Dispersion of emergent gravitons in local models

Can a massless spin-2 mode with linear dispersion emerge from local qubits or rotors? We consider harmonic theories of a symmetric tensor field with exponentially local couplings (real-analytic, periodic symbols), emergent linearised spatial diffeomorphisms and a stable Hamiltonian. The frequencies of the gapless modes then satisfy $\omega\le Ck^2$ at small momentum $k$ (Proposition 1). The bound remains when a scalar constraint also emerges and the kinetic term is exactly invariant under it (Proposition 2(a)), and it becomes $\omega\le Ck^3$ when the variables are compact in the sense of (H5) below (Proposition 2(b)). Linear dispersion therefore needs a kinetic term that is invariant under the scalar constraint only up to terms proportional to the constraints, as in general relativity. For bounded constraint operators on a finite-dimensional local space, and if the linearised generators are those operators (hypothesis (B) of section 3.3), such weak invariance is absent on the constraint surface (Proposition 3).

Sections 4 to 6 treat four other models of emergent gravity: composite tetrads, entanglement geometry, Gauss-law sectors and a tick-rate automaton. Strongly interacting models, such as the N-type model of Gu and Wen discussed in section 7, are outside Propositions 1 to 3.

Gu and Wen constructed a local bosonic lattice model with graviton excitations [Gu–Wen 2006] and two qubit models with emergent linearised diffeomorphisms [Gu–Wen 2009]. In the first qubit model, the L-type model, a semiclassical treatment gives gapless helicity-two modes with $\omega\propto k^3$ . In the second, the N-type model, the collective modes are strongly interacting. Boson liquids give $\omega\propto k^2$ and Lifshitz gravitons with $z=3$ and $z=2$ [Xu] [Xu–Hořava], and tensor gauge theories with fractons give linear dispersion with a gauge group smaller than the diffeomorphisms [Pretko]. Marolf showed that universal coupling to energy requires kinematic non-locality [Marolf]. Diakonov proposed spinor gravity with a composite tetrad, and Vladimirov and Diakonov analysed it by a lattice mean field [Diakonov] [Vladimirov–Diakonov]; Cao, Carroll and Michalakis proposed to recover space from entanglement [CCM]. The new result is the identification, in Propositions 1 to 3, of weak invariance of the kinetic term under the scalar constraint as the property that linear dispersion requires in this class. The page on a conserved lattice momentum treats a related locality question for gravitons with exact diffeomorphisms, and the page on exact universal graviton coupling the soft-theorem side.

1. Linearised gauge structure

The linearised spatial gauge transformation $A_{ij}\to A_{ij}+\partial_i\xi_j+\partial_j\xi_i$ of a symmetric tensor $A_{ij}$ (equation (3) of the page on a conserved lattice momentum) is generated by the momentum constraint $\partial_jE_{ij}=0$ on the conjugate momentum $E_{ij}$ . General relativity also has the Hamiltonian constraint $\partial_i\partial_jA_{ij}-\partial^2A_{ii}=0$ , a function of $A$ whose generator shifts the momentum, $E_{ij}\to E_{ij}+(\partial_i\partial_j-\delta_{ij}\partial^2)\theta$ . In Fourier space the two gauge directions are

g(k)\xi=i\,(k_i\xi_j+k_j\xi_i),\qquad s(k)=k\otimes k-\lvert k\rvert^2\mathbb 1. \tag{1}

We call them the vector and the scalar constraint. The vector one shifts $A$ along $g\xi$ and is imposed as $g^\dagger E=0$ . The scalar one shifts $E$ along $s\theta$ and is imposed as $s^\dagger A=0$ . The two generators commute because $k_is_{ij}=0$ , that is $g^\dagger s=0$ : a shift of $E$ along $s\theta$ does not change $g^\dagger E$ , and a shift of $A$ along $g\xi$ does not change $s^\dagger A$ .

The statements of sections 2 and 3 concern a harmonic Hamiltonian $H_2=\tfrac12\sum_k(E,A)^\dagger\mathbf H(k)(E,A)$ with

\mathbf H(k)=\begin{pmatrix}K(k)&X(k)\\ X(k)^\dagger&M(k)\end{pmatrix},

where $E$ and $A$ take values in the six-dimensional space of symmetric tensors with the Frobenius inner product, and $K,M$ are Hermitian. Two hypotheses are common to all propositions of section 3.

(H1) The entries of $\mathbf H(k)$ are real-analytic and $2\pi$ -periodic in $k$ , which is what exponentially local couplings on a lattice give.

(H2) The vector gauge transformation is $g(k)=g_1(k)+O(k^2)$ with $g_1(k)\xi=i(k_i\xi_j+k_j\xi_i)$ , and $\mathbf H(k)$ is invariant under it, $\mathbf H(k)(0,g\xi)=0$ , that is $Mg=0$ and $Xg=0$ . Physical states obey $g^\dagger E=0$ .

The quantities bounded in section 3 are the frequencies of the harmonic theory on physical states. With the gauge directions removed, the physical $(E,A)$ at fixed $k$ form a symplectic vector space, and $\mathbf H$ induces a quadratic form on it. If this form is positive definite, Williamson’s theorem gives a symplectic basis in which it is $\mathrm{diag}(\Omega,\Omega)$ with $\Omega=\mathrm{diag}(\omega_1,\omega_2,\dots)$ , and the $\omega_j$ are the mode frequencies. For a Hamiltonian $\tfrac12E^\dagger KE+\tfrac12A^\dagger MA$ without cross term they are the square roots of the eigenvalues of $KM$ . We say $z\ge z_0$ if $\omega(k)=O(\lvert k\rvert^{z_0})$ for all modes.

2. Worked examples

We take $k=\kappa(0,0,1)$ and write $K$ and $M$ as $6\times6$ matrices in an orthonormal basis of symmetric tensors. The Fierz–Pauli matrix $M_{\rm FP}$ is defined by $A^\dagger M_{\rm FP}A=2\,\mathcal U(A)$ , with $\mathcal U$ the potential of equation (2) of the page on a conserved lattice momentum and $\partial\to ik$ ; explicitly

A^\dagger M_{\rm FP}A=\tfrac12\lvert k\rvert^2\lvert A\rvert^2-\lvert kA\rvert^2+(k^{T}Ak)\,\mathrm{tr}A-\tfrac12\lvert k\rvert^2(\mathrm{tr}A)^2 .

The DeWitt kinetic matrix is $K_{\rm DW}=\mathbb 1-\tfrac12\,\mathbb 1\mathbb 1^T$ , half the kinetic form $\pi_{ij}\pi_{ij}-\tfrac12\pi_{kk}^2$ of that page. The two other matrices are

\mathrm{inc}(A)_{ij}=\varepsilon_{iab}\varepsilon_{jcd}k_ak_cA_{bd},\qquad \mathrm{symcurl}(E)_{ij}=\tfrac12(\varepsilon_{iab}k_aE_{bj}+\varepsilon_{jab}k_aE_{bi}),

the linearised Einstein-like operator, which is gauge invariant and of degree two in $k$ , and a curl acting on $E$ , which annihilates $s$ .

On the three-dimensional space of $A$ orthogonal to $g_1\xi$ , $M_{\rm FP}$ has eigenvalues $(+\tfrac12,+\tfrac12,-\tfrac12)\kappa^2$ : it is positive on the transverse traceless pair and negative on the transverse-trace direction $\mathrm{diag}(1,1,0)$ . The matrix $M_{\rm FP}$ is also the only gauge-invariant form of degree two. Writing a real symmetric $6\times6$ matrix $M(k)$ with entries that are homogeneous polynomials of degree $d$ in $k$ and imposing $M(k)g_1(k)=0$ for all $k$ is a linear system for the $21\times\binom{d+2}{2}$ coefficients, with no isotropy assumed. Its solution space has dimension $0$ for $d=0$ and $d=1$ and dimension $1$ for $d=2$ , spanned by $M_{\rm FP}$ . Hence no non-zero multiple of the degree-two solution is positive semidefinite on the physical $A$ .

