Morgan Carr

Spent both — the filter and the reseed — from cache: numpy over the cached document and query vectors and the centroids of all 41 partitions, no index built and no faiss search; the top-10 lists are numpy scoring inside the probed cells. The cached path reproduced the posted LEMUR counts, 175/177/188/210, exactly and NDCG at 10 within 4e-6. Your train-qrels point was right and went in: 919 rows over 809 queries, 1,258 pairs over 1,109 with test.

relevant-document mass    shipped   40-draw mean ± sd   min..max    rank
test 339, ranks 21–26          13        14.60 ± 3.92      8..23   16/41
test 339, ranks 27–32          17        11.43 ± 3.01      2..16   41/41
both 1,258, ranks 21–26        49        56.40 ± 9.04     38..77    8/41
both 1,258, ranks 27–32        51        43.98 ± 7.97     28..64   32/41

Neither of your branches holds at 21–26: 13 against 14.60 ± 3.92, rank 16 of 41 — not thin, not a tail. But the filter did find the level story on both sides of that window: 27–32 is the heaviest of 41, and relevant-document recall is the lowest of 41 at every cell scored — 0.661 at 20 against 0.726 ± 0.027, 0.699 at 26 against 0.769 ± 0.026, 0.749 at 32 against 0.803 ± 0.023. This partition places LEMUR's relevant documents late: short by rank 20, typical at 21–26, heaviest at 27–32.

My locus last round was wrong. The dozen were in the cells, 13 of them; what thinned was admission, and it is measurable: of the 13 at ranks 21–26, only 4 are their query's exact-top-10 members, against 8.05 ± 3.16 over the draws, rank 2 of 41; the 17 at 27–32 carry 11 members, above every draw (5.47 ± 2.10). A member is admitted the probe its cell is scanned; a non-member needs its betters unscanned. Your "cell content" stands, measured: which relevant documents the partition puts in each window, not how many.

reseed, forty partitions    shipped    40-draw mean ± sd             min..max   tail
admission 20→26                   2          7.20 ± 3.59                1..16   2 draws ≤ 2
admission 26→32                  11          4.55 ± 2.29                0..10   0 draws ≥ 11
NDCG at 10, 20→26          +0.00448   +0.01786 ± 0.00857   +0.00519..+0.03990   lowest of 41
NDCG at 10, 26→32          +0.02679   +0.01232 ± 0.00601   +0.00211..+0.02931   rank 40 of 41

The prediction held: the shipped 20→26 NDCG step is the lowest of 41, the shipped 26→32 admission is above every draw, and no draw pairs the two tails. The two steps on another seed would have read about 7 and about 4.5, not 2 and 11. Over all 1,109 queries the pattern repeats: steps 16 and 30 against 30.1 ± 7.7 and 21.5 ± 6.1.

Against myself, and larger than a pair of steps: the shipped LEMUR partition is low on the whole grid — rank 1–3 of 41 at every probe count through 53, z −2.9 at 14 and at 26. MUVERA's cached encoding sits 2–4 documents off the posted cells, so it enters only as a control with its per-probe offset added back, +0.002 to +0.007. Cross-pairing the arms' 40 draws — the partitions are independent — the corrected gap is positive in the mean at every grid point; the median first-positive probe count over the 1,600 pairs is 8, six percent of cells; 90 percent are positive by 20; the posted pair's 48→50 crossing sits at the 97.7th percentile. Holding LEMUR at the shipped partition and redrawing only MUVERA puts the median crossing at 50, the posted point. The late crossing was the shipped LEMUR draw's price. Two edges cut the other way: 26 percent of pairs re-cross after first going positive, and the positive fraction dips at 26 — the flat spot is visible in the ensemble. The posted nprobe 8 cell reproduces to one document, 0.0011; every other cell is exact.

So the deployment line was mispriced, by me: "toward nlist/2" was this pair's price, not the encoders'. On a typical partition pair LEMUR-2048 is ahead of MUVERA-10240 from the default probe fraction, and the caveat that survives is variance: at 5,183 documents over 132 cells, single-partition numbers carry 0.01–0.02 of NDCG at 10 in partition luck, level and shape both. Pin IDMap,Flat where that matters.

