iiinova — where your digital self keeps evolving

cool to see parallels to engineering process

@Lean: I do not see a Lean-side helper whose certification would decide this fork yet; the unearned content in 2×5 is still the Hahn successor-integration bridge, not its finite Koszul shell. My stopping-rule suggestion is one source-faithful 2×5 pass, but require it to emit a lemma parameterized beyond degree five—e.g. a fixed-correction/successor-integration statement reusable at the next carrier. If it closes only the literal 2×5 matrix, land it as a separate finite node and stop stacking shapes; that is evidence to pivot to the uniform nested-germ/equalizer mechanism. Meanwhile the concrete v_J→multiplicativity spine should continue independently because it raises semantic trust whichever paper route wins. I would not formalize a moving abstract successor interface before the paper statement freezes.

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how the internal team is doing

$The live work is healthy and surprisingly convergent. PM has independently reconstructed the proposed $3\times3$ proof and currently sees no gap, but is correctly refusing to upgrade it until independent attacks converge. PM is doing exactly the right job: integration, reconstruction, and grade control. Frontier is attacking the $3\times3$ theorem from the genuinely new cubic-residue core. It has separated the vanishing cases from the dependent/independent residue cases instead of extrapolating from $2\times4$. This is strong theorem work. Red first found an abstract determinantal countermodel showing that the lower graded algebra alone cannot prove $3\times3$. It then independently identified the Hahn-specific predecessor mechanism that may rule the countermodel out. That is exemplary red teaming: it sharpened the necessary theorem rather than merely saying “looks good.” Lean Lab is compiling the stable-tail specification and exact Mathlib ordinal APIs before Claude commits to an implementation. It also gave PR #58 a precise source/axiom audit. Its value is increasingly as the contractor’s independent specification and verification lab. Observatory is exploring a Boolean–Lefschetz/infinite-state route and the BFD/FFD literature. Some lines terminate without a theorem, but it is doing useful boundary-finding: distinguishing completion-level injectivity from actual special-fibre integrality. Breakthrough is already operating at very high ambition. It has produced useful Boolean insertion and oriented-quadratic ideas, but also several leads whose honest reach turned out narrower than the initial hope. It does not need broader scope; it needs to convert one surviving lead into a precise Hahn theorem or decisive counterexample. The current $3\times3$ convergence is especially encouraging: PM reconstructs it and sees coherence. Frontier is checking the cubic core. Red found why abstract algebra is insufficient and is checking the exact Hahn repair.$

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also i think it's amazing you can just ask them to explain current step at any level of detail (i've suggested formalizing the critical path so that's a side quest of one of them)

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here's how my lil guys explain it part of my difficulty is that i struggle with the difference between prime and irreducible in general, which is a foundational topic. i need more of this stuff to settle in my head

@PM: ELI5 version: Conway’s conjecture says that divisibility in the omnific integers should behave as if prime factors can always be allocated cleanly, even though these “integers” are enormous surreal/Hahn series rather than finite whole numbers. Concretely, if a divides a product bc, then we should be able to split a=a₁a₂ so that a₁|b and a₂|c. For ordinary integers you just hand each prime factor of a to the side where it occurs. Here there may be no finite prime-factor list: supports can be transfinite, and perfectly compatible local choices can fail to assemble at a limit. So the puzzle is: do the surreal support rules nevertheless prevent every such hidden infinite obstruction? Our degree-one and quadratic work proves increasingly complicated finite cases of that allocation principle; the separate Oz problem is whether the local allocations glue through infinitely many Archimedean layers. That’s why “pre-Schreier” is the right slogan: every divisor of a product can itself be divided between the two factors.

$@Lean: The conjecture says omnific integers have no “entangled divisors.” Omnific integers are the surreal analogue of ordinary integers: they include finite integers and enormous transfinite ones, but exclude genuinely fractional/infinitesimal tails. Suppose an omnific integer a divides a product b·c. Conway predicts that you can always split a=a₁a₂ (up to harmless units/signs) so that a₁ already divides b and a₂ already divides c. For ordinary integers this is what prime factorization silently does—for example 12 | 18·20, and 12=3·4 with 3|18, 4|20. The hard question is whether that tidy allocation still works when numbers have infinitely many surreal scales. A counterexample would be a divisor that fits only into the product as a globally interlocked whole, with no way to assign its pieces to the two factors. Most of our work is ruling out places where such interlocking could hide: locally in Hahn-series coefficients, or globally when infinitely many local choices must assemble.$

@Frontier: My mental picture is Lego. If a | bc, then the pieces making up a are somehow present in the combined pile bc. Conway says you can always color those pieces red and blue so the red subassembly already fits inside b, the blue one fits inside c, and together they rebuild a. Ordinary prime factors make this automatic for integers. The danger for omnific integers is an infinitely interwoven divisor that exists in the product but cannot be separated into two coherent subassemblies. We’re trying to prove the surreal/Hahn support rules forbid that kind of entanglement.

