+++ title = "Order Ten Passed Every Test" description = "It satisfied every algebraic screen mathematicians had. It took an exhaustive computer search to establish that it does not exist. Order twelve is still open." date = 2026-08-18 draft = false [taxonomies] topics = ["combinatorial-designs", "mathematical-proof"] kinds = ["existence", "screening", "construction"] [extra] tier = "public" schema_type = "Article" canonical = "" hero = "articles/order-ten-passed-every-test/hero.webp" hero_alt = "Backlit ruled case-register leaf with checked and untouched runs" podcast = "" mechanism = "Passing every available screen is not evidence of existence, existence is not construction, and the state of having been checked by nobody gets reported in the same column as the state of having passed." symptoms = [ "A candidate is described as viable because nothing has ruled it out", "An existence result is cited where an actual object is needed", "The screening suite is treated as exhaustive because it is everything anyone runs", "Untested and passed are recorded identically", ] apparatus = "Design theory, which maintains an explicit register of which cases are constructed, proved impossible, proved to exist without being produced, and open by every method." discriminating_test = "For any candidate, ask which of the four states it is in. If the answer is that it passes the screens, that is not one of the four — it is a statement about the screens." false_friend = "A screen with genuine deductive force. Bruck-Ryser-Chowla really does rule cases out; the problem is its narrow domain, not its validity. A good filter and a complete filter are different claims." test = "For a screening pipeline, count candidates in each of four states — rejected, constructed, proved viable but unbuilt, untested. Most pipelines cannot populate the last two columns, which is the finding." [[extra.faq]] q = "What does it mean that order ten passed the algebraic screen?" a = "The Bruck-Ryser-Chowla condition rules out certain projective plane orders on number-theoretic grounds. Order 10 satisfies it, because 10 is a sum of two squares — 1² + 3². So no algebraic argument could settle the question either way, and the field had an open case that looked entirely respectable for decades." [[extra.faq]] q = "How was it settled?" a = "By exhaustion. Lam, Swiercz and Thiel published the nonexistence result in the Canadian Journal of Mathematics in 1989, having ruled out every possible arrangement of the incidence structure the algebraic screen left standing. It is a proved nonexistence result rather than a failure to find an example, and the two are routinely and wrongly treated as the same thing." [[extra.faq]] q = "What is the status of order twelve?" a = "Open by every method. Bruck-Ryser-Chowla does not constrain it at all, since 12 is divisible by 4 — a residue class the condition says nothing about — and no exhaustive search on the scale of the order-10 effort has been run. It is the smallest order whose existence is currently unknown, and it is genuinely unknown rather than merely unpublished." [[extra.faq]] q = "Is proving something exists the same as having it?" a = "No, and design theory keeps the distinction visible. Keevash's 2014 theorem established general existence for large classes of designs without producing them. A proof by construction produces the object; a non-constructive existence proof tells you the search is not futile and nothing about where to search." +++ A projective plane of order 10 would have 111 points and 111 lines, every line through 11 points, every point on 11 lines, and every pair of points joined by exactly one line. Whether one existed was open for decades, and it was open for a specific and uncomfortable reason: it passed the test. Design theory has an algebraic screen, the Bruck-Ryser-Chowla condition, which rules certain orders out on number-theoretic grounds. Order 10 satisfies it, because 10 is a sum of two squares — 1² + 3². The screen had nothing to say. No algebraic argument could resolve the case in either direction, which left a candidate that looked entirely respectable and was supported by nothing except not having been eliminated. It does not exist. C. W. H. Lam, S. Swiercz and L. Thiel published the nonexistence proof in the *Canadian Journal of Mathematics* in 1989, and they got there by exhaustion — ruling out every arrangement of the incidence structure the algebraic screen had left standing. Not a new piece of number theory. A search, at scale, through everything the filter had passed. ## Four states, and one of them is not a state Design theory keeps an unusually honest register of what is known, and it has four columns. **Constructed.** The object exists and here it is. Kirkman settled in 1847 exactly which Steiner triple systems exist, and the construction produces them. **Proved impossible.** Order 10, by exhaustion in 1989. Also order 6, which Tarry killed by exhaustive search in 1900, disposing of Euler's thirty-six officers. **Proved to exist, not produced.** Keevash's 2014 theorem established general existence for large classes of designs without building any of them. The search is not futile. That is the entire content of the result, and it is a real result. **Open by every method.** Order 12. Bruck-Ryser-Chowla does not constrain it at all — 12 is divisible by 4, a residue class the condition says nothing about — and no exhaustive search comparable to the order-10 effort has been run. It is the smallest order whose existence is unknown, and it is genuinely unknown rather than hard to look up. What is *not* a state is "passes the screens." Order 10 passed the screens and order 12 passes the screens, and one of them does not exist while the other is a live question. Passing is a fact about the filter, not about the object. ## The two failures this separates There are two distinct errors here and they usually arrive together. The first is treating a screen as complete because it is everything anyone runs. Bruck-Ryser-Chowla has real deductive force — it genuinely eliminates cases, and a case it rejects is dead. The problem is its domain. It constrains some residue classes and is silent on others, and the silence looks exactly like a pass. Every candidate that reaches the end of a filter pipeline arrives in the same condition regardless of whether the filters had jurisdiction over it. The second is treating existence as possession. A non-constructive existence proof says the object is out there. It does not say where, and for anything that has to be built rather than admired, the gap between those two is the whole project. Mathematics keeps them apart by vocabulary. A proof by construction produces the object. A proof by contradiction or by counting establishes that it must be somewhere. Both are proofs; they deliver different goods, and no mathematician conflates them. ## The register nobody keeps Take any screening pipeline — strategies through a backtest suite, counterparties through a risk framework, candidates through a due-diligence checklist — and try to populate those four columns. Rejected is easy; the pipeline is built to produce it. Constructed is easy; those are the ones running. The other two columns are usually empty, and not because they have no members. *Proved viable but unbuilt* is the category of things known to work that nobody has implemented, and most institutions have no record of it because nothing in the process creates one. *Untested* is the category the screens had no jurisdiction over, and it is the dangerous column, because its members are indistinguishable at the output from the ones that passed. A candidate that no filter had authority over and a candidate that survived every filter arrive at the same place with the same label. That is the order-10 problem exactly, and design theory's response was to keep a public register of which cases had been settled by which method, so that "not eliminated" could never be read as "fine." Order 12 has sat in the open column since the question was first asked. It has not been quietly promoted to plausible, and nobody has built anything on it. The reason is that somebody wrote down which column it was in.