SolveKryptos K4 Helper Cards, Plain-Language Derivation
Date: 2026-05-28

Goal
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Show how helper cards are computed, not guessed.

Cards
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g_X, g_Y, g_Z1, g_Z2, Z2_delta

Alphabet order (KALPHA)
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K R Y P T O S A B C D E F G H I J L M N Q U V W X Z

Core rule (used everywhere)
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1) Compute total shift:      R = (C - P) mod 26
2) Remove gate bit:          r = (R - gate) mod 26
3) Solve helper value:       g_state(T) = (r - f(C)) mod 26

Where:
- C is the front ciphertext letter value.
- P is the solved plaintext letter value.
- T is the back-side helper letter at the same row and column.
- f(C) is the row-identity value from the solved f-table.
- state is X, Y, Z1, or Z2 depending on position range.

How each card is built
----------------------
g_X (positions 1-4):
- Use the cap letters on the back (W, X, Z, K).
- Apply the core rule at positions 1, 2, 3, and 4.
- Forced result:
  g_X(W)=21, g_X(X)=4, g_X(Z)=24, g_X(K)=11

g_Y (positions 5-35):
- Apply the core rule at each Y-pass position.
- Fill values for helper letters in KALPHA order.
- Repeated letters must return the same value, which is the check.

g_Z1:
- Derived from solved Y card using +3 internal drift and terminal +1 carry.

g_Z2:
- Derived from solved Y card using footer or basis-handoff collapse.
- Equivalent bridge form:
  g_Z2 = (g_Z1 + Z2_delta) mod 26

Z2_delta:
- Difference card:
  Z2_delta = (g_Z2 - g_Z1) mod 26

Solved card values
------------------
g_Y:
13 20 25 21 1 19 1 6 22 10 23 20 8 10 23 14 21 16 25 25 13 20 6 8 2 14

g_Z1:
13 4 13 13 24 11 14 17 13 9 19 13 13 17 24 4 13 24 13 5 13 14 25 6 13 24

g_Z2:
19 17 16 13 1 18 10 6 18 25 20 21 13 24 23 8 24 6 24 12 25 19 7 7 25 24

Z2_delta:
6 13 3 0 3 7 22 15 5 16 1 8 0 7 25 4 11 8 11 7 12 5 8 1 12 0

Sanity checks
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1) Every one of 97 positions satisfies:
   R = (C - P) mod 26
   R = (r + gate) mod 26

2) X pass cap is exact:
   OBKR -> THEC under the solved g_X values.

3) Z2 bridge is exact:
   g_Z2 = (g_Z1 + Z2_delta) mod 26 across all 26 KALPHA letters.