Perf T1-11: Pre-compute sparse pattern for use_sparse mode

[hmzh-khn]

Summary

Pre-compute SparseMatrixCSC sparsity patterns once per player during solver setup,
then reuse them across iterations by updating nzval in-place instead of calling
sparse(M) each time.

Changes

• src/MixedHierarchyGames.jl: Import SparseMatrixCSC, nonzeros, rowvals, nzrange from SparseArrays
• src/nonlinear_kkt.jl:
  • Add _update_sparse_from_dense!() helper for zero-alloc nzval update
  • Add sparse_cache kwarg to _solve_K!() to use cached sparsity patterns
  • Add sparse_M_caches kwarg to compute_K_evals() to thread caches through
  • Pre-compute patterns in run_nonlinear_solver() via probe evaluation at initial z_est
• test/test_sparse_pattern_cache.jl: New test file (12 tests) verifying:
  • Structural sparsity pattern consistency across different z values
  • Numerical equivalence of cached vs fresh sparse solve
  • run_nonlinear_solver works correctly with sparse caching
  • Allocation reduction from cached nzval update vs fresh sparse() (27x reduction: 1311→47 bytes/call)
• test/test_tiers.jl: Add new test file to slow test tier
• test/test_test_tiers.jl: Update test tier self-test expectations

Benchmark Results

3-player chain (T=3, s=2, c=2), 20 iterations:

• Sparse (cached): 1.01 MB alloc, 55.5 ms
• Dense: 1.27 MB alloc, 123.3 ms
• 20% allocation reduction, 55% time reduction

Sparse construction allocations (isolated):

• Fresh sparse(M): 1311 bytes/call
• Cached nzval update: 47 bytes/call
• 27x reduction per-call

Testing

• All 1353 tests pass (julia --project=. -e 'using Pkg; Pkg.test()')
• All existing sparse solve tests pass unchanged
• No tolerance changes

Changelog

• 2026-02-16: Initial PR with sparse pattern caching implementation

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