One Register:
A Common Origin for the Granularity of Hilbert Space and of Space

A qubit is a point on the Bloch sphere,

and in standard quantum mechanics every point is a state. Rational Quantum Mechanics (RaQM) [1, 2, 3, 5] keeps the Schrödinger equation and the Born rule but admits only those points, and only those measurement bases, for which

with $m\in \{0,\mathrm{\dots },L\}$, $n\in \{0,\mathrm{\dots },L-1\}$, and $L$ a large integer. A quantum state is then not a point on a continuum but a point of a very fine grid, and $L$ is how fine the grid is. Palmer represents such a state as a string of $L$ bits: the fraction of $+1$ bits is ${\cos }^2(\theta /2)$, and rotating the string by $n$ places is the phase [2]. Niven’s theorem [26] says that $\cos \alpha$ and $\alpha /2\pi$ are simultaneously rational only for a handful of angles, so Eq. (2) forbids the pair of counterfactual settings a Bell inequality needs, which is Palmer’s Impossible Triangle Corollary, and Bell’s theorem no longer implies nonlocality [1, 3].

A state of $N$ qubits spread across its Hilbert space has $2^N$ outcomes, and with probabilities read as frequencies over $L$ trials there is a largest $N$ for which such a state can be written at all, $N_{\max }\approx {\log }_{2}L$ [2]; the bit count $2^{N+1}-2\le NL$, Eq. (10) of Ref. [5], is a weaker condition. In Palmer’s version $L$ is a property of each qubit, set by its mass and energy, and ranges from ${10}^{64}$ to ${10}^{109}$.

In a previous paper [6] we made two changes and called them axioms: $L$ is a single constant of nature, and one step of quantum phase, $2\pi /L$, is one Planck length ${\mathrm{\ell }}_P=\sqrt{\mathrm{\hslash }G/c^3}$ on a great circle of the cosmological horizon at $R_{\mathrm{\Lambda }}=\sqrt{3/\mathrm{\Lambda }}=1.66\times {10}^{26}$ m [53],

From the two, Newton’s constant follows by Verlinde’s argument [50] with the pixel fixed, Einstein’s equations by Jacobson’s [31], rotation curves flatten at $a_0=c^2/L{\mathrm{\ell }}_P$, and $G\mathrm{\Lambda }=12{\pi }^2{c}^3/\mathrm{\hslash }{L}^2$. Milgrom’s $a_0$ [51, 52] gives $L=4.6\times {10}^{61}$ and the observed $\mathrm{\Lambda }$ gives $6.4\times {10}^{61}$, agreeing to a factor 1.4; both lie below every per-qubit value in Palmer’s range.

Equation (3) equates the granularity of two things introduced as separate, Hilbert space and space. Ref. [6] deferred to a separate paper the unification of its two axioms and the extension of the first to continuous variables; this is that paper. It takes the identification, with the horizon withheld, as its only postulate and draws out what it contains: a step that turns a phase and moves a length at once, whose repetitions close on a ring of $L$ Planck cells. The first axiom follows, the rational grid with $L$ universal; the second, placing the ring on the cosmological horizon, is taken over here, Sec. V, where it gives $L$ its value, and Ref. [17] derives it from the same discreteness with one postulate more, that a direction is the state of one qubit. Palmer’s phase step is what the step does to the slowest wave the closure admits, and the count of steps that close is $L$. Read for a qubit, the rotation is a phase; read for a particle, a translation; for light the two are one thing at every Planck tick.

Section II states the postulate and derives from it the first axiom of Ref. [6]. Sections III to V draw what follows from the step alone, with no direction chosen: its algebra and momenta, the capacity of the register, and, with the ring on the horizon, the quantities the horizon fixes. Section VI fixes a direction, where light, matter, and velocity appear and position in whole cells is one-dimensional; Sec. VI.3 says how far that reaches, several particles in several planes needing a three-dimensional space that the paper does not construct and whose dimension Ref. [17] derives. Section VII states what is predicted and what would falsify it, and Sec. VIII what the placement assumes and what no register holds.

