Part V · Dynamics and LimitsChapter XIX

Rulial Invariants

Plate XIX.1Descent through the join, on an invented example: a chain of confusions forces equal weights, leaving one weight per shaded class.

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

If the rule cannot be selected, what physics survives every choice of it?

Parts II to IV found the program’s kinematics rigid: the kernel graph, the celestial geometry of the observers, the spin double cover and the quark–lepton quartet follow from the observers’ structure under premises each chapter names, and no adjustable number enters them. The dynamics is different. A short list of numbers has resisted every attempt at selection: the weights of re-anchoring, of hops and of memory rewrites; the common cadence; the weight of a received exchange against a private append; the two branches of a promotion; the permission to meet across clocks; and the strengths and phases of ingredients added to make walks in space.

One response is to keep looking for the principle that selects. Another takes the freedom at its word. Chapter XI showed that the law is one family of rules read in seven charts; the proposal examined here extends that stance from charts to rules. An item that no observer can audit may be a coordinate on the space of rules, saying which rule describes the world and not which physics it has, and the physics is whatever stays invariant as the coordinate varies. The reading borrows Wolfram’s rulial relativity and applies it to the program’s own unselected parameters, a finite section of his space of rules.

The central result

Let several observers describe finitely many occurrences, each identifying those its description cannot tell apart. A positive weight per occurrence respects every observer’s blindness exactly when it is a function on the classes of the equivalence relation these identifications generate, so the admissible weights form a positive cone whose dimension is the number of classes. With the vantage kept as a literal record, hop, memory rewrite and re-anchoring stay in different classes and agreement fixes no ratio of their weights; under the program’s coarser working record, written in the observer’s moving frame with a relational vantage, a hop and a re-anchoring leave identical records, so their weights are equal. Which observables stay invariant across the unselected families is open.

Status

The descent theorem is exact and sharp in both directions; it transports a weight down to a quotient without choosing one. The selection of equal hop and re-anchoring weights is conditional on the working record clause, which is adopted as reversible. Beyond that there is a plan of work: an invariance test, specified and never run.

The data in hand already sort part of the list, and not every item sorts the way the reading hoped. A common scale is always free, and so are memory’s weight and the cadence under either record; but the recorder’s reception weight and the promotion branch weights are observable, so they behave as constants, to be fixed by a principle or measured. The lift has since fixed a promotion’s comparison and the form of a record’s weight.

Weights that every observer accepts

Plate XIX.1Descent through the join, on an invented example: a chain of confusions forces equal weights, leaving one weight per shaded class.

The world side of the program is a growing history of occurrences, each one completed update followed by one commit, leaving a permanent structural record while content is rewritten in place. The kernel principle is a premise, added and put under test: one occurrence carries one positive intensity weight, and that weight does not depend on which observer describes it. It does not say what the weight is; it says that no observer may see a difference in weight between occurrences its record cannot tell apart.

No probability, entropy, typicality or spectrum is used. The combinatorial content is the common function of Gács and Körner, what every party can compute from its own view, and the join is Aumann’s common knowledge. The theorem’s output is a count, the number of classes, which is the number of weights agreement leaves free: descent is exact, and selection becomes a question of how many classes the records leave.

Theorem(common weights descend through the join) proved

Let $E$ be a finite set of occurrences, each observer $o$ describing a subset $E_o$ by a map $\pi_o\colon E_o\to D_o$ , and let $J$ be the equivalence relation on $E$ generated by all the blindness relations $\pi_o(e)=\pi_o(f)$ , with quotient map $q\colon E\to E/J$ . A function $w\colon E\to\R_{>0}$ is constant on every blindness class of every observer if and only if $w=\bar w\circ q$ for a function $\bar w$ on $E/J$ . The admissible weights form the positive cone of a real vector space of dimension $\lvert E/J\rvert$ , and multiplying all of them by a common $\kappa>0$ preserves every constraint.

Proof

If $w=\bar w\circ q$ and $\pi_o(e)=\pi_o(f)$ , then $e\mathrel{J}f$ , so $w(e)=w(f)$ . Conversely, if $e\mathrel{J}f$ there is a chain from $e$ to $f$ in which consecutive occurrences are identified by some observer, and $w$ agrees across every link. The constraints are linear, so their solutions are the functions on $E/J$ ; the positive ones form its positive cone, and scaling preserves each equation.

What the literal records keep apart

Plate XIX.2Three kinds of move at the first memory-bearing depth, and two weightings that agreement both admits.

