Cosmological Tree Theorem

• Cosmological Tree Theorem is a dual concept that unifies unique atomic factorizations in the look-and-say map with loop-to-tree reductions in cosmological perturbation theory.
• In automata theory, the theorem shows that after a fixed number of iterations, any sequence decomposes uniquely into a finite set of stabilized atoms using finite-state machines.
• In cosmology, causal cuts and retarded propagators allow rewriting loop-level Feynman–Witten diagrams as phase-space integrals over simpler, tree-level diagrams.

The expression Cosmological Tree Theorem occurs in two distinct technical literatures. In automata theory and the study of the look-and-say map, it denotes Conway’s statement that sufficiently deep iterates of an audioactive sequence split uniquely into a fixed finite set of atomic constituents; a 2024 paper gives a finite-state, automata-theoretic proof of this result (Lairez et al., 2024). In cosmological perturbation theory, it denotes a set of cutting rules for the late-time wavefunctional that rewrites loop-level Feynman–Witten diagrams as phase-space integrals of tree diagrams, yielding a cosmological analogue of Feynman’s tree theorem (Agui-Salcedo et al., 2023). The shared terminology reflects a common tree decomposition motif, but the underlying objects, methods, and applications are otherwise different.

1. Terminological scope and disciplinary setting

In the Conway usage, the relevant dynamical system is the look-and-say, or audioactive, map on finite words of positive integers. The theorem is framed in terms of atoms, splittings, and a directed acyclic graph on the eventual atomic species. The 2024 exposition explicitly presents a “self-contained, automata–theoretic exposition of Conway’s Cosmological Tree Theorem,” emphasizing finite transducers, determinization, minimization, and stabilization phenomena at finite depth (Lairez et al., 2024).

In the cosmology usage, the theorem is formulated for the late-time wavefunctional

$\Psi[\phi] = \langle \phi |\, U \,| \Omega \rangle$

expanded in coefficient functions $\psi_{k_1\cdots k_n}$ computed by Feynman–Witten rules. Here the theorem is not about symbolic dynamics, but about perturbative representations of cosmological correlators: causality and unitarity imply that loop contributions can be opened into trees by cutting internal lines and replacing bulk-to-bulk propagators by bulk-to-boundary data and power spectra (Agui-Salcedo et al., 2023).

2. Conway’s theorem for audioactive sequences

Let $\mathbb{N}_+ = \{1,2,3,\ldots\}$, and let $x \in \mathbb{N}_+^*$ be a finite word. The look-and-say map $C(x)$ is obtained by reading off maximal runs of identical symbols. The standard example is

$x = 2222 \quad \Rightarrow \quad C(x)=42,$

read as “four 2’s” (Lairez et al., 2024).

The theorem is most cleanly stated after the day-one reduction. Define

$$\text{day-one} = \{x \in A^* \mid x \text{ has no four-in-a-row of any symbol}\},$$

and, because after the first step any symbol $\ge 4$ never appears in runs of length $\ge 2$, collapse all symbols $\ge 4$ into a single letter $\psi_{k_1\cdots k_n}$0, so that

$\psi_{k_1\cdots k_n}$1

The resulting map $\psi_{k_1\cdots k_n}$2 is well defined by a finite transducer (Lairez et al., 2024).

An atom is a nonempty word $\psi_{k_1\cdots k_n}$3 that cannot be nontrivially split: it never happens that $\psi_{k_1\cdots k_n}$4 with

$\psi_{k_1\cdots k_n}$5

Conway proved that there are exactly ninety-two common atoms, together with two infinite transuranic families, for a total of $\psi_{k_1\cdots k_n}$6 fundamental elements

$\psi_{k_1\cdots k_n}$7

The theorem then states that there exists $\psi_{k_1\cdots k_n}$8—in fact $\psi_{k_1\cdots k_n}$9—such that for every $\mathbb{N}_+ = \{1,2,3,\ldots\}$0 without four-in-a-row, and for every $\mathbb{N}_+ = \{1,2,3,\ldots\}$1, the word $\mathbb{N}_+ = \{1,2,3,\ldots\}$2 splits uniquely into a concatenation of atoms drawn from the fixed set $\mathbb{N}_+ = \{1,2,3,\ldots\}$3 (Lairez et al., 2024).

