# Kryptos K4 and the Berlin World Clock

The strongest result of this investigation is a narrow mathematical exclusion: **a 24-entry hour-number ring that advances between consecutive 24-letter laps cannot explain K4 under the registered substitution rules**. The calculation allows every sector advance and lap boundary, preserves all 97 letters, and was independently verified. The broader question of how the World Clock relates to K4 remains open. No authenticated new plaintext or encryption method was found.

The source work also improves the physical model. The original 1968 design sketch is accessible as an image; selected historical photographs and builder documentation distinguish the city plates, numbered ring, and planetary display. They do not establish a full original inscription sequence or a fixed relation between cardinal directions and clock sectors.

## The clue identifies an object, not an algorithm

The fixed plaintext consists of EAST at positions 22–25, NORTHEAST at 26–34, BERLIN at 64–69, and CLOCK at 70–74. Positions are one-based. These 24 letters are constraints on a proposed solution, not a recipe for deriving the other 73.

The 2014 CLOCK disclosure left the particular Berlin clock ambiguous in public coverage. Sanborn's November 2025 clarification identifies the World Clock at Alexanderplatz as the referent. This directs attention to the Urania/Weltzeituhr rather than treating the Mengenlehreuhr's lamp display as the confirmed object. It does not prove that the hour numbers are a key, that the cities specify an ordering, or that the clock is anything other than a location or object mentioned in the message. The accompanying Egypt, Berlin Wall, and message-delivery clues similarly lack a numerical conversion or complete procedure.[^1]

The [clue audit](clue_audit.md) separates the artist's reported statements from later interpretation. The [prior-method index](prior_methods.md) records related lamp-state, city-list, mechanical-error, geometric and reconstructed-plaintext approaches. Exact coverage of the nonzero lap-index model tested here was not established in the inspected public material; no claim that it has never been tried follows.

## Physical and historical evidence

Three components must be kept separate: fixed city/time-zone plates, the moving numbered hour ring, and the separately rotating planetary display. The official municipal description places the hour ring inside the city-zone cylinder; the builder provides its motion rate and construction details. Confusing the city plates with the moving numbers changes a proposed algorithm before any cryptanalysis begins.[^2]

| Evidence | What was recovered | What it does not establish |
|---|---|---|
| Erich John's 1968 pocket-calendar sketch | An accessible primary design image showing alternative stacked-ring forms | A final sector schedule, city transcription, rotation direction or cipher instruction |
| Dated 1970 Bundesarchiv photograph | A legible portion of the separate hour-number strip | Reliable city lettering or an original full ring |
| Dated 1989 photograph | A pre-restoration visual record | Readable World Clock sectors or orientation |
| Builder description and gallery | Distinct hour-ring and planetary motions | A K4 timestamp, letter-sampling interval or source-derived key |
| Official monument image and record | Wind-rose geometry and site metadata | A usable north reference, labeled sector zero or measured plate azimuth |

The [historical evidence report](historical_evidence.md) and [orientation check](orientation_check.md) record the particular images, observations and uncertainty. The [1968 sketch](https://museumderdinge.de/wp-content/uploads/2026/03/Weltzeituhr_1.jpg) is especially useful as pre-installation evidence, but it is a design study rather than a cipher plan.[^3]

The physical orientation remains a separate unknown. Twenty-four equal sectors imply 15-degree intervals; treating EASTNORTHEAST as the compass bearing ENE would give 67.5 degrees. That arithmetic does not select a plate unless an origin, sector boundary and bearing convention are also established. The mosaic image inspected here supplies none of those registrations. A displayed hour is a moving reading and cannot supply permanent north.

Historical versions also matter. An original count of eighty names and a report of twenty additions in 1997 do not determine the exact 1989 inventory. A complete intervening change log was not found. The modern list cannot silently replace an older list, and the older list cannot be reconstructed just by subtraction.[^4]

A timing-error model needs equally careful dating. The inspected maintainer report confirms post-1990 modernization but does not establish an original 54-minute run/six-minute stop cycle. The popular six-minute premise is also numerically different from a five-percent speed surplus: the latter would imply about 57.14 minutes running and 2.86 stopped. No error-timing key was admitted from these unverified parameters. The [mechanism audit](mechanism_audit.md) traces the discrepancy and the limits of the underlying sources.[^5]

## Testing a ring without guessing city names

Instead of filling an uncertain historical city list, the finite test gives the ring 24 arbitrary numeric entries. It asks whether *any* such ring can satisfy the disclosed letters when the lookup index advances between laps:

```text
slot(i) = (i + s × floor((i + b) / 24)) mod 24
```

Here i runs from 0 through 96. Both s, the sector advance, and b, the lap boundary, run from 0 through 23. The test uses the ordinary and KRYPTOS alphabets, with C = P + K or C = K − P modulo 26. These are declared research assumptions. The actual clock does not specify a letter-per-sector sampling rule.

