When an AI sys­tem solves a dif­fi­cult math­e­mat­i­cal prob­lem, the usual ex­pla­na­tion is that it has be­come more in­tel­li­gent.

Perhaps it has ab­sorbed mil­lions of math­e­mat­i­cal ex­am­ples. Perhaps re­in­force­ment learn­ing has taught it bet­ter rea­son­ing strate­gies. Perhaps it is be­gin­ning to de­velop some­thing re­sem­bling gen­uine math­e­mat­i­cal in­tu­ition.

All of these ex­pla­na­tions may con­tain some truth. But they over­look a sim­pler pos­si­bil­ity:

AI has ac­cess to a vastly larger work­ing mem­ory than the hu­man brain.

Or, more pre­cisely, it has ac­cess to an enor­mous ex­ter­nal sym­bolic work­space that per­forms many of the func­tions that work­ing mem­ory per­forms in hu­mans.

This dif­fer­ence may be es­pe­cially im­por­tant in math­e­mat­ics.

A hu­man math­e­mati­cian can hold only a small num­ber of un­fa­mil­iar el­e­ments in mind si­mul­ta­ne­ously. An AI model can keep the en­tire prob­lem state­ment, hun­dreds of in­ter­me­di­ate equa­tions, sev­eral aban­doned ap­proaches, de­f­i­n­i­tions, con­straints and ear­lier con­clu­sions in­side its con­text win­dow.

We nor­mally in­ter­pret the re­sult­ing per­for­mance as ev­i­dence of su­pe­rior rea­son­ing. But some of it may in­stead re­flect the re­moval of one of the most im­por­tant bi­o­log­i­cal lim­its on hu­man rea­son­ing: our ex­tremely re­stricted work­ing-mem­ory ca­pac­ity.

Working mem­ory is the men­tal sys­tem that al­lows us to hold and ma­nip­u­late in­for­ma­tion over short pe­ri­ods.

When solv­ing an equa­tion, you must re­mem­ber what each vari­able rep­re­sents, which op­er­a­tions have al­ready been per­formed and what the cur­rent goal is. During a proof, you may need to keep track of as­sump­tions, in­ter­me­di­ate lem­mas, ex­cep­tions and mul­ti­ple pos­si­ble cases.

Human work­ing mem­ory is re­mark­ably lim­ited.

Its ex­act ca­pac­ity de­pends on the task and on how in­for­ma­tion is or­ga­nized, but the gen­eral lim­i­ta­tion is ob­vi­ous from every­day ex­pe­ri­ence. Try mul­ti­ply­ing two three-digit num­bers in your head. The un­der­ly­ing op­er­a­tions are sim­ple. The dif­fi­culty comes largely from hav­ing to pre­serve par­tial re­sults while per­form­ing ad­di­tional cal­cu­la­tions.

Writing the num­bers down trans­forms the prob­lem.

Paper does not make you more in­tel­li­gent. It ex­pands your ef­fec­tive work­ing mem­ory.

The same prin­ci­ple ap­plies at higher lev­els of math­e­mat­ics. A math­e­mati­cian uses no­ta­tion, scratch pa­per, di­a­grams and pre­vi­ously writ­ten lem­mas not merely to com­mu­ni­cate the so­lu­tion, but to make the rea­son­ing cog­ni­tively pos­si­ble.

Experts com­pen­sate through “chunking.” A novice sees a long se­quence of sym­bols. An ex­pert rec­og­nizes a fa­mil­iar struc­ture and treats it as a sin­gle con­cep­tual ob­ject. This al­lows far more in­for­ma­tion to fit in­side the same bi­o­log­i­cal work­ing-mem­ory limit.

But chunk­ing does not elim­i­nate the limit. It merely com­presses the in­for­ma­tion.

An AI model faces a very dif­fer­ent con­straint.

The im­por­tance of work­ing mem­ory for math­e­mat­ics is not merely the­o­ret­i­cal. It is vis­i­ble in the dif­fer­ences be­tween hu­man be­ings.

Working mem­ory is strongly re­lated to gen­eral in­tel­li­gence, which raises an ob­vi­ous ques­tion: does it in­de­pen­dently pre­dict math­e­mat­i­cal per­for­mance, or is it merely an­other im­per­fect mea­sure of IQ?

Several stud­ies sug­gest that it con­tributes some­thing be­yond con­ven­tional in­tel­li­gence mea­sures. Alloway and Passolunghi (2011), for ex­am­ple, ex­am­ined work­ing mem­ory, ver­bal abil­ity and math­e­mat­i­cal skills in chil­dren. They found that work­ing-mem­ory mea­sures made a dis­tinct con­tri­bu­tion to math­e­mat­i­cal per­for­mance rather than sim­ply re­pro­duc­ing the as­so­ci­a­tion be­tween math­e­mat­ics and gen­eral ver­bal abil­ity.

In a sep­a­rate six-year lon­gi­tu­di­nal study, Alloway and Alloway (2010) mea­sured chil­dren at age five and then ex­am­ined their aca­d­e­mic achieve­ment six years later. Early work­ing-mem­ory per­for­mance pre­dicted later lit­er­acy and nu­mer­acy even af­ter IQ was in­cluded in the analy­sis. Indeed, work­ing mem­ory was a stronger pre­dic­tor of the later aca­d­e­mic out­comes than the IQ mea­sure used in the study.

Blankenship and col­leagues (2015) sim­i­larly re­ported that work­ing mem­ory ex­plained unique vari­a­tion in math­e­mat­i­cal flu­ency and cal­cu­la­tion af­ter sta­tis­ti­cally con­trol­ling for IQ and age. A large meta-analy­sis by Friso-van den Bos and col­leagues (2013) also found a con­sis­tent re­la­tion­ship be­tween work­ing mem­ory and math­e­mat­ics across pri­mary-school stud­ies, al­though the strength of the re­la­tion­ship var­ied ac­cord­ing to the type of work­ing-mem­ory and math­e­mat­i­cal task be­ing mea­sured.

These find­ings should not be ex­ag­ger­ated. Working mem­ory and in­tel­li­gence over­lap sub­stan­tially, and sta­tis­ti­cal con­trol can­not per­fectly iso­late them as in­de­pen­dent psy­cho­log­i­cal mech­a­nisms. Nor does the ev­i­dence im­ply that com­mer­cially train­ing work­ing mem­ory will nec­es­sar­ily pro­duce large im­prove­ments in in­tel­li­gence or math­e­mat­ics.

The nar­rower con­clu­sion is nev­er­the­less im­por­tant: among chil­dren with sim­i­lar mea­sured in­tel­li­gence, dif­fer­ences in the abil­ity to hold, up­date and ma­nip­u­late in­for­ma­tion still pre­dict dif­fer­ences in math­e­mat­i­cal per­for­mance.

This pro­vides a cru­cial clue for un­der­stand­ing AI. If hu­man math­e­mat­i­cal per­for­mance is partly capped by a work­ing-mem­ory bot­tle­neck, then giv­ing a ma­chine an enor­mous sym­bolic work­space changes the na­ture of the con­test. The ma­chine may ap­pear more math­e­mat­i­cally in­tel­li­gent partly be­cause it is much less con­strained by a cog­ni­tive lim­i­ta­tion that sup­presses hu­man per­for­mance.

A mod­ern lan­guage model can process an enor­mous se­quence of to­kens at once. This se­quence may in­clude the orig­i­nal ques­tion, de­f­i­n­i­tions, ex­am­ples, in­ter­me­di­ate cal­cu­la­tions and the mod­el’s own ear­lier rea­son­ing.

The con­text win­dow is not iden­ti­cal to hu­man work­ing mem­ory. It is bet­ter un­der­stood as a gi­gan­tic ex­ter­nal note­book com­bined with an im­per­fect sys­tem for search­ing and us­ing what has been writ­ten in it.

This dis­tinc­tion mat­ters.

Humans pos­sess a form of ac­tive in­ter­nal mem­ory. We can silently choose a num­ber, hold it in mind, trans­form it and re­place it with a new value with­out say­ing or writ­ing any­thing.

Standard lan­guage mod­els are much weaker at main­tain­ing this kind of pri­vate, con­tin­u­ously up­dated men­tal state. Their most sta­ble form of mem­ory is usu­ally the se­quence of to­kens that has al­ready been gen­er­ated.

If the model writes:

x=6

and later writes:

x+3=9,

those state­ments re­main in­side the con­text. The model can at­tend to them again when gen­er­at­ing the next step.

Its rea­son­ing is there­fore of­ten ex­ter­nal­ized. The text is not merely a re­port of a com­pleted thought process. The text is part of the mech­a­nism by which the rea­son­ing oc­curs.

Humans do some­thing sim­i­lar when us­ing scratch pa­per. The ma­jor dif­fer­ence is scale.

An un­aided hu­man may strug­gle to keep five un­fa­mil­iar con­di­tions ac­tive si­mul­ta­ne­ously. An AI can pre­serve dozens or hun­dreds of them in ex­plicit form.

This does not mean that every item in a long con­text is re­trieved per­fectly. Models can over­look rel­e­vant in­for­ma­tion, be­come dis­tracted or lose track of de­tails. Advertised con­text length is not the same as per­fectly us­able mem­ory.

Nevertheless, the dif­fer­ence in po­ten­tial ca­pac­ity is enor­mous.

