[jwr404]

Question

#4488 assumes ||B|| ≤ 10 p^2, ||C|| ≤ 10 p^2 + ε, and ||B - C|| < 2 ε on one axis, with ε = p^6 exp(-p^2). What follows from splitting B - C through the section polynomial?

Result

For vectors in a normed space,

B - C = (B - P) + (P - C),
||B - C|| ≤ ||B - P|| + ||P - C||,
||P|| ≤ ||B|| + ||P - B||.

If both ||B - P|| and ||C - P|| are strictly below ε, then ||B - C|| < 2 ε. If ||B|| ≤ 10 p^2 and ||P - B|| < ε, then ||P|| < 10 p^2 + ε.

The integer shadow uses errors of size 3: their sum is 6, and 6 < 2 · 4. The polynomial norm with ceiling 10 and error 2 covers the sample value 6. Dropping the error fails, since |13| > 10 + 1.

Claim type: one triangle split. No new κ. Neither section error is proved, and the inverse ceiling is not derived. The tensor sum and the gate below 2^{-C p} stay in #4488 and #4487. The #4100 notes are unchanged.

The locality audit already adds two p^{10} errors. This split is the same shape at ε = p^6 exp(-p^2), together with the polynomial norm under the ceiling 10 p^2.

This branch starts from canonical main. It does not replay #4488.

Test plan

python3.12 -B scripts/research_package.py --package research/gaussian-tensor-bridge

PASS_GAUSSIAN_TENSOR_BRIDGE and outcome=completed on this machine. Maintainer replay has not been run.

Base 2a13eb1827784206202db90e5dc8327cf8a250db, priority revision 7, selection refresh κ = 23/10000.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.