Holding Topology

I began by looking for a strict definition of juggling and its difference from object manipulation. The search narrowed to a physical question: what changes when bodily retention gives way to release?

Definitions and scope

In this article, object manipulation is the generic class of practices in which a performer intentionally changes or maintains the position, motion, or configuration of physical objects through bodily contact or a physical linkage. The comparison class is continuous-retention manipulation, where every active object remains directly retained by the body throughout the episode. The release-bearing structure studied here repeatedly includes an interval in which at least one active object is leased: no longer directly retained by the body, while its trajectory remains part of an intended capture, strike, pass, bounce, or subsequent movement.

Within this generic class, release transfers part of control from continuous contact to initial conditions, prediction, and renewed constraint. Under continuous retention, the body directly constrains motion at every moment; after release, the object state and environmental dynamics carry the motion. This difference changes the model’s state space: a retention snapshot describes an instant, while a release-bearing structure is described by a trajectory of releases and captures together with a continuation relation. A complete hold may occur as a pause and complete release as a flash.

The article is limited to the physical geometry of retention, release, and renewed constraint. Cultural classification belongs to another projection of a performance, shaped by practice, presentation, history, intention, and reception. The algebra takes an episode already chosen for study and classifies its retention structure. The interactive HTML simulator illustrates the formal model. Intention appears only as an external continuation condition supplied by a task, pattern, performer, or observer; the model assigns no psychological variables.

The system has a body, a finite set of objects, and an environment. Direct retention happens at contact points: a palm, a pinch, a press against the chest, a ball supported on a foot. Grip is the relation between such a point and an object. The body is the constraint on which of those points can exist, move, and retain objects together, including the moving balance of the stance. An object is retained when at least one contact point holds it; records that projection. Once released, an object follows physical dynamics set by its state at release and by gravity, air, collisions, and later interventions. Motion may be continuous between impacts while velocity changes at a collision, strike, or capture. Release therefore removes bodily retention; it does not by itself remove predictability, influence, or control.

Retention state

Let be the finite active object domain chosen for the description. Membership comes from the performance, task, or interval under study. Attention and an intended continuation may help an observer choose , but neither belongs to the physical retention state.

Once the body boundary, object granularity, temporal resolution, and criterion of direct retention have been fixed, classify each labelled object by when at least one contact point retains it and otherwise. Bivalence is an axiom of this projection. The held subset and its occupancy are

Two hands here are a modeling convention: two mobile contact points on one person, to keep the study domain small. Three or more held objects on those points share them as a multiplex. Passing gathers several people's contact points under one shared constraint, and is then the group's occupancy.

The complete discrete microstate is , equivalently a bit vector in . Occupancy forgets identity. Each object remains binary, while two existence predicates describe the whole domain:

The first coordinate says that some active object is unretained; the second says that some active object is retained. Reading the pair as “unheld present, held present” gives

The fourth pair occurs only for , where there is no active system. The occupancy sign of that empty domain is . For a nonempty domain, the same projection can be written by occupancy:

At the object level, and are complementary. Aggregation changes the quantifier: “some object is unretained” is an existential statement, while the negation of “some object is retained” says that every object is unretained,

The two system predicates can therefore be true together for different objects. Their conjunction is exactly retention heterogeneity:

The mixed sign is called Polymorphy here and is attainable exactly when . A singleton domain reaches only or ; with several objects, the system has a third sign while every object remains binary. The chain now separates three resolutions: which objects are retained, how many are retained, and whether retention is absent, heterogeneous, or total.

If an observer does not know the value of , the physical state still belongs to one of the signs attainable for that domain. Partial observation is defined over , for example as a set of compatible microstates or a probability distribution over them.

An object that is retained is tained; an unheld object whose capture continuation is still live is leased. A drop is a lease whose viable capture continuation becomes empty, and floor contact is evidence of that emptying. A dump never opens a lease. Objects outside are outside the episode.

Kratos carries senses of strength, power, rule, and mastery, while akrateia carries senses of powerlessness, debility, and want of self-control (LSJ κράτος; ἀκράτεια). Occupancy means direct bodily retention is present at the chosen grain, and means some active object is unheld. Control effectiveness, powerlessness, freedom, and moral self-control fall outside the occupancy projection. A throw and a drop both have at an instant; the throw is leased, the drop is a lease that has failed. The occupancy mathematics depends only on . The landing schedule of a siteswap supplies the scheduled return that keeps a flight leased.

