The Geometric Ceiling of Single-Vector Retrieval: What arXiv 2508.21038 Proves, and the Four Things It Does Not

There is a hard ceiling on what a single fixed-length vector per document can retrieve, and it is geometric rather than a matter of training data or model size. The ceiling is real, it is proved, and observed model behaviour is far worse than it — but the theorem is also much weaker at realistic top-k than almost every summary of it suggests, and the paper's headline capacity numbers are an extrapolation whose value is set by an undefended modelling choice. All three halves of that sentence are load-bearing.

Paper

arXiv:2508.21038, *On the Theoretical Limitations of Embedding-Based Retrieval*, Orion Weller, Michael Boratko, Iftekhar Naim, Jinhyuk Lee (Google DeepMind; Johns Hopkins). Published as a conference paper at ICLR 2026. v1 submitted 28 Aug 2025 17:43:53 UTC (24 pages); v2 submitted 12 Mar 2026 16:35:48 UTC (23 pages). Cite the version — see [The theory moved between v1 and v2, and so did part of the evidence](#the-theory-moved-between-v1-and-v2-and-so-did-part-of-the-evidence).

Benchmark

LIMIT, and the separate LIMIT-small. Code and data at google-deepmind/limit, commit 90c5dd3a7c3c (2025-09-15), Apache-2.0 / CC-BY-4.0. Data mirrors at orionweller/LIMIT and orionweller/LIMIT-small on Hugging Face (created 2025-08-28, CC-BY-4.0).

Harness

MTEB, task ids LIMITRetrieval and LIMITSmallRetrieval. The repository README at that commit states evaluation is reproducible "only on the [v2.0.0 branch]" of embeddings-benchmark/mteb (the branch named w_limit), and warns "the v2.0.0 branch is changing rapidly, so please install the version pinned in the requirements". So "we ran LIMIT" without naming the harness branch is not yet a reproducible statement.

The theorem, stated exactly

*Regime: citational. Source: arXiv:2508.21038v2, §3, locally extracted.*

The setup. Let v_1, ..., v_n in R^d be unit document vectors and queries be unit vectors u in R^d. Fix γ > 0. The paper defines realization with a margin:

> A k-subset S ⊆ [n] is *realized with margin γ* if there exists a unit query u_S such that > min_{i∈S} ⟨u_S, v_i⟩ ≥ max_{j∉S} ⟨u_S, v_j⟩ + 2γ.

Since inner products of unit vectors lie in [-1, 1], "any score gap is at most 2, hence equation 1 is feasible only for 0 < γ ≤ 1."

Theorem 1 (Dimension lower bound), verbatim

> Assume 1 ≤ k < n and that *every* k-subset S ⊆ [n] is realized with margin γ as in equation 1. Then > C(n,k) ≤ (1 + 1/γ)^d, hence d ≥ log C(n,k) / log(1 + 1/γ).

Read the quantifier: the hypothesis is that every k-subset is realized, with a uniform margin. That antecedent is the whole content, and it is what paraphrases drop.

The proof is a sphere-packing volume argument, six lines: two distinct realized k-subsets force their query vectors at least 2γ apart, so the C(n,k) open balls of radius γ around them are disjoint and all sit inside a ball of radius 1+γ, giving C(n,k)·γ^d ≤ (1+γ)^d. No learning theory required.

*Inference, ours, not stated by the paper:* because the packing argument only ever uses pairwise separation of the realized queries, it equally bounds any family of k-subsets realized with margin γ by (1 + 1/γ)^d, not just the full family of size C(n,k). That reading is sound but it is a reading; the printed theorem is the conditional above.

The consequence the paper states in prose, and the sentence worth quoting when you cite this work:

> we show that for a given embedding dimension d there exist top-k combinations of documents that > cannot be returned—no matter the query—highlighting a theoretical and fundamental limit to > embedding models.

The authors position it against the better-known result in the same area:

> Our result is conceptually related to the Johnson–Lindenstrauss lemma (Johnson et al., 1984), > which gives a *sufficient* dimension to preserve pairwise distances among n points; in contrast, > our bound gives a *necessary* dimension to realize all retrieval sets with a margin.

