Which of six requirements fail, or come at a cost, in twelve discrete, lattice and modified-continuum families of quantum gravity, with a register of 30 results and a table of cosmological constraints and routes to numbers.

A Poincaré-invariant, locally normal state of a bosonic field on spacetime with finite mean number of quanta in every bounded region is a mixture of constant-amplitude coherent states, so its counts are Poisson with a random intensity that is the same everywhere; the fermionic and determinantal analogues with finite density are trivial.

Additive, translation-covariant integer counts of finite variance have zero density in a positive-energy vacuum when their two-point function is real, or real on separated intervals of a light ray, and finite-dimensional local factors with weak additivity and locality are trivial there.

On exponential test fields, the mean of the four-dimensional causal-set d'Alembertian at an element of a positive, non-degenerate mixed Poisson sprinkling is the continuum operator times a factor below one, and a compactly supported signed mixing law that restores the factor in a non-Poisson process gives count probabilities outside [0,1].

On a lattice with exact translations and an exactly preserved Zariski-dense discrete Lorentz group, such as the integer Lorentz group, translation-invariant operators on finitely many field components that commute with the group have flat bands, and positive invariant scalar states have a uniform spectrum plus periodic parts.

Haar-random lattices are Poincaré invariant in law and not Poisson, invariant random sets of null lines or hyperplanes with locally finite mean measure are empty, Poisson points with frames known only modulo an arithmetic lattice have no equivariant finite non-empty neighbour rule, and an invariant random lattice can differ in law from its mirror image.

At the discreteness scale, the Benincasa–Dowker d'Alembertian on a Poisson sprinkling has standard deviation at least about one third of the cube of the width times the signal for timelike Gaussian packets (computed), and a layer operator with non-zero nearest-neighbour weight on the orbit lattice of one anisotropic integral form has no symbol (proved).

Poisson means of the causal-set action and of horizon counts leave the Newton constant undetermined when coupling, entropy unit and weight normalisation are free; the squared-mean-to-variance ratio of infinite-support exact weightings has no upper bound; the mean horizon count has no volume term for metrics conformal to a static one with round spatial sections.

In a classical perturbative Hamiltonian lattice theory with relativistic gravitons at low energy, an exactly conserved lattice momentum with an analytic symbol is incompatible with low-energy graviton scattering, and band-limited vertices conserve a momentum exactly only at the price of a preferred frame.

Under stated assumptions on the soft expansion of the amplitude, a graviton whose coupling to energy and momentum is exactly universal must have an exactly conical dispersion near zero momentum, and no band of a finite-dimensional real-analytic periodic Hermitian or unitary symbol is an exact cone.

In harmonic, analytic, stable models of local qubits or rotors with emergent linearised spatial diffeomorphisms, gapless spin-2 modes cannot disperse linearly unless a scalar constraint leaves the kinetic term invariant only up to constraint terms.

Under a hypothesis on the gradient expansion, a causal-set horizon count that follows the area law at every gradient order needs a weight kernel with infinitely many sign changes and a tail that decays more slowly than the exponential of any multiple of the square root of the linear size.

For the mean-field causal-set d'Alembertian in four dimensions, with a nonzero retarded Lorentz-invariant kernel bounded by a power of proper time, the Euclidean propagator is not a positive superposition of free propagators, and in two and four dimensions the same holds when the symbol agrees with the continuum one to all orders at low momentum.

For a rotation-invariant scalar propagator with a massless pole, positive Källén–Lehmann weight, no constant term and power-law growth of its inverse, the diffusion generated by the inverse on flat momentum space has a spectral dimension at least the dimension of space at every scale.

A compact residual symmetry group does not force gravitons, photons and scalars to share one limiting speed, no Lorentz-invariant probability measure on frames exists, and a frame in the scalar-line regulator of a Yukawa model shifts the fermion speed at one loop by a cutoff-independent amount when the scalar mass is small.

A four-dimensional Weyl spinor field with one global spinor space on a sprinkled causal set has no Lorentz-covariant kinetic term built from the order alone, and even-odd operators on the span-restricted order complex, self-adjoint for a degree-diagonal form, have more exact zero modes than elements (proved in expectation in two dimensions, measured in four).

