Part I · The Sentence and the KernelChapter I

The Founding Sentence

Plate I.1The four reports of the minimal register, written

(s,\ell)

, each with the shortest word that reaches it. No letters are drawn yet.

What does the program assume, and why is that enough to start?

The program assumes one sentence: “Something lawfully occurs; occurrence leaves records; records shape occurrence.” It names no space, no time, no particle and no Hilbert space. It speaks of events, of the marks they leave, and of the influence of those marks on later events. This chapter asks how much structure the sentence forces when it is given the smallest instantiation in which each clause does some work, and states with the same care what it leaves open.

The smallest instantiation is a register that carries one sign and remembers the last symbol written. The sign is blind: the register cannot tell which of two symbols produced it, only how often it has flipped. The remembered symbol has a consequence: a probe acts on the sign according to the last symbol written. Everything in the central result grows from the four classes of history that this register distinguishes.

The central result · What the sentence fixes

Instantiate the founding sentence with one blind bit and a probe that reads the last record. Then:

(i) the histories fall into exactly four classes of future indistinguishability, the reports, labelled by the blind bit and the consequence bit;

(ii) the six pairs of reports form an octahedron whose three antipodal pairs are the register’s three binary readings; the relabellings that respect its two fundamental readings, the sign and the probe ratio, form $D_4$ , and the record-controlled dressing of the third clause completes them to $S_4$ ;

(iii) the $S_4$ -invariant forms $\alpha(\sum x_i)^2+\beta\sum x_i^2$ on the report space are Lorentzian for $-4\alpha<\beta<0$ , and null reports select $\beta=-\alpha$ , where the alphabet’s time and space units coincide.

The sentence does not fix the weights of distinguishable alternatives, their cadence, what is written, or how two observers share an occurrence; these enter as named clauses.

Status

Items (i) and (ii) are proved from an explicit construction. Item (iii) is exact algebra; its physical reading is a separate question. The last sentence of the result is a theorem in one case, since the relational language of the sentence cannot express a rate, and a ledger of choices in the others; in the lift of Chapter XIII two of those choices, the weight of a record and the cadence, acquire a geometric reading. “Derived” in this chapter means proved from the sentence and the stated construction; it does not mean that a physical identification is established. The program now states its axioms as six premises, one of them a convention: P0, the founding sentence; P1, exact return; P2, blindness; P3, the world; P4, one update, one weight; P5, the orientation conventions. P5 has two parts, and they are independent. Which prime above seven carries the lift is a hand, since the lift’s own couplings tell it from its mirror image. Which of the octonion table and its mirror is the table, which is also which quartet is matter, is not, since exchanging the table with its mirror and the quartet with its conjugate is an exact symmetry of everything native.

The program has adopted four additions, all reversibly: the contact read P3b, by which a contact is read exactly when it spans a commit; the gauge-link clause, which makes the transports between neighbouring observers’ frames the links of a lattice gauge theory with the Standard Model’s group; three generations, three identical copies of one family that nothing native supplies or tells apart; and a colour-neutral contact, the one form colour allows for a collision of two registers that transfers momentum. The gauge-link clause also adds a breaking: that a pair of the family’s right-handed neutrinos condenses, by a force the law does not yet supply. Two working clauses stand beside them: a pair term, which mass needs, with Wigner’s particle states derived and a field still owed; and Fermi statistics for coexisting registers, although the swap of two registers is itself native. Matter’s spinor is the lift’s own, so matter’s hand is derived; the weak force’s hand needs only what the gauge-link clause adds, the point of the Cayley plane and a vacuum that breaks the symmetry between the quartets, and one naming: which quartet the vacuum keeps as weak. That matter’s interior keeps the octonion product and its unit is decided rather than derived. The generator of a register’s dynamics is an axiom, and its choice moves matter’s class constant but not radiation’s masslessness. The adoption proved nothing new. It records which statements the program now takes as premises and which it derives.

One blind bit and one consequence

Plate I.1The four reports of the minimal register, written

(s,\ell)

, each with the shortest word that reaches it. No letters are drawn yet.

