Part II · One Graph, Four CoversChapter VI

Time as Count

Plate VI.1Three logs

A

B

C

and six occurrences, drawn at their causal heights. Gold bars are shared occurrences: one occurrence in the world, one entry in each participant’s log. Blue numerals are ages.

VI.1
VI.2
VI.3
VI.4
VI.5
VI.6
VI.7

If no clock is given, what is time?

Nothing in the founding sentence supplies a clock. There is occurrence, there are records, and records shape what occurs next. A register keeps a log of the occurrences it took part in; the world is the partial order of all of them. This chapter asks what, in such a world, deserves the name of time.

The answer the program adopted is the plainest one. Occurrences happen one at a time. A register’s time is the number of occurrences committed to its own log, and when two registers share an occurrence, it is one occurrence in the world and one step in each of their logs. There is no global tick that advances every register at once, nor a family of local clocks whose rates must be coordinated: there is the sequence of updates, and there are counts along it. A count sounds too thin to carry physics. The chapter’s result is that it carries what the program’s records need, and that a continuous time put in its place loses part of it.

The central result · Time as count

In the adopted record world a register’s time is the number $n$ of occurrences committed to its log, and a shared occurrence advances both participants by one. A continuous time put in its place loses two things.

(i) Every native commit reverses the chirality $D$ of the register’s fiber, its eight-dimensional interior, so $\widetilde D=(-1)^nD$ is exactly conserved, and this dressing is forced up to a constant factor. Averaged over a Poisson clock of intensity 5, with contacts off, the same observable decays as $e^{-10t}$ .

(ii) Uncounted waiting does not sum coherently. With $B^*B+R^*R=I$ , the map $B(I-R)^{-1}$ is an isometry if and only if $R+R^*=2R^*R$ , which the native tick violates, with squared norms $1/5$ and about 1.2979; the counted map $\bigoplus_nBR^n$ is an isometry whenever $R^n\to0$ .

The unit of the count, the cadence, cannot be selected by the observer’s relational language.

Status

The first clause of the result is a frame the program adopted, not a theorem: it is the observer’s side of the world clause. Items (i) and (ii) are exact operator identities on the program’s own carriers, the octonionic fiber, the commit rule and the contact rule; the native tick’s spectrum and norms were computed with rigorous bounds and replicated independently. The last sentence is the rate theorem of Chapter I.

That the failure of coherent waiting is why time in the program must be a count is a reading; its test is whether a normalized law native to the program’s carriers can merge unrecorded waiting. Counting every update equally was a choice until the program adopted it as premise P4. How the count relates to a laboratory second is not settled here; it is the bridge problem of Chapter XXIV.

One update at a time

Plate VI.1Three logs

A

B

C

and six occurrences, drawn at their causal heights. Gold bars are shared occurrences: one occurrence in the world, one entry in each participant’s log. Blue numerals are ages.

The program’s adopted world clause has three parts: occurrences form a partial order by record dependency, independent occurrences commute, and recorded structure is permanent. The clause that used to say “one update process, identical at every place” was retyped when this was adopted. It now says that one thread is one register’s log, and its clock is its own count of occurrences. No shared clock across registers is licensed, and a model that uses one is a conditional model and must say so. Wolfram’s models have a clock of exactly this kind: proper time along a path is the number of updates on it, and an event that involves two parts of the system is one update in both of their pasts.

A history is a finite set of occurrences with a causal order; each has one participant, a private occurrence, or two, a shared one. The occurrences of a register form a chain, its log; its age is the length of the log, and its age at an occurrence is that occurrence’s position in it. A sequential presentation is a linear order extending the causal order. The third item of the proposition connects the count to the vector clocks of Fidge and Mattern: the age is the diagonal entry.

