Preprint series · Work in progress · 2026
Deriving the laws of gravity from the statistics of Planck-scale spacetime atoms
A theoretical programme deriving the laws of physics — gravity, spacetime, and particle physics observables — from the collective behaviour of discrete Planck-scale constituents governed by a single Hamiltonian constraint, with no free parameters fitted to observation. Current results include the cosmological constant and the Higgs boson mass.
Paper III
The Higgs Boson Mass from CP Violation and Baryon Asymmetry
The Standard Model offers no explanation for the Higgs boson mass. We show that the DEG framework predicts \(m_H\) from CP-violating observables alone, with no parameter fitted to electroweak data. In DEG, the Higgs doublet is a composite pseudo-Nambu-Goldstone boson of the family-sector chiral symmetry broken at \(\Lambda_{\mathrm{fam}} \sim 3\ \text{TeV}\). The single real parameter of the Higgs sector, the phase \(\varphi_{\mathrm{DEG}}\), is uniquely determined by the CKM angle \(\delta_{\mathrm{CKM}} = 1.20\ \text{rad}\) and the baryon-to-photon ratio \(\eta_B = (6.12\pm0.04)\times10^{-10}\). Propagating through the Coleman-Weinberg potential yields \(m_H = 124.8\pm 2.4\ \text{GeV}\), a deviation of \(0.17\,\sigma\) from the PDG value. Seven independent constraints on \(\varphi_{\mathrm{DEG}}\) are satisfied simultaneously. The primary falsification target is the triple Higgs coupling \(\kappa_\lambda^{\mathrm{DEG}} = -1.449\pm0.17\), accessible at HL-LHC and decisive at FCC-hh.
Paper IV
The Standard Model from SU(21) — Gauge Embedding, Unification, and Proton Decay
The technical foundation for Paper III. The full Standard Model gauge group is derived from an SU(21) embedding chain with no fitting; three generations arise from \(\mathrm{SU}(3)_{\mathrm{family}} \subset \mathrm{SU}(16)\) and all SM quantum numbers are verified. The paper proves in full the \(\xi\)-Independence and Decoupling Theorems stated in Paper III, derives the Coleman-Weinberg potential, and establishes doublet-triplet splitting algebraically. Proton decay via dimension-6 operators predicts \(\tau_p \sim 10^{33\text{--}34}\ \text{yr}\), testable at Hyper-Kamiokande. Fine-tuning is quantified explicitly: \(\Delta = 18.2\) at the canonical point, \(\Delta_{\min} = 5\pm2\) under optimal construction.
Paper II
The Cosmological Constant from Vacuum Atom Statistics
Using only the internal degeneracy \(g = 442\) of the spacetime atoms — fixed by the algebra of SU(21)×U(1) embedding the Standard Model gauge group, not fitted to any cosmological observable — the canonical partition function of the atom ensemble yields a cosmological constant consistent with the Planck 2018 observed value \((1.088\pm0.030)\times10^{-52}\ \text{m}^{-2}\) within the theoretical uncertainty of a two-loop renormalisation group derivation. The dark energy equation of state \(w = -1\) follows as an exact geometric corollary; any \(5\sigma\)-confirmed \(w\neq-1\) by DESI, Euclid, or the Nancy Grace Roman Space Telescope constitutes definitive falsification.
Paper I
Time, Matter Coupling, and Newton's Law
The foundational paper. A collection of \(N\) Planck-scale atoms governed by the Hamiltonian constraint \(H = \sum_n[p_n^2/2m - (\alpha/2)a_n^2] = 0\) gives rise, without additional assumptions, to York time as a collective degree of freedom, the thermodynamic arrow of time as an exact theorem, a consistent quantum matter coupling via wavefunction overlap, and Newton's law with its precise numerical coefficient. Paper II builds directly on these foundations.
Papers are made available here as preprints prior to journal submission. They have not been peer-reviewed. Comments and corrections are welcome.
Single postulate
\(H = \sum_n\!\left[\frac{p_n^2}{2m} - \frac{\alpha}{2}a_n^2\right] = 0\)
All results follow from this constraint alone. No background metric, no continuous fields, no additional assumptions.
