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 mark on 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 [23] 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. [2], 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 [48],

From the two, Newton’s constant follows by Verlinde’s argument [45] with the pixel fixed, Einstein’s equations by Jacobson’s [28], 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 $a_0$ [46, 47] 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. This paper shows they are two descriptions of a single process: a string on a dial of $L$ notches, turning at the rate its energy dictates, where one notch is a translation by one Planck length and Palmer’s phase step is what that translation does to the slowest wave the dial can hold. 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 recovers Eq. (2). Section III sets the ring in motion; the free dynamics is exact mode by mode in the plane, position in whole cells holds along one direction only, and Sec. III.4 says which states need a three-dimensional space; the paper does not construct it, and its dimension is derived in a following paper [17]. Section IV places the ring on the horizon and gives Eq. (3) a physical content. Section V states what is predicted and what would falsify it. Section VI states what the placement assumes.

Postulate. There is a dial of $L$ notches, with $L\in \mathbb{N}$ universal and each notch one Planck length long, which light crosses in one Planck time, closing on any great circle of the cosmological horizon. Every qubit is a pattern of marks on it. Turning the dial one notch is a translation by one Planck length; a wave with $\kappa$ wavelengths around the dial thereby gains $\kappa$ steps of phase, and Palmer’s phase step, $2\pi /L$, is the step of the slowest wave, whose wavelength is the circumference.

The picture is a clock face with $L$ marks. A qubit is a pattern on the face, which can be counted and turned; a wave is a pattern repeating $\kappa$ times around it. Advancing the face by one mark moves every wave one Planck length; a pattern with $\kappa$ repeats thereby turns through $\kappa$ marks of phase, the slowest wave through exactly one, which is Palmer’s unit. Time passing is a different operation: in one Planck tick each component of light moves one mark in its own direction, Sec. III.2. We call the notches cells, and the dial with its patterns the register. It is a ring of $L$ equal cells with no first cell, and two things can be done with a pattern on it: it can be counted, and it can be shifted by turning the dial. The count is unchanged by a shift and a shift does not depend on the count. Palmer’s bit string is such a pattern, written relative to a measurement basis; one string is one qubit. A free particle needs no clause of its own: quantum mechanics already gives it two-level structure, the right- and left-moving components at each momentum, so its chirality and its modes are patterns on the dial, its position is a distribution over cells, sharp when the pattern occupies one, and its orbital Hilbert space has dimension $L$, worth ${\log }_{2}L\approx 205$ qubits, to which the chirality of Sec. III.3 adds one, a general state being a superposition over $L$ modes with each mode a Palmer string. The integer $L$ therefore plays three roles: the length of Palmer’s string; the number of Planck lengths around a great circle of the horizon, equivalently the number of Planck times light takes to go once around it; and the dimension of a free particle’s orbital Hilbert space on the ring. The postulate is that all three are one number, and one symbol is used for that reason. Taken from RaQM are the Schrödinger equation and the reading of the Born probability as the frequency of marks [2]; assumed are $\mathrm{\hslash }$, $c$, and the numerical value of ${\mathrm{\ell }}_P$, from which $G={\mathrm{\ell }}_P^2{c}^3/\mathrm{\hslash }$ is derived in Ref. [6] rather than the reverse.

Read for a qubit, the count is the Born probability. A qubit’s latitude cuts the Bloch sphere into two caps whose fractional areas are ${\cos }^2(\theta /2)$ and ${\sin }^2(\theta /2)$; partition the sphere into $L$ equal-area bands and a state $m$ bands from the $|-1⟩$ pole has

since equal areas are equal steps in $\cos \theta$ by Archimedes’ theorem. The shift is the phase: laid around the equator, one cell is

and Palmer’s operator $\zeta$, which advances the phase by one step, is the rotation of the string by one place [2]; read on the ring, Sec. III, it is what one notch of translation does to the slowest wave, and faster waves gain more. Together these are Eq. (2). They do not collide: Niven’s theorem [23, 24] allows $\cos \alpha$ and $\alpha /2\pi$ to be simultaneously rational at only eight angles, but the count is a fraction of area and the shift a fraction of arc, and area and arc are incommensurate on a sphere, so the permitted set has of order $L^2$ points. The universal $L$ is the first axiom of Ref. [6], and everything Palmer derives from Eq. (2) carries over: the string, Born’s rule as a frequency, the Impossible Triangle Corollary, and the capacity $N_{\max }$.

The ring has circumference $L{\mathrm{\ell }}_P$ and radius $L{\mathrm{\ell }}_P/2\pi$, and the postulate places it on a great circle of the cosmological horizon,

which is Eq. (3) rearranged and the second axiom of Ref. [6]. It is not derived here, and the identification is one of structure, the same ${\mathbb{Z}}_L$ under a qubit’s phase and a horizon great circle; we do not claim a laboratory spin’s azimuth is a place on the sky. What the ring adds is the axiom’s content. Section III showed the circumference to be the longest wavelength a free particle can carry, so the second axiom reads: the longest wavelength available to a free particle is the circumference of the cosmological horizon. The momentum quantum 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 [44].

