Growing document · September 2026
An emergent picture of time, entropy, and matter coupling, derived from a single Hamiltonian constraint on discrete spacetime atoms.
Every claim tagged: postulate, definition, derivation, or proposed choice — nothing asserted by citation or analogy alone.
Key results — derived from the single postulate
| Result | Value | Status | Note |
|---|---|---|---|
| Emergent time–entropy relation | \(dS/d\tau = 9k_B N\) | Exact algebraic identity | Only a "second law" once a proposed choice fixes an expanding branch |
| Singularity avoidance (single atom) | \(h_n(0)\le0 \Leftrightarrow\) never singular | Exact condition, derived | Whether a given atom itself avoids the singularity isn't fixed by the global constraint alone — it depends on that atom's own individual starting condition, not one shared across the whole system |
| Dark energy equation of state | \(w = -1\) exactly | Exact, same argument as a cosmological constant | Testable now — see below |
| Wheeler–DeWitt unitarity | York-time evolution exactly unitary; \(S_{\rm vN}\) conserved | Exact, given \(H_{\rm York}\) self-adjoint | Self-adjointness is one proposed choice |
| Matter–spacetime overlap \(O_{mn}\) | Non-negative & normalized | Derived, given the Born-rule coupling | The coupling itself is a proposed choice, not forced |
| Population-level singularity statistics | 72.9% of 1,404 populations, \(p=1.7\times10^{-119}\) | Statistical, not deterministic | 12,000 simulated atoms; rests on disclosed astrophysical inputs |
| Matter's reverse coupling (force law) | Settles at \(r^{-2}\) only under one further proposed choice | Honest partial result | Coefficient off by \(\sim4.8\times\), uncalibrated — not a derivation of Newton's law |
The constraint's own \(\alpha_{\exp}\) term plays the same structural role as a cosmological constant; its magnitude is a calibrated input, not derived — \(w=-1\) is the exact consequence regardless of that value. This is also this document's one falsifiable, near-term prediction — see "On falsifiable predictions" below.
By section — derived, and what each result rests on
On falsifiable predictions
This document offers exactly one falsifiable, near-term prediction: the dark energy equation of state \(w=-1\) exactly — testable now, since any \(5\sigma\) detection of \(w\neq-1\) by DESI or Euclid would rule it out. Everything else here is a derivation, a proposed choice, or an honest partial result: real content, but not yet a falsifiable prediction in its own right. This is a foundational, derivational core, not a phenomenology programme, and isn't trying to be judged by how many predictions it offers.
The first three below are explicit exclusions stated in Section 9 of the document. The five after that are further open questions raised elsewhere in its body — real gaps, but not formal exclusions.
Gating open question
The \(g_{\rm eff}=1\) holographic-reduction question
Whether an atom's internal states fully decohere once information reaches a holographic screen is genuinely open, and it gates the recovery of both Newton's law and the Einstein field equations entirely — neither is derived or assumed anywhere in this document. The trend across successive, better-motivated coupling choices points toward less, not more, decoherence. This is the single most consequential open question in the current document.
Calibrated, not derived
The cosmological constant's magnitude, \(\alpha_{\exp}^{\rm cosm}\)
A fitted, calibrated input to the constraint's cosmological sector, not derived anywhere in this document. Multiple candidate first-principles derivations have been tried and found not to work; this document does not add another attempt.
Candidate — not yet decided
The area quantum, \(\Delta_A = 4\ln(g)\,\ell_P^2\)
Not excluded for being broken or for depending on the open \(g_{\rm eff}=1\) question — it is valid under the same bare postulate used throughout. What keeps it out is that its derivation imports one further external input, the Bekenstein–Hawking coefficient \(1/4\), not yet incorporated or tagged anywhere in this document. No decision has been made to add it.
Open question
The number of internal states, \(g\)
\(g\)'s value is left entirely free throughout this document — nothing derived depends on it, only on its being finite and positive. A group-theoretic origin for \(g\) from some internal symmetry structure is a separate, currently open question, not assumed or argued anywhere here.
Left symbolic
The lattice spacing, \(d\)
The spacing between neighbouring atoms is left fully symbolic everywhere it appears — not fixed by anything derived in this document. It is one of the two quantities (with the auxiliary mass parameter \(m_{\rm atom}\)) whose absence keeps the matter-dynamics investigation's force-law coefficient from being calibrated against real gravity.
