Information Theoretic Emergence of Spacetime and Matter - juhameskanen/abstract GitHub Wiki

Classical Statistical Model

Microscopic State Space & Discrete Dynamics

Consider a system composed of $n$ fundamental binary elements (bits). The total state space consists of $2^n$ configurations, corresponding to the vertices of an $n$-dimensional hypercube ${0, 1}^n$:

$$ \mathcal{H}_n = {0, 1}^n, \quad |\mathcal{H}_n| = 2^n $$

Time emerges as a stochastic traversal through these $2^n$ configurations. The minimal transition between two consecutive configurations involves flipping a single bit chosen uniformly at random:

$$ P(x \to x') = \begin{cases} \frac{1}{n}, & \text{if } d_H(x, x') = 1 \ 0, & \text{otherwise} \end{cases} $$

where $d_H(x, x')$ denotes the Hamming distance between configuration states $x, x' \in {0, 1}^n$. This uniform random walk on the $n$-hypercube graph is the Ehrenfest process.

Entropy Evolution & Thermal Saturation

Let the system start in a pure, zero-entropy state $x_0 = (0, 0, \dots, 0)^T$, where all bit values are known deterministically:

$$ S(0) = -\sum_{i=1}^{2^n} P_i(0) \ln P_i(0) = 0 $$

As single-bit flips accumulate over discrete steps $t$, the system diffuses away from the origin. The macroscopic state is parameterized by the Hamming weight $k(t) \in {0, 1, \dots, n}$, representing the count of active bits ( $1$ ).

Relaxation Dynamics

The expected Hamming weight evolves according to discrete exponential decay towards thermal equilibrium:

$$ \langle k(t) \rangle = \frac{n}{2} \left( 1 - \left( 1 - \frac{2}{n} \right)^t \right) \approx \frac{n}{2} \left( 1 - e^{-2t/n} \right) $$

As $t \to \infty$, the (parity-averaged) distribution over microstates becomes uniform over the $2^{n-1}$ configurations reachable at each parity class, yielding maximum thermodynamic entropy

$$ S_{\max} = \ln(2^{n-1}) = n\ln 2 - \ln 2 = n\ln 2 + O(1). $$

The $O(1)$ correction is a fixed constant independent of $n$ — it reflects the walk's bipartite structure, not a modeling choice — and is negligible relative to the $O(1/\sqrt n)$ fluctuation terms computed below for any $n$ of physical interest.

Constant flip rate, vanishing net drift. At every $k$, exactly one flip happens every tick: $P(k\to k+1) + P(k\to k-1) = 1$, always. What changes with $k$ is not whether a flip occurs but the net drift, $\mathbb{E}[\Delta k] = \frac{n-2k}{n}$, which is $+1$ at $k=0$, falls linearly, and is exactly $0$ at $k=n/2$. "Saturation" is a statement about vanishing net drift under constant flip rate, not about the system slowing down.

No hand-coded selection for "increasing entropy." Starting from $k=0$ (an extreme, atypical configuration), the drift above is a consequence of counting, not an imposed rule: with only 0s present, every flip must increase $k$, since $P(k=0\to 1)=1$ trivially. As $k$ grows toward $n/2$, the pool of available "either direction" flips balances out on its own. No selection of "increasing-entropy paths" needs to be added externally — it falls out of starting at an extreme point of a shape (the hypercube) whose overwhelming majority of volume lies near the middle.

Fuzziness of the saturation boundary

At equilibrium, $k$ is binomially distributed with mean $n/2$ and standard deviation $\sqrt n /2$. The relative width of the saturated region is therefore $\sim 1/\sqrt n$ — never exactly zero (the boundary is genuinely statistical, as it must be for a system with no hand-coded rules), but vanishingly small at macroscopic $n$:

$n$	relative spread (std/mean)
40	0.16
$10^6$	$1.0\times10^{-3}$
$1.5\times10^{77}$ (solar-mass BH entropy)	$2.6\times10^{-39}$

Spatial Resolution and Internal Degrees of Freedom

Spatial resolution is defined by the number of geometrically distinguishable positions an internal observer can resolve — the boundaries between adjacent unlike bits in the string. For a configuration $x \in {0,1}^n$, define the number of resolvable blocks as:

$$ B(x) = 1 + \sum_{i=1}^{n-1} \mathbb{1}[x_i \neq x_{i+1}] $$

Each maximal run of equal-valued bits is one undifferentiated region; each transition between unequal adjacent bits marks a detectable edge, i.e. a distinguishable spatial boundary.

