o10_069684: two reducible fillings, only one with a small summand

This bundle verifies the following census observation:

[ M=o10_069684,\qquad M(1,0)=L(6,1)#L(7,2),\qquad M(0,1)=L(3,1)#\mathbb{R}P^3. ]

The slopes are primitive, distinct, and have geometric distance one. The first connected sum has no summand homeomorphic to L(2,1), L(3,1), or L(4,1); the second has two such summands because RP3 = L(2,1).

The composite order of H1(L(6,1)) = Z/6 does not make L(6,1) a connected sum of L(2,1) and L(3,1): lens spaces are prime, and Regina returns L(6,1) itself as one prime summand.

Exact scope of the result

Dunfield 2020, Section 3.5 asks whether, when a one-cusped hyperbolic manifold has two connected-sum fillings, both fillings must each contain a summand among L(2,1), L(3,1), and L(4,1). This example gives a negative answer: the (1,0) filling does not.

Hoffman--Matignon 2003 asks the stronger existential question of whether an example exists with no small summand in either filling. This example does not answer that question because the (0,1) filling contains L(2,1) and L(3,1).

This is a consequence extracted from Li's already-public C10 dataset, not a new construction of the underlying manifold. No claim of publication novelty should be made without expert and literature review.

Reproduction

Use Python 3.12 or 3.13 with the pinned packages:

git clone https://github.com/infinityscroll/o10-069684.git
cd o10-069684
python -m venv .venv
.venv/bin/pip install -r requirements.txt
.venv/bin/python fetch_official_data.py data
.venv/bin/python reproduce.py > current_output.json
.venv/bin/python compare_semantics.py verified_output.json current_output.json

fetch_official_data.py takes the destination data directory as a positional argument. It downloads each exact Dataverse file ID in official_manifest.json to a temporary file, checks its byte count and SHA-256, and only then renames it atomically. A valid existing file is reused; a mismatched existing file causes an error rather than being overwritten. The downloaded data/ directory and current_output.json are ignored by git.

compare_semantics.py likewise takes two positional arguments: the reference report followed by the newly generated candidate report.

The second command compares the stable topological content and exits nonzero if it differs. Do not compare the two JSON files byte for byte. Regina and SnapPy can materialise different triangulations of the same filled manifold; closed-manifold and prime-summand isosigs, as well as their tetrahedron counts, are therefore labelled run_specific_* witnesses. Exact prime decomposition, lens-space recognition, homology, source hashes, slopes, and the census identity are included in the semantic comparison.

The script:

1. verifies SHA-256 and byte size for all four official Li files;
2. verifies the exact official rows and absence from the ambiguous-surface file;
3. checks the two slopes are primitive, distinct, and distance one;
4. checks the installed SnapPy C10 census identity, including decorated isosig and peripheral data hash;
5. materialises each filling through both Regina's SnapPea route and SnapPy's permanent-filling route;
6. runs Regina's exact irreducibility and prime-decomposition algorithms;
7. independently recognises every prime summand as a layered lens space.

The rigorous hyperbolicity input for the cusped base manifold is Li's certified census theorem. The pip build of SnapPy gives a positive numerical structure but cannot rerun verify_hyperbolicity() without Sage. The decomposition and irreducibility decisions here are exact Regina computations.

See source_notes.md for page-level source attribution and novelty_search.md for the bounded search already performed.