CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Lloyd can stop at a strictly suboptimal finite-data fixed point

Statement refuted

Every fixed point of the exact finite-data Lloyd iteration is a global minimizer of empirical squared-error distortion.

Facts & Assumptions

Given: Five equally weighted data points $0, 1, 4, 5, 9 \in R$ , two ordered centers, and the exact iteration of Exact finite-data Lloyd quantizer.

[F1]

Exact finite-data Lloyd quantizer: the iteration uses least-index nearest-center assignments, updates nonempty clusters to their weighted centroids, leaves empty centers unchanged, and stops when centers are unchanged.

[F2]

The weighted centroid minimizes finite-cluster squared error: each nonempty cluster's weighted squared error is minimized at its weighted centroid.

[F3]

Let $X_{5} : = {0, 1, 4, 5, 9}$ and $(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) : = (0, 1, 4, 5, 9)$ . Put $μ_{5} : = \frac{1}{5} \sum_{i = 1}^{5} δ_{x_{i}}$ on $X_{5}$ , where $δ_{x_{i}}$ is unit point mass at $x_{i}$ . This is a probability measure on the relative Borel sigma algebra of the finite Borel set $X_{5}$ , with finite second moment $\frac{1}{5} \sum_{i = 1}^{5} ∣ x_{i} ∣^{2} < \infty$ . For a center pair $c = (c_{1}, c_{2})$ , let $Q_{c} : X_{5} \to {1, 2}$ be its least-index nearest-center rule. Each fiber is Borel because $X_{5}$ is finite, so Finite measurable squared-error quantizer and nearest-center rule gives

$D_{μ_{5}} (Q_{c}; c_{1}, c_{2}) = \frac{1}{5} \sum_{i = 1}^{5} \min_{1 \leq j \leq 2} ∣ x_{i} - c_{j} ∣^{2},$

so the sample-average costs below are exactly the distortions for this explicitly specified empirical law.

Counterexample

1.1givenF1F3algebra

Give each point weight $1 / 5$ and initialize centers at $(5 / 2, 9)$ . The squared distances to these centers, in point order $0, 1, 4, 5, 9$ , are $(25 / 4, 81)$ , $(9 / 4, 64)$ , $(9 / 4, 25)$ , $(25 / 4, 16)$ , and $(169 / 4, 0)$ ; each comparison is strict, so the clusters are ${0, 1, 4, 5}$ and ${9}$ .

1.2givenF3algebra

The alternative centers $(1 / 2, 6)$ assign $0, 1$ to $1 / 2$ and $4, 5, 9$ to $6$ , with strict comparisons: the respective squared-distance pairs are $(1 / 4, 36)$ , $(1 / 4, 25)$ , $(49 / 4, 4)$ , $(81 / 4, 1)$ , and $(289 / 4, 9)$ . Their total squared error is $(1 / 2) + (4 + 1 + 9) = 29 / 2$ , hence their average distortion is $29 / 10$ .

2.1F1F2F3step 1.1algebra

The two weighted centroids are $(0 + 1 + 4 + 5) / 4 = 5 / 2$ and $9$ , so the center update leaves $(5 / 2, 9)$ unchanged by [F2]. The nearest assignment is unchanged as well by step 1.1, so [F1] stops at this tuple. Its distortion is $(25 / 4 + 9 / 4 + 9 / 4 + 25 / 4) / 5 = 17 / 5$ by [F3].

3.1step 2.1step 1.2algebra∎

Since $29 / 10 < 17 / 5$ , the alternative two-center quantizer has strictly lower distortion than the stopped Lloyd output in step 2.1. Thus that exact fixed point is not globally optimal, refuting the statement.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Robert Gallager, 6.450 Principles of Digital Communications I, Chapter 3: Quantization (standard reference, not scraped)
Robert M. Gray and David L. Neuhoff, Quantization, IEEE Transactions on Information Theory 44(6) (1998) (standard reference, not scraped)