Five small models with $X=0$ show how the exponent depends on the constraints. The frequencies are the square roots of the eigenvalues of $KM$ on the physical subspaces. The hypotheses (H1)–(H5) named in the discussion are stated in section 3. The symbols in the table are polynomials in $k$ , and only their small- $k$ behaviour is used.

model	$K$	$M$	constraints	physical modes	frequencies	$z$
A	$\mathbb 1$	$M_{\rm FP}$	vector	3	$\omega^2=\pm\kappa^2/2$ : two real, one imaginary	1, unstable
B	$\mathbb 1$	$\mathrm{inc}^\dagger\mathrm{inc}$	vector	3	$\omega=\kappa^2$ , three equal	2
C	$\mathrm{symcurl}^\dagger\mathrm{symcurl}$	$\mathrm{inc}^\dagger\mathrm{inc}$	vector and scalar	2	$\omega=\kappa^3$ , two equal	3
D	$K_{\rm DW}$	$M_{\rm FP}$	vector and scalar	2	$\omega=\kappa/\sqrt2$ , two equal	1
E	$\mathrm{symcurl}^\dagger\mathrm{symcurl}$	$M_{\rm FP}$	vector and scalar	2	$\omega=\kappa^2/\sqrt2$ , two equal	2

In model A the physical subspace contains the transverse-trace direction, where $M_{\rm FP}$ is negative, which gives one imaginary frequency. Model B satisfies (H1)–(H3a) and saturates Proposition 1. Model C has both constraints strongly invariant and satisfies the hypotheses of Proposition 2(b). Its $k^3$ dispersion is like that of the L-type model of Gu and Wen [Gu–Wen 2009]. Model D has the structure of general relativity with a rescaled kinetic term: its kinetic matrix does not annihilate $s$ (it gives $K_{\rm DW}s\propto\kappa^2e_{33}$ ), so it is invariant under the scalar constraint only up to a multiple of the vector constraint, and it is stable only because the scalar constraint removes the transverse-trace direction. Model E has the same potential as D and a kinetic term that does annihilate $s$ . It is stable on the constraint surface, not on the whole of the vector-constrained space, and its frequency is $\omega\propto\kappa^2$ . It satisfies (H3b) and (H4) but not (H3a), so Proposition 2(a) applies to it and Proposition 1 does not.

3. Qubit and rotor gravitons

3.1 Stability bounds the dispersion

Let $W=\{(E,A):g(k)^\dagger E=0\}$ be the vector-constrained subspace.

(H3a) $\mathbf H(k)$ is positive semidefinite on $W$ . In particular $M\ge0$ , because $(0,A)\in W$ for every $A$ .

Proposition 1 (stability gives $z\ge2$ ). Assume (H1), (H2) and (H3a). Then every physical mode has $\omega(k)\le C\lvert k\rvert^2$ for small $k$ .

Proof. Gauge invariance forces the low-order terms of $M$ to be the Fierz–Pauli form, which is indefinite on the physical subspace, so positivity removes them. The frequencies are then controlled by the sizes of $K$ and $M$ .

Expand $M=\sum_{n\ge0}M_n$ in homogeneous polynomials, each Hermitian. Order by order, $Mg=0$ reads $\sum_{a+b=n}M_ag_b=0$ with $g_b$ of degree $b\ge1$ . At $n=1$ , $M_0g_1=0$ ; the vectors $g_1(k)\xi$ span all symmetric tensors, so $M_0=0$ . At $n=2$ , $M_1g_1=0$ . The matrix $M_1(k)$ is odd in $k$ , and $M(k)/\lvert k\rvert\to M_1(\hat k)$ is positive semidefinite for $\hat k$ and for $-\hat k$ ; so $M_1(\hat k)$ and $-M_1(\hat k)$ are both positive semidefinite and $M_1=0$ . At $n=3$ , $M_2g_1=0$ . Because $g_1$ is a real matrix times $i$ , the real symmetric part $S$ and the imaginary antisymmetric part of $M_2$ annihilate $g_1$ separately. By the solution of the linear system in section 2, $S=\lambda M_{\rm FP}$ for a scalar $\lambda$ . The quadratic form of the positive semidefinite $M_2$ on real vectors is that of $S$ , so $\lambda M_{\rm FP}\ge0$ , and since $M_{\rm FP}$ is indefinite on the physical subspace, $\lambda=0$ . A Hermitian positive semidefinite matrix with zero real part has zero diagonal, hence vanishes: $M_2=0$ . At $n=4$ , $M_3g_1=0$ , and $M_3$ is odd, so as for $M_1$ it vanishes. Thus $\lVert M(k)\rVert=O(k^4)$ .

To pass to frequencies, let $\Theta(E,A)=(E,-A)$ . The map $\Theta$ preserves $W$ , and $\mathbf H+\Theta\mathbf H\Theta=2\,\mathrm{diag}(K,M)$ . On $W$ both $\mathbf H$ and $\Theta\mathbf H\Theta$ are positive semidefinite by (H3a), so $\mathbf H\le2\,\mathrm{diag}(K,M)$ on $W$ . The gauge directions $(0,g\xi)$ are null vectors of $\mathbf H$ in $W$ and lie in the kernel of the symplectic form restricted to $W$ , so everything descends to the symplectic quotient. Symplectic eigenvalues are monotone in the order of positive definite forms, which follows from the min–max principle for them [Bhatia–Jain]; for semidefinite forms add $t\mathbb 1$ and let $t\to0$ . The symplectic eigenvalues of $2\,\mathrm{diag}(K,M)$ are the square roots of the eigenvalues of $4KM$ , so $\omega\le2\sqrt{\lVert K\rVert\lVert M\rVert}$ , where $K$ is restricted to $\{g^\dagger E=0\}$ and is bounded because its entries are analytic and periodic. With $\lVert M\rVert=O(k^4)$ this gives $\omega=O(k^2)$ .

Positivity is used twice: to remove the terms of order $k$ and $k^2$ in $M$ , and to compare $\mathbf H$ with its diagonal part. General relativity evades the proposition because (H3a) fails there: its potential, restricted only by the vector constraint, is negative on the transverse-trace direction (model A), and stability holds only after the scalar constraint removes that direction. Within (H1), (H2) and (H3a) a mode with $\omega\propto\lvert k\rvert$ is impossible, so a graviton with linear dispersion needs one of the three to fail. The cases outside (H1), (H2) and (H3a) are taken up in sections 3.2, 3.3, 3.4 and 7.

3.2 Both constraints, strongly invariant

The scalar constraint changes what stability can be asked of. The physical space is now the transverse traceless pair, and the Hamiltonian only has to be positive there. Let $\Sigma=\{(E,A):g^\dagger E=0,\ s^\dagger A=0\}$ .

(H3b) $\mathbf H(k)$ is positive semidefinite on $\Sigma$ . This is implied by (H3a) and is weaker, because $\Sigma\subset W$ .

(H4) There is a scalar gauge direction $s(k)=s_2(k)+O(k^3)$ in $E$ with $s_2$ as in (1), and $\mathbf H(k)$ is invariant under it, $\mathbf H(k)(s\theta,0)=0$ , that is $Ks=0$ and $X^\dagger s=0$ . The two directions are orthogonal, $g(k)^\dagger s(k)=0$ (the generators commute). Physical $A$ obeys $s^\dagger A=0$ , and physical $E$ is taken modulo $s\theta$ . We call this strong invariance.

(H5) The potential is $V(A)=\sum_{x,\alpha}c_\alpha(1-\cos(n_\alpha\cdot A)_x)$ with finitely many integer vectors $n_\alpha$ of finite support and $c_\alpha>0$ , and the vacuum is $A=0$ , so $M(k)=\sum_\alpha c_\alpha\,n_\alpha(k)n_\alpha(k)^\dagger$ , where $n_\alpha(k)$ is the symbol of $n_\alpha$ .

