What cipher is it
Ranked by what survives the measurements.
A keyword-filled 5 × 5 square, rows written as 6–0 established
Each plaintext symbol becomes one two-digit unit: row plus five, then column. Twenty-five cells, rows labelled 6 7 8 9 0 and columns 1 2 3 4 5, so no separator is ever needed. The keyword is written in first and the rest of the alphabet after it, exactly as on page 118 — which is why the row totals spread and the column totals stay flat, and why the fifth row appears once (E3). The 1939 square almost certainly folded I and J together, as the book's Playfair does on page 123.
Followed by a transposition, unit by unit established
E7 leaves no room: the units sit exactly where a shuffle would put them and nowhere near where any language would. E8 adds that the shuffle was total — every unit moved on its own, which is what a columnar transposition does and what no coarser accident does. The frequency profile survives intact (E6), which is what a transposition does and what a polyalphabetic or homophonic layer would not.
Executed with a slip, or with a key beyond reach and these look identical
Every single columnar transposition in reach has been tried and rejected: widths 2–15 over whole units with the column order solved exactly, widths 3–13 over single digits by exhaustion, 300 unkeyed routes, double transposition at every reachable width, 6,535 real words and dates as keys, and the same widths again with the long columns allowed in the wrong places.
What is left is a key too long to search, a double key too large to search, or a mistake in the encipherment. Section 5 argues the third is at least as likely as the other two, and that no measurement available can tell them apart.
Or the plaintext is not English live
Every attack above scores candidate readings against English. If the message was in French, or in Russian transliteration — d'Agapeyeff was of Russian parentage — or was itself already encoded before the square was applied, all of this machinery is blind to it. The solver takes its corpus from a file; swapping that file is a five-minute experiment nobody has run.
A homophonic square ruled out
The one substitution family that can suppress repetition, and the numbers do not reach. Enough homophony to explain the missing triples costs 33 distinct symbols and a top-thirteen share of 59.5%; the cipher shows 18 and 95.9%. See E7.
A syllable table or nomenclator ruled out on its own
It fitted the frequency profile better than any alphabet, and page 128's invitation to build one is real. But E7 is invariant to what the symbols mean: a syllable stream still repeats itself, and this one does not. K3 adds that there is no periodic structure either, at any period from 2 to 8, so fixed-length code groups are out as well.
Plain Polybius substitution of English ruled out
Dead on E5 and E7 independently.
The book's own combined method, pages 124–125 ruled out
Everyone's first hypothesis. It fractionates — it transposes single coordinate letters and re-pairs them afterwards — and E4 shows the printed pairs were never broken apart that way. The narrow-width version that would survive E4 was then enumerated exhaustively and rejected: every one of the 641,000 keys at widths 3 to 13 that leaves the alternation intact.
Dictionary code in five-digit groups ruled out
Dead on E1: the alternation runs straight through the group boundaries, so the groups are not the units. A pity, since the book spends pages 153–156 teaching exactly how to break one.
Nihilist, Playfair, Vigenère, machine cipher ruled out
Nihilist addition produces sums from 22 to 110 and destroys the 6–9 / 1–5 split. Playfair is letter-to-letter and cannot give 18 symbols over 196 units. A polyalphabetic cipher flattens the index of coincidence; this text sits at 0.0697, squarely monoalphabetic, with no autocorrelation peak between lags 1 and 40 and no periodic structure at any period from 2 to 8. Rotor machines did not emit two-digit coordinates.