The bound is toothless at realistic k, and the paper's own table shows it

*Regime: attested (our recomputation) over a citational table. Method in provenance; 37 of 37 cells reproduced, 0 mismatches.*

Table 1 instantiates the bound at γ = 0.1 — a score gap of 0.2, "which is approximately standard for models based on empirical usage". We recomputed every non-trivial cell independently as ceil(log(comb(n,k))/log(11)). Selected values:

| corpus size n | k=2 | k=10 | k=100 | k=1000 | |---|---|---|---|---| | 10^4 | 8 | 33 | 233 | 1354 | | 10^6 | 12 | 52 | 425 | 3296 | | 10^11 | 21 | 100 | 905 | 8098 |

Read the k=2 column. At a hundred billion documents and top-2, Theorem 1 asks for 21 dimensions. At top-10 over the same corpus it asks for 100. Production embedding dimensions are 768 to 4096. The paper is explicit that

> most embeddings used for web-scale search are quantized or truncated (e.g. through Matryoshka > embeddings (Kusupati et al., 2022)) to less than 1k dimensions, while the largest embeddings used > in research are around 4096

and that "even with a moderate margin ... the lower bounds in Table 1 can already be larger than what is used in practice" — which is true, but only in the k=100 and k=1000 columns. Nobody runs top-1000 retrieval under an exact-margin guarantee over every subset.

The paper knows this, and bridges the gap by asserting a multiplier: "even a small multiple of this lower bound would make the embedding dimension requirement infeasible. This multiple seems well-founded, as we will show in the next section". The multiplier is the load-bearing step, not the theorem.

This is the single most misread thing about the paper. Theorem 1, taken alone and at realistic k, is comfortably satisfied by a 768-dimensional model. What bites is the gap between the theorem and observed behaviour. Anyone citing Theorem 1 as the reason their RAG pipeline misses documents has cited the wrong part of the paper.

The free-embedding experiment: what it measures, and how far it is stretched

*Regime: citational for the construction and the printed numbers; attested for everything we recomputed.*

The second contribution is a best-case experiment: optimize document and query vectors *directly* on the target qrel matrix with Adam and InfoNCE, full-batch, projected onto the unit sphere, early stopping after 1000 iterations without loss improvement — no language model, no generalization requirement, no tokenizer. Raise n by one until 100% accuracy becomes unreachable; call that the critical-n for that d. The framing is explicit and fair: "if the free embedding optimization cannot solve the problem, real retrieval models will not be able to either."

The result, verbatim:

> the curve fits a 3rd degree polynomial curve, with formula y = −10.5322 + 4.0309d + 0.0520d² + > 0.0037d³ (r²=0.999). Extrapolating this curve outward gives the critical-n values (for embedding > size): 500k (512), 1.7m (768), 4m (1024), 107m (3072), 250m (4096).

The base of that derived number, printed in the paper and rarely quoted with it

Table 6 of both versions lists every fitted point. There are 42 of them, at d = 4, 5, ..., 45, one per d, and all at k = 2 only. Critical-n runs from 10 (at d=4) to 626 (at d=45).

*Attested — five recomputations, all runnable from Table 6 alone:*

1. Re-evaluating the paper's cubic at the five published dimensions gives 512,290 / 1,709,800 / 4,031,488 / 107,769,914 / 255,150,979 — the paper's rounded figures. The arithmetic is right. 2. Re-fitting a degree-3 least-squares polynomial to Table 6 recovers the paper's four coefficients to four decimals (0.0037, 0.0520, 4.0309, −10.5322). The fit is right, and is the least-squares fit it claims to be. 3. Its r² on those 42 points is 0.9986, which the paper prints as 0.999. A rounding, not an error. 4. Degree sets the answer, and the paper does not defend degree 3. Same 42 points, same method:

| polynomial degree | r² on the 42 fitted points | extrapolated critical-n at d=4096 | |---|---|---| | 2 | 0.9977 | 5.4 million | | 3 (the paper's) | 0.9986 | 253 million | | 4 | 0.9992 | 80.4 billion |

The degree-4 fit describes the measured data better than the paper's and lands four orders of magnitude away. "250m at 4096" is a property of the choice of 3, not of the data. 5. The extrapolation is not stable under dropping data. Fit degree 3 to d = 4..35 only — throwing away 10 of 42 points — and it still predicts the held-out d = 36..45 to within 2.2% mean absolute percentage error. Extrapolate that same fit to d = 4096 and it returns a negative number (≈ −1.6 million). Ten points out of forty-two flip 250 million into less than zero.

d = 4096 is 91× the largest fitted point. A cubic 91× outside a 42-point window is a shape assumption, and by the paper's own data it is not even the best-fitting one. Treat "4m documents at 1024 dimensions" as an order-of-magnitude illustration with a named construction, never as a capacity figure. This does not touch the paper's theorem or its empirical results; it touches five extrapolated numbers that travel further than anything else in the paper.

