Reality’s record draws its own perspectives.

A history begins as a blend of one or more lineages. A lineage is a starting occupation of the two sites together with a starting loop flux, wherever the two clocks start. No native event of the ledger connects two lineages: however posting spreads the history over records, every lineage keeps exactly its starting weight. No symmetric rule picks one lineage.

Each square of the grid is a lineage: a starting occupation (● occupied, ○ empty) and a starting loop flux. Set the weights of a beginning and post. The figure posts the whole history, keeping every record, and reads each part of it back to the beginning it came from: from its occupation and loop flux after the last posting, back through its record. The weight it finds on each lineage, summed over all its records, is the starting weight at every stage, and all the weight reads back to a beginning. Each lineage starts its two clocks at the same position here; any clock start gives the same weights. The grid draws three loop values; there are lineages at every loop value.

The phase panel blends two starts, half each, with a phase between them. When the two are in different lineages, every record of three postings has the same probability as in the half-and-half mixture, at every phase: the record cannot tell the blend from the mixture, and cannot tell which loop flux a history started at. It can tell ●○ from ○●: the first posting from ●○ writes a current of +1 on the loop edge into the record, and the first from ○● writes −1. When both parts sit inside one lineage, the phase can show in the records: here, with two clock positions at ●○ and loop 0, it changes the record probabilities. It need not: a blend of the persistent loop current + and the oscillation at ●○ also sits inside one lineage, and its records ignore the phase.

The rule test: a rule that respects the posting’s symmetries (shifting the loop flux; filling the empty sites and emptying the occupied ones, or swapping the two sites with their clocks, each with the loop flux reflected) and picks any lineage must also pick every lineage those symmetries reach: every balanced lineage (○○, ●●) or every mixed one (●○, ○●), at every loop value. Such a rule picks none, all balanced, all mixed or all lineages, never a single lineage. The walls between lineages are drawn by the ledger’s own events: no outside environment is traced out, and the record is kept. A larger family of reads, which includes a read that recognition dynamics supplies (its retained read), crosses some of these walls, though never one between lineages with different starting loop fluxes; taking the native family as the operations inside the ledger is an identification, and which family describes them is open.

What is live, and what is drawn

The weights on the lineages, the record probabilities and the rule test are computed live, and so are the other counts, weights and differences the figure reports; the number of postings, the units of weight and the phase are what you set, and the loop values and the 72 phases are fixed text. It posts the whole history from the chosen beginning with the cell’s own step, up to five postings, every record kept, and reads each part back to its beginning to find the weight on each lineage. The phase panel computes the probability of every record of three postings for a half-and-half blend at the chosen phase and for the mixture. The rule test computes the closure of the picked lineages under the posting’s symmetries. The grid’s three loop columns are a window: there are lineages at every loop value.

Computed in your browser from the cell’s posting rule: the history is posted in full, with no sampling, every record kept, up to five postings, and weights are read back part by part; changes below one part in 10¹² show as 0. Record probabilities are computed for every record of three postings.

One cell of the ledger has 64 basic states, and every other state is built from them: two sites, each occupied or empty, and a clock at each site in one of four pole states. Every posting writes one label onto the record and steps the cell. Step through the postings: every one of the 64 except four holds neutral weight from the start or can, within three postings, write a record after which the state given that record has some neutral weight. The four that never leak, at any length, are four motions: two persistent loop currents of opposite direction, and the two states of one oscillation of occupation across the loop edge (at ●○ and at ○●, one posting apart). Every blend of them never leaks either; the work calls the four motions and every blend of them the cell’s participants.

The grid shows the 64 basic states of one cell, a basis from which every other state is built: four blocks for the four occupations of its two sites (● occupied, ○ empty), and in each block the four pole states p, a, P, A of the first site’s clock by row and of the second site’s clock by column. A basic state can start leaking at posting n when, for some record of n postings with positive probability, the state given that record has some neutral weight; the number in its square is the first such n, and 0 means it holds neutral weight from the start. Of the 64, 24 hold neutral weight from the start, 24 more can start leaking at the first posting, 8 at the second and 4 at the third. The other 4 never leak, the four motions: the persistent loop currents + and −, whose loop flux climbs or falls by one at every posting, and the oscillation at ●○ and at ○●, one posting apart, whose occupation hops across the loop edge, from one site to the other, at every posting, keeping the loop flux within one unit of its start. From three postings to four the set of states still free of neutral weight stays the same, and once one more posting changes nothing, no longer record can; so these four never leak, at any length. Neutral weight sits only on the occupations ●○ and ○●, on states where at least one clock is in p or a; the word refers to the clocks, not to electric charge.

