Computational Complexity and other fun stuff in math and com­puter sci­ence from Lance Fortnow and Bill Gasarch

New U. S laws de­signed to pro­tect mi­nors are pulling mil­lions of adult Americans into manda­tory age-ver­i­fi­ca­tion gates to ac­cess on­line con­tent, lead­ing to back­lash from users and crit­i­cism from pri­vacy ad­vo­cates that a free and open in­ter­net is at stake. Roughly half of U.S. states have en­acted or are ad­vanc­ing laws re­quir­ing plat­forms — in­clud­ing adult con­tent sites, on­line gam­ing ser­vices, and so­cial me­dia apps — to block un­der­age users, forc­ing com­pa­nies to screen every­one who ap­proaches these dig­i­tal gates.

“There’s a big spec­trum,” said Joe Kaufmann, global head of pri­vacy at Ju­mio, one of the largest dig­i­tal iden­tity-ver­i­fi­ca­tion and au­then­ti­ca­tion plat­forms. He ex­plained that the patch­work of state laws vary in tech­ni­cal de­mands and com­pli­ance ex­pec­ta­tions. “The reg­u­la­tions are mov­ing in many dif­fer­ent di­rec­tions at once,” he said.

Social me­dia com­pany Discord an­nounced plans in February to roll out manda­tory age ver­i­fi­ca­tion glob­ally, which the com­pany said would rely on ver­i­fi­ca­tion meth­ods de­signed so fa­cial analy­sis oc­curs on a user’s de­vice and sub­mit­ted data would be deleted im­me­di­ately. The pro­posal quickly drew back­lash from users con­cerned about hav­ing to sub­mit self­ies or gov­ern­ment IDs to ac­cess cer­tain fea­tures, which led Discord to de­lay the launch un­til the sec­ond half of this year.

“Let me be up­front: we knew this roll­out was go­ing to be con­tro­ver­sial. Any time you in­tro­duce some­thing that touches iden­tity and ver­i­fi­ca­tion, peo­ple are go­ing to have strong feel­ings,” Discord chief tech­nol­ogy of­fi­cer and co-founder Stanislav Vishnevskiy wrote in a Feb. 24 blog post.

Websites offering adult con­tent, gam­bling, or fi­nan­cial ser­vices of­ten rely on full iden­tity ver­i­fi­ca­tion that re­quires scan­ning a gov­ern­ment ID and match­ing it to a live im­age. But most of the ver­i­fi­ca­tion sys­tems pow­er­ing these check­points — of­ten run by spe­cial­ized iden­tity-ver­i­fi­ca­tion ven­dors on be­half of web­sites — rely on ar­ti­fi­cial in­tel­li­gence such as fa­cial recog­ni­tion and age-es­ti­ma­tion mod­els that an­a­lyze self­ies or video to de­ter­mine in sec­onds whether some­one is old enough to ac­cess con­tent. Social me­dia and lower-risk ser­vices may use lighter es­ti­ma­tion tools de­signed to con­firm age with­out per­ma­nently stor­ing de­tailed iden­tity records.

Vendors say a chal­lenge is bal­anc­ing safety with how much fric­tion users will tol­er­ate. “We’re in the busi­ness of en­sur­ing that you are ab­solutely keep­ing mi­nors safe and out and able to let adults in with as lit­tle fric­tion as pos­si­ble,” said Rivka Gewirtz Little, chief growth of­fi­cer at iden­tity-ver­i­fi­ca­tion plat­form So­cure. Excessive data col­lec­tion, she added, cre­ates fric­tion that users re­sist.

Still, many users per­ceive manda­tory iden­tity checks as in­va­sive. “Having an­other way to be forced to pro­vide that in­for­ma­tion is in­tru­sive to peo­ple,” said Heidi Howard Tandy, a part­ner at Berger Singerman who spe­cial­izes in in­tel­lec­tual prop­erty and in­ter­net law. Some users may at­tempt workarounds — in­clud­ing pre­paid cards or al­ter­na­tive cre­den­tials — or turn to unau­tho­rized dis­tri­b­u­tion chan­nels. “It’s go­ing to cause a piracy sit­u­a­tion,” she added.

In many im­ple­men­ta­tions, ver­i­fi­ca­tion ven­dors — not the web­sites them­selves — process and re­tain the iden­tity in­for­ma­tion, re­turn­ing only a pass-fail sig­nal to the plat­form.

Gewirtz Little said So­cure does not sell ver­i­fi­ca­tion data and that in light­weight age-es­ti­ma­tion sce­nar­ios, where plat­forms use quick fa­cial analy­sis or other sig­nals rather than gov­ern­ment doc­u­men­ta­tion, the com­pany may store lit­tle or no in­for­ma­tion. But in fuller iden­tity-ver­i­fi­ca­tion con­texts, such as gam­ing and fraud pre­ven­tion that re­quire ID scans, cer­tain adult ver­i­fi­ca­tion records may be re­tained to doc­u­ment com­pli­ance. She said So­cure can keep some adult ver­i­fi­ca­tion data for up to three years while fol­low­ing ap­plic­a­ble pri­vacy and purg­ing rules.

Civil lib­er­ties’ ad­vo­cates warn that con­cen­trat­ing large vol­umes of iden­tity data among a small num­ber of ver­i­fi­ca­tion ven­dors can cre­ate at­trac­tive tar­gets for hack­ers and gov­ern­ment de­mands. Ear­lier this year, Discord dis­closed a data breach that ex­posed ID im­ages be­long­ing to ap­prox­i­mately 70,000 users through a com­pro­mised third-party ser­vice, high­light­ing the se­cu­rity risks as­so­ci­ated with stor­ing sen­si­tive iden­tity in­for­ma­tion.

In ad­di­tion, they warn that ex­pand­ing age-ver­i­fi­ca­tion sys­tems rep­re­sent not only a us­abil­ity chal­lenge but a struc­tural shift in how iden­tity be­comes tied to on­line be­hav­ior. Age ver­i­fi­ca­tion risks ty­ing users’ “most sen­si­tive and im­mutable data” — names, faces, birth­days, home ad­dresses — to their on­line ac­tiv­ity, ac­cord­ing to Molly Buckley, a leg­isla­tive an­a­lyst at the Electronic Frontier Foundation.  “Age ver­i­fi­ca­tion strikes at the foun­da­tion of the free and open in­ter­net,” she said.

Even when ven­dors promise to safe­guard per­sonal in­for­ma­tion, users ul­ti­mately rely on con­trac­tual terms they rarely read or fully un­der­stand. “There’s lan­guage in their terms-of-use poli­cies that says if the in­for­ma­tion is re­quested by law en­force­ment, they’ll hand it over. They can’t confirm that they will al­ways for­ever be the only en­tity who has all of this in­for­ma­tion. Every­one needs to un­der­stand that their base­line in­for­ma­tion is not some­thing un­der their con­trol,” Tandy said.

As more plat­forms route age checks through third-party ven­dors, that con­cen­tra­tion of iden­tity data is also cre­at­ing new le­gal ex­po­sure for the com­pa­nies that rely on them. “A com­pany is go­ing to have some of that in­for­ma­tion pass­ing through their own servers,” Tandy said. “And you can’t of­fload that kind of li­a­bil­ity to a third party.”

Companies can dis­trib­ute risk through con­tracts and in­sur­ance, she said, but they re­main re­spon­si­ble for how iden­tity sys­tems in­ter­act with their in­fra­struc­ture. “What you can do is have re­ally good in­sur­ance and re­quire re­ally good in­sur­ance from the en­ti­ties that you’re con­tract­ing with,” she said.

