A map of the quantum-gravity families examined here

Which of six requirements can each of twelve discrete, lattice and modified-continuum families of quantum gravity keep? The requirements are Lorentz invariance, a massless universal spin-2 graviton with locality, positivity, chiral matter, horizon thermodynamics with a dynamical law, and a computed number (section 1). Table 1 gives the answer per family. In each family at least one requirement is excluded, or available only at a cost and no family has been shown to keep all six. Most of the grid is empty: most families have been tested against one or two requirements, the Poisson sprinkling against all six; an empty cell is not a finding. String theory, loop quantum gravity, spin foams and causal dynamical triangulations are not among the families (section 2). Eight further pages, on cosmology and the dark sector, test whether structures of the gravity sector fix or constrain cosmological quantities. None derives a number; Table 3 lists them.

Poisson sprinklings, the point sets behind most rows, are the causal-set model of Bombelli, Lee, Meyer and Sorkin [BLMS]; the causal-set d’Alembertian and action are due to Benincasa and Dowker [Benincasa–Dowker]. Table 2 includes two results. Among weights that depend only on the type of a horizon pair (the number of other points in the future of its lower point, below the slice), with finitely many types or an exponentially decaying kernel, none is non-negative and gives an area law for the expected count at linear order in curvature in three to twelve dimensions. Haar-random lattices are Poincaré-invariant and not Poisson.

1. The six requirements

Lorentz invariance means that the substrate selects no frame and that gravitons, photons and matter share one limiting speed.

A graviton and locality means a massless spin-2 graviton that couples with equal strength to the energy and momentum of every species, with an action that is local or whose non-locality is controlled. The column also takes the accuracy of a wave operator on the substrate against the continuum d’Alembertian, and the compatibility of finite-dimensional local algebras with positive-energy translations.

Positivity means non-negative weights in sums over configurations and, for an entropy built from horizon pairs, a non-negative amount per pair; positive states, so that probabilities are non-negative and the energy is bounded below; and a propagator that is a non-negative superposition of free propagators (a positive spectral weight, which excludes ghosts).

Chiral matter means one-handed fermions with one species per field and no doublers, the extra species that a naive lattice discretisation produces.

Horizon thermodynamics and a dynamical law means an entropy proportional to the horizon area, a temperature, and a law of change: an action, a sum over histories, a rule that selects a manifold-like geometry, or a rule that completes field values on a fixed sprinkling. The column also takes the boundary terms that carry the area term and the entropy of the vacuum on a region of a field theory. The pages test the area dependence of horizon counts, boundary terms and rules of change, not a temperature.

A computed number means Newton’s constant in units of the discreteness length, or a quantity of cosmology or the dark sector, obtained and not fitted.

The eight cosmology and dark-sector pages test whether structures of the gravity sector, such as a global constraint on the vacuum energy or the coupling of a modulus, fix or constrain a cosmological quantity. Table 3 says which constrain and which leave a quantity undetermined.

2. The families

A Poisson sprinkling is a random set of points of density $\ell^{-d}$ in $d$ -dimensional spacetime, with $\ell$ the discreteness length; the causal order of the manifold makes it a causal set. One realisation is one sampled point set. The interval count of a pair $x\prec y$ is the number of sprinkled points between them. The causal-set d’Alembertian at $x$ is a sum over the causal past of $x$ in which a point $y$ has a weight $w_m$ that depends on the interval count $m$ of the pair [Benincasa–Dowker]. Its mean over the sprinkling, the mean-field operator, is Lorentz-invariant and is described by a kernel, the weights as a function of the proper time of the interval.

A horizon count refers to a region $P$ below a spacelike slice, whose boundary is a horizon. It adds, over every pair $x\prec y$ with $x$ in $P$ and $y$ outside $P$ below the slice, a weight $w_m$ that depends on the number $m$ of other elements in the future of $x$ below the slice, the type of the pair (horizon counts). A completion rule predicts the value of a field at a sprinkled element from its values at finitely many other elements (local rules of change). In a sum over orders, almost all finite partial orders have three layers (levels) of sizes close to $n/4$ , $n/2$ and $n/4$ , the Kleitman–Rothschild orders, so the question is whether the action suppresses them.

