Vectorize I_nm/I_mk coupling integrals and remove per-element assembly loops

[Copilot]

I_nm and I_mk were scalar/branch-oriented and required nested Python loops in matrix assembly, limiting performance and making fast-path formulas harder to apply selectively. This change makes both integrals array-native and applies case-specific formulas via mask-based indexing so expensive/general formulas run only where required.

• Vectorized coupling APIs (multi_equations.py)

  • Added grid normalization to accept:
    • scalar mode indices,
    • 1D row/column mode vectors (implicit meshgrid),
    • pre-shaped 2D index arrays.
  • Refactored I_nm and I_mk to compute with logical masks for:
    • zero-mode fast paths,
    • mixed zero/nonzero cases,
    • general-case formulas,
    • equality/singularity edge handling only on relevant subsets.

• Assembly/cache integration (meem_engine.py)

  • Replaced nested (n,m) precompute loops for I_nm with a single vectorized call over mode grids.
  • Replaced nested (m,k) closure loop for I_mk with a single vectorized call, preserving existing cache interfaces and downstream block assembly behavior.

• Illustrative usage

  n_modes = np.arange(N)[:, None]
  m_modes = np.arange(M)[None, :]
  I_nm_vals = I_nm(n_modes, m_modes, i, d, h)
  
  m_modes = np.arange(Mi)[:, None]
  k_modes = np.arange(K)[None, :]
  I_mk_vals = I_mk(m_modes, k_modes, i, d, m0, h, m_k_arr, N_k_arr)

[codecov]

Codecov Report

❌ Patch coverage is 98.87640% with 1 line in your changes missing coverage. Please review.
✅ Project coverage is 98.53372%. Comparing base (0a1a4f0) to head (f36dbfb).

Files with missing lines	Patch %	Lines
package/src/openflash/multi_equations.py	98.79518%	1 Missing ⚠️

Additional details and impacted files

@@                 Coverage Diff                 @@
##                main         #99         +/-   ##
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- Coverage   98.56061%   98.53372%   -0.02688%     
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  Files             11          11                 
  Lines           1320        1364         +44     
===================================================
+ Hits            1301        1344         +43     
- Misses            19          20          +1     

Flag	Coverage Δ
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[rebeccamccabe]

@copilot look at the suggestions I have made which starts with nan and explicitly assigns zeros, instead of starting with zeros and having lack of assignment be the zeros. this is a change for readability to ensure the formulas are easier to check and it is clear exactly where there are zeros. apply these changes and the equivalent changes for the remainder of the function.

[Copilot]

@copilot look at the suggestions I have made which starts with nan and explicitly assigns zeros, instead of starting with zeros and having lack of assignment be the zeros. this is a change for readabi...

Implemented in commit 920f560. I_nm and I_mk now initialize outputs with NaN and explicitly assign zero-valued masked branches (including the dj == d[...] and m0 == inf cases), with equivalent explicit-zero handling applied across the remaining masked branches in both functions.

[rebeccamccabe]

restoring lost comments