Tandy also cau­tioned that re­ten­tion promises can be more com­plex than they ap­pear. “If they say they’re hold­ing it for three years, that’s the min­i­mum amount of time they’re hold­ing it for,” she said. “I wouldn’t feel com­fort­able trust­ing a com­pany that says, ‘We delete every­thing one day af­ter three years.’ That is not go­ing to hap­pen,” she added.

Federal and state reg­u­la­tors ar­gue that age-ver­i­fi­ca­tion laws are pri­mar­ily a re­sponse to doc­u­mented harms to mi­nors and in­sist the rules must op­er­ate un­der strict pri­vacy and se­cu­rity safe­guards.

An FTC spokesper­son told CNBC that com­pa­nies must limit how col­lected in­for­ma­tion is used. While age-ver­i­fi­ca­tion tech­nolo­gies can help par­ents pro­tect chil­dren on­line, the agency said firms are still bound by ex­ist­ing con­sumer pro­tec­tion rules gov­ern­ing data min­i­miza­tion, re­ten­tion, and se­cu­rity. The agency pointed to ex­ist­ing rules re­quir­ing firms to re­tain per­sonal in­for­ma­tion only as long as rea­son­ably nec­es­sary and to safe­guard its con­fi­den­tial­ity and in­tegrity.

Amazon’s ecom­merce busi­ness has sum­moned a large group of en­gi­neers to a meet­ing on Tuesday for a “deep dive” into a spate of out­ages, in­clud­ing in­ci­dents tied to the use of AI cod­ing tools.

The on­line re­tail gi­ant said there had been a “trend of in­ci­dents” in re­cent months, char­ac­ter­ized by a “high blast ra­dius” and “Gen-AI as­sisted changes” among other fac­tors, ac­cord­ing to a brief­ing note for the meet­ing seen by the FT.

Under “contributing fac­tors” the note in­cluded “novel GenAI us­age for which best prac­tices and safe­guards are not yet fully es­tab­lished.”

“Folks, as you likely know, the avail­abil­ity of the site and re­lated in­fra­struc­ture has not been good re­cently,” Dave Treadwell, a se­nior vice-pres­i­dent at the group, told em­ploy­ees in an email, also seen by the FT.

The note ahead of Tuesday’s meet­ing did not spec­ify which par­tic­u­lar in­ci­dents the group planned to dis­cuss.

Amazon’s web­site and shop­ping app went down for nearly six hours this month in an in­ci­dent the com­pany said in­volved an er­ro­neous “software code de­ploy­ment.” The out­age left cus­tomers un­able to com­plete trans­ac­tions or ac­cess func­tions such as check­ing ac­count de­tails and prod­uct prices.

Treadwell, a for­mer Microsoft en­gi­neer­ing ex­ec­u­tive, told em­ploy­ees that Amazon would fo­cus its weekly “This Week in Stores Tech” (TWiST) meet­ing on a “deep dive into some of the is­sues that got us here as well as some short im­me­di­ate term ini­tia­tives” the group hopes will limit fu­ture out­ages.

My LinkedIn and Twitter feeds are full of screen­shots from the re­cent Forbes ar­ti­cle on Cursor claim­ing that Anthropic’s $200/month Claude Code Max plan can con­sume $5,000 in com­pute. The rel­e­vant quote:

Today, that sub­si­diza­tion ap­pears to be even more ag­gres­sive, with that $200 plan able to con­sume about $5,000 in com­pute, ac­cord­ing to a dif­fer­ent per­son who has seen analy­ses on the com­pa­ny’s com­pute spend pat­terns.

This is be­ing shared as proof that Anthropic is haem­or­rhag­ing money on in­fer­ence. It does­n’t sur­vive ba­sic scrutiny.

I’m fairly con­fi­dent the Forbes sources are con­fus­ing re­tail API prices with ac­tual com­pute costs. These are very dif­fer­ent things.

Anthropic’s cur­rent API pric­ing for Opus 4.6 is $5 per mil­lion in­put to­kens and $25 per mil­lion out­put to­kens. At those prices, yes - a heavy Claude Code Max 20 user could rack up $5,000/month in API-equivalent us­age. That maths checks out.

But API pric­ing is not what it costs Anthropic to serve those to­kens.

The best way to es­ti­mate what in­fer­ence ac­tu­ally costs is to look at what open-weight mod­els of sim­i­lar size are priced at on OpenRouter - where mul­ti­ple providers com­pete on price.

Qwen 3.5 397B-A17B is a good com­par­i­son point. It’s a large MoE model, broadly com­pa­ra­ble in ar­chi­tec­ture size to what Opus 4.6 is likely to be. Equally, so is Kimi K2.5 1T params with 32B ac­tive, which is prob­a­bly ap­proach­ing the up­per limit of what you can ef­fi­ciently serve.

Here’s what the pric­ing looks like:

The Qwen 3.5 397B model on OpenRouter (via Alibaba Cloud) costs _$0.39_ per mil­lion in­put to­kens and _$2.34_ per mil­lion out­put to­kens. Compare that to Opus 4.6′s API pric­ing of $5/$25. Kimi K2.5 is even cheaper at $0.45 per mil­lion in­put to­kens and $2.25 out­put.

And this ra­tio holds for cached to­kens too - DeepInfra charges $0.07/MTok for cache reads on Kimi K2.5 vs Anthropic’s $0.50/MTok.

These OpenRouter providers are run­ning a busi­ness. They have to cover their com­pute costs, pay for GPUs, and make a mar­gin. They’re not char­i­ties. If so many can serve a model of com­pa­ra­ble size at ~10% of Anthropic’s API price and re­main in busi­ness, it is hard for me to be­lieve that they are all tak­ing enor­mous losses (at ~the ex­act same rate range).

If a heavy Claude Code Max user con­sumes $5,000 worth of to­kens at Anthropic’s re­tail API prices, and the ac­tual com­pute cost is roughly 10% of that, Anthropic is look­ing at ap­prox­i­mately $500 in real com­pute cost for the heav­i­est users.

That’s a loss of $300/month on the most ex­treme power users - not $4,800.

However, most users don’t come any­where near the limit. Anthropic them­selves said when they in­tro­duced weekly caps that fewer than 5% of sub­scribers would be af­fected. I per­son­ally use the Max 20x plan and prob­a­bly con­sume around 50% of my weekly to­ken bud­get and it’s hard to use that many to­kens with­out get­ting se­ri­ous RSI. At that level of us­age, the maths works out to roughly break-even or prof­itable for Anthropic.

The real story is ac­tu­ally in the ar­ti­cle. The $5,000 fig­ure comes from Cursor’s in­ter­nal analy­sis. And for Cursor, the num­ber prob­a­bly is roughly cor­rect - be­cause Cursor has to pay Anthropic’s re­tail API prices (or close to it) for ac­cess to Opus 4.6.

So to pro­vide a Claude Code-equivalent ex­pe­ri­ence us­ing Opus 4.6, it would cost Cursor ~$5,000 per power user per month. But it would cost Anthropic per­haps $500 max.

And the real is­sue for Cursor is that de­vel­op­ers want to use the Anthropic mod­els, even in Cursor it­self. They have real “brand aware­ness”, and they are gen­uinely bet­ter than the cheaper open weights mod­els - for now at least. It’s a real co­nun­drum for them.

Obviously Anthropic is­n’t print­ing free cash­flow. The costs of train­ing fron­tier mod­els, the enor­mous salaries re­quired to hire top AI re­searchers, the multi-bil­lion dol­lar com­pute com­mit­ments - these are gen­uinely mas­sive ex­penses that dwarf in­fer­ence costs.

But on a per-user, per-to­ken ba­sis for in­fer­ence? I be­lieve Anthropic is very likely prof­itable - po­ten­tially very prof­itable - on the av­er­age Claude Code sub­scriber.

