Entropy maximisers need not have balanced degrees

Pelekis, Le Matematiche 74(1), pp. 119–130, Conjecture 1.3, p. 122: “Let the positive integers m,n,r be such that m ≥ n > r ≥ 2. A hypergraph H = (V,ℰ) from D_{n,m,r} for which it holds H(X_H) ≥ H(X_G), for all G ∈ D_{n,m,r} is such that |deg_H(v₁) − deg_H(v₂)| ≤ 1, for all v₁,v₂ ∈ V.” Here $D_{n,m,r}$ includes repeated edges and isolated vertices. Each edge independently selects a uniformly random vertex, and $X$ counts the vertices selected at least once. Entropy uses logarithms of base two and $0\log 0=0$. The title's zero-in-degree count is $n-X$; that bijection preserves entropy.

Statement echo

| Source clause | Lean counterpart | Fidelity | | --- | --- | --- | | $m\ge n>r\ge2$ | claim quantifies over n m r, with n ≤ m, r < n, 2 ≤ r | All parameters are positive under these hypotheses. | | $r$-uniform multi-hypergraphs | E : Fin m → Finset (Fin n) and every edge has cardinality r | Ordered edges retain multiplicities; isolated vertices remain. | | Uniform independent orientations | The literal orientation filter inside entropyX r E, with denominator $(r:\mathbb R)^m$ | Mathlib Fintype.piFinset supplies the public correspondence. | | $X$ is the number with nonzero in-degree | X E o counts the image of o | The edge argument labels the family; the value depends on o. | | Shannon entropy | entropyX r E sums $-p\log_2p$ over $k=0,\ldots,n$ | r is explicit; zero-probability outcomes are included. | | Entropy at least every competitor | Every uniform F satisfies entropyX r F ≤ entropyX r E | Full competitor class, with no sampling restriction. | | Absolute degree difference at most one | Every ordered pair satisfies the integer difference bound | Both orders give the two signs of the absolute-value bound. | | Refutation | result : ¬ claim | Unconditional negation of the complete conjecture. |

The conjecture proposes degree balance as a necessary condition for entropy maximality. A counterexample must be globally maximal in the stated class; beating one balanced competitor would be insufficient.

The bounded source and literature checks recorded in #15995 and #15996 found no settlement in their inspected scope: arXiv v1, the journal version, the located citing work Silva–Andrade–Bazeia (arXiv:2101.05250), and eleven later arXiv works of the author. This does not establish exhaustive absence or worldwide priority. The full Pelekis–Schauer chapter remains unread.

Take $n=m=3$, $r=2$, with edges ${0,1},{0,1},{0,2}$. Its degrees are $(3,2,1)$ and its orientation counts for $X=0,1,2,3$ are $(0,1,5,2)$, with total eight. For counts $c_k$ with a common positive denominator $N$ and total $N$, the logarithm laws give $H=\log_2N-\log_2(\prod_k c_k^{c_k})/N$, with $0^0=1$. The certificate checks every one of the 27 ordered uniform edge tuples and proves that its product is at least $12500=5^5 2^2$, the witness product. Logarithm monotonicity proves maximality; vertices zero and two have degree difference two, contradicting claim.

A failure of uniformity, global maximality, the count certificate, or the degree difference would invalidate this witness. Any source convention excluding repeated edges would require a different target; the source explicitly allows them.

D5/S0/Certificates/Hypergraphs/OrientationEntropyBalance.lean contains the literal definitions, private finite certificates and the public result. The private certificates witness_counts, witness_degrees, all_counts_certificate and witness_maximises are on its proof path. The public helpers orientations_eq and sum_entropy_formula identify the orientation set and express entropy through the count product. Both are used by the three-vertex and circular comparisons. The axiom closure of every public declaration is contained in {propext, Classical.choice, Quot.sound}.

Their four-slot escape audits are unfinished: the source-to-law bridges elaborate without diagnostics, but variation, sensitivity and observational-dependence evidence is missing; see #16296.

The public result is the designated refutation result (basis=refutes) and is exempt from four-slot escape registration (CLAUDE.md §3.9).

Experiment

Fixed experiment directory. Command: python3 check.py; exit 0; SHA-256 of check.py: 33d20a7f5953e1b2e41631b562d584f91b7774ae2cb6c38024db385db60c7213. It enumerates loopless multigraphs and every orientation for exactly $(n,m)\in{3}\times{3,\ldots,10}\cup{4}\times{4,\ldots,7} \cup{(5,5),(5,6),(6,6)}$, with $r=2$. All fifteen cases have zero balanced maximisers according to its floating entropy comparisons, with comparison tolerance $10^{-12}$; the final reading is bad=0. At $(3,3)$ the maximum is $1.298795$ bits and a maximiser has sorted degrees $(3,2,1)$; at $(3,4)$ the displayed degrees are $(4,2,2)$. These finite numerical readings do not establish a uniform family theorem.