Undischarged choice
\(H_{\rm York}\)'s self-adjointness
Adopted as a disclosed assumption, not derived from \(H=0\), for Wheeler–DeWitt unitarity to go through. What remains a genuine, undischarged choice is an operator-ordering convention for quantizing the leftover relative degrees of freedom, needed before self-adjointness could even be posed as a question.
Real, but unresolved
Accretion/recession asymmetry doesn't unify with the expanding branch
Whether a matter-rich atom is rescued from singularity depends, directionally and consistently, on whether its local matter is accreting or receding — a real, mechanistically-understood effect. But it does not follow from the already-adopted expanding-branch choice, and if anything argues against that choice being the source of the rescuing direction. A preferred direction still needs its own separate proposed choice, or a genuinely missing physical mechanism this document does not yet have.
No equation of motion
Matter's reverse coupling onto spacetime
The full matter-dynamics investigation — internally consistent, energy-conserving, and detailed-balance-satisfying by construction — still supplies no equation of motion for matter, and every quantitative result in it rests on several proposed choices, none forced by the bare postulate. Reported here as a genuine partial success, not a claimed derivation of any known force law.
DEG postulates that spacetime consists of \(N\) discrete constituents — "spacetime atoms" — each with a size parameter \(a_n\), a conjugate momentum \(p_n\), and \(g\) internal quantum states, with \(g\)'s value left entirely free. The dynamics is governed by a single global Hamiltonian constraint — no background metric, no continuous fields at the fundamental level.
From this postulate alone, the document derives: an emergent notion of time (York time) and an exact algebraic link between that time and a Boltzmann entropy; the precise condition under which a spacetime atom avoids a finite-time singularity; that the dark energy equation of state \(w=-1\) follows by the same argument used for an ordinary cosmological constant; that the theory's time evolution is exactly unitary, so a closed DEG universe loses no information; and a proposed, Born-rule-motivated matter–spacetime coupling, together with its consequences for which atoms remain singular — including a large-scale, population-level statistical result.
What is not claimed. DEG is not a complete theory of quantum gravity, and this document is a small, self-checked core that grows only as further results meet its own stated standard — every claim an explicit postulate, definition, step-by-step derivation, or proposed choice, never an assertion resting on citation or analogy alone. One calibrated numeric input is used, \(\alpha_{\exp}^{\rm cosm}\), fit to the cosmological constant's observed magnitude, not derived. Newton's law and the Einstein field equations are not derived, or assumed, anywhere in this document — their recovery is contingent on the still-open \(g_{\rm eff}=1\) question.
Active research directions
This page is updated as the document develops.
Like loop quantum gravity and causal set theory, DEG takes discreteness as fundamental. It differs in using a statistical-mechanics approach: time, and its exact link to entropy, emerge from counting microstates of the atom configuration itself, not from geometric quantisation. Gravity's own recovery — Newton's law and the Einstein field equations — is not attempted here; it remains contingent on the open \(g_{\rm eff}=1\) question (see Open problems).
The dark energy equation of state \(w=-1\) follows as an exact consequence of the constraint's own structure, by the identical argument used for an ordinary cosmological constant in general relativity — independent of the calibrated input's numerical value. That input, \(\alpha_{\exp}^{\rm cosm}\), fixes only the magnitude; the exponent of state does not depend on it.
As noted above, \(w=-1\) is also the one result here with a near-term experimental handle.
The document's own standard. Every substantive claim is tagged as an explicit postulate, definition, derivation, or proposed choice. Results that don't yet meet this standard — including anything contingent on \(g_{\rm eff}=1\) — are tracked separately (Open problems) rather than discarded, and may be added here later once they do.
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.
This document is the primary reference for the programme. Not peer-reviewed.
Poster material will be made available here via Zenodo once uploaded.
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 document above contains full derivations, every claim tagged as postulate, definition, derivation, or proposed choice, and a complete, plainly-stated list of open problems. Nothing is behind a paywall or submission requirement.
The current document — every derivation, proposed choice, and open question — is listed under Document above; DOI: 10.5281/zenodo.22306976.
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 notes on specific points not fully developed in the document.