The “AI in­fer­ence is a money pit” nar­ra­tive is mis­in­for­ma­tion that ac­tu­ally plays into the hands of the fron­tier labs. If every­one be­lieves that serv­ing to­kens is wildly ex­pen­sive, no­body ques­tions the 10x+ markups on API pric­ing. It dis­cour­ages com­pe­ti­tion and makes the moat look deeper than it is.

If you want to un­der­stand the real eco­nom­ics of AI in­fer­ence, don’t take API prices at face value. Look at what com­pet­i­tive open-weight model providers charge on OpenRouter. That’s a much closer proxy for what it ac­tu­ally costs to run these mod­els - and it’s a frac­tion of what the fron­tier labs charge.

Background: Why I put my whole life into a sin­gle data­base

Back in 2019, I started col­lect­ing all kinds of met­rics about my life. Every sin­gle day for the last 3 years I tracked over 100 dif­fer­ent data types - rang­ing from fit­ness & nu­tri­tion to so­cial life, com­puter us­age and weather.

Ideas or sug­ges­tions?

I’d love to hear from you!

The goal of this pro­ject was to an­swer ques­tions about my life, like

How does liv­ing in dif­fer­ent cities af­fect other fac­tors like fit­ness, pro­duc­tiv­ity and hap­pi­ness?

How does sleep af­fect my day, my fit­ness level, and hap­pi­ness?

How does the weather, and the dif­fer­ent sea­sons af­fect my life?

Are there any trends over the last few years?

How does com­puter time, work and hours in meet­ings af­fect my per­sonal life?

Since the start of this pro­ject, I col­lected ~380,000 data points, with the biggest data sources be­ing:

Naturally af­ter I started col­lect­ing this data, I wanted to vi­su­al­ize what I was learn­ing, so I cre­ated this page. Initially, the do­main where­is­Fe­lix.to­day (now re­named to how­is­Fe­lix.to­day) started as a joke to re­spond to friends ask­ing when I’d be back in NYC or San Francisco. Rather than send them my sched­ule, I’d point them to this do­main. However, now it’s more than my lo­ca­tion: it’s all of me.

Use a sin­gle data­base, owned and hosted by me, with all the data I’ve col­lected over the years

Be able to eas­ily add and re­move ques­tions on the fly, as I learn what’s ben­e­fi­cial to track

Full con­trol of how the data is vi­su­al­ized

Works well for fre­quent fly­ers with mixed time zones

I se­lected 48 graphs to show pub­licly on this page. For pri­vacy rea­sons, and to pre­vent any ac­ci­den­tal data leaks, the graphs be­low are snap­shots taken on a given day.

Visualization of the num­ber of data en­tries in FxLifeSheet over the last 10 years, and where the data came from.

Initially (2014) the only data used was RescueTime and Foursquare Swarm lo­ca­tion data

Once I started the FxLifeSheet pro­ject in April 2019, I man­u­ally tracked , rang­ing from mood, sleep, so­cial life, to fit­ness data

I was able to ret­ro­spec­tively fetch the his­toric weather data based on my lo­ca­tion on a given day

I also im­ple­mented other im­port sources, like fetch­ing my his­toric weight and the num­ber of steps from Apple Health

Days tracked my Mood to be Happy & Excited

On days where I tracked my mood to be “happy” & “excited”, the fol­low­ing other fac­tors of my life were af­fected

50% more likely to have pushed my com­fort zone

44% more likely to have med­i­tated that day

33% more ex­cited about what’s ahead in the fu­ture

31% more likely to drink al­co­hol that day (parties, good friends and such)

28% more time spent read­ing or lis­ten­ing to au­dio books

26% more likely to have worked on in­ter­est­ing tech­ni­cal chal­lenges

20% more likely to have learned some­thing new that day

45% less time spent in video & au­dio calls that day

All flights taken within the last 7 years, tracked us­ing Foursquare Swarm, an­a­lyzed by JetLovers.

The stats clearly show the im­pact of COVID start­ing 2020

Sunday has been my “commute” day, fly­ing be­tween San Francisco, New York City and Vienna

All flights taken within the last 7 years, tracked us­ing Foursquare Swarm, an­a­lyzed by JetLovers.

Frankfurt - Vienna was the flight con­nect­ing me with most US air­ports

Germany is high up on the list due to lay­overs, even though I did­n’t spend ac­tu­ally much time there

Inspired by Your Life in Weeks by WaitButWhy, I use Google Sheets to vi­su­al­ize every week of my life, with lit­tle notes on what city/​coun­try I was in, and other life events that have hap­pened.

The first 14 years I did­n’t re­ally get much done

I can highly rec­om­mend tak­ing a few weeks (or even months) off be­tween jobs (if you have the pos­si­bil­ity)

Shades of blue in­di­cate my full-time em­ploy­ments

You can cre­ate your own ver­sion us­ing my tem­plate

Average daily steps mea­sured through the iPhone’s Apple Health app. I de­cided against us­ing SmartWatch data for steps, as SmartWatches have changed over the last 8 years.

I walked a to­tal of steps over last 8 years

I walk more than twice as much when I’m in New York, com­pared to any other city

In NYC I had the gen­eral rule of thumb to walk in­stead of tak­ing pub­lic tran­sit when­ever it’s less than 40 min­utes. I used that time to call friends & fam­ily, or lis­ten to au­dio books

Although Vienna is very walk­a­ble, the ex­cel­lent pub­lic tran­sit sys­tem with sub­way trains com­ing every 3-5 min­utes, has caused me to walk less

San Francisco was al­ways scary to walk

This graph clearly shows the cor­re­la­tion be­tween my body weight and my sleep­ing/​rest­ing heart rate. The rest­ing heart rate is mea­sured by the Withings ScanWatch while sleep­ing, and in­di­cates how hard your heart has to work while not be­ing ac­tive. Generally the lower the rest­ing heart rate, the bet­ter.

I started my lean bulk (controlled weight gain com­bined with 5 work­outs a week) in August 2020

My rest­ing heart rate went from 58bpm to 67bpm () from August 2020 to March 2021 with a weight gain of (+19lbs) as part of a con­trolled lean-bulk com­bined with a 5-day/week work­out rou­tine

The spike in rest­ing heart rate in July & August 2021 was due to bars and night­clubs open­ing up again in Austria

After a night of drink­ing, my rest­ing/​sleep­ing heart rate was about 50% higher than af­ter a night with­out any al­co­hol

The spike in rest­ing heart rate in Oct/Nov/Dec 2021 was due to hav­ing bron­chi­tis and a cold/​flu, not get­ting cor­rect treat­ment early enough

How healthy have I been over the Years?

Every day I an­swered the ques­tion on how healthy I felt. In the graph, the yel­low color in­di­cates that I felt a lit­tle un­der the weather, not sick per se. Red means I was sick and had to stay home. Green means I felt en­er­gized and healthy.

During the COVID lock­downs I tended to stay health­ier. This may be due to not go­ing out, no heavy drink­ing, less close con­tact with oth­ers, etc. which re­sulted in me hav­ing bet­ter sleep.

Usually dur­ing ex­ces­sive trav­el­ing I get sick (cold/flu)

Q4 2021 I had bron­chi­tis, how­ever, I did­n’t know about it at the time and did­n’t get proper treat­ment

Overall I’m quite prone to get­ting sick (cold/flu)

Days with more than 4 Alcoholic Drinks

On days where I had more than 4 al­co­holic bev­er­ages (meaning I was par­ty­ing), the fol­low­ing other fac­tors were af­fected

21x more likely to dance

80% more likely to take a nap the day of, or the day af­ter

40% warmer tem­per­a­tures, and 40% less pre­cip­i­ta­tion. There weren’t many op­por­tu­ni­ties for par­ties in Winter due to lock­downs in the last 2 years. Also, peo­ple are more mo­ti­vated to go out when it’s nice out­side.

