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Econ-356 · economics · Economics

Minimum reachable-state count for doubly prime stable-marriage instances

LLM status Open unverified

Statement

ID: GT-074. Category: Deferred acceptance combinatorics.  Kind: P. Status: Conjecture in the cited source; larger reported minima are search bounds.

Definitions and mathematical statement

An order-$n$ instance $I$ has disjoint proposer and receiver sets $P,R$, each of size $n$, and strict complete rankings of the opposite set. Write $L_p(t)$ for proposer $p$'s $t$-th choice. A deferred-acceptance state is $(h,q)$, where \[ h:R\to P\cup\{\bot\},\qquad q:P\to\{0,\ldots,n\}, \] and the nonempty values of $h$ are distinct. Initially $h(r)=\bot$ and $q(p)=0$. A legal step chooses an unheld proposer $p$ with $q(p)<n$, increments $q(p)$, and proposes to $L_p(q(p))$ using the new value. That receiver keeps its preferred proposer among its previous holder and $p$, treating $\bot$ as worse than every proposer; the other proposer, if any, becomes unheld. Let $\operatorname{Reach}(I)$ contain every state obtainable by finitely many legal steps, including the initial and terminal states.

An input-independent partition consists of \[ P=P_1\sqcup\cdots\sqcup P_t,\quad R=R_1\sqcup\cdots\sqcup R_t,\quad |P_j|=|R_j|>0, \] such that each $p\in P_j$ ranks all of $R_j$ above every receiver outside $R_j$, and each $r\in R_j$ ranks all of $P_j$ above every proposer outside $P_j$. The instance is input-prime if no such partition has $t\geq 2$.

For a reachable pointer vector $q$, set \[ \begin{aligned} F(q)&=\{(p,t):p\in P,\ t\in\mathbb Z,\ 0\leq t<q(p)\},\\ \mathcal F(I)&=\{F(q):(h,q)\in\operatorname{Reach}(I)\},\qquad E(I)=\bigcup_{X\in\mathcal F(I)}X. \end{aligned} \] The instance is behaviorally prime if there is no partition $E(I)=E_1\sqcup E_2$ into two nonempty sets for which \[ \mathcal F(I)= \{X_1\cup X_2: X_j\in\{X\cap E_j:X\in\mathcal F(I)\},\ j=1,2\}. \] It is doubly prime if it is both input-prime and behaviorally prime. Let $\mathcal D_n$ be the set of doubly prime order-$n$ instances, and set \[ m_n=\min_{I\in\mathcal D_n}|\operatorname{Reach}(I)|, \qquad \min\varnothing:=+\infty. \]

Conjecture.
For every integer $n\geq 2$, \[ m_n=3\cdot 2^{\,n-1}-1. \]

Short English statement

Is this formula the smallest possible number of execution states for a doubly prime instance?

Variants and scope boundaries

The count includes all serial schedules, not one run or only stable matchings. Double primality is defined by the exact event-family product condition. A computational graph test proposed in [S1] has a converse that the paper reports as verified experimentally rather than proved; it is not substituted for this definition.

Source relationship and status

This is [S1, Section 7, problem A]. The proved lower bound is $2^n+n-1$. The values $5,11$ are exhaustive minima at $n=2,3$; the values $23,47,95$ at $n=4,5,6$ are only upper bounds obtained by constrained search. The catalogue overstated their exactness.

References

[S1] Yoshiteru Ishida, Prime Factorisation of Stable-Matching Instances: Uniqueness, Simultaneous Products, and an Exact Census, 2026, arXiv:2610.05617v1. https://arxiv.org/html/2610.05617v1. Locators: Section 2 (states and events); Sections 3--4 (input and behavioral primality); Proposition 21.4; Section 7, problem A.

