A small companion · standard Wolfram encoding

Three bits.
Four futures.

Elementary cellular automata make the rule visible. One cell reads its left, center, and right neighbors; an eight-row lookup table decides the next color. This atlas lets Rule 110, its color-conjugate Rule 137, Rule 42, and Rule 30 share one honest control surface.

index = 4 × left + 2 × center + right
output = (rule >> index) & 1

64 cells · standard seedsame encoding · many worlds

01 / Controlled replay

Change the rule. Keep the question fixed.

Choose a rule, width, boundary, and outside background. The selected single board is independent; the paired 110 / 137 boards below are a separate exact color-conjugacy check.

Rule

Width

Boundary

Outside background

Rule 110 · width 64 · wrap · background 0single seed / generation 0

Rule: 110
Generation: 0
Live cells: 1
Boundary: wrap
Mapped pair: exact

02 / Exact relation

Rule 137 is Rule 110 in mirror colors.

For every local neighborhood, F137(1 − x) = 1 − F110(x). The paired boards begin as complements, use outside backgrounds 0 and 1, and step together at every tick.

Rule 110state x · background 0

Rule 137state 1 - x · background 1

Checking exact complement relation…Every tick compares Rule 137 to the complement of Rule 110.

03 / Eight local cases

Tables before adjectives.

Each card is the complete eight-input rule table. The shared index is explicit so the visual replay can be checked against the arithmetic.

Color conjugacy is a relation, not a benchmark. Rule 110 / 137 pass an exact mapped-state check by construction. Rule 42 and Rule 30 are shown as independent tables; this page makes no universality, computational-class, or biological claim about either one.

04 / Proof gates

Small tests, visible receipts.

The page runs deterministic checks on load: all eight table rows, the 110 / 137 relation, and random boards at both widths across wrap and dead boundaries.

Computed space-time atlas for Rules 110, 137, and 42 using a 256-cell standard-seed replay — Computed offline from the same standard encoding. The paper plate uses a 256-cell by 96-generation raster: Rule 110 starts with one central live cell on background 0; Rule 137 starts from its complemented state on background 1; Rule 42 is an independent standard-seed replay. The visual relationship is a finite state-conjugacy check, not a classification proof.

A tiny wink to the number, then back to the table. Rule 42 is included for comparison; this atlas makes no universality claim.