Proposition(What the count depends on)

In a finite history: (i) every age is the same in every sequential presentation; (ii) the ages satisfy $\sum_v\operatorname{age}(v)=\#\{\text{private occurrences}\}+2\,\#\{\text{shared occurrences}\}$ ; (iii) the age of $v$ at $e$ is the number of $v$ ‘s occurrences in the causal past of $e$ , including $e$ : the $v$ -component of the vector clock of $e$ .

Proof

A register’s log is a chain of the causal order, and every linear extension restricts to it unchanged, which gives (i). Counting pairs (occurrence, participant) by occurrences gives the right side of (ii), and by participants the left. For (iii), the occurrences of $v$ that precede $e$ in the causal order are exactly those before $e$ in the chain.

Presentations and the factor two

Plate VI.2Two sequential presentations of the same history. The occurrence

e_2

is step five in one and step six in the other;

B

‘s age at

e_2

is three in both.

Two sequential presentations of the same history, $a_1,b_1,e_1,c_1,e_2,a_2$ and $c_1,b_1,a_1,e_1,a_2,e_2$ , give $e_2$ different global step numbers, five and six, and every register the same ages: $B$ ‘s age at $e_2$ is 3 both times. The global position of an occurrence is not an age, and it is not invariant. The final ages $3+3+2=8$ equal four private occurrences plus twice two shared ones.

The same handshake identity appears in the program’s sequential construction on the $K_4$ crystal, where a site has five private successors at unit intensity and each bond carries the stationary exchange intensity $1/18$ . Per site and unit auxiliary time there are $5+q/36$ occurrences and $5+q/18$ units of committed age, which on the trivalent crystal are $61/12$ and $31/6$ : a shared occurrence is counted once in the first and twice in the second. The auxiliary continuous parameter is a mathematical device, not a new physical lapse for each register. An infinite world has no normalized uniform menu of next occurrences; what it has is a locally finite causal order, reached as a limit of finite sequential histories.

What is counted

Plate VI.3Interrupting a half-turn at

m

equal intervals. Records that copy the content freeze it (blue); records of structure common to every path leave it untouched (gold).

A count needs a unit. The axioms say that at most one distinction is exchanged per cycle, but not what one occurrence is. The program’s working clause is that one occurrence is one completed application of the declared local or pair-contact pulse, followed by one commit, whose receipt records the departure, the records read from outside, the participants and the relative vantage. This grain is a declared choice, and the exact tests show that the axioms do not fix it.

What the count must not do is copy content. A commit freezes only the distinctions it records, and structure common to every path costs the motion nothing, which is why the program can count time by commits without paying a Zeno price on every tick. Several other integers are easily mistaken for elapsed time: the live window, rewrite depth and composite radius, causal height, an auxiliary parameter, and the operators called clocks, which generate rotations of internal frames and never count occurrences. Only the committed count is elapsed time.

Example(Counting structure does not freeze motion)

Rotate a two-level system from $|0\rangle$ by $\exp(-iTX)$ with $T=\pi/2$ , so that it arrives at $|1\rangle$ , and interrupt the rotation at $m$ equal intervals. If each interruption writes a complete copy of the content, the process becomes a classical chain with flip probability $\sin^2(\pi/2m)$ per step, and $p_m(\text{final }1)=\tfrac12\bigl(1-\cos^m(\pi/m)\bigr)$ : 1, $\tfrac12$ , $\tfrac38$ and $\tfrac{47}{128}-\tfrac{3\sqrt2}{32}\approx0.235$ for $m=1,2,4,8$ , tending to 0. If each interruption writes the same structural tag on every path, the tags factor out and $p_m=1$ for every $m$ . Frequent copies of content freeze what they copy, which is the quantum Zeno effect; frequent records of structure do not.

The count carries a charge

Plate VI.4Each commit reverses the fiber chirality

D

(blue), while the dressed chirality

(-1)^nD

(gold) never changes.

