Gravitation from Hilbert-Space Granularity:
Pinning the Parameter $L$ of Rational Quantum Mechanics Across Scales

Rational Quantum Mechanics (RaQM) [2, 3, 4, 5] leaves the Schrödinger equation untouched and instead restricts the bases in which a quantum state is defined: a qubit

with $\theta$ and $\varphi$ the colatitude and azimuth of the state on the Bloch sphere, exists only in bases with

The state is then a length-$L$ bit string whose fraction of $+1$ entries is ${\cos }^2(\theta /2)$ and whose cyclic shift by $n$ places is the phase [3]. Born’s rule becomes a matter of counting. Bell’s inequality is evaded because, by Niven’s theorem [8, 9], the counterfactual measurement settings its derivation requires are not permitted bases [7, 4]. And because $N$ entangled qubits carry $NL$ bits against $2^{N+1}-2$ continuum degrees of freedom, there is a qubit information capacity

beyond which quantum algorithms that spread their state across all of Hilbert space lose their exponential advantage [3].

Palmer fixes $L$ from gravity. State reduction is a chaotic shift map that lowers $L$ by one each Planck time $t_P$ (in Palmer’s ordering, $t_P=\sqrt{\mathrm{\hslash }G/c^5}$), so the reduction time is ${\tau }^{\ast }=(L-1)t_P$; equating this to the Diósi–Penrose collapse time $\mathrm{\hslash }/E_G$ [10, 11] gives

where $E_P=\mathrm{\hslash }/t_P$ is the Planck energy, $E_G$ is the gravitational self-energy of the qubit’s superposition, and $\lceil x\rceil$ is the smallest integer not less than $x$. Both $E_P$ and $E_G$ contain $G$. Gravity is the input and $L$ the output, and $L$ varies with the qubit, from ${10}^{64}$ for a quantum-dot electron to ${10}^{109}$ for a hyperfine ion-trap qubit [3]. Palmer closes by noting that his $N_{\max }$ lands beside Davies’ cosmological information bound [12, 13] and that this is unlikely to be a coincidence.

We take that remark literally and reverse the causality. $L$ does not exist because of gravitation. Gravitation exists thanks to $L$, which is a single universal constant of nature. Such a claim is tested first of all by consistency, and then by prediction. Known gravitational phenomena on every scale, each used on its own to pin $L$, must return the same number. Section II states the two axioms and shows how Eq. (4) inverts. Section III shows how Newton’s constant follows from the pixel size ${\mathrm{\ell }}_P=2\pi R_{\mathrm{\Lambda }}/L$, i.e., from the ratio of the horizon radius to $L$. Section IV works through the scales in turn, from quantum to cosmological, deriving the familiar phenomena from $L$ on each and recording the value of $L$ each one pins. Section V asks whether $L$ holds. Section VI lists what would falsify the picture.

Axiom 1. Quantum states are defined only in bases satisfying Eq. (2), with a single universal $L\in \mathbb{N}$ for all systems.

This is Palmer’s discretization ansatz with one change: $L$ is not a function of the system’s mass or energy. In RaQM it is, through Eq. (4), and it differs from one qubit technology to another. Palmer’s earlier work constructed an arbitrarily dense discretization of the Bloch sphere in which states are bit strings of fixed finite length [2], the fineness of the discretization being determined by a finite prime-number parameter $p$ [6]; in the original, continuous version of the theory, heterogeneities in the fractal state-space geometry were proposed to be manifestations of gravity [1]. Everything Palmer derives from Eq. (2) at fixed $L$ carries over unchanged: the bit-string representation, Born’s rule, the Impossible Triangle Corollary, and the capacity bound (3).

Equation (2) is a two-level-system condition, and Axiom 1 is stated for such systems; the extension to continuous variables, and the question of whether Axioms 1 and 2 are two readings of a single postulate, will be treated separately.