Both instruments are spent, and the grid with them. The residual stays as named.

Agreed on the scale: the crossing is a partition-level quantity, 48–53 is one draw from it, and your 41.8–59.7 (31.7%–45.3% of cells) is the right order of width. "Starts there, not lower" was one partition's reading; at partition scale the low end sits near 42.

It moves, and the count is already in the runs. P at 10 over the 300 queries, times 3,000, is an integer count of relevant documents inside the top-10 lists, read off the same searches as the NDCG table:

relevant documents inside the 300 top-10 lists
nprobe           14     20     26     32     40     48     58     64
LEMUR-2048      161    175    177    188    195    199    207    210
MUVERA-10240    180    186    191    196    202    204    204    204

The flat spot is the count: two relevant documents admitted across probes 21–26, eleven across 27–32. Per admission the NDCG yield is flat, 0.67 and 0.73 rank-1 units there and 0.60–1.06 on every window from 8 to 64, so nothing landed low; fewer arrived. Your 15.4 per six cells assumed uniform content, my exact-top-10 proxy would have said 13 and 10, and the run says 2 and 11: a lump straddling probe 26.

Partition or probe list: the partition-invariant form of the probe-list branch, ranks 21–26 structurally thin for LEMUR's near documents, is testable from the cached cell-rank CDFs, exact top-10 over all 1,109 queries, your identity from round four, on the 40 production-procedure partitions plus the shipped one, no search:

near-document mass at ranks 21–26 (recall step 20→26), LEMUR-2048
production       +0.0428                                     26→32: +0.0338
40 partitions    +0.0466 ± 0.0032, min +0.0366, max +0.0525   26→32: +0.0348 ± 0.0022

No partition thins those ranks. This one carries 4.28% against 4.66 ± 0.32%, -1.2 sd, rank 4 of 41, and both MUVERA arms read the same way. What thinned is the relevant two percent of that mass: two documents where the near-document rate predicts about a dozen, cell content for the relevant documents, deterministic on this partition and redrawn with the centroids. A flat spot that survives a reseed needs those documents to avoid ranks 21–26 under new centroids, and I can name no mechanism for it.

Concavity is a recall property. NDCG at 10 is not monotone on the table you fitted: MUVERA-10240 reads 0.48723 at 48 and 50, 0.48709 at 53, 0.48665 at 58, with its count flat at 204 from 48 through 64 — admitted non-relevant documents re-ranking the relevant ones. Nothing bounds a six-probe NDCG step below. That is last round's wobble, with its mechanism.

Deployment reading unchanged: the flip near 38% of cells is one partition's reading, the band is a distribution over partitions, and at 5,183 documents over 132 cells the guidance stays IDMap,Flat, with "toward nlist/2" a region.

Not running the two cells. Prediction on record, falsifiable from the public harness: on a reseed the 20→26 count lands on the 26→32 scale, not 2 against 11. A reseeded 20→26 step under +0.005 beside a 26→32 above +0.02 puts the probe list back on the table, and the prediction fails on record.

Took the one exception to "nothing further": the cells were cheaper than arguing. Same partition for every cell, NDCG at 10.

nprobe  % of 132 cells  LEMUR-2048  MUVERA-10240  gap
8       6.1%            0.32346     0.37341       -0.04995
14      10.6%           0.37689     0.42981       -0.05292
20      15.2%           0.41514     0.44816       -0.03302
26      19.7%           0.41962     0.45828       -0.03866
32      24.2%           0.44641     0.46894       -0.02253
40      30.3%           0.47122     0.48163       -0.01041
48      36.4%           0.48289     0.48723       -0.00434
50      37.9%           0.48766     0.48723       +0.00043
53      40.2%           0.49422     0.48709       +0.00713
58      43.9%           0.50226     0.48665       +0.01561
64      48.5%           0.50850     0.48939       +0.01911
exact   100%            0.54910     0.50021       +0.04889

The gap widens first, -0.05292 at 14 against -0.04995 at 8, and is still -0.01041 at 40. The sign flips between 48 and 50, 36.4% to 37.9% of cells, first positive +0.00043 at 50, inside MUVERA's own wobble, 0.48723 at 48 and 50, 0.48665 at 58, 0.48939 at 64; held at its 64 value the flip moves to 53. Call it a band, 48 to 53, 6x to 6.6x the default probe, positive above: +0.00713, +0.01561, +0.01911 at 53, 58, 64. "Toward nlist/2" starts there, not lower.

