A repeat-distance construction in Z13 that yields HELLO

I want to present a structural observation in Z13 that produces HELLO.

I am not claiming that this proves a unique solution to Z13. Z13 is too short for that kind of claim. What I am claiming is narrower: there is a simple and reproducible construction based on the repeated symbols and their positions which, under a small set of explicit assumptions, yields HELLO.

The construction has also been tested against alternative rule choices and randomized Z13-like layouts. Those tests do not prove intent, but they indicate that HELLO is not simply an inevitable product of a highly flexible procedure.

The method does not require a proposed name, an anagram, a dictionary search, Grandpré, Gromark, Poe, Marsh, or any other proposed plaintext.

1. Start only with the symbol pattern

The 13 positions of Z13 can be represented as:

A – E – N – [crosshair] – [circled 8] – K – [circled 8] – M – [circled 8] – [upside-down T] – N – A – M

To study the repetition without assigning plaintext values, label each new symbol in order of first appearance:

1 2 3 4 5 6 5 7 5 8 3 1 7

These class numbers are only labels. They have no numerical role in the calculation.

There are eight distinct symbol classes in total.

Four repeat:

• class 1 at positions 1 and 12
• class 3 at positions 3 and 11
• class 5 at positions 5, 7 and 9
• class 7 at positions 8 and 13

The remaining four classes — 2, 4, 6 and 8 — occur only once.

2. Measure the repeat distances

Reading from left to right, the repeat intervals are:

1 → 12 = +11

3 → 11 = +8

5 → 7 = +2

7 → 9 = +2

8 → 13 = +5

Before assigning any plaintext, the distinct distances show a simple relationship:

11 + 2 = 13

8 + 5 = 13

So the four distinct repeat distances occur as complementary pairs around the total length of the cipher:

11 ↔ 2

8 ↔ 5

This suggested testing the positions cyclically modulo 13.

Representing each movement by its shortest signed residue modulo 13 gives:

+11 ≡ -2 mod 13

+8 ≡ -5 mod 13

while:

+2 = +2

+5 = +5

Preserving the ordinary left-to-right order of the repeat events by their starting positions — 1, 3, 5, 7 and 8 — gives:

-2, -5, +2, +2, +5

No sorting is performed afterward to create that sequence.

The second +2 exists because the symbol at positions 5, 7 and 9 occurs three times and therefore produces two consecutive repeat intervals:

5 → 7 = +2

7 → 9 = +2

This becomes important later.

3. Use one external anchor: N → E

The literal N symbol occurs at Z13 positions 3 and 11, so it is class 3 in the notation above.

The same cipher glyph N maps to plaintext E in both solved Zodiac ciphers Z408 and Z340.

I therefore test the explicit assumption that this mapping was retained in Z13:

N → E

Class 3 is the repeat class associated with:

-5

So the -5 member of the displacement sequence is anchored to plaintext E.

This anchor is not chosen because E produces a useful word. It is motivated independently by the two solved Zodiac ciphers.

4. Apply the same integer displacements to the alphabet

This is the central interpretive step.

The values:

-2, -5, +2, +2, +5

were obtained in the 13-position cipher space.

I am not claiming that arithmetic modulo 13 is mathematically identical to arithmetic modulo 26.

The proposed operation is instead:

1. use modulo 13 to obtain the shortest signed integer representing each positional movement;
2. take those resulting integers — -2, -5, +2, +2, +5 — and test them as alphabetic shifts from one common base letter.

Let the unknown common starting letter be called base.

Because the -5 displacement belongs to the N class, and N is being anchored to plaintext E:

base - 5 = E

Using alphabet arithmetic modulo 26 with A=0 and E=4:

base - 5 ≡ 4 (mod 26)

therefore:

base ≡ 9

Alphabet value 9 is J.

So once the N→E anchor and the common-shift rule are accepted, the base is not freely selected. It is forced to be J.

Applying the five Z13-derived displacements:

J - 2 = H

J - 5 = E

J + 2 = L

J + 2 = L

J + 5 = O

gives:

HELLO

5. Why the double L matters

The second L is not inserted afterward to turn the result into an English word.

It comes directly from the only symbol class occurring three times.

The repeated symbol at positions 5, 7 and 9 produces:

+2, +2

and therefore:

L, L

under the proposed alphabetic interpretation.

Without the third occurrence at position 9, the mechanism would produce:

HELO

The extra occurrence creates the extra L.

So the doubled consonant in HELLO has a direct structural counterpart in Z13.

6. Anchor sensitivity

A useful test is to hold the entire procedure fixed and vary only the assumed plaintext value of the N symbol.

