Righteousness is the opposite of iniquity which cancels righteousness, as exemplified in Ezekiel 33:13.
Accordingly, the righteous is someone whom God approves to live eternally but who still is susceptible to sin, to iniquity, to fall into temptation. The righteous man can sin to lose his salvation. The righteous is not shielded or excused by grace.
Ezekiel 33:13 does not picture the modern misleading grace of God which excuses everything because it is by grace that we are saved. Indeed, we are saved by grace. However, we must qualify for grace. Logically. If we do not qualify for grace, then every single person on earth, both unbelieving believers and unbelievers, will be saved because it is by grace not by our own actions.
Righteousness is being in tune with God 100% in everything because if we commit 1% iniquity, according to Ezekiel 33:13, we die eternally.
Righteousness is another word for justification.
Jesus God still has to stamp our righteousness because there is nothing that we can do that can pay for the default death penalty which is on the head of human being who has been born into this world. Besides the original sin (disagreement), the other iniquities that we will also accumulate, which we must learn to overcome, according to Ezekiel 33:13, must be paid for, and not by any human means.
What is righteousness? Righteousness is another word for justification, justice. Being righteous (righteousness) or just (justification). It is also a gift (Romans 5:17) brought about by the resurrection (Romans 4:25) which justifies us or makes us righteous before God Father.
Hence, we need the righteousness of Jesus (Philippians 3:9) to corroborate our own righteousness (Leviticus 20:7-8, Hebrews 12:14, 1 Peter 1:15-16).
Ezekiel 33:13 ASV
(13) When I say to the righteous, that he shall surely live; if he trust to his righteousness, and commit iniquity, none of his righteous deeds shall be remembered; but in his iniquity that he hath committed, therein shall he die.
1 Corinthians 1:29-31 ASV
(29) that no flesh should glory before God.
(30) But of him are ye in Christ Jesus, who was made unto us wisdom from God, and righteousness and sanctification, and redemption:
(31) that, according as it is written, He that glorieth, let him glory in the Lord [YHWH, Jeremiah 9:24].