Part I · The Sentence and the KernelChapter III

The Kernel

Plate III.1Two routes from the word

(a,b)

back to itself, through one reversible memory rewrite. A description that reads only where the register ends puts both routes in one fibre.

III.1
III.2
III.3
III.4
III.5
III.6

What does every description forget, and why is the forgotten part the source of the quantum?

Chapter II gave a blind observer one amplitude for each class of histories it cannot audit. Which histories share a class is decided by what is recorded. Every description forgets something: a report forgets the state behind it, a log forgets the private history of its partners, a tally forgets the order of its steps. What it forgets is a definite equivalence relation on histories, the kernel of the description. This chapter asks which kernel the world itself has, given that occurrences leave records, and finds that the quantum lives there.

The picture is two histories that leave a register in the same place. If the register’s permanent record distinguishes them, they are two classical alternatives and their probabilities add. If the record does not, they are two routes to one outcome and their amplitudes add, so they can interfere. Which case holds is decided by one thing only: whether the records agree.

The central result · Interference lives in the kernel of the record

Let each history $h$ of a process end with a live vector $z_h$ and leave a permanent record $R(h)$ , written as orthonormal labels in cells that never return to the live content. Then the observer’s live state is

\rho=\sum_c|\psi_c\rangle\langle\psi_c|,\qquad \psi_c=\sum_{R(h)=c}z_h:

amplitudes add within each fibre of the record map, probabilities add across fibres, and no later processing of the departed records changes any future live statistic. Under the adopted two-layer rule, which records occurrence structure and rewrites content in place, alternatives that differ only in content can reconverge and interfere, and alternatives that differ in recorded structure never do.

Status

The formula is exact; it is a partial trace. The statement about departed records is an exact theorem under a declared causal boundary, No Return. The two-layer rule is an adopted clause of the program’s frame, with two refinements still working clauses: what counts as one occurrence, and how the observer’s standpoint, its vantage, is recorded.

Three questions remain open: whether the native writer, which exports the departing letter, realizes the two-layer rule; whether one law schedules the reversible content dynamics and the commits together; and what history measure weights the alternatives. The Bell result below is conditional on two added choices, and it does not show that the native writer realizes the grain it needs.

What a description forgets

Plate III.1Two routes from the word

(a,b)

back to itself, through one reversible memory rewrite. A description that reads only where the register ends puts both routes in one fibre.

A description of histories is a map $D$ from histories to readings, and its kernel is the equivalence relation $D(h)=D(k)$ . Linearly, the kernel of the induced map on combinations of histories is spanned by the differences $h-k$ with $D(h)=D(k)$ : the combinations the description cannot see. A description is exact when its kernel survives every allowed later test, and the forcing theorem shows that such a kernel is the largest thing a description can forget without losing a prediction.

A kernel depends on the questions allowed. Let $A$ and $B$ flip two bits, and let the observer read $p=\frac1{\sqrt2}\left(\begin{smallmatrix}1&0&0&1\\0&1&1&0\end{smallmatrix}\right)$ on $\C^4$ . Both flips act as the same flip on the reading, $pA=Xp=pB$ , and $\ker p$ , spanned by $|00\rangle-|11\rangle$ and $|01\rangle-|10\rangle$ , is an exact observational kernel. Add the reading $pZ$ with $Z=\mathrm{diag}(1,1,-1,-1)$ and the kernel vanishes, because $p^*p+Z^*p^*pZ=I$ . The updates are unchanged; a new question made hidden distinctions visible.

Remark

The program uses the word kernel for three objects. The kernel graph of Chapter I is the fixed data of every frame. The kernel of a description is its equivalence relation. The kernel, without qualification, is the kernel of the world’s permanent record: the alternatives the record identifies.

Two layers

Plate III.2The commit between the two gates writes structure into a fresh cell of the tape. The order of the two records inside the live word is content, rewritten in place.

The founding sentence says that occurrences leave records, not what is recorded. The program’s answer, adopted as part of its frame, has two layers. A register’s live content is its current finite state, rewritten in place between occurrences by reversible moves. At each occurrence a commit writes a record into a fresh cell, and the growing collection of departed cells is the tape. Records hold only occurrence structure: the departure into a fresh cell, the participants of a shared occurrence, each naming the other, and what the occurrence read. Content is not recorded.