Quantum counts ask whether the number operators of a quantum field in a Poincaré-invariant state or a positive-energy vacuum can reproduce a sprinkling. Mixed-Poisson and signed sprinklings let the density be random, or the weights of configurations signed or complex.

Invariant random lattices are placed at random so that their law, the probability distribution of the point set, is Poincaré-invariant although no single realisation is: Haar-random lattices, and the orbit lattice of the integral quadratic form $Q_7=7x_0^2-x_1^2-x_2^2-x_3^2$ , whose shells are the sets of lattice vectors with equal $Q_7$ . A translation-invariant operator acts on plane waves by a multiplier, its symbol. A lattice with an exact discrete Lorentz group has exact translations and a Zariski-dense subgroup of the Lorentz group acting on it (contained in no proper subgroup cut out by polynomial equations), such as the integer Lorentz group. Invariant random sets of null lines or hyperplanes have Poincaré-invariant law.

Lattice gravitons are the gravitons of Hamiltonian lattice theories. A preferred frame or a compact residual symmetry group is what is left when a substrate selects a frame. Local qubit and rotor models have emergent linearised diffeomorphisms. Three continuum families complete the list: propagators with positive spectral weight, generalised free fields with a ghost (a negative spectral weight), and induced gravity, in which Newton’s constant is generated by the fluctuations of matter fields.

The thirteenth group is not a family. Its eight pages treat linear constraints on the vacuum energy, the sign of vacuum-matter energy exchange, structures that leave the cosmological constant undetermined, an apparent negative neutrino mass in baryon-acoustic-oscillation (BAO) distance data, the coupling of a dark-dimension modulus to baryons, an instability in Freezing Gravity (a dark-energy model with a freely prescribed expansion history) when the dark-energy equation of state is below $-1$ , the low multipoles of holographic cosmology, and a positivity bound on non-Gaussianity (section 6).

3. The map

The columns of Table 1 are the six requirements of section 1, headed Lorentz, Graviton, Positivity, Chiral, Horizon and Number. A filled cell holds one verdict word. Excluded means that a theorem rules the requirement out for the family. Limited means that the requirement is available only at a cost, in a weaker form or in a restricted case. Holds means that a construction or theorem achieves it in the regime named. Open means that the question was examined and is undecided. An empty cell means the combination was not examined.

Family and object	Lorentz	Graviton	Positivity	Chiral	Horizon	Number
Poisson: d’Alembertian	holds	limited	excluded
Poisson: horizon counts		limited	excluded		limited	excluded
Poisson: action, boundary terms					excluded
Poisson: completion rules					excluded
Causal sets: sums over orders					open
Poisson: order-only fermions				excluded
Poisson: overlap operators				excluded
Poisson: Euclidean overlap, time axis				limited
Poisson: link operators along a frame				excluded
Quantum counts: invariant states			limited
Quantum counts: integer counts			excluded
Quantum counts: finite-dimensional factors		excluded
Mixed Poisson		limited
Signed or complex			excluded
Signed Poisson plus Haar lattice		open
Haar-random lattice	holds	limited
Orbit lattice of $Q_7$	holds	excluded
Exact discrete Lorentz group	excluded
Null lines or hyperplanes	excluded
Lattice gravitons: conserved momentum		excluded
Lattice gravitons: soft coupling		excluded
Preferred frame	limited
Local qubit and rotor models		excluded
Positive spectral weight			limited
Ghost field			excluded		excluded
Induced gravity						limited

Table 1. Families against requirements.

Notes to the grid

Each item gives the requirement, the verdict word, what is excluded, the cost or what is achieved, and the page that shows it. Each verdict holds under the hypotheses of the linked page, which Table 2 lists.

