Part I · The Sentence and the KernelChapter IV
World, Kernel, Observer
How do a growing world, its invisible kernel and its observers fit into one architecture?
The first three chapters spoke about one observer: what its sentence forces, what its blindness costs, and where its alternatives interfere. An observer is not alone. It exchanges letters with others, and each exchange is one occurrence in two pasts. This chapter builds the world in which such occurrences happen and asks how the world, what each observer holds of it, and what no single observer holds fit together.
The picture is four registers exchanging letters in pairs. Two different schedules of exchanges can leave all four registers with identical words while the causes differ: in one, a late exchange depends on two earlier ones; in the other, it depends on a single earlier exchange and an unrelated one happens beside it. The words alone cannot tell the two worlds apart. If every exchange is written by both participants, each naming the other, the difference reappears in the records, and so does the entire causal order.
In the append/exchange world, registers anchored at reports grow words by private appends and by exchanges that write a received letter into both participants.
(i) If both participants record every exchange, each naming the other, the complete local logs determine the causal order of the history uniquely. Without the names, reconstruction first fails with four registers after three occurrences.
(ii) Independent occurrences commute, and alternatives that write different permanent records never reconverge: the world’s histories, taken up to the order of independent occurrences, form a multiway system that is non-confluent by construction.
(iii) Every word of the program’s tower, its model of one register’s memory, is the log of one register in this world, given partners at the three other reports, and each consecutive pair of its letters has a unique origin: persistence, an own move or a received exchange.
In the lift of Chapter XIII the three origins are changes of rest frame: no boost, a boost with , and a boost with .
Status
Part (i) is the vector-clock theorem of Fidge and Mattern in the program’s setting: timestamps that name partners capture causality exactly, and without them they do not. Its premises are complete logs, known initial words and persistent identities. Part (ii) places the world among Mazurkiewicz’s traces, and is Gorard’s precondition for quantum mechanics in the Wolfram model. Parts (i) and (iii) were established by exhaustive census at small size together with general arguments that extend them to all sizes. The rapidities and the first item of the lattice proposition are proved by hand, the rest of that proposition and the lift’s grammar were computed exactly, and the statements on binding combine exact criteria with numerical Green’s functions.
What the world does not supply is equally exact: it fixes each observer’s causal order, not space, and it supplies possibilities, not weights. The exchange itself is an added clause, the encounter rule. Partner names are what match one log to another, and the quanta of a field carry none; when registers are read as quanta, the working form of P3 records the partner’s state instead, which in every case the census tried fixes the order completely for registers that obey Fermi statistics, and those statistics are assumed, not derived. In the lift the arena and the crystal are one space read in two ways, on the condition that the lift’s symmetries are the symmetries of space, which the program’s axioms have not yet been restated to record. One adopted clause about contacts gives a bound composite; it binds but does not give gravity.
The world clause
Read on the world side, the founding sentence’s first clause gives a partial order and no clock. Occurrences are ordered by record dependency: an occurrence precedes another when the second reads a record the first wrote. Independent occurrences, which read and write disjoint records, commute. Records are permanent, so alternatives that write different records are different histories. There is no global clock; a register’s age is the number of occurrences in its own log. The forcing theorem proves what this structure is: if histories agree on the complete past of every occurrence they share, the local orders glue into one partial order, every history is a downward-closed part of it, and the admitted schedules are exactly its linear extensions. Histories up to the commutation of independent occurrences are Mazurkiewicz traces, the event-structure picture of concurrency theory.
In the append/exchange world a register has an anchor, one of the four reports, and a word of letters. A private append writes a new top letter equal or adjacent to the current top; the antipode is forbidden, so there are five targets. An exchange joins two registers anchored at different reports and . Their shared rod is the letter , and when their tops are the two opposite letters on its axis, each appends the other’s top, which is the antipode of its own. The exchange is the program’s encounter rule, an added clause; everything else is the tower’s own grammar read as a growth process.
This world is non-confluent by construction. Two different appends at one register write different letters into the same permanent position, and an append and an exchange write a non-antipodal and an antipodal letter there: no continuation can bring such branches back to a common word. Competing exchanges can reconverge to common words, but even then an earlier record can separate their causal pasts permanently: with three registers, in one branch every later event descends from an append, while in the other an exchange remains a second minimal event forever. The world thus meets Gorard’s precondition for quantum evolution at the world level, and asks for confluence only of what observers see.