The con­text-win­dow ad­van­tage is not equally use­ful in every kind of rea­son­ing.

It mat­ters es­pe­cially for math­e­mat­ics be­cause math­e­mat­i­cal rea­son­ing can be trans­lated un­usu­ally well into ex­plicit sym­bols.

Almost every rel­e­vant el­e­ment of a math­e­mat­i­cal prob­lem can be writ­ten down:

the as­sump­tions;

the as­sump­tions;

the de­f­i­n­i­tions;

the de­f­i­n­i­tions;

the known equa­tions;

the known equa­tions;

the cur­rent ob­jec­tive;

the cur­rent ob­jec­tive;

the re­sults al­ready proved;

the re­sults al­ready proved;

the cases that have been elim­i­nated;

the cases that have been elim­i­nated;

the con­di­tions un­der which each step re­mains valid.

the con­di­tions un­der which each step re­mains valid.

Once writ­ten, this in­for­ma­tion re­mains sta­ble.

If (x) is de­fined as an in­te­ger at the be­gin­ning of a proof, it re­mains an in­te­ger un­less the proof ex­plic­itly changes the de­f­i­n­i­tion. A strict in­equal­ity does not grad­u­ally be­come a non-strict in­equal­ity be­cause of changes in mood, con­text or in­ter­pre­ta­tion.

Mathematical sym­bols are de­signed to re­duce am­bi­gu­ity.

This makes math­e­mat­ics al­most per­fectly suited to an in­tel­li­gence that op­er­ates through a large tex­tual work­space.

Consider a prob­lem re­quir­ing the solver to re­mem­ber that:

n is odd;

n is odd;

p is prime;

p is prime;

x≠0

x≠0

and that one branch of the ar­gu­ment has al­ready pro­duced a con­tra­dic­tion.

A hu­man may un­der­stand the ba­sic strat­egy but di­vide by x be­fore es­tab­lish­ing that x≠0. The er­ror is not nec­es­sar­ily caused by a lack of in­tel­li­gence. It may be a fail­ure of book­keep­ing.

An AI can re­state the ac­tive con­straints at each stage:

We are work­ing un­der the as­sump­tions that (n) is odd, (p) is prime and (x≠0).

We are work­ing un­der the as­sump­tions that (n) is odd, (p) is prime and (x≠0).

The con­text be­comes a ledger of the rea­son­ing state.

Many dif­fi­cult math­e­mat­i­cal prob­lems con­tain a pro­found in­sight some­where near the be­gin­ning, but they also con­tain a large amount of less glam­orous work af­ter­ward: ex­pand­ing ex­pres­sions, check­ing cases, car­ry­ing con­di­tions through trans­for­ma­tions and mak­ing sure that the fi­nal con­clu­sion is com­pat­i­ble with every ear­lier as­sump­tion.

A ma­chine does not need to pos­sess deeper in­sight than a hu­man to gain an ad­van­tage here. It may sim­ply be bet­ter equipped to pre­serve the en­tire state of the prob­lem while com­plet­ing a long se­quence of op­er­a­tions.

Mathematics is highly com­po­si­tional.

A proof can of­ten be rep­re­sented as:

A → B → C → D

If each step is valid and the chain is pre­served ac­cu­rately, the con­clu­sion fol­lows.

A large work­ing space al­lows the model to con­struct much longer chains be­fore los­ing the thread.

This is im­por­tant be­cause the dif­fi­culty of a prob­lem does not de­pend only on the dif­fi­culty of each in­di­vid­ual step. It also de­pends on how many steps must be co­or­di­nated.

A per­son may be per­fectly ca­pa­ble of un­der­stand­ing every lo­cal in­fer­ence in a 100-step ar­gu­ment while still be­ing un­able to gen­er­ate the en­tire ar­gu­ment un­aided. The prob­lem ex­ceeds the per­son’s abil­ity to main­tain the global struc­ture.

AI can po­ten­tially com­pen­sate by writ­ing down nearly every­thing.

This may ex­plain why ad­di­tional “thinking time” of­ten im­proves model per­for­mance. More com­pu­ta­tion al­lows the sys­tem to pro­duce more in­ter­me­di­ate states, ex­am­ine al­ter­na­tive branches and pre­serve par­tial con­clu­sions.

What looks like deeper thought may some­times be broader search con­ducted in­side a much larger note­book.

Now com­pare a math­e­mat­i­cal prob­lem with a so­cial ques­tion:

Why has Maria sud­denly stopped re­ply­ing to my mes­sages?

Why has Maria sud­denly stopped re­ply­ing to my mes­sages?

A larger con­text win­dow might al­low an AI to ex­am­ine years of cor­re­spon­dence. It could iden­tify changes in tone, tim­ing and vo­cab­u­lary.

But the de­ci­sive in­for­ma­tion may still be miss­ing.

Perhaps Maria is an­gry. Perhaps she is busy. Perhaps she is ill. Perhaps she has lost her phone. Perhaps she is avoid­ing an un­re­lated prob­lem.

No amount of mem­ory can re­trieve facts that were never ob­served.

The chal­lenge is not sim­ply to pre­serve a known set of premises and de­rive their con­se­quences. It is to rea­son un­der un­cer­tainty about hid­den causes.

Informal rea­son­ing also de­pends heav­ily on con­cepts whose mean­ings are un­sta­ble.

Words such as “fair,” “successful,” “responsible,” “harmful” or “intelligent” do not pos­sess the ex­act­ness of math­e­mat­i­cal vari­ables. Their mean­ing de­pends on cul­ture, goals and con­text.

A model can re­mem­ber every sen­tence in a dis­cus­sion while still mis­un­der­stand­ing what the par­tic­i­pants mean.

The same ap­plies to po­lit­i­cal analy­sis, his­tor­i­cal in­ter­pre­ta­tion, busi­ness strat­egy and psy­cho­log­i­cal judg­ment. In these do­mains, the cen­tral prob­lem is of­ten not work­ing-mem­ory ca­pac­ity. It is iden­ti­fy­ing the cor­rect causal model when the ev­i­dence is in­com­plete and am­bigu­ous.

A larger note­book helps, but it does not solve the fun­da­men­tal prob­lem.

08 Jul, 2026

Table of Contents

Intro Contest in short Problem in­tro

Contest in short

Problem in­tro

Why this prob­lem is auto-re­search-able

Learning Enough to Ask Better Questions

(Optional) Math for QR de­com­po­si­tion: Householder re­flec­tions

Make se­r­ial work small with the help of the blocked Householder al­go­rithm

Other chal­lenges

Codex-maxxing Kernel progress break­throughs Breakthrough ideas

Kernel progress break­throughs

Breakthrough ideas

Introducing idea di­ver­sity to es­cape the lo­cal max­ima

Implementation Hints

What I could have done bet­ter

Conclusion

References

Acknowledgements

Intro

Contest in short

GPU Mode, in col­lab with Core Automation, re­cently hosted an auto-re­search themed con­test. The prob­lem state­ment was to im­ple­ment batched square com­pact-House­holder QR fac­tor­iza­tion aka QR de­com­po­si­tion. I placed 12th out of 183 par­tic­i­pants, end­ing up with a 232x speedup over the base­line so­lu­tion. This post is about how I got there. I will go through my ap­proach, learn­ings, and bot­tle­necks I ran into dur­ing the con­test. It was my first se­ri­ous at­tempt at auto-re­search. Some peo­ple will call this “loop en­gi­neer­ing”, and hon­estly that is fine too.

Note that you don’t need to go through the math­e­mat­ics or the prob­lem it­self in de­tail to fol­low most of this blog post. I have fo­cused on my ap­proach while keep­ing the math and the prob­lem it­self sec­ondary as most peo­ple who will read this won’t have par­tic­i­pated in the con­test.

You can check out the full con­test page here: Problem Link and Leaderboard

This con­test was part of GPU Mode’s Linear Algebra Kernels in the Age of Research se­ries.

Problem in­tro

We were given a batch of square FP32 CUDA ma­tri­ces A with shape batch x n x n, and had to re­turn the same com­pact Householder QR rep­re­sen­ta­tion as torch.geqrf(A): an H ma­trix whose up­per tri­an­gle is R and whose lower tri­an­gle stores Householder vec­tors, plus a tau vec­tor of re­flec­tor co­ef­fi­cients. The checker re­built Q with torch.linalg.house­hold­er_prod­uct(H, tau), took R = triu(H), and ver­i­fied:

A≈QR,Q⊤Q≈I,Q⊤A≈R

Among cor­rect sub­mis­sions, the leader­board ranked run­time by geo­met­ric mean across shapes and con­di­tion­ing cases. The im­por­tant sizes were batched square ma­tri­ces like 512 x 512, with larger 1024, 2048, and 4096 cases too. Low-bit FP16, FP8, or NVFP4 was al­lowed in­ter­nally, but re­turned fac­tors still had to sat­isfy FP32-style QR checks.

A tiny 3 x 3 ex­am­ple is:

A=[12−5146167−68−424−41]=[6/7−69/175−58/1753/7158/1756/175−2/76/35−33/35]⏟Q[1421−140175−700035]⏟R

Here Q is or­thog­o­nal, which means its columns are unit-length and per­pen­dic­u­lar to each other, and R is up­per tri­an­gu­lar, which means every­thing be­low the di­ag­o­nal is zero. The con­test was not ask­ing us to print dense Q and R di­rectly; it asked for the com­pact Householder ver­sion that lets the checker re­con­struct Q and read R from the up­per tri­an­gle.