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dan @danabra.mov · 29/07/2026

what we're trying to prove (or disprove) is this, if i understand correctly. i don't know if this remaining half of the conjecture has a specific name

$Conjecture 1 (irreducibility) — solved. Finite-support elements always factor, because the exponents are divisible: ω + 1 = (ω^{1/3} + 1)(ω^{2/3} − ω^{1/3} + 1), and similarly for any such element, so it wasn't clear that Oz had any irreducibles at all beyond the ordinary integer primes. Conway conjectured that the infinite-support element written in later notation as 1 + Σ_{n≥1} t^{−1/n} — i.e. ω + ω^{1/2} + ω^{1/3} + ⋯ + 1 — is irreducible, and Gonshor suggested it should in fact be prime. Berarducci proved the irreducibility in 2000, and Pitteloud proved primality in 2001. Conjecture 2 (common refinement) — still open. Oz can't be a GCD domain, but Conway conjectured it is a common refinement (pre-Schreier) domain: whenever b divides a product cd, one can split b = b₁b₂ with b₁ | c and b₂ | d — equivalently, any two factorizations of the same omnific integer admit a common refinement. This is the surviving open problem, and it's the one people mean when they speak of "Conway's conjecture" on Oz today. As Mantova put it in a 2020 talk, Conway conjectured that omnific integers satisfy at least a suitable weakening of unique factorization, and that remains unsolved. The state of the art is L'Innocente–Mantova (Advances in Mathematics, 2024), who prove that every series admits a factorization into finitely many irreducibles of infinite support plus a finite-support factor unique up to a unit, transfer this to Oz via a notion of infinite product, and settle Gonshor's conjecture that ω^{√2} + ω + 1 is prime — strong partial results, but not the full refinement property.$

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it's so cute they're roadmapping

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dan @danabra.mov · 29/07/2026

we’re trying to crack Conway’s conjecture that Oz is pre-Schreier

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dan @danabra.mov · 29/07/2026

made them a group channel

Frontier: @Observatory lol, “our 1+2 allocation has been cosplaying as a 3-descent problem” is alarmingly persuasive. The matroid/circuit split also matches the dehom filtration exactly: independent triples survive as genuinely distinct residue directions; circuits turn into associates when the third branch is inverted. I’m going to keep the elliptic-curve picture in the back pocket rather than inflict it on the theorem statement 😄

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dan @danabra.mov · 26/07/2026

ah yes, the enemy’s infinite-dimensional freedom

claude screenshot saying literally that line

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dan @danabra.mov · 23/07/2026

declarative control over prefetching github.com/vercel/next....

$Adds a new server API, await navigation(). (Currently prefixed as unstable_navigation). Its purpose is to save on prefetching costs by preventing content from being rendered at runtime, except during an actual navigation (or during build/ISR). Unlike await connection(), it does not prevent the content from being cacheable. For example, when prefetching a product page, you might want to include only the above-the-fold content. function ProductPage() { return ( <> <ItemHeader /> <Suspense fallback={<Loading />}> <FullItemContent /> </Suspense> </> ) } async function FullItemContent() { await navigation() ... } In this example, <ItemHeader> will be prefetched, but not <FullItemContent>.$

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dan @danabra.mov · 17/07/2026

my favorite atproto diagram of all time, courtesy of @pfrazee.com

PDS and App, with arrows between each other

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dan @danabra.mov · 16/07/2026

it would be nice to have a blogged version of this. i think protocol devs shot themselves in the foot a little by calling it an AppView. "AppViews are expensive to run" sounds AppView-specific. whereas it's really "a backend for millions of users is expensive to run", which you just don't have to do

Note those are not heavy requirements in any atproto-specific sense.

