[MatShep]

This answers the higher-rank invariant question in #4735. For ordered idempotents P=XY, Q=ZW with YX=I_r and WZ=I_s, set C=[X Z], D=[Y;W]. The ambient dimension together with the GL_r x GL_s orbit of (ker C, im D, DC) is a complete simultaneous-conjugacy invariant over every field. The written proof explicitly generalizes codyw912's rank-one extension argument, retaining its credit.

The five-file package also gives an exact realizability criterion and minimal active dimension
d_active = rank C + rank D - rank(DC) <= r+s+min(r,s), with sharp examples. It adds only research/idempotent-factor-invariant/ and imports no predecessor executable or external mathematical input.

Recorded local validation on CPython 3.12.10 / Windows:

• Registered ordinary and optimized replays pass, with explicit exceptions active under -O.
• Direct simultaneous conjugation agrees with the factor invariant on 1,355 complete fixed-first-projector cases over F2, F3, F4 and F5, including all 560 binary rank-two partners in dimension four (30 classes). Abstract reconstruction and independent block-rank active-dimension checks agree.
• Sixteen sharp fixtures and eleven semantic controls pass.
• Committed-byte preflight, official export integrity and a Git-free optimized exported replay with empty PATH pass; scientific source bytes remain unchanged.

Reproduce: python -B scripts/research_package.py --package research/idempotent-factor-invariant.

These are written algebraic claims with finite corroboration. No transition probabilities, header cover, physical caller, supplier admission or new kappa follows. Canonical base: b58ad9ef4d4e861b0eaa5dba41742bdc072dca51; selected kappa remains 60403/20000000. The optional source-history preflight rejects the shallow local checkout. Initial sandbox TEMP/export permission failures were resolved without scientific-source changes; external run logs retain those failures. No independent human or formal proof-kernel acceptance is claimed.

Credit codyw912 #4735; Mihir Srivastava #4689; ssafwann #4678/#4671; Nick Petalas's Schur/header lineage; Ryanson Jonathan; lee-phys; Cinna Davis and retained normalizer contributors. #4240 already covers the all-field terminal rank-one law. #4746 studies the distinct initial commuting identity-pivot minimum. Exact citations, scope and open next questions are in the package.

Prepared for MatShep by the authorized automated workflow with substantial OpenAI Codex GPT-6.1 Sol / xhigh assistance; Apache-2.0. GPT-6 Astra / high performed a brief obvious-error and scope check of the unchanged candidate and recorded evidence, plus fresh targeted open/closed PR, Issue and Discussion searches. This was a sanity check, not a deep theorem audit or independent replay. No equivalent contribution was located in that bounded search; universal novelty and exhaustive comment/diff coverage are unclaimed.