Read as mathematics, the three clauses ask for admissible events, stored marks, and a dependence of the first on the second. An occurrence is one application of a law to named inputs; a record is a mark it leaves that later occurrences can read. A register is a record-holding unit whose states are its histories modulo future indistinguishability: two histories are identified when every admitted continuation gives both the same readings. This is the Myhill–Nerode construction, and it makes a register’s state derived: it holds exactly the distinctions some future can reveal.

The minimal consequential register has the alphabet $\{a,b,p\}$ and a state of two signs $(s,\ell)$ , initially $(+1,+1)$ , with reading $s$ . The symbols act by $a:(s,\ell)\mapsto(s,+1)$ , $b:(s,\ell)\mapsto(-s,-1)$ and $p:(s,\ell)\mapsto(s\ell,+1)$ . The symbols $a$ and $b$ are a blind pair, differing only in the sign they leave; the probe $p$ is a consequence, acting on the sign according to the last symbol written. Remove the probe and the register needs only the two signs at every length of history: blindness without a consequence is a frozen two-state system.

Proposition(Four reports)

The histories, the nonempty words in $a$ and $b$ , fall into exactly four future-indistinguishability classes. The class of a word $w$ is the pair $(s(w),\ell(w))$ , where $s(w)=(-1)^{\#b(w)}$ and $\ell(w)=-1$ exactly when $w$ ends in $b$ . The register reads its own classes: the empty continuation reads $s$ , and the probe ratio $s(wp)/s(w)$ reads $\ell$ .

Proof

The rules act on $(s,\ell)$ alone, so words with equal pairs give equal readings after every continuation, by induction on its length: there are at most four classes. All four occur: the words $a$ , $b$ , $ba$ and $bb$ reach $(+,+)$ , $(-,-)$ , $(-,+)$ and $(+,-)$ . Words with different signs are separated by the empty continuation, and words with equal signs and different switches by $p$ , which reads $s\ell$ . Finally $s(wp)/s(w)=s\ell/s=\ell$ .

Letters, axes and three ways to forget

Plate I.2The six letters as the edges of

K_4

, coloured by axis: the

s

-axis in blue, the

\ell

-axis in gold, the

s\ell

-axis dashed. Each axis is the pair of fibres of one binary reading.

The four classes are the register’s reports, $r=(s,\ell)$ : $s$ is the blind bit and $\ell$ the consequence bit. A letter is an unordered pair of reports, and there are six. Two letters are adjacent when they share a report and antipodal when they are disjoint. An axis is a pair of antipodal letters, equivalently a perfect matching of the four reports, and there are three.

Under adjacency the six letters form the octahedron: each letter is adjacent to four letters and antipodal to exactly one. The reports form the group $\Z_2\times\Z_2$ under componentwise multiplication, whose three nontrivial characters are $s$ , $\ell$ and $s\ell$ , and the two fibres of each are a perfect matching. So a letter is one answer to one binary question about the reports, its antipode is the other answer, and the three axes are the three ways to forget one of the two bits.

The symmetry the third clause completes

Plate I.3The dressing

T

exchanges the two reports with

\ell=-1

. It carries the

s

-axis to the

s\ell

-axis and fixes the

\ell

-axis. The report

(+,+)

, at the centre, is fixed by

T

and by the swap

S

The register itself licenses only some relabellings of its reports. One that respects the two fundamental readings, the sign and the probe ratio, must carry the pair of their matchings to itself, so the first two clauses single out one axis, the mixed one, which pairs a blind datum with a record. The third clause removes the distinction. Let the register re-read its sign through its last record, $s'(w)=s(w)\,\ell(w)$ . This dressing changes no class, and it relabels the reports by $T:(s,\ell)\mapsto(s\ell,\ell)$ .

$T$ and the swap $S$ are automorphisms of the group $\Z_2\times\Z_2$ , so both fix its identity, the report $(+,+)$ , and together they generate its whole automorphism group $\GL(2,\F_2)\cong S_3$ , which permutes the other three reports, and the three axes, in all six ways. What moves $(+,+)$ are the sign relabellings $(s,\ell)\mapsto(\varepsilon s,\eta\ell)$ , multiplication by the report $(\varepsilon,\eta)$ , which besides the identity are the three double transpositions; every relabelling is an element of $\langle T,S\rangle$ followed by one of them. The reports are therefore places: the report group permutes them transitively, and none of them is an origin. The sign relabellings are displacements, steps between places, and conjugation by any relabelling fixes one of them, the identity, which is no step at all.

