Collins’ Conjecture Overview

• Collins’ Conjecture is a multi-faceted topic referring to distinct challenges in graph theory, matrix positivity, and quantum chemical correlation energy.
• In graph theory, it proposes that the canonical half-partition labeling distinguishes cyclic tournaments, with proofs for broad families including all Paley tournaments.
• In matrix analysis and quantum chemistry, it connects trace positivity and electronic correlation with entropy measures, supported by numerical and ab initio evidence yet not universally proven.

Searching arXiv for recent and directly relevant sources on “Collins’ Conjecture,” including the graph-theoretic Albertson–Collins conjecture, the Collins–Dykema–Torres-Ayala positivity variant related to BMV, and ab initio/electronic-structure formulations. “Collins’ Conjecture” is not a single universally fixed statement. In the arXiv literature represented here, the name designates multiple distinct conjectural programs associated with Collins: a graph-theoretic conjecture on canonical symmetry-breaking labelings of cyclic tournaments; a matrix-polynomial positivity conjecture of Collins, Dykema, and Torres-Ayala related to the Bessis–Moussa–Villani program; and a quantum-chemical conjecture linking electronic correlation energy to an entropy of occupation numbers or of the one-electron reduced density matrix. A precise treatment therefore requires disambiguation by domain, statement, and status (Meslem et al., 2016, Kim et al., 2021, Site, 2014, Zamani et al., 7 Apr 2025).

1. Nomenclature and domain-specific meanings

The most common ambiguity arises because the surname “Collins” appears in several unrelated conjectures. In graph theory, the relevant object is the Albertson–Collins conjecture for cyclic tournaments. In noncommutative and matrix positivity, the relevant statement is the Collins–Dykema–Torres-Ayala variant of the Lieb–Seiringer formulation of BMV. In electronic-structure theory, Collins’ conjecture refers to a proposed proportionality between correlation energy and an entropy built from occupation numbers or natural-orbital occupations. A separate arXiv record on the Collatz problem explicitly notes that confusing “Collins’ Conjecture” with “Collatz Conjecture” is a typo rather than a substantive identification (Wey, 2023).

Domain	Statement	Status in the cited sources
Cyclic tournaments	The canonical 2-labeling $\lambda^*(i)=1$ for $i\le q$, $\lambda^*(i)=2$ for $i>q$, is distinguishing for every cyclic tournament $T(2q+1;S)$	Proved for several broad families, including all Paley tournaments; open in full generality (Meslem et al., 2016)
Mixed trace coefficients	If $m$ and $r$ are even, then the coefficient $c_{m,r}(A,B)$ of $t^r$ in $\mathrm{Tr}[(A+tB)^m]$ is nonnegative for all symmetric $i\le q$0	Proved for $i\le q$1 for all $i\le q$2, and for $i\le q$3 when $i\le q$4 (Kim et al., 2021)
Electronic correlation	$i\le q$5, or in later form $i\le q$6	Supported by numerical and ab initio evidence; not established as a universal theorem (Site, 2014, Zamani et al., 7 Apr 2025)

A common misconception is that these are reformulations of one another. They are not. The three uses share only the eponym and, in two cases, an emphasis on symmetry or information, but they belong to different mathematical and physical frameworks.

2. The Albertson–Collins conjecture for cyclic tournaments

A tournament $i\le q$7 is a directed graph on a finite vertex set $i\le q$8 in which, for every unordered pair $i\le q$9 with $\lambda^*(i)=2$0, exactly one of the arcs $\lambda^*(i)=2$1 or $\lambda^*(i)=2$2 is present. For odd order $\lambda^*(i)=2$3, a cyclic tournament $\lambda^*(i)=2$4 is defined on $\lambda^*(i)=2$5 by choosing $\lambda^*(i)=2$6 such that for every nonzero residue $\lambda^*(i)=2$7, exactly one of $\lambda^*(i)=2$8 or $\lambda^*(i)=2$9 lies in $i>q$0, and orienting $i>q$1 iff $i>q$2. Every such cyclic tournament is regular: each vertex has indegree and outdegree $i>q$3 (Meslem et al., 2016).

