finding: the mask-tail optimisation is inert on every power-of-two population, including the 4096-row tile

[AdaWorldAPI]

Blackboard only, no code. This is the gate on the 11-function tail rewrite #315 argued for — established by arithmetic before writing it, and it changes whether that rewrite is worth doing at all.

The arithmetic

A mask over n rows is ceil(n/64) words. The facade's algebra ops walk U64x8, so a tail exists only when words % 8 != 0. For a power of two, words = 2^(k-6), and 2^(k-6) % 8 == 0 for every k >= 9.

rows	words	tail	tail share of the whole op
512	8	0	none
4096	64	0	none
10 000	157	5	73 %
65 536	1024	0	none
1 000 000	15 625	1	2.6 %
16 777 216	262 144	0	none

(share = a 16 ns padded tail against a body of groups × 0.31 ns, both from #315's own measurements)

Every power-of-two population at or above 512 rows has no tail at all — and this workspace's shapes are overwhelmingly powers of two: the 4096-row tile, the 64-word mask, the 512-byte node, the 4096-centroid codebook.

What this does to #315

#315 is not wrong. It is scoped narrower than it reads. Its sweep used base = 8 and base = 64 words — 512 and 4096 rows — plus k of 1..7, which is exactly the small end. Its "7 of 8 mask sizes have a tail" is a statement about arbitrary sizes, and consumer sizes are not arbitrary.

So the honest reading of the merged result:

• "the padded tail is routinely larger than the work it trails" — true at 8–71 words, i.e. 512–4544 rows.
• On a million-row table the same tail is 2.6 %.
• On the canonical 4096-row tile it is 0 % — there is no tail to pay for.

Consequence: the rewrite is not justified yet, and I had the gate wrong

I told the reviewer on #315 that the rewrite should land behind a quieter machine. That was the wrong gate. The right one is a census of real consumer population sizes — which populations actually reach mask_and and friends, at what row counts. If they are powers of two the work is inert; if arbitrary row counts dominate it is worth ~2.6 % there and a lot at sizes like 10 000 rows.

Ternlog points the other way, independently

For mask_ternlog the padded form is one instruction — a full-width vpternlogq zmm over the zero-padded register. The fixed-step peel needs three logic ops at 4-lane plus more at 2-lane plus a scalar step. So for that site the padded tail may well be cheaper, which is the opposite of the direction #315 proposed, and is a second reason not to sweep all 11 sites uniformly.

What still stands from #315, unchanged

The codegen witnesses are untouched: U64x4 & really is bit-identical to a hand-written loop (the assembler merges the symbols), fixed-width peels really are packed on aarch64 as well as x86, and the avx512vl gate really is unnecessary because LLVM selects the tier itself. What is withdrawn is only the implied priority of acting on them.

The class

#315's five corrections were each a label outrunning its measurement. This one is a level up: a measurement outrunning its regime. The numbers were right at the sizes probed and the conclusion was stated without them. The cheapest way to catch that is arithmetic on the size formula, not another benchmark.

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Update: the census this PR called for is now run, and it CLOSES step (b)

Pushed in 483b7c1e. Read-only against lance-graph-java's lgj-abi, the binding consumer of the facade's mask algebra.

1. There is no tiling of the population. lgj_pattern_open(n_rows, …) / lgj_rowstore_open(n_rows, …) take the row count directly, and a mask spans the whole population (abi.rs::mask_words_for = n_rows.div_ceil(64)). The only "tile" in that crate is the 12-byte facet register, not the rows. So the padded tail is paid once per op and amortizes over the entire body — the favourable direction for the status quo.

2. Every committed population size is blind to the tail, bimodally:

rows	words	can it see a tail?
4, 8, 10, 16, 32, 64	1	all tail — not one full group
70, 100, 128	2	all tail
200	4	all tail
500	8	none
1000, 1024	16	none
4096	64	none
64000	1000	none

The small ones are correctness fixtures where the op is its tail and speed is irrelevant; every size large enough to have a body has zero tail. No committed test or bench in the consumer can exercise the regime the optimisation targets — which is precisely why #315's probe had to choose base = 8 and 64 words to see it at all.

3. Where it does bite (16 ns tail, 0.31 ns/group): 85 % at 5 k rows, 35 % at 50 k, 21 % at 100 k, 5 % at ~502 k, 2.6 % at 1 M, 1 % at ~2.6 M, and 0 % on every power of two ≥ 512.

Verdict: step (b) is closed as measured-not-worth-doing, not deferred

It would touch a hot facade at 11 sites across six realizations to win something only for arbitrary populations in roughly the 5 k – 500 k row band — and it would make mask_ternlog worse, since its padded form is one full-width vpternlogq where the peel needs three ops plus more.

Reopening it needs a named consumer workload whose populations sit in that band. Not another kernel benchmark.

🤖 Generated with Claude Code

https://claude.ai/code/session_01DCfrD5y19cvFc4AoyydXYv

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