Kryptos K4 Appendix

This page collects the rigorous derivations, uniqueness tests, probability tests, and falsification statistics behind the K4 reconstruction, graded honestly between what public data forces and what is back-solved. None of it is required for the on-site reading. All of it is required to defend the reconstruction to a skeptic.

Full derivations Uniqueness proofs Falsification batteries Expert reference

Table of contents

  1. f-table derivation
  2. KRYPTOSA to IDQLNAME uniqueness
  3. Helper cards
  4. Z1 binary mask
  5. Y master template
  6. Z2 footer basis handoff
  7. SUB UMBRA FLOREO proof
  8. AMID GRAY / I MAP J
  9. KYPT / ROSA balance laws
  10. ANDER to KYARX
  11. K / C / S to T / Z / K (superseded)
  12. The rose-East rule: full evidence package
  13. Anchor frontier and the FPZ bridge
  14. Gate verification
  15. Falsification batteries
  16. Open problems

f-table derivation

The row identity card is partly forward-derived and partly reconstruction. Confirmed forward: the tail rule f = (4·kpos + 1) mod 26 at kpos ≥ 19 (N, Q, U, V, W, X, Z, 7 of 7, and none below the boundary), f(O) = 0 from the UNDERGRUUND misspelling, f(L) from the constant +7 misspelling signature, f(P) = 11 from the June anchor-frontier sweep (forced by the R-of-BERLIN anchor at position 66 with the gate and the derived Z1 scaffold; cross-checked through position 73), and f(T) = 13 from the July 1 transitive-closure sweep (position 67 forces the composite Z2 helper cell g_Z2(Z) = 24 from public data, and position 68 reads that same cell to close f(T)). That is 11 of 26 values from public data; the remaining entries are back-solved from the plaintext, and the A/E/I misspelling family closes only relationally.

The compass-seed attribution for the low-kpos entries is interpretive pending a blind test against the unseen letters. Per-entry grades are in the claim status summary in the downloads.

KRYPTOSA to IDQLNAME uniqueness

The identity signature KR → ID, YP → QL, TOSA → NAME remains a striking internal pattern, read from the back-solved table; status: working theory.

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, and forced means determined given the recovered and anchored plaintext: the recipe below uses P at step 1. That is internal consistency, not forward derivation, which is why the cards are labeled back-solved throughout the site.

  1. Compute R = (C - P) mod 26.
  2. Remove the gate bit: r = (R - gate) mod 26.
  3. Read the front letter C and the back letter T at that same grid cell.
  4. Use the solved f-table for C, then solve g_state(T) = (r - f(C)) mod 26.
  5. 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

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.

Provenance: the card values above are back-solved [F], the g_Z1 exception readings and Z2 transport families are witness-claimed [W], and the three hard Z2_delta cells (C, K, S) carry the A-class-candidate rose rule (below), with the K cell now publicly derived [D] (delta(K) = 6, July 2026). g_Y remains the largest underived object, though six of its 26 cells (I, L, M, N, V, Q) are public-forced by the June anchor-frontier sweep.

Raw values artifact: helper_cards.txt.

Public status artifact: claim_status_summary.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. Two caveats carried with it: the register is read from the back-solved cards, and the headline reflects one tested condition without accounting for how many card operations and bases were searched (the look-elsewhere effect). Status: working-theory authorial-signature candidate.

AMID GRAY / I MAP J

The internal instruction card is read from D = g_Y - g_Z1 in the KYPT/ROSA basis switch. It rides on the back-solved cards and is not uniquely forced; status: interpretive working theory.

KYPT / ROSA balance laws

The 26 letters partition into two 13-letter sets with zero overlap and balanced aggregate constraints. Read from the back-solved cards; status: working theory.

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

Superseded, June 2026. The earlier terminal-state reading of the three hard cells, and the May 2026 K = 6 balance reading, are retained for history only. Those cells now carry the rose-East rule (next section).

The rose-East rule: full evidence package (June 2026)

The publication statement: "The three Z2_delta hard residues are exactly the East ray of the standard alphabet wrapped clockwise around the 8-wind compass rose, with A at North. Their values are generated by ring count: delta = 16 × ring mod 26, giving C→16, K→6, S→22. No other wind admits any ring-multiple rule."