For the 3×3 ex­am­ple above, the very first re­flec­tor maps the first col­umn (12, 6, −4) straight onto (−14, 0, 0) in one shot. The −14 be­comes R11. How that works is in the math sec­tion.

Why this prob­lem is auto-re­search-able

GPU Mode pro­vides par­tic­i­pants with the pop­corn CLI mak­ing it agent-friendly. Agents can use this to test, bench­mark, and sub­mit to the leader­board di­rectly. The checker also pro­vided shape-wise feed­back along with the over­all geo­met­ric mean tim­ing.

Astute ob­servers will no­tice this is an apt setup for writ­ing a loop. Agents yearn for tight feed­back loops. They al­low them to hill-climb to their heart’s con­tent.

GPU Mode con­tests usu­ally give you some way to it­er­ate on ker­nels. Either you sub­mit di­rectly, or a spon­sor like Modal chips in cred­its. Here the or­ga­niz­ers ba­si­cally al­lowed un­lim­ited sub­mis­sions as long as you spaced them out. If you did­n’t, the queues got long and every­body’s runs timed out. At one point the work­space even ran out of Modal cred­its be­cause every­one had been ham­mer­ing sub­mis­sions. It’s a nice way to make learn­ing ac­ces­si­ble.

Over the course of 14 days, I made over 1500 sub­mis­sions.

Learning Enough to Ask Better Questions

I have known the ba­sics of GPU ker­nel op­ti­miza­tion (mainly in Triton with some un­der­stand­ing of CUDA) for a year, but haven’t worked in this do­main pro­fes­sion­ally. What I am try­ing to tell you is that I was an un­der­dog among the peo­ple around me on the leader­board. The per­son just above me on the leader­board (CUDA Colonel) is a prin­ci­pal en­gi­neer at NVIDIA.

Anyway aura farm­ing aside, since I know the ba­sics and had re­cently read about GatedDeltaNet, I was fresh on the gen­eral GPU ker­nel lingo.

The bet­ter you know some­thing, the bet­ter you can prompt the LLMs, be­cause you con­vert un­known un­knowns into known un­knowns.

At the same time, it’s worth not­ing that this con­test was doable with­out do­main knowl­edge - like you prob­a­bly won’t make it to the top 10, but you can get a re­spectable speedup over base­line by just re­ly­ing on your har­ness/​agent loop or what­ever.

My first steps in the con­test were to learn what QR de­com­po­si­tion is and how it can be done. There are a bunch of ways to do it - like Gram-Schmidt and Householder re­flec­tions. The con­test man­dated Householder re­flec­tions. I went back and forth with Claude and watched a few YouTube videos to build in­tu­ition. After my dis­cus­sions with Claude, it was clear that we needed to use the blocked Householder al­go­rithm as the main ar­chi­tec­ture with the trail­ing WY-update. As it turns out, GPT-5.5 also had a good idea about this. QR de­com­po­si­tion is a fairly well known prob­lem.

I found the con­cept in­ter­est­ing as ma­trix de­com­po­si­tions show up in sev­eral mod­ern op­ti­mizer vari­ants for LLM train­ing, es­pe­cially in meth­ods that use ma­trix pre­con­di­tion­ing, such as Shampoo-style op­ti­miz­ers and re­lated ap­proaches. Muon (used by Kimi) is an­other good ex­am­ple: in­stead of treat­ing a weight up­date as one gi­ant flat­tened vec­tor, it keeps the ma­trix struc­ture around and or­thog­o­nal­izes the mo­men­tum up­date, usu­ally through a few Newton-Schulz it­er­a­tions that ap­prox­i­mate the po­lar fac­tor.

(Optional) Math for QR de­com­po­si­tion: Householder re­flec­tions

I rec­om­mend skim­ming through this sec­tion if you are cu­ri­ous about the math oth­er­wise feel free to skip. The only thing to note is that there is a se­quen­tial de­pen­dency in Householder QR which makes it prob­lem­atic to do GEMM. We use blocked Householder to make it more ma­trix-mul­ti­pli­ca­tion shaped.

The con­tract

Quickly re­view­ing the con­tract: in­put is a batch of square FP32 ma­tri­ces A; out­put is the com­pact (H, tau) for­mat that torch.geqrf re­turns. The up­per tri­an­gle of H is R. Below the di­ag­o­nal, H stores the Householder vec­tors, and tau stores one scalar per col­umn. The checker re­builds Q from (H, tau) and ver­i­fies A ≈ QR.

Mirrors

Forget ma­tri­ces for a sec­ond. In a bath­room mir­ror, your re­flec­tion is ex­actly as far be­hind the glass as you are in front of it, straight through.

If x⟂ is the part of x stick­ing out per­pen­dic­u­lar to the glass, re­flec­tion just sub­tracts that part twice:

xre­flected=x−2x⟂

So a Householder re­flec­tion is about find­ing the per­pen­dic­u­lar part and sub­tract­ing it twice.

Storing the mir­ror

A Householder vec­tor is the mir­ror, stored com­pactly. In code, we don’t carry around the whole mir­ror plane. We store one vec­tor v stick­ing straight out of it. The mir­ror is every­thing per­pen­dic­u­lar to v, and the re­flec­tion moves along v.

The per­pen­dic­u­lar part is just the shadow of x along v, which is v⊤xv⊤v copies of v. Plug that into the sub­trac­tion above:

ℋx=x−τv(v⊤x),τ=2v⊤v

So tau is just 2v⊤v: the fac­tor of 2 and the length of v bun­dled into one pre­com­puted num­ber. v picks the mir­ror, tau scales the up­date. (I’ll write the math­e­mat­i­cal re­flec­tor as ℋj and re­serve H for the com­pact out­put ma­trix.)

Householder re­flec­tion in 2D tau = 0.00

Why QR cares about mir­rors

QR wants to turn A into an up­per-tri­an­gu­lar ma­trix R. Column 1 should be­come some­thing like (*, 0, 0), col­umn 2 should have ze­ros be­low row 2, and so on.

A Householder mir­ror is use­ful be­cause it can do that to a col­umn in one shot. Take the first col­umn of the 3×3 ex­am­ple above: (12, 6, -4). We want to send it to the x-axis so the lower en­tries be­come zero. A re­flec­tion can only change di­rec­tion, not length, so the tar­get must also have length 14. One valid tar­get is (-14, 0, 0). After that re­flec­tion, the 6 and -4 en­tries are gone, which is ex­actly what we wanted.

How do we find the mir­ror? It sits halfway be­tween the col­umn and its tar­get, so v, the vec­tor pok­ing through the mir­ror, is just the col­umn mi­nus its tar­get:

v = (12, 6, -4) - (-14, 0, 0) = (26, 6, -4)

The Householder up­date

Then com­pute tau = 2 / (vᵀv). The re­flec­tor it­self is:

ℋ=I−τvv⊤,τ=2v⊤v

ℋx=(I−τvv⊤)x

ℋx=x−τv(v⊤x)

ℋAactive=(I−τvv⊤)Aactive

ℋAactive=Aactive−τv(v⊤Aactive)

This turns the cur­rent col­umn into (-14, 0, 0) and rewrites the other columns con­sis­tently, so the next re­flec­tor is built from the up­dated ma­trix.

We keep do­ing this once per col­umn. In rough no­ta­tion, the re­peated up­dates look like:

A(1)=A(0)−τ1v1(v1⊤A(0))

A(2)=A(1)−τ2v2(v2⊤A(1))

A(3)=A(2)−τ3v3(v3⊤A(2))

⋯

R=A(n)

Each line uses the ma­trix pro­duced by the pre­vi­ous line. Each re­flec­tor ze­roes out every­thing be­low the di­ag­o­nal of its col­umn with­out dis­turb­ing the columns al­ready fin­ished. After the last one, A has walked down to an up­per-tri­an­gu­lar R:

A=ℋ1ℋ2⋯ℋn⏟QR

Mirrors don’t change lengths or an­gles, so each ℋj is or­thog­o­nal, and so is their prod­uct Q. That’s where the or­thog­o­nal­ity the checker ver­i­fies comes from for free.

What’s the com­pact for­mat

Once col­umn j is processed, every­thing be­low its di­ag­o­nal is dead space. geqrf reuses those slots to stash the tail of v_j (the lead­ing 1 is im­plicit). On and above the di­ag­o­nal you’re look­ing at R; be­low it, the re­flec­tors; and tau rides along as a sep­a­rate vec­tor. That’s why the checker needs both H and tau to re­build Q.

Compact geqrf stor­age (n = 6) hover a col­umn

H  (upper tri­an­gle = R, be­low di­ag­o­nal = re­flec­tor tails)

tau  (one scalar per re­flec­tor)

r en­tries of R tail of vj τj

Make se­r­ial work small with the help of the blocked Householder al­go­rithm

Householder QR ze­roes out A be­low the di­ag­o­nal one col­umn at a time. Each step builds a re­flec­tor from the cur­rent col­umn and ap­plies it to every­thing on the right. The prob­lem is re­flec­tor j+1 is built from the ma­trix af­ter re­flec­tor j has al­ready hit it. So you can’t re­order the steps and you can’t fuse them. It’s se­r­ial, and the se­r­ial ma­trix-vec­tor work runs in the slow vec­tor lanes of the SM while the ten­sor cores just sit there idle.

