[jwr404]

Question

#4482 leaves HALO's tensor error as g p^{10 g} exp(-p^2). Once the inverse ceiling is 10 p^2, does g log p = o(p) still put g (10 p^2)^g exp(-p^2), times a global factor exp(K p), below 2^{-C p} for every fixed C?

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)^g exp(K p - p^2) < 2^{-C p}.

Whenever g log p = o(p), every fixed pair K, C meets that gate for large p. The written exponent 10 is not required for this absorption. The gate is not optional for every g: at g = p^2 the substituted error is greater than 1 for every integer p ≥ 2.

Claim type: one absorption bound. No new κ. The per-axis ceiling 10 p^2 stays a hypothesis. The g-fold shape g (10 p^2)^{g-1} p^6 exp(-p^2) is not proved here, and the #4100 notes are unchanged.

Test plan

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

PASS_GAUSSIAN_TENSOR_ERROR 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.