The phrase Cosmological Tree refers to the subsequent decay structure of these atoms under one-step evolution. After sufficiently many iterations, arbitrary initial data no longer generate genuinely new species; they only rearrange and concatenate the stabilized atomic ones.

3. Automata-theoretic proof architecture

The 2024 proof is organized around explicit finite-state machines. For each $\mathbb{N}_+ = \{1,2,3,\ldots\}$4, there is a bounded-counter transducer

$\mathbb{N}_+ = \{1,2,3,\ldots\}$5

that handles runs of up to three $\mathbb{N}_+ = \{1,2,3,\ldots\}$6’s and outputs “$\mathbb{N}_+ = \{1,2,3,\ldots\}$7,” “$\mathbb{N}_+ = \{1,2,3,\ldots\}$8,” or “$\mathbb{N}_+ = \{1,2,3,\ldots\}$9.” Any fourth $x \in \mathbb{N}_+^*$0 in a row is excluded in day-one (Lairez et al., 2024).

The main one-step machine is the audioactive transducer

$x \in \mathbb{N}_+^*$1

which has 28 states and implements $x \in \mathbb{N}_+^*$2 on day-one. Its operation alternates between the parity classes $x \in \mathbb{N}_+^*$3 and $x \in \mathbb{N}_+^*$4, invoking the counters as macros and using $x \in \mathbb{N}_+^*$5-moves to shift reading phases (Lairez et al., 2024).

Three auxiliary transducers organize factorization:

• Multi inserts any number of markers $x \in \mathbb{N}_+^*$6 nondeterministically.
• Mark inserts exactly one marker.
• Scissors extracts the factor between two markers.

These are described as small 1–2-state transducers on $x \in \mathbb{N}_+^*$7 (Lairez et al., 2024).

To recognize legitimate splittings, the construction augments Audio to a transducer $x \in \mathbb{N}_+^*$8 on

$x \in \mathbb{N}_+^*$9

that forces any inserted $C(x)$0 to remain forever “between” two distinct final digits under all further iterations. The resulting splitting recognizer is

$C(x)$1

a 21-state DFA recognizing exactly the splittings $C(x)$2 over $C(x)$3 for which

$C(x)$4

A key stabilization occurs at $C(x)$5: the sequence of input languages of $C(x)$6 is nested and becomes constant there (Lairez et al., 2024).

The Atom recognizer is then obtained by collecting splittings of length $C(x)$7 using Mark and complementing, yielding a 26-state DFA that recognizes exactly the set of atoms in day-one (Lairez et al., 2024).

Composition is formalized by the theorem that if $C(x)$8 and $C(x)$9 are transducers, then

$x = 2222 \quad \Rightarrow \quad C(x)=42,$0

is again a transducer. From this, one builds iterates

$x = 2222 \quad \Rightarrow \quad C(x)=42,$1

and, by interleaving with Multi, Splitting, Scissors, and Atom, constructs

$x = 2222 \quad \Rightarrow \quad C(x)=42,$2

which nondeterministically extracts every atomic factor of an input word (Lairez et al., 2024).

The proof then uses determinization by subset construction and minimization by Brzozowski’s reversal–determinize–reverse–determinize procedure, or any standard Hopcroft–Ullman algorithm. Since minimal DFA’s are unique up to renaming, equivalence reduces to comparison of minimized state-transition tables (Lairez et al., 2024).

Two stabilization statements are decisive:

• At $x = 2222 \quad \Rightarrow \quad C(x)=42,$3,

$x = 2222 \quad \Rightarrow \quad C(x)=42,$4

• At $x = 2222 \quad \Rightarrow \quad C(x)=42,$5,

$x = 2222 \quad \Rightarrow \quad C(x)=42,$6

so the set of atoms $x = 2222 \quad \Rightarrow \quad C(x)=42,$7 that appear in outputs stabilizes with

$x = 2222 \quad \Rightarrow \quad C(x)=42,$8

4. The directed forest and the meaning of “tree”

Once the stabilized set $x = 2222 \quad \Rightarrow \quad C(x)=42,$9 is known, the directed graph

$$\text{day-one} = \{x \in A^* \mid x \text{ has no four-in-a-row of any symbol}\},$$0

is defined by

$$\text{day-one} = \{x \in A^* \mid x \text{ has no four-in-a-row of any symbol}\},$$1

The nodes are the $$\text{day-one} = \{x \in A^* \mid x \text{ has no four-in-a-row of any symbol}\},$$2 fundamental elements, and the edges encode one-step atomic decay transitions (Lairez et al., 2024).