For disjoint domains, adjoining the null sign gives a convenient composition. If a sign is represented by the pair “some object unheld, some object held,” then

Here is neutral and absorbs. This join records the coarse state of the union with three active regimes.

The smallest mixed composition is immediate:

One subsystem may be wholly retained and another wholly unretained. Neither subsystem is internally mixed, while their union is.

Paths through the cube

Let be the Boolean hypercube whose vertices are the held subsets . Join two vertices when one object changes retention:

A release changes to , and a capture changes to . A performance then writes an event word and traces a path through .

On an interval with a fixed domain , let and count elementary releases and captures. In a simultaneous event, each object change is counted separately. Then

Equal initial and final occupancies give . In particular, an episode returning to its initial contains equal numbers of captures and releases, while a transition has balance and contains at least releases. When and events change one object at a time, every such transition passes through : the integer moves from to in unit steps and must visit . Mixedness is the interior of a distributed transition.

For one object, the throw path is : tained, leased, tained. Two objects make polymorphy possible, yet mixed vertices and differ in two bits and are disconnected under single-bit edges. Any path between them made of single-object events touches either all-unheld occupancy or total retention . Sustained mixed occupancy at two objects therefore means remaining in one mixed vertex, or using a two-bit swap packet.

For , remove the two homogeneous vertices and call the remaining induced graph . Of the microstates in , one projects to , one to , and the remaining are the vertices of and project to . Thus the combinatorial fraction of mixed microstates is . This fraction assumes no probability distribution and predicts no fraction of performance time. For , the sole microstate is the empty sign .

A declared stochastic snapshot turns this count into a probability. Suppose the observation time is sampled from a stationary regime, the retention indicators are mutually independent at that instant, and . Then

These are g-cube occupancy shares. is time in all-unheld occupancy.

With a shared retention probability , these become , , and . At , the Polymorphy probability is , recovering the combinatorial fraction above. If the stationary process is also ergodic, these snapshot probabilities equal limiting time shares. Stationarity specifies only the distribution at an observation time; phase and coordination can break independence.

For , is connected exactly when . To see this, shrink any nonempty proper subset to one retained object, move between singleton subsets through a two-object subset, then reverse the shrinking process toward the target. The two-object intermediate remains proper only when a third object exists. For three objects the whole mixed region is already a cycle:

Three is therefore the first object count with internal mobility inside mixed retention under single-object events. A three-object cascade can follow this cycle while its macrostate remains . The mixed sign can persist while the held set rotates: polymorphy continues for as long as , and the objects that witness retention and release keep changing.

From total retention to total release, with each object released once and never recaptured, there are single-object orders. If several objects may leave together as one packet, each still leaving once, the histories are the ordered set-partitions of the objects, counted by the Fubini numbers : , , . Between two microstates the Hamming distance likewise gives shortest single-bit paths and shortest packet paths. Occupancy forgets those identities at once.

Because is connected for , its cycle rank is

Three objects supply one independent cycle; four supply eleven; five supply forty-one. That number is the dimension of the cycle space of the state graph.

Four marks a different threshold. The distance from a mixed microstate to a homogeneous boundary is

Every mixed state for three objects lies one event from a boundary. With four objects, a state holding two has and remains mixed after any single-object event. More generally, for a nonnegative integer , call buffered through events when every path of at most elementary changes starting at remains mixed. This is equivalent to . Such a state exists exactly when

because it needs at least retained and unretained objects. The buffered interior

is internally connected exactly when . For this interior has , , and states at , , and . At it has only the middle occupancy layer, and every elementary event leaves that layer. One additional object supplies two adjacent interior layers; shrinking to the lower layer, exchanging identities through the upper one, and expanding again connects any two states. The first pair of thresholds is polymorphy at two and circulation at three; the next is one-event buffering at four and buffered circulation at five. For , the greatest boundary distance among mixed states is . This distance measures topological redundancy: how many single-object events remain before a homogeneous vertex. Path redundancy in is a second quantity, already visible in the factorials and in . Mechanical robustness is a third, whether remaining contacts can carry and redistribute load. Juggling difficulty lies on other axes.

