The Universe as a Compositional Sequence of States

Modeling the cosmos as a sequence of functional states. See the entire concept

1. Introduction

Traditionally, physics treats time as a fundamental parameter: events unfold within a pre-existing temporal framework. Here, we explore an alternative interpretation: We propose a model in which the universe is not composed of objects evolving in time, but of states related by irreversible functional transitions. Time is not a fundamental parameter but an emergent ordering of these transitions. Causality is defined by composability, and physical laws describe constraints on allowable state transformations.

• The universe is a sequence of nested states.
• Time emerges from the succession of states, rather than existing independently.

$$ f_{n+1} = T(f_n) $$

2. Axioms

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Axiom 1 — Functional Ontology

The universe is a compositional structure of functions. What exists fundamentally are compositions of transitions, not fixed states or objects. Formally:

$$ fi​:(Si​→Si+1​)↦(Si+1​→Si+2​) $$

Consequence

• “Things” are stabilized patterns of transitions
• States are interfaces, not ontological primitives

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Axiom 2 — Minimal Transition Duration

There exists a universal, nonzero lower bound $d\tau_min$​ on the duration of any irreducible physical transition.

The value of this bound is an empirical parameter to be constrained by observation, not fixed by the axioms themselves.

Consequence

• No instantaneous change
• No infinite rates
• Zeno paradoxes are impossible
• Time must be emergent and discrete at base

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Axiom 3 — Entropy as Physical Quantity

Every irreducible transition carries a minimum entropy ΔSmin​ (e.g. one bit).

Consequence

• Energy = entropy flow rate
• Irreversibility is fundamental
• Arrow of time is built in
• Decoherence is not optional

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Axiom 4 — Causality as Composability

Causality is the ordered composability of transitions; only composable transitions can influence each other.

Consequence

• Causal structure precedes geometry
• Locality is emergent
• Causal cones are inevitable

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Axiom 5 — Invariant Speed of Causality

The minimum transition duration implies a universal upper bound on causal composition, denoted c, which is invariant.

Consequence

• Lorentz symmetry is forced
• No preferred frame
• c cannot vary without breaking causality

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3. Conceptual Framework

We represent the universe as a sequence of nested states:

$$ f_0, \quad f_1 = T(f_0), \quad f_2 = T(f_1), \quad \dots $$

where

$$ T(f_n) $$

is a successor function generating the next state.

• Each state $(f_n)$ encapsulates the physical properties of the universe at step $(n)$, and a successor function $(T)$ generates the next state:
• Observables are functions of the state: $(\mathcal{O}_n = \mathcal{O}(f_n))$.
• Time is emergent, measured as the accumulation of internal transitions along worldlines.
• Each transition between states has a finite duration $(d\tau$), reflecting the minimal physically allowed evolution (e.g., energy-limited quantum transitions).
• The evolution of the universe is compositional: each state is fully defined by applying the successor function to the previous state(s), with no reference to an external clock.

Note: Church Numeral Analogy

We can represent sequential states and their compositions using concepts analogous to Church numerals in lambda calculus. Each state can be seen as a function of the previous state, and iteration over these functions mirrors the counting structure of Church numerals.

Disclaimer: this analogy is purely mathematical and conceptual. It is not a claim about metaphysics, consciousness, or the universe literally being a computer. The purpose is to illustrate formal compositional structure in state evolution, not to make ontological or physical assertions about reality.

$$ f_0 = {}, \quad f_1 = {f_0}, \quad f_2 = {f_0, f_1}, \quad \dots $$

• Each numeral is a composition of a function applied $(n)$ times.
• Likewise, each universe state is a compositional application of the successor function, naturally encoding history and potential branching.
• Each composition step is associated with a transition duration $(d\tau)$, emphasizing that even functional evolution is physically mediated.

4. Application Examples

4.1. Successor Function and CMB Cooling Example

We can test this framework with a coarse-grained quantitative cosmology example: the cooling of the Cosmic Microwave Background (CMB) after recombination.

1. Physical Model
   After recombination $((t_0 \sim 3.8 \times 10^5) years)$, the CMB temperature evolves as:

$$ T(t) \propto a(t)^{-1} $$

where $(a(t))$ is the scale factor. In a matter-dominated universe::

$$ a(t) \propto t^{2/3} \quad \implies \quad T(t) \propto t^{-2/3}. $$

2. Discrete State Steps
   We define discrete states corresponding to successor function applications:

$$ f_n = T_n = T(f_n) $$

and choose each state step $(n \to n+1)$ to correspond to a doubling of cosmic time:

$$ t_n = t_0 \cdot 2^n. $$

Then the temperature at state $(n)$ is:

$$ T_n = T_0 \cdot 2^{-2n/3}. $$

Each state transition is associated with a minimal transition duration $(d\tau)$, reflecting the time it takes for physical changes to occur in the system.

3. Initial Conditions

$$ T_0 = 3000,\text{K}, \quad t_0 = 3.8\times 10^5,\text{yr}. $$

4. Successor Function (Discrete Update)

$$ f_{n+1} = T(f_n) = f_n \cdot 2^{-2/3} \approx 0.63 f_n $$

• The factor $(0.63 \approx 2^{-2/3})$ comes from the matter-dominated temperature scaling, not an arbitrary number.
• Each step represents one compositional application of the successor function.
• Each application takes a finite duration $(d\tau)$.

5. First Few States

n	t_n (yr)	T_n (K)
0	3.80×10⁵	3000
1	7.60×10⁵	1890
2	1.52×10⁶	1190
3	3.04×10⁶	750
4	6.08×10⁶	470
5	1.22×10⁷	296
…	…	…
16	2.50×10¹⁰	1.9

Observation: after ~16 compositional steps, the predicted temperature approaches the observed CMB temperature $(T_{\rm CMB} \approx 2.725)$ K.

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4.2.. Emergent Time and Transition Duration

• Each transition $(f_n \to _{n+1})$ represents a physically meaningful change: any evolution of a quantum state into a distinguishable new state
• Transitions are not instantaneou; they take a finite duration $(d\tau_n)$, reflecting the minimal physically allowed evolutio consistent with quantum mechanics and causality.
• The accumulated transition durations along a worldline define operational proper time:

$$ \tau_{\rm worldline} = \sum_n d\tau_n $$

• This formalizes the idea that time is emergent; only composition and transition counting matter.
• In this view, time dilation arises naturally: systems moving relative to one another, or in different gravitational potentials, accumulate fewer or more internal transitions per global state, reproducing the predictions of relativity without invoking a fundamental time parameter.

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Key Insights

1. The universe evolves compositionally, via repeated application of a successor function $(T)$.
2. Church numeral analogy makes nested, history-dependent evolution intuitive.
3. Each transition has a finite duration $(d\tau)$, giving rise to emergent proper time.
4. Discrete functional evolution reproduces real cosmological observables, like the cooling of the CMB.
5. Time dilation and spacetime geometry naturally emerge from transition counting and compositional structure.

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References

1. Margolus, N., & Levitin, L. B. "The maximum speed of dynamical evolution." Physica D, 1998.
2. Peebles, P. J. E. Principles of Physical Cosmology, 1993.
3. Sorkin, R. "Causal sets: Discrete gravity," 2003.