The conjectural issue is not whether cyclic tournaments are 2-distinguishable, but whether they admit a particular canonical 2-distinguishing labeling. A distinguishing $i>q$4-labeling is a labeling $i>q$5 such that no nontrivial automorphism preserves all labels; the distinguishing number $i>q$6 is the least such $i>q$7. By Gluck’s theorem on regular subsets for permutation groups of odd order, every cyclic tournament $i>q$8 satisfies $i>q$9. Albertson and Collins proposed that the canonical half-partition labeling

$T(2q+1;S)$0

is always distinguishing. Equivalently, $T(2q+1;S)$1 labels the subtournament $T(2q+1;S)$2 induced by $T(2q+1;S)$3 with label 1 and the subtournament $T(2q+1;S)$4 induced by $T(2q+1;S)$5 with label 2 (Meslem et al., 2016).

The significance of the conjecture is structural rather than existential. Gluck’s theorem already yields some distinguishing 2-labeling, but the conjecture asks whether the same canonical labeling works uniformly for all cyclic tournaments, independent of the detailed automorphism group. This reframes 2-distinguishability as a canonical symmetry-breaking problem in a highly regular class of digraphs.

3. Structural criteria, proved families, and the Paley case

The 2016 analysis develops a rigidity-based program. Its central sufficient condition is Proposition 8: if either half-subtournament $T(2q+1;S)$6 or $T(2q+1;S)$7 is rigid, then the canonical labeling $T(2q+1;S)$8 is distinguishing. The proof uses the fact that any $T(2q+1;S)$9-preserving automorphism restricts to automorphisms of the two halves; if one restriction is the identity, connector arithmetic and “agreement on fixed vertices” force the other restriction to be trivial as well. Proposition 10 complements this by showing that a nontrivial $m$0-preserving automorphism must have at least two orbits within each half, excluding single-orbit behavior and supplying many fixed-point arguments (Meslem et al., 2016).

These ideas yield several broad positive classes. The conjecture holds whenever one half is rigid on each indegree class $m$1 as in Theorem 24, and in particular when $m$2 as in Theorem 25. It also holds whenever $m$3 as in Theorem 26, hence also when $m$4 by Corollary 27. Theorem 31 proves the conjecture when $m$5 is an interval, and Corollary 32 extends this to the complementary union form $m$6. Lemma 22 handles additional cases determined by automorphism-group size and half-automorphism constraints (Meslem et al., 2016).

A particularly important family is furnished by Paley tournaments. For a prime $m$7, the Paley tournament $m$8 is defined by quadratic residues in $m$9, with automorphism group

$r$0

of size $r$1. Theorem 33 proves that every such Paley tournament satisfies the Albertson–Collins conjecture. The argument is affine and orbit-theoretic: any nontrivial $r$2-preserving automorphism would force incompatible agreement patterns on vertices inside the two halves, contradicting the residue structure of $r$3 (Meslem et al., 2016).

The conjecture remains unresolved in full generality. No counterexamples are known in the cited source. The open problem is therefore not whether $r$4, which is settled, but whether the canonical half-labeling is universally sufficient.

4. The Collins–Dykema–Torres-Ayala positivity conjecture

A different “Collins’ Conjecture” appears in the BMV orbit. The original Bessis–Moussa–Villani conjecture states that for Hermitian $r$5 and positive semidefinite $r$6, the function $r$7 is the Laplace transform of a positive measure on $r$8. An equivalent Lieb–Seiringer formulation says that for positive semidefinite $r$9, every coefficient $c_{m,r}(A,B)$0 in

$c_{m,r}(A,B)$1

is nonnegative. Collins, Dykema, and Torres-Ayala proposed a variant with no positive-semidefinite assumption: if $c_{m,r}(A,B)$2 and $c_{m,r}(A,B)$3 are even, then $c_{m,r}(A,B)$4 for all symmetric $c_{m,r}(A,B)$5. The 2021 paper stresses that this variant is neither stronger nor weaker than Lieb–Seiringer: it drops the psd hypothesis but restricts to even $c_{m,r}(A,B)$6 (Kim et al., 2021).

The technical setting is a transition from noncommutative to commutative sum-of-squares certification. In Hägele’s approach one studies the noncommutative word sum $c_{m,r}(A,B)$7, and trace positivity follows if $c_{m,r}(A,B)$8 is cyclically equivalent to a sum of Hermitian squares. Collins et al. had already shown that $c_{m,r}(A,B)$9 and $t^r$0 are cyclically equivalent to sums of Hermitian squares, while $t^r$1 is not. The 2021 contribution bypasses this obstruction by certifying nonnegativity directly for the scalar polynomial $t^r$2 in the commutative matrix-entry variables $t^r$3 (Kim et al., 2021).