Provenance
In a valid family of logs, the -th exchange with recorded by is the -th exchange with recorded by . Fuse those two entries into one event, keep each private append as a single event, and join each register’s events in order. The result is the read/write order, because an occurrence reads exactly the current records of its participants, which were written by their previous events. The program’s census checked the reconstruction on every history prefix of up to four occurrences, for one to four registers and all labelled one-letter initial states. Without partner names it fails: with four registers, 216 initial states already reach, within three occurrences, two histories whose words agree while their causal orders differ, and within four occurrences of the do.
So the world’s causal order is the observers’ logs glued along their shared occurrences. No single log holds it; together, with names, they hold all of it. This is why the program’s clause S8 was refined to its adopted form: one occurrence between two observers is one distinction recorded by both, each record naming the other. The names match one log to another, and the quanta of a field are identical and carry none. Records of the received letter alone lose the order. Records that copy the partner’s state fix it except where two registers in the same state could have exchanged roles, and in the census every ambiguity that survives is of that kind; by the blindness principle P2 such exchanges carry one amplitude, so they interfere, as identical particles do. Registers that obey Fermi statistics never share a state, and for them, in every case the census tried, records of states fix the order completely. A fresh mark written into both logs at each meeting would also recover it, as a detector records where and when a particle arrived, never which particle, but nothing in the program supplies such marks. So when registers are read as quanta, the working form of P3 records the partner’s state rather than its name.
Start from words and compare the schedules and , where is an exchange between registers and . Both end at the words . In the first, the last exchange reads records written by both earlier ones: a join of two parents. In the second, the two exchanges between 0 and 3 form a chain and the exchange between 1 and 2 stands alone. With partner names, register 0‘s log reads “received from 2, then from 3” in the first world and “from 3, then from 3” in the second, and the two worlds separate. Without names they do not: registers 0 and 1 start alike, and so do 2 and 3, so no copy of a partner’s state, however complete, separates the two worlds.
Words are logs
From a letter the other five letters are its four neighbours on the octahedron and its antipode, which shares no report with it. So for consecutive letters of a word exactly one of three holds: (persistence), shares exactly one report with (an own move), or (a received exchange). A pattern of origins of length occurs in exactly words of length , where is its number of own moves. The origins are readable from the word itself, so a register is never blind to whether a record was its own or received.
The tower’s own moves then take on a new meaning. A move that changes the newest letter replaces one possible last record by a sibling; a memory rewrite moves between possible logs of one age. In this precise sense the tower’s walk is branchial, a walk among the alternatives an observer’s log leaves open, while the world’s time is the append/exchange order. The identification is partial: a third of the free moves involve received alternatives, and the deepest memory rewrites join words with no common ancestor.
Fix a register anchored at one report, with one partner at each of the three other anchors. Every word over the six letters, of every length, is the log of this register in some history of the append/exchange world. With a single fixed partner the logs are a proper subset: at lengths 2,3,4 there are 32,172,924 of them, against 36,216,1296 words.
Build the word letter by letter. A repeated or adjacent letter is the register’s own append. For an antipodal step, choose the partner at the other end of the rod through the register’s anchor on that axis, prepare its top with at most two private appends, which suffices because the octahedron with persistence has diameter two, and perform the exchange. Partners’ earlier records are never reset, and the same three partners serve for every step.
The kernel between world and observer
The completion of an observer identifies all histories of the world that leave the same marked log for . Its fibres are the kernel of : the partners’ identities and private histories, their readiness, and the interleavings of occurrences never took part in. From its current letter an observer’s possible next records are exactly the five own letters equal or adjacent to , and the antipode of received from a partner at a different anchor whenever lies on their shared axis. Any hidden world compatible with the observer’s view can realize each of these, because a partner can be prepared privately in at most two steps: equality of views is a weak bisimulation of possibilities. The adopted frame reads this as making effective causal invariance, for each observer, a property of its records rather than an assumption. In Gorard’s analysis causal invariance buys relativity itself; the program’s world is not causally invariant, so that route is closed at the level of the web, and it is not needed: the report form supplies the signature, the report counts’ integral Lorentz group supplies the transformations, and the web is asked only for a unique recorded order for each observer and agreement on the order of measurements.
