Errata (typos).... I wrote:
Anyone with a modicum of math knowledge can look at the NPV formula and
see that if all the cash flows have the same sign, their sum cannot be
zero for valid IRRs greater than -1% (i.e. more positive).

The OP allows for IRRs less than -1%. As I have explained to him many
times, they are invalid IRRs because they flip the sign of the cash flows
for odd-numbered cash-flow periods.

Only then can the sum of the cash flows sum to zero. But in practical
terms, it is incorrect to flip the sign of some discounted cash flows. An
undiscounted inflow cannot turn into a discounted outflow, and vice versa.

But the OP constinues to insist that less-than-negative-1% is valid
because it can be derived algebraically.

"Obviously", I meant -100%, not -1%. I'm sure it is not "obvious" to the
OP, though.