Poisson sprinkling, d’Alembertian
- Lorentz invariance, holds: in law, no equivariant frame exists; species speeds untested (frame leakage).
- Graviton and locality, limited: one realisation has large fluctuations of the operator (kernel-scale kinematics).
- Positivity, excluded: the mean-field operator has no positive spectral representation, d = 4 (causal-set d’Alembertian).
Poisson sprinkling, horizon counts
- Graviton and locality, limited: an area law at all gradient orders needs a heavy-tailed kernel (horizon-count kernels).
- Positivity, excluded: no non-negative weights by pair type give the area law (horizon counts).
- Horizon thermodynamics, limited: area-law response only with signed weights (horizon counts).
- Computed number, excluded: the means leave Newton’s constant undetermined; the entropy unit is a free factor (dimensionless constants).
Poisson sprinkling, action and rules
- Horizon thermodynamics, excluded: flat static products, the finite part has no Gibbons–Hawking–York form (boundary and corner terms).
- Horizon thermodynamics, excluded: d ≥ 2, no exactly wave-consistent rule completes field values (local rules of change).
- Horizon thermodynamics, open: suppression of Kleitman–Rothschild orders, for actions of depth at least one (local rules of change).
Poisson sprinkling, fermions
- Chiral matter, excluded: d = 4, no covariant kinetic term from the order (fermions from the order).
- Chiral matter, excluded: overlap index zero or undefined (chirality obstructions).
- Chiral matter, limited: numerical, the expected index needs a time axis, a frame (chirality obstructions).
- Chiral matter, excluded: d = 4, a frame-cut Wilson term leaves exact zero modes at unpartnered points (Wilson terms).
Quantum counts
- Positivity, limited: invariant states give Poisson counts with a random intensity (quantum sprinklings).
- Positivity, excluded: integer counts of non-zero density need a non-real two-point function (integer counts).
- Graviton and locality, excluded: no non-trivial finite-dimensional local factors (integer counts).
Mixed-Poisson and signed sprinklings
- Graviton and locality, limited: d = 4, the mean d’Alembertian is the continuum one times a factor below one (mixed-Poisson and signed substrates).
- Positivity, excluded: restoring the normalisation in a non-Poisson process needs probabilities outside [0,1] (mixed-Poisson and signed substrates).
- Graviton and locality, open: d = 4, mean d’Alembertian of signed Poisson plus Haar lattice; the Haar mean is not computed (mixed-Poisson and signed substrates).
Invariant random lattices
- Lorentz invariance, holds: Haar-random lattices are Poincaré-invariant in law and not Poisson (Lorentz-invariant point sets).
- Graviton and locality, limited: two-dimensional only, weights on the first four interval counts have no finite mean (Lorentz-invariant point sets).
- Lorentz invariance, holds: the orbit lattice of $Q_7$ is Poincaré-invariant in law (kernel-scale kinematics).
- Graviton and locality, excluded: kernels on finitely many shells have no symbol in one realisation (kernel-scale kinematics).
Exact discrete Lorentz group, null lines and hyperplanes
- Lorentz invariance, excluded: a Zariski-dense group allows no dispersion relation (arithmetic Lorentz group).
- Lorentz invariance, excluded: invariant random sets with locally finite mean measure are empty (Lorentz-invariant point sets).
Lattice gravitons, preferred frame, local models
- Graviton and locality, excluded: no conserved lattice momentum with an analytic symbol (conserved lattice momentum).
- Graviton and locality, excluded: exactly universal soft coupling forces an exact cone, one step sketched (universal graviton coupling).
- Lorentz invariance, limited: with a compact residual group, species speeds are not forced equal (frame leakage).
- Graviton and locality, excluded: finite-dimensional local spaces give no linear spin-2 dispersion (emergent gravitons).
Continuum families
- Positivity, limited: the spectral dimension is at least the space dimension (spectral dimension).
- Positivity, excluded: a ghost field has no positive vacuum on any open region (ghost entropy).
- Horizon thermodynamics, excluded: a ghost field has no region vacuum entropy through a positive state (ghost entropy).
- Computed number, limited: induced Newton’s constant depends on the regulator shape (induced Newton constant).

The chiral, horizon and number columns are filled almost only for the Poisson sprinkling, which records where calculations were done. Haar-random lattices are parity-invariant in law and carry no hand; other invariant random lattices can (Lorentz-invariant point sets, Propositions 13 and 14), but no page builds a fermion operator on one. The sign of a weighted horizon count on a marked order, such as a chain of three elements, is not that of its mean (horizon counts).