If the -5 member is assigned plaintext letter P, then the five-letter output is forced to have the relative form:

P+3, P, P+7, P+7, P+10

The letters cannot be adjusted independently.

For example:

N → A gives DAHHK

N → E gives HELLO

N → I gives LIPPS

N → R gives URYYB

N → T gives WTAAD

Testing all 26 possible anchor values simply produces Caesar shifts of the same fixed relative pattern.

The important point is that E was not selected because it yielded HELLO. The reason for testing E comes from outside this construction: N→E is the mapping shared by the solved Z408 and Z340 keys.

This does not prove that Zodiac retained N→E in Z13, but it removes one obvious source of plaintext fitting.

7. Testing alternative rule choices

One legitimate objection to this type of analysis is the problem of forking paths: perhaps HELLO appears only because several reasonable-looking methodological choices were made after examining Z13.

To test that objection, a separate variant family was defined before inspecting its outputs.

The N class remained fixed as the anchor and N→E remained fixed.

Four methodological dimensions were varied:

Direction

• forward gap
• reverse gap

Treatment

• linear/raw displacement
• shortest signed residue modulo 13
• unsigned residue modulo 13

Event order

• start position ascending
• end position ascending
• start position descending
• end position descending

Mapping

• alphabetic shift +(v - vN)
• alphabetic shift -(v - vN)

This produces:

2 × 3 × 4 × 2 = 48

predefined rule variants.

On the actual Z13 layout, those 48 variants collapse to 16 distinct outputs because several variants are mathematically equivalent.

The distinct outputs were:

HELLO

HEYYB

BEKKH

YYEHB

KKEBH

BYYEH

HKKEB

BHEYY

HBEKK

BEXXU

LLEHO

XXEBU

OLLEH

UXXEB

OHELL

UBEXX

Across this family, HELLO is the only ordinary English word produced by the actual Z13 layout.

HELLO occurs in two of the 48 variants:

• forward / shortest signed / start ascending / positive mapping
• reverse / shortest signed / start ascending / negative mapping

These two are mathematically equivalent descriptions of the same underlying operation, since reversing the direction and reversing the sign cancel each other.

So there is effectively one HELLO-producing mechanism within this family, not two independent ones.

It is also worth noting that descending event order gives:

OLLEH

which is HELLO reversed.

That is consistent with the idea that left-to-right event ordering carries the plaintext order.

8. Randomized-layout null test

The same full 48-variant family was then tested on 200,000 randomized layouts preserving Z13’s symbol multiplicities:

3,2,2,2,1,1,1,1

The symbol positions were randomized, but the rules were not optimized separately for each sample.

Using a strict English word list:

• at least one of the 48 variants produced a five-letter word in approximately 0.78% of randomized layouts.

Using a somewhat more permissive word list:

• at least one variant produced a word in approximately 1.11% of layouts.

So approximately 99% of randomized layouts produced no English word at all, even when all 48 predefined rule variants were available.

The average individual variant produced a word in about:

0.085%

of random layouts.

Two or more distinct words appeared in only about:

0.04%

of layouts.

These dictionary-based figures depend on the precise word list used and should be treated as approximate calibration rather than exact probabilities.

This is not a formal probability that Zodiac intended HELLO. The rule family itself was designed by an analyst, and the analysis remains post-hoc with respect to the original discovery.

But the test does address a narrower objection:

HELLO is not simply an inevitable consequence of allowing a modest family of alternative gap, direction, ordering and sign conventions.

Within this particular predefined family, the real Z13 layout produces one English word:

HELLO

whereas a random Z13-like layout produces any English word only around 1% of the time.

9. Exact HELLO null test under the frozen original procedure

A separate exhaustive test was performed using only the original frozen procedure rather than the 48-variant family.

The relevant geometry is determined by the placements of the four repeated symbol classes, whose multiplicities are:

3,2,2,2

across 13 positions.

There are exactly:

5,405,400

distinct placements of those four repeated classes.

Under the frozen procedure, exactly:

100

of those repeat-class placements produce HELLO with N→E fixed.

The four distinct singleton labels occupy the remaining four positions. Their identities do not affect the repeat geometry, so they can be permuted among those four positions in:

4! = 24

ways.

This multiplies both numerator and denominator by the same factor:

5,405,400 × 24 = 129,729,600

fully labelled arrangements,

and:

100 × 24 = 2,400

HELLO-producing fully labelled arrangements.

Therefore the exact rate is the same either way:

100 / 5,405,400

or:

2,400 / 129,729,600

which equals approximately:

0.00185%

or:

1 in 54,054

randomized layouts.