Proposition 2 (the pair of constraints). Assume (H1), (H2), (H3b) and (H4).

(a) Every physical mode has $\omega\le C\lvert k\rvert^2$ , and the physical modes are the transverse traceless pair.

(b) If in addition (H5) holds, then $\omega\le C\lvert k\rvert^3$ .

Proof. At small $k\ne0$ the six-dimensional space of symmetric tensors contains the three directions $g\xi$ and the direction $s$ , which are mutually orthogonal by (H4); their orthogonal complement is a two-dimensional space $T(k)$ . At leading order it is the space $T_0(\hat k)$ of transverse traceless tensors, which is the same for $\hat k$ and $-\hat k$ , and the projector $P_T(k)$ onto $T(k)$ differs from the projector $P_0$ onto $T_0$ by $O(k)$ , because $g=g_1+O(k^2)$ and $s=s_2+O(k^3)$ . The symplectic form restricted to $\Sigma$ has as radical exactly the gauge directions $(s\theta,0)$ and $(0,g\xi)$ , since $E\cdot A'=0$ for all $A'$ with $s^\dagger A'=0$ means $E\propto s$ , and $E'\cdot A=0$ for all $E'$ with $g^\dagger E'=0$ means $A\in\mathrm{ran}\,g$ . By (H2) and (H4) these directions are null vectors of $\mathbf H$ . The quotient of $\Sigma$ by the radical is $T\oplus T$ , with the induced forms $K_T=P_TKP_T$ and $M_T=P_TMP_T$ ; they are well defined because $Ks=0$ and $Mg=0$ .

(a) We bound $K_T$ and $M_T$ . For $M$ , the argument of Proposition 1 gives $M_0=0$ . Next, $M(\pm t\hat k)=\pm tM_1(\hat k)+O(t^2)$ . Take $A\in T_0(\hat k)$ and subtract its component along $s$ , which is $O(t)$ because $s^\dagger A=(s-s_2)^\dagger A=O(t^3)$ while $\lvert s\rvert\sim t^2$ ; the result $A_t=A+O(t)$ satisfies $s^\dagger A_t=0$ , so $(0,A_t)\in\Sigma$ at $\pm t\hat k$ , and (H3b) gives $\pm t\,A^\dagger M_1(\hat k)A+O(t^2)\ge0$ , since $M=O(t)$ . Hence $A^\dagger M_1A=0$ on $T_0$ , that is $P_0M_1P_0=0$ . Since $M_1=O(k)$ and $P_T=P_0+O(k)$ , $M_T=O(k^2)$ . For $K$ : from $Ks=0$ and $s=s_2+O(k^3)$ , at order 2 $K_0s_2(k)=0$ for all $k$ ; the tensors $s_2(k)=k\otimes k-\lvert k\rvert^2\mathbb 1$ span all symmetric tensors (the diagonal ones for $k=e_1,e_2,e_3$ give all diagonal tensors, and $k=e_i+e_j$ the off-diagonal ones), so $K_0=0$ . For $E\in T_0(\hat k)$ subtracting the component along $\mathrm{ran}\,g$ , which is $O(t)$ because $g^\dagger E=O(t^2)$ while $\lvert g\rvert\sim t$ , gives $E_t=E+O(t)$ with $g^\dagger E_t=0$ , so $(E_t,0)\in\Sigma$ at both signs, and (H3b) gives $P_0K_1P_0=0$ and $K_T=O(k^2)$ .

For the comparison, $\Theta(E,A)=(E,-A)$ preserves $\Sigma$ and the radical, $\mathbf H$ and $\Theta\mathbf H\Theta$ are positive semidefinite on $\Sigma$ , and $\mathbf H+\Theta\mathbf H\Theta=2\,\mathrm{diag}(K,M)$ . So $\mathbf H\le2\,\mathrm{diag}(K,M)$ on $\Sigma$ , and on the quotient the monotonicity of symplectic eigenvalues used for Proposition 1 gives $\omega\le2\sqrt{\lVert K_T\rVert\lVert M_T\rVert}=O(k^2)$ .

(b) Under (H5), $Mg=0$ gives $0=\xi^\dagger g^\dagger Mg\xi=\sum_\alpha c_\alpha\lvert n_\alpha^\dagger g\xi\rvert^2$ , so $g(k)^\dagger n_\alpha(k)=0$ for each $\alpha$ because every $c_\alpha>0$ . The symbol $n(k)$ of a vector with finite support is a trigonometric polynomial with values in the symmetric tensors. Expand $n=n_0+n_1(k)+\dots$ in homogeneous terms. The condition $g^\dagger n=0$ has lowest order $k_in_{0,ij}=0$ for all $k$ , so $n_0=0$ , and then, since $g_2^\dagger n_0=0$ , the next order is $k_in_{1,ij}(k)=0$ . Write $n_{1,ij}=c_{ij,l}k_l$ with $c_{ij,l}=c_{ji,l}$ ; then $k_ik_lc_{ij,l}=0$ for all $k$ says that $c_{ij,l}+c_{lj,i}=0$ , and a tensor that is symmetric in its first two indices and antisymmetric under exchange of its first and third vanishes: $c_{ij,l}=-c_{lj,i}=-c_{jl,i}=c_{il,j}=c_{li,j}=-c_{ji,l}=-c_{ij,l}$ . Hence $n_\alpha(k)=O(k^2)$ , $M=\sum c_\alpha n_\alpha n_\alpha^\dagger=O(k^4)$ , and the comparison argument of (a) gives $\omega\le2\sqrt{\lVert K_T\rVert\lVert M_T\rVert}=O(k^3)$ .

Model E of section 2 satisfies the hypotheses of (a) and has $z=2$ , and model C satisfies those of (b) and has $z=3$ , so both bounds are attained. Model D satisfies (H1)–(H3b) and fails (H4), because $K_{\rm DW}s\ne0$ . Its frequency is linear in $\kappa$ . The proof of (b) shows what compactness adds: a local integer-valued gauge-invariant combination of the fields has at least two derivatives, as the linearised curvature does, so $M=O(k^4)$ . The same algebraic lemma appears in the proof of Theorem 2 of the page on a conserved lattice momentum, where it is applied to the rows of the symbol of the integer matrix $L$ . With the vector constraint alone, (H5) gives $M=O(k^4)$ as well, and with (H3a) the comparison of Proposition 1 gives $\omega=O(k^2)$ , which is Proposition 1 again. The exponent $3$ needs (H4) in addition.

3.3 Weak invariance in finite dimensions

General relativity reaches $z=1$ with both constraints because its kinetic term is invariant under the scalar constraint only weakly. The generator $\mathcal S_\theta$ of the scalar constraint shifts $\pi_{ij}$ by $s_{ij}\theta$ , and the kinetic term $\mathcal T=\pi_{ij}\pi_{ij}-\tfrac12\pi_{kk}^2$ changes by a multiple of $\theta\,\partial_i\partial_j\pi_{ij}$ , which is the divergence of the momentum constraint; this divergence commutes with $\mathcal S_\theta$ . The generator $\mathcal V_\xi$ of the vector constraint is a function of $\pi$ only and commutes with $\mathcal T$ . In the language of superoperators, $\mathrm{ad}_{\mathcal S}\mathcal T$ is proportional to the divergence of the momentum constraint, which is annihilated by $\mathrm{ad}_{\mathcal S}$ , so $\mathrm{ad}_{\mathcal S}$ acts on $\mathcal T$ as a $2\times2$ nilpotent Jordan block. The next proposition shows that such a block cannot occur on the constraint surface when the generators are bounded operators on a finite-dimensional space.

(H6) The local Hilbert spaces are finite-dimensional, and the constraints are commuting Hermitian operators $C_x$ supported near $x$ .