4. Register of results

Table 2 lists the 30 pages in the order of the home page, with hypotheses and kind of each main result. Proved means a theorem of the page under the hypotheses of the row; computer-assisted, proved with a last step of exact arithmetic; derived, obtained by calculation within a stated model or at a stated order; numerical, resting on computed values or fits; conditional, valid given a property named in the row and not proved. A parenthesis names a part of another kind, or a case left open.

Page	Hypotheses	Result	Kind
Quantum sprinklings	Poincaré-invariant, locally normal bosonic states of finite mean quanta; fermionic states of finite density on a ball; determinantal processes invariant under translations and a boost subgroup	Bosonic counts are Poisson with a random intensity; invariant fermionic states reduce to the vacuum; invariant determinantal processes are empty	proved
Integer counts	Additive, covariant integer counts of finite variance in a positive-energy vacuum; finite-dimensional local factors	Zero density if the two-point function is real, or on a light ray real on separated intervals; the factors are trivial	proved
Mixed-Poisson and signed substrates	Mixed-Poisson, signed and complex processes; four-dimensional causal-set d’Alembertian on exponential test fields	Mean is the continuum operator times a factor below one unless the density is constant; restoring it in a non-Poisson process needs probabilities outside the unit interval	proved
Arithmetic Lorentz group	Exact lattice translations, Zariski-dense arithmetic Lorentz group; finitely many components, or positive scalar states	Bands are flat; positive invariant states have a uniform spectrum plus periodic parts	proved (one case open)
Lorentz-invariant point sets	Haar-random lattices, non-zero weight on empty intervals for the divergence; invariant random sets of null lines or hyperplanes, locally finite mean measure	Haar-random lattices are Poincaré-invariant, not Poisson; in two dimensions a weighted sum over interval counts 0 to 3 diverges (Poisson: converges); null-line sets are empty	proved from cited theorems
Kernel-scale kinematics	One Poisson realisation, kernel scale equal to discreteness scale; orbit lattice of $Q_7$ , kernels on finitely many shells with non-zero nearest-neighbour weight	Standard deviation at least about $(L/\ell)^3/3$ of the signal for timelike Gaussian packets of width $L=2\ell$ to $32\ell$ ; no symbol on the orbit lattice	proved (bound numerical)
Dimensionless constants	Poisson sprinkling; free coupling and entropy unit; for the volume term, finite weightings, conformally static backgrounds, $d\ge3$	The means leave Newton’s constant in discreteness units undetermined; the mean horizon count has no volume term	proved
Conserved lattice momentum	Classical Hamiltonian gravitons on a cubic lattice, relativistic low-energy dispersion, non-vanishing low-energy scattering	No exactly conserved lattice momentum with a real-analytic symbol; band-limited (ball-shaped) vertices conserve one only with a frame, shifting the photon speed by 1.8 to 7.8 parts in $10^4$	proved
Universal graviton coupling	Finite-dimensional real-analytic periodic Bloch symbols; for the coupling, also a relativistic low-energy band, simple soft poles, soft gauge invariance, distinct hard velocities	No band of such a symbol is an exact cone; exactly universal soft coupling forces an exact cone	proved (one step sketched)
Emergent gravitons	Harmonic, analytic, stable models of local qubits or rotors with finite-dimensional local spaces and emergent spatial diffeomorphisms	Gapless spin-2 modes obey $\omega\le Ck^2$ ; linear dispersion needs weak scalar-constraint invariance, absent for local-constraint generators	proved
Horizon-count kernels	Poisson horizon count with the area law at every gradient order, each order assumed to add a new term; kernel integrable near zero	Kernel cannot decay like $e^{-c\sqrt x}$ or faster in linear size $x$ ; kernels with tails $e^{-cx^\kappa}$ , $\kappa<1/2$ , exist	conditional
Causal-set d’Alembertian	Mean-field d’Alembertian of a Poisson sprinkling; Lorentz-invariant retarded kernel bounded by a power of proper time	No positive Källén–Lehmann representation in four dimensions; operators with symbol equal to the continuum to all orders exist for tail exponent below 1/2 and for none above	proved (density computed)
Spectral dimension	Rotation-invariant scalar propagator with a massless pole and positive Stieltjes weight	Spectral dimension is at least the dimension of space at every scale	proved
Frame leakage	Compact residual group or no boosts, without a supersymmetry algebra of one speed constant; Yukawa model with a frame in the scalar regulator	Species speeds not forced equal; one-loop fermion speed shift $g^2/48\pi^2$ for small scalar mass	proved (one estimate not proved)
Fermions from the order	Four-dimensional Weyl spinors with one global spinor space on a Poisson causal set; self-adjoint even-odd Kähler–Dirac operators on the order complex	No covariant kinetic term in four dimensions; Kähler–Dirac zero modes number at least $\lvert\chi\rvert$ , the Euler characteristic	proved (growth numerical)
