According to https://studybible.info/strongs/G2596 "kata" means "according to"/ "against". While in English I can't think of one single word for this as it's almost an oxymoron, I believe a suitable English translation for "kata" could be "accordingly against."
It was the Lord Jesus himself in Matthew 6's sermon to the disciples on the mount, where He said "do not worry about [...] what you will wear." Lord Jesus' Eternal Father gives us Jesus as our clothing. https://www.biblestudytools.com/matthew/passage/?q=matthew+6:25-34
So to understand Paul's meaning of 1 Cor 11:4-5, we need to understand that Paul is talking about Genesis 3's Adam and Eve, the seat of Abraham's faith (in his head), and again, what it means to be fed by and Clothed in Christ.
Paul’s Torah-related verses should comply with Christ’s own teachings.
Jesus taught the Apostles, as God commanded Adam to not eat from the tree of knowledge of good and evil.
If Adam’s head has ears, then his head has memories.
The Apostles taught the church, as Adam taught Eve. When answering the snake, Eve surmised to neither eat nor touch the tree.
The snake hid that truth-claim with its question and Eve’s answer was thus against that mysterious hiddenness.
Remember in Greek, the word for truth is "ἀλήθεια" https://en.wikipedia.org/wiki/Aletheia which is a compound "not concealment" or "not forgetfulness" , or for the truth of light "not oblivion."
in 1 Corinthians 11:4
So when Adam sinned, and fell, he acted/prophesied against that which his head has.
Adam prophesied accordingly-against his head's having.
in 1 Corinthians 11:5
When Eve sinned and fell, though she answered, thus accordingly-against-hiddenness (katakalyptos) , she did/prophesied not + her own answer, (akatakalyptos).
Looking at the “Greek Concordance: κατὰ (kata),” https://biblehub.com/greek/kata_2596.htm , the Example of Luke 8:33, where Kata is used as a preposition regarding the topology of the hill-side. Well, running "accordingly against" the hillside is a much better topological description of running according to the land's contours, since if you went straight down, that would literally require a water-well as a passage.
Similarly with Acts 9:31, were the church parishes in the regions of "Judea" and "Samaria," accordingly-against the actual regional-definitions? since they were building the Kingdom of God, and are worshipping the resurrected Lord Jesus whom the state crucified. You couldn't contradict "kata" with "across/over" as some intrinsic partner with the region: The church of the first century wasn't peaceably & tolerably a tax-exempt part of the cultural state, as it is today.
Greco-Roman culture was male-dominated, and pertinently, held regional head-dress customs for married women and worshippers.
Jerome translated the Latin Vulgate from Greek, in the midst of this cultural dread.
If we look carefully, all translations of 1 Corinthians 11:4 which follow after Jerome’s Latin Vulgate translation uses a verb “to veil” in “velato capite” (to veil the head.) But the Greek which is preserved from Paul’s own words doesn’t use the verb “to veil”. Checking Matthew 27:51 the Latin uses the noun word “velum” for veil. And the Greek uses “katapetasma” where "petasma" means "spreading out"
anyway, i think this was the original meaning of Paul, because his discourse on shaved heads in the subsequent verses, has to do with Aristotelian metaphysics' concept of "identity" and in today's philosophical terminology, the 'sortal-concept', where more similarities and less distinctions mean a closer shared identity.
and the verse above (1 Corinthians 11:3) is also about the deity-of-prayers not being confused with a sortal-concept of the person-making-the-prayers. A praying woman can't confuse 'him' with herself, and a praying man can't confuse "Jesus of Nazareth' with himself. Nor can Lord Jesus ignore the full "God" (the good God, described by the doctrine of Trinity.)
check Matthew 9:29 excerpt "according to your faith, be it done unto you", by this version of kata "accordingly against your faith, be it done unto you".
Lord Jesus was touching their eyes when He said that.
Peace of Christ.