[jwr404]

Question

#4487 bounds the scalar monomial 2g (10 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(-p^2) and leaves open whether the tensor difference is at most that monomial. What does one axis of the product contribute?

Result

Let ε = p^6 exp(-p^2). If every axis satisfies ||B_i|| ≤ 10 p^2, ||C_i|| ≤ 10 p^2 + ε and ||B_i - C_i|| < 2 ε, and if ||X ⊗ Y|| = ||X|| ||Y||, then

⊗_i B_i - ⊗_i C_i = sum_k T_k,

with T_k changing only axis k. Each slice has norm strictly less than 2 (10 p^2 + ε)^{g-1} ε, and the sum of the slices is strictly less than 2g (10 p^2 + ε)^{g-1} ε.

The factor 2 is the sum of two section errors on the active axis. The scalar check is the one-dimensional case: for caps 10, 11 and 2, the three slices add to the product difference, and a g = 1 difference of 3 exceeds the cap.

Claim type: one telescoping bound. No new κ. The three axis inequalities stay hypotheses. The comparison with 2^{-C p} is #4487 and is not re-proved. The #4100 notes are unchanged.

The locality audit already records this shape with p^{10} in place of 10 p^2 and p^6. This branch starts from canonical main. It does not replay #4487.

Test plan

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

PASS_GAUSSIAN_TENSOR_AXIS 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.