Occurrence Theory

This repository contains a draft research paper and verification script for Occurrence Theory (OT), defined as an oriented form of Sedenion Settlement Dynamics (SSD).

The central object is the Aut-invariant settlement channel on the sedenion zero-divisor crack. The paper separates theorem, exact computation, measurement, interpretation, and conjecture using explicit ledger tags:

• [T] theorem
• [C] computation
• [M] measurement
• [I] interpretation
• [X] conjecture

Files

• topographo/ - reusable Python package (the library) for Cayley-Dickson algebra, validation gates, operators, SSD helpers, and the exceptional-algebra (Albert / F4 / G2) layer. Ships to PyPI with its own tests under topographo/tests/.
• verify/ - the consumer side: all Paper verification. Holds the canonical first-party audits, the tests that guard them, and independent reviewer results. See verify/README.md for the naming convention.
• occurrence-theory.md - main paper draft (Paper I).
• verify/occurrence_i_audit.py - numerical audit and verification script for the Paper I algebraic claims. Every printed [C]/[G] line is a computed number checked against a threshold; the script exits nonzero if any fails.
• verify/occurrence_i_cabarius.md - independent re-derivation of the paper's computational claims (reviewer: cabarius), and the corrections it produced.
• occurrence_theory_prompt.md - source prompt and writing constraints used to generate the paper.
• .github/workflows/topographo.yml - library CI: tests, builds, and releases the topographo package.
• .github/workflows/occurrence.yml - consumer CI: installs topographo, runs the audit exit-code gate and the verify/ tests.
• CHANGELOG.md - release history for the package and audit artifacts.
• LICENSE - MIT license.

Requirements

The audit script requires Python 3.11 or newer and NumPy, plus the topographo package (which it imports for the verified algebra). The exceptional-algebra (Albert / F4 / G2) reproduction now lives inside the package as topographo.exceptional.

uv is the preferred runner for local audit work:

uv run python verify/occurrence_i_audit.py

For editable package installation:

uv pip install -e .

After installation, the core math layer is importable without running the Occurrence Theory audit narrative:

from topographo.core import CayleyDicksonAlgebra, verify_gates
from topographo.ssd import SedenionAlgebra

For exact finite crack certificates, use basis_zero_divisors() to enumerate the full 84-point design. sample_crack(n) samples from that design with replacement and is intended for stochastic diagnostics, not machine-zero theorem gates.

API documentation is generated with pdoc and published to GitHub Pages: https://theswanfactory.github.io/occurrence/

To build it locally:

uv run pdoc \
  topographo \
  topographo.core \
  topographo.core.algebra \
  topographo.core.cayley_dickson \
  topographo.core.gates \
  topographo.ssd \
  topographo.ssd.channel \
  topographo.ssd.sedenion \
  topographo.exceptional \
  topographo.exceptional.lab \
  -o site

Run the Audit

From the repository root:

uv run python verify/occurrence_i_audit.py

To save the output:

uv run python verify/occurrence_i_audit.py > audit_results.txt

The audit exits 0 only if every certificate meets its threshold, and 1 otherwise, so it is safe to gate CI on it. A passing run means the paper's [C]-tagged claims reproduce on this implementation. It does not mean the paper's [I] interpretations are correct; those are not tested.

CI runs two workflows on pull requests and pushes to main: topographo.yml (library: tests, build, release) and occurrence.yml (consumer: installs the package, runs this audit as an exit-code gate, and runs the verify/ tests).

Status

This is a research workspace, not a packaged library. The paper is the primary artifact; the script is included to reproduce the computation-backed claims.

License

MIT. See LICENSE.