An approach that hasn't been mentioned so far is to encapsulate a function and its data-to-be-closed-over in a module. A my variable in a module is absolutely private unless a getter/setter is explicitly defined for it. All code in the module is executed at the point in script compilation at which the module is use-ed. Initialization and checking of any complexity can be done. (In some other languages, this is known as Compile Time Function Evaluation - CTFE - and is a Big Deal.) An extremely simple module with no exportation or OO can be used.

CloseOver.pm:

package CloseOver;

use strict;
use warnings;

my $FIXED_STRING = 'fixed_string';  # could be a constant
my %persistent = (42 => { $FIXED_STRING => 123 });

sub something {

    my ($x,
        ) = @_;

    $persistent{$x}{$FIXED_STRING} = rand;

    }

END {
    for my $k (keys %persistent) {
        print "$k: $persistent{$k}{$FIXED_STRING}\n";
        }
    }

1;
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Output:

Win8 Strawberry 5.8.9.5 (32)  Thu 02/18/2021 21:14:15
C:\@Work\Perl\monks\QM
>perl

use strict;
use warnings;

use CloseOver;

CloseOver::something($_) for 1 .. 3;

^Z
42: 123
1: 0.69775390625
3: 0.8636474609375
2: 0.555877685546875
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Update: Minor edit for clarity.