A match is only as surprising as the search that found it was narrow. Before judging any coincidence, count how many things could have matched, and how close counted as a match. This chapter does that for two numbers this series once called resonant, and it names the cases and the method before looking at the answers.
Evans and Rosenthal make the point with a model that is correct by construction (Example 9.3.1, p. 509). Draw n values from a standard normal distribution, test each one separately at the 5% level, and declare the model wrong if any single test fails. Each test is fair. Together they are not: the chance that at least one of n fair tests fails is
| tests run | chance of a false alarm | text prints |
|---|---|---|
| 10 | 0.401 | 0.40 |
| 20 | 0.642 | 0.64 |
| 100 | 0.994 | 0.99 |
At a hundred tries, a correct model is rejected almost surely COMPUTED. The textbook's remedy is procedural rather than clever: “before actually observing the data … we decide on a few relevant model-checking procedures to be carried out and implement only these” (p. 510). Fix the question first. That is why this volume's index named both cases below, and the quantity to measure for each, before this chapter's script was written.
A coincidence hunt is the multiple-comparisons problem run in reverse. Instead of many tests of one hypothesis, there are many candidate numbers on one side, many on the other, and a tolerance for “equal”. Every pair is a try. The question to ask of a striking match is not “how unlikely is this particular pair?” but “how many pairs were on offer, and how many would have looked just as striking?”
The observable fixed on the index for each case was: how many constants and combinations were available to match, and the chance of a match at the stated tolerance if nothing but that search were at work.
Volume V printed, among its verified constants, a Lean theorem named g6_equals_schumann, with a paragraph reading the number 33 — the layer count of the G6 crystal — as “simultaneously the fourth integer harmonic of the Schumann resonance (f4 ≈ 33.8 Hz). The coincidence is not ornamental.”
Two things are wrong with that, and they are different kinds of wrong.
Both sides of the theorem are definitions the author wrote, each equal to 33. The theorem is 33 = 33 CITED — the audit log records it under the class UNFALSIFIABLE GUARD, and G6Crystal.lean §4 withdrew the whole Schumann section on 2026-09-11, listing four independent failures. The first is enough: a count has no unit and a frequency is in hertz, and no physical quantity ever reached the kernel. A proof assistant certified that 33 is 33.
Set the proof aside and ask this chapter's question of the match itself, using only numbers the page prints. The corpus side is the g-series, a five-value taxonomy in GSeries.lean: 0, 2, 6, 33, 64. The physical side is a sequence of resonance modes, of which the page picks the fourth, at 33.8 Hz; the tolerance it uses is the integer part. So the first four modes all fall in the integer bins 1 to 34, and three g-values (2, 6, 33) sit in that range.
| if nothing but the search were at work | chance |
|---|---|
| one mode's integer part lands on a g-value | 3/34 = 0.088 |
| at least one of four modes does | 0.309 |
Roughly one in three MODEL. The model is deliberately crude — each mode's integer part equally likely to be any bin — and it is the one fixed before the run. It is also generous to the claim: it counts only the g-series and only four modes, while the page sits among eight “verified constants” and the physical side has further modes. A match that turns up one time in three under pure search is not evidence of a link between a crystal and the ionosphere.
The Lean was withdrawn on 2026-09-11. Two distributed pages still printed the theorem as a result: book5/chV-constants.html, with the paragraph above, and book5/index.html, in its list of theorems. Both now carry a correction pointing here, and the script fails if either loses it COMPUTED. This is Chapter 1's point again: the page a reader sees is the survivor, and a retraction recorded only in the Lean file has not reached them.
The Volume V hardback has a hard ceiling of 666 pages. Its source is practical: it is the most pages the printer will print for this edition, and another edition or another printer can change it. The number came from outside the series. Until 2026-09-27 the house rules added a reading on top of it: “666 = 6 × 111, resonant with the hexanacci/g6 threshold and the 111 Hz sacred frequency.”
That makes this a clean case for the chapter. A number arrived for an ordinary reason, and a meaning was then found in it. The question is the same as in case A: had the printer's ceiling been some other number, how likely is it that the same families would have supplied a meaning? Take the families the corpus itself invokes, the g-series and repdigits like 111. Count the page ceilings between 400 and 800 that can be written as a g-value times a repdigit, or as a repdigit alone:
On this narrow count, a ceiling that happens to factor this way is uncommon COMPUTED. That is a fact about 666. It says nothing about printing or resonance, and case A's lesson is what limits it. The count covers only two families. A reader allowed other favourite numbers — 33, 108, the Fibonacci numbers, any prime — would find a pattern in most ceilings. More to the point, the pattern does no work. The book is limited to 666 pages because that is where the printer stops, and it would be limited in the same way at 640 or 700. A property that plays no part in why the number is what it is cannot be evidence of anything. A meaning found after the fact belongs to the reader, not to the number.
The rule now states the fact and keeps the old sentence on record as history rather than deleting it. The limit stands as a production constraint for this edition. CITED — CLAUDE.md, “Hardback constraint”, corrected 2026-09-27.