Cosmological constant
\(\Lambda_{\text{pred}} \approx 1.1\times10^{-52}\ \text{m}^{-2}\)
Observed: \(1.088\times10^{-52}\ \text{m}^{-2}\) (Planck 2018). Derived from \(g=442\) alone — zero free parameters. Paper II.
Dark energy equation of state
\(w = -1.000\)
Exact geometric corollary. Any \(5\sigma\) detection of \(w\neq-1\) by DESI, Euclid, or Roman falsifies this sector.
Group-theoretic input
\(g = 442\)
Internal degeneracy fixed by SU(21)×U(1) embedding the SM gauge group. The sole discrete input; never fitted to any observable.
Higgs boson mass
\(m_H = 124.8 \pm 2.4\ \text{GeV}\)
Derived from \(\delta_{\mathrm{CKM}}\) and \(\eta_B\) alone. PDG: \(125.20\pm0.11\ \text{GeV}\) — \(0.17\,\sigma\) deviation. Paper III.
Triple Higgs coupling
\(\kappa_\lambda = -1.449 \pm 0.17\)
Primary falsification target of Paper III. Accessible at HL-LHC; decisive at FCC-hh.
DEG posits that spacetime is a statistical aggregate of \(N\) discrete Planck-scale atoms, each characterised by a size parameter \(a_n\) and an internal Hilbert space of dimension \(g = 442\). The dynamics is governed by a single Hamiltonian constraint — no background metric, no continuous fields at the fundamental level.
Papers I and II derive, without additional assumptions: the emergence of a macroscopic time coordinate; a consistent coupling of quantum matter to the discrete geometry; Newton's law with its correct numerical coefficient; and the observed cosmological constant as a consequence of vacuum atom statistics.
What is not claimed. DEG does not yet constitute a complete theory of quantum gravity. The derivations rest on statistical and thermodynamic assumptions whose justification at the full quantum level is an open problem. The programme is offered as a coherent and falsifiable research direction, not as a finished theory. Open problems and limitations are stated explicitly in each paper.
Programme roadmap — eleven papers, four phases
This page is updated as each paper becomes available.
Like loop quantum gravity and causal set theory, DEG takes discreteness as fundamental. It differs in using a statistical mechanics approach: gravity and time emerge thermodynamically rather than being quantised geometrically.
The cosmological constant result is, to our knowledge, the first zero-parameter derivation of the observed \(\Lambda\) from any discrete spacetime programme. The value of \(g = 442\) — a group-theoretic integer fixed by the Standard Model embedding — sets the scale of dark energy. The claim is specific and falsifiable; the uncertainty budget is given explicitly in Paper II.
The programme addresses the cosmological constant problem not by cancelling QFT vacuum contributions but by providing an alternative mechanism in which the relevant UV cutoff is the micrometre-scale atom spacing \(a \approx 0.74\,\mu\text{m}\) rather than the Planck length, suppressing vacuum energy by \((a/\ell_P)^3 \sim 10^{86}\).
Matteo Pinna is a theoretical physicist working independently on quantum gravity and emergent spacetime. His interest in emergent gravity began during his thesis work in 2018, shaped by a conviction that the foundations of physics should admit a simple, parameter-free description — and a specific dissatisfaction with the treatment of time in general relativity.
The starting point was a refusal to accept time as a curved fourth dimension behaving differently from the other three. If space is emergent, time should be too — and the arrow of time, rather than being imposed by initial conditions, should follow from the statistics of whatever is fundamental. DEG is the formalisation of that programme, developed over several years alongside a career in technology.
He is based in Madrid.
If you work in quantum gravity, emergent spacetime, or related areas and find this programme of interest, I would welcome correspondence — critical feedback especially.
matteo@deg-gravity.com
I am an independent researcher based in Madrid. Collaboration enquiries and comments from researchers with relevant expertise are very welcome.
The two papers above contain full derivations, explicit uncertainty budgets, and a list of open problems. Nothing is hidden behind a paywall or submission requirement.
Both preprints are available directly on this page as PDF. They represent work in progress and will be updated as revisions are made prior to submission.
If you are a physicist encountering DEG for the first time and would like to discuss the approach, its foundations, or its limitations, please feel free to write. I am also happy to share derivation notes on specific points not fully developed in the papers.