docs: complete euler beam planning gates

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+# 3D Euler-Bernoulli Beam Kernel Numerical Review
+
+## Metadata
+- feature_id: euler-beam-3d
+- source_formulation: docs/formulations/euler-beam-3d-formulation.md
+- status: pass-for-implementation-planning
+- owner_agent: numerical-review-agent
+- date: 2026-06-12
+
+## Review Verdict
+- verdict: pass-for-implementation-planning
+- reason: the formulation has a bounded kernel scope, explicit DOF order, closed-form local stiffness, unambiguous transform convention, and concrete invalid-input behavior.
+
+## Critical Findings
+- No blocking numerical defects were found for kernel implementation planning.
+- The formulation is not a full solver release contract. Parser, assembly, HDF5 output, reference comparison, and physics sanity remain downstream gates.
+
+## Numerical Risk Assessment
+- rigid_body_modes: expected six zero-energy modes for the unconstrained element. Implementation tests must include at least rigid global translation and local/global force consistency.
+- patch_test: solver-level patch testing is blocked until assembly and parser integration exist.
+- symmetry: local stiffness is symmetric by construction; global stiffness remains symmetric if `K_global = T^T K_local T` is used consistently.
+- positive_definiteness: isolated element stiffness is positive semidefinite, not positive definite. Positive definiteness requires enough constraints in a solver model.
+- hourglass: not applicable to closed-form Euler-Bernoulli beam stiffness.
+- shear_locking: Timoshenko shear terms are excluded, so shear locking is not introduced by this kernel increment.
+- volumetric_locking: not applicable.
+- distortion: curved geometry and offsets are out of scope; zero or near-zero length must be rejected.
+- singular_jacobian: zero length is the relevant singular mapping case.
+- conditioning: very slender elements or large stiffness ratios can be ill-conditioned; tests should use moderate deterministic values.
+- convergence: not applicable to this linear element kernel.
+
+## Consistency Checks
+- units: pass. `EA/L`, `GJ/L`, `EI/L^3`, `EI/L^2`, and `EI/L` have the expected force/displacement or moment/rotation dimensions for the matching DOF pairs.
+- dimensions: pass. The element vector is 12x1 and stiffness is 12x12.
+- signs: pass for the documented DOF order; local `v-rz` and `w-ry` coupling signs are explicitly stated.
+- dof_ordering: pass. The formulation keeps node 1 DOFs followed by node 2 DOFs.
+- coordinate_transforms: pass. `u_local = T*u_global`, `K_global = T^T*K_local*T`, and `f_global = T^T*f_local` are mutually consistent.
+- matrix_vector_dimensions: pass.
+- integration_weights: not applicable because the kernel uses closed-form stiffness.
+- output_locations: pass for element end forces only; stress/strain recovery is explicitly out of scope.
+
+## Verification Readiness
+- unit_tests:
+  - model topology accepts `ElementTopology::beam2`.
+  - local stiffness representative entries match named coefficients.
+  - local stiffness is symmetric within `1.0e-10`.
+  - local force recovery equals `K*u`.
+  - invalid length and nonpositive section constants throw `std::invalid_argument`.
+  - axis-aligned transform produces global stiffness equal to local stiffness.
+  - rotated global stiffness remains symmetric.
+  - rigid global translation produces near-zero global end forces.
+  - simple axial extension produces equal and opposite axial end forces.
+  - parallel orientation vector throws `std::invalid_argument`.
+- patch_tests: future solver integration gate.
+- mms_or_mes: not required for this closed-form kernel increment.
+- benchmark_reference_comparison: future gate after reference artifacts exist.
+- missing_evidence: no missing evidence blocks kernel implementation.
+
+## Required Revisions
+
+### Formulation Agent
+- None before implementation planning.
+
+### Research Agent
+- None before implementation planning.
+
+### Reference Model Agent
+- Future reference model artifacts remain required before release readiness, but they are not required for this kernel implementation.
+
+## Downstream Handoff
+
+### Implementation Planning Agent
+- Use the verification readiness list as the minimum C++ TDD checklist.
+- Keep tests deterministic and avoid ill-conditioned constants.
+
+### Reference Model Agent
+- Define future solution-verification models separately from these kernel-only unit tests.