Zero Entropy State ($S = 0$): Lock state $x_0 = 00\ldots0$. No adjacent pair differs, so $B(x_0) = 1$. There is exactly one resolvable region — a single, undifferentiated point. Zero internal distances are measurable.
Saturated Equilibrium State ($S = S_{\text{max}}$): At equilibrium the configuration is drawn (approximately) uniformly among strings of Hamming weight $k \approx n/2$. For $k$ ones placed uniformly at random among $n$ positions, the expected number of transitions is:

$$ \mathbb{E}[T \mid k] = \frac{2k(n-k)}{n} $$

so the expected number of resolvable blocks is:

$$ \mathbb{E}[B \mid k] = 1 + \frac{2k(n-k)}{n} $$

At $k = n/2$ this gives $\mathbb{E}[B] = 1 + n/2$: saturation yields, on average, on the order of $n/2$ resolvable spatial positions.

Crucially, $B(x)$ is not determined by Hamming weight alone — two configurations with identical $k$ can differ enormously in resolvable structure. For $n=8$, $k=4$: the sorted string $00001111$ has $B=2$ (a single boundary), while the alternating string $01010101$ has $B=8$ (maximal resolution). For $0 < k < n$, the bounds are:

$$ 2 ;\le; B(x) ;\le; \min\bigl(2\min(k,,n-k)+1,; n\bigr) $$

So even at fixed equilibrium entropy $S(k)$, spatial resolution is a fluctuating quantity, not a fixed number — $\sim n/2$ is the typical value, but individual saturated configurations range from barely-resolved ($B=2$) to maximally-resolved ($B=n$), depending on arrangement rather than entropy alone.

On the Choice of Bit-to-Geometry Mapping

The spatial resolution was defined in a geometry-, dimension- and topology-agnostic manner: a given budget of bits is distributed across any number of dimensions or topological spaces.

Run-counting is one specific choice of map from bitstring to geometric structure — treating a bit-value disagreement between neighbors as a resolvable spatial edge. It is not the only possible choice; a bitstring of length $n$ could equally be mapped to geometry via, e.g., Hamming-distance-as-metric between arbitrary (not just adjacent) index pairs, a graph/adjacency structure over bit positions, or a spectral decomposition of the string treated as a discrete signal.

Whatever mapping is adopted, it is constrained by self-containment: the mapping rule itself is part of the model's physical content, and so must be expressible using only the $n$-bit budget that constitutes the universe. It cannot invoke an external coordinate system, an external observer, or any information beyond what the $n$ bits themselves encode — there is no scaffolding outside the bitstring for the geometry to be read off of. This rules out mappings that implicitly assume a pre-existing embedding space (e.g., "place bit $i$ at position $i$ on a line") as anything more than a labeling convention; the physically meaningful geometric content must be recoverable from the bit values and their relations alone, not from an externally imposed index.

Topology

3-sphere

This is what GR implies. However, a theory from first principles should not postulate it, but derive it. Why 3, why spherical symmetry?

Hypercube

Instead of a rope, the emergent observer sees the bitstring as an $n$-cube. Every vertex of the $n$-cube is equidistant from the center (this is true for trivial reasons — vertex-transitivity of the hypercube graph — for every $n \ge 1$, not a discovery specific to $n=3$; it is a property of the topology choice, not evidence favoring it over the rope).

The full $n$-cube has exactly $n\cdot 2^{n-1}$ edges — this is a standard graph-theoretic fact.