The required caveat, never separated from the sentence above: two of sixteen wrap conventions place the class on East, the canonical one and its point-mirror, which is the theoretical minimum for any such claim; the engraved rose fixes the clockwise chirality, and North is the bearing origin the sculpture's own EAST NORTHEAST language uses. Conservative chance of the East ring-rule under value shuffling: 0.73% (0.108% granting East's a-priori role in the gate). Status: A-class candidate for these three cells only; the wrap procedure and the 16-per-ring step are precommitted for archive/K5 confirmation.

Update (July 2026): one cell now publicly derived. The July 1 transitive-closure sweep reaches the K cell without the fitted card: position 69 (N of BERLIN) with the derived f(T) = 13 forces g_Z2(K) = 19, and subtracting the publicly confirmed scaffold g_Z1(K) = 13 gives delta(K) = 6 [D], exactly the rule's ring-2 value (16 × 2 mod 26). delta(C) = 16 and delta(S) = 22 now stand as the rule's genuine out-of-sample predictions for archive/K5. The rule's overall grade is unchanged (A-class candidate pending an external witness); the K value itself is derived.

The evidence, item by item

The test it survived

A competing physical reading, the 30-character header-row count hypothesis, makes a natural fourth-term prediction: its next cell (27) is A, which would force delta(A) = 12. The actual value is 15, so it fails. The rose form self-terminates at exactly three cells and survives the same test. We publish this because a claim that has survived a falsification attempt is worth more than one that was never exposed to one.

Residuals, precommitted for archive/K5: the wrap procedure itself, and the step of 16. A declared witness scan for the 16 found clue-letter hits at chance level (non-promotable); the geometric readings (16 = two rose turns per ring; 16 = the wind count of the granularity that names EAST NORTHEAST) remain interpretive.

Gate verification

The gate is fixed by position under the declared rule set, plaintext-independent by construction. The complete 97-bit gate is reproduced as X cap + Y + Z1 + Z2.

The split is 69 ones and 28 zeros across the 97 active cells; the ray reading of the same map (69 E, 18 NE, 10 S) is a physical conjecture awaiting the blind on-site audit (protocol and field sheet in the downloads).

Falsification batteries

Null tests include block-preserving Monte Carlo controls, rotations, reversed templates, role swaps, and edge-bias probes. The June 2026 batteries added null-calibrated tests of every public-feature generator for the full Z2 card (all at or below chance), local cell rules, transport families, and exception-value compression.

Current summary used in publication: Y-template 31/31 with 0/100k null success, the rose rule's survived fourth-term test (above), and multiple structural controls that fail to reproduce the same fit family. The public download bundle includes negative_tests_summary.txt for reader-facing controls.

English-closure audit (July 1, 2026). A four-test pre-registered audit asked whether the plaintext could be recovered from the public spine plus an external English n-gram prior. It cannot: joint closure recovers at most 7 of 60 blocked letters under two model designs. The underdetermination result therefore extends to English-augmented recovery: language priors cannot substitute for an external witness, which is why the archive, K5, and physical audits are the precommitted closure path. A local-uniqueness variant (full-context conditioning, X calibrated to the K1–K3 separator rate) placed 8 of 13 multi-position components at rank 1, below the pre-registered bar; it is descriptive only and is not published as "locally unique."

Anchor frontier and the FPZ bridge (June 10 – July 1, 2026)

Every artist-confirmed anchor position whose cipher letter has a derived f value yields a single-unknown public equation: anchor plaintext, gate, and derived components in; one g-object out. Sweeping all 24 anchors public-forces six g_Y cells (I = 14, L = 16, M = 25, N = 25, V = 6, Q = 13), confirms the Z1 scaffold at K, forces g_Z2(Y) = 16 (promoting delta(Y) = 3 to derived) and g_Z2(Z) = 24 at value level, and closes f(P) = 11 outright. The July 1 transitive-closure sweep runs the anchor equations to graded fixpoint: it closes f(T) = 13 at pure D-chain grade (positions 67→68), forces g_Z2(K) = 19 and g_Z2(T) = 1 [D], and derives delta(K) = 6 [D], the first rose-East cell supported by public derivation rather than the fitted card.

