Exact card string — The Line Game | Surprise Is Not Meaning v7.2

AS AH AD AC 2S 2H 2D 3S 3H 3D 3C 4S 4H
4D 4C 5S 5H 5D 5C 6S 6H 6D 6C 7S 7H 7D
7C 8S 8H 8D 8C 9S 9H 9D 9C QD KH 10H KD
QH QS JS 10C JC QC KC 10S JD JH KS 10D 2C

Read left to right, then downward. First card is the top card.
S=spades; H=hearts; D=diamonds; C=clubs.
A=1; J=11; Q=12; K=13. Fixed null I=2C at position 52.
Reference order: A,2,...,10,J,Q,K, each with S,H,D,C, omitting 2C.
At active position i, d_i is the number of unused reference cards
preceding the displayed card. Rank R=sum(d_i*(51-i)!), i=1,...,51.

R = 14134725141735
Scale = 1000000000000
Decoded decimal = 14.134725141735
Unsigned binary = 11001101101011111111010001010110010011100111
Six bytes, big-endian = 0C DA FF 45 64 E7

Swap cards 50 and 51 (KS and 10D), leaving I fixed.
New R = 14134725141734; new decimal = 14.134725141734.

Under the companion's uniform-symbol model, the seven exact component-
weight class sizes are 2, 1, 9, 1, 9, 11, and 18. The complete profile
therefore admits 2!*1!*9!*1!*9!*11!*18! =
67305847895575919030981099520000000 compatible active orders. The swap
preserves that full position-by-position profile while changing the record.

Card faces are display labels, not required numerals. Any one-to-one
relabeling, paired with its matching inverse, recovers the same stored
heights, their order, and their gaps.

The original is the first stored numerical ordinate, at twelve decimal
places. Moving cards changes the record, not an actual zeta zero.
The arrangement is meaningful to War as an order; its interpretation
as a zero ordinate comes from the supplied source and decoding key.

This integer has 44 binary digits. The pack has 51! active orders,
about 219.88 bits of capacity. Neither figure includes every convention.
A direct linear-search decoder uses 51 lookups and at most 1326 card
comparisons, plus factorial arithmetic. This is an algorithmic budget,
not a minimum complexity result or the cost of computing the zero.