The graph is a disjoint union of rooted trees with no cycles, hence a forest. The cited proof justifies this by the fact that each atom decays into smaller atoms, so directed cycles cannot occur. The parent-to-child relation in the forest records the atomic structure of one step of the look-and-say evolution (Lairez et al., 2024).

The proof of the theorem then proceeds in three steps. First, $$\text{day-one} = \{x \in A^* \mid x \text{ has no four-in-a-row of any symbol}\},$$3 generates exactly $$\text{day-one} = \{x \in A^* \mid x \text{ has no four-in-a-row of any symbol}\},$$4, and stabilization at $$\text{day-one} = \{x \in A^* \mid x \text{ has no four-in-a-row of any symbol}\},$$5–$$\text{day-one} = \{x \in A^* \mid x \text{ has no four-in-a-row of any symbol}\},$$6 gives $$\text{day-one} = \{x \in A^* \mid x \text{ has no four-in-a-row of any symbol}\},$$7 for all $$\text{day-one} = \{x \in A^* \mid x \text{ has no four-in-a-row of any symbol}\},$$8. Second, any $$\text{day-one} = \{x \in A^* \mid x \text{ has no four-in-a-row of any symbol}\},$$9 splits uniquely into atoms $\ge 4$0, so

$\ge 4$1

and each $\ge 4$2 further splits into atoms from $\ge 4$3. Third, the graph $\ge 4$4 captures these relations exactly, establishing that after $\ge 4$5 steps every audioactive sequence factors into the same finite forest of $\ge 4$6 atoms and thereafter evolves only by permuting and concatenating those atoms (Lairez et al., 2024).

A common misunderstanding is to identify this theorem with a statement about cosmological correlators. The shared word “cosmological” is historical terminology in Conway’s work; the mathematical content here is a theorem about symbolic dynamics, finite automata, and unique eventual factorization.

5. The theorem in cosmological perturbation theory

In inflationary and more general time-dependent backgrounds, the basic observable is the late-time wavefunctional,

$\ge 4$7

The coefficients $\ge 4$8 are computed by Feynman–Witten rules using bulk-to-boundary propagators $\ge 4$9 and bulk-to-bulk propagators $\ge 2$0. Any loop in a Feynman–Witten diagram corresponds to a closed chain of time integrals over $\ge 2$1’s (Agui-Salcedo et al., 2023).

The cosmological theorem states that, by exploiting the retarded propagator

$\ge 2$2

and the fact that it obeys no closed timelike loops, one can rewrite any $\ge 2$3-loop contribution to $\ge 2$4 as a sum of phase-space integrals over tree diagrams, with each term obtained by cutting one or more internal lines and replacing $\ge 2$5’s by $\ge 2$6’s and power spectra $\ge 2$7 (Agui-Salcedo et al., 2023).

For a single-loop diagram $\ge 2$8 with loop lines $\ge 2$9, the paper states

$\ge 4$0

Here $\ge 4$1 is the diagram with the lines in $\ge 4$2 cut open, the $\ge 4$3 are the resulting disconnected subgraphs, and $\ge 4$4 extracts $\ge 4$5 from the corresponding external leg. Since each cut reduces the number of loops by at least one, repeated application opens all loops (Agui-Salcedo et al., 2023).

In this formulation, the word tree refers to ordinary tree diagrams in perturbation theory rather than to a graph on symbolic atoms. The theorem is explicitly presented as the cosmological analogue of Feynman’s tree theorem for amplitudes.

6. Causality cuts, loop–tree expansion, and applications

The derivation begins with unitarity cuts. Unitarity of the free evolution, together with Hermitian analyticity of $\ge 4$6 and $\ge 4$7, yields Cutkosky-like rules for $\ge 4$8:

$\ge 4$9

These determine the discontinuity, i.e. the imaginary part, of the diagram (Agui-Salcedo et al., 2023).

