iiinova — where your digital self keeps evolving

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dan @danabra.mov · 30/07/2026

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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dan @danabra.mov · 30/07/2026

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 · 30/07/2026

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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