A register has a vantage, the report it stands on, and a word of letters. A hop keeps the vantage and changes only the newest letter; a memory rewrite keeps it and changes at least two positions; a re-anchoring changes the vantage and leaves the word, in absolute labels, unchanged. In the literal record the vantage is kept, as the stipulation that the vantage is a record demands.

The ratio $\rho=w_{\mathrm R}/w_{\mathrm H}$ , the rate of rulial motion against motion within one vantage, is left free. At the first memory-bearing depth there are 144 hops, 144 memory rewrites and 108 re-anchorings: equal weights give orbit totals $\tfrac4{11}$ , $\tfrac4{11}$ , $\tfrac3{11}$ , doubling the re-anchoring weight gives $\tfrac27$ , $\tfrac27$ , $\tfrac37$ , and both laws are positive, normalized and uniform on every blind interior. The premise that would remove the free ratios, local move blindness or maximal ignorance, is an additional clause.

Proposition(the kinds are common knowledge) proved

For the declared descriptions of one register, the four report charts, all 24 report relabellings and the invertible whole-word chart, no description identifies two occurrences of different kinds, at any word depth. Consequently agreement imposes no relation among the kind weights $(w_{\mathrm H},w_{\mathrm M},w_{\mathrm R})$ : both $\kappa(1,1,1)$ and $\kappa(1,1,2)$ are admissible.

Four registers and two measures

Plate XIX.3Four registers at the four reports; registers 0 and 1 are ready when their tops are the shared rod 01 and its antipode 23.

Put four registers at the four reports of one clock, one at each. A private occurrence appends any letter but the antipode of the top; an exchange happens when two registers’ tops are their shared rod and its antipode. The path-counting measure $C$ gives one weight per occurrence, as in Wolfram and Gorard’s multiway counting; the class-uniform measure $S$ gives each auditable alternative an equal share, so an occurrence drawn from a menu of $d$ alternatives weighs in proportion to $1/d$ , with $d=5$ , or 6 when an exchange is ready.

With one register at each report a register has at most one ready partner, and at every ready exchange both participants have menus of six, so the two witnesses of a shared occurrence cannot tell $C$ from $S$ . Own-log blindness excludes using $1/d$ as an occurrence rate; it does not exclude equal raw weights normalized afterwards, and it does not select $C$ . The measure clause has survived every test meant to discriminate it.

Example(a blind class that the menu size splits)

Index the letters $0=01$ , $1=23$ , $2=02$ , $3=13$ , $4=03$ , $5=12$ , and let register 0, with top 01, append 01 again. In the top configurations $(0,0,0,0)$ and $(0,1,0,0)$ register 0‘s own log reads the same, but in the first no exchange is ready and $d=5$ , while in the second register 1 shows 23, the complement of 01, and $d=6$ . The rate $\kappa/d$ gives $\kappa/5$ and $\kappa/6$ on one blind class; the path-counting rate $\kappa/Z$ , with $Z=5+\tfrac16=\tfrac{31}6$ , is $\tfrac{6\kappa}{31}$ in both.

What a coarser record forces

Plate XIX.4The same two occurrences under two permanent records: the literal records differ, the moving-frame record does not.

The program’s working clause, adopted as reversible, replaces the literal vantage record by a coarser one: each commit describes the word in the observer’s moving frame, the chart that relabels the word whenever the vantage changes, and records the vantage only relative to encounter partners, as at the partner’s report or elsewhere; a solitary commit records no vantage, which becomes live content rewritten in place.

Under the kernel principle, uniform weights within each kind and this record, $w_{\mathrm H}=w_{\mathrm R}$ : $\rho=1$ , and the admissible kind weights are exactly $\kappa(1,\mu,1)$ with $\kappa,\mu>0$ free, since memory never joins the class. The selection depends precisely on the record: one bit saying whether the vantage changed separates the two moves again. Promotions show the same pattern, a receipt blind to the source report selecting the uniform mixture $a=\tfrac12$ of the two connections. Weights are forced equal exactly where the permanent record forgets a distinction, and nowhere else.

Example(one record for a hop and a re-anchoring)

Let a register at vantage 1 hold the word $(02,01)$ , with report 0 as reference partner, and compare the hop $(1;02,01)\to(1;02,03)$ with the re-anchoring $(1;02,01)\to(3;02,01)$ . In the moving chart from report 1 the source reads $(12,01)$ and the hop’s target $(12,13)$ ; at the re-anchoring’s target the chart’s last transposition $(0\,3)$ sends the displayed 01 to 13 and fixes 12, so it reads $(12,13)$ again. Neither vantage is the partner’s report, so the relational field reads elsewhere at both ends, and the two records coincide.

