Dictionary Collapse — The Dimensionality Illusion

The Experiment

We took terms from Black's Law Dictionary and cross-jurisdictional legal terminology — terms that carry different legal consequences despite surface similarity — and embedded them using nomic-embed-text (768 dimensions). Then we applied PCA at multiple compression levels and measured pairwise cosine similarity.

The question: at what dimensionality do legally distinct terms become indistinguishable to the embedding?

Experiment complete. 62 terms embedded via nomic-embed-text (768d), PCA-compressed to 16d. Variance explained at 16d: 63.5%. Model: nomic-embed-text v1.5.

Results: Two Failure Modes

PCA compression to 16 dimensions doesn't just lose information — it warps the space in two distinct ways:

Collapse (positive delta): Terms that are already similar become indistinguishable. Coercive contempt and punitive contempt, which determine whether you get a trial, merge into one point.
Distortion (negative delta): Terms that are meaningfully related become artificially distant. German Sorgerecht and Swedish gemensam vårdnad, which both mean "shared parental care," go from 0.640 similarity to 0.184.

Both failure modes are catastrophic for retrieval. The first means you can't distinguish concepts that produce different legal outcomes. The second means you can't find related concepts across jurisdictions.

Term Groups

1. Custody & Parental Rights

These terms produce radically different legal outcomes but describe overlapping concepts. In Illinois, the shift from "custody" to "allocation" in 2016 inverted the burden of proof.

Term A	Term B	768d	16d	Delta	Legal Consequence
custody	allocation	0.720	0.446	-0.273	Different burden of proof — PCA makes them more different than they are
guardianship	custody	0.762	0.519	-0.243	Guardian protects; custodian possesses — distorted apart
visitation	parenting time	0.767	0.758	-0.008	Nearly identical — these survive compression (both are English euphemisms)
joint custody	sole custody	0.836	0.932	+0.096	Opposite outcomes collapse together. One preserves, one eliminates.

2. Forced Heirship — The Missing Words

Cross-jurisdictional forced heirship terms that have no English equivalent. The big finding: legitieme portie, yuryubun, iryubun, and laglott all collapse toward each other at 16d — losing the tradition-specific legal distinctions.

Term A	Term B	768d	16d	Delta	Legal Consequence
legitieme portie	yuryubun	0.763	0.890	+0.127	Dutch vs. Korean — collapsed to near-identical
legitieme portie	iryubun	0.757	0.884	+0.127	Dutch vs. Japanese — collapsed
legitieme portie	laglott	0.762	0.872	+0.110	Dutch vs. Swedish — collapsed
Pflichtteil	elective share	0.709	0.418	-0.291	Pflichtteil protects children; elective share protects spouse only — distorted apart
Pflichtteil	réserve héréditaire	0.719	0.492	-0.227	Both protect children — but PCA destroys the relationship
faraid	intestate succession	0.554	0.026	-0.528	Divine mandate vs. legislative default — rendered unrelated
testamentary freedom	forced heirship	0.700	0.536	-0.163	Opposite concepts — PCA partially preserves the distinction but distorts it

The forced heirship collapse is the smoking gun. At 768d, the embedding correctly distinguishes Dutch legitieme portie from Korean yuryubun (0.763 — similar but distinct). At 16d, they merge to 0.890 — effectively identical. A retrieval system at 16d cannot tell you which country's law you're looking at. For comparative law research, this is catastrophic.

3. Contempt & Enforcement

The distinction between coercive and punitive contempt determines whether you get a jury trial. This is a constitutional rights question.

Term A	Term B	768d	16d	Delta	Legal Consequence
coercive contempt	punitive contempt	0.784	0.913	+0.129	Collapsed. Determines whether you get a trial or indefinite imprisonment.
civil contempt	criminal contempt	0.908	0.959	+0.050	Already similar at 768d — at 16d, nearly identical (0.96)
default	contumacy	0.569	0.202	-0.367	Neutral omission vs. willful defiance — distorted apart
ability to pay	willful refusal	0.638	0.317	-0.322	Turner v. Rogers hinges on this distinction — PCA destroys it

4. Death & Status

Term A	Term B	768d	16d	Delta	Legal Consequence
c.d. minima	c.d. maxima	0.911	0.970	+0.058	Loss of family status vs. total legal death — collapsed
civil death	death	0.670	0.637	-0.033	Legal fiction vs. biological fact — mostly preserved

5. Financial Death

Term A	Term B	768d	16d	Delta	Legal Consequence
nondischargeable	insolvency	0.703	0.814	+0.112	Permanent debt obligation merges with its opposite (debt relief)
domestic support obligation	discharge	0.655	0.288	-0.367	DSOs cannot be discharged — PCA destroys this critical relationship
discharge	nondischargeable	0.707	0.716	+0.009	Opposite outcomes — surprisingly preserved (shared vocabulary)
Restschuldbefreiung	discharge	0.725	0.632	-0.093	German "remaining-debt-liberation" drifts from English equivalent

6. Cross-Jurisdictional Parenting Terms

Term A	Term B	768d	16d	Delta	Legal Consequence
Sorgerecht	gemensam vårdnad	0.640	0.184	-0.456	Worst distortion. German and Swedish "shared care" become nearly unrelated.
hadanah	wilayah	0.722	0.630	-0.092	Islamic physical custody vs. legal authority — distorted apart
umgängessabotage	parental alienation	0.719	0.689	-0.030	Swedish statutory term vs. English clinical concept — mostly preserved
parental rights	termination of parental rights	0.728	0.565	-0.163	A right and its destruction — PCA weakens the connection

7. Control Group — Inter-Domain Terms

Term A	Term B	768d	16d	Delta	Result
medical malpractice	custody	0.560	-0.168	-0.728	Different domains — separation INCREASED (as expected)
patent infringement	forced heirship	0.565	0.168	-0.397	Different domains — separation increased

The Two Failure Modes Quantified

Metric	Intra-Domain	Inter-Domain
Mean similarity delta at 16d	-0.307	-0.634
Median similarity delta	-0.302	-0.640
Max positive delta (collapse)	+0.140	-0.023

Inter-domain pairs never collapse (max delta = -0.023). But intra-domain pairs collapse by up to +0.140. This is exactly what MTEB misses: inter-domain discrimination is preserved; intra-domain discrimination is destroyed.

The punchline: At 16 dimensions, your retrieval system can tell law from medicine. But it can't tell coercive contempt from punitive contempt — the distinction that determines whether you get a jury trial. It can't tell Dutch legitieme portie from Korean yuryubun — the distinction between two different countries' inheritance laws. It can't tell German Sorgerecht from Swedish gemensam vårdnad, even though it should recognize them as related concepts from different legal traditions. That's not compression. That's destruction.

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