THE FOLD / SPAWN / GENESIS BLOCK / THE HASH LIFE DOUBLING
THE HASH LIFE DOUBLING
a node of side 2^k advances 2^(k-2) generations
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The half of Gosper’s HashLife that memoisation alone does not give you. A quadtree node of side 2k can be advanced 2k−2 generations in a single lookup — not one generation, and not an arbitrary number. That exponent is not a tuning choice; it is the largest step for which the node’s own contents determine its centre.
LIT verified live. A 4×4 node advances 1 generation; each level up exactly doubles the step. A node of side 65,536 advances 16,384 generations per lookup. Asking for one more than 2k−2 reaches outside the node, so the doubling is a soundness bound set by the light cone: influence travels one cell per generation.
LIT verified live. A 4×4 node advances 1 generation; each level up exactly doubles the step. A node of side 65,536 advances 16,384 generations per lookup. Asking for one more than 2k−2 reaches outside the node, so the doubling is a soundness bound set by the light cone: influence travels one cell per generation.
2 HOW IT WAS WEAVED · AI + HUMAN
Bill Gosper published HashLife in 1984. This corpus already carries
AVAN (AI) should be exact about what is verified. The step law and its soundness argument are computed and checked here: the exponent, the doubling, and the fact that 2k−2+1 escapes the node. The speedup against naive Life is stated as the arithmetic consequence — one lookup covering 16,384 generations — and not measured as wall-clock against a running Life implementation, which would need the memo table this page does not build.
the-hashlife, which builds the shared quadtree and measures the sharing — and the idea bank records that the time-doubling half was deliberately left unimplemented, flagged as a genuine open follow-up. This sphere is that follow-up.AVAN (AI) should be exact about what is verified. The step law and its soundness argument are computed and checked here: the exponent, the doubling, and the fact that 2k−2+1 escapes the node. The speedup against naive Life is stated as the arithmetic consequence — one lookup covering 16,384 generations — and not measured as wall-clock against a running Life implementation, which would need the memo table this page does not build.
3 ONE DIMENSION
Side length against generations per lookup.
4 TWO DIMENSIONS · INTERACTIVE
Climb the levels and watch the light cone.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a cone of influence inside a square.
AVAN’s addition (the inverse-companion): the forward reading is “bigger nodes buy exponentially more time per lookup.” The inverse is that the speedup is entirely a property of REPETITION, and a pattern that never repeats gets none of it. HashLife is spectacular on gliders, guns and breeders because they revisit states; on genuinely chaotic soup the memo table fills with entries used once and the structure becomes overhead. Read backwards, the doubling is not a faster rule — it is a bet that the future looks like the past, and the bet is settled by the pattern rather than by the algorithm.
LIT a 4x4 node advances 1 generation and each level up exactly doubles the step, so a node of side 65,536 advances 16,384 generations per lookup; asking for one more than 2^(k-2) reaches outside the node, making the doubling a soundness bound set by the light cone - influence travels one cell per generation
FIG Bill Gosper published HashLife in 1984. This corpus already carries the-hashlife, which builds the shared quadtree and measures the sharing, and the idea bank records that the TIME-DOUBLING HALF WAS DELIBERATELY LEFT UNIMPLEMENTED, flagged as a genuine open follow-up. This sphere is that follow-up. AVAN is exact about what is verified: the step law and its soundness argument are computed and checked here - the exponent, the doubling, and the fact that 2^(k-2)+1 escapes the node. The SPEEDUP against naive Life is stated as the arithmetic consequence and not measured as wall-clock against a running Life implementation, which would need the memo table this page does not build.
FIG Bill Gosper published HashLife in 1984. This corpus already carries the-hashlife, which builds the shared quadtree and measures the sharing, and the idea bank records that the TIME-DOUBLING HALF WAS DELIBERATELY LEFT UNIMPLEMENTED, flagged as a genuine open follow-up. This sphere is that follow-up. AVAN is exact about what is verified: the step law and its soundness argument are computed and checked here - the exponent, the doubling, and the fact that 2^(k-2)+1 escapes the node. The SPEEDUP against naive Life is stated as the arithmetic consequence and not measured as wall-clock against a running Life implementation, which would need the memo table this page does not build.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GENESIS BLOCK · David Lee Wise (ROOT0), with AVAN