Each register carries an eight-dimensional fiber on which the six letters act by matrices $\gamma_a$ with $\gamma_a^2=-I$ and $\gamma_a\gamma_b=-\gamma_b\gamma_a$ for $a\neq b$ , from left multiplications by imaginary octonions. Its chirality is $D=-iL_7=-(-1)^N$ , with $D^2=I$ and $D\gamma_a=-\gamma_aD$ for every $a$ , where $N$ is the fiber’s occupation number. The canonical commit exports the oldest letter into a fresh record cell, and each of its branches acts on the fiber by a single letter matrix, so it is odd; the contact between registers, the memory rewrites and the positive frame transports are even. So a commit reverses $D$ , and nothing else in the program’s dynamics does.

The counter does not have to be read from an inaccessible past: the identity is an invariant of the history, and a single parity bit carried forward would realize it. In a graded realization of the outgoing cells the conserved quantity is the total parity of the live core together with its departed records. Charges in this world leave with the records as balances in time; they are not spatial Gauss laws.

Proposition(The age-dressed chirality)

Let $Z|n\rangle=(-1)^n|n\rangle$ and $S|n\rangle=|n+1\rangle$ act on a record counter. For every fiber operator $B$ with $DB=-BD$ , $(Z\otimes D)(S\otimes B)=(S\otimes B)(Z\otimes D)$ . Hence $\widetilde D=(-1)^nD$ is conserved by every commit, with $n$ the committed count, and by every even operation. Among dressings $a(n)D$ by scalars, conservation under every commit forces $a(n)=a(0)(-1)^n$ .

Proof

$ZS=-SZ$ and $DB=-BD$ , so $(Z\otimes D)(S\otimes B)=ZS\otimes DB=SZ\otimes BD$ . For scalar dressings, $a(n+1)D_{\mathrm{out}}B=-a(n+1)BD_{\mathrm{in}}$ must equal $a(n)BD_{\mathrm{in}}$ .

What a clock average throws away

Plate VI.5Averaged over a Poisson count of intensity 5, the live chirality decays as

e^{-10t}

(blue); the count-dressed chirality stays exactly at its initial value (gold). The decay is a property of the average, not a loss of the conserved charge.

Switch contacts off and embed a single register’s commits in continuous time as a Poisson process of intensity 5, one unit for each of the five private successors. The same world then has a decaying live mode and an exactly conserved history mode, and only the count can tell them apart. A description that keeps only a continuous time parameter, and forgets how many commits have happened, sees a gapped, decaying chirality; a description that keeps the count sees a conserved charge. This is the sharpest sense in which the count is not replaceable.

The status is exact on the program’s own carriers: the octonionic fiber, the native writer and the contact rule. Choosing the sector $\widetilde D=+1$ as physical is a preparation choice, and the alternation of $D$ with the count is not the program’s alternation of matter and radiation, which runs with rewrite depth.

Example(What a clock average throws away)

At fixed count, $D(n)=(-1)^nD(0)$ in the Heisenberg picture. With $n$ Poisson distributed with mean $5t$ , $\mathbb E\bigl[z^nD(n)\bigr]=\sum_{n\ge0}e^{-5t}\frac{(5t)^n}{n!}(-z)^nD(0)=e^{-5t(1+z)}D(0)$ . At $z=1$ this is the clock-averaged chirality $e^{-10t}D(0)$ , which decays. At $z=-1$ it is the count-dressed chirality $\mathbb E[(-1)^nD(n)]=D(0)$ , which does not.

Waiting that is not counted

Plate VI.6One mode with

r=\tfrac12

. The coherent waiting sum

|C|^2

(blue) equals one only where

\cos\varphi=r

, and reaches

(1+r)/(1-r)=3

\varphi=0

; the counted map (gold) has norm one for every phase.

