Informational Emergence of Spacetime

Blackhole Singularities demonstrated that gravitational collapse, analysed through the execution trace of a computational simulation, converges to a zero-entropy state at the classical singularity. The present paper asks what happens when we begin at that point and move in the opposite direction.

Introduction

Starting from the zero-entropy state and increasing entropy through minimal single-bit changes, we ask what geometric and structural features emerge? Do they resemble those of the observed universe?

When entropy is zero, no structures exist and the universe is a single unextended point. As entropy increases, minimal structures proliferate, and their uniform distribution across the bitstring is perceived by an internal observer as spatial expansion. Expansion is the geometric reading of entropy increase.

This reframing has three immediate consequences.

First, it resolves the initial conditions problem without appeal to quantum fluctuations or fine-tuning.

Second, it provides a native mechanism for matter formation. As entropy increases, higher-order structures emerge, and the qualitative shape of their abundance curve is fixed in advance by the combinatorial composition of the pattern being counted, not by any physical input.

Setup

Configuration Space and Entropy Evolution

The system is defined by a binary configuration string $S_t$ of fixed total length $L$. The initial state $S_0$ corresponds to the all-zero bitstring, establishing an initial boundary condition where Shannon entropy is minimised:

$$ H(S_0) = 0. $$

Consistent with the four axioms: rejection of any externally imposed clock or agent, the evolution is not driven by an external "bit-flipper." At each raw tick, one of the $L$ positions is selected uniformly at random --- a choice determined entirely by internal, countable facts about the current configuration (how many positions currently hold each value), not by any external rule --- and that position's value is toggled. Tracking $k(t_{raw})$, the number of $1$-bits in $S_t$ after $t_{raw}$ raw ticks, this process is exactly the \textbf{Ehrenfest urn model} \cite{ehrenfest1907}:

$$ P\bigl(k \to k+1\bigr) = \frac{L-k}{L}, \qquad P\bigl(k \to k-1\bigr) = \frac{k}{L}. $$

Writing $p(t_{raw}) = k(t_{raw})/L$ and the rescaled time $\tau = t_{raw}/L$, the mean trajectory obeys $dp/d\tau = 1-2p$, with exact solution

$$ p(\tau) = \frac12\left(1-e^{-2\tau}\right), $$

a closed-form, $L$-independent relaxation curve from $p=0$ toward the equilibrium value $p=1/2$. Note that $\tau$ is a rescaled variable, not a raw tick count: $\tau=O(1)$ corresponds to $O(L)$ actual bit-flip events having occurred. This distinction matters throughout, and especially in Section~\ref{sec:desitter}.

Two distinct timescales govern this process and should not be conflated: the relaxation time $L/2$ (in raw ticks), governing how fast the mean of $k$ approaches equilibrium, and the sharp mixing (cutoff) time $(L/4)\ln L$, governing when the distribution of $k$ becomes statistically indistinguishable from its equilibrium shape. We use the exact process above, not the mean-field curve alone, wherever the distinction matters.

Recursive Spatial Fabric and Structural Filtering

Level-0: The Spacetime Fabric

We define a minimal local target pattern string $P$ of length $w_0$, with $a$ ones and $b = w_0 - a$ zeros. The global string $S_t$ is parsed into non-overlapping blocks of size $w_0$, yielding a raw background capacity of $N_{\max} = L / w_0$ elements. A Level-0 ``space fabric'' token exists at index $m$ if the block matches the binary target integer value $V_P$:

$$ \sigma_G[m] = \begin{cases} 1 & \text{if } \sum_{j=0}^{w_0-1} S_t[m \cdot w_0 + j] \cdot 2^{w_0 - 1 - j} = V_P \ 0 & \text{otherwise} \end{cases} $$

The raw count of detected space fabric elements is $N_G = \sum \sigma_G[m]$.

Level-$k$: Hierarchical Matter Sub-Scales

Higher-order structural tiers are extracted by recursively filtering the underlying layer's output array. These higher-order structures can be thought representing Dark Matter, Hadrons, Atoms, and Compounds sequentially.

For a given structural level $k \in {1, 2, 3}$, the input array is scanned using a non-overlapping window of width $w_k$ subject to a density activation threshold $\tau_k$:

$$ \sigma_k[m] = \begin{cases} 1 & \text{if } \sum_{j=0}^{w_k-1} \sigma_{k-1}[m \cdot w_k + j] \geq \tau_k \ 0 & \text{otherwise} \end{cases} $$

The raw abundance of entities at level $k$ is given by $N_k = \sum \sigma_k[m]$. As discussed in Section~\ref{sec:results}, the qualitative shape of $N_k$ as a function of entropy is fixed by the effective composition $(a,b)$ implied by each level's threshold $\tau_k$, and should be checked explicitly rather than assumed.

