Part V · Dynamics and LimitsChapter XXI

Nonfinite Limits

Plate XXI.1The two controlled comparisons of the sequential law: complete histories at equal count, and active histories against a continuous semigroup.

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

What does it take for a finite law to have a controlled limit, and which limit is which?

Every construction in this volume is finite, and the physics it hopes to reach is not. A limit is the bridge, but “the limit” names at least five different operations. A thermodynamic limit takes the number of registers to infinity at a fixed local law; a temporal limit turns a count of occurrences into a continuous time; a coarse-graining limit passes to a forgetful description, or completes a tower of descriptions; a mesh limit sends a spatial step to zero; a low-energy limit keeps only what survives at scales much larger than the cutoff. These operations do not commute, and one cannot stand in for another.

The rule this chapter follows is to take a limit of the declared family, and to say which kind it is. The program has one controlled nonfinite limit of its own sequential law, and several conditional limits of constructions in which some structure was supplied.

The central result

Run the program’s sequential law, every elementary update weighted equally, on a declared dilute exhaustion: $s=4L^3$ connected sites of the $K_4$ crystal among $N=s^3$ registers that write by the program’s law, for $n=5NT$ occurrences. On the complete retained history, the event-only law and its waiting-resolved comparison differ in diamond norm by at most $9T/(160s)\to0$ ; on the active history, with the bath’s insertion positions forgotten, the law converges to a continuous recording semigroup $e^{T\mathcal{L}_A}$ at rate $O(1/s)$ . Both limits are temporal and scoped. Fixed-count and fixed-time kernels stay apart, the homogeneous world is open (its one-step overlap is $440/441$ at every size), and no spatial mesh limit or low-energy continuum follows.

Status

The limits of the program’s own sequential law are proved, with an exact scope. The surrounding population of disconnected writers is an added choice, not a homogeneous crystal; the equal weighting of occurrences is the program’s working counting rule, run at reception weight $\lambda=1$ ; the graph is supplied; and the recording process is one restricted instrument, with no in-place pulses, no active promotions and no encounters between clocks.

The thermodynamic limit exists in the local sense, and its ordinary sector is gapped. Every mesh and low-energy limit so far is a limit of a construction in which space, streaming, a return operator or a critical field was supplied: each is exact about its own family, and none is yet a limit of the program’s world.

One sequential law, two instruments

Plate XXI.1The two controlled comparisons of the sequential law: complete histories at equal count, and active histories against a continuous semigroup.

Take $N$ registers on a supplied graph, each a depth-two word with its $\C^8$ fiber. The elementary occurrences are the private five-way appends and the received exchanges, each writing a permanent receipt: the departed letter, the occurrence’s identity, the participants and the versions read. All raw jumps $J_r$ have weight one, with total intensity $Z=\sum_rJ_r^\dagger J_r=5N\,I+\sum_e\Pi_e$ , where $\Pi_e$ projects onto the words in which the rod $e$ is ready to exchange.

The two instruments agree on every classical question, since their next-event probabilities coincide at every size, and disagree on coherent ones: tracing the waiting cell gives $W=E\circ\Delta_Z$ , with $\Delta_Z(\lvert z\rangle\langle w\rvert)=\frac{2\sqrt{zw}}{z+w}\lvert z\rangle\langle w\rvert$ . In the smallest native witness, one register in a superposition of a ready and an unready word beside a partner, the two branches have intensities 11 and 10 and the same structural receipt, and acquire the relative overlap $2\sqrt{110}/21$ , whose square is $440/441$ . The waiting time carries information about readiness that the receipt does not.

Definition(event-only and waiting-resolved instruments)

The event-only step appends one receipt: $V_E=\sum_r\lvert r\rangle\otimes J_rZ^{-1/2}$ . The waiting-resolved step also writes the waiting time $\vartheta\ge0$ into a fresh kernel cell: $V_W=\sum_r\int_0^\infty\lvert r,\vartheta\rangle\otimes J_re^{-\vartheta Z/2}\,d\vartheta$ . Both are isometries, and both keep the entire prior tape in their output.