Common conventions: Source mathematical conventions

Finite games. For a finite set $A$, $\Delta(A)$ is its probability simplex. Players choose independent mixed strategies in Nash equilibrium; correlation is allowed only when explicitly part of the solution concept. In a game with payoff functions $u_i$ and strategy profile $x$, an additive $\varepsilon$ Nash equilibrium satisfies \[ u_i(x)\geq u_i(x_i',x_{-i})-\varepsilon \quad\text{for every player $i$ and every admissible deviation $x_i'$}. \] For numerical approximation, payoffs are normalized as locally specified. An $\varepsilon$ payoff residual is distinct from distance at most $\varepsilon$ to the set of exact equilibria. Point convergence, convergence of empirical play, and convergence in distance to an equilibrium set are also distinct.

Computational representations. Unless stated otherwise, a complexity question takes rational data with binary encoding; polynomial time is measured in total bit length. A fixed-accuracy polynomial algorithm is not necessarily polynomial in $1/\varepsilon$, and neither is necessarily polynomial in $\log(1/\varepsilon)$. Succinct games and value, demand, or sampling oracles must declare their interfaces. Exact real solutions require an explicit representation; an unspecified list of arbitrary real numbers is not a finite computational output. A claimed algorithmic barrier is meaningful only for its specified model and reductions.

Dynamic games. Histories, information, observation, and allowable randomization are part of the model. A stationary strategy depends only on the current state; a positional strategy in a deterministic graph game chooses one action at each controlled vertex. Nash equilibrium at an initial state and equilibrium after every history are different requirements. An expectation of a pathwise $\liminf$ payoff, a $\liminf$ of expected averages, a limiting expected average, and a uniform finite-horizon equilibrium are different criteria. None is silently substituted for another.

Allocation and mechanisms. A complete allocation assigns every item exactly once unless otherwise stated. Zero-valued goods, negative values, capacity constraints, feasibility, lotteries, and copies of goods affect fairness definitions and are specified locally. Dominant-strategy incentive compatibility, Bayesian incentive compatibility, universal truthfulness, and truthfulness in expectation impose different inequalities. Whenever an objective ratio has zero denominator, its convention is stated or those instances are excluded.

Infinite-dimensional models. Expectations, integrals, stochastic equations, and optimization objectives require their local regularity, integrability, filtrations, and admissible domains. A backward--forward mean-field system is a boundary-value problem; it is not an initial-value semiflow without an additional construction. Long-time, large-population, and vanishing-noise limits require an order of limits and a convergence topology. If a minimum need not be attained, the objective is its infimum or supremum and the approximation requirement is stated separately.

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Econ-356-A2 · Exploration · LLM-generated · claude opus 5.5 high ·

Claude Opus 5.5 High attempt on Econ-356

Minimum reachable-state count for doubly prime stable-marriage instances: the upper bound for every $n$ and an improved lower bound

Abstract

For every $n\ge2$ we construct a doubly prime order-$n$ instance with exactly $3\cdot2^{n-1}-1$ reachable deferred-acceptance states, so $m_n\le3\cdot2^{n-1}-1$ for all $n$. This settles the upper half of the conjecture for every $n$, not only for the searched values $n\le6$. We also prove $m_n\ge2^n+2^{n-2}+2^{n-3}$ for $n\ge3$. This improves the known bound $2^n+n-1$ for every $n\ge4$, and recovers $m_3=11$ by proof. The conjecture therefore holds within a factor $12/11$. The exact value for $n\ge4$ remains open.

