Fix #789: HC→StackerCrane connector cost prevents incorrect solution extraction

[isPANN]

Summary

Fixes #789: the reduction chain HamiltonianCircuit → StackerCrane → ILP returned Or(false) on the prism graph, which has a valid Hamiltonian circuit.

Root Cause

The HC→StackerCrane reduction splits each vertex $v_i$ into $v_i^{in}$ (= $2i$) and $v_i^{out}$ (= $2i+1$), adding:

• A mandatory directed arc ($2i → 2i+1$) of length 1 per vertex
• Two undirected connector edges per original edge ${u,v}$: ${2u+1, 2v}$ and ${2v+1, 2u}$, originally with length 0

The zero-cost connectors form a rich network where multi-hop paths are also zero-cost. On the prism graph, vertices 5 and 6 have no direct connector (original edge ${2,3}$ doesn't exist), but a 5-hop zero-cost path exists:

5 →(0) 0 →(0) 7 →(0) 10 →(0) 9 →(0) 6

This means the ILP can find an optimal SC permutation [1, 2, 3, 0, 5, 4] with cost $n = 6$ that does not correspond to a valid Hamiltonian circuit. When extracted back to HC, edges $2→3$ and $0→5$ don't exist → Or(false).

Fix

Change connector edge lengths from 0 to 1. This ensures:

• Adjacent vertices (direct connector edge): connector cost = 1
• Non-adjacent vertices (no direct connector): connector cost ≥ 2, because from an out-node $2i+1$ to a non-adjacent in-node $2j$, one step reaches an in-node $2v$ (neighbor of $i$), and reaching another in-node $2j$ from there requires at least two more steps through the bipartite out/in structure.

Correctness proof for all graphs:

Condition	Optimal SC cost	Reasoning
HC exists	$2n$	$n$ arcs × 1 + $n$ single-hop connectors × 1
HC does not exist	$≥ 2n + 1$	Every permutation has at least one non-adjacent pair (cost ≥ 2), so total connector cost ≥ $n + 1$

Since every optimal permutation at cost $2n$ must use only single-hop connectors (each costing exactly 1), every optimal SC permutation maps back to a valid HC. The extract_solution identity mapping is therefore correct.

Changes

• src/rules/hamiltoniancircuit_stackercrane.rs — connector edge lengths 0 → 1, updated doc comments
• src/unit_tests/rules/hamiltoniancircuit_stackercrane.rs — updated expected costs ($n → 2n$), added prism graph regression test

Test plan

• All existing HC→SC unit tests updated and passing
• New prism graph regression test (test_hamiltoniancircuit_to_stackercrane_prism_graph)
• CLI reproduction: pred reduce --to StackerCrane | pred solve now returns Or(true) on prism graph
• Full test suite with example-db and ilp features passing (3657 tests)
• clippy + fmt clean

🤖 Generated with Claude Code

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 97.85%. Comparing base (f29cfb1) to head (1ed4583).
⚠️ Report is 2 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #791      +/-   ##
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- Coverage   97.85%   97.85%   -0.01%     
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  Files         625      625              
  Lines       68311    68329      +18     
==========================================
+ Hits        66846    66863      +17     
- Misses       1465     1466       +1     

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