TCA_Explorer — a sparse 2D balanced-ternary cellular automaton

A 2D outer-totalistic CA over {-1, 0, +1} on a Moore neighborhood with toroidal boundaries. The rule has only 3 active entries out of 51 (λ ≈ 0.059), yet supports:

1. Four axis-aligned gliders (N, S, E, W) of velocity c and period 1.
2. Boolean universality via NAND and NOR gates built from glider collisions.
3. Native single-collision balanced-ternary trit-adder — all 9 single-digit cases resolved without gate composition.
4. Multi-trit ALU with carry propagation and free subtraction (balanced- ternary negation is a trit-wise sign flip, cost 0).

See PAPER_DRAFT.md for the full write-up.

All 9 balanced-ternary single-digit additions resolved in one CA collision each.

4-trit ALU computing 27 + 14 = 41 with carry propagation (stage 1: a+b, stage 2: d1+cin).

Repository layout

src/tca_sim/              simulator + canonicalization
verify_*.py               per-primitive empirical verifications
multi_bit_adder.py        4-bit ripple-carry adder (binary, composed)
multi_trit_adder.py       N-trit ALU (balanced-ternary, native)
gen_*.py                  GIF generators for paper figures
tests/                    canonicalization/metric unit tests
archive/                  earlier exploration, non-reproducible artifacts

Reproducibility map

Paper claim	Script	Artifact
INHIBIT primitive (E+N perpendicular)	verify_inhibit.py	—
AND gate via cascade	verify_and.py · gen_and_circuit_gif.py	and_circuit_verified.gif
NOR gate (functional completeness)	verify_nor.py	—
NAND + XOR depth-4 cascade	verify_nand.py	—
Half-adder	verify_half_adder.py · gen_half_adder_gif.py	half_adder_verified.gif
Full-adder (8/8 truth table)	verify_full_adder.py	—
4-bit ripple-carry adder	multi_bit_adder.py · gen_4bit_adder_gif.py	adder_4bit_11plus5.gif
Trit-adder (9/9 native)	verify_trit_adder.py · gen_trit_adder_gif.py	trit_adder_9cases.gif
Multi-trit ALU + free subtraction	multi_trit_adder.py · gen_multi_trit_alu_gif.py	multi_trit_alu.gif

Every script is self-contained — run directly with python <script>.py and it prints a truth table, asserts correctness, and (for gen_*.py) writes a GIF.

Quick start

pip install -r requirements.txt
python verify_trit_adder.py     # 9/9 balanced-ternary addition
python multi_trit_adder.py      # multi-trit ALU + subtraction
python -m pytest tests/ -q      # sanity checks

The rule

Cells take values in Σ = {-1, 0, +1}. For state σ and Moore neighbor sum s, the next state is looked up at index (σ+1)·17 + (s+8) in a 51-entry table. All entries are 0 except:

idx	(σ, s)	→
9	(-1, 0)	+1
23	( 0, -3)	-1
34	(+1, -8)	+1

Status

Pre-print draft. All numerical claims in PAPER_DRAFT.md reproduce directly from the scripts in this repo.