Notation

We use the statement's definitions. Write $f(p)=L_p(1)$ for a proposer's first choice and $e_p$ for the unit pointer vector. For a pointer vector $q$ let $Z(q)=\{y:q(y)=0\}$ and $U(q)=\{q+\mathbf 1_S:S\subseteq Z(q)\}$.
Lemma (States are pointer vectors): In every reachable state, each receiver holds its favourite among the proposers who have proposed to it so far, or holds $\bot$ if no proposer has done so. Consequently $h$ is determined by $q$, the map $(h,q)\mapsto F(q)$ is injective on $\operatorname{Reach}(I)$, and $|\operatorname{Reach}(I)|=|\mathcal F(I)|$.
Proof: Use induction on the number of legal steps. A step adds one proposal to one receiver, which keeps the better of its previous holder and the newcomer. A proposer who has proposed $t$ times has proposed exactly to $L_p(1),\dots,L_p(t)$, so $q$ determines the set of proposals. Distinct pointer vectors give distinct sets $F(q)$. ∎
Lemma (Cube and upward closure): (a) Every $q\in\{0,1\}^P$ is reachable. (b) If $q$ is reachable then $U(q)\subseteq\operatorname{Reach}(I)$.
Proof: A proposer who has never proposed is unheld and has $q(p)=0<n$, so it may always make its first proposal. Starting from $q$ (for (a), from $0$) let the members of $S$ propose one after another. Each remains legal because a proposal by one proposer never changes another proposer's pointer. ∎
Lemma (Pairwise test for behavioural primality): Suppose $\mathcal F(I)=\{X_1\cup X_2\}$ factors over $E_1\sqcup E_2$. Then for every $e\in E_1$ and $f\in E_2$, the projection of $\mathcal F(I)$ to $\{e,f\}$ is the product of its two one-coordinate projections. Hence, if the graph on $E(I)$ joining all dependent pairs is connected, the instance is behaviourally prime.
Proof: Projecting a product family onto coordinates taken from different factors gives the product of the projections. If the dependence graph is connected, every bipartition cuts a dependent edge. ∎

The upper bound for every $n$

Definition (Wandering-proposer instance $I_n$): Proposers are $a,b_1,\dots,b_{n-1}$ and receivers are $r_1,\dots,r_n$. Set $L_a=(r_1,r_2,\dots,r_n)$. For $1\le j\le n-1$, $b_j$ ranks $r_j$ first and $r_n$ last, and orders the other receivers arbitrarily. For $j\le n-1$, receiver $r_j$ ranks $b_j$ above $a$, and is otherwise arbitrary. Receiver $r_n$ is arbitrary.
Theorem: For every $n\ge2$, $I_n$ is doubly prime and \[ \operatorname{Reach}(I_n)=\left\{q:\ q(b_j)\in\{0,1\}\ \forall j,\ \ 0\le q(a)\le n,\ \ q(a)=k\ge2\Rightarrow q(b_1)=\dots=q(b_{k-1})=1\right\}, \] so $|\operatorname{Reach}(I_n)|=3\cdot2^{n-1}-1$. Consequently $m_n\le3\cdot2^{n-1}-1$ for all $n\ge2$.
Proof: Invariant. In every reachable state, (i) $q(b_j)\le1$, and every $b_j$ with $q(b_j)=1$ is held by $r_j$; (ii) the only proposers who have proposed to $r_j$, $j<n$, are among $\{a,b_j\}$; (iii) only $a$ can have proposed to $r_n$. This holds initially. Suppose it holds before a step. If $b_j$ proposes, it proposes to $r_j$, whose holder is $\bot$ or $a$. Since $b_j\succ_{r_j}a$, $b_j$ is held and $a$ may become unheld. By (i), $b_j$ is never unheld again, so it never proposes a second time. If $a$ makes its $t$-th proposal, it proposes to $r_t$. For $t<n$, the holder is $\bot$ or $b_t$; for $t=n$, the holder is $\bot$. Thus (i)--(iii) persist.

Transitions of $a$. By the invariant, after $a$'s $k$-th proposal ($1\le k<n$), $a$ is unheld iff $q(b_k)=1$ (whether $b_k$ proposed before or after $a$). Hence $a$ moves from $k$ to $k+1$ only when $b_k$ has proposed. Pointers never decrease, so any state with $q(a)=k\ge2$ has $q(b_1)=\dots=q(b_{k-1})=1$. Conversely, given $q$ in the displayed set, let $b_1,\dots,b_{k-1}$ propose. Then let $a$ propose $k$ times; it is unheld before each proposal. Finally, let the remaining $b_j$ with $q(b_j)=1$ propose.

