The Han–Wang–Yu gluing spectral hull conjecture is false

X. B. Han, S. Wang and C. Yu, PageRank of Gluing Networks and Corresponding Markov Chains, Mathematics 13 (2025), 2080, Conjecture 1, §4.3, page 17:

Glue Markov chains M_i = (S_i, A_i) to a Markov chain M = (S, A), then all the eigenvalues of A lie in the convex hull of the eigenvalues of all A_i on the complex plane.

The gluing is Definition 8, page 10. The Library note records its four entrywise cases. Preregistration: #19143.

Statement echo

| Source clause | Lean counterpart | Fidelity | | --- | --- | --- | | Finitely many Markov chains with selected states | RootedChain; families indexed by Fin k | States are Fin (n + 1), selected state 0; the stochastic field is Matrix.colStochastic | | Transition probability from $w$ to $v$ | Matrix entry A v w | Column convention of Definition 8 | | Common root and otherwise disjoint states | Option (Σ i, Fin (M i).n) | none is the common root, some ⟨i,v⟩ corresponds to v.succ | | Definition 8's four transition cases | glue | Root-root weighted sum, private-root weighted entry, root-private unchanged entry, same-chain unchanged entry and cross-chain zero | | $0<\theta_i<1$, $\sum_i\theta_i=1$ | Both antecedents of claim | No weight hypothesis is dropped | | All eigenvalues lie in the input spectral convex hull on the complex plane | claim | Complex spectrum after entrywise algebraMap ℝ ℂ, with convexHull ℝ in $\mathbb C$ | | Refutation | result : ¬ claim | Two identical 3-state chains give an admissible 5-state counterexample |

The state and spectrum expressions are expanded directly rather than introduced as forwarding definitions. Using labelled finite state sets loses no counterexample: these labelled matrices are themselves instances of the source's finite chains. The finite cardinality encoding and column convention introduce no additional hypothesis into the displayed conjecture.

Identifying the roots leaves private-block modes whose eigenvalues need not be input eigenvalues. Opposite private components can cancel at the root while retaining a root-deleted block eigenvalue.

The literature screen in #19143 reports the source, its citing works and fifteen later works of its authors, classical coalescence and lumpability references, MathDB, google-deepmind/formal-conjectures, QIQC, repository issues and the published problem register. No settlement was found in that searched scope. Completeness is ASSUMED-UNVERIFIED. The source and literature checks are orchestrator-reported; they are distinct from the kernel proof below.

Take two identical copies, rooted at 0, of

$$ B=\frac1{54}\begin{pmatrix}43&9&2\2&36&16\9&9&36\end{pmatrix}, \qquad \theta=(1/2,1/2). $$

The private helper B_stochastic certifies nonnegative entries and unit column sums. The helper glued_eigenvalue certifies the nonzero vector $(0,-4,3,4,-3)$ with glued eigenvalue $4/9$. The helper input_halfplane uses

$$ \det(\mu I-B)=(\mu-1)(\mu-17/27)(\mu-1/2) $$

to put the input spectrum in ${z\in\mathbb C:1/2\leq\operatorname{Re}z}$. The proof of result uses convexHull_min and convex_halfSpace_re_ge to put the input spectral hull in the same half-plane, excluding $4/9$. The eigenpair-to-spectrum step applies D5.S3.Combinatorics.NonBacktracking.LaplacianGapMinDegree.eigenpair_mem_spectrum to the glued matrix, eigenvalue $4/9$ and the displayed nonzero vector.

A discrepancy in Definition 8's root-column weights or column convention, a nonstochastic input, an incorrect eigenpair, or failure of the input half-plane bound would defeat this witness. The Lean declaration retains all strict weight bounds and their unit sum, and proves the literal negation of claim.

The module is D5/S3/Quantum/Matrix/HanWangYuGluingSpectrumRefutation.lean, under the namespace D5.S3.Quantum.Matrix.HanWangYuGluingSpectrumRefutation. Its only public theorem is result; its three theorem helpers are private and consumed. Every theorem has proof_shape: bind-only, with admission_basis: open-problem-resolution (#19143; Refuted) and certified-instance utility refuting claim. No strict escape witness is asserted.