Postulate. The granularity of state space and the granularity of space are one discreteness. Every qubit is a record of marks, one to each cell of a ring of $L$ cells, each cell one Planck length and one Planck time, each mark $+1$ or $-1$: the fraction of $+1$ marks is a Born probability and the cyclic position of the record is its phase, so that one step of quantum phase is one cell and a full turn of phase is one circuit of the ring. The global phase of a state is hidden: no law depends on it, and the dynamics does not expose it.

Phase is cyclic and space is not, so it is the phase that closes. A full turn returns a phase to itself, and the postulate says what that turn costs in cells. $L$ is their number: the cells in the ring, the marks in a record, and the steps in a full turn, since a record of $L$ marks has $L$ cyclic positions and one step carries it to the next. The second sentence read for the marks is the rationality condition of the first axiom of Ref. [6], Sec. II.2, with the universality clause a consequence, Sec. II.1; read for the phase it is that paper’s second axiom with the horizon withheld and the Planck time added; the third sentence is new. Nothing else is assumed. Where the ring lies, and so how large $L$ is, the postulate does not say: Sec. V places it on a great circle of the cosmological horizon, the second axiom of Ref. [6], which Ref. [17] derives from this postulate and one more.

The postulate in plain words.

One discreteness. Two things in physics are grainy for reasons usually thought unrelated. Quantum phase, on Palmer’s account, comes in a smallest step rather than varying continuously; and space, on holographic accounts [46, 47], carries at most one degree of freedom per Planck cell of its boundary. The postulate is that these are not two facts but one, and that the single grain, looked at one way, is a turn of phase and, looked at the other, a movement through space.

A qubit is a record of marks on the cells. What a qubit is, on the postulate, is a pattern of $+1$ and $-1$ marks around a ring of $L$ Planck cells, one mark to a cell. The fraction of $+1$ marks is its Born probability and the position of the pattern around the circle is its phase; a qubit that could not be so written does not exist, and neither does a measurement whose outcomes could not be. A state of $N$ qubits is $N$ records on the same cells, $NL$ marks read jointly, Sec. II.3, and is not a qubit.

One step of phase is one cell. The Planck length, $1.6\times {10}^{-35}$ m, is the shortest distance with operational meaning, and the cell is space’s grain at that scale. The clause says the smallest turn of phase and the smallest cell of space are the same grain counted two ways, so that asking how many steps make a turn is asking how many cells make the ring. It also says the grain is a Planck time as well as a Planck length, ${\mathrm{\ell }}_P=c{t}_P$, which is the tick of the dynamics of Sec. VI.

A full turn of phase is one circuit of the ring. Phase returns to itself after $2\pi$; the clause says that return corresponds to going once around the ring. Nothing here says space closes on itself. What closes is the phase, and the ring is what its closing costs. Because the expansion of the universe is accelerating, every observer has a sphere of radius $R_{\mathrm{\Lambda }}\approx 1.7\times {10}^{26}$ m beyond which nothing can ever be seen, and Sec. V places the ring on a great circle of it, the largest circle it has.

The global phase of a state is hidden. Every quantum state carries an overall phase that no measurement reads; that much is ordinary quantum mechanics. The postulate adds that it stays hidden: the phase advances, at the rate Sec. VI.2 gives, but no law depends on it and nothing the dynamics does exposes it. A state of several records read jointly, Sec. II.3, has one global phase; the relative arrangement of its records is its correlations, and it is data. Without this clause the gauge argument of Sec. VIII is incomplete, since a step with no privileged starting place still leaves open whether a state’s phase is referred to anything outside it, and the clause says it is not.