Proposition(The report group)

(i) The permutations of the reports preserving the pair of fundamental readings form a dihedral group $D_4$ of order eight. On the axes they act only by exchanging the $s$ -axis and the $\ell$ -axis; the $s\ell$ -axis is fixed.

(ii) The dressing $s\mapsto s\ell$ leaves the four classes unchanged and relabels them by $T$ , which exchanges the $s$ -axis and the $s\ell$ -axis and fixes the $\ell$ -axis.

(iii) $T$ and the swap $S:(s,\ell)\mapsto(\ell,s)$ generate a group of order six inducing all six permutations of the axes.

(iv) $D_4$ and $T$ generate the full symmetric group $S_4$ of the reports.

Proof

Parts (i) to (iii) are exhaustive finite checks over the twenty-four elements of $S_4$ . For (iv), $T$ carries the $s$ -axis to the $s\ell$ -axis, so $T\notin D_4$ . The subgroup $D_4$ has index three in $S_4$ , so a subgroup properly containing it has order divisible by eight and dividing twenty-four, hence twenty-four.

Why four

Plate I.4Four reports and the six pair records an exact-return apparatus keeps, one per letter. Its comparison algebra has blocks of sizes 1, 3 and 2; Chapter V finds them on the letters.

Four enters the program by more than one road, with different premises. The first is the construction above, whose premise is minimality; another update rule would give another count. The second is geometric: among $n$ reports permuted by $S_n$ , only $n=4$ gives each letter an antipode. The third is operational and the strongest. It asks what an apparatus must be if it stores comparisons of reports and must decode them exactly, and the program’s forcing theorem, proved in its companion paper, answers it.

None of these roads proves that the world has four reports; their agreement is the evidence the program offers. The selection theorem matters most, because its premises are operational and its output $\C\oplus M_3\oplus M_2$ is the report block plus the matching block of Chapter V.

Theorem(Selection by exact return)

Suppose an apparatus has an informationally complete $n$ -report measurement of a $k$ -level system $M_k(\C)$ , $k\ge2$ ; retains one amplitude record for each unordered pair of reports and no other accessible memory; and admits the coherent exchange tests, selective branches together with a nonreal relative phase. Its comparison algebra is $\C\oplus M_{n-1}(\C)\oplus M_{n(n-3)/2}(\C)$ . If a unital completely positive decoder onto $M_k$ , with a unital completely positive section, keeps its decoded equalities under every subsequent test, then $k=2$ , $n=4$ , and the algebra is $\C\oplus M_3(\C)\oplus M_2(\C)$ .

Proof

Completeness needs at least $k^2$ effects, so $n\ge k^2\ge4$ . Exactness forces the decoder to be a surjective $*$ -homomorphism, so $M_k$ is one of the blocks: $k\in\{1,\,n-1,\,n(n-3)/2\}$ . The first is excluded by $k\ge2$ ; the second needs $(n-1)^2\le n$ , false for $n\ge3$ ; the third gives $k\ge n$ , hence $k^2>n$ , once $n\ge5$ . At $n=4$ the blocks are 1,3,2, and only $k=2$ satisfies $k^2\le4$ .

The report form

Plate I.5The same four reports as a regular tetrahedron of null directions on the Bloch sphere, the report qubit’s celestial sphere. The midpoints of its edges, the pair sums less

I/2

, are the six vertices of an octahedron.

Give the four reports vectors $e_0,\dots,e_3$ in $\R^4$ , permuted by $S_4$ . The invariant quadratic forms are $Q_{\alpha,\beta}(x)=\alpha(\sum x_i)^2+\beta\sum x_i^2$ , with eigenvalue $4\alpha+\beta$ on the line of $\sum e_i$ and $\beta$ on the three-dimensional space of report differences. For $\alpha>0$ the form is Lorentzian, of signature $(1,3)$ , exactly when $-4\alpha<\beta<0$ , and the reports are null exactly when $\beta=-\alpha$ . There letter sums are time-like, letter differences space-like, and the three axes are mutually perpendicular spatial directions. Nullity is not what makes the signature Lorentzian: it selects the self-dual point of the Lorentzian range and fixes the exchange rate between the alphabet’s time unit and its space unit, $c=1$ .