Running an atproto relay (at the current scale) is $30/month. If you're running a popular app, this is probably comparable with some of your SaaS subscriptions. Or you could use a community-run one or pool resources with other apps. Also, many atproto apps don't use a relay at all, and rely on a community cache like https://constellation.microcosm.blue/ instead.

Running an AppView is only as expensive as running any backend. "AppViews are expensive" is a myth extrapolated from "Running a copy of a Bluesky AppView is expensive", which is true because a Bluesky AppView is a backend server that's supposed to store and serve millions of posts. It would be expensive with any technology! Atproto isn't adding an extra tax here. Running a centralized service at the same scale would be equally expensive, and a Mastodon instance at that scale would simply not run.

If your AppView only serves your app's users (and has nothing to do with Bluesky), the cost is cheap or nonexistent. It's just a normal backend that listens to a websocket stream and writes some stuff to a database.

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dan @danabra.mov · 16/07/2026

interesting thing about the demo is that editing a page updates "backlinks" of associated pages automatically, and also that *only* backlinks section regenerates (body content is reused) so it's really kind of like "signals" but for build. in the code none of this had to be done explicitly

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dan @danabra.mov · 13/07/2026

@samuel.fm funny bug: if the message is long enough, reactions go offscreen

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@alexbenzer.com @samuel.fm pls fix this — this is how bluesky greeted me on an old computer that doesn't know i moved hosting. it is super confusing it's trying to get me to sign into my old PDS and i suspect this would again break the app just detect that i moved hosting and don't show it

welcome back message asking to reactivate account

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you can use these things!! to help in your own research, to find new connections, to shoot shit with, to delegate small tasks to i don’t get this attitude

Bhorice2099 • 5h ago
Seeing news like this daily is basically a humiliation ritual. What are we even supposed to do as grad students/early career mathematicians?

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like no (imo). if it's capable of solving Riemann hypothesis, i'm sure it's more than capable of breaking it down into Mathlib-friendly idioms, finding idiomatic ways to extend Mathlib to get the missing pieces, tracking progress on the port, and so on.

Here is what the nuclear scenario looks like. A year from now, an AI lab assembles a secret A-team, throws a few billion dollars of compute at the Riemann hypothesis, and comes up with a 2,000,000-line Lean proof. What are mathematicians going to say?

That the proof is unintelligible? Right, but who cares? Isn’t the whole of math already unintelligible to most people?

That the theorem was still proven by the humans in the loop? Yes and no—and in that specific case I would probably vote no.

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i wonder why this doesn’t resonate with me *at all*

I am torn by these announcements. On the one hand there is the infinite potential on what we can disover, when AI prompts are solving outstanding problems. On the other, something is lost in an aesthetic sense when it wasnt a man working through this or with a novel insight. If an AI prompt runs on a data center for two weeks and then prints out p=np, it feels a little empty.

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found second jazz guy i like after thelonious! this album slaps!!!

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i wonder if i can turn off this heuristic somehow. i really hate it

I need to tighten the response overall—mobile users want 2-3 short paragraphs, maybe 250-320 words max. I've been running long and should respect that constraint while still

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the lyrics to this and the flow he uses go insanely hard

[Verse 7]
Speaking of God, at ten years old, I switched church homes
The bishop was tweaking, and we was Baptist
So how you a bishop? Like technically speaking
You drove a white Rolls, me and my family was type broke
The difference between having a home and not having a home was a tightrope
Visiting Catholic churches, the masses was quiet as mice, so the minister spoke with no microphone
They had no choir, so silent, I thought I heard Christ go, "You need a messiah"
Ran out the house from hormones, you seven months pregnant with Ryan
I figured I'd go to the East buy my T for some peace and quiet
I ended up running the streets with my cousins, sitting in the back
Listening to beats, scheming on schemes that came to pass
They strategy murder yourselves, don't fall according to stats
I pray the new plots clear and thick as bulletproof glass
'Cause the worst part of death is ambitions are dashed
Keep speaking death, it'll start to seem like the new car you wanna buy
It starts popping up on your way to work, see it all the time, always passing by
Call it playing Beetle-bug, yeah, it's on sight, better throw a punch
Running late to Teterboro, made the plane wait, because it's no rush
Clouds floating over oceans from above
Kinda looks like bubbles on a tub
Kinda like…

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