Ku's maximum-density conjecture for ordinary hypercubes

Kuo-Han Ku, Monochromatic k in a row, arXiv:2606.12880v1, section 5.1.2, states:

Conjecture 5.6. For every $k\in\mathbb{N}_{\geq 2}$ and $d\in\mathbb{N}$, $D(k,[k]^d)=1-\frac{2}{k}$.

Definitions 2.2, 2.4 and 2.6 fix the ordinary board $[k]^d={1,\ldots,k}^d$, all full signed integer lines, near-line avoidance, inclusion-maximal saturation, and the maximum saturated density. This universal assertion is refuted at $k=3,d=3$.

Statement echo

| Source clause | Formal counterpart | | --- | --- | | Every integer $k\geq2$ and natural dimension $d$ | KuGeometry.claim quantifies over natural k, the hypothesis $2\leq k$, and every natural d. | | Ordinary points in ${1,\ldots,k}^d$ | Board k d is Fin d → Fin k; a label is interpreted as its natural value plus one in the integers. | | Every full line $p+tv$, $0\leq t<k$, $0\ne v\in{-1,0,1}^d$, inside the board | GValid quantifies over every signed nonzero direction and every parameter and coordinate; GOn gives literal integer incidence. | | No $(k-1)$-subset of a line is selected | GSourceConfig; gsource_config_iff identifies it with GConfig for $k\geq2$. | | Saturated means maximal under inclusion | GSaturated: a containing configuration must equal the original one. | | $D$ is the attained maximum of $|S|/k^d$ over saturated configurations | density_maximum_attained supplies a saturated cardinality maximizer; density divides its cardinality by $(k:\mathbb{R})^d$. The denominator is positive for $k\geq2$. | | The claimed equality is $D=1-2/k$ | claim has type Prop; result has type Not claim. | | Refuting parameter pair $(3,3)$ | exact_real_event states density 3 3 < (1 : Real)/3. |

The fixed board has 27 points. source_config_iff and gconfig_33_iff connect the pairwise geometric predicate used in the bound to the general source predicate. Natural dimension zero is allowed by the formal universal quantifier; the refuting positive dimension three also lies in a convention that starts natural dimensions at one. No conclusion depends on dimension zero.

The paper's upper-density formula is suggested by configurations that attain the axis bound for certain parameters. The three-dimensional board shows why axis constraints alone cannot establish attainment: equality forces additional Latin-square structure that conflicts with genuine face diagonals.

The bounded literature reading on 2026-10-29 found the original assertion in arXiv v1; the arXiv record lists only v1. The primary HTML has 792385 bytes and SHA-256 6bc5cb16b8c6a9a71baf44f17e03c7361257aeba9c5c38ff4c3866283998dcb3. MathDB's exact problem entry displays "Solutions 0" and "No solutions have been posted yet". OpenAlex W7164573657 reports zero citations. A Crossref title query returned unrelated titles and supplied no matching DOI. These readings contain no matching prior resolution; they do not establish worldwide priority. General Google search returned a JavaScript redirect page, and the source-cited master's thesis was not read.

config_bound proves that every ordinary $3^3$ configuration has at most eight points. Projection onto the first two coordinates is injective, since two points in a projection fiber lie on an axis line. If nine points existed, projection would be bijective, giving a function assigning the third coordinate to each pair of first coordinates. The other two axis directions force its rows and columns to be permutations: a Latin square of order three.

latin_diagonal proves that such a square repeats a value on a main or opposite diagonal. diag_common turns this repetition into two selected points on a full ordinary face-diagonal line, contrary to the configuration condition. The contradiction rules out nine points. Applying the bound to the saturated maximizer defining density gives exact_real_event, and result contradicts the conjectured value $1-2/3=1/3$.

A configuration of nine points on this exact ordinary board would contradict config_bound. A mismatch between the source and formal line family, the saturation order, or maximum-density semantics would invalidate the external interpretation even if the finite combinatorial theorem remained true. The statement echo retains all of these conditions. A size-eight construction is not needed to refute the universal equality.

The mathematical source is D5/S3/Combinatorics/Geometry/KuNearLineRefutation.lean. KuGeometry.config_bound states the universal cardinality upper bound; KuGeometry.density_maximum_attained supplies the actual saturated maximum; KuGeometry.exact_real_event states the strict density inequality; KuGeometry.result is the closed negation of KuGeometry.claim. The axiom closure of every public declaration is contained in ${\mathrm{propext},\mathrm{Classical.choice},\mathrm{Quot.sound}}$. No numerical experiment or unverified search certificate is needed for the proof.

The reused order-three Latin predicate is D5.S3.Combinatorics.LatinEulerianMultiples.IsLatin, owned by D5/S3/Combinatorics/LatinEulerianDefs; the finite diagonal obstruction is proved in KuGeometry.latin_diagonal.

• Proved, kernel-checked: every ordinary $3^3$ near-three-line-avoiding configuration has cardinality at most eight (KuGeometry.config_bound, with gconfig_33_iff and source_config_iff giving the source correspondence).
• Proved, kernel-checked: the registered strict inequality $D(3,[3]^3)<1/3$ (KuGeometry.exact_real_event). Consequently the universally quantified Conjecture 5.6 is false (KuGeometry.result).
• Proved mechanism, kernel-checked: equality at nine would force a Latin square by axis projection (config_bound); the forced equal-symbol diagonal (latin_diagonal) produces an ordinary full-line collision (diag_common). This identifies an obstruction to attaining the axis upper bound, rather than changing the board or deleting directions.
• Open: the exact value of $D(3,[3]^3)$ and a size-eight saturated attainer. The source retained here proves an upper bound and strict inequality only; it does not prove $D(3,[3]^3)=8/27$.
• Open: determine $D(k,[k]^d)$ for every integer $k\geq2$ and $d\geq1$; in particular, classify the pairs where the source's upper bound is attained. Restrictions to larger $k$ or $d$, or other repaired equalities, are not established by this one refuting pair.
• Proved in the cited paper, not newly kernel-checked here: section 5.1.2 gives a separate $X_k$ construction for $d=2$, and invokes the Lemma 4.1 construction when $k$ is even and $k/2$ is coprime with $d!$. Those restricted source statements do not assume Conjecture 5.6. The refuting pair $k=3,d=3$ does not satisfy either restriction; no new certification of them is claimed.
• Literature reading; remaining questions open: the two occurrences of "Conjecture 5.6" in v1 are its statement and the subsequent explanation of the independent $d=2$ case. No later theorem is presented as conditional on Conjecture 5.6. Section 5.1.1's Conjectures 5.1--5.4 and Question 5.5 concern infinite or toric grids. Question 5.7 concerns minimum-density configurations, and section 5.2's winner discussion cites a separate pairing strategy. Their stated arguments do not use Conjecture 5.6; none is resolved or refuted by the maximum-density obstruction here. The universal equality itself is the claim withdrawn, without an asserted downstream theorem collapse.

The literature screen is bounded by the sources listed in Gap. The master's thesis and an unrestricted citation corpus were not checked; nondetection is not an exhaustive novelty claim.

The four-slot audit of KuGeometry.config_bound has a validated registration in Reg/D5/S3/Combinatorics/Geometry/KuNearLineRefutation.lean with an unknown continuation. The eleven other public audit targets remain unfinished as individually specified in the linked audit boundary. The designated negation KuGeometry.result is exempt. This audit boundary does not supply or replace any part of the mathematical counterexample proof.