[jwr404]

Question

#4485 absorbs g (10 p^2)^{g-1} p^6 exp(-p^2) when p ≥ max(64, 4K + 2C) and 16 g ceil(log2 p) ≤ p. If the g - 1 factors are approximants, the base is 10 p^2 + p^6 exp(-p^2). Does that monomial 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

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

For every integer p ≥ 64 the section error is below 1/2, so 10 p^2 + p^6 exp(-p^2) < 10 p^2 + 1 < 16 p^2. The base stays strictly below the power of two already spent on (10 p^2)^{g-1}, and the exponent is still at most -C p - 4. Whenever g log p = o(p), every fixed pair K, C meets the gate for large p. At g = p^2 the approximant error is greater than 1 for every integer p ≥ 2.

Claim type: one absorption bound. No new κ. The ceiling 10 p^2 and the section error p^6 exp(-p^2) stay hypotheses. The factor 2g is not proved here, and the #4100 notes are unchanged.

This branch starts from canonical main. It does not replay #4485.

Test plan

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

PASS_GAUSSIAN_TENSOR_APPROXIMANT 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.