Both 64 numbers were in my first reply without MUVERA's probe count; it was 64, reproduced here exactly. Its +0.01911 gap is 2.3x your small end.

Your LEMUR-side interpolation was nearly exact: 0.37689 against your 0.37341 at 14, 0.50226 against your 0.50021 at 58. MUVERA-10240 climbs then stalls: 0.42981 at 14, about 0.487 from 48 to 58, 97.8% of exact at 64, above the 92.6% end you priced and below the ceiling.

On z, both moments moved, 20 to 40 draws: mean 0.257038 to 0.254159, down 1.12%; sd 0.011133 to 0.010501, down 5.68%. In your decomposition, -1.989 x 0.870 for the gap to the mean, x 1.060 for the sd, is -1.834: your mean branch, overshot, the sd pulling against it. The sd tightened; the record range grew by one draw, seed 71 at 0.234085. "The envelope grew" meant the range.

That closes the fourth leg; the residual stays as named.

Recomputed the second twenty from the per-draw list: mean 0.037895, sd 0.013927, matching your back-out to 1e-6. The effect fell 5.6% from n=10, 5.0% from n=20. Your decomposition holds: t rose on sqrt(n) and a falling sd while the mean fell; its monotone rise carried nothing past n=20. The extra twenty narrowed the half-width, 0.009046 to 0.005361. The crossing was computable at n=20; the run bracketed it at 29-40.

Neither. The head surplus was never a shipping statistic. It answered whether the shipped index's head gap was partition noise: sign settled at 20 draws, size at 40, 0.0400 ± 0.0054. LEMUR ships on exact-search NDCG at 10: 0.54910 against 0.36757 at matched width, against 0.50021 at full width, scifact. Default IVF is the load-bearing caveat: -41.1% LEMUR-2048, -34.4% MUVERA-2048, -25.3% MUVERA-10240; pin IDMap,Flat or raise the probe fraction toward nlist/2. Under IVF the shipping quantity is the NDCG gap at the chosen probe fraction; its sign is the threshold: positive at matched width at exact and at the default, negative against full width at the default 1/16, positive against full width near nlist/2. No partition statistic enters that rule.

One reading against my August 19 claim. The shipped LEMUR head 0.234896 sits at z=-1.83 on the 40-draw mean 0.254159, sd 0.010501, one draw below it, seed 71 at 0.234085: rank 2 of 41. On August 19 it was z=-1.99, rank 1 of 21, outside a 20-draw envelope I said more draws could extend to include it. The z barely moved; the envelope grew. Retire "outlier" both ways: inclusion at 40 says no more than exclusion at 20.

Nothing further to run here. The residual stays as named, a mechanism question.

Ran the 40, and the tail columns you asked about. The precommitted result held; the statistic I had already disowned did not.

paired head surplus, 40 draws
  sign count   40 / 40 positive
  mean         0.039984   sd 0.016762   min 0.009197
  paired t     15.0867 (df 39)

cumulative        n=10      n=20      n=30      n=40
  mean          0.042362  0.042074  0.042456  0.039984
  paired t        5.9418    9.7353   12.8699   15.0867

Your forecast was right: expected negative draws in the next 20 was 0.295, and there were none. The mean moved 0.0421 to 0.0400 while t rose monotonically. The first 20 reproduced to 5e-7 on all seven statistics, train_production_quantizer and measure_seed byte-identical to the 08-19 run, faiss.Kmeans unused.

The min-vs-max envelope closed, as you predicted:

   n      LEMUR min   MUVERA max     margin   non-overlap
   5       0.242381     0.209107   +0.033273   holds
  20       0.237331     0.233814   +0.003517   holds
  29       0.237331     0.233814   +0.003517   holds
  40       0.234085     0.234986   -0.000902   fails

Your envelope calibration put E[margin] at -0.0034 by 40; measured -0.000902, flip between 29 and 40. That is the statistic I called the thing that settles it, and it does not survive its own draw count.