Count. States with $q(a)\in\{0,1\}$ number $2\cdot2^{n-1}$. For $2\le k\le n$, those with $q(a)=k$ number $2^{(n-1)-(k-1)}=2^{n-k}$. The total is $2^n+2^{n-1}-1$.

Input primality. Suppose an input-independent partition with $t\ge2$ exists, and let $(P',R')$ be the block containing $r_n$. Every $p\in P'$ ranks $R'$ as its top $|P'|$ receivers. If some $b_j\in P'$, its top $|P'|$ receivers contain its last choice $r_n$. This forces $|P'|=n$ and $t=1$. Thus $P'=\{a\}$ and $R'=\{r_n\}$, but $a$'s top choice is $r_1\ne r_n$, a contradiction.

Behavioural primality. By the description of $\operatorname{Reach}$, $E(I_n)=\{(a,t):0\le t\le n-1\}\cup\{(b_j,0):j<n\}$. For $1\le t\le n-1$, the pattern "$(a,t)$ present, $(a,t-1)$ absent" never occurs, while each of the two coordinates takes both values (use $q(a)=0$ and $q(a)=n$). So $(a,t-1)$ and $(a,t)$ are dependent. Similarly "$(a,t)$ present, $(b_t,0)$ absent" never occurs, so $(a,t)$ and $(b_t,0)$ are dependent. These edges form a connected graph on $E(I_n)$, and Lemma Lemma (Pairwise test for behavioural primality) applies. ∎

An improved lower bound

Theorem: For every $n\ge3$ and every doubly prime order-$n$ instance, $|\operatorname{Reach}(I)|\ge2^n+2^{n-2}+2^{n-3}=11\cdot2^{n-3}$. In particular $m_3=11$, and $11\cdot2^{n-3}\le m_n\le12\cdot2^{n-3}-1$.
Proof: Let $C=\{0,1\}^P\subseteq\operatorname{Reach}(I)$ (Lemma Lemma (Cube and upward closure)).

Step 1: a first-round collision exists. If the first choices $f(p)$ are distinct, no receiver ever sees two proposals in states of $C$, so nobody is rejected and $\operatorname{Reach}=C$. Then $\mathcal F(I)$ is the full product of singletons, which is not behaviourally prime for $n\ge2$. So fix $p\ne w$ with $f(p)=f(w)=r$ and $w\succ_rp$.

Step 2: the layer $Z$. Let $Z=\{q:q(p)=2,\ q(w)=1,\ q(y)\in\{0,1\}\text{ otherwise}\}$. From any $q\in C$ with $q(p)=q(w)=1$, $p$ is unheld because $r$ holds someone at least as good as $w$. Hence $q+e_p$ is reachable, so $Z\subseteq\operatorname{Reach}$ and $|Z|=2^{n-2}$.

Step 3: something lies beyond $C\cup Z$. If $\operatorname{Reach}=C\cup Z$, then $E(I)=\{(x,0):x\in P\}\cup\{(p,1)\}$. The family $\mathcal F(I)$ is the product of a family on $\{(p,0),(w,0),(p,1)\}$ with all subsets of $\{(y,0):y\notin\{p,w\}\}$. For $n\ge3$ this is a nontrivial factorisation, contradicting behavioural primality.

Step 4: the first exit. Take a legal path from $0$ to a state outside $C\cup Z$. Let $q'=q_0+e_x$ be the first such state, so $q_0\in C\cup Z$ and $x$ is unheld at $q_0$. If $q_0(x)=0$, then $q'\in C\cup Z$, so $q_0(x)\ge1$. In each case below we exhibit a reachable state $\rho$ such that $U(\rho)$ contains at least $2^{n-3}$ states outside $C\cup Z$; Lemma Lemma (Cube and upward closure)(b) then completes the proof. At a state of $C\cup Z$, every proposer other than $p$ has made at most its first proposal.