Proposition 3 (finite dimension forces strong invariance). Assume (H6). Let $H$ be any local Hamiltonian and $P$ the projector onto the common kernel of the $C_x$ . Then $PHP=PH_0P$ , where $H_0$ is the part of $H$ that commutes with every $C_x$ ; $H_0$ is local and strongly invariant. In particular, if $\mathrm{ad}_{C}^2H=0$ for a Hermitian $C=C_x$ , then $\mathrm{ad}_CH=0$ : no nilpotent Jordan block exists.

Proof. On the space of operators on a finite-dimensional Hilbert space with the Hilbert–Schmidt inner product $\langle A,B\rangle=\mathrm{tr}(A^\dagger B)$ , the superoperator $\mathrm{ad}_C$ is self-adjoint when $C$ is Hermitian, since $\langle A,[C,B]\rangle=\mathrm{tr}(A^\dagger CB-A^\dagger BC)=\langle[C,A],B\rangle$ . A self-adjoint operator is diagonalisable with real eigenvalues, so $\mathrm{ad}_C^2H=0$ implies $\mathrm{ad}_CH=0$ . For the commuting family, decompose $H=\sum_qH_q$ into joint eigen-operators, $[C_x,H_q]=q_xH_q$ . Then $0=P[C_x,H_q]P=q_xPH_qP$ because $C_xP=PC_x=0$ , so $PH_qP=0$ unless $q=0$ , and $PHP=PH_0P$ with $H_0=H_{q=0}$ . A term of $H$ supported in a region $Z$ commutes with $C_x$ for $x$ far from $Z$ , so its charge-zero part involves only the $C_x$ near $Z$ and is supported in a slightly larger region.

To pass from the exact lattice operators of Proposition 3 to the harmonic theory of sections 2 and 3.2, one more hypothesis is needed.

(B) The harmonic Hamiltonian $H_2$ is the long-wavelength limit of $PH_0P$ for a model satisfying (H6), and the linearised generators $\mathcal S_\theta$ and $\mathcal V_\xi$ act on the low-energy variables as the operators $C_x$ (or limits of their combinations), so that commutation of $H_0$ with the $C_x$ becomes invariance of $H_2$ under the shifts.

Under (H6) and (B), $H_2$ is invariant under both shifts: the invariance in (H2) and (H4) holds, in particular $Ks=0$ and $X^\dagger s=0$ , and the model has the strong invariance of Proposition 2. The combination of Proposition 3 with (B) therefore excludes the mechanism by which general relativity reaches $z=1$ with both constraints. Proposition 3 alone does not. A model whose constraints are generated by unbounded operators (rotors with non-compact angles) has an infinite-dimensional Hilbert space and is outside (H6). Its harmonic theory can have linear dispersion with both constraints, and Marolf showed that a nonlinear completion with universal coupling to energy requires kinematic non-locality [Marolf]. Gu and Wen impose their constraints as energy penalties on a finite-dimensional space. In their L-type model the invariance is emergent and the dispersion is cubic, as in model C.

3.4 A gauge redundancy that is not a symmetrised gradient

Remark (a gauge group smaller than the diffeomorphisms). The scalar-charge tensor gauge theory [Pretko] has the gauge redundancy $A_{ij}\to A_{ij}+\partial_i\partial_j\alpha$ and Gauss law $\partial_i\partial_jE_{ij}=\rho$ , which is not of the symmetrised-gradient type of (H2). Take $K=\mathbb 1$ and a potential built from the field strength $B_{ij}=\varepsilon_{iab}k_aA_{bj}$ , namely $V=b_1\lvert B_0\rvert^2+b_2\lvert B_A\rvert^2$ with $B_0$ the traceless symmetric part and $B_A$ the antisymmetric part of $B$ (the trace of $B$ vanishes for symmetric $A$ ). At $k=(0,0,1)$ the Hessian of $V$ has eigenvalue $0$ for the gauge direction $A_{33}$ , $2b_1$ twice (the transverse traceless pair $A_{11}-A_{22},A_{12}$ ), $2b_2$ once ( $A_{11}+A_{22}$ ) and $(b_1+b_2)/2$ twice ( $A_{13},A_{23}$ ). All five physical modes have $\omega\propto\lvert k\rvert$ , with $\omega^2=2b_1k^2$ for the pair of helicity two and $\omega^2=\tfrac12(b_1+b_2)k^2$ for the pair of helicity one, so the ratio of the squared speeds is $(b_1+b_2)/(4b_1)\ge\tfrac14$ for $b_2\ge0$ . The extra modes of helicity one and zero are present, and this gauge group does not contain the transport term of the diffeomorphisms.

4. Composite tetrads

Does a large-flavour mean field for fermions with a composite tetrad have a massless transverse traceless graviton at leading order? It does not, given a continuity assumption at zero momentum (T3) (Proposition 4). The section is a derivation in the mean-field model of section 4.1 and uses nothing from sections 1 to 3.

4.1 The mean-field model

In spinor gravity of the Akama–Diakonov type the tetrad is a composite of fermions, $e^A_\mu=\sum_f\bar\psi_f\gamma^A\nabla_\mu\psi_f$ over $N_f$ flavours, and the bare action is built from differential forms [Diakonov]. At zero derivatives the only such term is the volume $\det e$ . A Hubbard–Stratonovich field $J^\mu{}_A$ (the inverse-tetrad density, a $4\times4$ matrix per site) is introduced. For naive lattice fermions with action $\sum_\mu J^\mu{}_A\gamma^A i\sin p_\mu$ the determinant of the mode with lattice momentum $p$ is a power of $\lvert J^T\sin p\rvert^2=\lvert\sin p\rvert^2\,n^TJJ^Tn$ , where $n=\sin p/\lvert\sin p\rvert$ is a unit four-vector. At large $N_f$ this gives the mean-field action per site

\frac{\Gamma(J)}{N_f}=c\,\Phi(\det J)-2\int d\varrho(n)\,\log\big(n^TJJ^Tn\big)+\text{const}, \tag{2}

where $\Phi$ is an arbitrary function representing the bare volume term, $c$ a constant, and $\varrho$ a probability measure on unit four-vectors, the directions of the fermion modes: for naive fermions it is the distribution of $n$ for $p$ uniform in the Brillouin zone. The measure does not depend on $J$ because the number of modes is fixed per coordinate cell, not per proper volume. We call this a background-free regulator.

(T1) The effective action of the uniform field is (2), and $J_0=j\mathbb 1$ is a stationary point.

(T2) The measure $\varrho$ is a $J$ -independent probability measure.

(T3) The quadratic form of fluctuations of the shear $J=j(1+\delta\eta)$ , with $\eta$ a symmetric traceless $4\times4$ matrix, at wave vector $k\to0$ equals its value at $k=0$ (continuity at zero momentum).

Assumptions (T1) and (T2) define the model, and Proposition 4 is a derivation inside it. Assumption (T3) enters only the final consequence about masslessness. A transverse traceless polarisation is a shear $\eta$ whose only non-zero entries lie in the spatial block.

4.2 The shear form

Write $C_\varrho(\eta)=\langle\lvert\eta n\rvert^2-(n^T\eta n)^2\rangle_\varrho$ for a symmetric traceless $4\times4$ matrix $\eta$ . By Cauchy–Schwarz, $\lvert n^T\eta n\rvert\le\lvert\eta n\rvert$ for unit $n$ , so $C_\varrho\ge0$ .

Proposition 4 (shear no-go for condensed composite tetrads). Assume (T1) and (T2). Then:

$\langle nn^T\rangle_\varrho=\tfrac14\mathbb 1$ .
The coefficient of $\delta^2$ in $\Gamma(j(1+\delta\eta))/N_f$ is $Q(\eta)=-4\,C_\varrho(\eta)\le0$ for real $\eta$ (the sign is discussed after the proof), for every function $\Phi$ and every $c$ compatible with the gap equation. It vanishes only if $\varrho$ -almost every $n$ is an eigenvector of $\eta$ ; in particular $Q(\eta)<0$ for every $\eta\ne0$ if $\varrho$ gives zero weight to the unit spheres of the eigenspaces of every $\eta\ne0$ (for example if $\varrho$ has a density on the sphere).
For an orthonormal basis $\eta_1,\dots,\eta_9$ of the traceless symmetric matrices with $\mathrm{tr}(\eta_a\eta_b)=\delta_{ab}$ ,

\sum_{a=1}^9Q(\eta_a)=-6\qquad\text{for every }\varrho,\ \Phi\text{ and }c, \tag{3}

so some shear eigenvalue is at most $-2/3$ .