The rows follow the chosen start. Each row lists every record it can write in that many postings; a record’s width is its probability, and its colours split the state given that record: cyan on the four motions, grey hatched neutral, plain grey elsewhere. Never falling silent, which the colours do not draw, means writing the loop edge (a nonzero middle current in its label) at every posting. The four motions do it at every posting, with certainty. A cell wholly on neutral weight writes it at each of its first two postings with probability one half, and at each of its first three with one quarter; the Neutral column shows something else, the neutral weight itself. The ready vector is the basic state ○○ p, a. Each of the four motions writes one record of each length and stays wholly on the four motions; every state with some weight off the four motions can, within three postings, write a record after which the state given that record has some neutral weight. Blend two basic states with the Blend button to see that a blend of motions never leaks and a blend with anything else can start leaking. When one of the four motions is chosen, a Motion line steps it through the postings, and each record’s tag gives its occupied sites and how far it has moved the loop flux: the sum of the middle currents of its labels. The number at the end of each row, in the Neutral column, is the neutral weight of the state given each record, averaged over the row’s records by probability.

What is live, and what is drawn

Every record and grid in the figure is live, and so are the counts on the posting buttons and the two large readouts. The grids are computed in your browser from the cell’s posting rule when the figure loads; the records, their probabilities, the split of the state given each record and the weight on the four motions are computed from that rule each time you choose, blend or disturb a start. Nothing is a picture or a recording. The grids’ counts come from an exact rank computation; random, blended and disturbed starts use double precision. Each record’s tag is computed from that record too. The grey hatching marks neutral weight in the rows and, in the grid, the basic states that hold it, or can start leaking, within the postings shown.

Computed in your browser from the cell’s posting rule: 64 basic states, 15 record labels. The grid and the rows for basic states use exact arithmetic; random, blended and disturbed starts use double precision.

Each posting writes a label to the cell’s record, and given the record so far the cell’s weight on the four motions can change. Each square is one sampled record.

Weight on the four motions means weight on the four motions and every blend of them (the space they span). Here the four motions’ part of the state is the persistent loop current +, which writes the label (1, 1, −1) at every posting (a label is the current on the cell’s three edges; its middle entry is the current on the loop edge). A record is the list of labels the postings write. The state given the record is the cell’s state given that list, and the weight on the four motions is read from it. Its average over the next posting’s label, each label weighted by its probability given the record so far, equals its value given the record so far (it is a martingale). Writing never mixes the four motions’ part of the state with the rest, so the record law is the sum of the laws of the two parts, with no cross term. The loop current’s part writes (1, 1, −1) at every posting, so a record that writes any other label has weight 0 on the four motions from then on. So along every infinite record the weight tends to 0 or to 1. It can tend to 1 only along the one record that writes (1, 1, −1) forever, and it does so when the starting weight w on the four motions is above 0; that record has probability exactly w.

How fast is limited by the state’s neutral weight: weight where the sites are mixed (one occupied, one empty) but a clock sits on an undressed pole (p or a). Averaged over records, the part still undecided after n postings (n ≥ 2), the average of v(1 − v) where v is the weight given the record, is at most 2⁻⁽ⁿ⁻¹⁾ times the starting neutral weight, so at most (1/128)ʳ after 8r postings. A state with no neutral weight is decided by its first posting: with w = 9/25 and the rest ●○ A, P, the first label is (1, 1, −1) with probability 9/25, leaving weight 1, or (0, 1, −1) with probability 16/25, leaving weight 0. No fixed finite stretch of record decides for every state: ●○ a, P has no weight on the four motions, yet it writes (1, 1, −1) at each of its first n postings with chance 2⁻ⁿ (and the loop edge at each of them with chance 2⁻⁽ⁿ⁻¹⁾), which the loop current + does surely. The infinite record decides: a state’s chance of writing, forever, the list of labels one of the four motions writes is exactly its weight on the four motions.

Reading the state given the record as the cell’s present is an identification; read that way, the record settles whether the present tends to the four motions.

What is live, and what is drawn

Everything is live. Your browser computes every number from the cell’s posting rule. Each sampled record draws its labels with the probabilities set by the state given the record so far, so the counts out of 100 can change each time you press New records; the chance at each posting stays fixed until you change w or the rest (exact fractions for the two named rests, computed in double precision for a random rest). The sampling is the browser’s way to show the record law; the work does not claim that single occurrences sample it. The starting weight, the rest and the persistent loop current as the four motions’ part are chosen here, not derived. The squares and lines are drawings of computed values.

Computed in your browser from the cell’s posting rule: each sampled record draws its labels with the probabilities the state given the record assigns; the exact numbers apply the posting rule to the loop current’s labels, (1, 1, −1) at every posting, and are shown as fractions for the two named rests and computed in double precision for a random rest.

