[jwr404]

Question

#4481 leaves HALO's core side L = 2^{ceil(400 log2 p)} in place. The volume bound has two sparse powers. Both are negative for group exponents ε, β < 1 when A = p^{48} and L ≥ p^{400}. What is the smallest integer C that still does this when A = p^{26}?

Result

With A = p^{26} and L ≥ p^C, the powers are ε + β + 78 - C and ε + 28 - C. Both are negative for every ε < 1 and β < 1 precisely when C ≥ 80. At ε = β = 3/4, the exponent 79 gives 1/2. The second power is negative as soon as C ≥ 29, so the first power is the binding one.

The same 80 gives p^{80} > 4 p^{26} for every integer p ≥ 2, and g A/L has exponent -53 when g ≤ p. The written 400 still works and is larger. A uniform slack of 1 below zero needs 81; 80 is the smallest integer that makes both powers strictly negative.

Claim type: exponent comparison for the sparse volume. No new κ. The agreement distance p^{26} stays a hypothesis. The tensor error g p^{10 g} exp(-p^2) is not retuned, and the #4100 notes are unchanged.

Test plan

python3.12 -B scripts/research_package.py --package research/gaussian-core-side

PASS_GAUSSIAN_CORE_SIDE 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.