Under the Schrödinger equation the phase of a mode of energy $E$ advances by $Edt/\mathrm{\hslash }$. Palmer’s $\zeta$ advances it by $2\pi /L$. Schrödinger evolution, in Palmer’s representation, is therefore the string of a mode of energy $E$ rotating at

and in a superposition each mode’s string turns at its own rate, the differences between rates being the phases that show; a single mode’s rotation is the hidden phase of Sec. VI.2.

One cell is one Planck length and one Planck time, ${\mathrm{\ell }}_P=c{t}_P$, by the postulate; the step has one unit and $c$ converts it. Write $X$ for the step. On the $L$ cells of a record it is the shift $|j⟩\mapsto |j+1⟩$, and since a cell is a Planck length it is a translation,

which defines momentum with no continuum limit: momentum is what generates the step. Its eigenstates are the plane waves

with , whose eigenvalues return Eq. (5) in operator form: the phase advances by $2\pi \kappa /L$ per cell, a permitted phase with $n=j\kappa (modL)$, so $p_{\kappa }=h\kappa /L{\mathrm{\ell }}_P$ as before. The mode $\kappa =1$, of wavelength $L{\mathrm{\ell }}_P$, the ring, carries the smallest momentum there is. The states whose phase advances by a permitted amount per pixel are the momentum eigenstates; the phase step is the momentum quantum $h/L{\mathrm{\ell }}_P$; position and momentum each take exactly $L$ values; momentum is bounded at the zone edge $\pi \mathrm{\hslash }/{\mathrm{\ell }}_P$; and the phase-space cell is $L{\mathrm{\ell }}_P\cdot (2\pi \mathrm{\hslash }/{\mathrm{\ell }}_P)/L=h$.

Let $Z|j⟩=e^{2\pi ij/L}|j⟩$. Then $ZX=e^{2\pi i/L}XZ$, the finite Weyl relation, and Eqs. (9)–(10) are Schwinger’s finite quantum mechanics on ${\mathbb{Z}}_L$ [7, 8, 9]. A state on $N_x$ cells occupies at least $L/N_x$ momenta [11], so $\mathrm{\Delta }x\mathrm{\Delta }p\ge (N_x{\mathrm{\ell }}_P)\cdot (L/N_x)(2\pi \mathrm{\hslash }/L{\mathrm{\ell }}_P)=h$, and in variance form it is $\mathrm{\hslash }/2$ for ${\mathrm{\ell }}_P\ll \sigma \ll L{\mathrm{\ell }}_P$ [10]. Palmer reaches the same bound by averaging the spin relation over points uniform in $\cos \theta$ [5]; that measure is the count and this bound comes from the shift, and the two derivations are the register’s two operations applied to one relation.

The Hilbert space spanned by the $L$ cells has dimension $L$. The capacity of the register is fixed by room and not by bits. Read jointly, $N$ records give at each cell one of the $2^N$ joint outcomes, and the probability of an outcome is the fraction of cells reading it. A state spanning its Hilbert space gives every outcome a nonzero probability, so every outcome occupies at least one cell, and

205 at $L_{\cos }=6.4\times {10}^{61}$ and 204 at $L_{\mathrm{gal}}=4.6\times {10}^{61}$, the cosmological and galactic determinations of Sec. V. Branch by branch it is the same condition: a conditional strictly between 0 and 1 needs two cells, and the last record has $2^{N-1}$ branches. The bound is exact, and it is a ceiling any representation must respect and not a construction that attains it, which is the direction a falsifier needs. Its softness is in resolution rather than in the integer: a branch of two cells holds a conditional of one half and one of two phases, so a state whose deepest conditionals and phases must be told apart to one part in $r$ needs $r$ cells to a branch, $2^{N-1}r\le L$, and the ceiling falls by ${\log }_2(r/2)$. With $L$ known to a factor 1.4 besides, the ceiling sits between 203, for three bits of resolution, $r=8$, and 205. What the argument fixes is the scale, ${\log }_{2}L$, and that the ceiling is the same for every physical realization; the integer within it is not fixed to better than a unit, and nothing below turns on which.