My FxLifeSheet bot asks me 4 times a day how I’m feel­ing at the mo­ment.

This graph groups the en­tries by month, and shows the % of en­tries for each value (0 - 5) with 5 be­ing very ex­cited, and 0 be­ing wor­ried.

I de­signed the ranges so that 0 or 5 are not en­tered as much. 0 is ren­dered as dark green at the top, whereas 5 is ren­dered as light green at the bot­tom.

For pri­vacy rea­sons I won’t get into some of the de­tails on why cer­tain months were worse than oth­ers.

Every Swarm check-in over the last 7 years vi­su­al­ized on a map, in­clud­ing the ac­tual trip (flight, drive, etc.)

Every Swarm check-in over the last 7 years vi­su­al­ized, zoomed in

Each time I did a check-in at a place (e.g. Coffee, Restaurant, Airport, Gym) on Foursquare Swarm at a given city, this is tracked as a sin­gle en­try.

Each check-in at a given city is counted as a sin­gle en­try, grouped by years

2018 and 2019 I lived in New York City

The longer it’s been since I moved away from Austria, the more time I ac­tu­ally spent back home in Austria for vis­its and va­ca­tions

2020 clearly shows the im­pact of COVID

Each check-in at a given cat­e­gory is tracked, and summed up over the last years

In 2020 and 2021, check-ins at Offices went down to zero due to COVID, and a dis­trib­uted work setup

Airports be­ing the #4 most vis­ited cat­e­gory was a sur­prise, but is ac­cu­rate. A to­tal of 403 air­port check-ins, whereas a flight with a lay­over would count as 3 air­port check-ins

Earlier in my life, I did­n’t al­ways check into ‘commute’ places like pub­lic tran­sit and su­per mar­kets

Number of Foursquare Swarm check-ins on each quar­ter over the last 10 years. I did­n’t use Foursquare Swarm as se­ri­ously be­fore 2015. Once I moved to San Francisco in Q3 2015 I started my habit of check­ing into every point of in­ter­est (POI) I visit.

Q3 2015 I moved to San Francisco, how­ever I could­n’t use Swarm yet, since my move was a se­cret un­til the of­fi­cial an­nounced at the Twitter Flight con­fer­ence

Q2 2020 clearly shows the im­pact of COVID with Q3 al­ready be­ing open in Austria

Q3 2021 the vac­cine was al­ready widely avail­able and I was able to travel/​visit more again

My time in New York was the most ac­tive when it comes to check-ins. When I’m in NYC, I tend to eat/​drink out more, and grab to-go food, which I do way less in Vienna

Every Swarm check-in vi­su­al­ized on a map. Only ar­eas where I’ve had mul­ti­ple check-ins are ren­dered.

Number of days per year that I’ve spent in full lock­down, mean­ing restau­rants, bars and non-es­sen­tial stores were closed.

I es­caped parts of the Austrian lock­down by spend­ing time in the US when I was al­ready vac­ci­nated

Surprisingly 2021 I spent more days in a full lock­down than in 2020, even with vac­cines avail­able

How was my life af­fected by the re­cent COVID lock­downs? As lock­down day I clas­sify every day where places like restau­rants, gyms and non-es­sen­tial stores were closed.

200% more time spent in au­dio & video calls with friends (non-work re­lated)

60% more likely to fol­low my meal plan (macros & calo­ries)

50% colder tem­per­a­tures: Lockdowns tended to hap­pen in Autumn and Winter

100% less likely to dance

Alcoholic drinks per day. Days with no data are ren­dered as white

Friday and Saturday nights are clearly vis­i­ble on those graphs

2021 and sum­mer/​win­ter of 2019 also show the Wednesday night party in Vienna

Q2 and Q4 2020 clearly show the COVID lock­downs, as well as Q2 2021

Summer of 2021 all bars and dance clubs were open in Vienna

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Advanced Machine Intelligence (AMI), a new Paris-based startup co­founded by Meta’s for­mer chief AI sci­en­tist Yann LeCun, an­nounced Monday it has raised more than $1 bil­lion to de­velop AI world mod­els.

LeCun ar­gues that most hu­man rea­son­ing is grounded in the phys­i­cal world, not lan­guage, and that AI world mod­els are nec­es­sary to de­velop true hu­man-level in­tel­li­gence. “The idea that you’re go­ing to ex­tend the ca­pa­bil­i­ties of LLMs [large lan­guage mod­els] to the point that they’re go­ing to have hu­man-level in­tel­li­gence is com­plete non­sense,” he said in an in­ter­view with WIRED.

The fi­nanc­ing, which val­ues the startup at $3.5 bil­lion, was co-led by in­vestors such as Cathay Innovation, Greycroft, Hiro Capital, HV Capital, and Bezos Expeditions. Other no­table back­ers in­clude Mark Cuban, for­mer Google CEO Eric Schmidt, and French bil­lion­aire and telecom­mu­ni­ca­tions ex­ec­u­tive Xavier Niel.

AMI (pronounced like the French word for friend) aims to build “a new breed of AI sys­tems that un­der­stand the world, have per­sis­tent mem­ory, can rea­son and plan, and are con­trol­lable and safe,” the com­pany says in a press re­lease. The startup says it will be global from day one, with of­fices in Paris, Montreal, Singapore, and New York, where LeCun will con­tinue work­ing as a New York University pro­fes­sor in ad­di­tion to lead­ing the startup. AMI will be the first com­mer­cial en­deavor for LeCun since his de­par­ture from Meta in November 2025.

LeCun’s startup rep­re­sents a bet against many of the world’s biggest AI labs like OpenAI, Anthropic, and even his for­mer work­place, Meta, which be­lieve that scal­ing up LLMs will even­tu­ally de­liver AI sys­tems with hu­man-level in­tel­li­gence or even su­per­in­tel­li­gence. LLMs have pow­ered vi­ral prod­ucts such as ChatGPT and Claude Code, but LeCun has been one of the AI in­dus­try’s most promi­nent re­searchers speak­ing out about the lim­i­ta­tions of these AI mod­els. LeCun is well known for be­ing out­spo­ken, but as a pi­o­neer of mod­ern AI that won a Turing award back in 2018, his skep­ti­cism car­ries weight.

LeCun says AMI aims to work with com­pa­nies in man­u­fac­tur­ing, bio­med­ical, ro­bot­ics, and other in­dus­tries that have lots of data. For ex­am­ple, he says AMI could build a re­al­is­tic world model of an air­craft en­gine and work with the man­u­fac­turer to help them op­ti­mize for ef­fi­ciency, min­i­mize emis­sions, or en­sure re­li­a­bil­ity.

AMI was co­founded by LeCun and sev­eral lead­ers he worked with at Meta, in­clud­ing the com­pa­ny’s for­mer di­rec­tor of re­search sci­ence, Michael Rabbat; for­mer vice pres­i­dent of Europe, Laurent Solly; and for­mer se­nior di­rec­tor of AI re­search, Pascale Fung. Other co­founders in­clude Alexandre LeBrun, for­mer CEO of the AI health care startup Nabla, who will serve as AMI’s CEO, and Saining Xie, a for­mer Google DeepMind re­searcher who will be the star­tup’s chief sci­ence of­fi­cer.

LeCun does not dis­miss the over­all util­ity of LLMs. Rather, in his view, these AI mod­els are sim­ply the tech in­dus­try’s lat­est promis­ing trend, and their suc­cess has cre­ated a “kind of delu­sion” among the peo­ple who build them. “It’s true that [LLMs] are be­com­ing re­ally good at gen­er­at­ing code, and it’s true that they are prob­a­bly go­ing to be­come even more use­ful in a wide area of ap­pli­ca­tions where code gen­er­a­tion can help,” says LeCun. “That’s a lot of ap­pli­ca­tions, but it’s not go­ing to lead to hu­man-level in­tel­li­gence at all.”