Nullity has a concrete meaning: the reports of a qubit’s symmetric measurement are null. So stipulation S1 says that the observer’s reports are pure. The null rays of this Minkowski space form the Bloch sphere of the report qubit, which is its celestial sphere, and the four reports are a tetrahedral frame of null directions on it whose sum is the observer’s time. Its finite counterpart is the sky of Chapter X, the crystal’s null cone read modulo seven, and Chapter XIII shows that the sky is the shadow, modulo a prime above seven, of the integral Lorentz group of this very form, whose entries are Eisenstein integers. What is still to be built is dynamics: a law that moves states along these Lorentz orbits.

Proposition(The reports of a qubit are null)

Let $E_i=\tfrac14(I+r_i\cdot\sigma)$ , $i=0,\dots,3$ , with $r_i$ the vertices of a regular tetrahedron on the Bloch sphere, so that $\sum_iE_i=I$ . On the $2\times2$ Hermitian matrices, whose determinant is the Minkowski form, write $X=\sum_ix_iE_i$ . Then $\det X=\tfrac1{12}\bigl[(\sum_ix_i)^2-\sum_ix_i^2\bigr]$ , the null member of the invariant family. The reports are pure effects, hence null; the democratic direction $\sum_iE_i=I$ is time; the six pair sums $E_i+E_j$ are equally time-like, and after subtracting $I/2$ they are the six vertices of an octahedron.

Proof

The determinant $a^2-|v|^2$ of $aI+v\cdot\sigma$ is invariant under every orthogonal map of Bloch vectors, hence under the tetrahedral group permuting the $E_i$ ; it vanishes on each rank-one $E_i$ ; and $\det I=1$ . So it is the null member with $\alpha=\tfrac1{12}$ . The pair sums have scalar part $\tfrac12$ and vector parts $\tfrac14(r_i+r_j)$ , which are $\pm\tfrac1{2\sqrt3}$ times the three coordinate axes.

Clocks and anchored observers

Plate I.6The Fano plane with the clock 1 in gold. The four lines missing it, the circle 246 and the sides 257, 347 and 356, are its reports; the three lines through it are its axes. The base observer

(1,246)

stands on the circle: its rods are 2,4,6 and its odd letters 3,5,7, shaded.

Six letters and one further point make seven, and seven points with their lines form the Fano plane: its points are the seven nonzero vectors of $\F_2^3$ and its lines the seven triples summing to zero. A clock is a point $p$ . Relative to $p$ the letters are the six points $a\neq p$ , the antipode of $a$ is $a+p$ , the axes are the three lines through $p$ , and the reports are the four lines missing $p$ . An anchored observer is a pair $(p,L)$ with $L$ a line missing $p$ : a clock together with the report on which it stands, its anchor. The three letters on $L$ are the anchor’s rods; the other three are its odd letters. Time enters this vocabulary twice, and the two must not be confused: a clock is a marked direction, a choice of time unit among seven, while the age of a register is a count, the number of occurrences it has committed.

In the program’s octonion attachment the seven points are the imaginary units of the octonions and the clock is the unit chosen as time. Two letters on one axis multiply to $\pm$ the clock, and two letters on different axes to $\pm$ the third rod of their common report. Left multiplication by the clock sends every letter to $\pm$ its antipode and squares to $-1$ on the six letters, so the clock is a complex structure on the letter space whose complex lines are the axes: the first native appearance of an imaginary unit. With the octonion unit 1 the clock spans a fourth complex line, the unit’s own, which Chapter XVI identifies as the lepton’s; and because the stabilizer in $G_2$ of an imaginary unit is $\SU(3)$ , marking the clock is also the act that later makes colour a stabilizer (Chapter XIV).

Proposition(One clock is the kernel graph)

(i) Fix a clock $p$ . Each letter lies on exactly two report lines, and any two report lines meet in exactly one letter. The incidence is the complete graph $K_4$ , reports as vertices and letters as edges; antipodal letters are opposite edges and the axes are the three perfect matchings.

(ii) The stabilizer of $p$ in the collineation group $\GL(3,2)$ , of order 168, has order 24 and acts faithfully on the four report lines: it is the report group $S_4$ .