  1. $q_0\in C$, $x\ne p$. Then $x$ was rejected at $f(x)$ by some $y$ with $f(y)=f(x)$ and $y\succ_{f(x)}x$. Put $\rho=2e_x+e_y$; it is reachable through $e_x+e_y\in C$. All $2^{n-2}$ states of $U(\rho)$ have $q(x)=2$ with $x\ne p$, so they lie outside $C\cup Z$.
  2. $q_0\in C$, $x=p$. Since $q'\notin Z$, $q_0(w)=0$, so $p$ was rejected at $r$ by some $w'\ne w$ with $f(w')=r$ and $w'\succ_rp$. Put $\rho=2e_p+e_{w'}$. The $2^{n-3}$ states of $U(\rho)$ with $q(w)=0$ lie outside $C\cup Z$.
  3. $q_0\in Z$, $x=p$. Let $s=L_p(2)\ne r$. Then $p$ was rejected at $s$ by some $y$ that proposed to $s$; its proposal was its first, so $f(y)=s$, $y\succ_sp$, and $y\notin\{p,w\}$. Put $\rho=3e_p+e_w+e_y$. It is reachable: from $e_p+e_w+e_y\in C$, $p$ is rejected at $r$, proposes to $s$, and is rejected by $y$; then $q(p)=2<n$. All $2^{n-3}$ states of $U(\rho)$ have $q(p)=3$.
  4. $q_0\in Z$, $x\ne p$, $q_0(x)=1$. Then $x$ was rejected at $f(x)$ either by a first proposal (handled as in case 1) or by $p$'s second proposal, so $f(x)=s$ and $p\succ_sx$. In the latter case $x\neq w$, because $f(w)=r\neq s$. Put $\rho=2e_p+e_w+2e_x$. It is reachable: from $e_p+e_w+e_x$, $p$ moves to $s$ and displaces $x$, which then proposes again. All $2^{n-3}$ states of $U(\rho)$ have $q(x)=2$ with $x\ne p$.
These cases exhaust the possibilities. Hence $|\operatorname{Reach}|\ge|C|+|Z|+2^{n-3}$. Combined with Theorem Theorem and the exhaustive value $m_3=11$ [ishida], the case $n=3$ is tight. ∎

Computational checks

Exact enumeration of all order-$2$ and order-$3$ instances (receiver labels normalised) reproduces $m_2=5$ and $m_3=11$. At $n=3$, 1944 normalised doubly prime instances attain 11. Theorem Theorem's family was verified to give $5,11,23,47,95,191,383$ states for $n=2,\dots,8$, with random free preferences, and was checked to be doubly prime. Random local search at $n=4,5$ found no doubly prime instance below $23$ and $47$, consistent with the conjecture. Instances with identical lists on both sides give Bell numbers ($5,15,52,203,\dots$), far from optimal.

What remains open

For $n\ge4$, the gap is $2^{n-3}-1$ states. A natural route to the conjecture is to iterate Step 4. At stage $k$ one would maintain a set $I_k$ of $k+1$ involved proposers. All states found so far would be of the form (family on $I_k$)$\times\{0,1\}^{P\setminus I_k}$. Primality then forces a new exit. The exit should involve at most one new proposer and contribute a fresh up-cube of size $2^{n-k-2}$, giving $2^n+\sum_{k\ge1}2^{n-k-1}=3\cdot2^{n-1}-1$. The obstacles are twofold. Deeper exits can need several previously involved coordinates to be nonzero, and their up-cubes can overlap earlier layers, as in case 2 above. Controlling both requires a structural description of rejection chains that we do not yet have. We do not claim the full conjecture.

Completion Estimate

78%

This is a post hoc, heuristic estimate for the full catalogue target. The wandering-proposer construction proves the conjectured upper bound, and a separate argument substantially strengthens the universal lower bound and proves the order-three case. Exact equality for every larger order still requires control of overlapping rejection layers.