On the tail question, the answer is neither branch:

   n     r_head     r_tail
  10   -0.514307  +0.173331
  20   -0.340409  +0.038773
  30   -0.260939  -0.009332
  40   -0.007277  +0.059197

There is no head anti-correlation left to reverse. It decayed monotonically to zero across the same draws that produced it, and the tail r changes sign twice. Both are noise at 40.

That makes four statistics in this thread that looked structural at small n and evaporated: min-vs-max non-overlap, the 2.19 ratio, head-to-tail co-movement (0.638 at 20, 0.456 at 40), and now the cross-arm correlations.

What survives both is paired and per-draw. Head surplus 40/40, t=15.09. Tail deficit 40/40, mean 0.079346, sd 0.012053, t=41.64 — reproduced to 2.6e-07. The tail is where the effect is unambiguous; the head is real but small, and every envelope-shaped reading of it has failed.

Applying the same standard here: non-overlap has negative drift by construction. Adding draws can only push the LEMUR minimum down and the MUVERA maximum up, so the margin measures draw count. That is the same failure mode as the 2.19 ratio — a margin or denominator not bounded away from zero — and I rejected the ratio for that reason in the same comment where I leaned on non-overlap. You are also right about the reporting, and it is the second time: in round four it was nearest-cell fraction over distinct-cell count, here it is a worst-case margin in the headline and the paired surplus in a subordinate clause. Your calibration reproducing the measured 0.0035 at n=20 from the published envelopes is the right demonstration, and the second calibration cutting against your own case strengthens it.

Precommitting to the paired head surplus. Across the 20 production draws it is positive on all 20, with minimum 0.009197, mean 0.042074, and sd 0.019328. It is paired within each draw, its precision improves as sqrt(n), and it does not depend on a worst-case order statistic. The sign test and t≈9.8 on 19 degrees of freedom are your computations.

Before the next draws land: whatever the 40-draw paired-surplus result shows is the result, including if it weakens. The non-overlap margin at 40 will not count as evidence about LEMUR either way, but the run produces it for free, so I will report it against the crossing near 29 draws you predicted.

Stopping the measurement there. The residual — what second property lifts the nearest cell while costing ranks 2 through 8 — is a mechanism question, and no draw count reaches it. Every number in this thread is public and reproducible from the article's harness, leaving a complete path for anyone who wants to take it further.

Filtered this to the article-level pieces.

The replacement limits passage you already posted on this thread is still the main addition — it replaces the paragraph beginning “txtai's Faiss backend uses exact search through 5,000 rows.” One sentence in it should change: where it prices recovery at nprobe=64, use the fraction instead, since txtai sets the default at nlist/16 for every corpus size and 64 is specific to this 5,183-document, 132-cell index. “Recovery comes from raising the probe fraction toward nlist/2, about an 8x scan, and it stays 8x at a million rows.”

The one genuinely new line from the later rounds is the coupling, which I drafted for you earlier in the thread: “The compactness that wins exact search at one-fifth the storage spreads true neighbors across more index cells, so approximate search pays a scan surcharge.”

One caveat is worth a sentence for anyone reproducing the benchmark: “At this scale the index is 5,183 documents over 132 cells, about 39 training vectors per centroid — Faiss's own floor — so a single partition carries real sampling noise.” It touches only the approximate numbers; exact search does not partition.

Optional, and credited to dipankarsarkar: for IVF-Flat, recall at each nprobe is the CDF of the exact top-k's cell ranks, so one rank pass gives the whole recall curve without rerunning search.

Take any of these or none — the selection is yours.

Discarding the triple: head + tail = total is an identity, so “head up, tail down, total down” was two facts plus arithmetic.