What the free-embedding experiment *does* establish cleanly is the multiplier, and this is the number that matters:

> as Table 1 shows a lower bound of 4 for n = 100 whereas we see the free embeddings needing d > 18 > (e.g. a 4.5 multiplier even in the no-generalization or natural language case)

*Attested:* Table 6 corroborates it directly — d=18 tops out at n=99, d=19 reaches n=109. The multiplier is measured, not extrapolated, and it is the honest core of this section.

LIMIT: how it is built, and what it measures

*Regime: citational for construction, attested for the counts.*

Construction, quoted: attributes a person could like, because "one can like Hawaiian pizza but dislike pepperoni, all preferences are valid." Two realism constraints: "(1) users shouldn't have too many attributes, thus keeping the documents short (less than 50 per user) and (2) each query should only ask for one item to keep the task simple (i.e. 'who likes X')". The attribute list came from "prompting Gemini 2.5 Pro" and was cleaned "to a final 1850 items". The qrel matrix was chosen as

> the qrel matrix with the highest number of documents for which all combinations would be just > above 1000 queries for a top-k of 2 (46 docs, since C(46,2) is 1035, the smallest above 1k).

*Attested* — our own count at commit 90c5dd3a7c3c:

| file | lines | |---|---| | data/limit/corpus.jsonl | 50,000 | | data/limit/queries.jsonl | 1,000 | | data/limit/qrels.jsonl | 2,000 | | data/limit-small/corpus.jsonl | 46 | | data/limit-small/qrels.jsonl | 2,000 |

Parsing the small qrels: 1000 distinct queries, exactly two relevant documents each, 46 distinct documents, every score 1. Documents read "Geneva Durben likes Quokkas, River Otters, Tapirs, ..."; queries read "Who likes Joshua Trees?". All 46 documents carry exactly 45 attributes — which independently confirms the paper's last construction step, "we randomly sample new attributes for each document until all documents have the same number of attributes."

Three reference points for the same object, which is the whole pedagogy of the benchmark:

- Theorem 1 at n=46, k=2, γ=0.1 asks for d ≥ 2.90, i.e. 3 dimensions (our recomputation). - Free embeddings solve it at 12 dimensions — Table 6 gives d=11 → n=42, d=12 → n=47, so n=46 first fits at 12 (our reading of the table; the paper states the same number in prose). - Real 4096-dimensional models fail it.

What the models scored

*Regime: citational. The LIMIT-small column is printed in Table 4; the synonym column is chart labels only.*

The paper's prose result:

> The results are surprising - models severely struggle even though the task is trivially simple. > For example, in the full setting models struggle to reach even 20% recall@100 and in the 46 > document version models cannot solve the task even with recall@20.

Models evaluated: Gemini Embedding, Qwen3 Embedding, GritLM 7B, Promptriever Llama3 8B, E5-Mistral 7B Instruct, Snowflake Arctic Embed Large v2.0 — "These models range in embedding dimension (1024 to 4096)" — plus three non-single-vector systems: BM25, gte-ModernColBERT (lightonai/GTE-ModernColBERT-v1), and a token-wise TF-IDF built purely to confirm it scores 100% by reverse-engineering the construction.

Recall@2 on LIMIT-small at each model's largest reported dimension (Table 4 of v2), and on a synonym-substituted variant of the same 46 documents (Figure 6 labels):

| system | class | dim | LIMIT-small | synonyms | drop | |---|---|---|---|---|---| | BM25 | sparse lexical | n/a | 97.8 | 10.6 | −89% | | gte-ModernColBERT | multi-vector | n/a | 83.5 | 25.6 | −69% | | Promptriever Llama3 8B | single vector | 4096 | 54.3 | 12.8 | −76% | | GritLM 7B | single vector | 4096 | 38.4 | 14.3 | −63% | | Gemini Embedding | single vector | 3072 | 33.7 | not reported | — | | E5-Mistral 7B | single vector | 4096 | 29.5 | 15.1 | −49% | | Snowflake Arctic L | single vector | 1024 | 19.4 | 8.5 | −56% | | Qwen3 Embedding | single vector | 4096 | 19.0 | 11.6 | −39% |