It is the number of blocks and not the number of records that Eq. (11) constrains, and that difference is its entire content. A product state has one block per record, and the bound asks nothing of it at any $N$, as it must not for ${|+⟩}^{\otimes N}$. A stabilizer state has at most two blocks per record, one for each value of a parity of earlier marks, its conditional 0, 1, or one half, and a state whose correlations decay with a length $\xi$ has of order $2^{\xi }$. So a fault-tolerant code, a Slater determinant, a condensate, a nucleus, or a crystal clears the bound by scores of orders of magnitude, and the register limits neither how much matter exists nor the size of a code. What fills the last record with $2^{N-1}$ blocks is scrambling, a state whose every conditional depends at order one on every earlier record, which is what a random circuit produces once it anticoncentrates, at a depth logarithmic in $N$ [25]. No open system holds such a state, since a system entangled with its surroundings is described by a mixed state and a mixture costs no room beyond what its members need; a closed chaotic system isolated for a scrambling time does, which is what a quantum computer is. The one natural object that appears to exceed the bound, a black hole, is taken up in Sec. VIII.

Palmer’s bit count, $2^{N+1}-2\le NL$, gives 212 at $L_{\cos }$ [5]. It charges the $NL$ marks one bit for each real parameter, but what stops a state being written is not a shortage of bits: a record of $L$ cells holds at most $L$ bits whatever is on it, and a record whose $L$ blocks hold one cell each uses every bit and spans nothing, its conditionals all 0 or 1. The constraint is room, two cells to a branch, so the 205 stands and the 212 does not. Ref. [6] gave $N_{\max }\approx 205$ from ${\log }_{2}L$ and quoted the prediction as about 200; it also called the bit count the exact criterion and inverted a measured $N_{\max }$ into $G$ through it. The inversion keeps its form, Eq. (11) bracketing $L$ between $2^{N_{\max }}$ and $2^{N_{\max }+1}$. Equation (11) counts outcomes and not the systems that produce them, so a Hilbert space of any dimension $D$ spread across its outcomes obeys $D\le L$, for bosonic modes as for qubits.

The cells close in a circuit of circumference $L{\mathrm{\ell }}_P$ and radius $L{\mathrm{\ell }}_P/2\pi$. The second axiom of Ref. [6] places it on a great circle of the cosmological horizon,

which is Eq. (3). The postulate does not contain the placement and this paper does not derive it. Ref. [17] does, from the postulate together with one more, that a direction is the state of one qubit, under which the directions of space are a qubit’s sphere of states, a two-sphere of this radius. Here the placement is taken over, and it is what gives $L$ its value, $2\pi R_{\mathrm{\Lambda }}/{\mathrm{\ell }}_P$. The identification is one of structure, the same ${\mathbb{Z}}_L$ under a qubit’s phase and a horizon great circle; this paper does not need a laboratory spin’s azimuth to be a place on the sky, an identification Ref. [17] makes as its second postulate. Section II.1 gives the axiom its content: the circumference is the wavelength of the slowest wave the step admits, so the second axiom says that the longest wavelength available to a free particle is the circumference of the cosmological horizon. With the placement the momentum quantum $h/L{\mathrm{\ell }}_P$ is $\mathrm{\hslash }/R_{\mathrm{\Lambda }}=6.4\times {10}^{-61}$ kg m s^-1 and the energy quantum ${\epsilon }_0=\mathrm{\hslash }c/R_{\mathrm{\Lambda }}=1.2\times {10}^{-33}$ eV, which is $h$ divided by the lap time $L{t}_P=2\pi R_{\mathrm{\Lambda }}/c=3.5\times {10}^{18}$ s and $2\pi k_B$ times the Gibbons–Hawking temperature [49].