The complete history has a controlled limit

Plate XXI.2The complete-history bound at

T=1

along the declared exhaustion, falling as

1/s

The theorem needs an exhaustion in which the readiness range $b$ grows more slowly than $N$ . The declared one is dilute: a connected region of the $K_4$ crystal with $s=4L^3$ sites and $b=3s/2$ rods, among $N=s^3$ registers in all, the others genuine native private writers with no contacts. At $n=5NT$ occurrences $nb^2/(200N^2)=9T/(160s)\to0$ , which at $T=1$ is $9/640$ , $9/5120$ and $1/1920$ for $L=1,2,3$ , the last with more than six million occurrences. The interacting region grows without bound, and its rooted neighbourhoods exhaust the infinite crystal.

Theorem(complete-history bound) proved

For every $Z$ with spectrum in $[z_{\min},z_{\max}]$ ,

\lVert\Delta_Z-\id\rVert_\diamond\le\frac{(\sqrt{z_{\max}}-\sqrt{z_{\min}})^2}{2z_{\min}}\le\frac{(z_{\max}-z_{\min})^2}{8z_{\min}^2}\le\frac{b^2}{200N^2}.

After $n$ sequential occurrences, with all common receipts retained, $\lVert W_n-E_n\rVert_\diamond\le\min\bigl(2,\,nb^2/(200N^2)\bigr)$ . The same bound holds with arbitrary reference systems and with adaptive trace-preserving interventions that neither enlarge the readiness range nor feed departed tape back into the live core.

Proof

Write $S_Z(X)=\int_0^\infty e^{-uZ}Xe^{-uZ}\,du$ , a completely positive map of completely bounded norm $1/(2z_{\min})$ . Entrywise $\Delta_Z-\id=-S_Z\circ\mathrm{ad}_{\sqrt Z}^{\,2}$ , and after centring $\sqrt Z$ at the midpoint of its spectrum $\lVert\mathrm{ad}_{\sqrt Z}\rVert\le\sqrt{z_{\max}}-\sqrt{z_{\min}}$ , which gives the first bound. The others use $z_{\min}\ge5N$ and $z_{\max}-z_{\min}\le b$ . Neither route involves the Hilbert-space dimension, and every intermediate channel has diamond norm one, so the single-step errors telescope through the growing tape.

The active history, and two failures

Plate XXI.3Neither row identifies a global stopping time: fixed-count and fixed-time kernels stay apart.

Here $T=n/(5N)$ is a rescaled count, not an independently adjustable time at each site, and no lapse was fitted. Two nearby identifications fail exactly. A history stopped at time $T$ has a Poisson number of receipts with mean $n$ , an occurrence-stopped one exactly $n$ , and their total-variation distance is $1-e^{-n}n^n/n!\approx1-1/\sqrt{2\pi n}\to1$ : stopping at a count and stopping at a time are different instruments, and no limit identifies them.

And with contacts at finite density everywhere, a matching of $K_4$ axes makes $N/2$ rods ready while an all-equal-top configuration makes none, with intensities $\tfrac{11}2N$ and $5N$ , whose one-step waiting overlap squares to $440/441$ independent of $N$ . Global diamond convergence therefore fails for the homogeneous world. That is no obstruction to local convergence, which would need a local instrument norm with controlled boundary data; it is the next target, and it is open.

Theorem(a continuous recording limit) proved

Let $a=5(N-s)$ be the bath’s total private intensity and $Z_A$ the active intensity, with $M=\lVert Z_A\rVert\le5s+b$ , and let $\mathcal{Q}(X)=JXJ^\dagger$ , where $J$ appends the coherent receipt sum to the active histories. On the active observation algebra, which keeps the active region’s core, its ordered tape, its ages and provenance and forgets where each active event was inserted in the global sequence, one global occurrence acts as $T_N(X)=R_aXR_a+a^{-1}\mathcal{Q}(R_aXR_a)$ with $R_a=(I+Z_A/a)^{-1/2}$ , and with $\mathcal{L}_A=\mathcal{Q}-\tfrac12\{Z_A,\cdot\}$ ,

\lVert T_N^{\,n}-e^{T\mathcal{L}_A}\rVert_\diamond\le\min\Bigl(2,\;\frac{4nM^2}{a^2}+2M\Bigl\lvert\frac na-T\Bigr\rvert\Bigr).