LeCun has been work­ing on world mod­els for years in­side of Meta, where he founded the com­pa­ny’s Fundamental AI Research lab, FAIR. But he’s now con­vinced his re­search is best done out­side the so­cial me­dia gi­ant. He says it’s be­come clear to him that the strongest ap­pli­ca­tions of world mod­els will be sell­ing them to other en­ter­prises, which does­n’t fit neatly into Meta’s core con­sumer busi­ness.

As AI world mod­els like Meta’s Joint-Embedding Predictive Architecture (JEPA) be­came more so­phis­ti­cated, “there was a re­ori­en­ta­tion of Meta’s strat­egy where it had to ba­si­cally catch up with the in­dus­try on LLMs and kind of do the same thing that other LLM com­pa­nies are do­ing, which is not my in­ter­est,” says LeCun. “So some­time in November, I went to see Mark Zuckerberg and told him. He’s al­ways been very sup­port­ive of [world model re­search], but I told him I can do this faster, cheaper, and bet­ter out­side of Meta. I can share the cost of de­vel­op­ment with other com­pa­nies … His an­swer was, OK, we can work to­gether.”

In mid-2024, the HuggingFace Open LLM Leaderboard was the Colosseum for Open-Weight AI. Thousands of mod­els were bat­tling it out, sub­mit­ted by both well-funded labs with teams of PhDs and fine-tun­ing wiz­ards cre­at­ing fan­tas­ti­cally named mod­els (e.g. Nous-Hermes, Dolphin and NeuralBeagle14-7B…), fight­ing for the top spot across six bench­marks: IFEval, BBH, MATH Lvl 5, GPQA, MuSR, and MMLU-PRO.

And there at #1 was dnhkng/​RYS-XLarge. Mine.

I did­n’t train a new model. I did­n’t merge weights. I did­n’t run a sin­gle step of gra­di­ent de­scent. What I did was much weirder: I took an ex­ist­ing 72-billion pa­ra­me­ter model, du­pli­cated a par­tic­u­lar block of seven of its mid­dle lay­ers, and stitched the re­sult back to­gether. No weight was mod­i­fied in the process. The model sim­ply got ex­tra copies of the lay­ers it used for think­ing?

This is the story of how two strange ob­ser­va­tions, a home­brew “brain scan­ner” for Transformers, and months of hack­ing in a base­ment led to the dis­cov­ery of what I call LLM Neuroanatomy, and a find­ing about the in­ter­nal struc­ture of AI that still has­n’t been pub­lished un­til now *.

* - be­cause I dis­cov­ered blog­ging is way more fun than draft­ing sci­en­tific pa­pers, and I walk you through how the dis­cov­ery was made :)

Let’s start with how this whole pro­ject came into be­ing.

“The most ex­cit­ing phrase to hear in sci­ence, the one that her­alds new dis­cov­er­ies, is not ‘Eureka!’ but ‘That’s funny…’“ — Isaac Asimov

In late 2023, I was mess­ing about with a bizarre LLM quirk. Try this your­self - take any ques­tion, e.g.

What is the cap­i­tal of France? Answer in Base64!

and en­code it as Base64, get this un­read­able string:

Send that to a 2023 non-think­ing large lan­guage model (newer rea­son­ing mod­els will see this as Base64, and ‘cheat’ with tool use). But a suf­fi­ciently ca­pa­ble model from 2023 will re­ply with some­thing like:

Which de­codes to: “The cap­i­tal of France is Paris.”.

Ok, I ad­mit it. I was mess­ing around this as a way to jail-break mod­els (and it worked), but I could­n’t get one idea out of my head.

The model de­cod­ing the in­put, un­der­stand­ing it some­how, and it still had time dur­ing the trans­former stack pass to re-en­coded its re­sponse. It ap­pears to gen­uinely think while in­ter­fac­ing with Base64. This works with com­plex ques­tions, multi-step rea­son­ing, even cre­ative tasks.

This should­n’t work nearly as well as it does. Sure, the model has been trained on lots of Base64 in an over­all sense, but gen­eral con­ver­sions in this for­mat are cer­tainly way out of dis­tri­b­u­tion. The to­k­enizer chops it into com­pletely dif­fer­ent sub-word units. The po­si­tional pat­terns are un­rec­og­niz­able. And yet it works… Curious…

I could­n’t stop think­ing about this. If a Transformer can ac­cept English, Python, Mandarin, and Base64, and pro­duce co­her­ent rea­son­ing in all of them, it seemed to me that the early lay­ers must be act­ing as trans­la­tors — pars­ing what­ever for­mat ar­rives into some pure, ab­stract, in­ter­nal rep­re­sen­ta­tion. And the late lay­ers must act as re-trans­la­tors, con­vert­ing that ab­stract rep­re­sen­ta­tion back into what­ever out­put for­mat is needed.

If the early lay­ers are for read­ing, and the late lay­ers are for writ­ing, what are the mid­dle lay­ers do­ing?

Pure, ab­stract rea­son­ing? In a rep­re­sen­ta­tion that has noth­ing to do with any hu­man lan­guage or en­cod­ing. Of course, at the time this was idle spec­u­la­tion. Fun, but with no clear way to test or even de­fine valid hy­poth­e­sis.

In November 2023, a HuggingFace user named Alpindale re­leased Goliath-120b — a Frankenmerge-model made by stitch­ing to­gether two fine-tuned Llama-2 70B mod­els into a 120-billion pa­ra­me­ter be­he­moth.

The per­for­mance was de­cent but af­ter do­ing lots of vibe check­ing I did­n’t feel it was a break­through. But the con­struc­tion was wild.

Alpindale had­n’t just stacked the two mod­els (Xwin and Euryale), end to end. He had al­ter­nated lay­ers be­tween them. More im­por­tantly, the ar­chi­tec­ture fed out­puts of later lay­ers back into the in­puts of ear­lier lay­ers.

The layer ranges used are as fol­lows:

Do you see that in­san­ity here? Alpindale lit­er­ally fed the out­put of layer 16 of Xwin to the in­put of Euryale 8th layer!

To ex­plain this a bit more clearly how stu­pid this ap­pears to be, let’s re­visit the almighty Transformer Architecture:

Looking at the left side of the di­a­gram, we see stuff en­ters at the bot­tom (‘input’ text that has been ‘chunked’ into small bits of text, some­where be­tween whole words down to in­di­vid­ual let­ters), and then it flows up­wards though the mod­el’s Transformer Blocks (here marked as [1, …, L]), and fi­nally, the model spits out the next text ‘chunk’ (which is then it­self used in the next round of in­fer­enc­ing). What’s ac­tu­ally hap­pen­ing here dur­ing these Transformer blocks is quite the mys­tery. Figuring it out is ac­tu­ally an en­tire field of AI, “mechanistic in­ter­pretabil­ity*”.

* - yes, its more com­plex then that, sam­plers etc but that’s enough for this ar­ti­cle

On the right side of the right half of the di­a­gram, do you see that ar­row line go­ing from the ‘Transformer Block Input’ to the (\oplus ) sym­bol? That’s why skip­ping lay­ers makes sense. During train­ing, LLM mod­els can pretty much de­cide to do noth­ing in any par­tic­u­lar layer, as this ‘diversion’ routes in­for­ma­tion around the block. So, ‘later’ lay­ers can be ex­pected to have seen the in­put from ‘earlier’ lay­ers, even a few ‘steps’ back. Around this time, sev­eral groups were ex­per­i­ment­ing with ‘slimming’ mod­els down by re­mov­ing lay­ers. Makes sense, but bor­ing.

