Physics as a Stateless Computer

Consider the unfolding of physics as a hypothetical computer with complete instantaneous access to the present state of the universe, denoted \(S_t\). This computer possesses full knowledge of all particles, fields, and governing dynamics \(f\), but lacks any persistent memory of prior configurations \(S_{t-1}\). It functions as a stateless computational engine, applying physical laws directly to the current state to produce the next:

\[ S_{t+1} = f(S_t) \]

Crucially, this computer cannot reconstruct \(S_{t-1}\) from \(S_t\) unless \(f\) is bijective. In the general case, where \(f\) is non-injective (i.e., many-to-one), the mapping is irreversible:

\[ \exists S_{t-1}^{(1)} \neq S_{t-1}^{(2)} : f(S_{t-1}^{(1)}) = f(S_{t-1}^{(2)}) = S_t \]

This renders the inverse computation ill-defined, as the computer has no mechanism to distinguish or automatically select the correct precursor among multiple plausible candidates to the current state.

Irreversibility Across Scales

Irreversible dynamics manifest at multiple physical scales:

• Macroscopic systems: Diverse causal histories may lead to identical macrostates. Human decision-making, for instance, reflects this degeneracy: the same belief or action may result from numerous experiential trajectories.
• Neural systems: A neuron's spike is a discrete, lossy transformation of high-dimensional continuous input. Post-spike, the fine-grained synaptic configuration that triggered it becomes unrecoverable.
• Quantum systems: While unitary evolution is time-reversible, measurement collapses the wavefunction, destroying phase coherence and precluding full recovery of the prior superposition.

In each case, the forward transition \(f\) acts as a lossy compression operator; its inverse lacks a unique solution without external memory or auxiliary constraints.

Compression of Futures into a Single Present

What distinguishes the past from the future under this model is the asymmetry of representational compression. While the past may consist of many possible prior states that converge to the current one (i.e., many-to-one), the future evolves deterministically (or probabilistically, but still computably) into a single next state. In effect, forward transitions squeeze the plurality of potential futures into a unique present, collapsing uncertainty into actuality. This directional compression ensures that:

$$ \exists \, S_{t-1}^{(1)} \neq S_{t-1}^{(2)} : f(S_{t-1}^{(1)}) = f(S_{t-1}^{(2)}) = S_t, \quad \text{but} \quad f(S_t) \rightarrow S_{t+1} \text{ is unique} $$

The future is funneled into one consistent present, while the past branches remain computationally inaccessible without auxiliary constraints. This asymmetry, one-to-many in reverse but one-to-one (or deterministic/stochastic but computable) in forward time, explains why only forward simulation is feasible for the stateless, unbounded computer.

The Computational Arrow of Time

In a universe governed exclusively by reversible dynamics, temporal directionality would be arbitrary: both forward and backward evolution would be computationally equivalent. In such a system, the computer could compute:

\[ S_{t+1} = f(S_t) \quad \text{and} \quad S_{t-1} = f^{-1}(S_t) \]

However, with the ubiquity of irreversible transitions, only the forward direction admits a well-defined, computable update. The specific irreversible phenomena responsible for this asymmetry is quantum measurement. Because of the information loss inherent in measurement, when constrained to a single universe, the backward direction is fundamentally ill-posed: it entails inference over an unbounded, often unenumerable, space of past configurations. The other irreversible processes, while significant, are secondary to this fundamental asymmetry, as technically, those other processes could be reversed with perfect knowledge of microstates.

Hence, the arrow of time is not imposed externally by physical law, but internally selected by the computer according to computational feasibility.

Conclusion: Computability as Temporal Grounding

Temporal directionality, in this formulation, is a derived property of a system's informational structure and the computer’s computational constraints. For an unbounded computer devoid of memory, one limited to present-state evaluation of the universe, the direction of time corresponds to the axis along which simulation remains well-posed. The universe does not inherently flow toward the future; it is simply the only direction that remains accessible to forward modeling. This principle can be formalized as:

\[ \text{Time direction} = \arg\min_{d \in \{\text{forward}, \text{backward}\}} \text{Complexity}(\text{Compute}(S_d)) \]

This perspective suggests a unification of temporal asymmetry with complexity theory, where the apparent irreversibility of the world is not merely thermodynamic but computational in essence. It invites further inquiry into how informational constraints shape our most fundamental intuitions about time.