Chirality obstructions	Overlap operators on a causal set, retarded with chirally even on-site part or self-adjoint for a chirality-exchanging involution	Overlap index zero or undefined; a Euclidean overlap with a time axis gives the expected index in twelve two-dimensional cases	proved (Euclidean cases numerical)
Wilson terms and reflection positivity	Four-dimensional Poisson sprinkling; link operators cut along a frame with zero-row-sum Wilson term	Exact zero modes at a positive fraction of interior points; no non-empty translation-invariant random set is mirror symmetric in every realisation	proved
Horizon counts	Poisson sprinkling, expected count, weights depending only on pair type, finitely many types or an exponentially decaying kernel; linear order in curvature	Area-law response holds if a polynomial built from the weights vanishes at $d$ specified points, and only then for $3\le d\le12$ ; such weights take both signs	computer-assisted
Boundary and corner terms	Mean causal-set action on flat static four-dimensional product regions, embedded regime, kernels with zero conditions	Benincasa–Dowker kernel: divergence $\rho^{1/4}$ on a timelike boundary, removed by an extra zero; finite part has no Gibbons–Hawking–York form	proved (examples numerical)
Local rules of change	Poisson sprinkling of Minkowski space, $d\ge2$ ; completion rules on finitely many elements; three-layer orders	No exactly wave-consistent completion rule; three-layer orders are suppressed for fixed outer layers, and for the link action at Kleitman–Rothschild proportions under a phase condition	proved (depth at least one open)
Ghost entropy	Generalised free field with a ghost; Lorentz- and translation-invariant positive-energy vacuum	Vacuum is not positive on any open region, so defines no entropy through a positive state; a ghost removed from the algebra is outside the theorem	proved
Induced Newton constant	Free Lorentz-invariant fields, one proper-time regulator (four for the Standard Model cutoff), one loop	$G\Lambda^2$ , with $\Lambda$ the cutoff, depends on the regulator shape; the Standard Model needs a cutoff above the Planck mass unless the Higgs curvature coupling is below about $-1$	proved (Standard Model one-loop)
Linear constraints on the cosmological constant	Constraint linear in the stress tensor, causal-vector weights or sequestering weights summing to one; flat FRW, fluids of constant equation of state in $[0,1/3]$ and non-increasing density	Vacuum energy is non-positive; for a light-cone weight with separable measure and log-concave kernel, its magnitude does not grow along the worldline	proved (sequestering derived in the model)
Vacuum-matter sign	Vacuum energy at the global minimum of its effective potential (vacuum plus matter), or relaxing linearly from equilibrium; Standard Model plasma	Matter loses no energy (cumulatively, for linear relaxation) and particle masses do not fall; the electron is lighter at recombination than in the laboratory	proved (electron case partly numerical)
Undetermined cosmological constant	Free scalar on flat FRW; Kodama state (a wave functional of the Ashtekar connection) and Verlinde horizon-state conditions; group-field-theory condensate; clock Hamiltonian	The defining conditions leave the cosmological constant or the dark-matter mass undetermined	proved and derived (three inputs cited)
Negative neutrino mass	Equal cold dark matter in fit and truth; fit with equal masses and a smaller sum; extra components of any type, or conserved ones of constant density above redshift 300	Exact mimicry violates the null energy condition; constant-density components above redshift 300 leave the inferred sum at least 0.0638 eV	proved (floor numerical)
Dark-dimension coupling	Unwarped torus at tree level, point-like Standard Model brane, no brane Einstein–Hilbert term, Kaluza–Klein graviton dark matter	A field moving only the dark circle has baryon coupling one third of its dark-matter coupling, excluded by the Cassini solar-system bound at coupling 0.05	proved (warped case numerical)
Freezing Gravity	Freezing Gravity with dust and no radiation; leading order in a small lapse-gradient coefficient and kinetic detuning; zero tensor-speed excess	Sub-horizon instability in phantom phases at zero beyond-Horndeski coupling; no constant such coupling removes it on the phantom side of four DESI galaxy-survey fits	derived (checked numerically)
Holographic cosmology	Conformally coupled matter, large flavour number; model of the mixed diagrams; compressed likelihood, fixed background	Parameters fitted at multipole 30 and above fit the low multipoles worse than $\Lambda$ CDM by 18.5 in $-2\ln\mathcal L$ ; positivity limits non-conformal scalars to about 0.6 to 1.9	numerical (bound conditional)
Positivity wedge	Tree-level exchange of a unitary hidden sector with de Sitter-invariant coupling to an inflaton of sound speed 1; four proxy inner products	Equilateral and orthogonal amplitudes lie in a convex cone, a narrow wedge with negative equilateral amplitude	conditional (edges numerical)

