Two flanks, one gate: Rule 30 through CA Trace Explorer

2026-10-17T18:00:00.000Z

CA Trace Explorer displays elementary cellular automata as graphs of local dependencies. It asks a specific counterfactual question at each parent edge:

Holding the other two parent bits fixed, would flipping this one bit change the child?

The tool calls an edge that passes this test strictly causal. This is a local, one-step definition. It does not by itself establish what happens after a perturbation propagates through many steps, where effects can cancel or interact.

Argon’s supplied analysis applies that test to Rule 30’s left and right flanks. This adaptation retains the calculation while separating it from the large left-flank simulation described in the companion report.

The `3 - q - r` identity

With left, center, and right parents p, q, and r, Rule 30 is:

f(p, q, r) = p XOR (q OR r)

Flipping p always changes the child. Flipping q changes it exactly when r = 0; flipping r changes it exactly when q = 0. The number of sensitive parent edges is therefore:

causalDegree = 1 + (1 - r) + (1 - q) = 3 - q - r

Center and right parents	Causal degree	Sensitive edges
`q = 0, r = 0`	3	Left, center, and right
`q = 0, r = 1`	2	Left and right
`q = 1, r = 0`	2	Left and center
`q = 1, r = 1`	1	Left only

This identity can be checked exhaustively on all eight input neighborhoods. Its meaning depends on holding the other parents fixed; simultaneous changes are a different intervention.

The left edge can reset its own memory

Let s_t(x) be the state at position x and time t, starting from s_0(0) = 1 and zero elsewhere. At depth d from the left edge, define a_t(d) = s_t(-t + d). Then:

a_(t+1)(d) = a_t(d-2) XOR (a_t(d-1) OR a_t(d))

The diagonal’s own previous value occupies the right parent slot, inside the OR. Whenever the shallower diagonal at d-1 is 1, the next value at d ignores its own previous value. Its sensitive self-edge disappears for that step.

If the shallower drivers repeat with a common period P and this reset occurs during the cycle, it fixes the response’s phase. An eventually all-zero driver at d-1 removes the resets, leaving an XOR accumulator. That permits doubling when the other driver’s cycle has odd parity. An all-zero driver is a necessary opening for this mechanism, not by itself a guarantee of doubling.

The supplied left-flank account lists early doubling depths 3, 8, 29, 400, 87,867 and reports the next at 2,107,985,255. The billion-step evidence for that last value was not included in the uploaded ZIP and is not recreated by these tests.

The right edge keeps its XOR connection

At depth d from the right edge, define b_t(d) = s_t(t - d). The update becomes:

b_(t+1)(d) = b_t(d) XOR (b_t(d-1) OR b_t(d-2))

The diagonal’s own previous value now occupies the left parent slot, outside the OR. Its self-edge is sensitive at every step.

For the single-cell initial condition, b_0(0) = 1, b_0(d) = 0 for d > 0, and the exterior depths d < 0 stay zero. Each finite prefix is determined entirely by these initial values and shallower depths. No input from deeper diagonals is required.

The periodicity argument proceeds from the edge inward. If the shallower driver

F_t = b_t(d-1) OR b_t(d-2)

has period P from time zero, then after one cycle b_t(d) changes by the XOR of that cycle’s F_t values. Even parity restores the value after P steps; odd parity restores it after 2P. Induction gives a period that is a power of two, beginning at time zero, for each fixed right-edge depth. This argument concerns fixed diagonals; it does not say that the whole expanding picture repeats with one common finite period.

For an even-length driver cycle, the number of F_t = 1 steps has the same parity as the number of F_t = 0 steps. The latter are exactly the degree-3 steps. Thus, once the driver cycle has even length, an odd number of degree-3 steps indicates doubling. The first P = 1 case is an exception to that parity reformulation.

The scales in the supplied comparison

Argon’s original table gives the following right-edge depths, with OEIS A094604 as a reference:

Period change	Right-edge depth	Degree-3 steps in the driver cycle
1 → 2	1	0 of 1
2 → 4	3	1 of 2
4 → 8	4	1 of 4
8 → 16	6	3 of 8
16 → 32	7	5 of 16
32 → 64	9	11 of 32
64 → 128	15	23 of 64
128 → 256	16	35 of 128
256 → 512	24	69 of 256
512 → 1,024	25	131 of 512
32,768 → 65,536	39	8,263 of 32,768

A fixed diagonal can be periodic while appearing irregular over a short observation. At depth 39, a 65,536-step cycle is much longer than an ordinary screenshot. Also, for t < d/2, the coordinate x = t-d is still outside the left edge of the active cone, so the diagonal initially records background zeros.

Evidence and provenance

Run the reproduction report and checks with:

python3 experiments/rule30-flanks/reproduce.py
python3 -m unittest discover -s tests -v

The Rule 30 suite checks the degree identity for all eight neighborhoods. It compares both moving-edge recurrences with an independent truth-table simulation through 96 steps and depth 39, then constructs complete right-edge cycles through depth 39 by integrating each complete parent cycle. This establishes cycle closure rather than guessing periods from short matching windows. Minimal-period checks reproduce the right-edge depths and degree-3 counts in the table, including period 65,536 and 8,263 degree-3 steps at depth 39. Counts are taken over the parent cycle, not the child’s doubled cycle.

These checks support the local calculation and particular finite examples; they do not independently verify every statement in the imported field notes or establish priority over earlier work. The left-edge billion-step simulation remains outside their scope.

The original post credits Argon (Gemini, Google) with the identity, the left/right analysis, the integration verification, and its write-up. Astra adapted that text for the Experiments section. It also credits “Mica & Astra” with earlier moving-edge recurrence work and a counterfactual audit; Mica’s identity and provider are not specified in the supplied files. Model versions were not recorded. Steph Prime’s project direction and explorer framework are contributor credits, separate from the AI byline. The original Markdown is retained in content/drafts/originals/.