If in addition (T3) holds, the transverse traceless fluctuations of the composite metric have a non-zero value of their quadratic form at $k\to0$ whenever the condition in 2 holds, and no massless transverse traceless mode exists at order $N_f$ .

Proof. (1) Stationarity at $J_0=j\mathbb 1$ in a traceless direction: the bare term varies at order $\delta^2$ only (see below), and $\log(n^TJJ^Tn)$ varies at first order by $2\delta\,n^T\eta n$ . So $\langle n^T\eta n\rangle=0$ for all traceless symmetric $\eta$ , that is $\langle nn^T\rangle\propto\mathbb 1$ , and the trace of $nn^T$ is 1.

(2) Expand with $J=j(1+\delta\eta)$ and $\eta$ traceless symmetric. Then $JJ^T=j^2(1+2\delta\eta+\delta^2\eta^2)$ and

\log(n^TJJ^Tn)=2\log j+2\delta\,n^T\eta n+\delta^2\big(\lvert\eta n\rvert^2-2(n^T\eta n)^2\big)+O(\delta^3).

Also $\det J=j^4(1-\tfrac12\delta^2\mathrm{tr}\,\eta^2+O(\delta^3))$ . The gap equation, the stationarity of (2) in the scale $j$ at $J=j\mathbb 1$ , is $c\,\Phi'(w_0)w_0=1$ with $w_0=j^4$ , so the bare term contributes $-\tfrac12\mathrm{tr}\,\eta^2$ to the coefficient of $\delta^2$ . The fermion term contributes $-2\langle\lvert\eta n\rvert^2-2(n^T\eta n)^2\rangle$ . With (1), $\langle\lvert\eta n\rvert^2\rangle=\mathrm{tr}(\eta^2\langle nn^T\rangle)=\tfrac14\mathrm{tr}\,\eta^2$ , so the total is

Q(\eta)=-\tfrac12\mathrm{tr}\,\eta^2-\tfrac12\mathrm{tr}\,\eta^2+4\langle(n^T\eta n)^2\rangle=-4\Big[\tfrac14\mathrm{tr}\,\eta^2-\langle(n^T\eta n)^2\rangle\Big]=-4\,C_\varrho(\eta),

independent of $\Phi$ and $c$ . Equality $C_\varrho(\eta)=0$ requires $\lvert\eta n\rvert=\lvert n^T\eta n\rvert$ for $\varrho$ -almost every $n$ , that is $\eta n\parallel n$ . For $\eta\ne0$ traceless the eigenvectors of $\eta$ with a given eigenvalue span a proper subspace, so the unit eigenvectors form a finite union of lower-dimensional spheres.

(3) The completeness relation for traceless symmetric matrices is $\sum_a(\eta_a)_{bc}(\eta_a)_{de}=\tfrac12(\delta_{bd}\delta_{ce}+\delta_{be}\delta_{cd})-\tfrac14\delta_{bc}\delta_{de}$ . Hence $\sum_a\lvert\eta_an\rvert^2=\tfrac12(4+1)-\tfrac14=\tfrac94$ and $\sum_a(n^T\eta_an)^2=\tfrac12(1+1)-\tfrac14=\tfrac34$ for every unit $n$ , so $\sum_aC_\varrho(\eta_a)=\tfrac32$ and the sum of $Q$ is $-6$ . The largest eigenvalue in magnitude is at least $6/9$ .

For the consequence, a massless transverse traceless mode would need the quadratic form of shear fluctuations to vanish at $k\to0$ in the direction of its polarisation. By (T3) that form is $Q(\eta)$ for the polarisation $\eta$ , which is non-zero under the condition in 2.

For $\varrho$ uniform on the three-sphere, $\langle n_1^4\rangle=1/8$ and $\langle n_1^2n_2^2\rangle=1/24$ , which gives $Q=-2/3$ in every direction with $\mathrm{tr}\,\eta^2=1$ . For naive hypercubic fermions, averaging over the Brillouin zone gives $Q_E=-1+2\langle(n_1^2-n_2^2)^2\rangle=-0.826$ in the diagonal traceless channel (multiplicity three) and $Q_T=-1+8\langle n_1^2n_2^2\rangle=-0.587$ in the off-diagonal channel (multiplicity six).

The sign of $Q$ depends on the contour of the Hubbard–Stratonovich integral, which for a purely interacting Grassmann action has no canonical real form. Negative $Q$ is the reading for real $J$ . What does not depend on the contour is that the shear form is of order $N_f$ and non-zero in every direction under the condition in 2, which is all that the conclusion about masslessness uses. Where $Q$ is negative, the uniform tetrad is unstable at fixed volume: for the unit-determinant shape $\hat G=JJ^T/\lvert\det J\rvert^{1/2}=\mathrm{diag}(\lambda^3,\lambda^{-1},\lambda^{-1},\lambda^{-1})$ the fermion term behaves as $-6N_f\log\lambda$ at large $\lambda$ , which points to a runaway towards degenerate tetrads. The expansion is to leading order only, so higher orders may cut the runaway off at large $\lambda$ .

Remark (what is held fixed). Replacing $\varrho$ by a mixture of regulators (a quenched average over complexes) gives another probability measure and the same sum rule (3), because $Q$ is linear in $\varrho$ . Replacing $\Phi$ or $c$ changes nothing, since they cancel in $Q$ . Bare terms built from forms with derivatives (curvature or torsion) vanish on a constant $J$ with flat links, so they do not contribute to the uniform shear form either. This last statement is taken as part of the definition of the class of bare actions.

5. Entanglement geometry

Can gapped states in the entanglement-geometry framework carry gravitating mass? Not if the response of the mutual information to a local perturbation decays fast enough, given the locality assumption (L) (Proposition 6). Section 5.2 adds that the framework by itself does not restrict the emergent geometry (Proposition 5). The section uses only the relation (E1) of section 5.1 and nothing from sections 1 to 4.

5.1 The framework and a relation it uses

Cao, Carroll and Michalakis [CCM] decompose a Hilbert space into factors $A_p$ and consider states that are redundancy-constrained: the entropy of a union $B$ of factors is the sum of the mutual informations across its boundary,

S(B)=\tfrac12\sum_{p\in B,\ q\notin B}I(A_p:A_q), \tag{4}

where $I(A_p:A_q)=S(A_p)+S(A_q)-S(A_pA_q)$ . Mutual information defines distances on the graph of factors, and multidimensional scaling extracts an emergent manifold. They argue that a perturbation of the state that changes the entropy of a small region by $\delta S$ induces the scalar curvature $R_p=-\zeta\,\delta S_{A_p}=\zeta u_p$ , with $\zeta>0$ a constant fixed by the construction and $u_p=-\delta\langle K_{A_p}\rangle$ an effective modular energy density, using the first law of entanglement. Perturbations that entangle the region further have negative modular energy and give negative curvature, and they state that conditions limiting such effects would be needed. For non-local perturbations they find $R_p=\tilde\zeta\lvert u_p\rvert\ge0$ whatever the sign of $u_p$ . We use the relation in the form

(E1) in the linearised regime, $R^{(1)}=\zeta u$ with $\zeta>0$ ,

where $R^{(1)}=\partial_i\partial_jh_{ij}-\partial^2h_{ii}$ is the linearised scalar curvature of the emergent metric $\delta+h$ . Where a sign of $u$ is needed we say so: $u\ge0$ is the reading in which negative modular energy is excluded.

