This repository contains manuscripts in TeX and PDF, Wolfram notebooks, and the code and data needed to reproduce their computations. There are eight manuscripts, each with one PDF, and two notebooks.
| Source | Pages | Results | |
|---|---|---|---|
| Local Whitehead chain | 20 | The stationary law of the local walk on cubic multigraphs, spectral bounds, exact loop and bridge formulas, and mixing lower bounds. | |
| Field dynamics | 31 | Pair averaging on evolving cubic graphs, finite consensus schedules and contraction estimates. | |
| Computational companion | 8 | Exact finite calculations accompanying the field-dynamics paper and notebooks. | |
| Quartic period torsor | 16 | A rank-two period system, its local series and finite Hermitian matrix integrals; classification of positive representations and selection of the strongest moment bounds. | |
| Outer space | 8 | Cubic refinement transfer, Euler-characteristic observables under the Whitehead stationary law, and an explicit Morita cycle/cocycle certificate. | |
| Bessel recursion | 8 | Identification of the Broadhurst–Roberts recursion with the inverse de Rham pairing, including resonance, and factorial-scaled integrality. | |
| Histories and states | 11 | Permutation-torsor presentations, schedule weights, a criterion for observations to inherit the history median, event-bit repairs and confluent systems with nontrivial state topology. | |
| Curvature from causal orders | 9 | Recovery of latent null ranks from unlabelled two-dimensional orders, optimal curvature estimation for positive smooth densities, and stability under the choice of realizer. |
Page counts refer to the included PDFs.
The local Whitehead paper continues the diameter work of Rafi–Tao and the expansion and spectral work of Grave de Peralta–Kolpakov and Li. It determines the stationary law for the multiplicity chain and proves explicit spectral bounds and an exact conductance formula for the bridge cut. The field-dynamics paper and computational companion develop pair averaging on the evolving graphs.
The Outer space paper recovers the Euler characteristics of Smillie–Vogtmann and Borinsky–Vogtmann through expectations under the same stationary law, and gives a direct certificate for the first Morita class.
Each manuscript has a build target. The later papers have separate folders; the field-dynamics paper and its companion retain their original root paths. Bibliographies, TeX figures and table sources are included.
The Python files at the root enumerate Whitehead moves and analyse the resulting chains. Paper-specific checks and their exact data live beside the manuscripts or under research/. The scripts/ directory builds and checks the notebooks and their bundled models. Wolfram/ contains the local Wolfram package; lean/ contains the finite algebraic verification.
Manuscript PDFs and both notebooks are tracked. LaTeX intermediates, generated figure PDFs, submission archives, reference downloads, reasoning notes and exploratory scripts remain local. README.md is the repository guide and the only tracked Markdown file.
Python 3.11 or later, LaTeX with latexmk, pdflatex and BibTeX, and abbrvurl.bst are required. The tested Python package versions are recorded in requirements-repro.txt:
python3 -m pip install -r requirements-repro.txt
make papers
To build one paper:
make paper
make computational-notes
make rule-a-paper
make integral-torsor
make outer-space-paper
make bessel-recursion-paper
make history-state-paper
make causal-order-paper
The full check also needs Lean's lake command and the native nauty bridge. With nauty/Traces installed, build the bridge first:
python3 build_nauty_bridge.py
Then build the manuscripts, check the Lean identities and small-model resource, and run the paper-specific exact computations:
make check
Use the interpreter in which the dependencies were installed. It can be selected explicitly with make check PYTHON=/path/to/python3.
Individual checks are also available:
python3 rule-a-paper/verify_rule_a.py 8 --backend python --jobs 1
make integral-torsor-check
make outer-space-check
make bessel-recursion-check
make history-state-check
make causal-order-check
The history check compares independent incidence and execution enumerations, exhaustively tests 4,890 observation partitions, and checks the additive criteria, repair sets and confluent counterexamples. The Bessel checker compares the literal recursion, differential operator and full connection elimination for 3 through 32 Bessel factors. Quartic checks use exact rational arithmetic and Arb enclosures. The Outer space checks include orientation signs, refinement multiplicities and the cycle/cocycle certificate. The causal-order checks enumerate 7,456 two-realizers and independently verify the metric curvature, kernel identities and explicit examples.
Lean sources verify the specified finite averaging and permutation identities and consensus schedules over the rationals. They do not formalize the full graph or probability arguments. The toolchain is fixed in lean/lean-toolchain.
Rule A chooses one of the 3N/2 edges and one of its two Whitehead resolutions, retaining multiplicities; choosing a loop leaves the graph unchanged. Its stationary distribution is proportional to the reciprocal of the full incidence automorphism order, including loop flips and parallel-edge permutations. Rule B gives each distinct nonself neighbour probability 1/(3N) and stays put on the remaining ticks; its stationary distribution is uniform on unlabelled classes.
The complete unlabelled class counts for N=2,4,6,8,10,12,14 are 2, 5, 17, 71, 388, 2592, 21096. For a small complete run:
mkdir -p artifacts/csv
python3 gog_generate.py 8 --backend auto --jobs 1 \
--save artifacts/gog_N8.json \
--save-transitions artifacts/transitions_N8.json
python3 gog_edge_model_experiments.py artifacts/transitions_N8.json \
--csv-dir artifacts/csv --summary-json artifacts/edge-summary-N8.json
python3 gog_support_feature_analysis.py artifacts/transitions_N8.json \
--max-cycle 4 --eigenvalues 6 \
--summary-json artifacts/feature-summary-N8.json
python3 gog_ball_growth.py --load artifacts/gog_N8.json \
--max-radius 4 --sample 71
At N=8 the output contains 71 classes and 1,704 moves counted with multiplicity. Complete tables give finite evidence; a sampled trajectory does not establish an asymptotic claim. Truncated tables must not be used as complete enumerations.
The pure Python canonicalizer handles small examples. Larger runs use nauty/Traces, either through dreadnaut or the optional native bridge:
python3 build_nauty_bridge.py
python3 gog_degree_diagnostics.py artifacts/gog_N8.json --lowest 20
The degree diagnostic requires dreadnaut. Set NAUTY_PREFIX or pass --nauty-prefix when the library is installed outside the usual locations. The Outer backend-comparison check requires the compiled native bridge; run the bridge build before make outer-space-check or make check.
Both notebooks remain in the repository:
They use Wolfram Language 14.3 or later and the local GogFieldDynamics package. The bundled exact models cover N=2,4,6,8 and are checked with:
make notebook-data-check
make notebook-paclet-check
make notebook-check
To rebuild and check the computational-notes notebook:
make computational-notes-notebook
make computational-notes-notebook-check
The interactive notebook build uses a Mathematica front end and has an optional SetReplace path engine. SetReplace is not required by the computational-notes notebook. Pass WOLFRAM_KERNEL=/path/to/WolframKernel to make when needed. The scripts under scripts/ build the notebooks, reproduce their text outputs and check their evaluations; edit those sources for future notebook changes. For the computational companion's deterministic text transcript, run scripts/notebook_report_outputs.wls with WolframKernel.
make rule-a-paper-packages
This builds the Whitehead figures and manuscript, then writes the arXiv source and code archives under rule-a-paper/dist/. Submission archives, LaTeX intermediates, downloaded reference copies and working research notes remain local. The repository retains manuscript sources and PDFs, notebooks and reproducibility dependencies.