Boundary distance is a minimum over paths. An expected event count requires an event law. Let each step independently choose one of the bits uniformly and flip it. For , a step from goes to with probability and to with probability . Define

First-step conditioning gives

Here is a first-hit expectation from a mixed start. A boundary start has . Define its first return by . For the same uniform one-bit chain, starting from either boundary vertex gives , hence an expected mixed-state visits. From a central occupancy, the first-hit expectations are events for , for , and for . These values characterize the declared event law; a performance's event-selection law must be specified or estimated from observations.

The thresholds depend on the event alphabet. Simultaneous actions can be represented by packets such as or . If a two-object exchange packet is primitive, can occur without an observed intermediate macrostate. Synchronous throws, multiplexes, handoffs, and zero-duration catch–release events therefore need packet semantics that preserves simultaneity. Equal successive signs therefore do not name the same event. A stretch written can be a swap, a handoff, a change of which body holds, or a dwell in which is unchanged. One hand holds one object, the other flies: the sign is already . From that vertex, single-object events do not run every exchange .

Mixed retention describes one regime of juggling: tained and leased objects present together. A cascade may persist in , a total-release flash visits as all leased, and a complete hold visits . A three-object total-release flash from rest traces

The same signs also occur in other manipulation. The same configuration or trajectory may belong to an ongoing performance, a pause, or a completed action. A recovery keeps a lease's capture continuation live.

Fight Night Combat's 2025 referee rules decide when a player is still juggling three clubs in a match (Fight Night Combat, 27 July 2025). They distinguish a club in the air, a club in dynamic contact, and a club in non-dynamic contact. Dynamic contact includes a movement of one contact point that immediately reverses a descending club, an upward deflection from two or more ascending contacts, and a bounce or roll off a body or another club. Non-dynamic contact includes an ordinary catch or hold, a trap, a catch-and-throw with a non-hand body part, and a sustained balance. A player with one club in each hand and the third in non-dynamic contact is no longer considered to be juggling three clubs. Empty hands with all three airborne or dynamically contacting a body can constitute a live pattern. Occupancy records grip, and the chosen grain decides whether a body-touching club that remains in the pattern is retained.

Two clubs may sit in one hand while the third is airborne. Partial control of a three-club pattern remains when one of the held pair is thrown before the airborne club is caught or otherwise comes into contact with the player or opponent. A high throw remains in the player's pattern until it is dropped or discarded. The match rule is one cultural projection over that physical trajectory.

Siteswap beside retention

Siteswap records which hand an object is thrown to and how many beats pass before it is thrown again. Juggling Lab allows a to be interpreted as a hold, uses to force a normally held throw, and uses for no throw (Juggling Lab). Chung and Graham’s siteswap state is a landing schedule: a binary sequence marking the future beats on which airborne balls will land (Chung and Graham, 2008). A complete hold of three objects on two hands places two objects in one hand. The two-beat writing of that occupancy is ; a string of vanilla s averages two objects. Leaving the hold on an alternating one-throw-per-beat flash writes at the loaded hand: one object stays as a , one leaves as a . Juggling Lab’s generator treats a multiplex that contains only held s as outside “true multiplexing” (Juggling Lab generator). Daniel Simu lists starts and stops among the places where siteswap enters multiplex notation, and writes that siteswaps assume the middle of a pattern (Existing notations; dNote). Beatmap records the same occupancies as , , and (Beatmap).

Siteswap and retention are two projections of a richer physical performance :

describes scheduled throws and future landings; describes present bodily retention. The landing bits of supply the scheduled return that keeps an unheld object leased. Neither projection is generally injective. Different physical performances can share a siteswap, and different siteswaps can share a coarse retention trajectory. A numerical siteswap alone also leaves dwell and the physical reading of a open. A grip-aware pattern search would therefore ask for a legal siteswap, a legal retention path, and compatibility between them.

The projections also use different clocks. A siteswap beat has an integer event index , while retention unfolds in physical time . A realization assigns physical event times and fills the intervals with catch times, dwell, releases, and flight. The siteswap can stay fixed while the sequence and change.

Simulator

Under the hold convention, is a one-object pattern that stays in . The periodic repeatedly enters total release, while contains a flash-to-all-held segment that walks before the repeating schedule continues. Asynchronous cycles , , and start from a multiplex hold, throw a classic -high flash, collect, and return to hold. Synchronous also releases all three, passes through empty hands, and retains all three again.