The two determinations of $L$ are the two scales on which this circuit appears: from the horizon, $2\pi R_{\mathrm{\Lambda }}/{\mathrm{\ell }}_P=6.4\times {10}^{61}$; from galaxies, $c^2/a_0{\mathrm{\ell }}_P=4.6\times {10}^{61}$ using $a_0=c^2/L{\mathrm{\ell }}_P$ of Ref. [6]. Only the product $L{\mathrm{\ell }}_P$ enters $a_0$, so their agreement to 1.4 is Milgrom’s $a_0\approx c^2/2\pi R_{\mathrm{\Lambda }}$ restated; a derivation of $a_0$ from the circuit alone would make the placement a prediction, and we do not have one.

The horizon has $A/{\mathrm{\ell }}_P^2=L^2/\pi$ pixels, a register being the $L$ along one great circle, and its Bekenstein–Hawking entropy [48] is

the value of Bousso’s $N$-bound [28, 30], which is the empty de Sitter horizon entropy written as $\pi R_{\mathrm{\Lambda }}^2/{\mathrm{\ell }}_P^2$. Equation (13) is that number with $R_{\mathrm{\Lambda }}=L{\mathrm{\ell }}_P/2\pi$, an identity and not a derivation; what the register adds is that the integer fixing the log-dimension of a static patch is the $L$ that governs the qubit, Banks’ proposal that positive $\mathrm{\Lambda }$ and a finite Hilbert space go together [34] with its integer named.

The step translates, so it has a direction, but nothing so far has required one to be named: the algebra of Sec. III, the bound of Sec. IV, and the horizon quantities of Sec. V are the same along any direction. Dynamics is where a direction must be fixed, because a direction and its reverse make a line, and it is on a line that light has two components and matter has a chirality. Fix one. Section V placed the ring on a great circle of the horizon, and every great circle has the same circumference, so which one the direction selects changes nothing and $L$ is unaffected. Along the direction the same ${\mathbb{Z}}_L$ carries the momenta $\kappa$ of Eq. (10), and the basis $|j⟩$ conjugate to them is what position in whole cells means below: a coordinate on the register, periodic with the circumference because no wave is longer, and not a pixelation of the line. What follows is what the postulate gives on one line; several particles in several planes need a three-dimensional space, which is not constructed here and whose dimension Ref. [17] derives.

The register inherits the predictions of Ref. [6], corrects one criterion, and adds no test.

Inherited. The two predictions of Ref. [6] stand, since this paper rests on the same $L$. First, a state written on $N$ records gives cells to at most $L$ joint outcomes, Eq. (11): a ceiling of about 200 usefully entangled qubits, which Sec. IV places between 203 and 205 and does not claim is attained, the ${\log }_{2}L$ of Ref. [6] in place of the 212 it called exact. The ceiling holds for any physical realization, $M$ bosonic modes with photon-number cutoff $n$ reaching it at $n^M\approx L$. Operationally: run a random circuit on $N$ fault-tolerant logical qubits, deep enough to anticoncentrate, which is logarithmic depth [25], and then its inverse. In standard quantum mechanics the state returns to the initial one with the gate fidelity. On the register the return probability falls to about ${(L/2^N)}^2$ for $N$ beyond $\lfloor {\log }_{2}L\rfloor$, since the intermediate state has $2^N$ branches and the register has cells for $L$. The same circuit built from Clifford gates, whose intermediate states have at most three blocks to a record, returns at any $N$. The test needs no classical simulation, only the check of a return; no such machine exists. Second, $\mathrm{\Lambda }$ is constant, so the dark-energy equation of state is $w=-1$ exactly, against the DESI DR2 preference for evolving dark energy [40], which the framework predicts will not survive.

