> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification.

# WSFC R12 Package Reconstruction Notes

**Date:** 2026-10-08 · reconstructed-v1

The R12 paper and State Crystal 12 are previously recovered originals. The missing program package is newly implemented for this reconstruction and is not presented as the original 2026-09-23 package.

## Input and execution

Requires Python 3.11+ and SymPy 1.14. Extract `WSFC_Round12_Reconstructed_Package_2026-10-08.zip` and run these commands in the package directory:

```powershell
python -m pip install -r requirements.txt
python wsfc_point_segment_compiler.py examples/canonical.json --output canonical-result.json
python wsfc_point_segment_compiler.py examples/irrational.json --output irrational-result.json
python verify_r12.py
python wsfc_round13_trusted_root_checker.py canonical-result.json
python wsfc_round13_trusted_root_checker.py irrational-result.json
```

All geometric coefficients are integers or rational-number strings; floating-point input is rejected. Input consists of the fixed endpoint P0, affine moving endpoint Q0+tD, circle centre C, positive radius r and a finite parameter interval.

## Restored workflow

The program generates the six polynomials A, N, S, H0, L and H1 from the paper's definitions, merges their roots, performs rational isolation, constructs root-free atomic intervals, and selects the actual safety polynomial by nearest feature. Numerical approximations are not used as authoritative predicates.

The canonical example recovers the nearest-feature switch at 1/2, supporting-line-only tangency at 6/7, and endpoint contact at t=1. R11's polygon additionally has coincident third vertices at t=1; this generic segment program reports endpoint contact and does not misclassify it as zero length of the segment.

The irrational example recovers `2 - sqrt(3)/2` with a rational isolating interval. Additional checks cover a static degenerate segment, an actual zero-length event, initial contact and a no-event case. Existing initial contact is recorded in first_actual_event rather than silently treated as strict safety.

## Verification boundaries

The Fraction/Sturm root-coverage checker from the R13 package is retained byte-for-byte as another route for checking the root set. It verifies complete root coverage, not every geometric semantic condition or generic formal soundness. The paper's complete-square identities and phase conditions are separately checked by the new program.

This reconstruction's test PASS does not fill the unknown execution history of the original R12 with a PASS, nor change `GENERIC FORMAL SOUNDNESS OPEN` to CLOSED.

MANIFEST.json records every package member's SHA-256. Original copies, the retained R13 checker, and the newly written compiler/tests each carry separate provenance labels.

===END OF RECONSTRUCTED PACKAGE NOTES===