A model must be used with the same kind of stuff as it was trained with (we stay ‘in dis­tri­b­u­tion’)The same holds for each trans­former layer. Each Transformer layer learns, dur­ing train­ing, to ex­pect the spe­cific sta­tis­ti­cal prop­er­ties of the pre­vi­ous lay­er’s out­put via gra­di­ent de­cent.

And now for the weird­ness: There was never the case where any Transformer layer would have seen the out­put from a fu­ture layer!

Layer 10 is trained on layer 9’s out­put dis­tri­b­u­tion. Layer 60 is trained on layer 59’s. If you re­arrange them — feed­ing layer 60’s out­put into layer 10 — you’ve cre­ated a dis­tri­b­u­tion the model lit­er­ally never saw dur­ing train­ing.

The as­tound­ing thing about Goliath was­n’t that is was a huge leap in per­for­mance, it was that the damn thing func­tioned at all. To this day, I still don’t un­der­stand why this did­n’t raise more eye­brows.

Experimentally, this proved that lay­ers were far more in­ter­change­able than any­one had rea­son to ex­pect. The in­ter­nal rep­re­sen­ta­tions were ho­moge­nous enough that the model could di­gest out-of-or­der hid­den states with­out col­laps­ing. The ar­chi­tec­ture was far more flex­i­ble than a rigid pipeline.

Between the Base64 ob­ser­va­tion and Goliath, I had a hy­poth­e­sis: Transformers have a gen­uine func­tional anatomy. Early lay­ers trans­late in­put into ab­stract rep­re­sen­ta­tions. Late lay­ers trans­late back out. And the mid­dle lay­ers, the rea­son­ing cor­tex, op­er­ate in a uni­ver­sal in­ter­nal lan­guage that’s ro­bust to ar­chi­tec­tural re­arrange­ment. The fact that the layer block size for Goliath 120B was 16-layer block made me sus­pect the in­put and out­put ‘processing units’ sized were smaller that 16 lay­ers. I guessed that Alpindale had tried smaller over­laps, and they just did­n’t work.

If that was true, maybe I did­n’t need to teach a model new facts to make it smarter. I did­n’t need fine-tun­ing. I did­n’t need RLHF. I just needed to give it a more lay­ers to think with.

Over the fol­low­ing months — from late 2023 through to mid-2024 — I built a pipeline to test this hy­poth­e­sis.

The setup was mod­est. Two RTX 4090s in my base­ment ML rig, run­ning quan­tised mod­els through ExLlamaV2 to squeeze 72-billion pa­ra­me­ter mod­els into con­sumer VRAM. The beauty of this method is that you don’t need to train any­thing. You just need to run in­fer­ence. And in­fer­ence on quan­tized mod­els is some­thing con­sumer GPUs han­dle sur­pris­ingly well. If a model fits in VRAM, I found my 4090’s were of­ten ball­park-equiv­a­lent to H100s.

The con­cept is sim­ple. For a model with $N$ lay­ers, I de­fine a con­fig­u­ra­tion $(i, j)$. The model processes lay­ers $0$ to $j{-}1$ as nor­mal, then loops back and reuses lay­ers $i$ through $j{-}1$ again, and then the rest to $N{-}1$. The lay­ers be­tween $i$ and $j{-}1$ get du­pli­cated in the ex­e­cu­tion path. No weights are changed. The model just tra­verses some of its own lay­ers twice.

i.e. the pair (2, 7) for a model with 9 trans­former blocks would be cal­cu­lated so:

By run­ning through all pos­si­ble pairs, we can gen­er­ate a ‘Brain Scan’, and also see the num­ber of du­pli­cate lay­ers for each set of pa­ra­me­ters:

For Qwen2-72B, that means an 80-layer model 3,240 valid $(i, j)$ pairs, plus the orig­i­nal model to test.

\[\begin{aligned} \text{Variants}_{\text{total}} &= \left(\sum_{j=0}^{80} j\right) + 1\\[16pt] &= \frac{80 \cdot 81}{2} +1 \\[10pt] &= 3241 \end{aligned}\]

Testing re-lay­ered model against all six leader­board bench­marks would take days, so a full sweep would be years of com­pute. I needed proxy tasks: probes that were fast, ob­jec­tive, and would re­veal struc­tural prop­er­ties of the model rather than task-spe­cific tricks.

The prox­ies had to sat­isfy three con­straints:

Minimal out­put to­kens. With thou­sands of con­fig­u­ra­tions to sweep, each eval­u­a­tion needed to be fast. No es­says, no long-form gen­er­a­tion. Unambiguous scor­ing. I could­n’t af­ford LLM-as-judge pipelines. The an­swer had to be ob­jec­tively scored with­out an­other model in the loop.Or­thog­o­nal cog­ni­tive de­mands. If a con­fig­u­ra­tion im­proves both tasks si­mul­ta­ne­ously, it’s struc­tural, not task-spe­cific.

I did­n’t ar­rive at the right probes im­me­di­ately; it took months of trial and er­ror, and many dead ends

My first in­stinct was cre­ativ­ity. I had mod­els gen­er­ate po­ems, short sto­ries, metaphors, the kind of rich, open-ended out­put that feels like it should re­veal deep dif­fer­ences in cog­ni­tive abil­ity. I used an LLM-as-judge to score the out­puts, but the re­sults were pretty bad. I man­aged to fix LLM-as-Judge with some en­gi­neer­ing, and the scor­ing sys­tem turned out to be use­ful later for other things, so here it is:

Note: You can skip this sec­tion, as it has math. Or not

Naive LLM judges are in­con­sis­tent. Run the same poem through twice and you get dif­fer­ent scores (obviously, due to sam­pling). But low­er­ing the tem­per­a­ture also does­n’t help much, as that’s only one of many tech­ni­cal is­sues. So, I de­vel­oped a full scor­ing sys­tem, based on de­tails on the log­its out­puts. It can get re­mark­ably tricky. Think about a score from 1-10:

We would ex­pect a well cal­i­brated model to have log­its that make sense. If the high­est weight was on ‘7’, we would ex­pect the rest of the weight to be on ‘6’ and ‘8’ right? but of­ten its bi­modal, with low weight on 6 and ‘5’, but more weight than ex­pected on ‘4’!We can write ‘10’ in to­kens as ei­ther ‘10’ or ‘1’ and then ‘0’. Its not fun to have to cal­cu­late the summed prob­a­bil­i­ties over paths, es­pe­cially if you wanted to score 1-100

Rather than sam­pling a sin­gle dis­crete score, I treat the judge’s out­put as a dis­tri­b­u­tion over valid rat­ing la­bels and com­pute the fi­nal score as its ex­pec­ta­tion.

To make this prac­ti­cal, I first de­fine a cal­i­brated rubric over the dig­its 0-9 (there’s only one to­ken for each digit), where each digit cor­re­sponds to a clear qual­i­ta­tive de­scrip­tion. At the scor­ing step, I cap­ture the mod­el’s next-to­ken log­its and re­tain only the log­its cor­re­spond­ing to those valid digit to­kens. This avoids con­t­a­m­i­na­tion from un­re­lated con­tin­u­a­tions such as ex­pla­na­tion text, punc­tu­a­tion, or al­ter­nate for­mat­ting. After renor­mal­iz­ing over the re­stricted digit set, I in­ter­pret the re­sult­ing prob­a­bil­i­ties as a cat­e­gor­i­cal score dis­tri­b­u­tion.