The pattern field compiles asynchronous beats, synchronous pairs, multiplexes, x suffixes that toggle the ordinary parity rule, and mixed packet streams on one base-beat clock. An asynchronous token advances one beat. An ordinary synchronous pair such as advances two; a trailing ! suppresses that spacer, as in Juggling Lab notation. This admits hybrids such as 5(2,4)1 without adding an intermediate object-state. The interface's own mask symbol ? stands for one unknown compressed throw while every fixed token remains fixed. Rhythm family and written period construct a random mask; for either random or manual-mask completion, the mask, object count, maximum throw, and seed define the bounded traversal. It checks at most search states and displays at most results, reporting whether the search completed or met either bound. Its randomized order is reproducible, but it is neither a uniform draw from all legal siteswaps nor necessarily exhaustive. Random hybrid generation adds one filter: both rhythm families must contain a positive action. Manual-mask completion interprets the submitted mask literally. Legal matches appear in a collapsible list and can be loaded back into the court.

The comparison reel below the court recalculates its points under the current dwell, tempo, and reading of a non-crossing . It compares either the curated examples or the current mask matches; clicking a point loads that siteswap. Its numerical intervals use the exact relation . Physical-metric comparisons omit any pattern–dwell pair that would give a non-hold throw negative flight; structural comparisons may retain it. The court may still draw that pair with a short visible pass, solely as an animation accommodation. The displayed correlation is an uncontrolled description of the visible, search-biased set and invites inspection of a candidate relation. It performs no matching or causal test. Estimating a population of patterns or performers would require a declared sampling frame and empirical observations.

Measurements

For a fixed nonempty domain over an interval of duration , mixed-state persistence is

Average normalized retention is

These quantities answer different questions. A system spending half its time in and half in has and ; a two-object system retaining exactly one object throughout has and . More detail lives in the occupancy signature

An entropy becomes meaningful only after choosing a distribution over occupancies or microstates, for example . It would measure diversity of visited retention configurations. Environmental uncertainty requires a separate stochastic model.

An earlier performance vocabulary also turns on dwell. Denis Paumier reported in 2004 that Jérôme Thomas called short-dwell juggling binaire and long-dwell juggling ternaire, within a practice that varied temps de préhension (Kaskade 73). That vocabulary distinguishes duration; the present topology distinguishes which retention configuration exists. Dwell controls how long a trajectory occupies its regions.

For a uniform juggle, Shannon related flight , dwell , vacant time , balls , and hands by

If is the dwell ratio, then in the ordinary one-object-per-hand model without joint retention the mean number retained is . Under the constant-timing assumptions of an alternating two-hand three-object cascade, the two hand cycles are half a cycle apart. This gives an exact occupancy signature, apart from isolated event instants:

Consequently , , and . Below , the cascade alternates between and . At and above the boundary, at least one hand is occupied and at least one ball is airborne, so the path remains in . At the boundary the two hand-contact intervals merely meet; packet semantics decides the state at that isolated instant, which has zero duration in these measurements.

Phase can be isolated in a second exact law. Let each of two objects be retained on one contiguous circular interval of normalized length , with circular offset . Apart from interval endpoints, their retained-overlap share is

and the three macrostate shares are

For the tested value , this becomes

The second branch is a share plateau. At , occupies one positive-duration bout per cycle; for , it occupies two. Equal per-object retention shares and equal macrostate shares can therefore coexist with different fragmentation.

Beek and Turvey reported , , and as prominent dwell ratios for three-ball cascade juggling and about for five and seven balls (Beek and Turvey, 1992). In this ideal cascade, means one ball is held for of the time and two for ; at the shares are and ; at they are equal. All three modes remain in . In the experiments, the three-ball dwell ratio decreased as juggling frequency increased and did not vary systematically with ball mass; five- and seven-ball ratios remained near across frequencies, while three-scarf ratios often exceeded and varied inversely with frequency. A 2024 robot-planning paper cites human preference near while choosing for its simulated hardware constraints (Gomez Andreu, Ploeger, and Peters, 2024).

The ratio can be measured on every hand cycle. If is a catch by hand , its following release, and the next catch by that hand, then

When every active object and every relevant bodily retention channel is recorded, with simultaneous channels assigned and deduplicated by object, the complete contact record reconstructs . It reconstructs the identity-bearing when object identity or an unambiguous assignment is recorded as well. A study can then report hand-specific dwell distributions, phase offset, , macrostate persistence, and the duration of each boundary visit. A mean cannot determine the path once hand phasing and cycle-to-cycle variation are allowed. Yamamoto, Tsutsui, and Yamamoto predicted learning modes near , , , and ; their observations strongly supported the first two, gave only a hint of , and did not realize (Yamamoto, Tsutsui, and Yamamoto, 2015). Later work with eighteen jugglers, seven of them experts, found between dwell ratio and a hand-motion frequency measure separating smoother rhythmic movement from movement with a stop phase (Yamamoto, Shinya, and Kudo, 2018).

