Metadata

slug: pelekis-2018-circular-orientation-entropy
bibkey: pelekis2019orientationentropy
doi: 10.4418/2019.74.1.8
url: https://arxiv.org/abs/1808.04166
triage: theorem
motivation_gids:
  - D5/S0/Certificates/Hypergraphs/CircularOrientationEntropy

Circular hypergraphs do not maximise orientation-count entropy

Problem

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 $r \geq 3$ 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
$r \geq 3$ 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 $(i + j) mod n$ , $0 \leq j < r$	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 $(r : R)^{n}$ , 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.

Motivation

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.

Gap

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.

Route

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 $2^{2} 2 7^{27} 4 4^{44} 8^{8} < 2 4^{24} 4 8^{48} 9^{9}$ , together with $H = lo g_{2} 81 - lo g_{2} (\prod c^{c}) /81$ , gives a strict entropy advantage. The claim applied to this uniform witness gives the opposite weak inequality.

Falsifier

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.

Evidence

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 $1 0^{- 12}$ to update a maximiser and $1 0^{- 9}$ for the entropy advantage test. Numerical maximisation is not a uniform classification theorem.

Triage

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 $m \geq 3$	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.

ASSUMED-UNVERIFIED

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.

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