Householder QR, one re­flec­tor at a time (n = 5) math view: ze­ros ap­pear, trail­ing block gets rewrit­ten

orig­i­nal a fi­nal en­try of R rewrit­ten by this re­flec­tor 0 zeroed (vj gets stashed here)

The clas­sic fix is the blocked al­go­rithm. You pick a nar­row panel of b columns (say 32 or 64) and do all the se­r­ial work in­side it. That’s fine, be­cause the panel is only b columns wide, so it stays cheap. Then, in­stead of ap­ply­ing the pan­el’s b re­flec­tors to the rest of the ma­trix one at a time, you com­press them into a sin­gle rank-b up­date (the “WY rep­re­sen­ta­tion”) and hit the en­tire trail­ing block in one shot with three back-to-back ma­trix mul­ti­plies. The se­r­ial work stays con­fined to the panel, and every­thing else turns into GEMMs, which is ex­actly the shape the ten­sor cores want.

Concretely, the WY rep­re­sen­ta­tion col­lapses a pan­el’s b re­flec­tors into a sin­gle rank-b up­date. Stack the pan­el’s Householder vec­tors as columns of V=[v1,v2,…,vb], build a small b×b up­per-tri­an­gu­lar T, then:

ℋ1ℋ2⋯ℋb=I−VTV⊤

and the trail­ing-block up­date be­comes three GEMM-shaped steps:

W=V⊤Atrail

Z=T⊤W

Atrail←Atrail−VZ

Blocked Householder (n = 6, panel width b = 2) con­fine the se­r­ial work, GEMM the rest

Earlier this year Apple de­nied me ac­cess to App Store Connect af­ter de­cid­ing that I matched some­one on a U.S. gov­ern­ment re­stricted-party list.

Their ex­pla­na­tion was fairly de­fin­i­tive:

“The in­for­ma­tion you pro­vided fully matches one or more re­stricted par­ties on the U.S. gov­ern­ment con­sol­i­dated screen­ing list or an­other gov­ern­men­t’s sanc­tions list.”

“The in­for­ma­tion you pro­vided fully matches one or more re­stricted par­ties on the U.S. gov­ern­ment con­sol­i­dated screen­ing list or an­other gov­ern­men­t’s sanc­tions list.”

They al­ready had my pass­port.

I replied with my full le­gal name, Sean Joseph Byrne, up­loaded my dri­ver’s li­cense, and pointed out the ad­dress on the gov­ern­ment record they ap­peared to be match­ing me against. I’ve never lived at that ad­dress, never lived in County Sligo, and have no con­nec­tion to the com­pany in­volved. I asked them to es­ca­late it to their sanc­tions com­pli­ance folks and make a proper non-match de­ter­mi­na­tion.

Apple still has­n’t replied.

Apple’s re­sponse af­ter re­view­ing my iden­tity in­for­ma­tion.

I knew what had hap­pened be­cause this was­n’t the first time.

Cloonmull House

Search the U.S. gov­ern­men­t’s Consolidated Screening List for Sean Byrne and you get ex­actly one re­sult:

Sean Byrne Cloonmull House Drumcliffe, County Sligo Ireland

Source: Entity List, Bureau of Industry and Security Added: July 21, 2009 License re­quire­ment: All items sub­ject to the EAR License pol­icy: Presumption of de­nial

Sean Byrne Cloonmull House Drumcliffe, County Sligo Ireland

Source: Entity List, Bureau of Industry and Security Added: July 21, 2009 License re­quire­ment: All items sub­ject to the EAR License pol­icy: Presumption of de­nial

The Consolidated Screening List is­n’t it­self a sanc­tions list. It’s a U.S. gov­ern­ment screen­ing tool that com­bines a num­ber of ex­port-con­trol, sanc­tions and other re­stricted-party lists main­tained by the Departments of Commerce, State and Treasury.

The re­sult comes from the Commerce Department’s Bureau of Industry and Security Entity List. “All items sub­ject to the EAR” means the Export Administration Regulations, the rules gov­ern­ing what U.S. com­pa­nies can ship abroad. “Presumption of de­nial” is a li­cens­ing pos­ture: if some­one ap­plies for a li­cence to ex­port some­thing to this per­son, the de­fault an­swer is no. That’s the en­tire pur­pose. It’s an ex­port-con­trol in­stru­ment but it says noth­ing about who can be em­ployed, or who can sell shares.

The per­son in the search re­sult is­n’t me. More in­ter­est­ingly, it does­n’t ap­pear to be any­one.

The en­try came out of the pros­e­cu­tion of an Irish air­craft-parts busi­ness called Mac Aviation. In 2009, the Department of Justice de­scribed Sean Byrne as Mac Aviation’s com­mer­cial man­ager and charged him along­side Thomas and Sean McGuinn over the il­le­gal ex­port of U.S. air­craft equip­ment to Iran.

Except Mac Aviation had ap­par­ently in­vented em­ploy­ees to make the com­pany look big­ger than it was.

Mac Aviation was a fa­ther and son work­ing out of a cot­tage on the edge of Drumcliffe vil­lage, Ben Bulben be­hind it. A Rolls-Royce of­fi­cial who came to visit was re­port­edly speech­less. The com­pany he had been sell­ing he­li­copter en­gines to, and had taken for a global op­er­a­tion em­ploy­ing hun­dreds of pro­fes­sion­als, was a house in Sligo.

Drumcliffe, County Sligo, with Ben Bulben in the back­ground.

To keep up the im­pres­sion of a much larger firm, the McGuinns signed doc­u­ments with false names. Sean Byrne was one of them. John Mooney re­ported in the Sunday Times that the name ap­peared on so much com­pany pa­per­work that the American au­thor­i­ties “became con­vinced Byrne ex­isted and tried to in­dict him.”

When DOJ filed a su­per­sed­ing in­dict­ment in 2010, re­plac­ing the orig­i­nal, Sean Byrne was no longer a de­fen­dant. The de­fen­dants were Mac Aviation and Thomas and Sean McGuinn.

More im­por­tantly, the su­per­sed­ing in­dict­ment re­peat­edly de­scribes Sean Byrne as an alias used by one or more co-con­spir­a­tors. The phrase ap­pears more than fif­teen times, at­tached to spe­cific in­voices, emails and an own­er­ship state­ment. Mac Aviation staff used the name with sup­pli­ers in the U.S. and cus­tomers in Iran.

At some stage the U.S. gov­ern­ment ap­pears to have worked out that Sean Byrne was­n’t ac­tu­ally a sep­a­rate per­son. And yet the en­try in the Entity List sur­vived. Sixteen years later it still has no date of birth, pass­port num­ber, mid­dle name or other use­ful per­sonal iden­ti­fier. It’s ba­si­cally a com­mon Irish name, an ad­dress in Sligo and Ireland.

Cloonmull House in Sligo was Thomas McGuinn’s home, and the in­dict­ment gives it as Mac Aviation’s reg­is­tered mail­ing ad­dress. So the en­try is­n’t a record of a man in Sligo. It’s a name at­tached to some­body else’s house. And I’ve never lived in Sligo.

This has hap­pened be­fore

Years be­fore I moved back to Ireland, I was sell­ing stock through a ten­der of­fer when Nasdaq stopped my or­der af­ter a back­ground check re­turned a match on my name.

Their Head of Account Management emailed me say­ing that the check had found a match as­so­ci­ated with a pre­vi­ous in­ci­dent and that he was con­fi­dent it was a false pos­i­tive, but com­pli­ance wanted ad­di­tional proof of my California ad­dress. On the phone he gave me more de­tail and specif­i­cally asked me about Mac Aviation. I ex­plained that I’d never had any­thing to do with the com­pany, pro­vided the ex­tra doc­u­men­ta­tion they wanted and the sale went through with an en­ter­tain­ing story to tell peo­ple.

Nasdaq’s re­sponse af­ter a back­ground check matched my name.

More re­cently I or­dered a Starlink mount­ing pole from California. DHL, ship­ping on be­half of SpaceX, told me the prob­lem was a re­stricted-party match and asked for my pass­port. I sent it, they cleared it and the pole ar­rived. DHL also would­n’t send a hat I’d or­dered in the U.S. on to me in Ireland with­out a copy of my pass­port.

The fake Sean Byrne was as­so­ci­ated with at­tempts to pro­cure he­li­copter en­gines, fighter-air­craft parts and other U.S. equip­ment for Iran. The real Sean Byrne oc­ca­sion­ally needs to pro­duce a pass­port be­fore some­one will send him a hat.

Apple is the odd one out. Nasdaq and the ship­pers both gen­er­ated false pos­i­tives, asked for enough in­for­ma­tion to re­solve them, and then re­solved them. Apple al­ready had my pass­port, re­ceived my dri­ver’s li­cense and a fairly de­tailed ex­pla­na­tion of ex­actly why the match was wrong, and still told me that I “fully” matched a re­stricted party.

Twelve men named Robert Johnson

None of this is novel. In October 2006, 60 Minutes found twelve American men named Robert Johnson who all had trou­ble board­ing flights, and brought them to New York to­gether. A politi­cian, a soc­cer coach, busi­ness­men and a serv­ing mem­ber of the mil­i­tary.