Axiom 2. The granularity of state space and the granularity of space are the same discreteness: one step of quantum phase, $2\pi /L$, is one pixel on a great circle of the cosmological horizon of radius $R_{\mathrm{\Lambda }}$. The pixel is therefore

Axiom 2 says that a phase rotation through a full turn counts $L$ pixels, which is the natural reading of Palmer’s construction: his phase operator $\zeta$ is a cyclic permutation of the $L$-bit string, and each application is a rotation by exactly $2\pi /L$ about the polar axis [3]. The axiom refers to the cosmological horizon without saying which one. Nothing in it fixes the radius in time; but $L$ is a constant, so ${\mathrm{\ell }}_P$ and hence $G$ scale with the radius, and the observed constancy of $G$ then requires the radius to be constant too (Sec. IV). In an expanding universe the only constant horizon is the de Sitter one, whose radius is set by the cosmological constant. Whether the value of $L$ this gives agrees with the value from galactic dynamics is the consistency check of Sec. IV, and the alternative, a horizon that grows with the Hubble radius, is excluded there on observational grounds. Because every observer’s de Sitter horizon has the same radius, $L$ is observer-independent, as a constant of nature must be.

In this form the constants of the theory are $\mathrm{\hslash }$, $c$, the integer $L$, and one length, $R_{\mathrm{\Lambda }}$. The pixel ${\mathrm{\ell }}_P$ is derived, and so are the other Planck units, $t_P\equiv {\mathrm{\ell }}_P/c$ and $E_P\equiv \mathrm{\hslash }c/{\mathrm{\ell }}_P$, none of which involves $G$; Sec. III shows that $G$ is derived too. Because $L$ has not yet been measured, we run the chain backwards throughout this paper: we take ${\mathrm{\ell }}_P$ at its accepted value, $1.616\times {10}^{-35}$ m, and use Eq. (5) to predict $L$. The values of $L$ obtained this way scale linearly with ${\mathrm{\ell }}_P$; the quantum prediction $N_{\max }$ depends on it only logarithmically.

The reversal. Keep Palmer’s picture of state reduction as a shift map at one step per Planck time, so that ${\tau }^{\ast }=L{t}_P$, but with $L$ universal. Then, using Eq. (5) and ${\mathrm{\ell }}_P=c{t}_P$,

The state-reduction clock is the de Sitter period $2\pi /H_{\mathrm{\infty }}$, where $H_{\mathrm{\infty }}=c/R_{\mathrm{\Lambda }}$ is the asymptotic Hubble rate. It is the time for one full turn of the horizon at the speed of light. The universal energy that replaces $E_G$ is then

the Gibbons–Hawking temperature of the horizon [30]. Palmer’s Eq. (4), read backwards, says that gravity’s characteristic energy in the reduction process is the thermal energy of the cosmological horizon, and that every qubit shares it. Three consequences of the reversal are worth stating at once. First, ${\tau }^{\ast }\sim {10}^{11}$ yr, so an isolated qubit never completes reduction. Palmer makes the same observation: collapse is irrelevant as a practical constraint. Second, $L$ no longer depends on the qubit, so $N_{\max }$ is the same for every technology. That is the sharpest experimental difference between the two causal orderings. Third, the reversal discards mass-dependent gravitational collapse. In Palmer’s ordering a heavy superposition has small $L$ and reduces in milliseconds; here every system shares the ${10}^{11}$-yr clock, so the definiteness of macroscopic bodies is not supplied by gravity and must come, as in standard quantum theory, from environmental decoherence. The reversed theory therefore predicts no mass-dependent collapse signal of Diósi–Penrose type, consistent with the null result of the underground search of Ref. [14]. Palmer’s shift map, however, is non-dissipative by construction and is equally consistent with that null result, so the null result does not discriminate between the two orderings. $N_{\max }$ does.