What the settlement shows

| Item | Evidence kind | Result and boundary | | --- | --- | --- | | Three-vertex mechanism | Kernel-checked private certificates on result's proof path | witness_counts gives $(0,1,5,2)$; witness_degrees gives $(3,2,1)$; all_counts_certificate and witness_maximises prove maximality over all 27 ordered tuples. result refutes Conjecture 1.3. | | Why entropy rewards the spread | Open explanation | Repeated edges retain the rare outcome $X=1$ alongside the other two outcomes. The exact product comparison establishes the finite advantage; no general monotonicity of entropy under an increase of support size is proved here. | | Three-vertex family for every $m\ge3$ | Open | Lemmas 1–3 in #15995 propose counts $(\varepsilon,\frac12+\varepsilon,\frac12-2\varepsilon)$ for $\varepsilon=2^{-m}$ and an entropy gap. This family is not proved here. | | Beyond three vertices | Computed | The experiment entry covers exactly the fifteen cases above; no balanced maximiser is located under its comparison tolerance. A uniform extension and classification of all maximisers remain open. | | Simple connected five-vertex case | Proved in prose | The simple graph with edges $01,02,03,04,12$ has degrees $(4,2,2,1,1)$ and greater entropy than every balanced simple graph with five vertices and five edges; the derivation below gives both count vectors and the exact product comparison. | | Restrictions under which balance remains optimal | Open | Neither this instance nor the finite experiments characterize such restrictions. | | Pelekis–Schauer chapter | Open literature check (ASSUMED-UNVERIFIED) | Subscription preview only; supplied Firecrawl HTTP 402; see the final section. |

Simple connected five-vertex comparison

Let $G$ have edges $01,02,03,04,12$. It is simple and connected, and its degrees are $(4,2,2,1,1)$. Condition on edge $12$ choosing vertex $1$. Among edges $02,03,04$, let $B$ count those choosing their outer endpoint. For $B<3$, at least one of these edges chooses $0$, so the selected-vertex count is $X=2+B$, independently of edge $01$. The corresponding numbers of orientations are $2,6,6$ for $B=0,1,2$. For $B=3$, edge $01$ choosing $0$ gives $X=5$, while choosing $1$ gives $X=4$. Thus the conditional counts for $X=2,3,4,5$ are $(2,6,7,1)$. Conditioning on vertex $2$ instead has the same counts by the symmetry swapping $1$ and $2$. The full counts are therefore $(4,12,14,2)$ out of $32$.

A balanced simple graph on five vertices with five edges has every degree $2$: the average degree is $2$, and integral degrees differing by at most one cannot have that average unless they are all $2$. Its components are cycles of length at least three; on five vertices there is room for only one component, so it is $C_5$. Describe an orientation of $C_5$ by a cyclic binary word of length five. A vertex of in-degree zero corresponds to one $0$-to-$1$ transition. There are $2$ constant words, $2\binom{5}{2}=20$ words with two transition locations, and $2\binom{5}{4}=10$ with four. The selected vertex counts are respectively $5,4,3$, giving counts $(10,20,2)$.

The two entropy denominators are $32$. Their count products satisfy $4^4 12^{12}14^{14}2^2<10^{10}20^{20}2^2$. After cancellation this is $3^{12}7^{14}<16\cdot5^{30}$: use $21<25$ and $49<16\cdot5^6$ to obtain $3^{12}7^{14}=21^{12}\cdot49<5^{24}\cdot49<16\cdot5^{30}$. The count-product entropy identity then gives $H(X_G)>H(X_{C_5})$. This is a prose proof of a separate five-vertex comparison; the settling Lean module certifies the three-vertex multigraph instance.

The failure concerns degree balance as a necessary condition for a maximiser; it supplies no classification of the unrestricted optimisers. The source's proved entropy bounds do not acquire new hypotheses or lose their proofs. Its circular maximality statement is separately refuted by D5.S0.Certificates.Hypergraphs.CircularOrientationEntropy.result. The asymptotic Conjecture 4.1 remains open here.

The Pelekis–Schauer chapter, “Network Coloring and Colored Coin Games”, Springer (2013), DOI 10.1007/978-1-4614-6825-7_4, was accessible only as a subscription preview; the supplied Firecrawl reading returned HTTP 402. Its full contents and worldwide prior-art clearance are ASSUMED-UNVERIFIED.