Infinite Hex on the full infinite plane and Non dyadic Laver algebras in 4D: Real-Time Neural Weight Rearrangement Through Nonlinear Ray-Tracing Operators, a novel RSI Method

We develop an executable architecture that learns favorable parameter geometry and checks its mathematical scope before using it. The main result is a complete recognition theorem for a regular polynomial family on R2n: an unknown affine symplectic transformation hides a potential shear over two positive-definite quadratic factors. The highest-degree objective coefficients reveal the hidden Lagrangian subspace. A rational canonical completion then recovers the shear and verifies a global polynomial identity. Acceptance yields a unique rational global minimizer for rational input data, with polynomial cost in the dense coefficient input size. Neither favorable coordinates nor a descent direction are supplied to the recognizer. Once learned, the chart solves every new shared-chart, shared-metric target task from one fresh gradient at an arbitrary current point. This statement is exact and conditional on task-family membership. For quartic objectives, arbitrary physical linear tilts admit a separate complete classification into unique minima, nonunique minima, or unboundedness through a small affine matrix pencil. A finite-query alias shows why observations alone cannot establish an unrestricted global class promise. The local mechanism retains a sharp symplectic direction compiler: the least condition number subject to SSTg = d is c+√c2 −1, where c = ( g 2+d 2)/(2gTd). Optimal scaling reduces it to secϑ + tanϑ, bounded sharply by cond2(H) for an SPD Newton system. A further minimum tensor construction realizes any requested update acceleration using at most seven complete cubic Hamiltonian shears. Curvature can be inferred from residual measurements; its established geodesic baseline is included. Exact rational tests recover 18 hidden objectives through dimension eight and degree six. An independent evaluator verifies their global identities. Learned geometry solves 256 changing target numerical tasks in one update, using 64 setup and 256 online gradients; an equivalent cached polynomial baseline matches it. On 24 nonlinear-neuron tasks, the ray chart takes 115 updates versus 118 for direct geodesic acceleration, but is slower in Python. An integrated exact controller certifies 12 supported tasks and rejects two altered objectives before bounded descent. The non-dyadic Laver and full-plane Hex contracts remain explicit: invalid algebra rewrites are guarded, and finite boundary signatures preserve specified exterior reachability and geometric winning paths. Their roles do not imply an AGI capability. The result is a proved solvable family, reusable learned representation, and verified bounded optimization architecture. Historical priority of the combined recognition criterion, broad optimizer superiority, and autonomous AGI remain unestablished.