Definite, but not discriminating. Momentum is quantized in units of $\mathrm{\hslash }/R_{\mathrm{\Lambda }}$ and bounded at $\pi \mathrm{\hslash }/{\mathrm{\ell }}_P$. Light is a record at every Planck tick, and the right–left reading of matter at rest is a permitted setting once a quarter Compton period. The settings in which any two-mode subspace of a free particle may be read are restricted by Palmer’s Impossible Triangle Corollary. The maximum group velocity of a particle of mass $M$ is $c\cos (M/m_P)$, reached at $p=\pi \mathrm{\hslash }/2{\mathrm{\ell }}_P$ and below $c$ by ${(M/m_P)}^2/2=9\times {10}^{-46}$ for an electron, the group velocity of matter vanishing at the zone edge while light’s is $c$ at every momentum. The lattice fixes what Lorentz violation the register entails: the tick’s exact dispersion, Eq. (19), is $E=|p|c$ for light at every momentum and, for matter,

a fractional correction, of order ${(M{c}^2/E_P)}^2\approx 6\times {10}^{-39}$ for a proton, that does not grow with energy. The register therefore predicts no energy-dependent speed of light at any order and would be falsified by one; the correction for matter is beyond any measurement. The lattice likewise deforms the commutator: with the lattice momentum ${\widehat{p}}_{\mathrm{lat}}=(\mathrm{\hslash }/{\mathrm{\ell }}_P)\sin (\widehat{p}{\mathrm{\ell }}_P/\mathrm{\hslash })$, $[\widehat{x},{\widehat{p}}_{\mathrm{lat}}]=i\mathrm{\hslash }\cos (\widehat{p}{\mathrm{\ell }}_P/\mathrm{\hslash })\approx i\mathrm{\hslash }[1-\frac{1}{2}{(\widehat{p}{\mathrm{\ell }}_P/\mathrm{\hslash })}^2]$, a generalized uncertainty principle of the maximal-momentum type with coefficient ${\beta }_0=-1/2$ for one particle. Each constituent of a body carries its own record, so for the center of mass of $N$ of them the coefficient is $-1/2N^2$, the suppression Amelino-Camelia argued any composite must show [19]. The mechanical-resonator bounds,  [20], $5\times {10}^6$ [21], and now 62 from a cryogenic quartz resonator [22], by the route Pikovski et al. proposed [18], are on center-of-mass modes of $N\approx {10}^{20}$ constituents: two orders of magnitude above one particle’s coefficient and forty above the body’s. Because a two-path superposition of a composite’s center of mass is one qubit whatever the mass, and the register has no collapse mechanism, it predicts no intrinsic, mass-dependent loss of matter-wave contrast, in agreement with experiment to $2.5\times {10}^4$ atomic mass units [23] and unlike collapse models. None of these can distinguish the register from standard quantum mechanics.

The register’s intrinsic scales are $m_P{(2\pi /L)}^{1/n}$: $1.2\times {10}^{-33}$ eV, $3.8$ meV, $56$ MeV, and $6.8$ TeV. The second is, to a factor 1.7, the dark-energy scale ${\rho }_{\mathrm{\Lambda }}^{1/4}$, which follows from Eq. (12) as ${(3\pi /2)}^{1/4}{E}_P/\sqrt{L}=2.2$ meV, the fourth root of the $L^2$ suppression of the Planck density counted in Ref. [16]; the third is Weinberg’s ${({\mathrm{\hslash }}^2{H}_0/Gc)}^{1/3}$ [15]. That the neutrino and the pion sit near the second and third rungs is the set of large-number coincidences known since Dirac, in one integer; the register offers no mechanism for them.

Placing every register on the horizon can mean two things. The first, that the cells have a definite orientation in space, costs nothing. A cyclic register has no pole and no first cell; both are coordinates on the Bloch sphere, and every quantity in Eq. (2) is relational, an overlap of two states or a phase relative to a reference, which Sec. VI.2 shows to be a position relative to a reference. Palmer’s Bell conditions are on angles between settings [3], which are rotation-invariant; he fixes the gauge by placing the pole at his middle setting [5], and his distinction between a nominal setting and the exact one within it [4] says the gauge cannot be probed.

