[jwr404]

Summary

On one axis,

B - C = (B - P) + (P - C).

If both pieces are strictly below ε, then ||B - C|| < 2 ε. If ||B|| ≤ 9 p^2 and ||P - B|| < ε, then ||P|| < 9 p^2 + ε. The integer shadow uses ceiling 9 and error 2: |10| ≤ 9 + 2 while |10| > 9, and |12| > 9 + 1.

The ceiling ||B|| ≤ 9 p^2 and each section error stay hypotheses. The factor 2 is the sum of the two errors. HALO's 2g rectangles stay a volume partition. No new κ. The selected record remains 23/10000.

This answers the open question of #4508 for the difference and the approximant norm. It does not replay #4508. #4490 is the same split at ceiling 10 p^2. Which hypothesis a core row past ceil(p^{48}) inherits is left open.

Test plan

• python3.12 -B scripts/research_package.py --package research/gaussian-tensor-nine-bridge
• Prints PASS_GAUSSIAN_TENSOR_NINE_BRIDGE and outcome=completed
• Maintainer replay was not run

Base f9f85c1e4fa1e955ff39f10fca447e3e2a4e4adc, priority revision 7. Upstream fetched at 2026-10-22T02:53:38Z was that commit, and the pre-submit fetch was unchanged. κ remains 23/10000.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.