Four facts from the program’s research shape the rule. In-place rewrites are reversible: each rewrite block that survives in the generator is orthogonal, so forgetting in place is not losing. Records cannot simply be merged: identifying histories with the same final letter is not a law, since at depth four the resulting maps sum to $\tfrac{141}{125}$ times the identity. Records need room: twenty-seven branches cannot be orthogonal in twelve dimensions, which is why commits write to fresh cells. And records must include what was read, since a private move followed by a neighbour’s private move that read it defeats reconstruction from partner names alone.

Remark

Two refinements remain working clauses, adopted for their consequences and reversible. The grain: one completed application of a declared local or pair pulse to its named inputs, followed by one commit, is one occurrence, and the paths inside the pulse are not further occurrences. The vantage: a commit records where the observer stands only relative to its encounter partners, in the observer’s moving frame.

The record decides

Plate III.3If the commit records nothing that distinguishes the routes, they lie in one fibre and their amplitudes add. If it records the axis of the top letter, which differs between

|ab\rangle

and

|ba\rangle

, they lie in two fibres and their probabilities add.

Part (ii) of the theorem below is the exact content of the two-layer rule. Two histories in the same fibre of the record map are one outcome reached by two routes, and their amplitudes add; two histories in different fibres are different outcomes. It is the rescue form of Chapter II read on histories: one amplitude per class that the observer’s records can audit, summed coherently over the interior of the class.

The word “return” matters. A record that is still live content, even a perfectly distinguishable one, can be rewritten back, and then the alternatives it distinguished interfere again. In the program’s exchange-only control, a branch encoding kept inside the live register gives return probability 0, as if unrecorded; the same encoding written to an outgoing cell gives $\tfrac12$ . “Recorded anywhere” is too strong a condition; “recorded in a cell that never returns” is the right one.

Theorem(The record decides)

Let a live core and a tape evolve by an isometry, so that the joint state after a process is $\sum_hz_h\otimes r_h$ , where $z_h$ is the live vector reached along history $h$ and $r_h$ the record it wrote.

(i) The observer’s live state is $\rho=\sum_{h,k}\langle r_k,r_h\rangle\,|z_h\rangle\langle z_k|$ .

(ii) If each of the $n$ commits of the process copies a structural class $f$ of its step into orthonormal labels, then $\langle r_k,r_h\rangle=\prod_j\mathbf 1[f(h_j)=f(k_j)]$ , and $\rho=\sum_c|\psi_c\rangle\langle\psi_c|$ with $\psi_c=\sum_{R(h)=c}z_h$ , where $R(h)=(f(h_1),f(h_2),\dots)$ is the record of $h$ .

(iii) (No Return.) Let a trace-preserving map $\Psi$ act on the departed tape $T$ alone, with nothing flowing from $T$ or its environment back into the live content or its controls. Then $\operatorname{Tr}_T[(\id\otimes\Psi)(X)]=\operatorname{Tr}_TX$ for every operator $X$ on live content and tape, so every later unconditional live statistic is unchanged.

Proof

Part (i) is the partial trace of $\sum_{h,k}|z_h\rangle\langle z_k|\otimes|r_h\rangle\langle r_k|$ . In (ii) the records are tensor products of orthonormal labels, so their inner product is the product of the label overlaps, one or zero; grouping the histories by record gives the stated form. For (iii) write $\Psi(Y)=\sum_jA_jYA_j^\dagger$ with $\sum_jA_j^\dagger A_j=I$ ; the partial trace over $T$ is unchanged by conjugating the $T$ factor with the $A_j$ and summing. An induction through later live operations, each of which touches no departed cell, gives the statement about statistics.

The smallest interferometer

Plate III.4The amplitudes on the two routes are

\cos\theta\cdot\cos\theta

and

i\sin\theta\cdot i\sin\theta

. Unrecorded, the return probability is

\cos^22\theta

; recorded,

\cos^4\theta+\sin^4\theta

. At

\theta=\pi/4

that is 0 against

\tfrac12

Let two letters $a$ and $b$ lie on different axes. The depth-two words $(a,b)$ and $(b,a)$ , the same two records in the opposite order, are joined by a native memory rewrite, a reversible exchange of the two records. On this pair the generator is the real symmetric coupling $\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$ , up to sign, times the identity on the fiber, the observer’s eight-dimensional interior, and a gate of angle $\theta$ acts by $|ab\rangle\mapsto\cos\theta\,|ab\rangle+i\sin\theta\,|ba\rangle$ .

This is the two-layer law in its smallest instance. The two routes differ only in content, the order of two records inside the live word, and the reversible rewrite brings the content back together. Nothing in the permanent record distinguishes them unless the writer is made to record the axis, and then they cannot interfere. The coupling is the first coherent memory rewrite of words of length two, and both probabilities were computed exactly.

