Quantum Gravity

The goal is a new physical principle for quantum gravity that derives, without fitting, a measured number nobody can currently explain: the neutrino masses, the dark matter density, or the size of the cosmological constant. The notes here are results found along the way.

1. Weighted horizon-molecule counts in causal sets

   Weighting causal-set horizon molecules by the number of elements to their future can make the expected count respond to curvature exactly as the horizon area does, at every diamond size, at linear order and for uniform curvature. The weights are unique at minimal degree and must take both signs. If every order in the curvature gradients brings new terms, exactness to all orders requires weight kernels with very slowly decaying tails.

2. A proof of the coth joint conjecture for regions between two spacelike or null polygonal curves

   For causal sets sprinkled into a flat two-dimensional region bounded below and above by piecewise-linear spacelike or null curves, or into the product of such a region with a flat plane in four dimensions, the mean Benincasa–Dowker action tends to the hyperbolic cotangent of the boost angle summed over the two joints. The same expansion gives the exact linear-order mean action of a curved two-dimensional causal interval.

3. The Cauchy law of the sprinkled Pauli–Jordan operator in 1+1 dimensions

   For a causal set sprinkled into a two-dimensional causal diamond, the massless Pauli–Jordan matrix is exactly a sum of two copies of one fixed matrix, one per null direction, related by a uniformly random permutation. Its bulk eigenvalue distribution tends to a Cauchy law of width one half, and the volume law of the untruncated entanglement entropy is reproduced to within ten per cent by one null chain thinned at random.

4. Lattice gravitons, exact diffeomorphisms and lattice momentum

   In a Hamiltonian lattice theory of gravitons, a conserved lattice momentum built from exponentially local terms is incompatible with graviton scattering at low energy, which excludes exact deformed diffeomorphism invariance if such a symmetry supplies that charge. Switching off interactions near the zone boundary conserves lattice momentum exactly, but by a heuristic closure criterion is not preserved by the nonlinear gauge transformations, and at one loop it shifts the speed of light by a few parts in ten thousand.

5. A positivity wedge for inflationary non-Gaussianity

   When the primordial bispectrum comes from tree-level exchange with a unitary heavy hidden sector, the reported equilateral and orthogonal amplitudes lie in a narrow wedge, with negative equilateral and positive orthogonal amplitude. Three hypotheses carry the result, and current data are consistent with it.

6. What a negative neutrino mass in the BAO data would require

   In a flat universe whose cold dark matter is stable, no component obeying the null energy condition can make the expansion history mimic a lighter neutrino. Under DESI DR2 weights the BAO-inferred mass sum has a floor of 0.0638 eV.

7. The low multipoles of holographic cosmology at large flavour number

   In holographic cosmology with conformally coupled matter, the leading large-flavour resummation of the dual stress-tensor correlator fixes the power spectrum at the largest angular scales with no new parameter. Given numerically established properties of the resummed correlator, positivity of the spectrum bounds the number of non-conformally coupled fields and excludes the explicit solution published with the model, and a fit to Planck temperature data is about two standard deviations worse than the standard cosmological model.

8. How a dark-dimension modulus couples to baryons

   In an unwarped dark-dimension compactification with the Standard Model on a brane, a field that changes only the size of the dark dimension couples to baryons with one third of its coupling to dark matter, which the Cassini bound excludes at the coupling preferred by cosmological fits. Fields that leave baryons uncoupled must preserve the internal volume, and their dark-matter coupling is then capped at an order-one value.

9. A matter-induced instability in Freezing Gravity

   Adding pressureless matter to the Freezing Gravity dark-energy model produces a sub-horizon instability whose growth rate does not depend on wavenumber. With constant parameters it affects every phantom phase and every nonzero beyond-Horndeski coupling, even on a cosmological-constant background, and no constant choice of that coupling stabilises the DESI DR2 fits.

10. A sign rule for an early change in the electron mass

    A scalar field that follows the minimum of its effective potential in the Standard Model plasma can only make the electron lighter in the past, while cosmological fits prefer it about one percent heavier at recombination. A field with a linear coupling allowed by the MICROSCOPE test of the equivalence principle would have to move further than its kinetic energy allows.