient conditions, in terms of the original landing times modulo p+q and a forbidden residue interval, for the extended sequence to be a valid b-ball siteswap of period p+q. For ground-state sequences these conditions always hold (Corollaries 2.4 and 2.10). The paper also reviews the standard siteswap background and concludes with several worked examples, including a longer constructed routine.

Significance. The central results are correct. The proofs are complete and reduce the extension problem directly to the permutation test: in the forward case all old throws land at T_k+q and the new throws occupy the consecutive residues b,...,b+q−1; in the backward case the selected throws shift by q and the appended throws occupy p+b−i,...,p+b−i+q−1. The conditions are explicit, parameter-free, and derived from the externally defined permutation test rather than from the target validity statement, so there is no circularity. The restriction to simple siteswaps is acknowledged in Definition 1.2 and in the Conclusion, so the scope limitation is not a gap. The supplementary visualizer and video are useful and the worked examples illustrate the criteria well. The novelty is modest, but the paper is honest and the treatment is rigorous.

minor comments (4)
  1. [Definition 1.7 / Example 1.8] The flattening algorithm uses 'shift' in both directions: 552 is shifted to 525 (a left shift) and 345 is shifted to 534 (a right shift), while Definition 1.7 only defines the cyclic right shift. Please clarify that any cyclic shift is permitted, or define the left shift as well.
  2. [Theorem 1.9] The permutation test is load-bearing for both main theorems but is stated without proof or reference. Since the proof strategy in Theorems 2.3 and 2.9 relies entirely on this criterion, adding a citation to [Pol03] or a one-sentence proof would make the paper more self-contained.
  3. [Example 1.6] The arrow diagram '642 swap 0 and 1 → 552, 642 swap 0 and 2 → 444' is visually confusing because the two site swaps are written on one line. Separating them into two lines would improve readability.
  4. [Section 2 tables] The tables for forward and backward extensions include a q=0 row, although the definitions require q≥1. Consider marking this row as the base sequence rather than as an instance of the construction, to avoid any ambiguity.

Circularity Check

0 steps flagged

No significant circularity identified; both main theorems are direct applications of the permutation test to the explicitly defined extension operations.

full rationale

The paper's two main results, Theorem 2.3 and Theorem 2.9, are derived by computing the landing times of the constructed sequences and applying the independent permutation test (Theorem 1.9). No parameter is fitted to the target validity statement; the conditions in the theorems are shown to be exactly the permutation-test conditions on the shifted or modified landing times. The constructions F_{i,q}(s) and B_{i,q}(s) are explicitly defined in terms of a given valid siteswap s, and the proofs do not assume the extension is valid except when applying the permutation test to verify it. The only citations by the author are to the auxiliary interactive visualizer [Para] and a video [Parb], which are not load-bearing for the mathematical claims. The ground-state corollaries use an external cited result (Proposition 1.19 from Chung–Graham) and follow by direct interval union. The paper explicitly restricts to simple siteswaps, so reliance on the permutation test is a stated modeling boundary, not a circular step. Overall, the derivation chain is self-contained relative to the standard siteswap validity criterion and contains no fitted-input-called-prediction or self-citation-circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper introduces no fitted constants and no new physical or mathematical entities. It relies on standard siteswap characterizations (permutation test, average theorem) and a cited ground-state criterion. Parameters p, b, i, q are inputs, not inferred from data.

axioms (3)
  • standard math Permutation Test (Theorem 1.9): s is valid iff i+a_i mod p is a permutation of Z_p.
    Used in both main proofs; this is the external benchmark converting the definition of validity into modular arithmetic.
  • standard math Average Theorem (Theorem 1.3): the number of balls equals the average of the entries.
    Used to show that the extensions remain b-ball patterns; follows from the permutation test and is stated as a known theorem.
  • domain assumption Ground-state landing-time criterion (Proposition 1.19, from [CG08]): a ground-state sequence has landing times exactly {b, b+1, ..., b+p-1}.
    Used only in Corollaries 2.4 and 2.10, not in the main necessary-and-sufficient theorems.

pith-pipeline@v1.3.0-daily-deepseek · 10393 in / 13052 out tokens · 130819 ms · 2026-08-03T10:41:58.958473+00:00 · methodology

0 comments
read the original abstract

In this note, we study two simple operations for extending juggling (siteswap) sequences, called first throws and last catches. Both constructions come from a natural question for a juggler: how can one add throws at the beginning or catches at the end of a pattern without creating a collision? Using landing times and the permutation test, we give necessary and sufficient conditions under which these constructions, when applied to an arbitrary valid siteswap, produce another valid siteswap. We complement our analysis with several examples. An interactive visualization of these extensions is available in [Para].

Figures

Figures reproduced from arXiv: 2607.29255 by Hugo Parada (SPHINX, IECL).

