German USt-IdNr

What it covers

Returns a valid German USt-IdNr (VAT identification number) with a correct ISO 7064 MOD 11,10 check digit. Use `format=national` for the bare nine-digit core. Set `invalid=true` for a number with a deliberately wrong check digit.

How it’s calculated

The German USt-IdNr (EU VAT number) appends a single ISO/IEC 7064 MOD 11,10 check digit computed by a pure recursive recurrence over the first eight payload digits.

Start product = 10.
d=1: sum=(1+10) mod 10 = 1; sum!=0; product=(1*2) mod 11 = 2
d=3: sum=(3+2) mod 10 = 5; product=(5*2) mod 11 = 10
d=5: sum=(5+10) mod 10 = 5; product=(5*2) mod 11 = 10
d=7: sum=(7+10) mod 10 = 7; product=(7*2) mod 11 = 14 mod 11 = 3
d=9: sum=(9+3) mod 10 = 2; product=(2*2) mod 11 = 4
d=2: sum=(2+4) mod 10 = 6; product=(6*2) mod 11 = 12 mod 11 = 1
d=4: sum=(4+1) mod 10 = 5; product=(5*2) mod 11 = 10
d=6: sum=(6+10) mod 10 = 6; product=(6*2) mod 11 = 12 mod 11 = 1
checkDigit = (11 - 1) mod 10 = 10 mod 10 = 0

Edge cases

• A check digit of 0 is valid and occurs (the `zero-check-digit` edge class generates cores whose ninth digit is 0), so naive filters assuming a non-zero trailing digit will wrongly reject valid numbers.
• The first payload digit is always 1-9 (never 0); the smallest legal leading digit is 1 (the `leading-small` edge class exercises exactly this).

Gotchas

• This is ISO 7064 MOD 11,10 (a pure recursive recurrence), NOT a weighted-sum mod-11 scheme — do not confuse it with the mod11.ts primitive. The recurrence maps an intermediate sum of 0 to 10 before doubling mod 11.
• Because the system always produces a single decimal digit, there is never a 'residue 10 / redraw' case as with plain mod-11 identifiers.
• The generator's value is synthetic: the check digit is mathematically valid but the number is not guaranteed to belong to a registered business.