Before testing the hypothesis at widely separated scales, we derive the relation between $G$ and ${\mathrm{\ell }}_P$ by assuming that gravity arises from a pixelated holographic screen: a two-dimensional surface enclosing a region of space, on which all the information about that region is stored one bit per pixel. Space itself is not pixelated in this picture; only its boundaries are, and that is what makes the count an area law rather than a volume law. The argument is Verlinde’s [15]; we repeat it here with the pixel size written out, so that it is clear that $G$ is derived from ${\mathrm{\ell }}_P$. Entropic gravity in this form is not uncontested [16, 17]. We use it only as the existing route from a pixelated screen to an inverse-square law, and rely on nothing beyond the scaling $G\propto {\mathrm{\ell }}_P^2$ that it shares with every holographic account. Consider a spatial sphere of radius $r$ enclosing mass-energy $M=E/c^2$.

On each scale we derive the known phenomena from $L$ and $R_{\mathrm{\Lambda }}$ using the existing mathematics of horizon thermodynamics (Verlinde’s for Newton [15], Jacobson’s for Einstein [18], and Verlinde’s later work for the galactic regime [19]), and we record what each scale pins.

Table 1 collects the results. Two scales pin $L$ independently. The galactic scale, through the acceleration at which rotation curves flatten, gives $4.6\times {10}^{61}$. The cosmological scale, through the size of the de Sitter horizon, gives $6.4\times {10}^{61}$. They agree to a factor of $1.4$, which is half a bit in ${\log }_{2}L$, the scale the quantum prediction lives on: $204.9$ against $205.3$. The two measurements come from scales five orders of magnitude apart and use entirely different methods, yet they give the same order of magnitude for a $62$-digit number.

We do not pretend this agreement is new. It is the coincidence $a_0\sim c{H}_0$ that Milgrom noted in 1983 [33], and one may fairly object that the two pins are one old observation in new clothes. What the framework adds is twofold. It turns the coincidence into a fixed relation, Eq. (17) combined with Eq. (12), so that it is a coincidence no longer. And it ties the same $L$ to a third quantity that has nothing to do with either galaxies or cosmology: the number of qubits a quantum computer can usefully entangle. That third pin is where the theory is falsifiable.

The terrestrial and solar scales do not pin $L$, and we have said why. They pin ${\mathrm{\ell }}_P$, the combination $2\pi R_{\mathrm{\Lambda }}/L$. For the reversed theory they establish that gravity exists at all: a pixelated screen with a Planck pixel produces Newton’s law and Einstein’s equations with the observed strength. This checks that the pixel is Planckian (Sec. III); it is not a check on $L$, and it is passed by construction once ${\mathrm{\ell }}_P$ is the Planck length. We do not count it as evidence for $L$.

The quantum scale will pin $L$ directly. Inserting either galactic or cosmological value into Eq. (3) gives $N_{\max }\approx 205$, and the residual disagreement between the two pins shifts this by half a unit. We quote the prediction as $N_{\max }\approx 200$, since the leading approximation ${\log }_{2}L$ and Palmer’s exact criterion $2^{N+1}-2\le NL$ differ by several units. That ambiguity does not touch the point that matters, which is that the ceiling is universal. A measurement of $N_{\max }$ near $200$ for one technology and near $400$ for another would restore Palmer’s causal order; the same value for all would confirm ours.

That measurement also closes the loop in the direction the title promises. With $L$ known, Eq. (5) gives the Planck length from the size of the universe, and Eq. (11) gives Newton’s constant:

Gravity’s strength is then a derived quantity: the number of bits that fit around the horizon. Converting a measured $N_{\max }$ into $L$ requires the exact capacity condition rather than its logarithmic approximation: $N_{\max }$ is the largest $N$ with $2^{N+1}-2\le NL$, which brackets $L$ between $2^{N_{\max }+1}/N_{\max }$ and $2^{N_{\max }+2}/(N_{\max }+1)$, a factor of two. Hence ${\mathrm{\ell }}_P$ follows to a factor two and $G$ to a factor four. The observed $G$ corresponds to $N_{\max }=212$ under this criterion (Appendix A); a count of $211$ or $213$ would return $G$ four times too large or too small. Nothing else in the theory can do this: in $a_0=c^2/L{\mathrm{\ell }}_P$ only the product $L{\mathrm{\ell }}_P=2\pi R_{\mathrm{\Lambda }}$ appears, so galactic and cosmological data fix each other and leave ${\mathrm{\ell }}_P$ untouched. Only the quantum computer separates $L$ from ${\mathrm{\ell }}_P$.

