[petalas]

Summary

homogeneous inverse-generator targets, both positive i/j labels, arbitrary restored dirt
- lower bound n >= 2t; four identical targets could still use eight
+ 2ni+nj >= 4t and 2ni+3nj >= 6t, hence n >= 5t/2
+ four identical targets require ten or twelve; ten forces counts 6/4

The author-written proof charges both boundary ranks using the homogeneous zero-product block and both orientations of PR3570's quadratic/cubic identities. It closes the homogeneous eight-call handoff. The ten-call decision and mixed 2/2 boundary remain open. No physical supplier, new kappa or selected-frontier change is claimed.

Exact used PR3570/3536/3558 proof bytes, origins, attribution and source closure are retained. Canonical investigation.

Evidence

• Before: PR3570's count relaxation admitted four homogeneous targets at eight balanced-label calls.
  After: the new inequalities reject eight and force six target-positive and four opposite calls at ten. Twelve is attained by direct words.
• At actual commit 4395a9cf3edf9ba36918c48477f36dcde110f45a, CPython 3.12 registered science and optimized fault controls pass, as does scripts/check_research_commit.py --package research/q8-homogeneous-mixed-floor --check-source-history for all 15 files. Scoped Ruff E/F/B/UP passes on the exact committed scientific bytes.
• Scientific replay checks 390 generic words on 9,360 independent dirty columns, 1,593 triangular rank fixtures, 186 zero-product pairs and 36 direct endpoints with 378 omissions. Exact normal/optimized scientific and optimized control commands also pass in an owned Git-free committed-byte export.
• Reproduce with python3.12 -Werror -B scripts/research_package.py --package research/q8-homogeneous-mixed-floor. PROOF.md, RESULT.json and VALIDATION.md separate the universal proof, finite evidence and unrun obligations. Independent human/formal proof-kernel acceptance, the selected cold replay, Lean, Docker and hosted completion are unclaimed.

Merge Danger

Door: two-way

Blast Radius: isolated

One new opt-in research package. No shared engine, selected record, frozen predecessor or supplier changes.