Two explicit positive results are established. For $t^r$4, the paper proves for all $t^r$5 that

$t^r$6

with $t^r$7, yielding a commutative SoS certificate. For $t^r$8, it proves nonnegativity for $t^r$9 through

$\mathrm{Tr}[(A+tB)^m]$0

with $\mathrm{Tr}[(A+tB)^m]$1 and $\mathrm{Tr}[(A+tB)^m]$2 constrained by explicit linear coefficient-matching equations in parameters $\mathrm{Tr}[(A+tB)^m]$3. These constructions rely on Gram-matrix methods, necklace enumeration, symmetry reduction, Schur complements, and semidefinite programming (Kim et al., 2021).

The status is therefore mixed. The even-parameter positivity conjecture is supported by exact SoS certificates in low cases, but the general case $\mathrm{Tr}[(A+tB)^m]$4 for all $\mathrm{Tr}[(A+tB)^m]$5, and larger even pairs $\mathrm{Tr}[(A+tB)^m]$6, remain open.

5. The entropy-based conjecture in many-electron theory

In quantum chemistry, Collins’ conjecture is a 1993 proposal that the electronic correlation energy is proportional to a Shannon-type entropy of occupation numbers: $\mathrm{Tr}[(A+tB)^m]$7 Here $\mathrm{Tr}[(A+tB)^m]$8 are spin-orbital occupation numbers with $\mathrm{Tr}[(A+tB)^m]$9 and $i\le q$00. The motivation is that increasing correlation makes occupations more fractional and more dispersed relative to the idempotent Hartree–Fock limit, so an entropy-like quantity should track correlation strength. Ziesche later reformulated the idea for the uniform electron gas in momentum space, using

$i\le q$01

and the cited account summarizes the interpretation as: “s measures, at least for the uniform electron gas, the correlation strength” (Site, 2014).

The same source develops a position-space extension motivated by Quantum Monte Carlo data. With unnormalized one-electron density $i\le q$02, it uses

$i\le q$03

related to the normalized entropy $i\le q$04 by $i\le q$05. For a nonuniform system, the proposed leading-order behavior is

$i\le q$06

$i\le q$07

and hence

$i\le q$08

For the uniform gas this reduces to $i\le q$09 in an intermediate-density regime (Site, 2014).

The empirical basis reported there comes from ground-state Reptation Quantum Monte Carlo for the uniform electron gas over densities $i\le q$10 to $i\le q$11 e/bohr$i\le q$12, corresponding in the paper’s own mapping to $i\le q$13 from $i\le q$14 to $i\le q$15 bohr. Over $i\le q$16–$i\le q$17 e/bohr$i\le q$18, both the kinetic and Coulomb correlation energies per particle, $i\le q$19 and $i\le q$20, exhibit a nearly linear dependence on $i\le q$21. At $i\le q$22 e/bohr$i\le q$23, those components deviate individually, but their sum $i\le q$24 remains approximately linear in $i\le q$25. No explicit fit coefficients or uncertainties are given (Site, 2014).

The same work is explicit about limitations. The functional $i\le q$26 does not satisfy the required uniform coordinate scaling behavior for an energy functional, and an entropy-exponential ansatz

$i\le q$27

is suggested as a scaling-consistent alternative whose Taylor expansion could recover the observed $i\le q$28 term at intermediate densities. The paper also notes open issues concerning representation dependence, extensivity, spin, temperature, low- and high-density limits, and the relation between Löwdin and Kohn–Sham correlation definitions (Site, 2014).

6. Ab initio realizations via Jaynes entropy and MP2 repartitioning

The 2025 paper “Towards ab initio Realizations of Collins’ Conjecture” reformulates the entropy-based idea in terms of the Jaynes entropy of the one-electron density matrix: $i\le q$29 where $i\le q$30 are natural-orbital occupation numbers. At the Hartree–Fock level, the density matrix is idempotent and $i\le q$31. Collins’ conjecture is then invoked as

$i\le q$32

with $i\le q$33 a system-dependent proportionality constant (Zamani et al., 7 Apr 2025).

