Folks, I know you've been doing quite some work around FastCholesky.jl, but the constant warnings I experience during inference recently have become a bit annoying. So, I decided to look into the code again, using Gemini 3.0.

It flagged something that looks like a potential bug in the fallback logic. I know AI code analysis can often be "slop," but I ran the reproduction script it suggested, and the numbers actually don't match up.

The potential issue:
Gemini claims that because the initial cholesky!(A) is in-place, it modifies the matrix A (partially factorizing it) even if it fails. When the code then enters the fallback block (if fallback_gmw81), it passes this partially modified/corrupted A to PositiveFactorizations, rather than the original input.

Here is the reproduction script:
I compared the result of FastCholesky against running PositiveFactorizations directly on a clean matrix.

using LinearAlgebra
using FastCholesky
using PositiveFactorizations
using Test

# [ 1  2 ] -> Determinant is -3 (Indefinite).
# Standard Cholesky fails; PositiveFactorizations modifies diagonal to handle it.
A_orig = [1.0 2.0; 2.0 1.0] 

# 1. Get the "Correct" Answer (GMW81 on clean data)
# We copy A_orig to ensure it's clean.
C_correct = cholesky(PositiveFactorizations.Positive, Hermitian(copy(A_orig)))

# 2. Run FastCholesky! 
# If the hypothesis is true, this will try standard Cholesky -> Fail (modifying A) -> Retry on dirty A
A_fast = copy(A_orig)
C_fast = fastcholesky!(A_fast)

println("--- Correct L Factor (GMW81 on clean data) ---")
display(C_correct.L)

println("\n--- FastCholesky L Factor (GMW81 on potentially dirty data) ---")
display(C_fast.L)

println("\n--- Do they match? ---")
println(C_correct.L ≈ C_fast.L) 

Output:

--- Correct L Factor (GMW81 on clean data) ---
2×2 LowerTriangular{Float64, Matrix{Float64}}:
 1.0   ⋅ 
 2.0  1.73205

--- FastCholesky L Factor (GMW81 on potentially dirty data) ---
2×2 LowerTriangular{Float64, Matrix{Float64}}:
 1.0   ⋅ 
 2.0  2.64575

--- Do they match? ---
false

Is this a real bug regarding the state of A being carried over to the fallback, or am I missing something about how cholesky! handles failures?

Thanks! (sorry if slop)