Nineteen over eighteen should have been ten over nine

This is the archive’s smallest correction and its most contagious: a five-term sum, mis-summed once in the published literature, copied for years by papers including this programme’s, and worth a post because the whole failure fits in a paragraph and the whole lesson fits in a sentence.

The sum

The eta invariant of a lens space S cubed over Z6, the order-6 quotient of the three-sphere, reduces by the APS theorem to minus one sixth times the sum over k from 1 to 5 of cotangent squared of pi k over 6. The sum: cot squared of 30 degrees is 3, of 60 degrees is one third, of 90 degrees is 0, of 120 degrees is one third, of 150 degrees is 3. Total: 3 plus a third plus 0 plus a third plus 3, which is 20 over 3.

The published value in the literature was minus 19 over 18, which corresponds to a sum of 19 over 3. The difference between 20 over 3 and 19 over 3 is one third: exactly one term, mis-entered. Someone, somewhere, added five numbers and dropped one bracket, and the slip propagated through papers, including this programme’s early drafts, because copying a published value is easier than re-adding five terms, and nobody re-adds five terms.

The correction

Two independent checks settle it. The classical identity: the sum of cotangent squared from k equals 1 to p minus 1 of pi k over p equals p minus 1 times p minus 2 over 3, a theorem with several published proofs. At p equals 6: 5 times 4 over 3, which is 20 over 3. And direct evaluation, term by term, also 20 over 3. The correct eta invariant is minus one sixth of 20 over 3, which is minus 10 over 9, about minus 1.111, and the literature’s minus 19 over 18 is a single-term arithmetic slip.

The corrected value matters downstream, which is the reason a five-term sum gets a status entry at all: the eta invariant feeds the vacuum-energy calculation’s topological sector, and a wrong eta shifts a coefficient that other numbers lean on. The framework’s papers now carry minus 10 over 9, with the identity and the direct evaluation both cited, and the correction entry is the shortest in the ledger.

The lesson

Copying is how science scales, and it is also how arithmetic errors live forever. The defence is mechanical, not moral: every finite sum gets summed by a second route before use, and the second route is not re-reading, it is recomputing by a different method, an identity, a symmetry, a direct evaluation. The cost is minutes. The alternative is a wrong coefficient propagating through a literature for years, discovered, if ever, by someone re-adding five numbers that anyone could have re-added on the day the slip was made.

The next entry is a divisor described as exact that was reverse-engineered from the measurement it was meant to predict: the efficiency I solved for.