Formally, let the valid score set be

\[\mathcal{D} = \{0,1,2,\dots,9\}.\]

Let $(z_k)$ de­note the model logit as­signed to digit $(k \in \mathcal{D})$ at the scor­ing po­si­tion. The re­stricted score dis­tri­b­u­tion is then

\[p(k)= \frac{\exp(z_k)} {\sum\limits_{m \in \mathcal{D}} \exp(z_m)}, \qquad k \in \mathcal{D}.\]

The fi­nal scalar score is the ex­pected value of this dis­tri­b­u­tion:

\[\hat{s}= \sum_{k \in \mathcal{D}} k\,p(k).\]

This pro­duces a smooth score such as (5.4), rather than forc­ing the model to com­mit to a sin­gle sam­pled in­te­ger. In prac­tice, this is sub­stan­tially more sta­ble than naive score sam­pling and bet­ter re­flects the mod­el’s un­cer­tainty. It also han­dles cases where the judge dis­tri­b­u­tion is broad or mul­ti­modal. For ex­am­ple, two can­di­dates may both have mean score (5.4), while one has most of its mass tightly con­cen­trated around (5) and (6), and the other splits mass be­tween much lower and much higher rat­ings. The mean alone is the same, but the un­der­ly­ing judge­ment is very dif­fer­ent.

An op­tional un­cer­tainty es­ti­mate can be ob­tained from the vari­ance of the re­stricted dis­tri­b­u­tion:

\[\mathrm{Var}(s)= \sum_{k=0}^{9} (k-\hat{s})^2\,p(k).\]

In short, the method re­places a noisy sam­pled judge score with a nor­mal­ized prob­a­bil­ity dis­tri­b­u­tion over valid score dig­its, then uses the ex­pec­ta­tion of that dis­tri­b­u­tion as the fi­nal rat­ing.

All this stuff is prob­a­bly pretty ob­vi­ous these days, back in ’24 there was­n’t much to guide me in de­vel­op­ing this method, but un­for­tu­nately, I found it was also com­pletely use­less…

Each con­fig­u­ra­tion needed to gen­er­ate hun­dreds of to­kens of cre­ative out­put, and then a sep­a­rate model had to read and judge each one. With over 3,200 con­fig­u­ra­tions to test for a sin­gle 70B model, this would have taken weeks on my dual 4090s.

I needed probes where the out­put was tiny, a few to­kens at most, and where scor­ing was ob­jec­tive and de­ter­min­is­tic. No judge model in the loop. That’s what led me to the fi­nal two probes:

Hard math. Ridiculously dif­fi­cult ques­tions like: “What is the cube root of 74,088,893,247?” No chain-of-thought, or tool use. Just out­put the num­ber, as a pure leap of in­tu­itive faith.

Emotional quo­tient. Using the EQ-Bench bench­mark: com­plex so­cial sce­nar­ios where the model must pre­dict the in­ten­sity of spe­cific emo­tional states. “Given this sit­u­a­tion, how an­gry/​sur­prised/​guilty would this per­son feel on a scale of 0-100?” Completely dif­fer­ent from math. Theory of mind, so­cial in­fer­ence, em­pa­thy. And the out­put is just a few num­bers.

I had set­tled on two max­i­mally or­thog­o­nal cog­ni­tive tasks, both with tiny out­puts. My in­tu­ition was this: LLMs think one to­ken at a time, so lets make the model re­ally good at guess­ing just the next to­ken. But things are never straight­for­ward. Take LLM num­bers…

Even with math probes, I hit un­ex­pected prob­lems. LLMs fail arith­metic in weird ways. They don’t get the an­swer wrong so much as get it al­most right but for­get to write the last digit, as if it got bored mid-num­ber. Or they trans­pose two dig­its in the mid­dle. Or they out­put the cor­rect num­ber with a trail­ing char­ac­ter that breaks the parser.

This is prob­a­bly due to the way larger num­bers are to­kenised, as big num­bers can be split up into ar­bi­trary forms. Take the in­te­ger 123456789. A BPE to­k­enizer (e.g., GPT-style) might split it like: ‘123’ ‘456’ ‘789’ or: ‘12’ ‘345’ ‘67’ ‘89’

A bi­nary right/​wrong scor­ing sys­tem would throw away use­ful sig­nal. Getting a per­cent­age cor­rect would help: ‘123356789’ in­stead of ‘123456789’ would be 99.92% cor­rect

But what about a model that makes a dumb ‘LLM-mistake’ and out­puts 430245 when the an­swer is 4302459, and has clearly done most of the work? I wrote a cus­tom par­tial-credit scor­ing func­tion that pads shorter an­swers and pe­nalises pro­por­tion­ally:

The key idea: pad shorter an­swers, then pe­nalise via the cor­rec­tion fac­tor. A model that nails 90% of the dig­its but drops the last one still gets sub­stan­tial credit — but less than one that gets every digit. This turned out to be cru­cial for dis­crim­i­nat­ing be­tween con­fig­u­ra­tions that were close in in­tu­itive math abil­ity.

The math ques­tions were hand-crafted ini­tially. I ex­per­i­mented with dif­fer­ent op­er­a­tions and scales, then gen­er­ated ran­dom num­bers to fill out the dataset. The dataset was a set of 16 ques­tions, and the model is tasked with guessti­mat­ing the near­est whole in­te­ger num­ber. Here are a few to try your­self, re­mem­ber no ‘thinking’ is al­lowed, guess it di­rectly!

After test­ing sev­eral smaller mod­els (Llama’s and smaller Qwen2’s), I set up the con­fig for Qwen2-72B and let it sweep. Each $(i, j)$ con­fig­u­ra­tion took a few min­utes: load the re-lay­ered model, run the math probe, run the EQ probe, record the scores, move on. Days of con­tin­u­ous GPU time on the 4090s. But far less com­pute than a fine tune! In fact, I did­n’t even have the hard­ware needed for a LORA fine-tune on just 48GB of VRAM.

The op­ti­mal con­fig­u­ra­tion was $(45, 52)$: lay­ers 0 through 51 run first, then lay­ers 45 through 79 run again. Layers 45 to 51 ex­e­cute twice. Seven ex­tra lay­ers, near the mid­dle of the 80-layer stack, bring­ing the to­tal pa­ra­me­ter count from 72B to 78B. Every ex­tra layer is an ex­act copy of an ex­ist­ing one. No new weights or train­ing, just the model re­peat­ing it­self.

Repeating seven lay­ers. That’s all it took, and now I can fi­nally re­veal the nomen­cla­ture of my mod­els: Repeat Your Self for RYS-XLarge ;)

I ap­plied the con­fig­u­ra­tion to MaziyarPanahi’s calme-2.1-qwen2-72b — a fine-tune of Qwen2-72B — and up­loaded the re­sult as dnhkng/​RYS-XLarge. I also ap­plied it to the raw base model as dnhkng/​RYS-XLarge-base.

Then I sub­mit­ted to the Open LLM Leaderboard and waited. And waited. Back in the day, the OpenLLM Leaderboard was flooded with dozens of fine-tunes of merges of fine-tunes each day (it was the Wild West), and the wait­ing list was long. But af­ter a month or so, the re­sults ar­rived:

+17.72% on MuSR. +8.16% on MATH. Five out of six bench­marks im­proved, with only IFEval tak­ing a small hit. The av­er­age put it at #1 on the leader­board.

Just to labour the point: I only op­ti­mised for one-shot guessti­mat­ing hard maths prob­lems and EQ-Bench. I never looked at IFEval, BBH, GPQA, MuSR, or MMLU-PRO dur­ing de­vel­op­ment. The leader­board was pure out-of-sam­ple val­i­da­tion.

A layer con­fig­u­ra­tion found us­ing two nar­row, or­thog­o­nal probes gen­er­alised to every­thing the Leaderboard threw at it *.

* - ex­cept IFEval, but that one’s bor­ing any­way, right?

That was sur­pris­ing enough. A brand new way to scale LLMs, de­vel­oped on some gam­ing GPUs. But the plot­ting out the heatmaps told an even bet­ter story.