The stronger step is the causality cut. Using

$\psi_{k_1\cdots k_n}$00

one finds

$\psi_{k_1\cdots k_n}$01

Because the left-hand side is a difference of retarded and advanced propagators, each proportional to step functions, the relation is expressible entirely in terms of $\psi_{k_1\cdots k_n}$02’s. The absence of closed $\psi_{k_1\cdots k_n}$03-loops gives the complete complex cutting relation, not just its imaginary part. Graphically, reversing the arrow on the retarded line and subtracting yields a sum of tree diagrams with that line cut (Agui-Salcedo et al., 2023).

The paper gives the general loop–tree formula

$\psi_{k_1\cdots k_n}$04

with

$\psi_{k_1\cdots k_n}$05

Repeated application removes all loops (Agui-Salcedo et al., 2023).

Two principal applications are emphasized.

First, the theorem leads to a cosmological KLN theorem for equal-time correlators. At one loop, the power spectrum is schematically

$\psi_{k_1\cdots k_n}$06

Applying the theorem to $\psi_{k_1\cdots k_n}$07 trades it for cuts of tree-level objects, and the would-be new branch cuts in the total energy cancel between “virtual” and “real” contributions. For massless fields, any integrand is scale-free, of the form $\psi_{k_1\cdots k_n}$08, and vanishes in dimensional regularization (Agui-Salcedo et al., 2023).

Second, the theorem bootstraps tree-level exchanges from contact diagrams. Causality cuts plus unitarity determine the discontinuity across each partial energy channel, leaving only an even function of the exchanged momentum unfixed. Imposing the Bunch–Davies condition—the absence of “folded” singularities when a partial energy tends to zero—fully fixes that even part. Consequently every exchange $\psi_{k_1\cdots k_n}$09 is algebraically reconstructed from contact $\psi_{k_1\cdots k_n}$10’s plus their discontinuities, with no further time integrals needed (Agui-Salcedo et al., 2023).

The worked examples make the procedure concrete. For a massless scalar with interaction

$\psi_{k_1\cdots k_n}$11

the contact three-point function is

$\psi_{k_1\cdots k_n}$12

and the $\psi_{k_1\cdots k_n}$13-channel exchange is reconstructed as

$\psi_{k_1\cdots k_n}$14

The one-loop “fish” example similarly gives

$\psi_{k_1\cdots k_n}$15

with no time integrals. The overall conclusion is that these rules remove all nested time integrals from perturbative cosmological correlators once the single-vertex contact diagrams are known (Agui-Salcedo et al., 2023).

7. Comparison, interpretation, and historical relation of the two usages

The two theorems share a name but not a domain. In the Conway setting, the primitive operation is the iteration of the look-and-say map on words, the core structures are atoms and finite transducers, and the final object is a forest on $\psi_{k_1\cdots k_n}$16 stabilized elements after $\psi_{k_1\cdots k_n}$17 steps (Lairez et al., 2024). In the cosmological setting, the primitive operation is the cutting of internal lines in Feynman–Witten diagrams, the core structures are propagators, discontinuities, power spectra, and causality constraints, and the final outcome is a systematic reduction of loop diagrams to trees (Agui-Salcedo et al., 2023).

This suggests that the common term tree is structural rather than substantive. In one case it names the directed acyclic decay graph of atomic species; in the other it names the tree-level diagrams that remain after loops are opened by cuts. The common term cosmological is likewise context-dependent: in Conway’s terminology it belongs to the look-and-say literature, whereas in quantum field theory it refers to cosmological correlators and the wavefunction of the Universe.

The modern literature reflects both traditions. The automata-theoretic exposition is explicitly framed as a new proof of Conway’s theorem using finite-state machines and computer-assisted composition and minimization (Lairez et al., 2024). The cosmological formulation is presented as a self-contained derivation with applications to exchange bootstrapping and loop reductions, and it explicitly cites Baumann and Pimentel’s “The Cosmological Tree Theorem” (Kritos et al., 2021). Taken together, these works show that the phrase Cosmological Tree Theorem now names two technically mature, but entirely different, results: one in symbolic and automata-theoretic dynamics, the other in the perturbative analysis of cosmological observables.