Table 2. Results by page.

5. How the results depend on each other

Several results rest on the non-compactness of the Lorentz group. No Lorentz-invariant probability measure exists on the hyperboloid of future unit timelike vectors, so a frame can be neither averaged away nor read off a Lorentz-invariant point set (frame leakage, Proposition 6 and Corollary 7). Invariant locally finite measures also fail to exist on the spaces of null lines and null hyperplanes, which empties invariant random sets of such lines (Theorem 4 of Lorentz-invariant point sets). Haar-random lattices escape this because the space of unimodular lattices (of unit covolume) carries a finite $SL(d,\mathbb R)$ -invariant measure (Theorem 1 there).

The Zariski density of an arithmetic Lorentz group acts on lattices. Exact lattice translations together with such a group make the orbit of every non-torsion momentum dense in the Brillouin torus (lattice momenta modulo $2\pi$ ), which leaves flat bands (arithmetic Lorentz group, Corollary 1 and Theorem 1). Its Proposition 1 says that a bounded invariant function on finitely many shells has no dispersion on any open set; kernel-scale kinematics, Proposition 4, applies it to the stabiliser of the orbit lattice (the Lorentz transformations preserving it).

A propagator with positive Stieltjes weight has spectral dimension at least the dimension of space (spectral dimension). The mean-field d’Alembertian in four dimensions has no such weight (causal-set d’Alembertian, which cites that result), and mixed-Poisson and signed substrates extends the statement to matrix-valued kernels that depend only on the interval (Theorem 17). For horizon counts, an area-law response needs weights of both signs (horizon counts), which horizon-count kernels, dimensionless constants and boundary terms cite.

The d’Alembertian and horizon-count results share a threshold in the tail exponent of the kernel. The Mellin transform of a kernel (its integral against a power of the interval volume or proper time) fixes both the low-momentum symbol of the d’Alembertian and the gradient expansion of the horizon count, and exactness to all orders means zeros of that transform. A d’Alembertian kernel bounded by $e^{-\lambda\tau^{\beta}}$ in proper time $\tau$ , with symbol equal to the continuum one to all orders, needs $\beta\le1/2$ , and such kernels exist for every $\beta<1/2$ (Theorems 6 and 7 of the d’Alembertian page). Under the gradient-order condition of the horizon-count kernel page, a kernel integrable near zero with the area law at every gradient order cannot decay like $e^{-c\sqrt x}$ or faster in the linear size $x$ , and kernels with tails $e^{-cx^{\kappa}}$ exist for every $\kappa<1/2$ (Theorems 1 and 4 there). The exponent 1/2 itself is excluded in the second case and open in the first.

