Two famous cracks — a cipher called unbreakable for three centuries, and a script that sat unread for three thousand years — turn out to be the same move. A code or a script assigns arbitrary labels to an underlying system. The labels are free. But some structure among the labels is forced by the system, and so it cannot be moved by relabeling. You grab the system by that invariant (before you can read a single symbol), and then the anchors break the rest.
This page joins two earlier layers and tries to make the whole exceed the stack:
◆ What the Cipher Couldn't Hide — the index of coincidence breaks Vigenère.
◆ The Grid That Spoke Greek: Linear B deciphered through structural analysis, phonetic guesses and place-name guesses.
Nothing here is an original finding; the new thing is the join — and one number it lets you state exactly.
A simple substitution cipher swaps every letter for another symbol — A always becomes Q, B always W, and so on. It can change which letter wears which mask. It cannot change how often the masks recur. Below is one English passage. Scramble its alphabet as hard as you like and watch the two histograms: the one sorted by letter jumps; the one sorted by height never moves.
This composed passage is kept verbatim as the cipher's input. Its closing claim that a single name finishes the reading overstates what the grid shows: one pin leaves four assignments, and a second pin in a fresh row and column leaves one in this 3×3 toy. The historical decipherment also used multiple guesses and deductions.
The index of coincidence is the chance that two letters picked from the text are the same — a one-number summary of that sorted shape. It is identical, to twelve decimal places, under every one of the 26! ≈ 4×10²⁶ ways to relabel the alphabet. That invariance is the moat: the cipher hides the letters; it cannot hide the language.
The Vigenère cipher defeats the attack above by rotating L alphabets in turn, so the same letter becomes different symbols at different positions. That genuinely flattens the lumpiness toward random. But the flattening is periodic — it repeats every L letters — and that period is a shadow the hiding casts. Slide the key length and watch the shape collapse; then read the period back off the columns.
At L=1 one Caesar shift preserves the English frequency shape and IC.
As L grows, the whole-text IC tends toward the random benchmark, with fluctuations.
Cut the ciphertext (enciphered with a real 7-letter key) into L columns — every L-th letter together. At the fundamental period and its multiples, each column uses a single shift, so its IC recovers the English-like shape. Nonmultiples generally mix alphabets. Here 7 and 14 both read English-like; the first such column-set gives the fundamental period, 7.
The key is VENTRIS — seven letters, for the man who broke Linear B, here hiding a cipher.
The scan finds 7 with no help. Once the period is known, each column is a plain Caesar shift and a
frequency score can help identify its letter, an anchor for the break.
Now the harder case. In this cipher example the language is known; only the labels are hidden. In a lost script, the labels and the language are unknown. And yet the same handle is there. Alice Kober showed that Linear B's signs fall on a grid — you can tell which signs share a consonant and which share a vowel from how words inflect, with no idea what any sign says. That relational structure is an invariant of whatever sounds you assign later.
Here is the principle as a toy you can run: a 3×3 grid of signs, the rows consonants, the columns vowels. The structure alone — same-row, same-column — can't tell you which sounds go where. It leaves the grid fixed up to relabeling the rows among themselves and the columns among themselves. Pin a sign to a sound — guess a name — and the freedom collapses.
Structure alone: 36 ways to read the grid. None of them is wrong yet — the signs simply don't say which sound they are. (Counted by brute force, every relation-preserving permutation enumerated — the same routine the verifier runs.)
That is the whole shape of the 1952 decipherment, in miniature. Michael Ventris guessed that
often-repeated sign-groups were Cretan place-names, including Knossos, Amnisos, Tylissos and Phaistos. Each pinned
a few rows and columns; signs that appeared in no place-name then came out forced, spelling real
Greek — ti-ri-po, "tripod." The anchors broke the symmetry the structure had left.
relabelings the structure can't distinguish — every way to swap the alphabet.
The invariant: the index of coincidence, fixed over all of them.
An anchor: English frequencies can help identify a substitution. A known-period Vigenère column instead has 26 possible shifts.
relabelings the structure can't distinguish — rename the consonants, rename the vowels.
