Following Wolfram, Gorard, et al. (2020–), we model blockchain state evolution as a multiway rewriting system on a hypergraph.
Let $\mathcal{H} = (V, E)$ where $V$ is a set of atoms (UTXO bits / account state atoms) and $E \subseteq \mathcal{P}(V)$ is a set of hyperedges (relations among atoms). A blockchain state is a snapshot of $\mathcal{H}$ at a given block height.
A transaction $\tau$ is a rewriting rule $r: L \hookrightarrow R$ where $L, R \subseteq E$ are the consumed and produced hyperedge patterns. Applying $\tau$ to $\mathcal{H}$ yields a new hypergraph $\mathcal{H}' = (\mathcal{H} \setminus L) \cup R$.
The multiway system $\mathcal{M}$ generates all possible orderings of transaction application simultaneously, producing a directed graph of hypergraph states:
The causal graph $\mathcal{C} = (\mathcal{E}, \prec)$ has nodes $\mathcal{E}$ = rewriting events and edges $e_i \prec e_j$ iff the output hyperedges of event $e_i$ are consumed as input by event $e_j$. This is Lamport's happened-before relation $\rightarrow$, now grounded in physical rewriting.
The causal graph induces a natural metric. Define the causal distance:
where $|\gamma|$ is the number of edges along path $\gamma$ in $\mathcal{C}$. This is the discrete analog of proper time in special relativity.
The causal ball of radius $r$ centered at event $e$:
$$B(e, r) = \{ e' \in \mathcal{E} \mid d_{\mathcal{C}}(e, e') \leq r \}$$The effective dimension of the causal graph:
$$d_{\text{eff}} = \lim_{r \to \infty} \frac{\log |B(e, r)|}{\log r}$$For a $(3+1)$-dimensional spacetime to emerge, we require $d_{\text{eff}} \to 4$.
Causal Invariance (Wolfram 2020): All branches of $\mathcal{M}$ eventually produce isomorphic causal graphs. Formally:
In the continuum limit, causal invariance in the multiway system is equivalent to Lorentz invariance of the emergent spacetime. Different orderings of event processing (different reference frames) yield the same causal structure.
Blockchain translation: Different nodes processing transactions in different orders converge to the same causal graph. Fork resolution is guaranteed by causal invariance, not by protocol agreement. This is why PoW dissolves FLP rather than solving it.
Standard Wolfram Physics assigns unit weight to causal edges. We introduce an entropy weight to capture PoW thermodynamics.
For a rewriting event $e_i$ (a block application), the entropy production is:
$$\sigma(e_i) = k_B \ln\left(\frac{\Omega_{\text{after}}}{\Omega_{\text{before}}}\right) \geq 0$$In PoW terms, $\sigma(e_i)$ is proportional to the work performed — the number of hash attempts required to find a valid nonce. By Landauer's principle, each irreversible bit erasure in the hash function produces $k_B T \ln 2$ joules of heat.
Reweight the causal graph edges by entropy production:
$$d_S(e_i, e_j) = \min_{\gamma: e_i \to e_j} \sum_{e_k \in \gamma} \sigma(e_k)$$This is the thermoeconomic distance between two consensus events.
The full spacetime interval in thermoeconomic spacetime, with Qi as the invariant speed:
where $d\tau$ is the number of causal steps (block time), $d\ell$ is the state-space distance between UTXO sets, and $Q_i$ is the energy cost per unit computation — Qi, the invariant.
A path $\gamma^*$ in the entropy-weighted causal graph is a geodesic if it extremizes the action:
$$S[\gamma] = \int_\gamma \sigma\, d\lambda = \sum_{e_k \in \gamma} \sigma(e_k)$$The geodesic equation in the discrete causal graph:
$$\frac{\delta S[\gamma]}{\delta \gamma} = 0 \quad \Longrightarrow \quad \nabla_{\dot\gamma} \dot\gamma = 0$$In the continuum limit, using the causal graph metric $g_{\mu\nu}$:
$$\frac{d^2 x^\mu}{d\lambda^2} + \Gamma^\mu_{\nu\rho}\, \frac{dx^\nu}{d\lambda}\, \frac{dx^\rho}{d\lambda} = 0$$where $\Gamma^\mu_{\nu\rho}$ are the Christoffel symbols of the emergent metric.