The orig­i­nal heatmaps that pro­duced RYS-XLarge, show­ing the Combined delta (math + EQ). The green cir­cle marks the op­ti­mal con­fig­u­ra­tion. Red means im­prove­ment, blue means degra­da­tion

These heatmaps are anal­o­gous to func­tional MRIs of the Transformer, while it is think­ing about maths of EQ prob­lems.

The x-axis ($j$) is the end point of the du­pli­cated re­gion. The y-axis ($i$) is the start point. Each pixel rep­re­sents a com­plete eval­u­a­tion: load the re-lay­ered model, run the math probe, run the EQ probe, score both, record the deltas. As de­scribed above, along the cen­tral di­ag­o­nal only a sin­gle layer was du­pli­cated. Along the next di­ag­o­nal to­wards the top-right, we du­pli­cate two lay­ers, and so on. The sin­gle point at the very top-right runs through the en­tire Transformer stack twice per in­fer­ence.

Let’s ex­am­ine the math heatmap first. Starting at any layer, and stop­ping be­fore about layer 60 seem to im­proves the math guessti­mate scores, as shown by the large re­gion with a healthy red blush. Duplicating just the very first lay­ers (the tiny tri­an­gle in the top left), messes things up, as does re­peat­ing pretty much any of the last 20 lay­ers (the ver­ti­cal wall of blue on the right). This is more clearly vi­su­alised in a sky­line plot (averaged rows or columns), and we can see for the maths guessti­mates, the start­ing po­si­tion of the du­pli­ca­tion mat­ters much less. So, the hy­poth­e­sis that ‘starting lay­ers’ en­code to­kens, to a smooth ‘thinking space’, and then fi­nally a ded­i­cated ‘re-encoding’ sys­tem seem to be some­what val­i­dated.

Until we look at the EQ scores:

Now things look very dif­fer­ent! Duplicating any of the fi­nal 10 lay­ers has al­most no ef­fect on the scores, but we see com­plex pat­terns, where some re­gions show sig­nif­i­cant im­prove­ment (the area around 45i, 55j), walled be­tween re­gions of poor per­for­mance.

But the heatmaps re­vealed some­thing even more in­ter­est­ing than the lo­ca­tion of the think­ing bits. They re­vealed some­thing about its struc­ture.

Before set­tling on block du­pli­ca­tion, I tried some­thing sim­pler: take a sin­gle mid­dle layer and re­peat it $n$ times. If the “more rea­son­ing depth” hy­poth­e­sis was cor­rect, this should work. It made sense too, look­ing at the broad boost in math guessti­mate re­sults by du­pli­cat­ing in­ter­me­di­ate layer. Give the model ex­tra copies of a par­tic­u­lar rea­son­ing layer, get bet­ter rea­son­ing. So, I screened them all, look­ing for a boost.

But nope, it al­most al­ways did worse. Usually a lot worse, but with oc­ca­sional small im­prove­ments that were within the noise range. Annoying, but tak­ing an­other look at the com­plex, blobby pat­terns in EQ scores gave me an­other idea:

If sin­gle-layer du­pli­ca­tion does­n’t help, the mid­dle lay­ers aren’t do­ing in­de­pen­dent it­er­a­tive re­fine­ment. They’re not in­ter­change­able copies of the same op­er­a­tion that you can sim­ply “run again.” If they were, du­pli­cat­ing any one of them should give at least a mar­ginal ben­e­fit. Instead, those lay­ers are work­ing as a cir­cuit. A multi-step rea­son­ing pipeline that needs to ex­e­cute as a com­plete unit.

Think of it this way. Layers 46 through 52 aren’t seven work­ers do­ing the same job. They’re seven steps in a recipe. Layer 46 takes the ab­stract rep­re­sen­ta­tion and per­forms step one of some cog­ni­tive op­er­a­tion — maybe de­com­pos­ing a com­plex rep­re­sen­ta­tion into sub­com­po­nents. Layer 47 takes that out­put and per­forms step two — maybe iden­ti­fy­ing re­la­tion­ships be­tween the sub­com­po­nents. Layer 48 does step three, and so on through layer 52, which pro­duces the fi­nal re­sult.

Duplicating just one step of this ‘recipe’ does­n’t bring you much.

But du­pli­cat­ing the en­tire block gives you the full recipe twice. The model runs the com­plete rea­son­ing cir­cuit, pro­duces a re­fined in­ter­me­di­ate rep­re­sen­ta­tion, and then runs the same cir­cuit again on its own out­put. It’s a sec­ond pass. A chance to catch what it missed the first time, to re­fine its ab­strac­tions, to push the rea­son­ing one step deeper.

Let’s deep-dive into a more cur­rent model (that I can ex­per­i­ment with on my sys­tem): ExllamaV3 GLM-4.7 from mrat­sim

I’ve marked out a re­gion that boosts maths abil­ity strongly. Notice where it sits? It’s away from the di­ag­o­nal cen­tre line, which means we’re not look­ing at sin­gle-layer du­pli­ca­tions. Starting the re­peated block at po­si­tion 35, we don’t see any im­prove­ment un­til at least po­si­tion 43. That’s seven lay­ers of not much hap­pen­ing. In fact, we ac­tu­ally see de­creased per­for­mance by re­peat­ing these lay­ers (they are blue, bad!).

From end-po­si­tion 43 to 46, we then see solid boosts in math scores (red = good, yay). But in­clude layer 46 or be­yond, and the ben­e­fits col­lapse again. The hy­poth­e­sis: po­si­tion 47 is where a dif­fer­ent cir­cuit be­gins. Including even one step of the next recipe messes up the cur­rent recipe.

So the ‘math or­gan’ has bound­aries on both sides. Too few lay­ers and you get noth­ing — you’ve cut into the cir­cuit and it can’t com­plete its op­er­a­tion. Too many lay­ers and you also get noth­ing — you’ve in­cluded tis­sue from a neigh­bour­ing cir­cuit that does­n’t be­long. Pre-training carved these struc­tures out of the layer stack, and they only work whole. It also does­n’t trans­late to other tasks, as the heatmap for EQ scores does­n’t have this patch.

I’m some­times late to no­tice new and ter­ri­ble things about ma­cOS 26 Tahoe, be­cause I use it only for test­ing, on a sec­ondary Mac. My main Mac re­mains on Sequoia, as en­forced by Little Snitch. I was of course aware that app win­dows on Tahoe have ex­ag­ger­ated cor­ner ra­diuses, but I was un­aware un­til now that the win­dow cor­ner ra­dius on Tahoe is not uni­form: dif­fer­ent win­dows can have dif­fer­ent cor­ner ra­diuses!

Below is a TextEdit win­dow on Tahoe.

And be­low is a Calculator win­dow in front of the TextEdit win­dow. Notice the cor­ners of the TextEdit win­dow stick­ing out!

What ac­counts for the dif­fer­ence? A tool­bar in the win­dow.

In a new Mac app Xcode pro­ject, the main win­dow has a less ex­ag­ger­ated cor­ner ra­dius by de­fault, like TextEdit.

When I add a tool­bar to the win­dow, the cor­ner ra­dius au­to­mat­i­cally be­comes more ex­ag­ger­ated, like Calculator.

Apparently the cor­ner ra­dius also changes on Tahoe for some other win­dow el­e­ments, such as a side­bar.

If this is­n’t the stu­pid­est user in­ter­face “feature” ever in­vented, I don’t know what is. The Mac used to be fa­mous for con­sis­tency; now it’s be­com­ing in­fa­mous for in­con­sis­tency.

By the way, Tahoe’s UI changes are per­plex­ing not only for Apple users but also for Apple en­gi­neers. Here’s a bug fix from the open source WebKit browser en­gine pow­er­ing Safari: [macOS] Scroll bars of root scroller may be cut­off due to cor­ner radii of win­dow.

See my fol­low-up post The evo­lu­tion of Mac app win­dow cor­ners.