Again, this should not be interpreted as a p-value for Zodiac’s intention, because the procedure itself was discovered through prior exploration.

Its narrower meaning is simply that, once the procedure is frozen, the exact HELLO-producing repeat geometry is uncommon among layouts with the same repeat multiplicities.

10. How the 13 positions contribute

All 13 positions define the positional geometry, but the decoding statistic is generated only by transitions between successive occurrences of the four repeated classes.

The four singleton classes — 2, 4, 6 and 8 — do not individually generate plaintext letters, and their individual identities are not used by this construction.

What matters is which positions are occupied by singleton classes rather than by repeated classes, because those occupied positions help determine the distances between successive occurrences of the repeated classes.

Permuting the four singleton labels among the same four singleton positions leaves the repeat geometry and the resulting output unchanged.

Changing which positions are occupied by singleton classes, however, changes the placement of the repeated classes and can therefore change the transition distances from which the decoding statistic is derived.

The proposal is therefore better described as:

13-position geometry → 5 transitions between repeated-class occurrences → 5 plaintext letters

rather than:

13 ciphertext symbols → 13 plaintext letters

11. What is observation, and what is assumption?

This distinction is essential.

Directly observable in Z13

• there are 13 positions;

• there are eight distinct symbol classes;

• the normalized pattern is:

  1 2 3 4 5 6 5 7 5 8 3 1 7

• four classes repeat: 1, 3, 5 and 7;

• four classes are singletons: 2, 4, 6 and 8;

• the repeat positions are 1/12, 3/11, 5/7/9 and 8/13;

• the forward repeat gaps are:

  11, 8, 2, 2, 5

• the distinct gap sizes contain the complementary pairs:

  11 + 2 = 13

  8 + 5 = 13

• under shortest signed residues modulo 13, the repeat sequence in left-to-right event order is:

  -2, -5, +2, +2, +5

• the duplicated +2 comes from the only three-occurrence class.

The construction makes three explicit assumptions

Assumption 1

The 13 cipher positions may legitimately be treated cyclically, and the shortest signed residues modulo 13 are meaningful representations of the repeat movements.

Assumption 2

The N→E mapping shared by Z408 and Z340 was retained for the N symbol in Z13.

Assumption 3

The signed integer values obtained from the positional movements are also intended to function as signed alphabetic shifts from one common base letter.

Once those assumptions are accepted, HELLO is no longer freely selectable.

The -5 member being fixed to E forces the base to J, and then:

-2 → H

-5 → E

+2 → L

+2 → L

+5 → O

giving:

HELLO

12. What I am not claiming

I am not claiming that HELLO has been proven to be the unique solution of Z13.

A cipher this short is inherently underdetermined, and many possible systems can be proposed.

I am also not claiming to know Zodiac’s intention.

The narrower claim is:

Z13 contains a repeat-distance construction which, when represented by shortest signed movements modulo 13, anchored by the independently motivated N→E mapping, and reused as alphabetic offsets from a common base, produces HELLO.

The construction is reproducible.

It has also survived two attempts to test whether the output is merely an artifact of flexible rules:

1. under the frozen original procedure, HELLO occurs in exactly 100 of 5,405,400 repeat-class layouts — equivalent to 2,400 of 129,729,600 fully labelled arrangements — or about 1 in 54,054;
2. across one predefined family of 48 plausible methodological variants, HELLO is the only English word produced by the actual Z13 layout, while randomized layouts produce any English word in only about 0.8–1.1% of cases under the tested dictionaries.

Neither result proves intention.

They do, however, make the weaker explanation — that HELLO appears trivially because almost any reasonable manipulation of the Z13 repeat structure produces a word — less convincing.

Finally, the result appears immediately after Zodiac’s introduction:

“MY NAME IS —”

which gives the possible reading:

“MY NAME IS — HELLO”

That contextual fit is interesting, but it should remain supporting context rather than evidence used to construct the result.

The remaining question is therefore quite specific:

Did Zodiac intend the signed repeat movements in the 13-position cipher to be reused as alphabetic shifts?

That is the principal unresolved interpretive step in determining whether this construction reflects Zodiac’s intended mechanism rather than a reproducible structural coincidence.

Reproducibility note

The variant and null tests described above were implemented independently from the proposed plaintext and can be reproduced from the stated rules.

For transparency, the test code and word-list definitions can be provided on request.

The dictionary-based rates depend on the particular English word list used. The exact HELLO enumeration:

100 / 5,405,400

equivalently:

2,400 / 129,729,600

does not depend on a dictionary.