(iii) There are seven clocks and twenty-eight anchored observers.

Proof

A point $a\neq p$ lies on three lines, one of which is $\{p,a,a+p\}$ ; the other two miss $p$ . Two distinct lines meet in one point, not $p$ when both miss it. The group $\GL(3,2)$ is transitive on the seven points, so the stabilizer has order $168/7=24$ ; faithfulness is a finite check. Each point is missed by four lines.

What the sentence does not fix

Plate I.7Two worlds with one description: the same support on three occasions, with probabilities

(\tfrac12,\tfrac12)

and

(\tfrac13,\tfrac23)

out of 0. Their stationary laws differ, and the relational language cannot tell them apart.

Three kinds of quantity are absent from the sentence: the weight of each distinguishable alternative, the rate of occurrence, and the content of what is written; a fourth absence concerns two observers. The sentence speaks in finite sets and relations, and no rate or probability can be defined in that language: every strictly positive reweighting of a transition support has the same actual and possible edges, reach closure, recurrent classes, backed possibilities and re-readable records. Multiplying every rate by a common factor changes only the unit, so the absolute cadence is free at every grade of description, like the choice of a second.

The same separation occurs in the program’s own dynamics: the rates $\rho=1$ and $\rho=2$ of re-anchoring, the move by which an observer changes the report it stands on, give identical supports, move counts, reports and reach, and different Hamiltonians. Blindness narrows the freedom to one weight per auditable kind of move, and choosing among those weights needs a fourth clause, a measure; the program’s candidate, maximal ignorance, is recorded as an addition. In the lift of Chapter XIII every record is one unit of proper time, so a weight local to one occurrence and covariant under the lift’s Lorentz group is the same for every record. That fixes the weight of a record, not the weights of an observer’s descriptive moves, and the program has adopted it as premise P4, one update, one weight: an auditable history’s weight is a product over its occurrences of a factor depending on each occurrence alone, which exact return makes a pure phase. In the lift the cadence is exactly the conversion of that unit of proper time into seconds. What is written is the two-layer rule of Chapter III, adopted. How one occurrence belongs to two observers is clause S8, adopted in its provenance form, and the encounter rule itself is an addition. When registers are read as the quanta of a field, P3’s working form records the other’s state instead of its name, which for fermions keeps the same order.

Example(Two worlds with one description)

Take three occasions. From 0 the process moves to 1 or 2, and from each of these it returns to 0. Give one world the probabilities $(\tfrac12,\tfrac12)$ out of 0 and the other $(\tfrac13,\tfrac23)$ . Both have the same strongly connected support, hence the same relational description. Stationarity gives $\pi_{\mathrm{equal}}=(\tfrac12,\tfrac14,\tfrac14)$ and $\pi_{\mathrm{tilted}}=(\tfrac12,\tfrac16,\tfrac13)$ . The two worlds differ in what any long record shows, and nothing in the sentence can say which one it describes.

What the restatement leaves open is as definite. Matter’s value transport is product-preserving, decided at every scale and exact observer by observer up to colour signs, with spin carried by a separate factor, the report qubit; at a meeting it is fixed up to colour, and its flat form is the gauge links’ vacuum within a scale, and between scales once the tower of scales is given a direction, which Chapter XII shows can always be done. The meeting rule itself is open, and across all observers the spin factor is a bundle over the seven clocks in two forms that nothing yet distinguishes. The composition law, which would turn records into shared positions, is where gravity would enter. P4 does not reach the weights of an observer’s descriptive moves. Where memory lives is half answered: what a record is, a report, a letter or an axis, is read at the lift’s prime above three, but the way records accumulate, a binary counter on any two axes, is structure at the prime two, and the mass number depends on the rest of the answer. The relative helicities of the lift’s sectors are open.

The rest of Part I follows the architecture the sentence implies. Its observers are blind to most of what occurs, and Chapter II asks what a lawful observer can say about an occurrence it cannot audit. Between observers and world lies a kernel, what every description forgets and where interference lives (Chapter III); the world itself, the occurrences and their permanent records, is built in Chapter IV. Part II then finds one graph beneath the levels, the kernel graph of one clock: time is the count of the occurrences that write a history on it, and branching, space and what space forgets are its covers.