The overlap index of an operator on a causal set, with any weights, is zero or undefined when the operator is retarded with a chirally even on-site part or self-adjoint for a matrix that exchanges the two chiralities; a finite boundary-projected index is nonzero in general but depends on boundary data alone.

On a four-dimensional Poisson sprinkling, a link operator cut off along a frame, with a Wilson term whose rows sum to zero, has exact zero modes at a positive fraction of the interior points, and no non-empty translation-invariant random point set is mirror symmetric in every realisation.

In Poisson sprinklings in three to twelve dimensions, weights that depend only on the type of a causal-set horizon molecule, with finitely many types or an exponentially decaying kernel, make the expected count respond to uniform curvature like the horizon area at every ball size only if they take both signs.

In four-dimensional flat space, on product regions in the embedded regime, the mean causal-set action diverges like the inverse discreteness length on a timelike boundary, kernels with an extra Mellin zero remove the divergence, and on static regions the finite part that remains is not of Gibbons–Hawking–York form.

On the two-dimensional null lattice, rectangle rules whose result does not depend on how a rectangle is cut are heap operations of groups, but no rule using finitely many sprinkled elements is exactly wave-consistent; for three-layer orders with fixed outer layers, the suppression bound depends on the action only through the link phase.

For a generalised free field with a ghost, the Lorentz-invariant positive-energy vacuum is not positive on the algebra of any non-empty open region and so defines no entropy through a positive state, while positive states remain if the ghost has negative energy or is absent from the algebra.

For free fields with a proper-time regulator, the Standard Model reproduces the observed Newton constant only with a cutoff above the Planck mass, unless the Higgs curvature coupling is below about -1; with the quadratic divergence cancelled (one loop, 't Hooft–Feynman gauge) the induced inverse Newton constant is negative, 4e-35 to 2e-33 times the observed modulus.

In a flat dust-and-radiation universe, a linear global constraint gives a non-positive vacuum energy for causal-vector weights and sequestering weights summing to one, and a past-light-cone weight with separable measure, log-concave kernel and constant energy-to-trace ratio not between minus a quarter and zero gives a value that does not grow along the worldline.

A vacuum energy without kinetic term that follows the global minimum of its effective potential transfers energy to matter, and a scalar that changes only the electron mass and follows its minimum in a Standard Model plasma makes the electron lighter at recombination than in a laboratory environment less dense than the plasma.

In the structures examined here, conditions on a renormalised vacuum energy, a horizon integrality condition, a group-field-theory condensate, a clock Hamiltonian, a causal-set count, a Standard Model degeneracy condition or a number-density relation leave the cosmological constant or the dark-matter mass undetermined, under the model assumptions stated.

In a flat universe with standard neutrinos, components obeying the null energy condition cannot reproduce the exact expansion history of a fit with a smaller equal-mass neutrino sum when fit and truth have the same stable cold dark matter density, and can if the fit has more cold matter by a threshold amount.

In an unwarped dark-dimension compactification at tree level, with the Standard Model on a point-like brane carrying no Einstein-Hilbert term, a light field that moves only the dark circle among the mass-setting quantities couples to baryons at one third of its dark-matter coupling, which the Cassini bound excludes at the fitted value 0.05.

At leading order for small lapse-gradient coefficient and kinetic detuning, and at zero tensor-speed excess, Freezing Gravity with minimally coupled dust has a sub-horizon scalar instability with wavenumber-independent growth rate in every phantom phase with zero beyond-Horndeski coupling, and no constant coupling removes it on the phantom side of four DESI DR2 fits.

With conformally coupled matter, resumming the gluon self-energy of holographic cosmology at large flavour number expresses the low-multipole spectrum through the three high-multipole parameters; values fitted at multipole 30 and above fit the low multipoles worse than LambdaCDM, and a fit to all multipoles is worse than LambdaCDM by about 5 in minus twice the log likelihood.

Tree-level exchange of a unitary hidden sector with a de Sitter-invariant coupling to an inflaton of sound speed 1 reaches a convex cone of equilateral and orthogonal bispectrum amplitudes, which for four proxy inner products is a narrow wedge with negative equilateral and positive orthogonal amplitude.