This leaves a measurable programme open. Experiments can vary dwell together with left-right phase, tempo, object count, siteswap, and prop type; they can test whether local dwell variability predicts brief visits to , whether a catch error changes the next release time, and whether distance from relates to run length, drop probability, or spatial variance. At fixed , a pure change of tempo preserves the ideal occupancy fractions while increasing the number of retention events per second. These are hypotheses about physical realizations of the algebra; the topology alone supplies no performance law.

Siteswap comparison hypotheses

Let index the event packets in one written cycle. Let count its positive throw or hold actions, count actual releases after the chosen hold convention, and name its packet rhythm. Four structural coordinates are

Here is release concentration, defined as zero when there are no releases. If an elementary release is drawn uniformly from all releases in the cycle, is the expected number of other releases sharing its packet: isolated releases give , strict pairs give , strict triples give . is empty-packet share; preserves clustering that discards; and is asynchronous–synchronous switching density per written beat. The reel also measures the variance of positive siteswap values, notation period , the compiled labelled-prop-and-hand routing cycle , and microstate change rate. The ratio matters because a short written pattern need not immediately return every named prop to the same hand route.

Nine comparisons expose candidate pairs in the article. The reel shows their raw associations; matching and stratification belong to a subsequent generated sweep. In that designed analysis, the first hypothesis predicts that larger coincides with more time in after object count, period, dwell, tempo, and positive-value distribution are matched. The empty-packet hypotheses separate as a predictor of from as a predictor of the longest bout after is held fixed. A fragmentation comparison permits equal with different entry rates . Further tests would estimate conditional associations of with after matching the mean positive value and of with boundary-entry rate after matching occupancy marginals. The final two compare with and ask whether equal can conceal different rates of identity-bearing microstate turnover. The airborne-pair reel adds an exact relation for the distribution of :

Pair exposure can therefore rise while mean airborne count stays fixed if its variance rises. It measures occupancy; collision probability would additionally require spatial trajectories and collision geometry. Tempo supplies another exact scaling check. When all event times scale with the beat and dwell ratio stays fixed, the macrostate shares stay fixed while entry rates scale with beats per second. The hold- switch is a semantic sensitivity test; changing it can alter without altering the legality of the written siteswap.

One fixed comparison makes the distinctions concrete. At a beat, dwell ratio , and the hold- convention, independently implemented exact interval calculations in Ruby and Julia give 3, 441, and 531 the same and . Their compiled prop-and-hand routing cycles are nevertheless , , and beats. Under the same protocol, 55500 gives ; 2[22] remains at ; ([44],4)(0,0)([22],2) gives with release and catch packets of size three; and 5(2,4)1 gives with a maximum catch packet of two. These are exact consequences of the declared interval protocol; the independent Ruby and Julia implementations agree on all 31 shared deterministic metrics for all seven fixtures. Difficulty, error, and effort remain unmeasured. Their performer-level extensions become empirical hypotheses: packet size may predict correlated catch failure, empty-run length may predict correction demand, and microstate turnover may predict coordination load after the mechanical controls are matched.

Each visible divergence can seed the next protocol: hold one coordinate fixed, vary another with the mask, carry the selected notations into Ruby or Julia, and decide whether the remaining relation deserves a physical or human experiment.

A total-release load hypothesis

The informal phrase constant flash needs a fixed protocol. A periodic , separate flashes from rest with catch, hold, and restart, and a low-dwell cascade that enters are different trajectories. Call a periodic path a repeated total-release cycle when it returns to the same after a cycle of duration and enters at least once within that cycle. Over an observation of duration , keep the rate of entries and the time spent there separate:

Neither quantity is an effort measure. The originating observation is that sustaining repeated flashes feels physically draining. Its protocol was not recorded, so this remains testimony about . A plausible mechanical contributor is repeated launch, arrest, and restart, but the retention sign alone cannot identify it.