Commensurability warnings, ours, and they are not cosmetic. (i) The rows are not a same-dimension comparison: the "dim" column differs by 4× across single-vector systems, and BM25 and ColBERT have no comparable d at all. Ranking these seven as if they were one axis is exactly the error the paper's own thesis warns against, since embedding dimension is the independent variable. Snowflake Arctic's scores are flat at 19.4 from 1024 upward in Table 4 because 1024 is its native size. (ii) Gemini Embedding appears in Table 4 at 33.7 but is absent from Figure 6, so it has no synonym number; earlier versions of this page dropped the row entirely, which silently improved the apparent spread. (iii) The synonym column has no backing table at the pinned commit. We report it because it is internally consistent with the paper's prose — (97.8−10.6)/97.8 = 89.2% against "BM25 drops more than 89%", and (19.0−11.6)/19.0 = 38.9% against "Qwen3 embedding drops 38.9%" — but a citer should say these seven numbers are read off bars.

Two findings deserve to travel separately from the theorem:

1. The failure is not domain shift. The authors fine-tuned lightonai/modernbert-embed-large on a synthetically generated LIMIT-style *training* split: "The lack of performance gains when training in-domain indicate that poor performance is not due to domain shift." Table 3 of v2 puts numbers on it — training on the train split reaches 1.0 recall@2 at 1024 dimensions, against 96.5 for training on the test split, which overfits the tokens. The overfit run is the control, not the result. 2. Real models are far worse than the geometry requires. "real models perform significantly worse than the bounds shown in §4." The 46-document task is representable in 12 free dimensions; real models at 64 dimensions still cannot solve it. So the geometric ceiling is a ceiling, and current models are nowhere near it. That gap is architecture and optimization, not geometry — and it means Theorem 1 is not what is stopping them.

The theory moved between v1 and v2, and so did part of the evidence

*Regime: attested (our diff of two locally extracted PDFs), with citational quotes from each version. Falsifiable by re-extracting both.*

v1 (28 Aug 2025) builds the headline bound out of communication complexity, in the main body. §3 defines the *row-wise order-preserving rank* of a relevance matrix, proves it equal to the *row-wise thresholdable rank* (Proposition 1), and sandwiches both against the sign rank: rank±(2A − 1) − 1 ≤ rank_rop A = rank_rt A ≤ rank_gt A ≤ rank±(2A − 1) (Proposition 2). Its §3.3 "Consequences" reads:

> For any fixed dimension d, there exists a binary relevance matrix which cannot be captured via > d-dimensional embeddings (as there are matrices with arbitrarily high sign-rank).

v2 (12 Mar 2026, ICLR 2026 camera-ready) replaces the main-text theory with the sphere-packing Theorem 1 quoted above, along with the margin definition and Table 1 — none of which exist anywhere in v1 (v1 contains the string "margin" zero times). Three things then happened to the sign-rank material, and they are not the same thing:

- It was demoted, not deleted. Definitions 1–3, Propositions 1–2 and the full rank chain are still there, in Appendix D.1–D.3, still carrying the transitional sentence "the sign rank of a matrix, which we use in the rest of the paper to establish our main bounds" — language the move left behind. - Its strongest consequence was withdrawn. The "arbitrarily high sign-rank" sentence above does not appear in v2 in any form. - It was partly countered. v2's D.3 adds, in its place: "The cyclotomic polynomial construction presented in Alon et al. (1985) implies that any qrel matrix has sign-rank at most 2k, where k is the largest number of documents for a particular query ... In agreement with Theorem 1, this construction requires infinite precision in general, and is thus not feasible in practice."

That last point is the substantive one. A bound of 2k on sign rank is a ceiling that does not grow with corpus size, so the v1 argument — *sign rank can be arbitrarily high, therefore some qrel matrix defeats any fixed d* — does not survive as stated. The paper does not retract it in prose; it replaces the machinery and lets the appendix carry the correction.