Two further remarks. The universal $L$ found here, $\sim 6\times {10}^{61}$, lies below every per-qubit value Palmer obtains from Eq. (4), so if both mechanisms were at work, the cosmological one would be the binding constraint. And Davies’ bound [12], which Palmer flagged as suspiciously close to his own numbers, is explained rather than merely matched. Since $L$ counts pixels around the horizon’s circumference, the horizon’s area holds $4\pi R_{\mathrm{\Lambda }}^2/{\mathrm{\ell }}_P^2=L^2/\pi =1.3\times {10}^{123}$ of them, which is Lloyd’s estimate of ${10}^{122}$ bits for the observable universe [13]. Davies’ $N={\log }_2{10}^{122}\approx 405$ is the logarithm of that pixel count, i.e. $2{\log }_{2}L$. Palmer’s capacity bound counts the bit-string length $L$, not the number of pixels, so $N_{\max }={\log }_{2}L\approx 205$, half Davies’ exponent, and the factor of two now has a definite origin.

1. 1.

   A universal entanglement ceiling near 200 qubits, the same for photonic, trapped-ion, superconducting and semiconductor qubits, insensitive to isolation and to logical error correction. Technology dependence falsifies the reversal and restores Eq. (4).

2. 2.

   Flat rotation curves from baryons alone, with $v_c^4=a_{0}G{M}_{\mathrm{baryon}}$, Eq. (14), and $a_0=c^2/2\pi R_{\mathrm{\Lambda }}$, Eq. (12). Robust detection of a dark-matter particle, or a galaxy whose rotation curve requires one, falsifies the galactic sector.

3. 3.

   No redshift evolution of $a_{\mathrm{𝟎}}$. The acceleration scale, Eq. (12), is tied to the constant $R_{\mathrm{\Lambda }}$ of Axiom 2, not to $H(z)$; proposals with $a_0\propto cH(z)$ predict a scale $1.8$ times larger at redshift $z=1$, and this framework predicts none.

4. 4.

   Constancy of $𝚲$. With $L$ fixed, Eq. (18) gives $G\propto R_{\mathrm{\Lambda }}^2$, so the lunar-laser-ranging bound on $\dot{G}/G$ [39] requires the horizon radius, and with it $\mathrm{\Lambda }$, to be constant to one part in ${10}^{13}$ per year, and their product $G\mathrm{\Lambda }=12{\pi }^2{c}^3/\mathrm{\hslash }{L}^2$, Eq. (17), is fixed by $L$ alone. Since a dark-energy component with equation of state $w$ evolves as $\dot{\rho }/\rho =-3H(1+w)$, this is a bound on the present-day equation of state, $|1+w_0|\lesssim 5\times {10}^{-4}$ (Appendix A). It applies to the present epoch, the only one lunar laser ranging probes. A dark-energy density that is still evolving on the Hubble timescale today, as some current data prefer [36], is incompatible with the framework as stated.

Palmer built a discrete quantum mechanics and used gravity to set its granularity. We have argued that the granularity comes first. One universal integer $L$, one length $R_{\mathrm{\Lambda }}$, and the identification of a full turn of quantum phase with the circumference of the cosmological horizon are enough to make the Planck length and Newton’s constant derived quantities, to obtain general relativity through Jacobson’s argument, to produce flat galactic rotation curves with no dark matter, and to bind Newton’s constant and the cosmological constant into one. Palmer’s collapse formula inverts into a universal reduction clock equal to the de Sitter period, with the horizon temperature as the energy scale, and mass-dependent gravitational collapse drops out of the theory. None of these successes tests the framework on its own, since most are inherited from horizon thermodynamics once the pixel is Planckian. The real test is whether the two scales on which $L$ appears on its own in gravity agree. They do, to a factor of $1.4$ on a number of $62$ digits, and that agreement is Milgrom’s coincidence explained. The third scale on which $L$ appears on its own, the quantum computer, will report within the decade.