[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"S8Yd7dEGf1":3},"# PlanarMidpoint\n\nA Lean proof of the **dimension-two case of Nielsen–Okamura Conjecture 9.1**:\non a connected open subset of the Euclidean plane, complementary canonical\nmidpoints for a pair of dual torsion-free connections force their symmetric\ncubic tensor to be constant.\n\nThe theorem concerns dimension two; it does not resolve the conjecture\nin arbitrary dimension.\n\n## Statements\n\nWrite a totally symmetric cubic tensor as its four coefficients\n$C=(a,b,c,d)=(C_{111},C_{112},C_{122},C_{222})$. Raising an index with the\nEuclidean metric gives the symmetric bilinear map\n\n```math\nK_C(u,v)=\\bigl(a u_1v_1+b(u_1v_2+u_2v_1)+c u_2v_2,\n b u_1v_1+c(u_1v_2+u_2v_1)+d u_2v_2\\bigr).\n```\n\nThus $\\langle K_C(u,v),w\\rangle=C(u,v,w)$ and the dual connections are\n$D+K_C$ and $D-K_C$. Total symmetry is built into these four coordinates;\nthe derivative of the field retains all eight independent entries.\n`TensorSemantics.lean` proves that this representation exhausts symmetric\ntrilinear tensors on the Euclidean plane.\n\nThe independent, Mathlib-only statement file is\n[`PlanarMidpointChallenge.lean`](PlanarMidpointChallenge.lean).\n[`PlanarMidpointSolution.lean`](PlanarMidpointSolution.lean) imports the proofs.\nThe three Comparator entries are:\n\n* `PlanarMidpoint.planar_dual_midpoint_rigidity`: a smooth field on an open\n  preconnected domain is constant if the canonical local midpoint maps of\n  $D+K_C$ and $D-K_C$ satisfy $A_+(P,Q)+A_-(P,Q)=P+Q$ near every diagonal point.\n* `PlanarMidpoint.planar_obstruction_rigidity`: an everywhere Fréchet\n  differentiable field on such a domain is constant if, at every point and in\n  every direction $h$, the following expression vanishes, with $J=DC$:\n\n```math\nE(C,J;h)=2K_C\\bigl(h,K_{Jh}(h,h)\\bigr)\n +5K_{J(K_C(h,h))}(h,h)\n -4K_{Jh}\\bigl(h,K_C(h,h)\\bigr).\n```\n\n* `PlanarMidpoint.planar_five_direction_rigidity`: in the differential theorem,\n  the five directions $h=(j,1)$ for $j=0,1,2,3,4$ suffice, at every spatial point.\n\nThe geometric proof in fact needs only $C^2$ regularity, as recorded by\n`planar_midpoint_rigidity_C2`. Preconnectedness includes the empty domain,\nwhere the constant-field conclusion is vacuous. Convexity is not assumed.\n\n## Canonical midpoint meaning\n\n`IsGeodesicSegment` uses genuine curves with\n$\\gamma'=v$ and $v'=-\\sigma K_{C(\\gamma)}(v,v)$ on $[0,1]$.\n`IsCanonicalLocalMidpoint` requires prescribed endpoints, position and velocity\nbounds near the diagonal, and uniqueness among these short curves; the midpoint\nis $\\gamma(1/2)$. Both signs use the same endpoint neighborhood. The neighborhood\nand shortness radius may depend on the basepoint.\n\n`exists_unique_short_geodesic` constructs this branch by a Banach contraction\nwith the Dirichlet Green operator. `exists_canonical_local_midpoint` proves\nexistence of the resulting germs, and `canonical_midpoint_germs_agree` proves\nindependence of the construction after shrinking the common neighborhood.\nNo endpoint smoothness or obstruction identity is assumed in the geometric\ntheorem. The short branch is the usual canonical local geodesic branch:\nsmall initial-data geodesics lie in the short class, and uniqueness identifies\ntheir midpoint germs.\n\n## Proof\n\nFor endpoints $p\\pm\\varepsilon h/2$, an explicit fourth-order approximate\ngeodesic satisfies the endpoints exactly. Spatial $C^2$ Taylor estimates and a\nquantitative Dirichlet stability estimate compare it with the actual short\ngeodesic. Exact polynomial identities give the sum of the two approximate\nmidpoints as $2p+\\varepsilon^4E(C(p),DC(p);h)/192$. The error is\n$o(\\varepsilon^4)$. Exact complementarity therefore forces $E=0$.\nThe formal proof uses explicit norm bounds to make this limit argument;\nit does not presume a smoothly parameterized family of boundary solutions.\n\nEach component of $E$ is a homogeneous quartic in $h$. Its ten coefficients\nform a linear system in the eight entries of $DC$. Exact polynomial\ncertificates show that a nonzero cubic with nontrivial kernel must belong\nto one of two exceptional graphs. With $\\tau=(a+c,b+d)=(s,t)\\ne0$, these are\n\n```math\nQ(s,t)=\\frac{(s^3,s^2t,st^2,t^3)}{s^2+t^2},\\qquad\nR(s,t)=(3s,t,s,3t)-3Q(s,t).\n```\n\nBoth extend continuously by zero at the origin. Away from these graphs,\nthe linear system is injective. On either nonzero graph, separate exact\ncertificates show that the space of derivatives tangent to the graph\nintersects the obstruction kernel only in zero. The normalizing coordinate change is fixed at a point; no\nvarying frame is differentiated as a constant. A relative clopen level-set\nargument handles transitions and zero values on arbitrary connected open\ndomains using only differentiability. Finally, quartic interpolation yields\nthe five-direction criterion.\n\nThe finite certificates are ordinary Lean algebra proofs, not external\noracle calls or numerical tests. All proof dependencies use only\n`propext`, `Quot.sound`, and `Classical.choice`. Deliberate `sorry` placeholders\noccur only in the independent Challenge statement file.\n\n## Reproduce\n\nThe committed `lean-toolchain`, `lakefile.toml` and `lake-manifest.json` pin\nLean and every dependency. With Elan installed:\n\n```sh\nlake exe cache get\nlake build\nlake comparator --config comparator.json\n```\n\nComparator requires Linux with bubblewrap. The GitHub Actions workflow\nbuilds the exact checked-out commit, audits the theorem axioms, compares all\nthree statements and definitions, and replays the exported proofs through\nLean, NanoDa and con-ron. A separate fresh-runner build checks reproducibility.\nThe runtime configuration enabling the extra kernels is generated in the\nrunner's temporary directory; the committed configuration follows Palomar's\nstatement format.\n\n## Sources and authorship\n\nFrank Nielsen and Kazuki Okamura,\n[arXiv:2609.07551v2, Section 9](https://arxiv.org/html/2609.07551v2#S9),\npose the conjecture and supply the midpoint-expansion framework and\nconstant-cubic sufficiency. This project proves planar necessity, including\nthe exceptional tensors, and strengthens the differential regularity.\nThe nearby [two-dimensional Matkowski–Sutô work](https://arxiv.org/html/2609.11102v1)\nconcerns coordinate generators and does not supply the Euclidean-dual theorem\nproved here. No global priority or external peer-review claim is made.\n\nJD Jones is the responsible human maintainer. AI assistance and review are\ndescribed in [Disclosure.md](Disclosure.md); structured provenance is recorded\nin [`formalization.yaml`](formalization.yaml).\n",1791742521432]