Whitening Reveals Cluster Commitment as the Geometric Separator of Hallucination Types

Matic Korun

cs.CLMar 8, 2026

Whitening Reveals Cluster Commitment as the Geometric Separator of Hallucination Types

Authors: Matic Korun

Organizations: Independent Researcher, Ljubljana, Slovenia

Abstract

A geometric hallucination taxonomy distinguishes three failure types -- center-drift (Type~~1), wrong-well convergence (Type~~2), and coverage gaps (Type~~3) -- by their signatures in embedding cluster space. Prior work found Types~~1 and2 indistinguishable in full-dimensional contextual measurement. We address this through PCA-whitening and eigenspectrum decomposition on GPT-2-small, using multi-run stability analysis (20 seeds) with prompt-level aggregation. Whitening transforms the micro-signal regime into a space where peak cluster alignment (max_sim) separates Type2 from Type~~3 at Holm-corrected significance, with condition means following the taxonomy's predicted ordering: Type~~2 (highest commitment) $>$ Type~~1 (intermediate) $>$ Type~~3 (lowest). A first directionally stable but underpowered hint of Type1/2 separation emerges via the same metric, generating a capacity prediction for larger models. Prompt diversification from 15 to 30 prompts per group eliminates a false positive in whitened entropy that appeared robust at the smaller set, demonstrating prompt-set sensitivity in the micro-signal regime. Eigenspectrum decomposition localizes this artifact to the dominant principal components and confirms that Type1/2 separation does not emerge in any spectral band, rejecting the spectral mixing hypothesis. The contribution is threefold: whitening as preprocessing that reveals cluster commitment as the theoretically correct separating metric, evidence that the Type~1/2 boundary is a capacity limitation rather than a measurement artifact, and a methodological finding about prompt-set fragility in near-saturated representation spaces.

Figures & tables

Band	PCs	Variance
Dominant	1–16	98.0%
Transition	17–48	0.7%
Mid-range A	49–128	0.6%
Mid-range B	129–256	0.4%
Lower	257–512	0.3%
Tail	513–768	$<$ 0.1%

Table 1: Spectral bands defined across the eigenspectrum.

Pair	Sig rate	Holm	Med $r$	Dir	Pseudo
T2–T3	60%	40%	$-0.31$	20/20	$0.3\times$
T1–T2	35%	15%	$+0.21$	17/20	$1.4\times$
T1–T3	5%	5%	$-0.15$	18/20 -	$12\times$

Table 2: Whitened max_sim pairwise results (

N=30

/group, 20 seeds). Sig rate: fraction of runs with prompt-level MW

p<0.05

. Holm: fraction surviving Holm-Bonferroni correction. Med

r

: median rank-biserial effect size. Dir: directional stability (runs with consistent sign). Pseudo: pseudoreplication ratio (token sig rate / prompt sig rate); values

<1

indicate the prompt-level effect is stronger than token-level—a genuine between-condition effect; values

>1

indicate token-level inflation (standard pseudoreplication).

Figure 1: Whitened max_sim results. (a) Per-prompt mean max_sim by condition from the representative seed ( $N=30$ /group). The ordering T2 $>$ T1 $>$ T3 matches the taxonomy’s core prediction: wrong-well commits to a cluster (highest), center-drift drifts without commitment (intermediate), coverage gap aligns with nothing (lowest). Grand means annotated. (b) Rank-biserial effect size $r$ across 20 independent generation seeds, with median and IQR bars. T2–T3 (green): 20/20 directional stability, 8/20 Holm. T1–T2 (blue): 17/20 directional, 3/20 Holm—the emerging signal. T1–T3 (gray): directionally stable but weak, reflecting the combined gap.

Figure 2: The $H(\mathbf{v})$ collapse and max_sim emergence. Each dot is one seed’s prompt-level Mann-Whitney $p$ -value (20 seeds total). Vertical ticks: median. Red dashed line: $\alpha=0.05$ . (a) Whitened $H(\mathbf{v})$ at $N=30$ : $p$ -values scattered widely for all pairs. The strongest $N=15$ signals—T1–T3 (65% Holm) and T2–T3 (30% Holm)—collapse to 5% and 0% significance respectively. (b) Whitened max_sim at $N=30$ : T2–T3 (green) clusters near zero (12/20 sig, 8/20 Holm). T1–T2 (blue) shows a partial cluster below $\alpha$ (7/20 sig, 3/20 Holm)—the emerging signal.

Experiment	Metric	Holm	$r$	Dir
Static (Paper 2)	$H(\mathbf{v})$	0/20	$-0.44^{*}$	—
Raw contextual (Paper 2)	norm	0/20	n.s.	—
Full-spectrum whitened	max_sim	3/20	$+0.21$	17/20
Best spectral band (tail)	max_sim	1/20	$+0.23$	16/20

Table 3: Type 1/2 separation across experiments. Paper 2 results from single run; Paper 3 results from 20-seed multi-run. The static result (*

p=0.042

, uncorrected) does not survive Holm correction. The whitened max_sim result at

r=+0.21

with 17/20 directional stability is the first hint of separation—underpowered but sign-consistent.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: Spectral band decomposition ( $N=15$ /group, 20 seeds). Cell color: fraction of 20 runs where prompt-level Mann-Whitney $p<0.05$ . Cell annotations (where sig rate $\geq 15$ %): median rank-biserial $r$ (top) and Holm survival rate (bottom). T1–T2 columns are uniformly pale across all bands—Type 1/2 separation does not exist in any spectral region. Dominant band $H(\mathbf{v})$ (PCs 1–16) shows the artifact that collapses at $N=30$ . Tail band effects (PCs 513–768) are genuine but in $<$ 0.1% variance dimensions.

Figure 4: Token-level vs. prompt-level significance ( $-\log_{10}p$ , median across 20 seeds) for all experiment–metric–pair combinations. Dashed lines: $p=0.05$ . Circles: whitened ( $N=30$ ). Squares: raw norm ( $N=30$ ). Diamonds: spectral bands ( $N=15$ ). Color: pair type (blue = T1–T2, gray = T1–T3, green = T2–T3). Points in the “token only” quadrant (lower right) are pseudoreplication artifacts. The spectral tail produces the most extreme both-levels results.

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From Prerequisites to Predictions: Validating a Geometric Hallucination Taxonomy Through Controlled Induction

Matic Korun

Independent Researcher Ljubljana, Slovenia

Hallucination in Language Models Neural Representation Geometry

Open paper AskPDF arXiv

We test whether a geometric hallucination taxonomy -- classifying failures as center-drift (Type~~1), wrong-well convergence (Type~~2), or coverage gaps (Type3) -- can distinguish hallucination types through controlled induction in GPT-2. Using a two-level statistical design with prompts ( $N = 15$ /group) as the unit of inference, we run each experiment 20 times with different generation seeds to quantify result stability. In static embeddings, Type3 norm separation is robust (significant in 18/20 runs, Holm-corrected in 14/20, median

r = +0.61

). In contextual hidden states, the Type~~3 norm effect direction is stable (19/20 runs) but underpowered at $N = 15$ (significant in 4/20, median $r = -0.28$ ). Types~~1 and2 do not separate in either space ( ${\leq}\,3/20$ runs). Token-level tests inflate significance by 4--16 $\times$ through pseudoreplication -- a finding replicated across all 20 runs. The results establish coverage-gap hallucinations as the most geometrically distinctive failure mode, carried by magnitude rather than direction, and confirm the Type1/2 non-separation as genuine at 124M parameters.

Abstract

Feature Whitening

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