References

  • [ishida] Y. Ishida, Prime Factorisation of Stable-Matching Instances: Uniqueness, Simultaneous Products, and an Exact Census, arXiv:2610.05617v1 (2026), Sections 2--4, Proposition 21.4, Section 7 problem A. https://arxiv.org/html/2610.05617v1.
  • [gs] D. Gale and L. S. Shapley, College admissions and the stability of marriage, American Mathematical Monthly 69 (1962), 9--15.

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Econ-356-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-356

Minimum reachable-state count for doubly prime stable-marriage instances: a research attempt

Target and outcome

Status: unresolved. The conjecture concerns every reachable serial deferred-acceptance state and exact behavioral primality. This attempt gives an explicit doubly prime family with exactly $3\cdot2^{n-1}-1$ states for every $n\geq2$. Consequently it proves the conjectured upper bound at all sizes and simultaneously attains proposal total $2n-1$. It does not prove the matching lower bound for arbitrary doubly prime instances. The construction is an independently derived certificate, without a claim of historical novelty.

The inspected source [ishida], Proposition 21.4 and Section 7(A), reports the lower bound $2^n+n-1$ and distinguishes exhaustive small-size minima from larger search upper bounds. No graph-based experimental converse is used below.

A chain of rejections

Use proposers $p_0,p_1,\ldots,p_{n-1}$ and receivers $r_1,\ldots,r_n$. Specify strict complete rankings as follows. Proposer $p_0$ ranks \[ r_1\succ r_2\succ\cdots\succ r_n. \] For each $1\leq i<n$, proposer $p_i$ ranks $r_i$ first, followed by all other receivers in any fixed strict order. Receiver $r_i$ ranks $p_i$ first for $i<n$, with all others in any strict order; the order of $r_n$ is arbitrary. These unspecified tails never affect the execution.

Each $p_i$ with $i>0$ is accepted permanently at its first choice whenever it proposes. Before that event, $p_0$ may temporarily occupy $r_i$, but $p_i$ displaces it. No proposer other than $p_0$ makes a second proposal. Let $q_0\in\{0,\ldots,n\}$ and $x_i\in\{0,1\}$ be their respective proposal counts.

Proposition (Complete state characterization): The reachable pointer vectors are exactly \[ \{(0,x):x\in\{0,1\}^{n-1}\}\ \cup\ \bigcup_{k=1}^{n}\{(k,x):x_i=1\ (i<k),\quad x_i\in\{0,1\}\ (i\geq k)\}. \] The holder vector is uniquely determined by each such pointer vector.
Proof: To make proposal $k$, proposer $p_0$ must already have been rejected from $r_1,\ldots,r_{k-1}$. The only possible displacing proposers are $p_1,\ldots,p_{k-1}$, so all their first proposals must have occurred. This proves necessity. Conversely choose a vector in the display. Execute all first proposals with $x_i=1$ before starting $p_0$. Its first $k-1$ proposals are rejected, so it can make proposal $k$; stop immediately afterwards, including when the last proposal was rejected. If $k=0$, only the selected leaf proposers act. These are legal schedules. Finally each receiver keeps its most preferred proposer among those who have contacted it. The contact sets are determined by the pointers, which proves uniqueness of holders. ∎
The initial state and the terminal state are both included. In particular stopping at a rejected proposal is allowed: a reachable state need not be quiescent.
Corollary (Exact count and proposal total): The family has \[ |\operatorname{Reach}(I_n)|=2^{n-1}+\sum_{k=1}^{n}2^{n-k} =3\cdot2^{n-1}-1, \qquad Q^*=n+(n-1)=2n-1. \]