Proof of Entropy Minima (PoEM) selects the canonical chain by minimizing total entropy production across competing forks. In the thermoeconomic spacetime language:
This is the geodesic condition. PoEM doesn't select the longest chain (Nakamoto) or the chain with highest total difficulty (Bitcoin) — it selects the path of minimum entropy production through causal spacetime. Free fall, not vote.
Let two competing chain forks $\gamma_1, \gamma_2$ diverge at block $b_0$ and converge at merge block $b_*$. PoEM selects $\gamma_1$ iff:
$$\sum_{b \in \gamma_1} \sigma(b) < \sum_{b \in \gamma_2} \sigma(b)$$This is equivalent to selecting the geodesic in the entropy-weighted causal graph between $b_0$ and $b_*$. The selected path is the one along which the metric distance $d_S$ is minimized — i.e., the path of extremal proper thermoeconomic time.
Following Gorard (2020) and Ollivier (2009), we compute curvature of the causal graph using Ollivier-Ricci curvature — a coarse notion of Ricci curvature defined for metric measure spaces.
For adjacent events $e_i, e_j \in \mathcal{C}$, let $\mu_{e_i}, \mu_{e_j}$ be the uniform probability measures on their causal neighborhoods. The Ollivier-Ricci curvature is:
$$\kappa(e_i, e_j) = 1 - \frac{W_1(\mu_{e_i},\, \mu_{e_j})}{d_S(e_i, e_j)}$$where $W_1$ is the Wasserstein-1 (Earth Mover's) distance between the two neighborhood distributions.
Interpretation of $\kappa$:
A network with $R > 0$ globally is geometrically stable. Selfish mining, eclipse attacks, and long-range attacks all manifest as local regions of $\kappa < 0$.
In GR, Einstein's equations relate the geometry of spacetime (Einstein tensor $G_{\mu\nu}$) to the distribution of energy and momentum (stress-energy tensor $T_{\mu\nu}$):
Gorard (2020) shows that in the continuum limit of a causal graph satisfying causal invariance, the Ollivier-Ricci curvature produces exactly the Einstein equations with:
We substitute the thermoeconomic quantities:
Where:
$G_{\mu\nu}[\mathcal{C}]$ — Einstein tensor computed from Ollivier-Ricci curvature of the entropy-weighted causal graph $\mathcal{C}$
$\Lambda_d$ — base mining difficulty (cosmological constant analog: sets the vacuum energy of the consensus field)
$Q_i$ — Qi, playing the role of $c$ (note $c^4$ in denominator of GR $\to$ $Q_i^4$ here)
$T^{\text{hash}}_{\mu\nu}$ — the hashrate stress-energy tensor (defined below)
Let $\rho_H$ = hashrate density (hashes per second per unit causal volume), $\vec{J}_H$ = hashrate current (directional flow of work through the network). Then:
$$T^{\text{hash}}_{\mu\nu} = \begin{pmatrix} \rho_H Q_i^2 & Q_i J^x_H & Q_i J^y_H & Q_i J^z_H \\ Q_i J^x_H & \Pi_{xx} & \Pi_{xy} & \Pi_{xz} \\ Q_i J^y_H & \Pi_{yx} & \Pi_{yy} & \Pi_{yz} \\ Q_i J^z_H & \Pi_{zx} & \Pi_{zy} & \Pi_{zz} \end{pmatrix}$$where $\Pi_{ij}$ is the mining pressure tensor — the flux of the $i$-th component of hashrate momentum in the $j$-th spatial direction.
The $T^{00}$ component is the energy density of the mining field: $T^{00} = \rho_H Q_i^2$. This is the thermoeconomic analog of $E = mc^2$ — hashrate density times the square of the invariant speed gives the energy density of the consensus field.
Every continuous symmetry of the thermoeconomic action $S[\gamma]$ produces a conserved quantity.