The Robert Johnson they kept be­ing con­fused with was­n’t a man named Robert Johnson. It was a known alias of some­one con­victed of plot­ting to bomb a Hindu tem­ple and a cin­ema in Toronto, who by then had served his sen­tence and been de­ported to Trinidad. The air­line agents check­ing the twelve real ones against it had a name and noth­ing else. Not even a date of birth.

Asked about it, the head of the FBI’s Terrorist Screening Center said Robert Johnson would never get off the list, and that any­one with the name would be in­con­ve­nienced every time they tried to check in.

I’m not on the Entity List. I’m be­ing misiden­ti­fied as an en­try on it. Apple did­n’t make that dis­tinc­tion.

The con­sumer ver­sion of this has been lit­i­gated. In 2005 Sandra Cortez was held up buy­ing a car in Colorado be­cause TransUnion matched her against a woman on the Treasury Department’s sanc­tions list who was born 27 years af­ter her. The credit bu­reau had com­pared first and last names only, not dates of birth. A jury awarded her dam­ages and the Third Circuit up­held it, de­scrib­ing the fail­ure to com­pare birth dates as rep­re­hen­si­ble. Sergio Ramirez had the same ex­pe­ri­ence at a car deal­er­ship six years later, and his case reached the Supreme Court in 2021.

In both cases the courts called for bet­ter match­ing. Compare the date of birth. Compare the mid­dle name.

There is no ver­sion of that avail­able to me. The list­ing has no date of birth to com­pare. No mid­dle name and no pass­port num­ber, be­cause the per­son does­n’t ex­ist. A screen­ing sys­tem that does its job per­fectly will still flag me, for­ever, on the only two facts the record con­tains: a com­mon Irish name and a coun­try.

Which is why ar­gu­ing with com­pa­nies one at a time is the wrong ap­proach.

The real fake Sean Byrne

Remote hir­ing has de­vel­oped a se­ri­ous iden­tity-fraud prob­lem. It has also de­vel­oped, some­what un­be­liev­ably, a North Korea prob­lem.

North Korean IT work­ers have been get­ting re­mote jobs at U.S. com­pa­nies us­ing stolen or fab­ri­cated iden­ti­ties, proxy in­ter­view­ers and U.S.-based “laptop farms” that make work­ers over­seas ap­pear to be con­nect­ing from in­side the United States. The FBI has been warn­ing com­pa­nies about it, and the DOJ has pros­e­cuted schemes that suc­cess­fully placed work­ers at more than 100 U.S. com­pa­nies.

So if you’re hir­ing re­mote en­gi­neers, “is this per­son ac­tu­ally who they claim to be?” is now a le­git­i­mate se­cu­rity prob­lem.

A new class of re­cruit­ing prod­ucts is be­ing built around that prob­lem. They sit in­side the soft­ware com­pa­nies use to man­age job ap­pli­ca­tions, the ap­pli­cant track­ing sys­tem or ATS.

Tofu is one of them. Their pitch is that they screen every ap­pli­cant across more than forty sig­nals be­fore a re­cruiter opens a ré­sumé, and that when a screened ap­pli­cant trig­gers a sanc­tions match the sig­nal routes straight to com­pli­ance re­view. They are ex­plicit that this has to hap­pen early: screen­ing at the back­ground-check stage is, on their ac­count, al­ready too late, so it should run at ap­pli­ca­tion sub­mis­sion be­fore any re­cruiter makes con­tact. They also say a can­di­date flagged by one of their cus­tomers is flagged across their whole cus­tomer net­work through their API. Brainner makes a sim­i­lar case, check­ing ap­pli­cants against 3.5 bil­lion data points and flag­ging high-risk pro­files be­fore a re­cruiter re­views them.

Tofu says its data­base is built from analysed ap­pli­cant pro­files. Their home­page says more than 18 mil­lion. Most of their other pages say more than 5 mil­lion.

The sanc­tions screen­ing these ven­dors de­scribe is OFAC and the Specially Designated Nationals list, which is the right list for the risk they’re sell­ing against: pay­ing wages to a sanc­tioned per­son. I’ve no ev­i­dence that ei­ther com­pany queries the BIS Entity List, and no idea whether any com­pany I’ve ap­plied to uses ei­ther prod­uct.

But screen­ing my name against U.S. re­stricted-party data pro­duces a false pos­i­tive. It did at Nasdaq, at SpaceX, at DHL and at Apple. Four times in six years. And the in­dus­try’s an­swer to re­mote-hir­ing fraud is to run that class of check ear­lier, be­fore a hu­man is in­volved, and prop­a­gate the re­sult across a net­work of em­ploy­ers.

Mac Aviation fab­ri­cated an em­ployee to make it­self look like a big­ger com­pany. That fab­ri­cated em­ployee ended up in an au­thor­i­ta­tive U.S. gov­ern­ment data­base. Sixteen years later, com­pa­nies are build­ing sys­tems us­ing that data­base to de­tect fake ap­pli­cants, fraud­sters, crim­i­nals and state-backed ac­tors be­fore they get through the hir­ing process.

Has this cost me a job? I don’t know. I’ve spent my ca­reer in in­for­ma­tion se­cu­rity, much of it in the U.S., and I’m now ap­ply­ing for roles from Ireland. There have been jobs where I’ve a back­ground that should at least get a con­ver­sa­tion and I’ve heard noth­ing. That’s hardly re­mark­able on its own. Hiring is messy, roles get frozen, re­cruiters dis­ap­pear and com­pa­nies re­ject per­fectly good can­di­dates for rea­sons the can­di­date will never know. There is an en­tire web­site called Did They Ghost You?, so we’re not deal­ing with an un­ex­plained phe­nom­e­non.

But Nasdaq told me it was Mac Aviation. DHL ship­ping on be­half of SpaceX asked for a pass­port and told me it was due to a hit against the re­stricted par­ties on the U.S. gov­ern­ment con­sol­i­dated screen­ing list. Apple at least told me I’d matched some­thing, even if it then stopped talk­ing. An ap­pli­cant track­ing sys­tem will tell me noth­ing at all.

Upstream

I asked the Bureau of Industry and Security’s End-User Review Committee to re­view the orig­i­nal Entity List en­try. I’m not sure any­thing will come of it. The nor­mal process is de­signed for a listed per­son ask­ing to be re­moved, which cre­ates an in­ter­est­ing prob­lem here. I’m not the listed Sean Byrne, and the avail­able ev­i­dence sug­gests that per­son may never have ex­isted.

For now the U.S. gov­ern­men­t’s screen­ing data still says Sean Byrne, Cloonmull House, Drumcliffe, County Sligo.

I’ve still never lived in Sligo.

allen@bargi:~/​notes$ cat work­ing-with-ai.md

For most of my ca­reer, code gave me cer­tainty. A pro­gram did what its in­struc­tions told it to do. If the same in­put pro­duced a dif­fer­ent re­sult, we called it a bug.

People were never like that. As a leader, I can ex­plain a task and get ex­actly what I asked for. I can also get some­thing bet­ter be­cause a col­league un­der­stood the in­tent be­hind the re­quest. Sometimes the re­sult shows that I was not as clear as I thought.

Working with AI feels closer to the sec­ond ex­pe­ri­ence.

AI runs on soft­ware, but work­ing with it is not fully pre­dictable. The same re­quest can pro­duce a dif­fer­ent an­swer. It can make a use­ful con­nec­tion, miss an ob­vi­ous point, or sur­prise me with an ap­proach I had not con­sid­ered.

This is frus­trat­ing when I treat AI like a com­piler. It be­comes more use­ful when I treat the in­ter­ac­tion as a form of col­lab­o­ra­tion.

That does not make AI a per­son. It has no lived ex­pe­ri­ence, ac­count­abil­ity, or hu­man judg­ment. The com­par­i­son is about how we work. Good lead­ers do more than is­sue in­struc­tions. They share con­text, ex­plain the de­sired out­come, set bound­aries, and re­spond to what comes back.

The same habits im­prove my work with AI. A good prompt helps, but a shared work­ing con­text helps more. Examples, cor­rec­tions, and reusable in­struc­tions re­duce mis­un­der­stand­ings. Over time, the sys­tem be­comes bet­ter aligned with how I think and what I need from it.

The in­vest­ment is not in pre­tend­ing that AI is hu­man. It is in be­com­ing bet­ter at ex­press­ing in­tent.

We spent years learn­ing how to tell com­put­ers ex­actly what to do. Now we also need to ex­plain why the work mat­ters, what a good re­sult looks like, and where judg­ment is needed.

For me, that is the shift. AI is mak­ing soft­ware work less like is­su­ing com­mands to a ma­chine and more like lead­ing through a con­ver­sa­tion. The tech­nol­ogy is new. The lead­er­ship skills are not.