The model includes the old fixed period-24 case as s = 0. A nonzero advance changes which ring entry is consulted on the next lap; it is not simply a numerical increment added to a repeating key. Duplicate schedules are identified by their entire 97-position equality pattern.

Each known plaintext letter forces a value in one ring slot. If another known letter forces a different value in that slot, the entire case fails. This eliminates all possible rings for that case without enumerating 26^24 keys.

| Measure | Result |
|---|---:|
| Raw advance/boundary schedules | 576 |
| Unique full-97 schedules | 553 |
| Unique alphabet/law/schedule cases | 2,212 |
| Cases with contradictory forced ring values | 2,210 |
| Compatible arbitrary-entry cases | 2 |
| Compatible cases with distinct hour-number entries | **0** |

Both remaining unrestricted cases use 23 constrained ring entries to fit the 24 disclosed letters, leaving one entry free. Consequently, each checks only one redundant equality beyond its freely fitted values. Neither is an independently derived historical key. A single deterministic full-97 witness is retained to test reversibility, not as a proposed solution.

The distinct-hour-number restriction closes the last two cases. One forces slots 0 and 9 to share value 14; the other forces slots 0 and 3 to share value 2. A ring with 24 distinct numerals cannot do that under an injective encoding. This corollary applies to the same registered index law. It does not exclude repeated city initials, additional transposition, nonlinear combinations, or clock-based methods generally.

The [mathematical report](lap_ring_report.md) contains the model, certificates, controls and scope. The [frozen registration](lap_ring_preregistration.json) predates evaluation; the numerical-ring restriction and statistical expectation are explicitly labeled as subsequent analytical corollaries of that unchanged case set.

## Why the two loose fits are weak evidence

Under an explicitly stated null model of independent uniform ciphertext letters, a case using r different ring slots passes the fixed 24-letter constraints with probability 26^(r−24). Summing across the tested unique cases gives an expected 4.369993723275752 compatible cases. Independence between cases is unnecessary for that expectation. The two observed fits provide no excess over it.

This is not a p-value, a posterior probability, or an English-language model. It is a useful check against mistaking a flexible key fitted to known text for a prediction. The hour-number specialization has a stronger result: exact contradictions in every case.

A separate implementation independently reconstructed all cases. All 2,210 first-contradiction certificates were compared field by field with zero mismatches. Each implementation passed 2,304 synthetic full-97 round trips. The expectation and the two remaining injectivity contradictions also agree. [Verification record](verification.json)

## Where further work can add evidence

The next priorities are specific:

1. **A dated plate schedule or a better view of a particular relevant historical sector.** Acquire only the cells required by a declared extraction rule. A complete city inventory is necessary for an all-city rule, but not for every clock hypothesis.
2. **A measured physical registration.** An original drawing or suitable overhead image tying a labeled city plate and the wind rose to true north would make a compass-sector interpretation testable. The present material does not supply that relation.
3. **An instruction that selects the clock's role.** A dated Sanborn note or authenticated disclosure distinguishing place, time, operation and message-delivery context would be more discriminating than a larger collection of guessed keys.

The lap-advanced numeric-ring branch is closed within its stated bounds. The clock's broader role is unresolved. The [reproduction guide](REPRODUCE.md), [source ledgers](coordinator_sources.json), [artifact manifest](manifest.json), and [native usage audit](goal_completion.json) preserve the evidence needed to continue without repeating this work.

## Sources

[^1]: Scientific American, [“CIA Kryptos Puzzle Creator Releases Final Clues”](https://www.scientificamerican.com/article/cia-kryptos-puzzle-creator-releases-final-clues/), November 2025. Contemporary report of Sanborn's World Clock identification and accompanying clues; full access distinctions are in clue_sources.json.
[^2]: Berlin.de, [“World Clock”](https://www.berlin.de/en/attractions-and-sights/3561749-3104052-world-clock.en.html); Kunsch Metallbau, [“Weltzeituhr Berlin Alexanderplatz”](https://kunsch-metallbau.de/referenzen-/weltzeituhr-berlin-alexanderplatz/). Official and builder descriptions, checked September 2026.
[^3]: Werkbundarchiv–Museum der Dinge, [“Gestalten für Berlin”](https://museumderdinge.de/ausstellungen/gestalten-fuer-berlin-design-aus-der-kunsthochschule-berlin-weissensee/), including the photographed 1968 pocket calendar; historical_sources.json supplies the individual photograph records.
[^4]: Weltzeituhr Berlin, [“The Urania Worldtimeclock”](https://www.weltzeituhr-berlin.de/en/urania-worldtimeclock); Berliner Zeitung, [restoration report](https://www.berliner-zeitung.de/article/20-staedtenamen-wurden-bei-der-restaurierung-ergaenzt-frisch-poliert-die-weltzeituhr-dreht-sich-bald-7079), December 12, 1997.
[^5]: Berliner Zeitung, [“Treffpunkt mit Innenleben”](https://www.berliner-zeitung.de/article/treffpunkt-mit-innenleben-7077), January 30, 2007. Report with a clock maintainer; exact original correction timing is not supplied.