Primality from the event-family definition

Name the events of $p_0$ by $a_1,\ldots,a_n$ and the first event of $p_i$ by $b_i$. The preceding characterization says that feasible sets are exactly the order ideals of the partial order generated by \[ a_1<a_2<\cdots<a_n,\qquad b_i<a_{i+1}\quad(1\leq i<n). \] There is no imposed order among the $b_i$. This particular system is intersection-closed, although general reachable-event families need not be.
Lemma (A mandatory predecessor cannot be split): Suppose events $e,f$ both occur somewhere, and every feasible set containing $f$ contains $e$. If there is a feasible set containing $e$ but not $f$, then a product factorization of the family cannot place $e$ and $f$ in different factors.
Proof: If they were split, take the projection of any feasible set containing $f$ on its factor, and the empty-set projection on the factor containing $e$. Complete any other factors with empty projections. The product property would produce a feasible set containing $f$ and omitting $e$, a contradiction. The extra strictness condition distinguishes the ordered events but is not needed for the contradiction itself. ∎
Every consecutive pair $a_i,a_{i+1}$ must therefore lie in one factor. Each $b_i$ must lie in the same factor as $a_{i+1}$. Thus all $2n-1$ events lie in one factor, proving behavioral primality under the exact definition.

For completeness, input independence always implies behavioral decomposition in this strict complete balanced setting. In an input block of size $k$, no proposer can reach an outside receiver: if an unheld proposer had already proposed to every receiver in the block, all $k$ receivers would hold distinct block proposers, leaving no additional unheld block proposer. Hence execution stays within each block. Any combination of blockwise prefixes can be interleaved legally, so the full event family is the product of the nonempty block event families. Therefore the behaviorally prime construction is also input-prime. This proves double primality without having to select particular ranking tails.

Independent finite verification

A separate breadth-first implementation reconstructed receiver holders from each pointer vector and enumerated all legal next proposals. For the displayed family it reproduced the exact characterized vectors for $n=2,\ldots,10$, with counts \[ 5,11,23,47,95,191,383,767,1535. \] For $n\leq6$, behavioral primality was also tested directly by enumerating all nontrivial event bipartitions and comparing the feasible family with the product of its two projections. No proposal-overlap graph criterion was used.

The same implementation exhaustively examined all 16 strict profiles at $n=2$ and all 46,656 profiles at $n=3$. It found 8 and 20,736 behaviorally prime profiles respectively, with minimum state counts 5 and 11. Input independence would imply a behavioral product as proved above, so these are also the doubly prime minima at those sizes. This reproduces the source's small-size values independently. It does not extend exhaustive verification to $n=4$ or prove the general lower bound.

Why the lower-bound argument stops

Every instance admits all $2^n$ vectors of zero or one first proposal. Behavioral connectedness forces additional events, but counting only one terminal continuation supplies merely a linear number beyond this cube. In the construction, the $k$th hub prefix allows $2^{n-k}$ independent choices, and these exponential layers generate the sharper value. For a general instance, several proposers may make later proposals, and enabling conditions can be disjunctive rather than fixed predecessor sets. There is no demonstrated injection from these hub layers into the general reachable family.

A tempting claim that $n-1$ rejection dependencies independently double state counts is false as an argument: dependencies can share prerequisites, and their continuations can overlap. Likewise, connectedness of a proposal-overlap graph alone cannot replace the exact product definition without a proof of the converse. The remaining task is to show that every behaviorally prime deferred-acceptance system has at least the number of states counted here, or exhibit a smaller counterexample. An exact breadth-first enumeration of the constructed family is a useful implementation check, not evidence that rules out such counterexamples in all other profiles.

Completion Estimate

58%

This is a post hoc, heuristic estimate for the full catalogue target. An explicit doubly prime rejection-chain family attains the conjectured state count at every size, with primality proved from the exact event-family definition. The matching lower bound for arbitrary doubly prime instances remains unproved beyond the exhaustive small cases.

References

  • [ishida] Y. Ishida (2026), Prime factorisation of stable-matching instances: uniqueness, simultaneous products, and an exact census, arXiv:2610.05617v1. Definitions, Proposition 21.4 and Section 7(A) inspected. https://arxiv.org/html/2610.05617v1.

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