Contacts can share sites. The parity law is about configurations in which every clock on a site with contacts is one of the two dressed pole states, P or A; a configuration with a clock on p or a at a site with contacts is never kept. Here every clock is P or A. The Star network is the example of step 5: a centre with three partners, every clock A, bits 0 at the centre and 1, 0, 1 at the partners in the order their contacts post.

Each site has an occupation bit, 1 when occupied (●) and 0 when empty (○), and a clock; the small numbers name the sites. The clocks here are the two dressed pole states: P, which keeps its site’s bit when a posting meets a mixed pair, and A, which flips it. A contact posts on its two sites with the cell’s own step, contact after contact in the round order shown. A posting that meets a balanced pair (both sites occupied or both empty) branches: with these clocks it can write any of four labels, and the configuration is not kept.

The parity law: with these clocks, every posting meets a mixed pair, in every round, exactly when (i) at every contact u–v, f(u)·deg(u) = f(v)·deg(v) mod 2, where f is 1 for A and 0 for P and deg counts a site’s contacts; and (ii) at every position i of the order, the two starting bits (as set, before the run) add to γ = 1 + f(u)·ν(u) + f(v)·ν(v) mod 2, where ν (the Greek letter nu) counts the contacts through that site that come before position i. The parity-law box checks both conditions; the strip runs the actual postings. In its round order, the states the postings keep on a network are spanned by these configurations; a site with no contact is free to carry any bit and any of its four pole states. With every clock P, a configuration is kept exactly when its bits alternate across every contact (each contact of a kept configuration carries a persistent current); some bits do this exactly on networks with no odd cycle, and the triangle frustrates them. With every clock A, the six contacts of four fully joined sites keep the joint oscillation in some orders and nothing in others: 192 of the 720 orders keep it.

What is live, and what is drawn

The postings and the counts are live. Each posting is the cell’s own native step applied to the contact’s two sites, and the counts are computed in your browser. The contacts, their orientation and their round order are chosen here; none is derived for these networks, and no posting order is forced.

Computed in your browser from the cell’s posting rule: every posting applies it to the contact’s two sites, and the counts run every choice of P or A clocks and bits on the sites with contacts through three rounds and compare the result with the law. The contacts, their orientation and their round order are chosen here; none is derived for these networks, and no posting order is forced.

A pointer line is an occupation of the two sites and, for each clock, a pole (p, a, P, A) or a standing wave (p+a, p−a, P+A, P−A). Once on a line, the present moves at each posting to a next line, with probabilities that depend only on the line it is on: a classical chain.

The chain is computed from the cell’s own step: one posting carries each pointer line to one line for each label it can write, with that label’s probability (1/4, 1/2 or 1). The figure reads the cell’s state given its record as the present; that reading is an identification. From every start, on almost every record, this state comes as close as you like to the 256 pointer lines in the long run; averaged over records it does so at a proved but far-from-sharp rate, and no collapse is assumed.

Five closed classes, sets of lines the chain never leaves, inside which every line can reach every other: the three that hold the four motions (the persistent loop current +, the persistent loop current −, and the two states of the oscillation); a class of 64 lines, in each of which exactly one clock sits on its zero plane; and a class of 44 lines, in each of which both clocks sit on their zero planes or neither does. (A clock’s zero plane is P, A and their standing waves when the two sites agree, and p, a and theirs when they differ; lines outside the classes can share either property.) The other 144 lines are left for good with probability 1: after 2m postings at most 2⁻ᵐ of the weight remains on them. Each class has a conserved weight, the present’s weight on the space spanned by the class’s lines (lines outside the classes can carry part of it): averaged over records by probability, it is the same after every posting. On the three classes of the four motions the chain, in its steady state, runs the same backward as forward: its stationary current is zero. On the 64-line and 44-line classes the stationary current is not zero; it is divergence-free (as much flows into each line as out of it), and around the two closed walks shown it adds to 1/24 and 3/112, so no potential explains it: a current made of differences of a potential adds to zero around every closed walk. Every reverse step on those walks has probability 0.

What is live, and what is drawn

Every computed number in the figure is live; the class sizes and line counts in the key and header are fixed labels that match the computation. It applies the cell’s step to each of the 256 lines, finds the closed classes, solves each class’s long-run law in exact fractions, and computes the current on every step and its sum around each of the two closed walks, which are the manuscript’s (Theorem 11.4), chosen rather than searched for. The present’s moves are sampled at the chain’s probabilities. The walk drawings are drawings of the computed walks; the moving stroke shows the current’s direction only.

Computed in your browser from the cell’s posting rule: the 256-line chain, its closed classes, their stationary laws and currents in exact fractions, and the weight not yet in a closed class; the present’s moves are sampled at the chain’s probabilities.