5.2 Which states are redundancy-constrained

Proposition 5 (pair states). Let a finite graph with vertices $p$ and weights $w_{pq}\in[0,\ln\chi]$ on its edges be given. Place on each edge a pure two-party state with entanglement entropy $w_{pq}$ , one party in $A_p$ and the other in $A_q$ (each factor a tensor product of the parties it holds, each party of dimension $\chi$ ). Then the product state satisfies (4) exactly for every subset $B$ , with $I(A_p:A_q)=2w_{pq}$ on edges and $0$ elsewhere.

Proof. The state is a product over edges, so the entropy of any union of factors is the sum of $w_{pq}$ over the edges cut by its boundary. For two factors, $S(A_p)=\sum_{q}w_{pq}$ and $S(A_pA_q)=S(A_p)+S(A_q)-2w_{pq}$ , since the pure state on the edge $pq$ is inside $A_pA_q$ ; so $I=2w_{pq}$ . The sum in (4) is then $\sum_{\rm cut}w_{pq}=S(B)$ .

Every weighted graph with weights in $[0,2\ln\chi]$ therefore arises as the mutual-information graph of a redundancy-constrained state, and the condition (4), the dimension of the Hilbert space and the entropies of the factors do not by themselves restrict the emergent geometry. For a graph that is the one-skeleton of a triangulation of a closed three-manifold with equilateral edges, the sign of the curvature is set by the mean degree. Euler’s relation $f_0-f_1+f_2-f_3=0$ for the numbers $f_0,\dots,f_3$ of vertices, edges, triangles and tetrahedra, with $f_2=2f_3$ , gives $f_3=f_1-f_0$ , and since each tetrahedron has six edges the mean number of tetrahedra per edge is $\bar n=6f_3/f_1=6(1-2/\langle\deg\rangle)$ , where $\langle\deg\rangle=2f_1/f_0$ . The deficit angle per edge for regular tetrahedra is $2\pi-\bar n\arccos(1/3)$ , which vanishes for $\langle\deg\rangle=2/(1-2\pi/(6\arccos(1/3)))=13.397$ and is positive below it. If the emergent curvature of such a graph is the Regge deficit angle of its triangulation, a curvature of order one in units of the factor size, against a cosmological scale, would require $\langle\deg\rangle$ to be tuned to the corresponding accuracy. The dimension of the Hilbert space does not enter this estimate.

5.3 Short-range response carries no mass

Proposition 6 (short-range metric response). Let $D\ge3$ and consider the metric $\delta+h$ on $\mathbb R^D$ with $h$ smooth and $\partial h=o(r^{-(D-1)})$ as $r\to\infty$ .

(a) The integrals of $R^{(1)}$ over balls of radius $\mathcal R$ tend to $0$ as $\mathcal R\to\infty$ : the linearised ADM mass vanishes.

(b) If $R^{(1)}\ge0$ everywhere then $R^{(1)}\equiv0$ . Consequently, under (E1) with $u\ge0$ , such a metric has $u\equiv0$ . Under (E1) without a sign condition on $u$ , the integrals of $u$ over balls tend to $0$ .

Proof. The linearised scalar curvature is a divergence, $R^{(1)}=\partial_i(\partial_jh_{ij}-\partial_ih_{jj})$ , so its integral is a surface term that the assumed decay makes vanish. By the divergence theorem, $\int_{r<\mathcal R}R^{(1)}=\oint_{r=\mathcal R}(\partial_jh_{ij}-\partial_ih_{jj})n^i\,dS$ . The surface has area proportional to $\mathcal R^{D-1}$ and $\partial h=o(\mathcal R^{-(D-1)})$ , so the right side tends to zero. This is (a), the surface term being the linearised ADM mass. For (b), $R^{(1)}\ge0$ makes the integral over the ball increase to $\int R^{(1)}$ , which is therefore zero, and the continuous non-negative function $R^{(1)}$ vanishes. The statements about $u$ follow from (E1).

The linearised mass of a metric perturbation is read off from its far field. A source of positive mass $M$ has $h\sim G_NM/r^{D-2}$ , so it requires a far-field response of the mutual information of order $r^{-(D-2)}$ [CCM]. Which states respond faster than this depends on the state. For gapped ground states the argument below rests on one unproved assumption.

(L) If the change of the mutual information of any two fixed factors under a localised perturbation decays faster than any power of their distance from the perturbation, then the linearised metric $h$ extracted from the mutual-information distances and its derivative satisfy the decay hypotheses of Proposition 6.

Ground states of weakly interacting spin systems with gapped on-site terms respond to a localised perturbation, even one that closes the gap, only locally: expectation values of observables at distance $d$ change by at most $\mathrm{const}\,e^{-c\,d}$ [Henheik–Teufel–Wessel]. For general gapped systems, when the gap stays open along the perturbation, the locality of the response is the principle of local perturbations perturbing locally, with decay faster than any power [BMNS]. The mutual information of a finite number of finite-dimensional factors is a continuous function of their reduced density matrices, so the change of the mutual information of fixed factors far from a perturbation inherits this decay. The passage to the metric is where (L) enters: multidimensional scaling is a global embedding, the number of pairs of factors that determine $h$ and $\partial h$ at a distant point grows with the distance, and the constants in the locality bounds grow with the size of the region, so the decay of $h$ and $\partial h$ does not follow from the decay of single pairs. If (L) holds, Proposition 6(a) gives zero total gravitating mass for such states: any positive modular energy is cancelled by negative modular energy elsewhere, and under $u\ge0$ none exists.

In the ring $H=-\sum_iZ_iZ_{i+1}-\nu\sum_iX_i$ of $N=16$ qubits, exact diagonalisation gives the logarithmic derivative $\lvert d\ln I/d\nu_0\rvert$ of the mutual information of the neighbouring qubits $(r,r+1)$ with respect to the field on qubit 0. In the gapped phase $\nu=1.5$ it is $0.015$ at $r=1$ and $4\times10^{-4}$ from $r=4$ on; at the critical field $\nu=1$ it is still $0.041$ at $r=7$ , a slow decay consistent with a power law at this size. This tests only the premise of (L), the decay of single pairs; the passage to the metric, and the relation (4) for these states, are not tested.

6. Gauss-law sectors and a tick-rate automaton

Section 6.1 uses the finite-dimensional local spaces of (H6) and nothing else from sections 1 to 5. Section 6.2 uses Theorem 2(b) of the page on exact universal graviton coupling.

6.1 Gauss-law sectors in the maximally mixed state

A closed gauge-constrained qubit model has its total charge fixed. The question is whether this constraint is visible in the weights with which the charge sectors of the matter at one vertex appear in the maximally mixed state on the physical space. For a finite group in which the identity is the only element acting as a scalar on the vertex representation, the weights agree with those of the unconstrained space up to corrections that are exponentially small in the number of vertices and, when the group has a trivial centre, also in the number of independent loops (Proposition 7).

Take a finite group $G$ , a finite connected graph with $N$ vertices and $L=\lvert\text{edges}\rvert-N+1$ independent loops, a copy of a unitary representation $H$ of $G$ at each vertex (dimension $d_H$ , character $\varphi_H$ ) and the group algebra $\mathbb C[G]$ on each edge. The Hilbert space is $\bigotimes_vH_v\otimes\bigotimes_e\mathbb C[G]$ , with $y\in G^V$ acting on matter at $v$ by $R_v(y_v)$ and on an edge $(v,v')$ by $a\mapsto y_va\,y_{v'}^{-1}$ . The physical space is the subspace invariant under all $y$ (Gauss law at every vertex), with projector $P_{\rm phys}$ .

Proposition 7 (the maximally mixed physical state). (a) For a gauge-invariant operator $X$ on $H_v$ ,

\frac{\mathrm{tr}(P_{\rm phys}X)}{\mathrm{tr}\,P_{\rm phys}}=\frac{\sum_\gamma\mathrm{tr}_{H}(R(\gamma)X)\,\varphi_H(\gamma)^{N-1}\,\lvert C_G(\gamma)\rvert^{L}}{\sum_\gamma\varphi_H(\gamma)^{N}\,\lvert C_G(\gamma)\rvert^{L}}, \tag{5}

with $C_G(\gamma)$ the centraliser of $\gamma$ , and $\dim H_{\rm phys}=\lvert G\rvert^{-1}\sum_\gamma\varphi_H(\gamma)^N\lvert C_G(\gamma)\rvert^L$ .

