story identification - Future society with life span depending upon achievements

I seek some help identifying a short book I read in my childbook. It was a science fiction book about a society where due to scarcity of resources, you were allowed a certain life span depending on your achievements and contributions to society. If you exceed the amount before doing something that causes you to get promoted to a higher life span, you will be taken in the night by some sort of police and vanish.

For extraordinary achievements the highest status you could achieve was immortality (I seem to dimly remember the usage of the word amaranth for those people).

The protagonist of the novel was someone who had once been granted immortaility but then had it stripped away from him again due to some misdeed or mistake. Now he tries to get that status again (maybe under a fake identity?), but one of the immortal people is suspicious of him so he murders her. She gets revived later, because that's part of the immortality package.

It ends with him being apprehended, but society decides to invest heavily into space again so that hopefully in the future there is space for everyone to be immortal. The book ends with the protagonists being sent of an explorer into space alone.

Answer

That sounds like Jack Vance's To Live Forever.

• Medical technology advanced to a point where they can lengthen a humans livespan considerably


• There are five different categories of people, where each of the first four categories grants another 20 years of life.


• The ultimate category is called "Amaranth", and it offers true immortality.


• The hero (the Grayven Warlock) was an Amaranth, but has become a fugitive after a feud with another Amaranth resulting in the latters death.

Excerpt:

Eventually the Fair-Play Act was drafted, and won a grudging approval. In essence, the system rewarded public service with years of extended life. Five phyle, or levels of achievement, were stipulated: Base, Second, Third, Fourth, Fifth. Base became known as Brood; Second, the Wedge; Third, less frequently, Arrant; and Fourth, Verge. When the original Grand-Union group organized the Amaranth Society, Fifth became Amaranth.

The Fair-Play Act carefully defined the conditions of advance. A child was born without phyle identification. At any time after the age of sixteen he might register in the Brood, thus submitting to the provisions of the Fair-Play Act.

If he chose not to register, he suffered no penalty and lived a natural life without benefit of the Grand-Union treatments, to an average age of 82. These persons were the "glarks," and commanded only small social status.

The Fair-Play Act established the life span of the Brood equal to the average life span of a non-participator, roughly 82 years. Attaining Wedge, a man underwent the Grand-Union process halting bodily degeneration, and was allowed an added ten years of life. Reaching Third, he won sixteen more years; Verge, another twenty years. Breaking through into Amaranth brought the ultimate reward.