This note led to a thought­ful dis­cus­sion on Hacker News. I rec­om­mend read­ing through all the com­ments; the agree­ment, crit­i­cism, and dif­fer­ent ex­pe­ri­ences add more to the idea than I could fit here.

how it works

A drum­head clamped at its rim can only vi­brate in cer­tain shapes, at cer­tain fre­quen­cies. Those shapes and fre­quen­cies are the so­lu­tions of

−∇²u = λu  inside the shape,  u = 0 on the edge

Each so­lu­tion u is a mode, a stand­ing wave, and each λ gives a fre­quency pro­por­tional to √λ. This is an eigen­value prob­lem, and for al­most every shape it has no for­mula. So Eigendrum solves it nu­mer­i­cally: it cov­ers your shape with a mesh of tri­an­gles, builds the fi­nite el­e­ment stiff­ness and mass ma­tri­ces, and finds the small­est eigen­val­ues of Kφ = λMφ.

why you can trust the num­bers

A few shapes have spec­tra that can be writ­ten down ex­actly, and the solver is tested against them on every change. A cir­cle’s fre­quen­cies are the ze­ros of Bessel func­tions; a rec­tan­gle’s are π²(m²/​a² + n²/​b²). The solver re­pro­duces both to bet­ter than a tenth of a per­cent, and be­cause a con­form­ing fi­nite el­e­ment method min­imises en­ergy over a re­stricted space, its an­swers are guar­an­teed slight over­es­ti­mates, never un­der. The mea­sured er­ror is in “the num­bers”.

where you strike it mat­ters

Striking a spot dri­ves each mode in pro­por­tion to how much that mode moves there. Hit a line where a mode stands still and you can­not ex­cite it at all. That was not pro­grammed in; it falls out of pro­ject­ing the mal­let onto the modes.

So a strike is never one mode: it is every mode at once, in a mix­ture set by where your mal­let landed. The rules along the mode list are that mix­ture, and the modes marked with a square were the ones your mal­let could not reach. Pressing a row in­stead plays that sin­gle mode alone - some­thing no mal­let can do, and the only way to hear what one fre­quency of a shape ac­tu­ally sounds like.

drums from equa­tions

Besides trac­ing an out­line you can write one. r(t) gives the ra­dius as t sweeps one full turn, so 1 + 0.3cos(5t) is a five-lobed flower; a para­met­ric x(t), y(t) pair reaches the closed curves po­lar can­not, like a nephroid or an egg. This is not a short­cut for draw­ing. It reaches shapes no hand traces ac­cu­rately - eleven even lobes, a super­el­lipse part­way be­tween a cir­cle and a square - and it makes a shape some­thing you vary: change one num­ber and hear what moved.

A writ­ten shape trav­els as its own text. The link for a for­mula holds the for­mula, so it is some­thing you can read and re­type rather than a few hun­dred char­ac­ters of en­coded out­line, and edit­ing it in the ad­dress bar works. Anything too thin to mesh hon­estly is re­fused rather than an­swered, be­cause a sliver would still re­turn num­bers and they would be wrong.

can one hear the shape of a drum?

Mark Kac asked ex­actly that in 1966. In 1992 Carolyn Gordon, David Webb and Scott Wolpert an­swered no, by build­ing two dif­fer­ent shapes with iden­ti­cal spec­tra. Both are in the form list as Kac drum I and II. Each is made from the same seven tri­an­gles, re­arranged. They en­close the same area and the same perime­ter, and every fre­quency matches. Switch be­tween them and lis­ten: the out­lines are plainly dif­fer­ent and the sound is not.

what’s physics, and what’s just a slider

The fre­quency ra­tios, the mode shapes, and the pitch of the fun­da­men­tal are all baked into the out­line - you can’t touch them. What you can move is the wave speed (tension and den­sity): the pitch slider names the note a cir­cle of this area would sound, and your ac­tual shape lands above or be­low that on its own. Since every shape gets scaled to the same area first, that off­set is gen­uinely about the shape - roughly six semi­tones of spread across the built-in pre­sets, with the cir­cle al­ways low­est thanks to Faber-Krahn’s in­equal­ity. Fade time is a real slider too, since that’s ma­te­r­ial and air, not some­thing the maths pins down.

The mal­let works the same way. Its width is a slider; its con­tact time is fixed at a few mil­lisec­onds, be­cause no real beater is in­stant, and one that was would slam every mode equally hard. Both change how much of a mode a strike can reach - nei­ther one can shift where a mode ac­tu­ally sits. Damping is Rayleigh damp­ing, so loss rises with the square of fre­quency: the high over­tones die away first, which is why a drum dark­ens as it rings.

where it lives, and how to reach me

Eigendrum is hosted at eigen­drum.com. That is the ad­dress to link to and to cite; the older base­lashraf81.github.io/​eigen­drum is a mir­ror that now redi­rects there.

For ad­ver­tis­ing or part­ner­ship en­quiries, write to u2679054@uel.ac.uk. For any­thing wrong with the maths or the in­ter­face, an is­sue on the repos­i­tory is bet­ter, be­cause then the fix is pub­lic.

colophon

No build step and no ap­pli­ca­tion back­end: the mesh, the solve and the au­dio all run on your own ma­chine. The de­ployed site uses Vercel Analytics and Google Analytics; it car­ries no ad­ver­tis­ing net­work and no con­sent ban­ner. Support to­ward the do­main and host­ing is vol­un­tary, via the link above. The shape you draw lives in the ad­dress bar af­ter the #, which browsers never send to a server, and an­a­lyt­ics is con­fig­ured not to record it. Details in the pri­vacy no­tice. Set in Jost* by in­de­struc­tible type*. After Kac, Can One Hear the Shape of a Drum? (1966); Gordon, Webb and Wolpert (1992); and Driscoll, Eigenmodes of Isospectral Drums (1997), whose co­or­di­nates the two Kac drums use.

Source, in­clud­ing the solver and the tests that check it against the closed-form spec­tra: github.com/​Base­lAshraf81/​eigen­drum

Free to use, with no ac­count and noth­ing to in­stall. If you would like to put some­thing to­wards it, or would rather it were not ad-sup­ported: ko-fi.com/​base­lashraf

Raph Levien, August 8, 2026

I have for many decades been fas­ci­nated by the prospect of a curve fam­ily bet­ter suited for in­ter­ac­tive de­sign than cu­bic Béziers. In that search, I have come to a new-found re­spect for those Béziers. In par­tic­u­lar, though other curves like Euler spi­rals are bet­ter at rep­re­sent­ing smooth curves, they fall short at rep­re­sent­ing re­gions large cur­va­ture vari­a­tion, where cu­bic Béziers ex­cel. The great strength of Béziers is their ver­sa­til­ity.

So, to find a strictly bet­ter curve fam­ily, one re­quire­ment is clear: the fam­ily should con­tain both smooth cur­va­ture vari­a­tion and higher-ten­sion re­gions where cur­va­ture peaks. Polynomial spi­rals, or Spiro curves, the sub­ject of my PhD the­sis, fail to achieve this goal.

After con­sid­er­able search and re­ject­ing a num­ber of can­di­dates, I now bring a pro­posal for a curve fam­ily which I think is a very strong can­di­date for sup­plant­ing cu­bic Béziers in 2D vec­tor graphic de­sign.

Without fur­ther ado, the curve fam­ily is rep­re­sented by the Cesàro equa­tion, spec­i­fy­ing cur­va­ture as a func­tion of arc length:

$$\kappa(s) = \frac{as+b}{(cs^2+ds+1)^{1.5}}$$

This curve be­haves sur­pris­ingly sim­i­larly to a cu­bic Bézier, es­pe­cially at smaller an­gles, but when pushed has very dif­fer­ent be­hav­ior. Overall it has smoother cur­va­ture vari­a­tion and is more likely to have mo­not­o­nic cur­va­ture. It con­tains within it a few valu­able an­a­lytic curves, and is also good at ap­prox­i­mat­ing a wide range of oth­ers. The re­main­der of this blog post is de­voted to show­ing its be­hav­ior in a wide range of con­texts.

Approximation of cu­bic Béziers

The hy­per­bezier closely ap­prox­i­mates cu­bic Béziers at low de­flec­tion an­gles at the end­points. I will show rather than try to pre­sent math­e­matic rea­son­ing. Below is an in­ter­ac­tive tester that maps cu­bic Bézier con­trol points to a cor­re­spond­ing hy­per­bezier. The Bézier is shown in gray for com­par­i­son.

op­tions

cu­bic Bézier cur­va­ture comb

At larger an­gles, and also when the “arm lengths” of the con­trol han­dles get larger, the fit with the cu­bic Bézier is not par­tic­u­larly close, but the con­trol scheme is still a use­ful way of set­ting the pa­ra­me­ters for the hy­per­bezier curve (setting the poly­no­mial co­ef­fi­cients di­rectly is not at all in­tu­itive).

Note that this pa­ra­me­ter map­ping is a first us­able draft, and may not be the fi­nal ver­sion. That said, in an in­ter­ac­tive edit­ing con­text, a per­fect map­ping is not re­quired, as it’s al­ways pos­si­ble to tweak the con­trol points to reach any de­sired curve shape.

The de­tails of the map­ping can be found in the JavaScript source for this page, but the ba­sic prin­ci­ple is that the arm lengths set the de­nom­i­na­tor, then the nu­mer­a­tor is solved to make the end­point tan­gents match.

Exact an­a­lyt­i­cal curves

The most ob­vi­ous an­a­lyt­i­cal curve con­tained in the fam­ily is the Euler spi­ral, which is clearly at­tained when $c == d == 0$. The Euler spi­ral has smooth cur­va­ture vari­a­tion (it is a so­lu­tion to the Minimum Variation Curve prob­lem) and mo­not­o­nic cur­va­ture. An ex­act cir­cu­lar arc is also within the pa­ra­me­ter space, sim­ply when $a == 0$ as well. Cubic Béziers, by com­par­i­son, only ap­prox­i­mate cir­cu­lar arcs.