The postulate’s last clause carries this from the step to the states written on it. Not only is there no first cell; there is no phase reference outside a state, its global phase being compared with nothing. Nothing in Eq. (2) reads a state’s global phase and no dynamics above does either, so the extension costs nothing and closes the one place where an unprobed convention could have passed for data. It has no effect on the capacity: a state’s global phase is the origin of its cell labels, which no count and no block reads.

The second, that different observers’ horizons are the same cells, is de Sitter complementarity [32, 29, 43, 44, 45]: distinct static patches describe one finite set of degrees of freedom, not independent copies. The shared reading follows from unitarity within a patch by the no-cloning argument used for black holes [33, 42], taken over without proof; whether that unitarity is consistent with a smooth horizon is the firewall question [41], open in de Sitter and not addressed here. Nothing in Secs. II to VII depends on it. A finite Hilbert space of log-dimension $L^2/4\pi$ entails recurrences and a low-entropy past that requires explanation [43], to which Palmer’s nonergodic Invariant Set Postulate [39] is RaQM’s response; it entails positive $\mathrm{\Lambda }$ [34] and no exact asymptotic observables [35]; and finite-dimensional Hilbert spaces for quantum gravity have been advocated independently [36, 37, 38].

A finite register also says what it does not hold. The bound of Sec. IV is on a pure state written on one register. An observer outside a black hole describes it by a mixed state, whose entropy counts the records the description ranges over, as Eq. (13) counts them for the cosmological horizon; Sec. IV counts the outcomes to which one record can give cells, and the two are not the same quantity. What the register does not admit is a scrambled pure state above $\lfloor {\log }_{2}L\rfloor$ qubits, and the pure state of a hole together with its radiation, which unitarity of the pair asserts and which scrambling makes generic [24], is one. No observer’s register writes it. That is the register’s position on the information question, recorded here and not resolved; it agrees with there being no exact observables in de Sitter space [35].

The rational states Ref. [6] took from Palmer and the universality it had to stipulate are one statement about one ring. A string of $L$ cells turns at the rate its energy sets. Counted, the pattern is a Born probability; for the right- and left-moving components that probability sets the velocity. Shifted, it is a phase; on the ring read as space, one cell is a translation by one Planck length, the slowest wave gains one step of phase, Palmer’s $2\pi /L$, per translation, and for light the phase and the translation are the same thing at every tick. Mass is the amplitude per tick to reverse direction, a particle at rest is a standing wave of trapped light, and the Compton clock is the global phase that Palmer hides and that, having no gradient, displaces nothing. The circumference is the longest wavelength a particle can have, and placing it on the horizon is the second axiom with that content attached.

Less is stipulated than in Ref. [6], though not less is assumed. Its first axiom is a consequence of the postulate, Eq. (2) with $L$ universal, since the cell belongs to space rather than to any system in it; its second, the placement of the ring on the horizon, is taken over here and derived in Ref. [17], and it is what gives $L$ its value, Eq. (3). What changes is what must be defended: a universal $L$ no longer has to be imposed against Palmer’s mass-dependent one, because a grain that varies with the system cannot be the grain that does not. The other gains are structural. RaQM’s phase step is momentum quantization on a closed ring, its qubit is the right–left doublet of a Dirac particle, and its hidden global phase is the Compton clock. Its capacity bound, two cells to every branch of a state, is $\lfloor {\log }_{2}L\rfloor$, soft by a unit in resolution: a ceiling of about 200 that no representation may exceed and that holds whatever the qubits are made of. What remains open is what was open before, the coefficient in $a_0$ and complementarity; the three-dimensional bulk is the subject of Ref. [17]. Ref. [6] remains the statement the framework rests on, its exact capacity criterion corrected here. Palmer has left the discretization of space and time to a future paper and announced work on vacuum energy [5]; this paper and Ref. [16] attempt those questions with a universal $L$.