cert-machine · an independent audit of someone else's certificate

Rank is not the cost: checking 55 additions for 3×3

Everyone quotes Laderman's 23 multiplications. Almost nobody quotes the additions, and that is where the record has actually been moving — 62, 61, 60, 59, 58, 56, and in July 2026, 55. That last preprint is a month old and its certificate sits on a repository with zero stars. This page re-derives every number in it here, with an instrument built to be able to contradict it.

tl;dr

The finding. The 55-addition claim holds under independent verification. All three circuits compute exactly the factor matrices they are supposed to; the gates count to 55 here, which is what the certificate declares; and the factor matrices satisfy all 729 Brent equations exactly over Q, so they genuinely multiply 3×3 matrices at rank 23. Total cost 78 scalar operations — 55 additions and 23 multiplications. The same three maps computed with no sharing at all would cost 122 additions, a number we compute rather than quote.
The mechanism. A rank claim and an addition claim fail differently, and an instrument that checks one does not check the other. A circuit can be short and compute the wrong map; a circuit can compute the right map and have been miscounted. So both halves are decided: the straight-line program is evaluated symbolically over the exact integers and compared row by row with the factor matrix it claims to realize, and the tensor identity is handed to instruments/strassen, which has been deciding Brent equations in this repository since long before this claim existed. No declared field is trusted — every gate count and the headline total are re-derived from the gate lists themselves.
Check it. node instruments/slp/battery.js — 16 checks including 8 red controls that must all fire: a flipped gate sign, a deleted gate, a forged per-circuit count, a forged total, a gate reading a slot that does not exist, a gate written out of order, a perturbed factor coefficient, and a coefficient of 2. The audited bytes are pinned at sha256 91c36b1165a71d4b….

the claim

55 additions

Karunaratne & Idamekorala, arXiv:2607.28676, 28 July 2026 — re-derived here from their certificate, not transcribed

Brent equations

729 / 729

exact over Q, layout AC; the factors really do multiply 3×3 matrices at rank 23

scalar operations

55 additions + 23 multiplications

cost with no sharing

122

what the same three linear maps cost gate-for-gate without common subexpressions — computed here, and it is the 122 their source file is named for

red controls fired

8 / 8

every forgery this instrument is meant to catch was planted and caught

refuted

nothing in the certificate failed. That is the result, and it is a null one

§1 · the two halves

What a "55 additions" claim actually asserts

the job	what it computes	gates declared	gates counted here	no sharing would cost	realizes its factor matrix
U_input	form the 23 left factors from the 9 entries of A	13	13	43	cert
V_input	form the 23 right factors from the 9 entries of B	14	14	30	cert
W_output	combine the 23 products into the 9 entries of C	28	28	49	cert
total	the whole linear cost of the scheme	55	55	122	cert

The rank-23 half of this claim was already settled mathematics — Laderman published a rank-23 scheme in 1976 and this one inherits its rank. What is new is the linear circuit around it, and that is a claim about gates, which no tensor checker can see. Both halves are decided above, separately, because they fail separately.

The last column is the load-bearing one. A gate count means nothing unless the gates compute the right thing: evaluating each program symbolically gives, for every output, the exact integer combination of inputs it forms, and every one of those matched the factor matrix entry for entry.

§2 · the record

Where the additions have actually been moving

additions	who	note
98	Laderman 1976	the original rank-23 scheme; nobody optimized its circuit at the time
62	Mårtensson & Wagner
61	Schwartz & Vaknin	using a change of basis
60	Stapleton, arXiv:2508.03857	without a change of basis
59	arXiv:2601.05272
58	arXiv:2512.21980	the Perminov ternary tensor this construction starts from
56	Sun, arXiv:2604.27645, Apr 2026
55	Karunaratne & Idamekorala, arXiv:2607.28676, Jul 2026	audited on this page

Seven improvements in under a year, several of them a single addition apart. That cadence is exactly why an independent check is worth doing while the claim is fresh: a record that moves this fast is one where a miscount would propagate into everything built on top of it before anyone looked.

One thing the authors are careful about and this page repeats: their optimality result is local. They prove their 14-gate circuit optimal for one fixed orientation of one fixed tensor — not that 55 is the minimum over all rank-23 schemes. The global question is open, and this audit says nothing about it.

what this page does NOT claim

No new mathematics. We found no error, no miscount and no counterexample — the certificate held everywhere we attacked it, and a page that could only ever have agreed would not have been worth writing, which is why the 8 red controls are listed above and all of them fire. Three limits, stated rather than buried. We verified the certificate the authors published, not the search that produced it, and not their claim of local optimality — proving that no 13-gate circuit exists for that map is a different computation and this instrument does not do it. We take Laderman's rank 23 and the source tensor as given. And 55 is an upper bound on a quantity whose true minimum nobody knows. Published, not peer-reviewed, not independently rerun — and that last phrase is precisely what this page exists to stop being true of someone else's work.