Circular hypergraphs do not maximise orientation-count entropy

Pelekis, Le Matematiche 74(1), pp. 128–129, Section 4: “We conjecture that circular hypergraphs are such that H(X_{C_{n,r}}) ≥ H(X_H), for all H ∈ D_{n,n,r}.” The preceding definition assumes $r\ge3$ and $n>r$; $C_{n,r}$ has all $n$ cyclic paths of $r$ vertices as edges. The competitor class consists of multi-hypergraphs and permits repeated edges and isolated vertices. Every edge independently selects one uniformly random vertex, and $X$ counts the selected vertices. The title's complementary zero-in-degree count has the same entropy.

Statement echo

| Source clause | Lean counterpart | Fidelity | | --- | --- | --- | | $r\ge3$ and $n>r$ | claim quantifies over r n, with 3 ≤ r and r < n | No additional hypothesis. | | All cyclic paths of size $r$ | circular n r maps each starting vertex to the image of natural residues $(i+j)\bmod n$, $0\le j<r$ | Its bound derives $n>0$ from the actual index i : Fin n; for $n=0$ there is no index. | | $D_{n,n,r}$ | E : Fin n → Finset (Fin n) with all cardinalities equal to r | Exactly $n$ labelled edges; multiplicities retained. | | Uniform independent orientations and Shannon entropy | Shared X and entropyX r E from OrientationEntropyBalance | Literal filter, denominator $(r:\mathbb R)^n$, base-two logarithm and zero-probability convention. | | Circular entropy at least every competitor | entropyX r E ≤ entropyX r (circular n r) for every uniform E | No restriction of the competitor class. | | Refutation | result : ¬ claim | Unconditional negation; the numbered asymptotic Conjecture 4.1 is a separate statement. |

Uniform placement of edges around a circle is proposed to maximise entropy. A single admissible competitor with strictly larger entropy disproves that maximality without any need to classify all optimisers.

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.

At $n=4$, $r=3$, take ${0,1,2},{0,1,2},{0,1,3},{0,1,3}$. Its counts for $X=0,1,2,3,4$ are $(0,2,27,44,8)$; those of $C_{4,3}$ are $(0,0,24,48,9)$. Both total 81. The exact integer inequality $2^2 27^{27}44^{44}8^8<24^{24}48^{48}9^9$, together with $H=\log_2 81-\log_2(\prod c^c)/81$, gives a strict entropy advantage. The claim applied to this uniform witness gives the opposite weak inequality.

A failure of edge uniformity, either exact count vector, or the strict product comparison would invalidate this witness. Replacing the literal modular construction or excluding source-permitted multiplicities changes the problem.

D5/S0/Certificates/Hypergraphs/CircularOrientationEntropy.lean imports OrientationEntropyBalance for X, entropyX, orientations_eq and sum_entropy_formula. Its entropy comparison directly applies the shared count-product formula. Its private witness_counts, circular_counts, counts_certificate and witness_entropy_gt establish the finite comparison on the public result's proof path. The axiom closure of every public declaration is contained in {propext, Classical.choice, Quot.sound}.

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_circular.py; exit 0; SHA-256 of check_circular.py: 6631b05d7b320fb08b39fa72e9541e4ad5e5a6d30e01edb240553ad3cdba35ca. It enumerates every multi-hypergraph and orientation for exactly $(n,r)=(4,3),(5,3),(6,3),(5,4)$. The final reading is bad=0, including a true exact integer comparison at $(4,3)$. Floating maximum and circular entropies, respectively, are $1.468283$ and $1.319521$, $1.703670$ and $1.450639$, $1.907803$ and $1.583841$, $1.594842$ and $1.522207$ bits. The script uses tolerance $10^{-12}$ to update a maximiser and $10^{-9}$ for the entropy advantage test. Numerical maximisation is not a uniform classification theorem.

What the settlement shows

| Item | Evidence kind | Result and boundary | | --- | --- | --- | | Circular failure mechanism | Kernel-checked private certificates on result's proof path | witness_counts gives $(0,2,27,44,8)$, circular_counts gives $(0,0,24,48,9)$; counts_certificate and witness_entropy_gt give the strict comparison at $(4,3)$. result refutes circular maximality. | | Relation to the three-vertex mechanism | Kernel-checked sibling result | OrientationEntropyBalance's private witness_counts, witness_degrees and witness_maximises, used by its result, establish counts $(0,1,5,2)$, degrees $(3,2,1)$ and maximality over all 27 ordered tuples at $n=m=3$, $r=2$. | | Why concentration can increase entropy | Open explanation | A common core retains the rare outcome $X=1$, absent in $C_{4,3}$. Support size alone does not order entropy; the exact product inequality establishes this advantage. | | What maximises in the tested cases | Computed | The experiment displays a maximiser with a common core in each of exactly four cases above. It does not classify every maximiser or any unbounded family. | | Further circular counterexamples | Computed | Strict numerical advantages occur at $(5,3),(6,3),(5,4)$ in the fixed experiment. Only $(4,3)$ is certified by this Lean module. | | Three-vertex family for every $m\ge3$ | Open | The Lemmas 1–3 candidate of #15995 is not proved here. | | Restrictions preserving balance optimality | Open | No characterization follows from the two finite refutations. | | Asymptotic Conjecture 4.1 | Open | Its lower bound is logically distinct from maximality and is not settled here. | | Pelekis–Schauer chapter | Open literature check (ASSUMED-UNVERIFIED) | Subscription preview only; supplied Firecrawl HTTP 402; see the final section. |

The counterexample breaks the proposed optimum, without contradicting the source's separate proved entropy bounds. Any inference using the maximality conjecture lacks that premise; the source's numbered asymptotic question retains its stated open boundary.

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.