Results

Starting from the pristine $H=0$ boundary state, the following macroscopic phenomena emerge consistently as entropy increases:

Zero entropy yields a structureless state. The all-zero configuration contains no detectable structural matches ($N_G = 0$), mapping naturally under any relational decoding scheme to a single, unextended spatial point. This is the initial state -- the singularity.

Spatial expansion. As entropy grows, the density of $1$-bits climbs toward $0.5$ following the exact relaxation curve $p(\tau) = \tfrac12(1-e^{-2\tau})$ as derived in the Setup section, precipitating a freeze-out of background $L_0$ fabrics. The decoded spatial extent expands, following this closed-form approach to equilibrium.

Three regimes of structural abundance. For a fixed pattern of length $k=a+b$ (with $a$ ones and $b$ zeros) counted against a bitstring whose global $1$-density $p(\tau)$ sweeps monotonically from $0$ to $1/2$, the expected count is proportional to $p^a(1-p)^b$. This function's unconstrained maximiser is $p^\ast = a/(a+b)$. Since $p(\tau)$ only ever reaches the range $[0, 1/2]$, the observed shape depends entirely on where $p^\ast$ falls relative to that range:

• $a > b$ (so $p^\ast > 1/2$): the abundance curve is \emph{monotonically increasing} over the entire reachable range, with no interior peak.
• $a = b$ (so $p^\ast = 1/2$): the abundance curve is monotonically increasing and flattens smoothly as it approaches its maximum exactly at equilibrium.
• $a < b$ (so $p^\ast < 1/2$): the abundance curve exhibits a genuine interior maximum at $p=p^\ast$, i.e.\ a true rise-and-fall hump, reachable within $[0,1/2]$, closely approximated by a hump-like shape.

Hump-Class Dominance Across the Relaxation Trajectory

The classification above answers which shape a given composition $(a,b)$ produces. A distinct question is which class is typical: if a window of fixed length $k=a+b$ is drawn at time $\tau$, which regime does its observed composition tend to fall into? Answering this requires weighting each composition by the number of arrangements that share it, $\binom{k}{a}$, rather than examining a single representative pattern in isolation.

The number of $1$-bits in a length-$k$ window is distributed as $\mathrm{Binomial}\bigl(k,,p(\tau)\bigr)$. Since $p(\tau)\in[0,\tfrac12)$ for every finite $\tau$, the mean and mode of this distribution lie strictly inside the hump region $a<b$ throughout the relaxation, approaching the $a=b$ boundary only as $\tau\to\infty$. Formally, define the composition-weighted probability mass in each regime:

$$ M_{\text{hump}}(\tau) ;=; \sum_{a \lt b} \binom{k}{a} p(\tau)^a \bigl(1-p(\tau)\bigr)^b,\qquad M_{\text{non-hump}}(\tau) ;=; \sum_{a\ge b} \binom{k}{a} p(\tau)^a \bigl(1-p(\tau)\bigr)^b, $$

with $M_{\mathrm{hump}}+M_{\mathrm{non\text{-}hump}}=1$ by construction.

Direct evaluation, with the crossover $\tau_c(k)$ located by root-finding on $M_{\mathrm{hump}}(\tau)-M_{\mathrm{non\text{-}hump}}(\tau)=0$, shows that $M_{\mathrm{hump}}(\tau)$ holds the overwhelming majority of the mass from just above $\tau=0$ up to $\tau_c(k)$, and that $\tau_c(k)$ grows with $k$, with $p(\tau_c)\to\tfrac12$ as $k\to\infty$:

The hump regime is therefore not merely one of three equally weighted possibilities: it is the typical outcome of the relaxation process at any window length, dominating essentially the entire trajectory. The non-hump regimes only attain majority mass in the immediate approach to final equilibrium --- a fraction of the trajectory that shrinks as $k$ grows, vanishing entirely in the $k\to\infty$ limit except at $\tau\to\infty$ itself.

This is a fully predictive, closed-form classification: the qualitative shape of any given filter's abundance curve can be determined in advance from $(a,b)$ alone, without running the simulation.

Filter Independence

The most significant result of this paper is the universality of the classification rule above, not any single curve shape in isolation.

Expansion as entropy increase

This has a sharp implication. The arrow of time, the thermodynamic second law, and the expansion of the universe are not three independent features of the observed world. They are one feature.