Which rule, and which physics

Plate XIX.5Two chains with one support and different weights: the same relational description, different statistics.

A three-occasion chain shows what is invisible: from occasion 0 move to 1 with probability $p$ or to 2, and return. The stationary law $(\tfrac12,\tfrac p2,\tfrac{1-p}2)$ is $(\tfrac12,\tfrac14,\tfrac14)$ at $p=\tfrac12$ and $(\tfrac12,\tfrac16,\tfrac13)$ at $p=\tfrac13$ , while the relational description is the same. And multiplying every jump operator by $\sqrt r$ preserves all normalized branching probabilities, supports, charts and covariance while the generator scales by $r$ : the common cadence is a unit.

So the proposal must name its observers. An item $\theta$ is a rulial coordinate relative to a class $\mathcal{O}$ of auditable observables if every observable in $\mathcal{O}$ is invariant along $\theta$ , and the reading is that the unselected items are rulial coordinates relative to what observers can actually audit, the physics being the invariants across all the families at once. Its test, computing the established observables as functions of each parameter, has not been run. For the seven clocks such passive invariance is a theorem, and its invariant content includes the curvature.

Theorem(the observer language cannot select a rate) proved

Every strictly positive reweighting of one transition support is elementary-equivalent to the original in the relational language of the observer axioms: same actual and possible edges, reach closure, recurrent classes, backed possibilities and re-readable records.

What the existing data already sort

Plate XIX.6The recorder’s ready probability rises strictly with the reception weight

\lambda

, so

\lambda

is observable.

Several results already compute an observable along an unselected family. The recorder’s reception weight $\lambda$ is not a coordinate: the probability that two registers are ready is $(10+\lambda)/(180+\lambda)$ , strictly increasing, and even a single register’s own log shows it, received exchanges standing to own appends as $1/90$ at $\lambda=1$ and $1/15$ at $\lambda=6$ . The promotion weight is visible to spin holonomy, which differs between $F_0$ and $F_1$ on every one of the $2{,}016$ triangles, and invisible to the committed equilibrium of the synchronized two-register promotion; and the cross-clock exchange on Coxeter edges is the same for every connection.

The composites’ rates give the first invariant with physical content: requiring light and matter to be massless together cuts the rate ratios to the line $(\sqrt2/2,\sqrt2/2,m)$ , pinning two ratios and leaving one modulus, and leaving the line gives matter the gap $2\lvert\delta\rho\rvert/33$ . Redistributing memory’s weights covariantly leaves the depth law and the ultrametric exponent $\alpha=1$ unchanged, while the probability that the newest letter lies at the register’s report moves from $25/43$ to $31/43$ . What looks invariant so far is structural: a holonomy class up to frame gauge, a zero locus, a depth exponent, an exchange law.

What the lift fixes

Plate XIX.7Rod holonomy of the

2{,}016

three-clock promotion loops, for the two finite connections and the lift’s transport.

Two rows have since been settled by geometry. The comparison along a promotion: the lift’s parallel transport $T$ , the identity on $1{,}680$ rooted triangles and a half-turn on 336, is the one comparison covariant under every isometry of the lift, with $F_0$ and $F_1$ as $T$ composed with cell rotations; it settles the comparison of frames and, through the double cover, the transport of spin, while matter’s labels are carried by relabellings that keep the octonion product. And the weight of a record: read in the lift, every record is one unit of proper time, so a weight local to one occurrence and Lorentz-invariant is the same for every record, and exact return makes it a pure phase $e^{-im}$ per record, with loss of norm entering only through blind sums.

The measure between classes is a separate question. Read strictly, the blindness premise gives what an observer cannot audit one amplitude, the mean over the alternatives, which is idempotent and so favours the class-uniform measure; the program’s restated axioms take this as premise P4. No test has discriminated the two measures, so the choice is one of premise.

Open questionopen

Which observables are invariant across all the unselected families at once, relative to observers who can audit only the permanent record? Is any number among them, or are the program’s rulial invariants exactly its symmetry and zero-locus data?

The descent theorem is the kernel’s quantitative content: weights live on what observers commonly know and are forced equal exactly where the permanent record forgets. Numbers move while structures survive: the class constant of Chapter XXIV changes from 0.0214 to 0.0038 under a legal change of private energy, while the circular $S_3$ cone and the $\mathbf{4}\oplus\overline{\mathbf{4}}$ structure survive. On this reading, a number that nature could refute must be either a rulial invariant of every family the program leaves open or a constant fixed by a principle about the record.

The next chapter asks the same question level by level: whether evolving and describing agree.

Concepts: absencestatus
Objects: the twenty-eightthe seven points