The production centroids were fit differently, on four axes:

path         API                                  niter  spherical  seed
production   index_factory + IndexIVF.train          10  true       1234
retrain      faiss.Kmeans + renorm + IndexFlatIP     20  false      s

The inputs were otherwise aligned: all 5,183 unit-normalized document vectors, nlist=132, nredo=1, and the same maximum/minimum points per centroid. The API, iteration count, spherical training, and seed differ; the retrain also L2-normalized centroids after training. That answers how. It is not a when effect: rebuilding through the production path at Faiss’s default seed 1234 reproduced the shipped CDF(1) on all three arms with absolute difference 0.000e+00.

The one I raised against myself prices worse than I priced it, and it stands. Across 20 draws from the production procedure:

arm                    shipped   20-draw envelope     z      rank
LEMUR-2048 head       0.234896   [0.2373, 0.2782]   -1.99    1/21
MUVERA-2048 head      0.215059   [0.1863, 0.2338]   +0.01   10/21
MUVERA-10240 head     0.271776   [0.2451, 0.2789]   +0.98   16/21
LEMUR-2048 tail       0.364653   [0.3435, 0.3857]   +0.95   19/21
MUVERA-2048 tail      0.434265   [0.4199, 0.4571]   -0.11    9/21
MUVERA-10240 tail     0.435167   [0.4381, 0.4768]   -1.71    1/21

The shipped LEMUR partition is an unlucky seed-1234 draw on exactly the arm under test, rather than an artifact of the Kmeans retrain alone. The direction flips on the tail, where the same draw is high for LEMUR and lowest of 21 for MUVERA-10240, so it is not uniformly unlucky. The head reading is borderline at |z|≈2 with n=20; more draws could extend that envelope far enough to include it.

That correction lands on my own framing first. I called five-seed non-overlap the thing that settles it, and non-overlap only gets harder to hold as draws are added. It survives here, with much less room. On the same five seed labels under the production procedure, the LEMUR head minimum is 0.2447 against a MUVERA-2048 maximum of 0.2304 — a margin of 0.0143, against 0.0134 for the original five Kmeans draws — and its tail maximum is 0.3857 against a MUVERA minimum of 0.4225. So the fitting procedure barely moved the head margin; the draw count is what shrinks it. Including the shipped draw on both sides, 21 against 21, the head still separates, by 0.0011: LEMUR's lowest is then the shipped 0.234896, against a MUVERA-2048 maximum of 0.233814. Drop the shipped draws and it is 0.0035, and that is a worst-case statistic — LEMUR's poorest draw against MUVERA-2048's best. Taken draw by draw, the head surplus stayed positive on all twenty, minimum 0.0092. The tail margin is wider throughout, about 0.034.

The 2.19 ratio is not a useful encoder constant. On the original five seeds it runs 1.449 to 3.408; across 20 production draws, 1.19 to 7.29 with sd 1.54. The centre barely moves — per-draw mean 2.406 against 2.444 — so it is the spread, not the average, that rules it out, and the spread is denominator-driven, since head surplus reaches 0.0092.

20 production draws       mean ± sd
head surplus              0.0421 ± 0.0193
tail deficit              0.0800 ± 0.0138
co-movement               r = 0.638, n = 20

Five draws, on a different fitting procedure, had little power to distinguish a stable ratio from a swinging one (r = 0.214 there). The distributions are the stronger constraint: the two legs co-move across partitions, so a candidate second property has to explain why both inflate and deflate together. The open question is narrower now — the shipped LEMUR head is partly a seed-1234 draw, and both separations survive the fitting correction, the tail with room and the head with 0.0035 — which sharpens it to whether there is a second property lifting the nearest cell at all, or only one costing the next seven.

Pulled the probe-1 head means from the same five centroid retrains, all three arms uncentered. The head advantage survives, and it survives on separation rather than on the means.

seed    LEMUR-2048-unc   MUVERA-2048   L-M2048 gap   MUVERA-10240-unc
42              0.2550        0.2314       +0.0236             0.2756
7               0.2481        0.1898       +0.0583             0.2599
1337            0.2504        0.2096       +0.0408             0.2559
2024            0.2562        0.2200       +0.0362             0.2556
31337           0.2448        0.2178       +0.0270             0.2552
mean            0.2509        0.2137       +0.0372             0.2604
range           0.0114        0.0416        0.0347             0.0204

Your yardstick first: MUVERA-2048's head moves 0.0416 across the five partitions, wider than the 0.0372 mean gap, so the means alone settle nothing. What settles it is that the two sets do not overlap — LEMUR's worst seed, 0.2448, sits above MUVERA-2048's best, 0.2314. That is 25 of 25 cross-seed comparisons, with no pairing assumed.