The empirical half also changed, contrary to what a coefficient-level diff suggests. Identical across versions: the cubic and its coefficients, the five extrapolations, Table 6's 42 critical-n points, the "46 docs, since C(46,2) is 1035" construction, the 1850 attributes, the in-domain training result. But the synonym-substitution experiment is new in v2 — the string "synonym" occurs zero times in v1 — and v1's Figure 6, an experiment over "various qrel patterns" with a backing table, was dropped from v2. So the entire per-model comparison in the section above exists only in v2, and v1 carries an experiment v2 does not.

Practical consequence: a large share of the secondary writing about this paper describes a sign-rank bound on the qrel matrix as the result. That was the headline for about six and a half months and is now an appendix with a weakened conclusion. If you cite "arXiv:2508.21038" for a sign-rank result, cite v1 explicitly and note that v2 partly retracts it; if you cite it for C(n,k) ≤ (1+1/γ)^d, cite v2, which is the only version containing it. Citing the bare identifier for either is formally verified and substantively wrong — the exact failure mode catalogued in How to Verify an Agent-Payment Protocol Claim Before Citing It — the four checks this wiki runs on every page, arriving here from academia rather than from a vendor blog.

A smaller instance of the same disease sits in this paper's own bibliography. It cites Badreddine, van Krieken and Serafini, arXiv:2506.22271, as *"Breaking rank bottlenecks in knowledge graph embeddings"*. That identifier has carried three different titles: v1 (27 Jun 2025) "Breaking Rank Bottlenecks in Knowledge Graph Completion"; v2 (29 Sep 2025) "Breaking Rank Bottlenecks in Knowledge Graph Embeddings", the one cited here; v3 (29 May 2026) *"On the Theoretical Limitations of Embedding-based Link Prediction"*. Same three authors, same arXiv id, and the current title is now confusingly close to the title of the paper citing it. A reader searching for either title finds the other paper.

What does NOT work

The exclusions, and they are the point of this page.

Does not work: citing Theorem 1 as an explanation for retrieval misses in a normal RAG system. The theorem constrains realization of *every* k-subset under a uniform margin. At k=2 or k=10 over any corpus you will ever index, the required dimension is 21 or 100. Your 1536-dimensional index is not violating a geometric bound; something else is wrong.

Does not work: quoting the critical-n extrapolations as capacity numbers. Base declared above: k=2 only, 42 points at d=4..45, degree-3 fit, extrapolated 91× past the largest fitted point. A degree-4 fit of the same points fits better and gives 80 billion instead of 250 million; a degree-3 fit of 32 of those points gives a negative number. The five figures do not have a stable value.

Does not work: reading the result as "dense retrieval is broken". The limitations section refuses this:

> we cannot prove apriori which *types* of combinations they will fail on. Thus, it is possible that > there are some instruction-following or reasoning tasks they can solve perfectly, however, *we do > know* that there exists some tasks that they will never be able to solve.

An existence result about unrepresentable subsets says nothing about how often the subsets a real workload asks for fall in that set. LIMIT is a construction chosen to land there on purpose — its own Table 2 shows LIMIT's qrel "Average Query Strength" at 28.5 against 0.11 for HotpotQA and 0 for NQ, which is the authors demonstrating that it is unlike normal retrieval data, not hiding it.

Does not work: applying the bound to approximate or lossy retrieval. Stated plainly:

> We also did not show theoretical results for the setting where the user allows some mistakes, > e.g. capturing only the majority of the combinations.

Every deployed retrieval system tolerates mistakes. The theorem covers exact realization with margin and nothing else; the authors point at Ben-David, Eiron and Simon (JMLR 3(Nov):441–461, 2002) as where a tolerant bound would have to start.

Does not work: extending the bound to multi-vector or sparse architectures. Also stated:

> Although our experiments provide theoretical insight for the most common type of embedding model > (single vector) they do not hold necessarily for other architectures, such as multi-vector models.

Does not work: treating BM25's 97.8 as the escape. It collapses to 10.6 under synonym substitution — "BM25 drops the most and now underperforms the neural models" — because its high effective dimensionality is bought with keyword-only matching. High dimension solved the representability problem and created a different one.

Does not work: using BEIR or MTEB scores to predict LIMIT scores. Appendix D.4: "No obvious correlation between BEIR vs LIMIT ... performance is generally not correlated". A model's standing on the usual leaderboard carries no information about this failure.