The public result is the designated refutation result (basis=refutes) and is exempt from four-slot escape registration (CLAUDE.md §3.9). The published owner D5.S3.Combinatorics.NonBacktracking.LaplacianGapMinDegree.eigenpair_mem_spectrum has an unfinished four-slot escape audit under #19143.

Experiment

Pinned experiment directory. Command, from that directory: python3 check.py; exit code $0$, result PASS. The script's SHA-256 is 909c9d85d9a165992829444f84a0c8818f58f6757476bb83accbb66747d9bdf6. The command uses the pinned script bytes.

The six seed cases have $k\in{2,3,4}$, each with uniform weights, and with nonuniform weights $(1/3,2/3)$, $(1/2,1/4,1/4)$ and $(1/2,1/6,1/6,1/6)$ respectively. Exact rational arithmetic checks column stochasticity and the eigenvector $(0,-4,3,4,-3,0,\ldots)$ at $4/9$; floating point checks corroborate the seed input eigenvalues and that glued eigenvalue. The script also checks the three stated seed determinant roots in exact rational arithmetic. It does not compute a symbolic characteristic polynomial for arbitrary $k$.

The nine family samples use $r\in{1/9,1/20,1/100}$ and $\delta\in{1/2,3/4,9/10}$, $h=2r+3r^2$, and

$$ B_{r,\delta}=\delta I+(1-\delta) \begin{pmatrix} 1/3+h&1/3&1/3-h\ 1/3-h&1/3&1/3+h\ 1/3&1/3&1/3 \end{pmatrix}. $$

For these samples, exact arithmetic checks strict positivity and column stochasticity; the two-copy, equal-weight glued eigenvalue $\delta-(1-\delta)r$ is checked numerically to tolerance $10^{-9}$, with its strict inequality below $\delta$ checked exactly. The script does not verify the general input spectral formula or a similarity decomposition.

Tier 1: an explicit published conjecture. Settlement: Refuted. Settlement kind: small case with family mechanism. Literature completeness: ASSUMED-UNVERIFIED.

What the settlement shows

| Item | Status and evidence | Scope | | --- | --- | --- | | Conjecture 1 | Proved false, kernel-checked result | Two identical 3-state inputs, root 0 and weights $(1/2,1/2)$ | | Failure mechanism | Kernel-checked glued_eigenvalue and input_halfplane, used by result | Zero root coordinate and opposite private blocks yield $4/9$; all input spectral real parts are at least $1/2$ | | Six seed gluings | Computed by the pinned experiment above, exit 0 | Exactly the six stated $k$/weight choices; rational stochasticity and eigenpair checks | | Nine family samples | Computed by the same experiment, exit 0 | Exactly the nine $(r,\delta)$ pairs and two equal-weight copies; glued eigenvalue check is numerical | | General identical-copy similarity $G\sim B\oplus D^{\oplus(k-1)}$ and invariant-subspace description | Open; unformalized search-seat assertion in #19143 | All admissible weights and arbitrary $k\geq2$; no uniform theorem is claimed | | General positive doubly stochastic family, input spectrum and hull separation | Open; unformalized beyond the tested properties above | $0<r\leq1/9$, $1/2\leq\delta<1$ and arbitrary $k\geq2$ | | Reversible input chains as a sufficient condition for containment | Open; unformalized search-seat assertion in #19143 | A separate candidate, with irreducibility and reversible symmetrization to be checked |

The refutation isolates cancellation of private-block modes as the obstruction to universal spectral-hull containment. The exact six seed checks extend the observed obstruction to the stated finite choices; they do not prove the uniform family or similarity assertions. Reversibility is the nearest proposed surviving restriction, with no positive theorem claimed here.

The conclusion negates Conjecture 1. It does not negate the source's Theorem 2, peripheral-eigenvalue statements, or Examples 1 and 2. An argument using Conjecture 1 as a universal premise cannot retain that premise; determining which additional hypotheses recover such arguments remains open.

Literature completeness and priority remain ASSUMED-UNVERIFIED. Rymkhanova et al. (2026), Schwenk (1974), Laffey–Šmigoc and Belardo–Li Marzi–Simić (2010) were not accessed in the recorded literature screen. General families, identical-copy similarity, invariant-subspace classification and the reversible-chain containment assertion are not formalized here. Numerical family eigenvalue checks are not kernel certificates.