fix(identification): return the ordered Cholesky factor in data row order

[thomaspinder]

Summary

Fixes the P1 mislabelling bug #184 and implements #81's QR-based reordering in the same eight lines.

The bug: IdentifiedVAR.shock_matrix labels response rows with var_names (data order) and multiplies by MA coefficients built in data order — for every scheme. Cholesky.identify with a non-identity ordering returned chol(ΠΣΠ') with rows left in ordering order: mislabelled response rows and a mixed-coordinate Φ @ P product in IRF/FEVD/HD. The red-first proof is in the test commit — pre-fix, the structural zero sat at 2.04 in the cell its labels claimed, and reconstructed variances were transposed between variables.

The fix (#81's construction): the permuted factor is now obtained by LQ-via-QR of the row-permuted factor — qr((ΠL)') gives G = R' with GG' = ΠΣΠ' exactly, sign-fixed to a positive diagonal, then un-permuted back to data row order. Σ is never formed, so input conditioning is not squared (verified at cond 1.4e12: max relative error 1.5e-16, structural zeros exactly zero), and one batched decomposition per call is saved. Columns stay in ordering order, matching shock_coords — the contract is now stated at both the producer and consumer ends.

Breaking (pre-v0.1): any analysis using a custom Cholesky(ordering=...) will produce different — previously mislabelled — numbers. Identity orderings are byte-identical (fast path returns L unchanged; the pinned IRF/FEVD/HD regression file is untouched and green). A side effect worth knowing: rank-deficient input now yields a zero diagonal instead of LinAlgError — unreachable through the pipeline (volatility adapters always produce valid factors), accepted deliberately.

Notable blast radius: the monetary-policy tutorial's prose ("orderings A and B give identical policy IRFs") was false in the rendered notebook and becomes true on the next render — no docs edit needed.

Fixes #184
Closes #81

Review

Independently reviewed (verdict: approve, zero findings): the LQ construction re-derived from scratch, column-vs-row sign scaling explicitly discriminated numerically (row-scaling error 5.3 vs column 1.8e-15), a non-involution 4-cycle permutation stress-tested beyond the committed tests, and test discrimination proven by monkeypatching the old implementation back in — all six behavioural tests go red while the pin file stays green under both implementations.

Merge note

Lands cleanly before #183 (feat/143) — the only contact point is the _linalg import line (mechanical rebase; sigma_from_cholesky remains used by fitted.py).

ruff / ty / fast suite: all green (533 passed, 29 deselected).

🤖 Generated with Claude Code

https://claude.ai/code/session_01Egjd7ToFeb9TQqFnfRQZxV

[codecov-commenter]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 94.8%. Comparing base (a378a8f) to head (e2305ac).

Additional details and impacted files

@@          Coverage Diff          @@
##            main    #208   +/-   ##
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  Coverage   94.8%   94.8%           
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  Files         40      40           
  Lines       2416    2419    +3     
  Branches     280     280           
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+ Hits        2292    2295    +3     
  Misses        81      81           
  Partials      43      43           

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