Lamport (1978) proved there is no global clock in a distributed system. Quai's response is not to find a global clock but to decompose the problem into a spectrum of clocks.
Quai's three-tier zone structure defines a set of PoW oscillators at different frequencies:
$$\{ \omega_{\text{prime}},\, \omega_{\text{region}},\, \omega_{\text{zone}} \} \quad \text{with} \quad \omega_{\text{prime}} \ll \omega_{\text{region}} \ll \omega_{\text{zone}}$$Each chain is a separate PoW oscillator. Their causal relationships define the merge topology. The ensemble forms a multirate Lamport clock system grounded in physical work.
In the multiway hypergraph, each zone's chain is a branch. The coincident blocks (blocks valid at multiple hierarchy levels) are the merge events. The Wolfram multiway graph for Quai is:
where $\sim_{\text{coincident}}$ identifies states that are valid across multiple zone levels. Causal invariance of $\mathcal{M}_{\text{Quai}}$ guarantees that the merge structure is path-independent — the canonical chain is the same regardless of which zone's perspective you start from.
Fischer, Lynch, Paterson (1985): No deterministic protocol can solve consensus in an asynchronous system if even one process can fail.
The proof's key step: you cannot distinguish a slow node from a dead node. Therefore, no finite waiting time suffices to declare consensus.
PoW changes the type signature of the consensus problem:
FLP applies to protocols where consensus is a logical agreement among processes. PoW defines consensus as the causal geodesic in a thermodynamically-weighted hypergraph. The canonical chain is not decided by nodes — it is the minimum-entropy path through the causal graph, which exists and is unique (given causal invariance) regardless of which nodes are online.
Nodes don't agree on the chain. They measure it. Measurement requires no synchrony. $\square$
The ruliad (Wolfram 2021) is the entangled limit of all possible computations — the space of all possible causal graphs generated by all possible rewriting rules. Each observer occupies a thread in the ruliad — a path through computational history that defines their subjective experience of the system.
An observer $\mathcal{O}$ at rulial coordinates $\xi_\mathcal{O}$ perceives a cross-section of the multiway system. Different observers have different $\xi$, corresponding to different:
Crucially, sub-rulial observers share causal structure even when they disagree on state. The causal graph $\mathcal{C}$ is the shared invariant — the intersubjective object that allows coordination without a global clock or global state.
This isomorphism $\phi$ is exactly causal invariance. It says: even though observers $\mathcal{O}_1$ and $\mathcal{O}_2$ disagree on local state, they will always agree on what caused what. The causal graph is the sub-rulial coordination medium.
Qi's role: since $Q_i$ is the invariant speed in the thermoeconomic metric, it is the same for all sub-rulial observers — just as $c$ is the same in all inertial frames. Converting any local price, any local hashrate, any local value into energy units removes all frame-dependence. Qi is the Lorentz factor that makes economic spacetime intervals objective.
FLP — Impossibility of deterministic consensus in logical message-passing systems.
↓ resolved by changing domain: thermodynamic self-evidence replaces logical agreement.
Lamport — No global clock; only partial order from message receipt.
↓ resolved by PoW: partial order becomes irreversible causal order (second law).
Quai clock ensemble — Multirate PoW oscillators at zone/region/prime frequencies.
↓ generates a multiway hypergraph with multiple causal branches.
Causal invariance — All branches yield isomorphic causal graphs.
↓ gives Lorentz invariance in the continuum limit (Gorard 2020).
PoEM — Fork choice selects minimum entropy production path.
↓ this is the geodesic equation in entropy-weighted causal spacetime.
Ollivier-Ricci curvature — Curvature of causal graph = distribution of hashrate.
↓ yields thermoeconomic Einstein equations $G_{\mu\nu} = \frac{8\pi}{Q_i^4} T^{\text{hash}}_{\mu\nu}$.
Qi — Energy cost per unit computation. Invariant across all economic reference frames.
↓ plays the role of $c$: makes economic spacetime intervals observer-independent.
Ruliad — Each observer is a thread in the space of all computations.
↓ Sub-rulial coordination is possible because the causal graph is the shared invariant, and Qi is the shared metric constant.