The two determinations of $L$ are the two scales on which this ring 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 ring 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, of which a register is the $L$ along one great circle, and its Bekenstein–Hawking entropy [43] is

the value of Bousso’s $N$-bound [25, 27], which is the empty de Sitter horizon entropy written as $\pi R_{\mathrm{\Lambda }}^2/{\mathrm{\ell }}_P^2$. Equation (17) is that number with $R_{\mathrm{\Lambda }}=L{\mathrm{\ell }}_P/2\pi$: an identity, 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, which is Banks’ proposal that positive $\mathrm{\Lambda }$ and a finite Hilbert space go together [31] with its integer named.

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, no physical state has more than $L$ independently specified outcome probabilities, Eq. (15), a ceiling of about 200 usefully entangled qubits, which Sec. III.4 fixes at $N_{\max }=205$ at $L_{\cos }$, the ${\log }_{2}L$ of Ref. [6], in place of the 212 it called exact, and extends from qubits to any physical realization, $M$ bosonic modes with photon-number cutoff $n$ reaching it at $n^M\approx L$. Operationally: a random circuit on $N$ fault-tolerant logical qubits followed by its inverse returns to the initial state, in standard quantum mechanics, with the gate fidelity; on the register the return probability falls to about ${(L/2^N)}^2$ for $N$ beyond $\lfloor {\log }_{2}L\rfloor$, because the intermediate state has $2^N$ independent amplitudes and the register holds $L$, while the same circuit built from Clifford gates, whose intermediate states have polynomially many, 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 [34], 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 permitted at every Planck tick and matter of definite momentum, at rest, only every quarter Compton period; a Bell test on any two-mode subspace of a free particle is subject to Palmer’s Impossible Triangle Corollary; and the maximum speed of a particle of mass $M$ is $c/\sqrt{1+{(M/\pi m_P)}^2}$, below $c$ by ${10}^{-47}$ for an electron. The lattice also fixes what Lorentz violation the register entails. The tick’s exact dispersion, Eq. (14), is $E=|p|c$ for light at every momentum and, for matter,

a fractional correction of order ${(M{c}^2/E_P)}^2\approx {10}^{-39}$ for a proton that does not grow with energy. The register therefore predicts no energy-dependent speed of light at any order, in one dimension, 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 $-1/2$, six to eleven orders of magnitude below the present optomechanical bounds on that coefficient [20]. Because a two-path superposition of a composite’s center of mass is one qubit whatever the mass, and the register contains no collapse mechanism, it predicts no intrinsic, mass-dependent loss of matter-wave interference contrast, in agreement with experiment to $2.5\times {10}^4$ atomic mass units [21] and in contrast to 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, $6.8$ TeV. The second is, to a factor 1.7, the dark-energy scale ${\rho }_{\mathrm{\Lambda }}^{1/4}$, which follows from Eq. (16) 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, written 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, and a phase relative to a reference, which Sec. III.3 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 second, that different observers’ horizons are the same cells, is de Sitter complementarity [29, 26, 37, 38, 39]: 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 [30, 36], taken over without proof; whether that unitarity is consistent with a smooth horizon is the firewall question [35], open in de Sitter and not addressed here. Nothing in Secs. II to V depends on it. A finite Hilbert space of log-dimension $L^2/4\pi$ entails recurrences and a low-entropy past requiring explanation [37], to which Palmer’s nonergodic Invariant Set Postulate [33] is RaQM’s response, positive $\mathrm{\Lambda }$ [31], and no exact asymptotic observables [32]; finite-dimensional Hilbert spaces for quantum gravity have been advocated independently [40, 41, 42].

The two axioms of Ref. [6] are two descriptions of one rotation. 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 notch is a translation by one Planck length, Palmer’s phase step is what that translation does to the slowest wave, and for light the phase and the translation are the same thing at every tick. Mass is the amplitude per tick for a right-mover to become a left-mover, 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.

Nothing is postulated beyond the two axioms of Ref. [6], restated as one. The gains are structural: RaQM’s phase grid is momentum quantization on a closed ring; its qubit is the right–left doublet of a Dirac particle; its hidden global phase is the Compton clock; and its capacity bound, read off Born’s rule as a frequency, is the Hilbert-space dimension of one particle on the ring, so the ceiling of about 200 qubits is $\lfloor {\log }_{2}L\rfloor$ exactly and holds whatever the qubits are made of. What remains open is what was open before: the coefficient in $a_0$, the three-dimensional bulk, and complementarity. Ref. [6] remains the load-bearing statement of the framework, its exact capacity criterion corrected here. Palmer has left the discretization of space and time to a future paper and announced work on why vacuum energy does not gravitate [5]; this paper and Ref. [16] attempt those questions with a universal $L$.

We record one coincidence without a mechanism. The heaviest observationally stable nucleus is ²⁰⁸Pb; ²⁰⁹Bi decays with a half-life of $2\times {10}^{19}$ years [19, 18]; and $\lfloor {\log }_{2}L\rfloor =205$. For the capacity bound to bear on where the table ends, four things would be needed, and the framework supplies none: a representation in which a state spanning more than ${\log }_{2}L$ two-level systems can be written, which the frequency reading of Born’s rule forbids; a two-level degree of freedom per nucleon that the framework selects, which it does not, a particle’s chirality being fixed by its momentum through Eq. (11) and nuclear interactions acting on positions, spins, and isospins; a statement within RaQM’s dynamics about a composite whose spanning states exceed capacity, which the Invariant Set Postulate does not provide, since the dynamics never leaves the permitted set; and a rate, since no nucleus above $A\approx 150$ is absolutely stable (²⁰⁸Pb has $Q_{\alpha }=+0.52$ MeV) and only half-lives are observable.