Appendix
This page collects the rigorous derivations, uniqueness proofs, probability tests, and falsification statistics that underpin the K4 solution. None of it is required for the on-site solve. All of it is required to defend the solve to a skeptic.
Table of contents
f-table derivation
The row identity card is fully derived: 24 entries from chain rules, and 2 entries from unique closure constraints.
Result: no free 26-parameter fit remains in the row map.
KRYPTOSA to IDQLNAME uniqueness
The identity signature KR → ID, YP → QL, TOSA → NAME remains one of the strongest internal checks.
TOSA → NAME is unique over 358,800 ordered 4-letter sources in the 26-letter alphabet.
Helper cards (g_X, g_Y, g_Z1, g_Z2, Z2_delta)
Simple version: these cards are not guessed. Each number is forced by arithmetic at real K4 positions.
- Compute R = (C - P) mod 26.
- Remove the gate bit: r = (R - gate) mod 26.
- Read the front letter C and the back letter T at that same grid cell.
- Use the solved f-table for C, then solve g_state(T) = (r - f(C)) mod 26.
- Write the value onto the active pass card and check repeats match.
Note: mod 26 means wrap around the alphabet, after 25 comes 0.
How each card is derived
- g_X (cap pass): Use positions 1-4 only. The back letters are W, X, Z, K. Solving the same equation gives the four forced values below.
- g_Y (Y pass): Use positions 5-35. Solve the equation row by row. Repeated letters must return the same number, which acts like a built-in check.
- g_Z1: Start from Y and apply the solved transform, +3 internal drift and terminal +1 carry.
- g_Z2: Start from Y and apply the solved footer or basis-handoff collapse transform.
- Z2_delta: This is the bridge from Z1 to Z2, Z2_delta = (g_Z2 - g_Z1) mod 26. So g_Z2 = (g_Z1 + Z2_delta) mod 26.
KALPHA: 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 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
Cap-pass values: g_X(W)=21, g_X(X)=4, g_X(Z)=24, g_X(K)=11.
Raw values artifact: helper_cards.txt.
Plain-language artifact: helper_cards_explained.txt.
Z1 binary mask
The Z1 mask is derivable from KALPHA class structure (kpos mod 8 plus three constrained exceptions).
Y master template (rev 16)
Let L = Y_ROW[p-1] for p in {1..31}
gate(p) = 0 iff
L in {A, F, G, N, Q, T, X}
OR (p in {1, 2, 28, 29, 30} AND L in {Y, X, Z, K})
Verification: 31/31 exact match. Null control: 0 of 100,000 random rules pass.
This section also absorbs the older zero-template stub by treating Z1 and Z2 as transforms of Y: +3 phase drift, +1 terminal carry, and six footer-collapse operations.
SUB UMBRA FLOREO proof
The 14-letter phrase aligns to the 14-lane register and survives strict boundary tests.
Random-string probability headline remains 6.82 × 10-8 for the source-letter condition tested in the battery.
AMID GRAY / I MAP J
The internal instruction card is recovered from D = g_Y - g_Z1 in the KYPT/ROSA basis switch.
KYPT / ROSA balance laws
The 26 letters partition into two 13-letter sets with zero overlap and balanced aggregate constraints.
ANDER to KYARX
ANDER (Alexanderplatz witness) maps through the card relation to KYARX, surfacing YAR in the output.
K / C / S to T / Z / K
Terminal-state closure remains consistent under g_Z2 and standard alphabet indexing.
Gate verification
The gate is fixed by position and fully verifiable. The complete 97-bit gate is solved as X cap + Y + Z1 + Z2.
Current solved split is 69 ones and 28 zeros across the 97 active cells.
Falsification batteries
Null tests include block-preserving Monte Carlo controls, rotations, reversed templates, role swaps, and edge-bias probes.
Current summary used in publication: Y-template 31/31 with 0/100k null success, plus multiple structural controls that fail to reproduce the same fit family.