One mechanical lower bound does follow under explicit assumptions. Consider a free-toss realization on spanning one or more complete mechanical periods, so the props' total vertical momentum satisfies . Let their total mass be , let gravity be the only non-body vertical force, assume a finite continuous body–prop force, and classify every interval with nonzero body–prop force as retention. The bound then follows from momentum balance on intervals already named retention. Let be the signed vertical component of the total body-on-prop force, positive upward. Momentum balance gives

Because during , force can act for at most . Therefore, for ,

For the uniform three-ball cascade above, with equal ball mass and , , so

More time in total release compresses the required net upward body impulse into the remaining retention windows. Exact constant cannot form such a finite-force periodic cycle. This is a bound on external contact-load concentration. Fatigue, oxygen demand, and perceived exertion require their own measurements. Phase also blocks a simple energy reading: two one-ball columns can retain identical per-object trajectory shapes and dwell while a change in relative phase changes and the timing of their aggregate load.

A symmetric ballistic calculation supplies one launch-energy scale. For a throw of mass with flight time , equal release and capture heights, and negligible drag,

The sum ranges over throw events released during and counts vertical kinetic energy carried at release. It is not actual positive body work, because incoming energy may be redirected or recovered, and it is not metabolic cost. A richer model can add prop momentum-change magnitude and positive joint work . Body-contact impulse follows from only after accounting for gravity and other external impulses over the contact interval. The hypothesis becomes testable only after matching or controlling object count and mass, throw height, tempo, dwell, relative phase, catch compliance, and correction rate. Simulation can compare the mechanical terms; a human study must measure any claim about fatigue, oxygen demand, or perceived exertion. The useful question is whether or predicts cost after those controls.

Beyond a binary grip

The pair , where contains positions, velocities, rotations, and other continuous variables, is a hybrid state representation. A full hybrid dynamical model also specifies continuous flow laws, event guards, and impact or capture resets; position may stay continuous while velocity jumps. Robotics already models juggling systems with continuous flows and impact jumps, and nonprehensile manipulation studies planning and control without stable grasping (Sanfelice, Teel, and Sepulchre, 2007; Ruggiero, Lippiello, and Siciliano, 2018; Ramadani et al., 2026).

Binary bodily retention remains a quotient of a richer relation. In a basic poi configuration, the working grain is the poi as a unit, retained at the handle, while a tether constrains the head. Occupancy bits record occurrent retention. A constitutive tether is a standing constraint: marked held, occupancy is unreachable and mixed occupancy is almost automatic. Spinning with both handles held stays ; releasing one handle while the other stays parked is ; releasing both is ; both parked is . Varpanen’s poi state graph records the hand–tether braid as a spatial layer beside the siteswap (Varpanen, 2014; Fontanov, 2025). Bounce juggling lets the floor constrain an object. Passing distinguishes release by one body from capture by another, while a handoff may include joint retention. A pass is a lease whose catcher is the other body; a self-throw keeps the catcher on the thrower. A time-varying constraint relation between agents, components, environment, and objects can represent these cases. The set keeps only the direct body–object retention edge. When several people are combined into one body, this projection erases who released or captured an object, how many hands participated, and whether retention was joint.

Stance-balance is a further global constraint on which contacts may exist together. Each hand can take a club while the pair dumps the stance; a release can restore a viable balance. Holding two equal clubs can keep that stance; holding only one of them can dump it, so a subset of a feasible held set can be infeasible. Mixed occupancy is held and unheld objects present together. In the practice names that is tained and leased together. How evenly those roles are shared is a further measure. A cascade's mixed sign continues by exchanging which objects occupy the contact points.

This quotient keeps release separate from loss of control, state separate from intention, and siteswap separate from physical realization. It also gives structural definitions without deciding the cultural name of a performance. A mixed-retention episode spends positive duration in ; retention circulation visits different microstates along a path inside ; a fully participating cycle returns to its initial after every active object has visited both binary values. Under fixed-domain single-object events, two objects first permit heterogeneity, three first permit circulation, four first permit one-event buffering, and five first permit circulation within that buffered interior.

Note

I wanted a three-object flash from rest, then a hold of all three, then the flash again, written into . Direct does not close. After the three fives the landing sites still need before the held cycle sits. The restart from that hold is . Short cycle averages , and the court walks , , and . Repeating lengthens the hold, as in . Four and five follow the same peel from the heavier hand: and . A synchronous lattice is a different clock from .