A Spin-Sector Obstruction to ((7,3,3))₂ Quantum Codes

A Spin-Sector Obstruction to (((7,3,3))_2) Quantum Codes This record contains a focused research manuscript and an extended research version investigating structural obstructions to the existence of binary quantum error-correcting codes with parameters (((7,3,3))_2). The primary manuscript establishes a sector-specific obstruction for codes contained entirely in the total-spin (j=5/2) sector. The proof uses pair-root projectors, weighted quadrupole identities, compression arguments, Cauchy interlacing, and the total-spin Casimir identity. Additional results exclude the sector-pure (j=7/2) and (j=1/2) cases using collective-pair correlation bounds. The record contains two complementary manuscript versions: - Primary manuscript (10 pages): A focused presentation of the principal sector-obstruction results and their proofs.- Extended research manuscript (23 pages): A fuller treatment containing additional derivations, correlation bounds, moment analyses, and supporting mathematical investigations. The deposit also includes the corresponding LaTeX source files and Python scripts for exact-arithmetic verification of selected operator identities and quadrupole-coefficient uniqueness calculations. Scope of the results: These results exclude the specified sector-pure cases. They do not resolve the unrestricted existence problem for (((7,3,3))_2) quantum codes. The sector-pure (j=3/2) case and cases involving mixed total-spin sectors remain unresolved in this work. The verification scripts support reproducibility of the stated algebraic calculations; they should not be interpreted as an independent computer-assisted proof of every theorem in the manuscripts. The primary manuscript is intended as the concise research presentation, while the extended manuscript preserves additional mathematical development and supporting analysis. Both are included to facilitate scholarly scrutiny, reproducibility, and accurate citation. AI assistance : This work was produced with substantial assistance from an AI system (ChatGPT, Claude, Gemini). The proofs have not yet been independently checked by a human specialist. Corrections are welcome.