Example(One memory rewrite, recorded or not)

Start at $|ab\rangle$ and apply two gates. The register returns to $|ab\rangle$ by two routes, staying (amplitude $\cos^2\theta$ ) or going through $|ba\rangle$ and back (amplitude $(i\sin\theta)^2=-\sin^2\theta$ ). If nothing is recorded between the gates, the routes are one fibre and $P_{\mathrm{return}}=(\cos^2\theta-\sin^2\theta)^2=\cos^22\theta$ . If the commit between the gates records the axis of the top letter, the routes are two fibres and $P_{\mathrm{return}}=\cos^4\theta+\sin^4\theta$ . At $\theta=\pi/4$ the probabilities are 0 and $\tfrac12$ ; at $\theta=\pi/6$ they are $\tfrac14$ and $\tfrac58$ .

No Return

Plate III.5No Return in three circuits. Copying the live value into a tape cell removes the live interference. Resetting the tape does not bring it back; only coherently undoing the copy, which sends the record back into contact with the live content, restores it.

Part (iii) of the theorem says that a departed record can be edited, copied or scrambled at will, provided nothing comes back into the live content. Erasing the visible record does not restore interference; returning it does, and returning is exactly what No Return forbids.

No Return is a causal condition on the wiring, not a property of the symbols on the tape, and it is weaker than it may look. It does not make records readable: a tape reset destroys every visible receipt while keeping the live statistics. It does not keep charges on the tape: a conserved quantity written out with the records stays balanced only if the processing respects it. And it is not a definition of Markovian dynamics.

Example(Copy, reset, return)

Let the live qubit be $|+\rangle$ and a fresh tape cell $|0\rangle$ . Copying the live value into the cell gives $\tfrac1{\sqrt2}(|00\rangle+|11\rangle)$ , and the live state is $\tfrac12I$ : the visibility of $X$ is 0. Resetting the tape cell, the trace-preserving map sending both $|0\rangle\langle0|$ and $|1\rangle\langle1|$ to $|0\rangle\langle0|$ , sends $|0\rangle\langle1|$ to zero; the live state stays $\tfrac12I$ and the visibility stays 0. Undoing the copy coherently, before any tape processing, restores $|+\rangle|0\rangle$ and visibility 1.

The quantum in the kernel

Plate III.6The CHSH score of two registers in contact, against the classical and quantum bounds. Records of structure leave it unchanged; one record that copies the content brings every local reading down to at most 2.

Two of the program’s registers, in contact for a fixed time with no record written during the contact and then read by complete local reports, violate the Clauser–Horne–Shimony–Holt inequality: by exact interval arithmetic the score lies between 2.54745362600 and 2.54745362601, against the classical bound 2 and the quantum bound $2\sqrt2$ . Records that carry only structure common to all the interfering paths, the departure, the partner and the read set, leave the violation unchanged for any finite number of records; a single record that copies the content makes the source separable, and every local reading gives at most 2. The deciding object is the overlap of the records the interfering alternatives leave. The result is conditional on two added choices, a fixed-buffer writer and the treatment of the whole contact pulse as one occurrence.

Because content forgets and structure does not, the program’s world cannot be confluent in the sense of the Wolfram model; the smallest witness has two registers, and a census of more than four million competing pairs found sixteen further all-future obstructions. In Gorard’s analysis relativistic evolution requires confluence while quantum evolution requires its failure, so the two-layer world meets the precondition for quantum mechanics by construction. Relativity is then asked of observers: each must see one order for its own records, which Chapter IV supplies through provenance.

Remark

Wolfram’s multiway causal graph has three kinds of separation, and the program has all three. Independent occurrences commute exactly, which is the spacelike case. Alternatives are branchlike, and the two-layer law splits them in two: those that differ only in content may reconverge and interfere, and those that differ in recorded structure never do. A change of chart is rulelike.

The kernel returns throughout the volume. Each cover of the kernel graph in Part II keeps part of a history and forgets the rest, and what it forgets is a kernel in this chapter’s sense. The sharpest instance is the kernel of the map from ordered histories to their tally, whose shortest loops in space are decagons (Chapter IX). Whether native alternatives interfere exactly when they have equal tally and equal recorded structure, and never otherwise, is the kernel square of Chapter XX. First the world whose record this is must be built, with its registers and their exchanges: that is the work of Chapter IV.

Words defined here: kernel
In the Esquisse: 1Un objet, plusieurs noms