(b) Suppose the identity $1$ is the only element of $G$ that acts as a scalar on $H$ , and let $\beta=\max_{\gamma\ne1}\lvert\varphi_H(\gamma)/d_H\rvert^{N-1}(\lvert C_G(\gamma)\rvert/\lvert G\rvert)^L$ with $(\lvert G\rvert-1)\beta<1$ . Then for every irreducible representation $\lambda$ of $G$ , with multiplicity $m_\lambda$ and dimension $d_\lambda$ in $H$ , the weight $p_v(\lambda)=\mathrm{tr}(P_{\rm phys}\Pi^v_\lambda)/\mathrm{tr}P_{\rm phys}$ of the $\lambda$ -isotypic component at a vertex satisfies

\Big\lvert\frac{p_v(\lambda)}{m_\lambda d_\lambda/d_H}-1\Big\rvert\le\frac{2(\lvert G\rvert-1)\beta}{1-(\lvert G\rvert-1)\beta}. \tag{6}

Proof. (a) Choose a spanning tree and a root. The gauge transformations at the $N-1$ non-root vertices act freely and transitively on the configurations of the tree edges, so setting every tree edge to the identity is a unitary isomorphism of the physical space with the invariants of the residual diagonal action of $G$ on $\bigotimes_vH_v\otimes\mathbb C[G]^{\otimes L}$ , where $G$ acts on matter by $\bigotimes_vR_v$ and on each of the $L$ non-tree edges by conjugation. The projector onto invariants is $\lvert G\rvert^{-1}\sum_\gamma U(\gamma)$ , the trace of conjugation on $\mathbb C[G]$ at $\gamma$ is $\lvert C_G(\gamma)\rvert$ , and an operator $X$ on $H_v$ that commutes with $R_v$ is unchanged by the isomorphism. Taking the trace gives (5) and, with $X=\mathbb 1$ , the dimension.

(b) Divide numerator and denominator of (5) by $d_H^N\lvert G\rvert^L$ and put $X=\Pi_\lambda$ , for which $\mathrm{tr}_H(R(\gamma)\Pi_\lambda)=m_\lambda\varphi_\lambda(\gamma)$ . The term $\gamma=1$ gives $\bar p_\lambda:=m_\lambda d_\lambda/d_H$ in the numerator and $1$ in the denominator. The terms $\gamma\ne1$ are bounded by $\bar p_\lambda\beta$ each in the numerator (using $\lvert\varphi_\lambda(\gamma)\rvert\le d_\lambda$ ) and by $\beta$ each in the denominator, so numerator $=\bar p_\lambda(1+\epsilon_1)$ and denominator $=1+\epsilon_2$ with $\lvert\epsilon_1\rvert,\lvert\epsilon_2\rvert\le(\lvert G\rvert-1)\beta$ , and $\lvert(1+\epsilon_1)/(1+\epsilon_2)-1\rvert\le\lvert\epsilon_1-\epsilon_2\rvert/(1-\lvert\epsilon_2\rvert)$ .

The weights tend to the values $m_\lambda d_\lambda/d_H$ of the unconstrained space exponentially in $N$ , because $\lvert\varphi_H(\gamma)\rvert<d_H$ for every $\gamma\ne1$ when no such $\gamma$ acts as a scalar. A loop adds the factor $\lvert C_G(\gamma)\rvert/\lvert G\rvert\le\tfrac12$ for non-central $\gamma$ and no decay for central $\gamma$ , so $\beta$ also decays in $L$ when the centre of $G$ is trivial. Elements that act as scalars carry the only memory of the constraint, namely the class of the total central charge (a term of (5) for such an element $z$ equals the identity term times the $N$ -th power of the phase by which $z$ acts on $H$ ), and (6) does not cover that case. The maximally mixed state on the physical space is the Haar average of pure physical states. For an individual random physical state to be close to it when $\dim H_{\rm phys}$ is large, a concentration-of-measure statement is needed, which is not shown.

For $G=S_3$ and $H=\mathrm{triv}\oplus\mathrm{sign}\oplus\mathrm{std}$ (dimension 4, isotypic weights $\tfrac14,\tfrac14,\tfrac12$ ) the classes have sizes $1,3,2$ , $\varphi_H=4,0,1$ and centraliser orders $6,2,3$ . The formula (5) gives the following weights of the trivial and the standard component at one vertex. They depend on the graph only through $N$ and $L$ ; the first column names one graph with each pair.

graph	$N$	$L$	$\dim H_{\rm phys}$	$p_v(\mathrm{triv})$	$p_v(\mathrm{std})$
path	3	0	11	0.27273	0.45455
triangle	3	1	65	0.26154	0.47692
square	4	1	257	0.25292	0.49416
theta	4	2	1539	0.25146	0.49708
$K_4$	4	3	9225	0.25073	0.49854

For the path, $p_v(\mathrm{triv})=(4^{N-1}+2)/(4^N+2)$ , which is $(16+2\cdot1)/(64+0+2)=3/11$ at $N=3$ . On any tree ( $L=0$ ) the same formula gives $\lvert p_v(\mathrm{triv})-\tfrac14\rvert=1.5/(4^N+2)$ , a decay by a factor $\tfrac14$ per vertex.

6.2 Automata: a Lorentz-violating band and a lapse coupling

A quantum cellular automaton is a local unitary on a lattice, and its Bloch symbol is a finite unitary matrix, so Theorem 2(b) of the page on exact universal graviton coupling shows that none of its bands is exactly relativistic on an open set, that is, has $\Omega^2$ equal to a positive-definite quadratic form in $k$ plus a constant. A Weyl automaton on the cubic lattice has a Lorentz-violating term already at cubic order in $k$ , and a Dirac automaton coupled to a static field through a local tick rate satisfies the equivalence principle to first order in the mass parameter only (Proposition 8).

For the Weyl automaton take the $2\times2$ Bloch matrix $A_\sigma(k)=d_I-i\,\vec d\cdot\vec\sigma$ with $c_i=\cos k_i$ , $s_i=\sin k_i$ , $d_I=c_xc_yc_z-\sigma s_xs_ys_z$ and $\vec d=(s_xc_yc_z+\sigma c_xs_ys_z,\ c_xs_yc_z-\sigma s_xc_ys_z,\ c_xc_ys_z+\sigma s_xs_yc_z)$ , $\sigma=\pm1$ . Automata of this type on the body-centred cubic lattice are due to D’Ariano and Perinotti [D’Ariano–Perinotti]. This form need not coincide with their conventions. The matrix is unitary, $d_I^2+\lvert\vec d\rvert^2=1$ identically, and its eigenphases $\pm\omega$ satisfy $\cos\omega=d_I$ . A series expansion along a unit vector $n$ gives

\omega^2=\lvert k\rvert^2+2\sigma\,k_xk_yk_z-\tfrac13\big(k_x^2k_y^2+k_y^2k_z^2+k_z^2k_x^2\big)+O(k^5). \tag{7}

The leading correction is cubic in $k$ , anisotropic and odd under $k\to-k$ : $\omega=\lvert k\rvert+\sigma\,n_xn_yn_z\lvert k\rvert^2+\dots$ , a relative correction linear in the energy. In the notation of the page on exact universal graviton coupling, the lowest non-zero term of $\omega^2-\lvert k\rvert^2$ has degree $d=3$ . A spin-2 band with a leading correction of this form that satisfies hypothesis (B) of that page would therefore have order of the defect $r=d-2=1$ (Proposition 4 there). In one space dimension the corresponding massless walk is exactly linear. Bounds on the lattice spacing of automata of this kind from Lorentz-violation data are derived in [Mlodinow–Brun].

