Abstract
A final digit is said to “protect” a number. This paper counts that the score runs from $0$ to $1$ depending on what it protects against. No new theorem or law is claimed.
First, single-digit slips are caught by every scheme: all three catch them at a share of $1.000000$ (Section 2).
Second, and this is the core. On swaps the schemes split: the plain sum catches $0.000000$ and the weighted scheme $1.000000$: both ends, for one and the same single check digit (Section 3).
Third, the plain sum’s $0$ is no accident. It never uses the order of the terms, so rearranging leaves the sum untouched (Section 3).
Fourth, Luhn sits between them: catching a share of $\mathbf{0.979503}$ and letting the rest through (Section 3).
Fifth, only one kind gets through: all $52$ escapes were swaps of $0$ against $9$ (Section 3).
Sixth, the separator is whether one asks about a single-digit slip or about a swap: “protects” fixes no meaning until it says against what (Section 4).
Keywords: check digit, error detection, adjacent swap, weights, modulus, exhaustive enumeration.
Scope (what is not claimed)
No new theorem or law is claimed: that a check digit catches all single-digit errors, that catching swaps requires weights that vary by position, that the Luhn scheme misses only $0$ against $9$, and that a weighted scheme modulo $11$ catches both kinds, are all known. The use of real numbering systems is not discussed, which standard to adopt, and how to migrate, are not stated. Error correction is not entered: repairing the wrong digit is not treated; only detection. The frequency of slips is not treated: how often people make each kind of mistake is not included; all kinds are counted equally. Simultaneous errors are not counted: cases where two or more digits change at once were not counted. Relation to earlier papers: Paper 752 showed that the Huffman code is optimal, but only within the constraint that code lengths be whole numbers. This paper stands beside it: there which constraint hides behind “optimal”, here which error hides behind “protects”. Paper 300 showed that whether two things share a root is decidable: single-digit slips and swaps are distinct roots by that criterion. What is added is laying three schemes over one sample and measuring both kinds of error, counting that the scores split from $0.000000$ to $1.000000$, pinning Luhn’s escapes to the single kind $0$ against $9$, and placing the separator on slip versus swap.
1. Notation and the Question
Setting: for a string of eight digits, three schemes compute a check digit.
$$\text{plain sum}=\sum_i d_i \bmod 10,\qquad \text{weighted}=\sum_i w_i d_i \bmod 11$$The Luhn scheme doubles every second digit from the right, folding two-digit results by adding their digits, then sums. An error counts as caught when the check digit changes.
Question: “protects” against what?
2. Single-Digit Slips Are Caught by All
Computation 2.1: $400$ eight-digit numbers were built and every digit replaced by each of the other nine ($28800$ cases).
| scheme | share caught |
|---|---|
| plain sum (modulo $10$) | $1.000000$ |
| Luhn (modulo $10$) | $1.000000$ |
| weighted (modulo $11$) | $1.000000$ |
Observation 2.2 all three caught everything. Changing one digit always changes that digit’s contribution: with no weight equal to zero, the sum must move.
Observation 2.3 so looking here alone the three appear identical. “Detects single-digit errors” does not separate them: the difference only appears on another kind of error.
3. On Swaps They Split
Computation 3.1: on the same $400$ numbers, every swap of adjacent digits was applied.
| scheme | share caught |
|---|---|
| plain sum (modulo $10$) | $0.000000$ |
| Luhn (modulo $10$) | $0.979503$ |
| weighted (modulo $11$) | $1.000000$ |
Observation 3.2 this is the core. One single check digit produces both ends, $0$ and $1$. The plain sum catches nothing and the weighted scheme lets nothing through, because the plain sum never uses the order of the terms, so rearranging leaves it unchanged.
Observation 3.3 this $0.000000$ is not an empty swing. On the same sample, single-digit slips were caught at $1.000000$: same scheme, same sample; only the kind of error changed, and $1$ became $0$.
Computation 3.4: Luhn’s escapes were counted by which pair of digits was swapped.
| pair that escaped | cases |
|---|---|
| $0$ against $9$ | $52$ |
| anything else | $0$ |
Observation 3.5 only one kind gets through. Only for $0$ against $9$ does doubling-and-folding leave the sum unchanged: the awkward $0.979503$ is just the share this one kind occupies.
Observation 3.6 the hole can be named because the score can be measured. The share alone says “nearly everything”; splitting the escapes narrows it to one kind, where it is weak and how weak it is are different pieces of information.
4. The Separator
Observation 4.1 the separator is whether one asks about a single-digit slip or about a swap. On the first the three schemes stand level; on the second they split from $0$ to $1$: “protects” fixes no meaning until it says against what.
Observation 4.2 placed beside Paper 752. 752 showed that the Huffman code is optimal, but only within the constraint that code lengths be whole numbers. There which constraint hides behind “optimal”; here which error hides behind “protects”. Both bring out a condition buried under an adjective. Paper 300 showed that whether two things share a root is decidable: the two kinds of error are distinct roots by that criterion.
Observation 4.3 what is not said. Choosing a real standard, error correction and the frequency of slips are not treated. What is shown is that one digit scores $0$ or $1$ according to the kind of error.
5. Census and Closing
- slips: all three at $1.000000$ (Computation 2.1): indistinguishable.
- $28800$ cases, all of them (Computation 2.1): not a sample.
- plain sum on swaps: $0.000000$ (Computation 3.1): the core.
- weighted: $1.000000$ (Computation 3.1): the other end.
- Luhn: $\mathbf{0.979503}$ (Computation 3.1): in between.
- same sample, $1$ became $0$ (Observation 3.3): no empty swing.
- $52$ escapes (Computation 3.4): only $0$ against $9$.
- everything else: $0$ (Computation 3.4): the hole is named.
- where weak against how weak (Observation 3.6): different information.
- slip, or swap (Observation 4.1): the separator; Papers 752, 300.
Closing: One and the same check digit scores anywhere from $0$ to $1$ depending on the kind of error. Applying all $28800$ single-digit slips to $400$ eight-digit numbers, all three schemes catch them at $1.000000$ and cannot be told apart, because no weight is zero, so changing a digit must move the sum. Yet on swaps of adjacent digits the plain sum catches $0.000000$, Luhn $0.979503$ and the weighted scheme $1.000000$, because the plain sum never uses the order of the terms, so rearranging leaves it unchanged. All $52$ of Luhn’s escapes were swaps of $0$ against $9$, so the hole narrows to one kind. Placed among the earlier papers: beside Paper 752’s “optimal, but within which constraint”, as “protects, but against which error”, and treating the two kinds of error as distinct roots by the criterion of Paper 300. To be honest: all kinds were counted equally, with nothing included about how often people actually make each mistake.
References
- Kirtland, J. (2001). Identification Numbers and Check Digit Schemes. Mathematical Association of America.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. YOI: https://www.yamagishiyuuki.com/yoi/300; DOI: 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). The Huffman Code Is Optimal, but Only Within the Constraint That Code Lengths Be Whole Numbers [Paper 752]. YOI: https://www.yamagishiyuuki.com/yoi/752; DOI: 10.5281/zenodo.23015146
Reference
Yamagishi, Y. (2026). One and the Same Check Digit Scores Anywhere From $0$ to $1$ Depending on the Kind of Error [Paper 845]. YOI: https://www.yamagishiyuuki.com/yoi/845
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 845. This paper is a preprint and has not been peer reviewed.