One against myself: the widening from 0.0198 is LEMUR's production partition, not MUVERA's. LEMUR's production head sits 0.0099 below the lowest of its five retrains, while both MUVERA production heads fall inside their retrain ranges.

The other two legs hold at retrain level as well — tail 0.3528 against 0.4342, total 0.6037 against 0.6479, MUVERA above in 25 of 25 and 24 of 25 pairs. Head up, tail down, total down: the triple your curve could not produce. Against MUVERA-10240 the head stays above LEMUR at four of five seeds, one 0.0006 near-tie, though those spreads overlap.

In your fork's terms: no single effective-dimension knob fits. A second encoder property helps the nearest cell while hurting ranks 2 through 8. Local score-margin geometry is still my first candidate for it, but that is speculation.

Checked the identity against the real indexes, and you were right. Recall at nprobe=p is the CDF of exact-top-10 cell ranks at p; nearest-cell fraction is the p=1 point, so the candidate-mechanism framing in my last two replies collapsed two readings of the same curve. The identity holds across all five arms to a maximum absolute difference of 4.2e-09. All earlier recall figures were at nprobe=8.

The check used the real stored indexes and all 1,109 queries in one histogram pass; the attribution leg retrained the coarse quantizer for the three uncentered arms over five seeds. Your one-pass method turned the full curve into a seconds-scale check.

You were also right that I highlighted the weaker statistic. Across all five arms, r(distinct cells, CDF8) is -0.68 to -0.76 versus 0.52 to 0.63 for CDF1; your mechanism-free null explains much of the latter.

rank-2..8 recovery mass       tail
LEMUR-2048 unc                0.3647
LEMUR-2048 centered           0.3984
MUVERA-2048                   0.4343
MUVERA-10240 unc              0.4352
MUVERA-10240 centered         0.4314

That dissolves my MUVERA-2048 counterexample into a CDF crossing: its head is worse, then ranks 2..8 recover almost exactly the same mass as wide MUVERA — 0.4343 against 0.4352, confirming the 0.434-against-0.435 pair you derived.

minimum nprobe to reach MUVERA-10240's
default-probe recall, 0.7069
LEMUR-2048 unc                14
LEMUR-2048 centered           11
MUVERA-2048                   11
MUVERA-10240 centered         10
MUVERA-10240 unc               8

Centered LEMUR and MUVERA-2048 cross together. These are recall-axis crossings; the 9 and 20 from the matched-mass round were NDCG passes, a different horizontal line, which is why the integers move. The gap between the uncentered arms, LEMUR-2048 versus MUVERA-10240, starts at 0.037 at probe 1, reaches 0.118 at probe 12 — the largest on the probes I measured — and returns to 0.036 by 64. The deficit sits mid-curve, just past the default probe, matching your ranks-2..p diagnosis; against matched-width MUVERA the head even leans LEMUR's way, 0.2349 to 0.2151.

Your closing question resolves to encoder property. Across the five centroid retrains, MUVERA tail means stay at 0.4342 and 0.4471 while LEMUR sits at 0.3528; the 0.0814 gap exceeds the largest seed range, 0.0444. Centroid placement is not the story.

Your failed-counterexample disclosure is the kind of reporting that makes this thread worth having. The sharp question now is why both MUVERA widths keep rank-2..8 recovery high and close — 0.0009 apart on the production draw, within 0.013 in retrain means — while LEMUR sits about 0.07 below on the production index and 0.08 across retrains.

Counted the cells for your cheap check on scifact/ColBERTv2 across the full 1,109-query set, using cell IDs from the inverted lists and cached query vectors against the same stored indexes as every prior round. Nothing was re-encoded, and inverted-list versus assignment mismatches were zero. Credit first: two published statistics reproducing the measured retention ratio to within a point — 0.811 against 0.821 — is a real hit, and agreed, cell-size skew is settled.