There a few other log-aes­thetic curves lurk­ing in the pa­ra­me­ter space, mean­ing cur­va­ture is sim­ply the ar­clength raised to a par­tic­u­lar power. Reachable ex­po­nents in­clude -3, -2, -1.5, and -0.5 (alternatively, us­ing the con­ven­tion from the log-aes­thetic pa­pers, α can be 1/3, 1/2, 2/3, and 2). The last of these is the cir­cle in­vo­lute, which is in­ter­est­ing be­cause it is its own par­al­lel curve, among other things. And the first of these is the evo­lute of the Euler spi­ral.

The pa­ra­me­ter map­ping de­scribed above is de­signed so that when the con­trol points lie on the “double parabola” in my Euler spi­ral par­al­lel curve blog post, then the re­sult is an ex­act Euler spi­ral.

Superellipses and squir­cles

A cu­bic Bézier can do super­el­lipses up to a cer­tain point, but does not ap­proach a sharp cor­ner; some­where be­fore then it starts de­vel­op­ing ad­di­tional in­flec­tion points, while a true super­el­lipse or squir­cle is of course con­vex.

The hy­per­bezier can go all the way to a sharp cor­ner, and vi­su­ally looks pretty close to the super­el­lipse. It’s not in­cred­i­bly ac­cu­rate (for mod­er­ate ex­po­nents like 5, the best cu­bic Bézier fit is slightly bet­ter, in fact), but vi­su­ally does the right thing. It’s prob­a­bly best to say that the hy­per­bezier is a sub­tly dif­fer­ent squir­cle than the super­el­lipse, nei­ther bet­ter nor worse.

Hyperbola

Another fun­da­men­tal curve is the hy­per­bola. It is round at the turn, but cur­va­ture tails off and the curve reaches a lin­ear as­ymp­tote on both sides. Cubic Béziers do not fit this curve well, and do not ex­hibit that as­ymp­totic be­hav­ior. But the hy­per­bezier fits it nat­u­rally and with very high ac­cu­racy.

We can also ap­pre­ci­ate the re­la­tion­ship math­e­mat­i­cally. In both, the cur­va­ture tails off as $\kappa \approx s^{-3}$ as­ymp­tot­i­cally. In ad­di­tion, us­ing the small an­gle ap­prox­i­ma­tion $\sin \theta \approx \theta, \cos \theta \approx 1$, in­te­grat­ing the Whewell equa­tion (see be­low) of the even-sym­met­ric hy­per­bezier yields the hy­per­bola ex­actly. This close re­la­tion­ship is one of the in­spi­ra­tions of the name; it is some­thing of a fu­sion be­tween a hy­per­bola and a Bézier. Another in­spi­ra­tion, in­ci­den­tally, is that the log-aes­thetic so­lu­tions, in­clud­ing the Euler spi­ral, are rep­re­sented by hy­per­ge­o­met­ric func­tions.

Elastica

An im­por­tant curve fam­ily to con­sider is the elas­tica, an ide­al­ized thin flex­i­ble strip. The math be­hind the elas­tica has hun­dreds of years of his­tory, but more to the point it is both a smooth curve (it can be char­ac­ter­ized as the ex­act so­lu­tion to the Minimum Energy Curve prob­lem) and can ex­hibit large cur­va­ture vari­a­tion when placed un­der ten­sion. To my eyes, it’s a more pleas­ing and nat­ural curve, as it’s based in phys­i­cal re­al­ity in a way that cu­bic Béziers aren’t. Ideally we’d be able to ap­prox­i­mate it well, and the fact that Spiro curves can’t is a strong case against them. Fortunately, our hy­per­bezier does rea­son­ably well. As in the case of the super­el­lipse, it’s a fairly de­cent vi­sual match, though again not a pre­cise ap­prox­i­ma­tion.

Some math­e­mat­i­cal prop­er­ties

This sec­tion is a grab-bag of some math­e­mat­i­cal prop­er­ties of the hy­per­bezier curve.

An ap­peal­ing prop­erty is that the Cesàro equa­tion is eas­ily in­te­grated, yield­ing a Whewell equa­tion:

$$\theta(s) = \frac{a’s+b’}{\sqrt{(cs^2+ds+1)}}$$

Here, the pa­ra­me­ters in the nu­mer­a­tor aren’t the same as the Cesàro equa­tion (though they are read­ily de­rived), but the qua­dratic poly­no­mial in the de­nom­i­na­tor is the same. Another round of in­te­gra­tion (which I be­lieve will be es­pe­cially use­ful for curve fit­ting) also yields sim­ple an­a­lyt­i­cal equa­tions, re­veal­ing that a close re­la­tion to trig func­tions.

The de­riva­tion of pa­ra­me­ters for the Whewell equa­tion fol­lows fairly straight­for­wardly from the fol­low­ing in­te­gral, which has a pleas­ing sym­me­try:

$$\int \frac{as+b}{(cs^2 + 1)^{1.5}} ds = \frac{bs-a/c}{\sqrt{cs^2 + 1}} + C$$

Going in the op­po­site di­rec­tion, the de­riv­a­tive is also straight­for­ward, yield­ing a qua­dratic over the same poly­no­mial in the de­nom­i­na­tor, this time with an ex­po­nent of 2.5. To com­pute roots, iden­ti­fy­ing the cur­va­ture ex­trema, in­volves merely solv­ing the qua­dratic.

Because the de­nom­i­na­tor is pos­i­tive, there is at most one in­flec­tion point, at $s = -b/a$, when it is in range [0, 1]. That clearly sig­nals that it can’t ac­cu­rately ap­prox­i­mate all cu­bic Béziers, as those can have two in­flec­tion points. In my ex­pe­ri­ence, in­stances of cu­bic Béziers with two in­flec­tion points are rarely used in de­signs (it might be in­ter­est­ing to look at a cor­pus of vec­tor graph­ics to quan­tify this).

I have ex­per­i­mented with other val­ues for the ex­po­nent than 1.5. A value of 1 would be ap­peal­ingly sim­ple, as it would re­sem­ble Padé ap­prox­i­ma­tion, but I found it can­not rep­re­sent high-ten­sion curves in a nu­mer­i­cally sta­ble way. I also con­sid­ered an ex­po­nent of 2, but the in­te­grals don’t come out as cleanly and over­all it did­n’t match cu­bic Béziers as closely. Perhaps this is not sur­pris­ing, as the ex­po­nent of 1.5 re­sem­bles the equa­tion for cur­va­ture of a para­met­ric curve $\mathbf{x}(t)$:

$$\kappa = \frac{\mathbf{x}′ \times \mathbf{x}‘’}{(\mathbf{x}′ \cdot \mathbf{x}’)^{1.5}}$$

In gen­eral, arc length pa­ram­e­trized curves are in many ways more pleas­ant to work with than gen­eral para­met­ric curves; there’s a non­triv­ial cal­cu­la­tion to solve the in­verse arc prob­lem for Béziers.

Future work

In this post, I pre­sent the curve fam­ily to the world. I wel­come and en­cour­age ex­per­i­men­ta­tion. Of course, there is lots to be done to make the math­e­mat­i­cal idea into a prac­ti­cal tool. For one, it should be wired up into a spline. I ex­pect to fol­low the ideas in The hy­per­bezier pen tool by Colin Rofls, which was based on an ear­lier draft of the hy­per­bezier math. This ear­lier draft had some of the de­sired be­hav­ior of the cur­rent pro­posal, but also some flaws, in­clud­ing not be­ing able to ap­prox­i­mate sharp super­el­lipses, and also not be­ing closed un­der sub­di­vi­sion. On the other hand, it had a straight­for­ward so­lu­tion to the prob­lem of map­ping Bézier-like con­trol points to the pa­ra­me­ters.

One idea from the above spline that I think is worth pre­serv­ing is mak­ing splines G2 con­tin­u­ous by con­struc­tion. Such ap­proaches don’t work well with cu­bic Béziers be­cause there are mul­ti­ple lo­cal min­ima, cre­at­ing sad­dle points in the op­ti­miza­tion ter­rain.

Another promis­ing di­rec­tion is work­ing out curve fit­ting. While I haven’t got­ten deep into it yet, I ex­pect curve fit­ting to be fairly straight­for­ward, es­pe­cially com­pared with cu­bic Béziers. I’m es­pe­cially in­ter­ested in fit­ting touch or pen data. As usual for this prob­lem, a cen­tral part of the chal­lenge is find­ing good sub­di­vi­sion points.

I hope to land bet­ter im­ple­men­ta­tions in the spline repos­i­tory, but make no guar­an­tees how fast that will go.

Acknowledgements

I’ve been work­ing on this for years, and have ben­e­fited from dis­cus­sions with many peo­ple, in­clud­ing Colin Rofls, Alex Ionuț, Jacob Rus, and Trevor David Black (apologies if I’ve missed some­one; just let me know). Special credit goes to Muhammad Ragib Hasin, who im­proved many of the nu­mer­i­cal and solv­ing tech­niques for pa­ra­me­ter map­ping. AI as­sis­tance was used to re­fine the cur­rent ver­sion of pa­ra­me­ter map­ping and to pre­pare the vi­su­als for this post.