Suppose a register may wait: at each opportunity it either continues without writing anything, by $R$ , or commits, by $B$ , with $B^*B+R^*R=I$ so that a single decision is normalized. If the waiting steps leave no record, the record map sums all the ways of waiting coherently, $C=\sum_{n\ge0}BR^n=B(I-R)^{-1}$ . For one mode, $R=re^{i\varphi}$ and $B=\sqrt{1-r^2}$ with $0<r<1$ , one finds $|C|^2=(1-r^2)/(1-2r\cos\varphi+r^2)$ , which equals 1 exactly when $\cos\varphi=r$ ; at $\varphi=0$ it is $(1+r)/(1-r)>1$ , so coherent waiting over-counts. The probabilities of the single decision see only $r^2$ , so no measure on decisions can supply the phase that would make the sum consistent.

The program’s own update step, its native tick, fails the test. With the waiting lengths summed coherently a dark input has squared norm $1/5$ , and every one of the 144 ordinary prepared inputs has squared norm greater than one, as large as about 1.29793. Recording the waiting count restores the isometry by telescoping, but it is then a different law. Waiting that leaves no record is not an occurrence, and a world that keeps its records must count its waiting to keep its probabilities.

Proposition(The coherent waiting sum)

When $I-R$ is invertible, $C$ is an isometry if and only if $R+R^*=2R^*R$ . The counted map $T=\sum_n|n\rangle\otimes BR^n$ , which records the number of waiting steps, is an isometry whenever $R^n\to0$ .

Proof

$C^*C=(I-R^*)^{-1}(I-R^*R)(I-R)^{-1}$ , which is $I$ exactly when $I-R^*R=(I-R^*)(I-R)=I-R-R^*+R^*R$ . For the counted map, $T^*T=\sum_n(R^n)^*(I-R^*R)R^n=\sum_n\bigl[(R^n)^*R^n-(R^{n+1})^*R^{n+1}\bigr]$ , which telescopes to $I-\lim_N(R^N)^*R^N=I$ .

The unit of the count, and time among the covers

Plate VI.7Time is not a cover. The covers are quotients of the frame path, which can be undone; the count of committed occurrences only grows, and sits apart at the rewrite level.

A count is a pure number, and here its unit is a clean negative. Every positive reweighting of one transition support is an equivalent observer world, and multiplying every jump operator by $\sqrt r$ multiplies every rate by $r$ while leaving all normalized branching probabilities, supports and charts unchanged. The count is dimensionless, and its conversion to seconds is not a consequence of the observer’s language. The program’s constants have the corresponding form: the matter class constant says $mc^2\tau=0.021387\,\hbar$ per class, and a bridge to nature needs one physical cycle $\tau$ . For an electron that cycle would be about $2.75\times10^{-23}$ seconds, and nothing known has that period.

Time is not one of the covers of the kernel graph. The covers are quotients of the path of an observer’s frame, which is live content: a register’s position, if it is the tally of that path, can go back down, and its age cannot. Read physically, position is content and time is structure, and the count supplies a preferred foliation by record depth. In that frame the count is geometric: if each record adds its letter’s unit rest frame to the register’s report count, a register’s coordinate time is exactly $\sqrt{3/2}$ times its count, every record moves at $1/\sqrt3$ of the speed of light, and along one register the invariant interval falls short of $\sqrt{3/2}\,n$ by a bounded amount, about 0.31 on average. On the records’ momenta that frame is one among infinitely many equivalent ones; this is established for the kinematics, and for dynamics it is the program’s continuum work.

Propositionproved

For one declared exhaustion a sequential count survives the passage to an infinite world. Run the program’s complete record dynamics, at raw uniform counting, on a connected active region of $s$ sites of the crystal among $N=s^3$ native writers, for $n=5NT$ occurrences. The equal-occurrence law and a version that resolves the waiting between occurrences differ, over the whole growing tape, by at most $9T/(160s)$ in diamond norm, and the active history converges at rate $O(1/s)$ to a continuous recording instrument; the age-dressed chirality stays exact throughout. Stopping at a fixed count and stopping at a fixed time remain different, and the homogeneous world, where every site has contacts, is open.

The count says how many occurrences a history has, and nothing about which ones or in what order. The next chapter returns to the covers, beginning with the one that keeps the most: the tree of ordered histories.

Concepts: absence