Knowledge Is Compute: Recursive Knowledge Compilation in an Exact Discovery System

Abstract

Search is the default way to buy capability: a deeper search, a larger budget, more samples. We show a second axis that is cheaper and compounds. A solver that certifies its own results exactly can compile them back into the machine that searches, as new instructions and as a learned search order, and the compiled machine then finds results that the next compilation is built from. We call this recursive knowledge compilation and measure it in Wirkung, an exact formula engine that searches modulo a library of observationally distinct functions. Four instructions learned from the engine's own exact solutions raise the solved part of an untouched test set from 19 to 36 of 179 targets at 3.6 times the search work of the primitive search; deeper primitive search reaches 26 at depth six for 7.7 times the work and 21 at depth seven for 71 times. The gain transfers: 19 test targets solved for 2 lost. When the loop stalled at six instructions, every one of nine candidates losing on validation, the stall was a limit of the engine's language, not of its compute: instantiating the constant holes of its two-hole templates and widening the candidate cut produced two new instructions, h+cos x and h/x, the second worth -1.0 on its own and +5.8 after the first, and raised validation from 56 to 65 of 187 and test from 43 to 46 of 179. Compiled search experience is the second channel: a learned order of the meet-in-the-middle joins solves 184 more of 20,354 learning targets at unchanged work and lifts validation from 56 to 60 at work equal to four digits. Each compilation produced the finds the next one was learned from; the sleep that proposes the third round abstracts 23,700 exact solutions and the near misses of 15,367 unsolved targets, 561 of them solvable only with the compiled machine, and in it the templates that lost by two validation targets at six instructions reach the threshold of admission. Every number is a count of machine-exact solutions with the search work spent on them, on a Mac mini with 16 GB and rented CPU boxes. The same move, replacing search by a compiled exact form, carries the author's constant-state replacement of attention and the resolution of the self-similar blow-up rates of the Córdoba–Córdoba–Fontelos model. The complete record of the loop, including every failure, is public.