Read-backs

Throughout, a board is a finite multiset of natural numbers (repetitions allowed and counted, order irrelevant). For a board $s$ and a value $v$, $\#_v(s)$ denotes the multiplicity of $v$ in $s$, $\prod s$ the product of all entries of $s$ (with repetition), and $|s|$ the number of entries of $s$ (with repetition). For $x, p \in \mathbb{N}$, $v_p(x)$ denotes the exponent of $p$ in the prime factorization of $x$; this is $0$ whenever $p$ is not prime, and $0$ for every $p$ when $x = 0$ or $x = 1$.

Three auxiliary notions are used and are expanded inline below wherever they occur; they are recorded here once.

One move. $s \to t$ ("$t$ is obtained from $s$ by one move") means: there exist natural numbers $m$ and $n$ with

$$m > 1, \qquad n > 1, \qquad m \in s, \qquad n \in (s \setminus \{m\}),$$

where $s \setminus \{m\}$ is $s$ with exactly one copy of $m$ deleted, and such that

$$t \;=\; \{\gcd(m,n)\} \;\uplus\; \{\operatorname{lcm}(m,n) \,/\, \gcd(m,n)\} \;\uplus\; \big((s \setminus \{m\}) \setminus \{n\}\big).$$

That is: delete one copy of $m$ and one copy of $n$ from $s$ and insert the two values $\gcd(m,n)$ and $\operatorname{lcm}(m,n)/\gcd(m,n)$. The division is truncated natural-number division. Note that $m$ and $n$ are values: the condition $n \in s \setminus \{m\}$ permits $m = n$ when that value occurs at least twice in $s$, and it forces $s$ to contain at least two entries greater than $1$ (counted with multiplicity). The two inserted values are not required to exceed $1$. Every move preserves the number of entries: $|t| = |s|$.

Reachability. $s \rightsquigarrow t$ means $t$ is obtained from $s$ by a finite (possibly empty) chain of moves — the reflexive–transitive closure of $\to$. In particular $s \rightsquigarrow s$ always holds.

Terminal. $t$ is terminal means there is no board $u$ with $t \to u$. Since the target board in the definition of a move is produced by an explicit equation, this is the same as saying no admissible pair $m, n$ can be selected at all: $t$ does not contain two entries (with multiplicity) both greater than $1$.

Big part. $\operatorname{big}(t)$ is the sub-multiset of $t$ consisting of exactly those entries that are greater than $1$, with their multiplicities.

Exponent gcd. For $p \in \mathbb{N}$ and a board $s$, $G_p(s)$ is obtained by replacing each entry $x$ of $s$ by $v_p(x)$ and folding the resulting multiset of natural numbers with $\gcd$, starting from the value $0$; since $\gcd(0,a) = a$, this is the greatest common divisor of the numbers $v_p(x)$ over all entries $x$ of $s$, with $G_p(\emptyset) = 0$. Entries equal to $0$ or $1$ contribute the value $0$ and therefore do not change the result. If $p$ is not prime then $v_p(x) = 0$ for every $x$ and $G_p(s) = 0$ for every board $s$.

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IMO2026P1.imo2026_p1

Let $s$ be a board (explicit argument: a finite multiset of natural numbers) satisfying two hypotheses:

• $|s| = 2026$: the number of entries of $s$, counted with multiplicity, is exactly $2026$;
• every entry $x$ of $s$ satisfies $x > 1$.

Then the conjunction of the following two claims holds.

(A) Existence of a terminal descendant. There exists a board $t$ that is reachable from $s$ by a finite (possibly empty) chain of moves and is terminal, i.e. admits no move: $t$ contains no two entries (with multiplicity) both greater than $1$.

(B) A single value $M$ serving every terminal descendant. There exists a natural number $M$ such that $M > 1$ and such that, for every board $t$, if $t$ is reachable from $s$ and $t$ is terminal, then

$$\operatorname{big}(t) = \{M\},$$

i.e. the sub-multiset of entries of $t$ that exceed $1$ is exactly the one-element multiset $\{M\}$ — exactly one entry of $t$ is greater than $1$ (multiplicity one, so no repetition of $M$ above $1$), and that entry equals $M$; all other entries of $t$ are $0$ or $1$.

