A projective plane of order 12 would have 157 points and 157 lines. Every line would contain 13 points, every point would lie on 13 lines, and each pair of points would determine one line. Equivalently, it would be a symmetric \(2\text{-}(157,13,1)\) design.

Akiyama, Suetake, and Tanaka state in their 2019 primary paper that the order-12 existence question is still unknown. Their exhaustive computation excludes collineation groups of order 9. Their 2023 sequel sharpens the symmetry restriction to collineation-group orders 1, 2, or 3 while leaving the plane itself unresolved. Kharaghani and Suda give a 2023 equivalence with a balancedly multi-splittable quaternary Hadamard matrix of order 144, which supplies another exact target rather than a resolution.

The research check through 2026-07-25 found no later accepted construction or nonexistence theorem. The answer recorded here is therefore open.

Bruck and Ryser prove that if a projective plane of order \(n\) exists and \(n\equiv1\) or \(2\pmod4\), then \(n\) must be a sum of two integer squares. Since \[ 12\equiv0\pmod4, \] the theorem is silent here.

The elementary determinant condition is also consistent. If \(B\) is a \(157\times157\) incidence matrix, then \[ BB^{\mathsf T}=12I_{157}+J_{157}. \] The eigenvalues on the right are \(169=13^2\) once and 12 with multiplicity 156. Hence \[ \det(B)^2=13^2\,12^{156} \quad\text{and}\quad |\det(B)|=13\,12^{78}. \] This is an integer square identity, so it supplies no contradiction. Any order-12 exclusion needs information beyond this arithmetic test.

Akiyama, Suetake, and Tanaka prove that a projective plane of order 12 cannot admit a collineation group of order 4. Their paper combines this result with the earlier exclusions and states the resulting classification: \[ |G|\in\{1,2,3\} \] for every collineation group \(G\) of a hypothetical plane of order 12. In particular, its full collineation group would have one of these three orders.

This theorem sharply limits symmetry-based enumeration. A search restricted to automorphism groups of order at least 4 covers an empty class. A complete computation must still handle planes with a trivial group or a group of order 2 or 3.

Let \(N(12)\) be the largest size of a pairwise mutually orthogonal family of Latin squares of order 12. Bose, Chakravarti, and Knuth explicitly construct two families of five in their 1960 paper, proving \[ N(12)\ge5. \] The universal bound gives \(N(12)\le11\). A family of 11 is a complete family, equivalent to an affine plane and hence to a projective plane of order 12.

Thus the published five-square construction supplies a concrete partial object. Extending any compatible family to 11 would prove existence. A proof that all complete-family branches fail would prove nonexistence.

The audit checked the 1949 Bruck-Ryser paper, the 1960 five-MOLS construction, the 2019 order-9 collineation exclusion, the 2023 order-4 exclusion, and the 2023 Hadamard equivalence. It also checked Eric Moorhouse's Projective Planes of Small Order catalogue, which describes itself as a current list of known small planes and includes every known plane below order 34 known to its maintainer. The table has no order-12 incidence record.

The catalogue states that its completeness guarantee ends at order 10. Its empty order-12 slot therefore serves as a database cross-check rather than a mathematical exclusion. The strongest global restriction found is the 2023 theorem that every collineation group has order 1, 2, or 3. The strongest explicit partial Latin-square object found remains a family of five MOLS. The complete family of 11 remains unconstructed.

Use integer variables \[ L_{k,r,c}\in\{0,\ldots,11\}, \quad 0\le k<11,\quad0\le r,c<12. \] For each \(k\), impose `AllDifferent` on every row and every column of \(L_k\). This gives 264 Latin constraints. For each pair \(k<\ell\), impose `AllDifferent` on the 144 values \[ 12L_{k,r,c}+L_{\ell,r,c}. \] There are 55 such constraints. Since every encoded value lies between 0 and 143, each constraint forces every ordered symbol pair to occur once. A satisfying assignment is exactly a complete family of 11 MOLS, or equivalently an \(OA_1(144,13,12,2)\).

Safe normalization fixes \(L_{k,0,c}=c\) for every \(k,c\), then fixes \(L_{0,r,0}=r\) for \(1\le r<12\). These are 143 fixed cells. They come from row and column permutations plus independent symbol relabeling. Any satisfying assignment constructs the plane. A complete unsatisfiability certificate for this normalized model proves nonexistence.

The inline program checks the design identities, the determinant value, the orthogonal-array dimensions, and the full constraint inventory. Its canonical report has SHA-256 digest `c980fe81ebe7fa46510d87f9c4c8f000f681b13eaff128ebb54bdfab7797feb1`.