What completion does not supply is a weight. The natural uniform counts over hidden partners disagree, one-sixth against one-seventh for reception against an own move, and the count over unbounded hidden histories cannot be normalized, so the observer’s weights are its law, not the world’s multiplicities; the measure between classes is a clause, now adopted as premise P4. The kernel of Chapter III and the kernel of an observer are two faces of one thing: there, what the permanent record does not keep, content rewritten in place; here, the rest of the world behind an observer’s receptions. Clause B joins them, with one amplitude per auditable class.
Provenance, read as a rule about which records an occurrence may later rewrite, does further work. Suppose a record may be rewritten only on an axis carried by one of the two records that produced it. Then, among the relational memory rules that reproduce four records at every scale and respect inversion, the tower’s rule is the unique survivor. Locality in axes, the last assumption of the tower’s memory rule, is derived from provenance; what remains assumed is the provenance clause itself. The rule it selects has a simple arithmetic shape: on any two axes its rewrites count in binary, adding or subtracting one with carries.
Records are boosts
Provenance fixes order and no geometry. Part III finds a geometry for the same records without a new clause. The report form of Chapter I has an integral Lorentz group, the transformations that carry report counts to report counts. Dividing velocity space, the hyperbolic space of rest frames, by the elements of that group that become the identity modulo a prime above seven gives the lift of Chapter XIII, a hyperbolic three-manifold whose eight ends, its cusps, are the eight points of the finite sky. In the lift a report is a null vector and a letter is the unit rest frame proportional to , so every record is a frame, and consecutive records are related by a boost.
The grammar of these records is the geometry of the lift’s cells. Each clock owns a cube of the lift, made of two tetrahedra; its four reports are the four ends of one of them and its six letters are that tetrahedron’s six edges. Each edge is an anchored observer: the letter of clock is an observer of clock , the third point of its axis, so a change of frame at one clock is a boost along the line of sight of an observer at another. An own move replaces an edge by one sharing a cusp with it, and a reception by the disjoint edge. Cycles of three own moves go around a corner of the tetrahedron, where the lift’s transport is trivial, or around a face, where it is a half-turn. The rapidities are per record; how many records occur per unit of time, the cadence, is not fixed by them.
Read each record as the rest frame of its letter. Then persistence is no boost, an own move is a boost with , speed , and a received exchange is a boost with , speed .
Let every report be null and every pair of distinct reports have the same product . Then . Two letters sharing the report have , since three of the four cross terms are products of distinct reports, and two antipodal letters have . Dividing by gives the Lorentz factors and 2 between the unit frames, and speeds of and .
One lattice, two readings
The lift also offers positions, as a reading rather than a clause. If each record adds its letter’s rest frame to its register’s report count, a register traces a worldline of unit steps of proper time, one per record. In the frame in which the four reports are symmetric, coordinate time is times the count for every register, and the six letters’ spatial parts are along three orthogonal axes: in this arena a register walks on a cubic lattice at of the speed of light. The arena is flat and has a preferred frame. The lift’s own transport does not bend it: carrying each letter’s frame to the next by the lift’s spinor transport develops any path into exactly its report count.
At first sight the arena is a rival to the crystal of Chapter VIII: its letters sit on three axes with antipodes opposite, and an odd relabelling acts on them as a mirror, while every relabelling acts on the crystal’s periods as a rotation. The lift removes the choice. There the arena and the crystal are one space, read as positions, where the records have put a register, and as circulations, how its vantage has turned. The mirror that seemed to separate them belongs to the report qubit’s abstract symmetry group, whose odd elements are reflections; the lift acts on both readings by proper rotations. The conclusion is conditional on the lift’s symmetries being the symmetries of space; it does not say which reading a given law uses, and it says nothing about dynamics. A meeting would record a position and a circulation, both in this one lattice, and how such records combine is the composition law.
(i) The spatial parts of the report counts form a body-centred cubic lattice. A count made of an even number of reports lies on the cubic lattice of corners, where the letters’ steps are; a count made of an odd number lies at the cube centres, where the single reports are. The crystal’s periods form the same lattice, with its square loops at the corners and its triangles at the centres.
