Sandpiles

Open, AI-produced research on unconventional computation in the ordinary two-dimensional Abelian sandpile.

Current results

Packet 71: local parity AND and an eight-terminal transducer

From the stable core

1 1
2 2

add 71a and 71b grains at the two top cells. For every a,b in {0,1,2,3}, two exterior sites have odometer parity

(a mod 2) AND (b mod 2).

An exhaustive audit of all 256 stable 2 x 2 cores, every common packet 1 <= p <= 71, all sixteen inputs, and every reached exterior site finds no such tap below 71 and exactly sixteen at 71.

The Boolean restriction supplies an exact hybrid interface. Eight initially empty boundary cells receive exactly one grain iff a=b=1, without toppling. Precharge those cells and arbitrary prescribed finite outward fuse leads to height three before supplying the inputs. In the three false cases no terminal or fuse cell topples; in the true case every terminal and fuse cell topples exactly once. This is a one-stabilization, packet-input/presence-output one-shot AND module with eight unloaded terminals and no separate read pulse.

Fuse signaling, presence-AND, and branching are classical sandpile-circuit ingredients. The result here is the exact certified packet-to-presence interface. Its one-toppling output encoding does not match its 71-grain input encoding, so self-cascade and arbitrary downstream loading are not claimed.

See the packet-71 audit, its independent Python certificate, and the independent C++ exhaustive audit.

Packet 672: a local odometer-residue half-adder

From the stable core

0 3
3 2

add 672a and 672b grains at the two top cells. At the exterior sites SUM=(-3,3) and CARRY=(-1,3),

u(SUM)   mod 2 = (a mod 2) XOR (b mod 2)
u(CARRY) mod 2 = (a mod 2) AND (b mod 2)

for all sixteen a,b in {0,1,2,3}. Thus the two taps form a half-adder of the input parity bits over a four-symbol packet-multiplicity alphabet; this is not a base-four adder. Two independent exhaustive engines find no such two-tap half-adder in the same 256-core, equal-packet class below 672; at 672 there are ten cores and 28 role-labelled output pairs.

This is a decisive nonlinear two-output local observable, but its integer output counts are not yet normalized packets and attaching a receiver can feed back into the same avalanche. See the packet-672 audit.

Packet 925: a crossed parity identity

The repository began with an exact four-terminal odometer-parity identity on the infinite square lattice. From the stable 2 x 2 seed

0 0
2 2

add 925a and 925b grains at the two top cells, with a,b in {0,1,2,3}. After stabilization, the odometer parities at the oppositely paired bottom cells are exactly (a mod 2, b mod 2) for all sixteen inputs.

This is a crossed local readout, not yet a classical composable crossing gate. The distinction, exact table, scoped minimality result, analytic no-propagation theorem, literature context, and failed composition audits are in the paper:

A Four-Terminal Odometer-Parity Identity in the Planar Abelian Sandpile

Contribution statement

OpenAI Codex produced the specific construction, searches, proofs, code, verification, literature framing, and manuscript in a ChatGPT work session on 23 July 2026.

Sophie (heartpunk) had not encountered this sandpile problem before that session. Her contribution was to pose a broad challenge, filter proposed directions by whether they seemed interesting, encourage the system to keep trying, request a publishable write-up, and make the work public. She did not derive or independently verify the technical results.

Hosting this repository under heartpunk denotes publication stewardship, not technical authorship. The technical claims are unreviewed AI-produced research and should be checked independently.

Reproduce the results

The packet-71 certificates and scoped exhaustive audit:

python3 generate_packet71_and_latch_certificate.py
python3 verify_packet71_and_latch_certificate.py

c++ -O3 -std=c++20 sandpile_packet71_and_latch_audit.cpp -o audit71
./audit71 packet71_and_latch_cpp_audit.json

The packet-672 half-adder certificates and two independent scoped exhaustive audits:

python3 generate_halfadder672_certificate.py
python3 verify_halfadder672_certificate.py

c++ -O3 -std=c++20 audit_halfadder672_exhaustive.cpp -o audit672
./audit672

c++ -O3 -std=c++20 audit_halfadder672_unit_exhaustive.cpp -o audit672-unit
./audit672-unit

The original packet-925 checks require only Python 3:

python3 verify_packet925_full_alphabet_certificate.py
python3 audit_full_alphabet_925_witness.py

The scoped minimality audit requires a C++20 compiler:

c++ -O3 -std=c++20 audit_full_alphabet_925.cpp -o audit925
./audit925 --maximum-pulse 925

The exact packet-family scan through 100000 is:

c++ -O3 -std=c++20 -pthread scan_full_alphabet_0022.cpp -o scan100k
./scan100k --maximum-pulse 100000

The PDF also embeds its certificate, verifiers, proof notes, scan audit, and typesetting source as file attachments.

Repository contents

The root intentionally preserves the research scripts and their relative imports. It includes:

• exact certificates and independent verifiers;
• dense and sparse exhaustive searches;
• packet-family scans and replay audits;
• analytic no-go proofs;
• parity-wire, normalizer, reset, and interface experiments;
• bounded negative composition results;
• exploratory Python, C++, Z3, and JavaScript programs.

The unsuccessful searches are included because they delimit the result and make the path to later claims auditable.

Ongoing direction

Further work will be committed publicly. The immediate target is to close the encoding gap between packet inputs, parity observables, and ordinary fuse-wire outputs without losing exactness under physical attachment. No claim of functional completeness, a crossover gate, P-completeness, or universality is made.

License

Use this however you want.

Everything in the repository is released under CC0-1.0: use, copy, modify, publish, redistribute, sell, or build on it for any purpose. Attribution is appreciated but not required.

Source code is additionally available under the MIT License. You may choose either license for code.