[jwr404]

Question

#4486 absorbs g (10 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(-p^2) when p ≥ max(64, 4K + 2C) and 16 g ceil(log2 p) ≤ p, with exponent at most -C p - 4. Does the extra factor 2, as in 2g, stay below 2^{-C p} under the same gate?

Result

Let p ≥ 2, g ≥ 1, K ≥ 0 and C ≥ 0 be integers, and let L = ceil(log2 p). If p ≥ max(64, 4K + 2C) and 16 g L ≤ p, then

2 g (10 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(K p - p^2) < 2^{-C p}.

The factor 2 is one bit. The exponent is at most -C p - 3. The section error is still below 1/2 for p ≥ 64, so the approximant base remains below 16 p^2. Whenever g log p = o(p), every fixed pair K, C meets the gate for large p. At g = p^2 the error is greater than 1 for every integer p ≥ 2.

Claim type: one absorption bound. No new κ. This does not identify HALO's 2g rectangles with the tensor difference. The ceiling 10 p^2 and the section error stay hypotheses, and the #4100 notes are unchanged.

The locality audit already records that a factor 2 on g p^{10 g} exp(-p^2) is absorbed by 2^{-C p}. The explicit gate and the slack 3 are for the approximant monomial.

This branch starts from canonical main. It does not replay #4486.

Test plan

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

PASS_GAUSSIAN_TENSOR_FACES 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.