(ii) The Hodge star of Chapter V induces an isometry between the two lattices. It sends each report to the triangle opposite it and each letter’s step to a square loop of the crystal.
(iii) On the report qubit’s own symmetry the isometry commutes with relabellings of the reports only up to the sign of the permutation: the report counts carry the tetrahedral group, mirrors included, and the periods the rotations of a cube. In the lift a clock’s relabellings act on the lift’s eight ends, the clock’s four reports and their four antipodal anti-reports, as the rotations of a cube, the odd ones exchanging reports with anti-reports, and on the report counts they act by proper rotations. There the isometry commutes with all twenty-four relabellings, and by Schur’s lemma it is the only such isometry, up to the one sign that the orientation convention P5 fixes.
For (i): in suitable units the reports’ spatial parts are four alternate corners of a cube, , , and . Their integer combinations are exactly the integer triples whose three coordinates have the same parity, and the coordinates are odd exactly when the number of reports used is odd; two reports sum to a vector , a letter’s step. The periods give the same set: the square has period , and the four triangle periods are odd points . Items (ii) and (iii) are exact finite computations over the twenty-four relabellings.
Meetings, and the one added clause
In the arena a meeting that requires coincidence is rare. Two free registers that share one occurrence drift apart like the square root of their count, coincide again with probability decaying like the count to the power , and so meet only finitely often, as Pólya’s theorem requires in three dimensions. A steady rate of meetings needs binding, a closed arena, or an unbounded uniform crowd, and no contact rule clusters such a crowd, since every record moves its register at of the speed of light and a contact can change only directions. Binding comes from what a meeting records. Matter’s hop carries two blind sums, and reading the shared letter at a contact audits one of them, which lowers matter’s mass in every tested branch. The native writer does not read it: a reception exports the oldest letters, so the letters of a contact are read only when they depart, too late for an audit to act, since clause B works at each occurrence. The program has therefore adopted one clause, reversibly and as an addition still to be unpacked: at a committed shared occurrence, for a fraction of contacts, the receipt includes the shared letter, and the other contacts stay coherent until commit.
With the clause, two registers in the arena bind once exceeds about 0.75; for from 0.8 to 1 the binding is 0.03 to 0.29 per record and the pair’s size 5.6 to 1.8 lattice units. The bound state is a co-moving trap, two registers in one place with one letter, turning together. The unread contacts, heralded by receipts that lack the shared letter, keep the full Bell violation at rate , so binding and Bell correlations coexist and compete for the same bit. Binding saturates, at most per register and record, and a bound pair couples to an external audit about five times less, per unit of rest leakage, than a free register, so the clause binds but does not give gravity. The lift has a candidate mediator, its massless helicity-two field, which couples to every massless field by one rule, the equivalence principle’s form, but with no strength fixed. What fixes is left open here; on the reading in which a commit writes the shared letter of every contact live at it, in the arena, since every encounter there spans a commit, and Chapter I lists that reading as the adopted clause P3b. The trap as computed has no exchange statistics: swapping two registers, with their tapes and the names their records give each other, is an exact symmetry of the law, but the law does not fix the sign of the swap. The program assumes Fermi statistics for registers, and then the trap would need an antisymmetric state of their fibers or of their memories.
The world is the growing, non-confluent record of occurrences, ordered by what each occurrence read, with no global clock. An observer is a register: what it holds is its log, a word of the program’s tower, and what it knows of the world is its completion. The kernel lies between: invisible to any one observer, held jointly by all of them, and the place where alternatives that no record separates can interfere. Provenance fixes order, not space. Part II builds space, as it builds every level, from one graph: the four reports and six letters on which every register’s log is written.
The missing pieces are equally definite. The world supplies possibilities and an order, not a history measure, which Part V takes up as the question of rulial invariants. Clause S8 is adopted in its provenance form, in which each observer’s world stays in its own frame; its strong form, a record state that two observers address at once in their own frames, is where continuous structure first appears, and it has not been adopted. Which observers at different clocks can meet at all is the subject of Chapter XII. The lift gives the records a kinematics, their positions are the crystal’s lattice read another way, and the contact clause gives them a bound composite; the composition law that would turn records into shared positions, and supply gravity, is still missing.
- Words defined here
- arena
- Concepts
- objectincarnationseamcontinuum