Figure 1
Figure 1. Figure 1: Juggling diagram for the siteswap sequence 441. Date: August 3, 2026. Hugo Parada is funded by the Agence Nationale de la Recherche by the QuBiCCS project (ANR-24-CE40-3008). 1 arXiv:2607.29255v1 [cs.DM] 31 Jul 2026 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Visualizing throws. Each number encodes not only the throw height, but also implicitly determines which hand performs the next catch and at what future beat it will occur. This convention reflects what the audience perceives during an actual juggling performance: a rhythmic alternation of throws of varying heights, each corresponding to a distinct visual trajectory in space and time. Disclaimer. Throughout… view at source ↗
Figure 3
Figure 3. Figure 3: The landing permutation of 6451: 0 7→ 2, 1 7→ 1, 2 7→ 3, 3 7→ 0. Theorem 1.9 (Permutation Test). Let s = (a0, a1, . . . , ap−1) ∈ N p . Define the map φs : Zp → Zp, φs(i) = (i + ai) mod p. Then the sequence s defines a valid juggling pattern if and only if φs is a permutation of Zp. Example 1.10. • 6451: (0 + 6, 1 + 4, 2 + 5, 3 + 1) mod 4 = (2, 1, 3, 0) ⇒ valid. • 6145: (0 + 6, 1 + 1, 2 + 4, 3 + 5) mod 4 =… view at source ↗
Figure 4
Figure 4. Figure 4: State evolution along the siteswap 531. A throw of height j is allowed if the corresponding landing site is unoccupied. Thus, the current state determines which throws are valid at any moment. The set of all juggling states for b balls and maximum throw height h forms the vertex set of a state graph. Each vertex corresponds to a binary word of length h with exactly b ones. Definition 1.15. Let Vert(b, h) d… view at source ↗
Figure 5
Figure 5. Figure 5: State graph for 3 balls and height 5 (partial view) [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Ground-state cycles 3, 42, 531 and excited-state cycle 51 in the state graph. Theorem 1.16. Every closed loop (cycle) in the state graph corresponds to a valid juggling sequence. We now recall Polster’s terminology for ground and excited states and sequences [Pol03, Sec￾tion 2.8.2]. Definition 1.17. • The ground state is the unique state 11 · · · 1 | {z } b 00 · · · 0 | {z } h−b , with the b ones placed at… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

12 extracted references

  1. [1]

    Juggling drops and descents

    Joe Buhler, David Eisenbud, Ron Graham, and Colin Wright. Juggling drops and descents. The American Mathematical Monthly , 101(6):507--519, 1994

    1994

  2. [2]

    Juggling lab

    Jack Boyce and Juggling Lab contributors . Juggling lab. Computer software, version 1.7.5, 2026. Available at https://jugglinglab.org/. Accessed July 28, 2026

    2026

  3. [3]

    Beek and Arthur Lewbel

    Peter J. Beek and Arthur Lewbel. The science of juggling. Scientific American , 273(5):92--97, 1995

    1995

  4. [4]

    Primitive juggling sequences

    Fan Chung and Ron Graham. Primitive juggling sequences. The American Mathematical Monthly , 115(3):185--194, 2008

    2008

  5. [5]

    The physics of juggling

    Bengt Magnusson and Bruce Tiemann. The physics of juggling. The Physics Teacher , 27(8):584--589, 1989

    1989

  6. [6]

    Siteswap extension visualizer

    Hugo Parada. Siteswap extension visualizer. https://sites.google.com/view/hugo-parada/juggling-outreach/siteswap-extensions

  7. [7]

    Siteswap a8531 771 88177700

    Hugo Parada. Siteswap a8531 771 88177700 . https://drive.google.com/file/d/1agN4oUrzV3i3ucQqtWmatr7iOcLwWUE1/view?usp=drive_link

  8. [8]

    The Mathematics of Juggling

    Burkard Polster. The Mathematics of Juggling . Springer, New York, 2003

    2003

  9. [9]

    Claude E. Shannon. Scientific aspects of juggling. In N. J. A. Sloane and Aaron D. Wyner, editors, Claude Elwood Shannon: Collected Papers , pages 850--864. IEEE Press, New York, 1993. Manuscript written circa 1980

    1993

  10. [10]

    A notation for juggling tricks: A lot of juggling tricks

    Bruce Tiemann and Bengt Magnusson. A notation for juggling tricks: A lot of juggling tricks. Juggler's World , 43(2):31--33, 1991. Available at https://www.dev.juggle.org/history/archives/jugmags/43-2/43-2,p31.htm

    1991

  11. [11]

    Variations for numbers jugglers

    Jeff Walker. Variations for numbers jugglers. Juggler's World , 34(1):11, 1982. Available at https://www.jonglage.net/theorie/notation/ladder/refs/Jeff Walker - Variations for numbers jugglers - JW-vol34no1-p11.pdf

    1982

  12. [12]

    Wright and Mike Day

    Colin D. Wright and Mike Day. Responses in ``all site swap original inventors sought''. Posts to the rec.juggling newsgroup, 1995. Messages posted on 23--24 May 1995. Archived by Google Groups at https://groups.google.com/g/rec.juggling/c/lB-jy9wcgNs

    1995