Under the declared exhaustion, with $n=5NT$ , the right side is $O(1/s)$ .

The thermodynamic limit

Plate XXI.4The single-site spectrum on top letters: 0 once,

-4

three times,

-6

twice; exchanges only widen the gap.

The waiting-resolved core process has a local generator, five-way private terms at each site and exchange terms on each rod, with completely bounded norms at most 10 per site and 2 per rod, and balls on the $K_4$ crystal grow polynomially, of degree three. The theorem of Nachtergaele, Vershynina and Zagrebnov then supplies a unital completely positive local dynamics in infinite volume, with a volume-independent Lieb–Robinson bound. A spreading bound is not a propagating mode, and in the ordinary sector there is none.

Taking this sector to infinite size creates no light. The full quantum-history sector, with its fiber and off-diagonal modes, was not classified, and it is where any gapless collective mode would have to live.

Proposition(the top sector is gapped at every size) proved

On observables of the registers’ top letters, the full law has spectral gap exactly 4 on every finite graph, and at least 4 in infinite volume.

Proof

At one site the private generator on top letters is $L_i=J_6-P-5I$ , with $J_6$ the all-ones matrix and $P$ the exchange of each letter with its antipode, which commute. It gives 0 on the constant vector, $-4$ on the three vectors odd under $P$ , and $-6$ on the two even ones with zero sum. Each exchange is an involution on top configurations and adds a nonnegative Dirichlet form, so the gap stays at least 4, and the function with value 1 on 01, $-1$ on 23 and 0 elsewhere, summed over sites, is an eigenvector with eigenvalue $-4$ .

Mesh limits: the scaling is part of the physics

Plate XXI.5Two paths to the same origin: along

\tau=a

the coherent echo gives a Weyl field and the read walk freezes; along

\tau\propto a^2

the read walk diffuses.

The time step is not a unit to be chosen afterwards: the path by which space and time are refined changes the equation obtained. The crystal’s lowest band, $\lVert k\rVert^2/4$ at small $k$ , supports three time laws, diffusion, a Schrödinger limit and a second-order wave of speed $1/2$ , and the graph does not choose among them. At mesh $h$ a fixed contraction per step becomes ultralocal, and a finite inverse correlation length $m$ needs $r(h)=1-mh+o(h)$ : every finite level still forgets, but not uniformly.

Example(one encounter, two limits)

Take four tetrahedral preparations $P_s=(I+r_s\cdot\sigma)/2$ of a qubit, with $r_s\cdot r_t=-\tfrac13$ , and let report $s$ carry a displacement $a\,r_s$ in a supplied space. Reading every encounter gives a Markov chain $T=\tfrac13I+\tfrac16J_4$ , with diffusion constant $D=a^2/(3\tau)$ : along $\tau=a$ it vanishes, and along $\tau=a^2$ the walk converges to the heat equation $\partial_tf=\tfrac13\Delta f$ . Keeping every encounter coherent and echoing it with the reflection of the encoding isometry gives the compression $\tfrac13\sigma\cdot p$ , and along $\tau=a$ the limit is the Weyl equation $i\partial_t\psi=-\tfrac i3\sigma\cdot\nabla\psi$ , with speed $1/3$ .