The invariant: the grid of shared rows and columns, fixed over all of them.
The anchors: one guessed place-name pins specific sounds and cuts 36 readings to 4; a second forces the grid.
Same move, different groups in the models. Unrestricted alphabet substitution has 26! keys; a known-period Vigenère column has only 26 possible shifts. Frequency scoring can help choose a shift, but a known plaintext letter and its ciphertext letter also determine it. The 3×3 grid toy begins with 36 assignments, then four after one pin and one after two. That illustrates constraint propagation; group size alone does not explain the historical decipherment.
r!·c! sound-assignments, and that
anchors break that symmetry — and nothing more. Real Linear B has ~90 signs, gaps in the grid, signs
that resist it, and a script so lossy one spelling can be two words. The actual decipherment is in
its own layer; this page borrows only its logic.
research/invariant-of-relabeling/verify.mjs (§4). It agrees with
r!·c! because the math says it must, but the page shows the count, not the claim.
Corrected 2026-09-28 (found by a scan of the site's pages for verdicts printed whatever the page computed): the grid's messages typed "36 ways" and "the structure is solved" by how many anchors were pinned, not by the brute-force count beside them. Both were true, but the messages did not check them; they now use the counted value.
Corrected 2026-10-14: This page said Linear B was cracked from pure structure. Structural analysis was joined by phonetic and place-name guesses (found by the assay line, 2026-10-14).
Corrected 2026-10-14: This page said every wrong candidate period left mixed, flat columns. Multiples of the fundamental period also use a single shift per column: this scan gives mean IC 0.07255 at 7 and 0.07134 at 14 (found by the assay line, 2026-10-14).
Corrected 2026-10-14: This page and its verifier called the ambiguity 26! per Vigenère column. A known-period Vigenère column has 26 possible shifts; 26! counts unrestricted alphabet substitutions, under all of which IC is invariant (found by the assay line, 2026-10-14).
Corrected 2026-10-14: This page said a whole frequency table was needed to break the cipher. Frequency scoring is one available method; a known plaintext letter and its ciphertext letter also determine a Caesar shift (found by the assay line, 2026-10-14).
Corrected 2026-10-14: The descriptions assigned the 36 → 4 → 1 counts to Linear B and said group size alone explained why a name or two sufficed. Those counts and the two-pin result belong to this 3×3 toy. The group comparison is between the stated models, and does not establish historical sufficiency (found by the assay line, 2026-10-14).
Corrected 2026-10-14: The plain description, markdown account and demonstration passage said one anchor or single name could finish the reading. In this toy one pin leaves four assignments; a second in a fresh row and column leaves one. The demonstration passage is preserved verbatim as cipher input, with its overclaim explained beside it (found by the assay line, 2026-10-14).
Corrected 2026-10-14: This page said the slider text was in the clear at L=1. Its key is T, a Caesar shift of 19, which preserves the plaintext IC without leaving the text in the clear (found by the assay line, 2026-10-14).
Written 2026-06-19. Claims re-read against their sources on 2026-10-14: 138 checked, 98 confirmed, 37 wrong, 0 unverifiable, 3 true when written and overtaken since (dated, not corrected). By the assay line (research/assay-line/queue-1014c/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex and a Claude agent.
This was the directed repair pass for the-invariant-of-relabeling at f5e5f84f34, using the queue-1014c triage after the Codex and Claude source checks. The fixer reread the evidence before editing, reran the page calculations, and carried forward the checkers' distinct CONFIRMED claims, excluding verdicts superseded by triage. There is one line for each of the 42 triage items, including repeated findings; all 17 MINOR lines count in wrong, while the 20 WRONG lines retain their level. The seven dated corrections cover WRONG findings only. The quotation remains the cipher input, with its closing factual overclaim explicitly corrected beside it. The two Shannon findings were rejected on the printed formula. The JSON-LD and verification dates use the triage's exceptions. No other layer or shared file was edited.
This record is long (138 claim lines and 40 findings), so it is not part of the page until you open it. It loads here now if your browser runs scripts; otherwise, or to keep it, read the whole record as plain text.
The assay office: what a record is, and every layer re-read so far.