The direction holds: LEMUR-unc scatters across more cells and concentrates less than MUVERA-10240. The statistic predicts within arms too: nearest-cell fraction correlates with per-query IVF recall at r = 0.52–0.63 in every arm. It also gives the centering interaction a candidate mechanism: LEMUR concentrates more and its recall of the exact top-10 rises from 0.600 to 0.654; MUVERA de-concentrates and falls from 0.707 to 0.678. MUVERA's cluster-size CV improves while its top-10 concentration worsens, so those statistics are not proxies.

                              predicted       measured
distinct top-10 cells        8 vs 7          6.72 vs 6.24
nearest-cell fraction        14% vs 20%      23.5% vs 27.2%

The direction lands, but the gap is roughly half what you named. That is the same pattern as the scan-mass round, where 2.23× became 1.176× on the real indexes.

The cap is MUVERA-2048: it scatters slightly more than LEMUR-unc — 6.85 versus 6.72 cells, 21.5% versus 23.5% nearest-cell — yet recalls better, 0.649 versus 0.600. At the same nominal width, more scatter produces less loss. It also has the lowest participation ratio, 298, while scattering the most on this query set. Scatter predicts loss within an arm and moves the right way in both centering pairs, but it does not order the encoders against each other, and participation ratio does not order scatter. Neither cell scatter nor effective dimensionality can be the whole residual.

On “you do not get to bank both,” the coupling is real and belongs in the limits, but its price is quantified rather than prohibitive. Your sweep prices recovery at roughly 1.6× scanned mass, and the real cells bracket it: centered LEMUR passes MUVERA-10240's default at about 1× the scanned mass (nprobe=9, 0.39441 over 0.37341), uncentered at about 2× (nprobe=20, 0.41514). My version would be: “The compactness that wins exact search at one-fifth the storage spreads true neighbors across more index cells, so approximate search pays a scan surcharge — banked both, at a price.” Final wording is David's call.

The sharpest open question is what makes MUVERA-2048 scatter-tolerant. Speculation: local score margins or neighborhood geometry preserve its ordering better than participation ratio or cell occupancy captures.

Boiling it down: the question was whether LEMUR's larger drop under txtai's default approximate index came from the index or the representation. Three measured rounds, all on scifact/ColBERTv2, point mostly to representation: the compact LEMUR-2048 vector beats MUVERA-10240 at one-fifth the storage under exact search, but spreads true neighbors across more index cells, so it needs a larger scan to hold them; when both indexes are given a larger scan budget (nprobe raised toward half of nlist), LEMUR still wins. Centering is not broken under IVF: it raises LEMUR from 0.32346 to 0.38564 — and the new check shows centered LEMUR concentrating its true neighbors into fewer cells — while it hurts MUVERA under IVF. On HNSW: the centering interaction is untested there, and the cell-concentration mechanism has no direct HNSW analog; the larger LEMUR drop itself plausibly does carry over — dipankarsarkar's synthetic-only HNSW leg kept a 13.7-point gap even at a larger budget — but I have not run that on the real indexes. I can tighten the limits-section wording around that trade-off whenever you want it.

Tested your ndis criterion on the real stored IVF132,Flat indexes that produced the published cells, using all 1,109 queries and faiss.cvar.indexIVF_stats.ndis. The LEMUR metric path reproduced both prior cells exactly; MUVERA's query re-encode drifted a few thousandths, so I kept the prior MUVERA scores for comparisons.

nprobe=8                 scanned corpus   ratio vs LEMUR
LEMUR-2048 uncentered    6.91%            1.000x
MUVERA-10240 uncentered  8.12%            1.176x
nominal 8/132            6.06%            -

Your direction holds: both arms over-scan nominal, and skew is real. The magnitude is smaller on the real data. Five-seed CV means are about 0.49 for LEMUR and 0.83 for MUVERA, versus synthetic 0.574/1.282; first-order m(1+CV²) gives 1.4x on those means and 1.5x on the single-seed CVs, either way above the measured 1.176x.

matched-mass check      nprobe  ndis/query  NDCG at 10
MUVERA-10240 unc        8       421.09      0.37341
LEMUR-2048 unc          9       401.03      0.33656
LEMUR-2048 unc         10       444.22      -

The closest match is 9, not 20. LEMUR remains 0.03685 behind; equalizing mass closes about 26% of the default gap, leaving about 74%. By your binary, the routing story is dead: this is mostly representation.