Contact: Olivia Walther, owalther@acc.org ,

WASHINGTON (Aug 11, 2026) -

The size of a per­son’s mid­sec­tion is a bet­ter in­di­ca­tor of car­dio­vas­cu­lar dis­ease risk when com­pared to their body mass in­dex (BMI) alone, ac­cord­ing to a study pub­lished in JACC, the flag­ship jour­nal of the American College of Cardiology. BMI has tra­di­tion­ally been used to de­ter­mine over­weight or obe­sity, com­mon risk fac­tors for heart dis­ease, but this study shows that not ac­count­ing for waist cir­cum­fer­ence (WC) or waist-to-hip ra­tio (WHR) can lead to mis­clas­si­fi­ca­tion of car­dio­vas­cu­lar dis­ease risk.

“Indeed, it ap­pears that WC and WHR re­clas­sify risk de­fined by tra­di­tional BMI thresh­olds,” said Michael J. Blaha, MD, MPH, se­nior au­thor of the study and di­rec­tor of clin­i­cal re­search at Johns Hopkins Ciccarone Center for the Prevention of Cardiovascular Disease. “We saw in­di­vid­u­als with clin­i­cally de­ter­mined nor­mal weight who had el­e­vated cen­tral adi­pos­ity and high WHR, as­so­ci­at­ing them with higher risk across most out­comes.”

BMI is cal­cu­lated by di­vid­ing weight in kilo­grams by height in me­ters squared and is a com­mon prac­tice for di­ag­nos­ing obe­sity. However, BMI can­not ac­count for dis­tri­b­u­tion of body fat. Studies have shown that vis­ceral fat, which is fat that sur­rounds the in­ter­nal or­gans in the ab­dom­i­nal area, is as­so­ci­ated with chronic dis­eases like heart dis­ease and di­a­betes, while sub­cu­ta­neous fat, which is lo­cated di­rectly un­der the skin, is not as strongly as­so­ci­ated.

Despite ev­i­dence link­ing cen­tral adi­pos­ity, the ac­cu­mu­la­tion of both vis­ceral and sub­cu­ta­neous fat in the ab­dom­i­nal area, to ad­verse car­dio­vas­cu­lar health out­comes, BMI is still the most used met­ric to de­ter­mine over­weight and obe­sity and fu­ture car­dio­vas­cu­lar risk.

This study ex­am­ines whether adding WC and WHR to BMI bet­ter pre­dicts fu­ture car­dio­vas­cu­lar risk. Researchers from the Cross Cohort Collaboration looked at over 260,000 peo­ple over an av­er­age of 20 years who had ei­ther WC or WHR data and at least one of nine out­comes: time to first fa­tal and non-fa­tal my­ocar­dial in­farc­tion, fa­tal and non-fa­tal stroke, heart fail­ure, atrial fib­ril­la­tion, to­tal coro­nary heart dis­ease (CHD), to­tal car­dio­vas­cu­lar dis­ease (CVD), CHD mor­tal­ity, CVD mor­tal­ity and/​or all-cause mor­tal­ity.

In in­di­vid­u­als with nor­mal weight as de­ter­mined by BMI, 5% had high WC and 18% had high WHR; among those with over­weight, 39% had high WC and 40% had high WHR. Among those with obe­sity, 9% had low WC and 45% had low WHR.

Those in­di­vid­u­als with nor­mal weight or over­weight and clin­i­cally de­fined high WC or WHR were as­so­ci­ated with a 15% - 50% greater risk for most of the nine stud­ied out­comes. Those with obe­sity and low WC were not found to be as­so­ci­ated with a sig­nif­i­cantly dif­fer­ent risk of out­comes com­pared with those who had nor­mal weight and low WC, ex­cept for all-cause mor­tal­ity, for which risk was sig­nif­i­cantly lower.

“Our find­ings em­pha­size the crit­i­cal role of iden­ti­fy­ing el­e­vated cen­tral adi­pos­ity, even in in­di­vid­u­als with a nor­mal BMI or with a BMI in the over­weight range. Relying solely on BMI may re­sult in mis­clas­si­fi­ca­tion of car­dio­vas­cu­lar risk across a wide range of car­dio­vas­cu­lar out­comes,” Zeina A. Dardari, PhD, MS, lead au­thor of the study, said. “We en­cour­age clin­i­cians to con­sider cen­tral adi­pos­ity dis­tri­b­u­tion across the en­tire BMI spec­trum when eval­u­at­ing car­dio­vas­cu­lar risk in pri­mary pre­ven­tion set­tings.”

Limitations of the study in­clude that it did not have mea­sures of phys­i­cal ac­tiv­ity, diet or ge­netic obe­sity risk, which have all been shown to play a role in the de­vel­op­ment of CVD. It also in­cluded only one as­sess­ment of WC and WHR, which could limit un­der­stand­ing of how changes in cen­tral fat ac­cu­mu­la­tion over time in­flu­ences CVD risk.

“It is time to aban­don a sole fo­cus on body mass in­dex,” said Harlan M. Krumholz, MD, SM, MACC, FAHA, Editor-in-Chief of JACC and the Harold H. Hines, Jr Professor at the Yale School of Medicine. “This enor­mously im­por­tant study, based on data from hun­dreds of thou­sands of par­tic­i­pants in large-scale co­hort stud­ies, au­thor­i­ta­tively shows that waist cir­cum­fer­ence and waist-to-hip ra­tio pro­vide crit­i­cal in­for­ma­tion about car­dio­vas­cu­lar risk, even among peo­ple with a BMI con­sid­ered nor­mal. Where fat is dis­trib­uted mat­ters, and these sim­ple mea­sures should be part of rou­tine car­dio­vas­cu­lar risk as­sess­ment.”

For an em­bar­goed copy of the study “Risk Reclassification Beyond BMI by Waist Circumference and Waist-to-Hip Ratio Across Nine Cardiovascular Outcomes: Results from the Cross-Cohort Collaboration,” con­tact JACC Media Relations Manager Olivia Walther at owalther@acc.org.

The American College of Cardiology (ACC) is a global leader ded­i­cated to trans­form­ing car­dio­vas­cu­lar care and im­prov­ing heart health for all. For more than 75 years, the ACC has em­pow­ered a com­mu­nity of over 60,000 car­dio­vas­cu­lar pro­fes­sion­als across more than 140 coun­tries with cut­ting-edge ed­u­ca­tion and ad­vo­cacy, rig­or­ous pro­fes­sional cre­den­tials, and trusted clin­i­cal guid­ance. From its world-class JACC Journals and NCDR reg­istries to its Accreditation Services, global net­work of Chapters and Sections, and CardioSmart pa­tient ini­tia­tives, the College is com­mit­ted to cre­at­ing a world where sci­ence, knowl­edge and in­no­va­tion op­ti­mize pa­tient care and out­comes. Learn more at www.ACC.org or con­nect on so­cial me­dia at @ACCinTouch.

The ACC’s JACC Journals rank among the top car­dio­vas­cu­lar jour­nals in the world for sci­en­tific im­pact. The flag­ship jour­nal, the Journal of the American College of Cardiology (JACC) — and spe­cialty jour­nals con­sist­ing of JACC: Advances, JACC: Asia, JACC: Basic to Translational Science, JACC: CardioOncology, JACC: Cardiovascular Imaging, JACC: Cardiovascular Interventions, JACC: Case Reports, JACC: Clinical Electrophysiology and JACC: Heart Failure — pride them­selves on pub­lish­ing the top peer-re­viewed re­search on all as­pects of car­dio­vas­cu­lar dis­ease. Learn more at JACC.org.

###

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ASUS Oxiis E250G1

Ride Easy. Explore More.

Oxiis is a uni­ver­sal, fric­tion-drive mo­tor sys­tem that trans­forms any con­ven­tional bi­cy­cle into a smart e-bike. Its name fuses the Greek Oxis (agility) and Axis (pivot)—perfectly re­flect­ing how its high-per­for­mance drive core de­liv­ers ul­ti­mate ac­cel­er­a­tion and in­stant, nim­ble re­spon­sive­ness right to your ride.

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Compatible with tire sizes from 16 to 29 inches, plus 700C.

Seat post

Fits 25.4 – 34.9 mm posts (spacers in­cluded).

Installation Guide

Easy in­stal­la­tion

No com­plex tools are needed to in­stall ASUS Oxiis E250G1.

App

ASUS Oxiis app

Three modes. One seam­less ex­pe­ri­ence.

Switch modes ef­fort­lessly us­ing the in­te­grated but­ton or ded­i­cated app.

Eco mode: Efficient sup­port for flat ter­rain and tail­winds.

Normal mode: Smooth, bal­anced power for every­day ad­ven­tures.

Sport mode: Instant boost for steep climbs and high-speed per­for­mance.

Spec

Product Specifications

Motor Power

250W (Rated) / 500W (Peak)

Range

50 km (31 miles) in eco mode

Battery

158.4 Wh / 36 V

Weight

3.7 kg (including bat­tery)

Dimensions

400 x 84 x 128 mm (L x W x H)

Max Speed

32 km/​h (Limited to 25 km/​h in spe­cific re­gions)

Charging Time

2 hours (100W PD Charger re­quired)

Waterproof Rating

IPX4

App Compatibility

Android & iOS

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