Which part is existence and which is uniqueness-style, and where the quantifiers sit. Claim (A) is a pure existence claim: $\exists t$. In claim (B) the existential $\exists M$ comes first, and the universal $\forall t$ is nested inside it: the order is

$$\exists M \in \mathbb{N}. \; \big( M > 1 \;\wedge\; \forall t. \; (s \rightsquigarrow t) \to \operatorname{terminal}(t) \to \operatorname{big}(t) = \{M\} \big).$$

So $M$ is chosen once, before $t$ is considered, and is therefore not allowed to depend on the terminal board $t$; it may depend only on $s$ (and on the hypotheses about $s$). The existence half of (B) is "$\exists M$, $M > 1$"; the uniqueness-style half is the inner universal statement, which says that all reachable terminal boards have the same big part, and that this common big part is a singleton. Had the quantifiers been ordered the other way — $\forall t, (s \rightsquigarrow t) \to \operatorname{terminal}(t) \to \exists M > 1, \operatorname{big}(t) = \{M\}$ — the claim would only be that each reachable terminal board has exactly one entry above $1$, with different terminal boards permitted to carry different such values. The order as written rules that out: one value $M$ must work for all of them. Note also that $M$ is introduced with $\exists$, not $\exists!$; no uniqueness of $M$ itself is written down, though claim (A) guarantees at least one reachable terminal board exists, so the inner universal is not vacuous.

Use of the cardinality hypothesis. The number $2026$ appears only in the hypothesis $|s| = 2026$. It occurs nowhere in the conclusion: neither (A) nor (B) mentions $|s|$, $|t|$, or any numeral. The hypothesis therefore restricts which boards the statement speaks about, but the asserted conclusion is word-for-word the same sentence about $s$ as it would be for a board of any other size; the statement conveys no information about how the conclusion depends on $2026$, and in particular does not assert that $2026$ is necessary, sufficient, or special.

Conditionality. The statement is conditional on the two hypotheses about the starting board $s$; it is not conditional on a move existing from $s$, and $s$ is not assumed reachable from anything. Inside (B), the claim about a board $t$ is conditional on $t$ being reachable from $s$ and terminal; for any $t$ failing either condition, that implication is vacuously true and (B) says nothing about such a $t$ — in particular nothing about reachable boards that are not terminal, and nothing about terminal boards not reachable from $s$.

Degenerate cases. The hypotheses exclude the empty board ($|s| = 2026 \ne 0$) and any board containing a $1$ or a $0$ (every entry must exceed $1$), so the empty board and the all-$1$s board fall outside the statement entirely. The two hypotheses are jointly satisfiable — for instance $s$ consisting of $2026$ copies of $2$ — so the statement is not vacuous. A board satisfying both hypotheses has at least two entries greater than $1$, so the defining condition of a move is met and such an $s$ is never itself terminal; the reflexive case $t = s$ of (B) therefore does not arise under these hypotheses, and no board for which "already terminal" and the hypotheses hold simultaneously exists. Boards of $2026$ entries containing a $0$ or a $1$ are excluded by the second hypothesis; boards of entries all greater than $1$ but of a size other than $2026$ are excluded by the first. Nothing is claimed about any excluded board.

Not asserted. No formula, construction, or characterization of $M$ is given — not in terms of the entries of $s$, their gcd, their lcm, their product, or anything else; only $M > 1$ is stated about it. No claim that $M$ is an entry of $s$, or divides or is divisible by anything. No claim about the non-big part of the reachable terminal boards: the number of $1$s, the presence or absence of $0$s, and $|t|$ are all unconstrained by (B), so (B) does not assert that reachable terminal boards are equal — only that their entries above $1$ coincide. No uniqueness of the terminal board $t$ in (A), no bound on the number of moves needed, no claim that every sequence of moves terminates, and no claim of confluence beyond what the shared value $M$ states. No claim that $M$ is unique as an $\exists!$, no claim about boards reachable from $s$ that are not terminal, and no claim about what happens for starting boards violating either hypothesis.