Conserved lattice momentum is cited by universal graviton coupling, and both by emergent gravitons; quantum sprinklings is cited by integer counts. The matter pages share one question, whether a frame is needed: the order-only page identifies a global spinor space with a frame, the chirality and reflection-positivity pages study operators with and without one, and the frame-leakage page, which all three cite, computes its cost. Five of the dark-sector pages are connected by links among them: linear constraints, vacuum-matter sign, negative neutrino mass in BAO data, Freezing Gravity and the undetermined cosmological constant, the last of which also cites the dimensionless-constants page. Three pages link to no other page and are linked from none in Table 2: dark-dimension coupling, holographic cosmology and the positivity wedge.

6. Cosmological constraints and routes to numbers

In Table 3, excluded means that the route or scenario named is ruled out under the hypotheses of the row, with the kind of result given in Table 2.

Quantity	Route and hypotheses	Result	Page
Cosmological constant	Constraint linear in the stress tensor; flat universe with pressure between zero and a third of the density	excluded: a positive value	Linear constraints
Cosmological constant	Free scalar on flat FRW; Kodama and Verlinde conditions	excluded: a value fixed by these conditions	Undetermined cosmological constant
Cosmological constant	Mean horizon count of a Poisson sprinkling, $d\ge3$	excluded: a volume term	Dimensionless constants
Dark-matter to baryon ratio	Relations between number densities alone	excluded: a value fixed by them	Undetermined cosmological constant
Newton’s constant in discreteness units	Means of the causal-set action and of horizon counts; free entropy unit	excluded: a value fixed by the means	Dimensionless constants
Newton’s constant	Induced gravity, free fields with a proper-time regulator; Standard Model content, one loop	$1/G$ is negative and smaller in modulus than observed by a factor of about $4\times10^{-35}$ to $2\times10^{-33}$	Induced Newton constant
Sign of vacuum-matter exchange	Vacuum energy tracking the global minimum of its effective potential	excluded: net energy loss by matter	Vacuum-matter sign
Neutrino mass sum	Extra components mimicking a smaller equal-mass sum	excluded: exact mimicry by components obeying the null energy condition	Negative neutrino mass
Dark-matter coupling	Light modulus of an unwarped dark-dimension torus at tree level	Coupling 0.05 conflicts with the Cassini bound	Dark-dimension coupling
Dark-energy behaviour	Freezing Gravity with dust, leading order	excluded: phantom phases with zero beyond-Horndeski coupling and tensor-speed excess	Freezing Gravity
Low-multipole spectrum	Holographic cosmology at large flavour number, compressed likelihood	Values fitted at multipole 30 and above fit the low multipoles worse than $\Lambda$ CDM by 18.5 in $-2\ln\mathcal L$	Holographic cosmology
Non-Gaussianity	Tree-level exchange of a unitary hidden sector, inflaton sound speed 1	excluded: a positive equilateral amplitude, conditional on two numerically checked properties	Positivity wedge

Table 3. Routes and constraints.

No row of Table 3 derives a measured number. Each page names the cases outside its result.

7. Open questions

Lattice gravitons: whether a lattice amplitude exists whose soft factor is gauge invariant up to high order in the product of energy and lattice spacing (universal graviton coupling).
Chiral matter on a Poisson sprinkling: operators that complete a retarded block to be Hermitian for the chirality operator, and Wilson terms without chiral symmetry, lie outside the theorems; reflection positivity of the disorder average of the unsymmetrised ensemble is open (chirality obstructions, Wilson terms and reflection positivity).
The d’Alembertian: whether a kernel with a non-locality scale much larger than the discreteness scale reduces the single-realisation fluctuation, and how the bound depends on that scale (kernel-scale kinematics); which weights realise a heavy-tailed kernel (horizon-count kernels).
Laws of change: those that read common predecessors, interval contents or links are not covered, and suppression of three-layer orders at Kleitman–Rothschild proportions is undecided for actions of depth at least one (local rules of change).
Counts: scalar volume counts that obey Einstein locality and whose two-point function is neither real nor real on separated light-ray intervals (integer counts).

References

Bombelli, Lee, Meyer, Sorkin, “Space-time as a causal set,” Phys. Rev. Lett. 59 (1987) 521. Used for: the causal set as a discrete substrate.
Benincasa, Dowker, “The scalar curvature of a causal set,” Phys. Rev. Lett. 104 (2010) 181301, arXiv:1001.2725. Used for: the causal-set d’Alembertian and action.