In one space dimension the Dirac automaton $U(k)=\begin{pmatrix}\cos\mu\,e^{ik}&-i\sin\mu\\-i\sin\mu&\cos\mu\,e^{-ik}\end{pmatrix}$ with mass parameter $m=\sin\mu$ has $\cos\omega=\cos\mu\cos k$ [Bisio–D’Ariano–Tosini]. Then $\omega(0)=\mu$ and $\omega''(0)=\cot\mu$ .

Proposition 8 (equivalence principle for a tick-rate coupling). Model a static gravitational field by a local tick rate: the cell at $x$ performs $\tau(x)$ updates per coordinate time, so the generator is $H=\tfrac12\{\tau(x),\Omega\}$ with $\Omega=i\log U$ . Assume a wave packet narrow in $k$ and a slowly varying $\tau$ , so that the ray equations $\dot x=\tau\omega'(k)$ , $\dot k=-\tau'\omega(k)$ hold. A packet at rest ( $k=0$ ) at a point where $\tau=1$ , $\tau'=\varpi$ accelerates with

a=-\varpi\,\omega(0)\,\omega''(0)=-\varpi\,\mu\cot\mu=-\varpi\Big(1-\frac{\mu^2}3-\frac{\mu^4}{45}-\dots\Big). \tag{8}

Proof. The ray Hamiltonian is $H=\tau(x)\omega(k)$ . At $k=0$ , $\omega'(0)=0$ (the band is even), so $\ddot x=\tau\omega''(k)\dot k+\tau'\dot x\omega'=-\tau\tau'\omega(0)\omega''(0)$ , which is (8) with $\omega(0)=\mu$ and $\omega''(0)=\cot\mu$ . The force $\dot k=-\varpi\mu$ defines a gravitational mass $\mu$ and the curvature of the band an inertial mass $\tan\mu$ , whose ratio is $\mu\cot\mu$ .

The weak equivalence principle requires the ratio $a/(-\varpi)$ to be $1$ for every mass. By (8) it holds to first order in $\mu$ and fails at second order, where the ratio is $1-\mu^2/3$ . At $\mu\to\pi/2$ the band is flat: the inertial mass diverges while the gravitational mass stays finite. Evolving a Gaussian wave packet in the positive band in the tick rate $\tau=1+\varpi x$ tests the ray equations. The table compares the rate of change of momentum and the acceleration extracted from the evolved packet, each divided by $-\varpi$ , and the band curvature at its momentum, with the formulas.

$m$	momentum rate	$\mu$	curvature	$\cot\mu$	acceleration	$\mu\cot\mu$
0.5	0.5237	0.5236	1.7305	1.7321	0.9059	0.9069
0.9	1.1197	1.1198	0.4842	0.4843	0.5422	0.5423

Table. Packet of width 40 lattice sites evolved for time 500 with $\varpi=2\times10^{-5}$ .

For a lattice spacing equal to the Planck length, a proton has $\mu\approx m_p/M_{\rm Pl}$ with $M_{\rm Pl}=1.22\times10^{19}\,\mathrm{GeV}$ , so $\mu\approx7.7\times10^{-20}$ and $\mu^2/3\approx2\times10^{-39}$ . The MICROSCOPE test gives the Eötvös parameter of titanium and platinum as $(-1.5\pm2.3\,(\mathrm{stat})\pm1.5\,(\mathrm{syst}))\times10^{-15}$ [Touboul et al.], so $\mu^2/3$ lies more than twenty-three orders of magnitude below its uncertainty. Applying the formula to composites whose mass is mostly binding energy is an assumption.

7. Linear dispersion outside the hypotheses, and open questions

Linear dispersion with extra modes of helicity one and zero occurs where (H2) fails, in the scalar-charge tensor gauge theory of section 3.4 and in other fracton tensor gauge theories [Pretko]. Whether any of them couples universally to energy is an open question. Effective symbols that are non-analytic violate (H1), and the proofs above do not apply to them.

For the N-type model of Gu and Wen, the authors state that no reliable approach to its low-energy dynamics is available, and with a spin-wave and quantum-freeze argument they argue that it may contain helicity-two modes with linear dispersion as its only low-energy excitations [Gu–Wen 2009]. The argument does not provide a harmonic limit with analytic symbols and the constraints of (H2) and (H4), so Propositions 1 to 3 neither confirm nor exclude it.

A model built from finite-dimensional local spaces, with a stable potential and the full set of constraints of general relativity at the harmonic level, has not been constructed. Non-compact rotors lie outside this class (section 3.3).

The one-loop fluctuation of $J$ in the composite-tetrad model could shift the trace of the shear form by a positive amount $\Delta_1$ , in which case the trace vanishes at $N_f=\Delta_1/6$ ; the corrections of order $N_f^0$ to the sum rule (3) are not computed. Dynamical gauge links, simplicial actions with several composite channels and phases in which the metric condenses without the tetrad (vestigial phases, see [Volovik]) add channels with zero-derivative shape dependence and are not covered by Proposition 4, which uses a fermion operator linear in a single site-valued $J$ and a $J$ -independent mode set.

A family of states that are redundancy-constrained at a coarse scale $\ell$ and have a mutual-information response of order $(\ell/r)^{D-2}$ would carry mass consistently with Proposition 6; whether such a family exists is not known.

References

Gu, Wen, “A lattice bosonic model as a quantum theory of gravity,” arXiv:gr-qc/0606100.
Gu, Wen, “Emergence of helicity $\pm2$ modes (gravitons) from qubit models,” Nucl. Phys. B 863, 90 (2012), arXiv:0907.1203.
Xu, “Novel algebraic boson liquid phase with soft graviton excitations,” arXiv:cond-mat/0602443.
Xu, Hořava, “Emergent gravity at a Lifshitz point from a Bose liquid on the lattice,” Phys. Rev. D 81, 104033 (2010), arXiv:1003.0009.
Pretko, “Emergent gravity of fractons: Mach’s principle revisited,” Phys. Rev. D 96, 024051 (2017), arXiv:1702.07613.
Marolf, “Emergent gravity requires kinematic non-locality,” Phys. Rev. Lett. 114, 031104 (2015), arXiv:1409.2509.
Bhatia, Jain, “On symplectic eigenvalues of positive definite matrices,” J. Math. Phys. 56, 112201 (2015), arXiv:1803.04647.
Diakonov, “Towards lattice-regularized quantum gravity,” arXiv:1109.0091.
Vladimirov, Diakonov, “Phase transitions in spinor quantum gravity on a lattice,” arXiv:1208.1254.
Volovik, “Fermionic quartet and vestigial gravity,” arXiv:2312.09435.
Cao, Carroll, Michalakis, “Space from Hilbert space: recovering geometry from bulk entanglement,” Phys. Rev. D 95, 024031 (2017), arXiv:1606.08444.
Henheik, Teufel, Wessel, “Local stability of ground states in locally gapped and weakly interacting quantum spin systems,” Lett. Math. Phys. 112, 9 (2022), arXiv:2106.13780.
Bachmann, Michalakis, Nachtergaele, Sims, “Automorphic equivalence within gapped phases of quantum lattice systems,” Commun. Math. Phys. 309, 835 (2012), arXiv:1102.0842.
D’Ariano, Perinotti, “Derivation of the Dirac equation from principles of information processing,” Phys. Rev. A 90, 062106 (2014), arXiv:1306.1934.
Bisio, D’Ariano, Tosini, “Quantum field as a quantum cellular automaton: the Dirac free evolution in one dimension,” Ann. Phys. 354, 244 (2015), arXiv:1212.2839.
Mlodinow, Brun, “Bounds on quantum cellular automaton lattice spacing from data on Lorentz violation,” Phys. Rev. D 112, 074513 (2025), arXiv:2506.20136.
Touboul et al. (MICROSCOPE Collaboration), “MICROSCOPE mission: final results of the test of the Equivalence Principle,” Phys. Rev. Lett. 129, 121102 (2022), arXiv:2209.15487.