The central technical move is an entropy-inspired repartitioning of the Møller–Plesset Hamiltonian. Writing

$i\le q$34

the paper shifts one-particle energies in $i\le q$35 by terms tied to subsets of MP2 pair-correlation contributions: $i\le q$36 The resulting dressed-amplitude equations retain MP2 tensor structure but replace bare denominators by dressed orbital energies $i\le q$37. The orbital-invariant $i\le q$38MP2 scheme is stated to be orbital-invariant, size-consistent, and size-extensive, and becomes equivalent to a mosaic-CCD truncation when $i\le q$39 (Zamani et al., 7 Apr 2025).

Parameter determination is tied to density quality rather than direct energetic fitting. At large separation, $i\le q$40 Å, the HOMO and LUMO natural-orbital occupations in each fragment are tuned toward $i\le q$41, enforcing a diradical-like dissociation limit and avoiding RMP2 pathologies such as negative occupations and a vanishing HOMO–LUMO gap. Once $i\le q$42 is fixed, the energy is corrected by an additional static-correlation term,

$i\le q$43

The preferred orbital-invariant scaling is $i\le q$44; a non-invariant formulation uses $i\le q$45 but produces non-smooth potential-energy surfaces (Zamani et al., 7 Apr 2025).

The reported results are specific. For LiH at $i\le q$46 Å, oi-RI-$i\le q$47MP2 gives $i\le q$48 and $i\le q$49, compared with CASSCF/CASCI values $i\le q$50, and dissociation entropies $i\le q$51–$i\le q$52. For bond dissociation energies, orbital-invariant $i\le q$53MP2$i\le q$54 yields mean absolute errors of approximately $i\le q$55 kcal/mol with respect to experiment and CCSD(T); standalone $i\le q$56MP2 gives roughly twice the error; unscaled $i\le q$57 is unsatisfactory. The non-invariant variant reduces BDE errors to approximately $i\le q$58–$i\le q$59 kcal/mol but is not recommended for geometries or dynamics because of non-smooth PES behavior. For Cu$i\le q$60, the $i\le q$61MP2$i\le q$62 dissociation-energy error is $i\le q$63 kcal/mol relative to CCSD and $i\le q$64 kcal/mol relative to experiment. A generic orbital-invariant parameter set $i\le q$65 mHa and $i\le q$66 mHa is proposed, with approximately $i\le q$67 average BDE accuracy across the single-bond test set (Zamani et al., 7 Apr 2025).

These results do not prove the entropy–correlation proportionality as a universal law. They do, however, provide a concrete ab initio realization in which entropy derived from a correlated one-electron density matrix is used both to regularize densities and to restore static correlation omitted by regularized perturbative treatments.

7. Open problems, interpretive cautions, and comparative status

Across the three domains, “Collins’ Conjecture” has markedly different logical status. In cyclic tournaments, the full Albertson–Collins conjecture remains open, though the 2016 paper proves it for several expansive structural families and for all Paley tournaments, with no known counterexamples reported there (Meslem et al., 2016). In matrix positivity, the Collins–Dykema–Torres-Ayala conjecture is established only in low even cases by explicit commutative SoS certificates, and the general even-even regime remains unresolved (Kim et al., 2021). In electronic structure, the entropy-based conjecture is not a theorem but a heuristic or phenomenological principle supported by QMC trends, earlier numerical studies on small molecules, and more recent ab initio constructions at the MP2 level (Site, 2014, Zamani et al., 7 Apr 2025).

A second caution concerns the meaning of “entropy.” The graph-theoretic conjecture has no entropic content. The positivity conjecture concerns traces, words, cyclic equivalence, and Gram matrices, not occupation-number uncertainty. Only the quantum-chemical usage identifies correlation with Shannon or Jaynes entropy. Even there, the cited literature distinguishes occupation-number entropy, momentum-space entropy, position-space entropy, Jaynes entropy of the 1-RDM, and Fermi–Dirac entropy, and emphasizes that scaling, normalization, and representation dependence are nontrivial (Site, 2014, Zamani et al., 7 Apr 2025).

A third caution is nomenclatural. The Collatz conjecture is unrelated. One cited arXiv paper explicitly frames “Collins’ Conjecture” in that context as a typo requiring correction to “Collatz Conjecture” (Wey, 2023).

Taken together, the eponym “Collins’ Conjecture” names a family of unrelated conjectural enterprises rather than a single object. In graph theory it is a canonical labeling problem for cyclic tournaments; in matrix analysis it is an even-parameter trace-positivity problem; in quantum chemistry it is an information-theoretic ansatz for correlation energy. Their only commonality is historical naming, not shared mathematical substance.