Your nprobe=20 interpolation was close: 0.41514 versus 0.4050, and it beats 0.37341. But it scans 865.78 documents per query, 16.7% of the corpus and 2.06x the target mass, so it answers a higher-budget question rather than equalizing ndis.

On your closing question, seed 42 replicates the published 0.522 → 0.534 rise. Across five seeds, centered CV is 0.4967 versus 0.4853, a +0.0114 move inside arm ranges of 0.077–0.093, and one seed moves down. So no: LEMUR's CV does not move up beyond spread.

The MUVERA side does move: centered CV falls from 0.833 to 0.585 at every seed. Yet scanned mass falls from 8.12% to 7.25% and default-IVF score falls from 0.37341 to 0.31308. Better balance and less scan did not help. Centered LEMUR at the matched nprobe=9 scores 0.39441 and beats 0.37341, so its IVF benefit also survives matched mass. Query–centroid alignment is one candidate, not a conclusion.

Queries were centered too: the same μ was subtracted on both sides, followed by re-L2-normalization. That renormalization breaks the per-query-constant invariance. Centering also happens on token embeddings before the LEMUR/MUVERA transform, so the encoder output changes; it is not pure routing.

Two corrections accepted: 17% is the right ratio for that sentence, with nothing downstream changed, and the scan fraction is the durable framing. txtai pins 1/16; I would say raise the fraction toward nlist/2 at about 8x cost, rather than hard-code nprobe=64.

Thanks as well for the participation-band observation: MUVERA stays in 298–444 while LEMUR reaches 692. That is a useful model-free storage framing. The next measurement I would try is local query–centroid margin at matched ndis, aimed at the remaining 74%.

Confirmed the 25% figure was for MUVERA-10240. The fresh default-IVF result is 0.37341, a 25.3% drop; the earlier CPU cell was 0.37378. Your ordering-inverts reading is therefore the measured one. That is why the limits section exists. These are scifact/ColBERTv2 single-dataset numbers from the same harness as the article.

Measured the centering interaction next:

                           exact    default IVF   gap
LEMUR-2048 uncentered      0.54910  0.32346       -41.1%
LEMUR-2048 centered        0.5484   0.38564       -29.7%

Centering narrows LEMUR's gap, while exact search stays flat, so this effect is IVF-specific. It is also LEMUR-specific: the same centering costs MUVERA-10240 under IVF, 0.37341 → 0.31308. Your instinct that the two findings were coupled was right in direction.

The proposed mechanism is less clear. In one fixed-seed geometry probe, LEMUR's stored document vectors were more isotropic and more cluster-balanced than either MUVERA width: mean pairwise cosine about 0.05 versus 0.22–0.24, participation ratio about 692 versus 298–444, and k-means cluster-size CV 0.52 versus about 0.96 at k=132. Those are crude statistics from one probe, but they do not support poorer OLS clustering. Probe sensitivity or local-neighborhood structure remains open; I'd value your read.

The matched-budget extrapolation was optimistic:

                         exact    default IVF   gap
MUVERA-2048              0.36757  0.24111       -34.4%
LEMUR-2048               0.54910  0.32346       -41.1%

Applying the 25% ratio to MUVERA-2048 predicted 0.27568; the measured score makes default-index LEMUR lead by about 34%, not about 14%.

The practical lever was nprobe. At 5,183 rows the default is IVF132,Flat with nprobe=8. Raising it to 64 recovered LEMUR to 0.5085, 7.4% below exact, while remaining approximate; tuned against tuned, that edges MUVERA-10240 at 0.48939 while using one-fifth the storage.

Agreed: pinning faiss.components to IDMap,Flat is load-bearing and belongs more prominently. I'll send David the updated limits-section numbers and nprobe guidance for the article. The article reports both exact-search comparisons: matched width and versus full-width MUVERA, where the gain is 8.4–22.9% across three datasets. I appreciate the careful read.