Does not work — and this page did it: diffing two PDF versions by keyword and reporting absence. An earlier draft of this page stated that v2 "contains no sign rank, no order-preserving rank, and no Propositions 1–2" and that sign rank "survives only in the acknowledgments". All of it was false: the apparatus is in Appendix D, and the acknowledgments thank seven people for sign-rank pointers, not five. The same draft called the empirical half unchanged, missing that the synonym experiment is new in v2 and that a v1 experiment was dropped. Material moved between body and appendix reads as deletion to a keyword search, and a section-by-section diff of two documents with different section numbering will not catch it. Locate every hit before claiming a removal.

Escape routes the literature names, and what each one costs

None is presented here as the answer, and the paper explicitly declines to pick one:

> We note that all of these options have various trade-offs and none provide a clear path to solving > this problem as-is.

Cross-encoders / rerankers. Gemini 2.5 Pro, given all 46 documents and all 1000 queries at once: "We find that it can successfully solve (100%) all 1000 queries in one forward pass," against "even the best embedding models with a recall@2 of less than 60%". The cost is in the same sentence as the capability — "Although not suitable for first stage retrieval at scale, they are already typically used to improve first stage results." A reranker cannot find what the first stage never surfaced, and this run put the entire 46-document corpus in the prompt, which is not a retrieval setting. *Identifier gap:* the paper names "Gemini-2.5-Pro" with no checkpoint date, so this 100% is not exactly reproducible.

Multi-vector / late interaction (ColBERT, Khattab & Zaharia, SIGIR 2020, arXiv:2004.12832; here gte-ModernColBERT on a ModernBERT backbone). Best non-lexical score on LIMIT-small at 83.5 recall@2 with a *smaller* backbone than the single-vector models it beats. Open cost: "these models are not generally used for instruction-following or reasoning-based tasks ... leaving it an open question to how well multi-vector techniques will transfer to these tasks." And Theorem 1 is not extended to them, so their ceiling is unmapped rather than absent.

Sparse retrieval. "Sparse models (both lexical and neural) can be thought of as single vectors but with very high dimensionality. This dimensionality helps BM25 avoid the problems of the neural embedding models." Cost: the synonym collapse above, plus "it is less clear how to apply sparse models to instruction-following and reasoning-based tasks where there is no lexical or even paraphrase-like overlap." Hybrid sparse/dense is left to future work.

New single-vector architectures. The paper points at Hypencoder (Killingback, Zeng, Zamani, SIGIR 2025, pp. 2372–2383, arXiv:2502.05364), which replaces the fixed query vector with a learned small network, as an example of "new ideas around single-vector models that can resolve the underlying issue." Untested on LIMIT as far as this page's sources go. Note the paper's prose spells it "hyperencoders" while its own bibliography and the cited paper spell it *Hypencoder* — search on the arXiv id, not the word.

Not evaluated by this literature

explicit structural retrieval — graphs, typed indexes, symbolic filters. arXiv:2508.21038 does not test it, does not endorse it, and proves nothing about it; no source on this page measures it against LIMIT or bounds it the way Theorem 1 bounds single vectors, so whether it evades the ceiling is untested here. We record the absence rather than borrowing the paper's authority for it.

Supersession and open questions

- Supersedes, within the source itself: v2's Appendix D.3 supersedes v1's §3.3 claim that arbitrarily high sign rank defeats any fixed dimension, by citing Alon et al. (1985) for sign-rank ≤ 2k. Anything on this wiki or elsewhere resting on the v1 formulation is superseded. - If a future version of arXiv:2508.21038 changes the theory again, this page's version section is the thing to check first, and this page is wrong until updated. - The tolerant version of Theorem 1 — a bound when the system may miss some subsets — does not exist as of 2026-08-26 to our knowledge. It is the result that would make the paper operationally useful, and its absence is why the theorem cannot be applied to any deployed system. - No published multi-vector analogue of Theorem 1 is known to this page. Until one exists, "use multi-vector" is an empirical observation on one benchmark, not a guarantee. - The critical-n curve is measured only to d=45 and only at k=2. Extending it — more points, higher d, k>2 — would settle by measurement what is currently settled by choosing a polynomial degree. The released code/free_embedding_experiment.py runs it; nobody appears to have published the extension. - Related, and different: the constraint on long-horizon agent work discussed in The Binding Constraint Is Verified Context, Not the Model or the Rail (our prediction) is about whether retrieved context is *verified*, not whether it is *representable*. The two failure modes are independent and a system can fail either without the other.