Nothing internal to the bitstring dictates which topology it should be read under.

Temporal Resolution & Non-Equilibrium Windows

Temporal resolution corresponds to the characteristic mixing timescale required to drive the system from a low-entropy state to thermal equilibrium.

Time & The Arrow of Time

The characteristic relaxation time $t_{\text{relax}}$ to reach a $99%$ equilibrium threshold scales as:

$$ t_{\text{relax}} \sim \mathcal{O}(n \ln n) $$

Transient Non-Equilibrium Regime ( $0 \le S < S_{\text{max}}$ ): The non-zero entropy gradient ( $\frac{dS}{dt} > 0$ ) defines a macroscopic arrow of time. This transient window is where directional dynamics, information flow, and macro-structures exist.
Equilibrium Domain ( $S \approx S_{\text{max}}$ ): Once $S$ saturates, macro-evolution ceases. The system enters a stationary state with time-symmetric noise, effectively stopping directional macro-time.

Emergent Micro-Structures

An emergent micro-structure is defined as a specific sub-string pattern of length $m \le n$ (e.g., 10110) embedded within the $n$-bit configuration.

The probability $P(\text{pattern}, t)$ typically follows a characteristic hump-shaped curve, as described in Entropy-and-Emergent-Structures.

In short:

$t = 0$ (Zero Entropy): $P(\text{pattern}, 0) = 0$ for any pattern requiring a mix of $1$s and $0$s.
Transient Growth: As bit flips diffuse through the hypercube, local clusters assemble into the target sub-string, causing $P(\text{pattern}, t)$ to peak.
Thermal Decay & Residual: As mixing completes, position-specific ordering is thermalized, and the probability asymptotically approaches its uniform baseline.

6. Local Time Dilation & Coarse-Grained Lapse Rate

Consider a coherent emergent micro-structure spanning $w$ bits.

An observer internal to this structure cannot register single bit-flip events occurring outside or faster than the structure's own internal refresh cycle. The structure is blind to updates at scales finer than $w$ raw updates.

Local Clock and Lapse Equation

Defining the fundamental Planck-level update step as $t_P$, the local clock period $\Delta t_w$ and effective lapse rate $N_w$ for a $w$-bit structure are given by:

$$ \Delta t_w = w \cdot t_P, \qquad N_w = \frac{1}{w} \quad \text{(lapse)} $$

As structural complexity $w$ grows, internal local time ticks slower relative to the fundamental microscopic clock ($N_w < 1$).

Matter Formation & Spatial Contraction

Emergent micro-structures are not external bits added into the system; they are formed by binding existing bits from the background fabric ${0, 1}^n$.

When $w$ microscopic bits lock into a single coherent micro-structure (representing matter or bound state):

Unbound Vacuum Capacity: $w$ independent bits provide $w$ spatial degrees of freedom ($2^w$ states).
Bound Structure: The $w$ bits act as 1 single composite degree of freedom to an internal observer.
Spatial Resolution Contraction: The available free background degrees of freedom drop by:

$$ \Delta \text{Bits} = w - 1 $$

Consider $n$ bits, all initially free fabric, giving resolution $n$. When one composite structure of width $w$ bits emerges, $w$ fabric tokens are consumed and one matter entity is created. The new resolution is:

$$ (n - w) + 1 = n - (w - 1). $$

Resolution decreases by $w - 1$ for every composite entity formed. Generalising, if matter structures collectively consume $m(t)$ bits and produce $k(t)$ composite entities, the total resolution at time $t$ is:

$$ \text{Resolution}(t) = \bigl(n - m(t)\bigr) + k(t). $$

Physical Analogy to Gravitational Geometry

Because matter formation consumes $w$ microscopic bits into $1$ collective entity, the surrounding background spatial resolution contracts by $w - 1$ bits. To an internal observer using these micro-structures as measuring tools, space undergoes a local metric contraction, offering a purely information-theoretic foundation for gravity and relativistic spacetime curvature.

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