Quadratic transfers and an infinite memory

Plate XXI.6The joint limit of memory depth and mesh: every finite

\ell

is quadratic near zero, and only a memory infinite in physical units gives the linear law

E=\sqrt\lambda

By the lemma, near the top of a finite positive transfer the energies $-\log\lambda_j$ are $O(\lVert k\rVert^2)$ ; with $k=ap$ and one physical step $a$ , divided by $a$ they vanish, and no conversion of the time unit turns a quadratic minimum into $c\lvert p\rvert$ . An infinite critical memory, eliminated exactly, does what the lemma excludes. For a memory of ages $0,\dots,L$ , eliminating every age but the root, with the energy $E=F_L(a^2\lambda)/a$ and $a(L+\tfrac12)\to\ell$ , gives $E\to0$ at $\ell=0$ , $E\to\sqrt\lambda\tanh(\ell\sqrt\lambda)$ for finite $\ell$ , and $E\to\sqrt\lambda$ at $\ell=\infty$ .

The relativistic square root appears only when the memory is infinite in physical units, $aL\to\infty$ ; a memory with $L\sim\log(1/a)$ levels still gives zero. The return operator and the spatial operator are supplied, and whether the native process selects that critical behaviour is open.

Lemma(finite positive transfers are quadratic) proved

Let $T(k)$ be a twice continuously differentiable family of Hermitian matrices of fixed finite size with $0\le T(k)\le I$ , and suppose 1 is an eigenvalue of $T(0)$ of multiplicity $r$ . Then the $r$ eigenvalues of $T(k)$ near 1 obey $0\le-\log\lambda_j(T(k))=O(\lVert k\rVert^2)$ .

Proof

Put $B(k)=I-T(k)\ge0$ and let $P$ project onto $\ker B(0)$ . For $v$ in that kernel the nonnegative function $v^\dagger B(k)v$ has a minimum at $k=0$ , so its first derivatives vanish, and by polarization $P\,\partial_iB(0)\,P=0$ . Hence $PB(k)P=O(\lVert k\rVert^2)$ , and the min–max principle bounds the $r$ lowest eigenvalues of $B(k)$ by the same order; the others stay away from zero, and $-\log(1-x)=x+O(x^2)$ .

Low energies, completions, and which limit is which

Plate XXI.7The program’s nonfinite limits by kind; only the first three use its own sequential law.

An interacting cutoff family on a supplied lattice has continuum limit points with a nonzero fermion algebra, positive energy and a finite propagation cone: existence of a continuum process, not of an interaction, which may be free. If such a limit had unitary four-dimensional conformal covariance with an unchanged canonical spinor as a primary, the canonical anticommutator would force its dimension to $\tfrac32$ , saturating the unitarity bound and making it free, so an interacting limit must be carried by other, renormalized fields. The common speed has a chapter of its own, the next: the locking flow that makes coupled sectors share a speed is too slow to supply one from below, and what fixes the speed at the cutoff is the part of the lift’s group that the finite sky cannot see.

Completions form the last family. A register whose successive covers each double the one before completes to the solenoid, where irreversibility is a property of the completion and not of any finite stage, and two readings of the memory tower’s completion give two limits. Gorard’s route to gravity rests on assumptions the program does not meet, since its world is not a hypergraph and not causally invariant; the lift offers instead a kinematic helicity-two sector whose couplings are known in form, not in strength. Of the causal-set results tested against the program’s causal order, Sorkin’s fluctuating cosmological constant overshoots by about thirty orders of magnitude for a crowd of registers, whose count noise is worldline noise, and Rideout and Sorkin’s Bell causality fails for the native exchange, which depends on the partner’s state.

Open questionopen

Does the sequential law, run with finite-density contacts on the whole crystal and with the address retained, have a local nonfinite limit on complete record tests, and does any retained-history mode of that limit close its gap in a way that fixes the relation between the spatial step and the occurrence count?

A limit of a description is a limit of the world only on a family where that description’s square commutes: a spatial mesh limit must be taken of the lifted law with its retention clause, and since the path by which space and time are refined changes the equation, a derived continuum needs the scaling itself to be derived. The class constant of Chapter XXIV is a fixed, non-critical gap in the register’s own units, so a bridge from it to a physical mass must supply either a critical scaling or one physical cycle that sets the unit.

The common speed is taken up in the next chapter; Chapter XXIII then turns to what any such world must supply first: light, a vacuum and a preferred hand.

Concepts: status