Coherence · the instrument, not the result
A verification script prints PASS and a reader concludes something. This chapter is about the gap between what was printed and what was concluded — measured, because the corpus supplied four instances of it in a day.
1 · A near-integer is a claim about the digit you did not look at
The class invariant at $N = 58$ computes to
$$g_58^{12} = 19601.99999489847\ldots$$
which was asserted to be 19602, then 19601. It is neither. Any tolerance chosen after seeing the number passes whichever assertion you prefer, and both assertions are false.
Rule 1
When a computed quantity lands near an integer, the fix is never a looser tolerance. It is the exact algebraic form. Here $g_{58}^{12} = 9801 + 1820\sqrt{29}$, and the exact integer belongs to a different quantity — $g^{12} + g^{-12} = 19602$ — which is exact only because $9801^2 - 29\cdot 1820^2 = 1$.
A near-integer never becomes an integer by being measured harder.
2 · A damaged source is read by enumeration, not reconstruction
The scan of the theorem defining $x_N$ is unusable: x2Nn;1, (1 + k2 )2. Transcribing it would be guessing with a citation attached — the most dangerous object in a corpus, because it inherits the source's authority and none of its content.
What was done instead: five candidate readings, each evaluated at $N = 58$ and compared against $1/396^4$, the value the series independently forces. One closed exactly. The near-miss agreed to eight significant figures and was wrong.
Rule 2
Do not reconstruct a damaged source. Enumerate the readings and let an independently forced value choose. This requires that something other than the damaged text pins the answer — here, an integer identity, $64\cdot 19602^2 = 396^4$. Without such an anchor the method does not apply, and saying so is part of the method.
3 · Two instruments over one corpus
This is the rule the corpus paid most for. On 17 September four defects surfaced in the measuring tools themselves, and every one was caught by comparison; none by inspection. Each script ran clean and exited zero throughout.
| Defect | What made it visible |
|---|
| Working-tree walk not recursive | HEAD reported four fewer scripts than a commit it postdates. Scripts do not vanish. |
| Same defect in a second tool | the two tools disagreed about one corpus |
| Gap counter double-counting | 31 against 21 on an identical tree; the published ratio was wrong by 40% |
| Detector blind to prose | a third reading of the same scripts; the headline was wrong by an order of magnitude |
Rule 3
A system cannot certify itself — that is not an engineering limit, it is Gödel's. The buildable substitute is two independent instruments over the same material, required to agree. Disagreement is information; agreement is weak evidence; a single instrument reporting success is neither.
This is why α58 is computed from theta series and from radicals, and why π is built by Machin before any series is tested against it.
4 · Recording a limit is not heeding one
The hardest of the four, because it looks like virtue. WP-125 measured the corpus's gap records and published a headline of 4.1%. Its own gap [1] read: “the gap detector recognises two syntactic shapes, not the idea of a gap … the true count is a floor.”
The true figure was 51%. The limitation was identified, written down, published, and reasoned past in the same document.
Rule 4
A limits section is not a defence. Writing “this may be an undercount” and then quoting the undercount as a finding is worse than not writing it, because it converts a known weakness into apparent rigour.
The test: does any conclusion in the document change if the gap is real? If not, the gap was not heeded. In WP-125's case the headline inverted.
5 · What a check does establish
The chapter would be dishonest if it only subtracted. Verification in this corpus has:
- Overturned published claims, including several of this project's own — the Sn overclaim, the Grönwall radius, the μmax calibration, and WP-125's own headline.
- Recovered structure nobody was looking for: 9801 sat in Chapter R for months before anything connected it to the series it belongs to.
- Made every number in these volumes checkable without trusting the author, which is the floor the whole series rests on.
What it has never done is explain anything. Watson's proof of the notebook algorithm is a verification; its authors looked for a proof that showed how Ramanujan found the formula, and did not find one. A script that confirms $\alpha_{190}$ to seventy-five digits knows nothing about why it was written down, and neither does the person who runs it.
The position
Verification is the floor, not the ceiling. It makes a claim safe to build on. It does not make it understood, and a corpus that confuses those two has mistaken its foundations for its building.
6 · Known limits of this chapter
- Four rules from one afternoon is a small sample. Three of the four came from a single line of work, so they may be specific to numerical verification rather than to checking in general. OPEN
- Rule 3 requires genuine independence, and the chapter offers no test for it. Two instruments written by the same author on the same day share assumptions — all four defects above were mine, and two were the same defect twice.
- Rule 4's test (“does a conclusion change?”) is stated and not automated. Nothing here detects a gap that was written and ignored.
7 · Sources
- WP-125 — the corpus-wide measurement and its correction.
- Vol VII, Chapter R §VI(e)–(f) — the near-integer and the enumeration method as they occurred.
- Vol IV — the same material worked as a usable procedure.
- book7/ch-feynman-verify.py — the HONESTY block that overturned WP-125's first conclusion.