Comparison with Penrose's Weyl curvature hypothesis

Penrose's Weyl curvature hypothesis proposes that the thermodynamic arrow of time originates in the asymmetry between initial and final singularities: the initial singularity had vanishing Weyl curvature (smooth, ordered), while final singularities (black holes) have diverging Weyl curvature (chaotic, disordered). This is a geometric statement about the initial state, motivated by the need to explain the arrow of time.

The present results suggest that the initial state is not merely smooth within quantum limits but perfectly smooth: zero entropy state admitting no fluctuations. Yet qualitative features of matter organisation and spatial expansion can emerge. No residual quantum irregularities are required.

An exact correspondence with de~Sitter expansion

The exact solution gives $1-2p(\tau) = e^{-2\tau}$. Its reciprocal,

$$ X(\tau) ;\equiv; \frac{1}{1-2p(\tau)} ;=; e^{2\tau}, $$

is an exponentially growing quantity, unbounded as $\tau\to\infty$ (as $p\to1/2$). Identifying $\tau =\tfrac{H}{2},t$ --- a strictly \emph{linear} relationship between rescaled bit-flip time and cosmological time $t$ --- gives

$$ X(\tau) = e^{Ht} = a(t), $$

an exact match to the deSitter scale factor, valid at every $\tau$, not merely asymptotically or in some restricted regime. This resolves what would otherwise be a structural mismatch between this model and deSitter expansion: entropy itself, $S(\tau)=H_2(p(\tau))$, is bounded and cannot correspond to an unbounded scale factor under any reparametrisation of time, but $X(\tau)$ is a different, unbounded transform of the same underlying relaxation variable $p(\tau)$, and it is $X(\tau)$, not $S(\tau)$, that plays the role of scale factor here.

The correspondence has a clean physical reading: as $\tau\to\infty$ (equilibrium, i.e.\ heat death), $X\to\infty$, so the model predicts unbounded expansion that never actually completes in finite $\tau$ --- the same asymptotic structure as the usual heat-death-as-limit picture, recovered here without being assumed.

Conclusions

Cosmic expansion is shown to be the geometric interpretation of increasing informational entropy, with the underlying entropy trajectory given by the exact, closed-form Ehrenfest relaxation curve $p(\tau)=\tfrac12(1-e^{-2\tau})$.

Beginning from the zero-entropy state as the informational singularity, this process drives the systematic expansion of the spatial manifold without requiring a cosmological constant, ad-hoc inflaton fields, or hard-coded equations of motion.

Second, the abundance of any emergent micro-structure follows one of exactly three determined shapes --- monotone increasing, monotone-flattening, or a genuine interior peak --- fully predictable in advance from the composition $(a,b)$ of the pattern being counted. Moreover, the hump-shaped regime ($a<b$) is not merely one of three equally likely outcomes but the typical one: weighting compositions by their multiplicities shows it dominates the composition-weighted probability mass across essentially the entire relaxation trajectory, ceding majority status to the non-hump regimes only in the immediate approach to final equilibrium, a window that narrows as pattern length grows. Physically-motivated matter-detection filters should have their composition checked against this classification to confirm they fall in the peak-producing regime, which the present result shows is the generic, expected case rather than a special one.

Third, the reciprocal transform $X(\tau)=(1-2p(\tau))^{-1}=e^{2\tau}$ reproduces the de~Sitter scale factor exactly under a linear identification of bit-flip time with cosmological time, resolving the apparent shape incompatibility between this model's (bounded) entropy curve and cosmological expansion.

Open Questions and Future Directions

The emergence of localized microstructures in the $a<b$ regime naturally invites the hypothesis that these configurations serve as the informational correlates of physical particles. A primary theoretical challenge is to verify explicitly that the specific recursive filters used for Hadron/Atom/Compound-level detection do in fact satisfy $a<b$ under their actual thresholds $\tau_k$.

The de~Sitter correspondence raises two concrete follow-up questions. First, whether $X(\tau)=(1-2p)^{-1}$ can be derived directly from the spatial decoding schemes rather than introduced as the transform that happens to reproduce the target curve. Second, whether the implied relation $H=2/L$ can be reconciled with the observed inflationary Hubble rate --- either by revisiting the identification of one raw tick with one Planck time, or by finding that $L$ itself should be measured differently during the inflationary regime than the horizon-entropy definition used there.

Statistical Models, like this one, tend to suffer from Bolzzman Brain problem.

Furthermore, within a static, timeless informational framework, the constraints governing the dimensionality and boundaries of spatial and temporal domains remain to be fully defined.

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