Of the original eight plaintext-free relational invariants (fitted f cancels between anchor pairs), four are now absorbed into absolute derived values (g_Y(Q) − g_Z1(Y) = 0, g_Y(Q) − g_Z2(T) = 12, g_Z1(Y) − g_Z2(T) = 12, g_Z2(Z) − g_Z2(K) = 5, all verified exactly, an internal audit of the anchor web). Four live invariants remain as hard constraints any candidate chart must satisfy: g_Y(H) − g_Z2(P) = 10, g_Y(J) − g_Y(U) = 1, g_Y(X) − g_Z2(O) = 10, and g_Y(Z) − g_Y(K) = 1.

The conditional ladder simplifies once f(T) is unconditional [D]: the rose step now grants zero positions. The remaining rungs are the FPZ bridge [B] → f(F) = 3 (positions 62, 81), and the witnessed exception families [W] → f(K) = 8 (six positions) and f(Y) = 16; f(M) = 19 stays parked on delta(R). Each closure carries the grade of its weakest input, never higher. The transitive-closure baseline of back-solved positions is 45 (July 1: position 38 flips to a fully public decode, 36 and 51 to [W], 81 to [B], all via f(T) = 13); under the full ladder that count falls to 37.

The FPZ bridge. The zero-delta class {F, P, Z} is exactly the ring-aligned shadow of the rose-East class {C, K, S}: C(2)→F(5), K(10)→P(15), S(18)→Z(25). Caveat carried with the claim: both sets are arithmetic progressions, so the (3, 5, 7) step pattern is automatic once the ordered sets are fixed; the evidence is the exact identity and ring alignment, not the rule's form. Null hierarchy: 0.0023 under a principled bridge family (the citable figure); 0.12 under a free affine bridge (non-promotable form); progression structure alone is common (null 0.17). Grade: B-class candidate, promotable only by archive or K5 evidence of zero-stencil handling.

Parked: delta(R) = 13. Internally consistent and it would close f(M) = 19 at the CLOCK anchor, but it is not in the witnessed stencil families and is basis-dependent: the helper-R cells carry physical footer letter C, so under footer-active labeling the cell names C rather than R. It stays an unresolved dependency with written closure conditions, not a witnessed cell.

Open problems

The live register, in priority order. Each item has a written closure condition.

  1. The g_Y derivation. The largest open object in the system: 18 of the Y pass's 31 positions rest on values that exist only as an authored worksheet. Six of the card's 26 cells are now public-forced (anchor frontier) and serve as out-of-sample tests for any proposed generator. Closure: an archive document, K5, or a verified physical feature supplying an external witness.
  2. The step-16 witness. The rose rule's multiplier has geometric readings but no promotable witness. Closure: archive/K5 confirmation of the 16-per-ring step.
  3. The wrap procedure. That an alphabet wrap around the rose was the authored procedure at all is precommitted, not established. Closure: archive/K5.
  4. The blind gate/ray field read. The pre-registered two-stage protocol and the 97-row field sheet are in the downloads; the decisive minimum subset is the lane-10 spine (7 cells, all must read East) followed by the 10 S-cells. Awaits site access.
  5. The delta(R) / f(M) dependency. Parked, ungraded. Promotion requires an independent source for g_Z2(R) = 17 or f(M) = 19, a physical witness forcing the repeated-Z labeling at this cell over the footer-active labeling, or the archive chart. Falsified by an archive chart showing a different value at this cell. Details in the anchor frontier section.
  6. Scoring the precommitments. The timestamped archive/K5 predictions, each with a written pass/fail condition, get scored publicly when the material becomes available. If they fail, this site will say so.

Note on scope: the three hard Z2_delta cells are not on this list. They carry the A-class-candidate rose rule; what remains open about them is the witness for its step and procedure, items 2 and 3 above.

Continue

How It Works → Core mechanism, then linked derivations. Verify It Yourself → Position checks, grids, and download bundle. Resources → Changelog, references, and downloadable artifacts.