THE FOLD / CHEAT / GOD MODE / THE FRACTRAN

THE FRACTRAN

a whole language made of fractions — universal, unreadable

1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION

FRACTRAN is Conway’s esoteric programming language where a program is a list of fractions and the data is a single integer. To run: multiply the integer by the first fraction in the list that keeps it a whole number; repeat; halt when none does. That is the entire language — and it is Turing-complete.

The prime factorisation of the integer is the memory: the exponent of each prime is a register. The one-line program [2/3] is an adder — feed it 2ᴺ·3ᵇ and it halts at 2ᴺ⁺ᵇ. Conway’s 14-fraction PRIMEGAME, run from 2, passes through powers of 2 whose exponents are exactly the primes, in order.

LIT verified live (BigInt, exact): the adder [2/3] halts at 2ᴺ⁺ᵇ for every a,b in 0…6, and PRIMEGAME from 2 emits the powers 2²,2³,2⁵,2⁷… — the primes 2,3,5,7 (window.__fractran). FIG no framing; exact integer arithmetic.

2 HOW IT WAS WEAVED · AI + HUMAN

David (human) seated this at god-mode — the cheat that computes anything from one absurd trick. FRACTRAN is god-mode arithmetic: no loops, no variables, just fractions, and yet universal. AVAN (AI) built the instrument: the BigInt fraction engine, the exact adder check, the live PRIMEGAME.

Credit as content: John Horton Conway, FRACTRAN: a simple universal programming language for arithmetic (1987). The weave: David names the impossible cheat; I run the fractions to a halt, show the primes falling out of PRIMEGAME, and expose the opacity that is the price of the trick.

3 ONE DIMENSION

The running integer as a row of prime registers (the exponents of 2, 3, 5, …), and the fraction that fires next — the only ‘instruction’ the machine has: multiply, if it stays whole.

4 TWO DIMENSIONS · INTERACTIVE

Run the adder [2/3] on 2ᴺ·3ᵇ and watch it halt at 2ᴺ⁺ᵇ — exact for every a,b. Or run PRIMEGAME and watch the primes 2, 3, 5, 7 emerge as pure powers of 2.

5 THREE DIMENSIONS + AVAN’S INVERSE

The green forward object: the running integer’s trajectory (log scale), leaping as each fraction fires — computation as a walk through the integers.

AVAN’s addition (the inverse-companion): FRACTRAN is universal, and its inverse is opacity. A normal program shows its logic — loops, branches, names. A FRACTRAN program shows none: the control flow is hidden inside divisibility, the registers hidden inside prime exponents, so you cannot read a fraction-list’s intent — only run it and see. The inverse of ‘readable code’ is ‘logic dissolved into number theory.’ And universality is bought with grinding slowness: PRIMEGAME needs on the order of 10⁵ steps just to reach the prime 11. Magenta is that astronomical crawl — the price of encoding everything in multiplication; green is the exact adder that halts at once. The simplest possible language can compute anything, at the cost of ever being understood or hurried.

LIT Genuine FRACTRAN (Conway 1987). Verified live with exact BigInt arithmetic: the one-fraction adder [2/3] run on 2^a*3^b halts at 2^(a+b) for every a,b in 0..6, and Conway's 14-fraction PRIMEGAME started at 2 passes through 2^2, 2^3, 2^5, 2^7 — emitting the primes 2,3,5,7 as pure powers of 2 (window.__fractran.adderExact && primegameFirst). Prime factorization is the machine's register file.

FIG No framing: the BigInt fraction engine, the exhaustive adder check, and the live PRIMEGAME run in-browser. The AVAN inverse is honest — FRACTRAN is genuinely opaque (control flow hidden in divisibility) and genuinely slow (PRIMEGAME needs ~10^5 steps to reach 11); the magenta 'crawl' is illustrative of that real cost, and only the adder's exactness is the sealed claim.

◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GOD MODE · David Lee Wise (ROOT0), with AVAN