docs: approve MITC4 implementation handoff

docs/numerical-reviews/linear-static-mitc4-shell-review.md

@@ -1,6 +1,6 @@
 # Linear Static MITC4 Shell Numerical Review
 
-## Metadata
+## 1. Metadata
 
 - feature_id: `linear-static-mitc4-shell`
 - source_formulation: `docs/formulations/mitc4-shell-formulation.md`
@@ -8,339 +8,357 @@
 - source_research: `docs/research/linear-static-mitc4-shell-research.md`
 - source_io_definition: `docs/io-definitions/linear-static-mitc4-shell-io.md`
 - source_reference_inventory: `docs/reference-models/linear-static-mitc4-shell-reference-models.md`
-- repository_policy: `AGENTS.md`, `docs/ADR.md`, `docs/ARCHITECTURE.md`,
-  `docs/SOLVER_AGENT_DESIGN.md`
-- reviewed_revisions: `73df844`, `22a3238`
+- repository_policy: `AGENTS.md`, `docs/SOLVER_AGENT_DESIGN.md`,
+  `docs/numerical-reviews/README.md`
+- reviewed_head: `a058ef7`
+- prior_pass_commit: `60b42f4` (`context-only; verdict not inherited`)
 - status: `pass-for-implementation-planning`
 - owner_agent: `numerical-review-agent`
 - date: `2026-08-12`
 - implementation_planning_authorized: `true`
 - implementation_complete: `false`
+- build_test_complete: `false`
 - reference_comparison_complete: `false`
+- physics_evaluation_complete: `false`
+- release_ready: `false`
 
-## Review Verdict
+이번 재검토는 현재 HEAD의 요구조건, 연구, 정식화, I/O 및 reference-case 계약을
+처음부터 상호 대조했다. 기존 review의 판정과 artifact 관찰 결과는 결론의 전제로
+사용하지 않았고, 이전 finding은 현 문서의 수식으로 다시 검산한 뒤 disposition만
+기록했다. 원 MITC4 local paper는 tying 위치와 covariant shear 보간을 확인하는 데
+read-only로 사용했다.
+
+이 단계에서는 Abaqus, Harness, C++ build/test, FESA 실행 및 reference comparison을
+수행하지 않았다. Reference artifact를 생성, 수정, 복원 또는 정규화하지 않았다.
+
+## 2. Review Verdict
 
 - verdict: `pass-for-implementation-planning`
-- reason: The current linear-static formulation closes the physical 20-DOF MITC4
-  kernel, its global 24-DOF embedding, fixed drilling regularization, Jacobian and
-  quadrature rules, residual/stiffness equations, recovery signs, and verification
-  invariants without a mathematical inconsistency in the approved feature scope.
 - critical_blockers: `none`
-- remaining_formulation_revisions: `none for the current linear-static scope`
-- downstream_boundary: Implementation Planning may begin. This verdict does not
-  claim implementation, build/test, reference-comparison, physics-sanity, or release
-  completion.
+- confirmed_defects: `none in the approved current linear-static scope`
+- open_blocking_questions: `none`
+- reason: 현재 정식화는 24 global DOF와 20 physical DOF의 관계, MITC4 shear
+  tying, plane-stress section law, 공통 `2 x 2 x 2` quadrature, residual/stiffness,
+  고정 drilling 안정화, 물리 recovery 및 검증 불변식을 구현계획으로 옮길 수 있을
+  만큼 명시한다. 요구조건, I/O 및 reference 계약과 모순되는 차원, 부호, 위치 또는
+  pass/fail 의미도 발견되지 않았다.
+- downstream_boundary: 이 판정은 Implementation Planning 진입만 허용한다. 구현,
+  MSVC build/CTest, reference comparison, physics sanity 또는 release를 승인하지 않는다.
 
-The future geometrically nonlinear material in Formulation Section 15 remains
-explicitly non-executable. Its unresolved global finite-rotation map and objective
-drilling potential do not block the current linear-static implementation plan.
+정식화 Section 15의 geometrically nonlinear residual/tangent는 future-only다. 완전한
+`Phi: R24 -> R20`, map Hessian, objective drilling potential 및 finite-rotation load work가
+미정인 사실은 미래 nonlinear 구현을 막지만 현재 linear-static 판정은 막지 않는다.
 
-## Critical Findings
+## 3. Critical Findings
 
-No confirmed mathematical defect remains in the approved linear-static formulation.
-The previous review's `needs-reference-model` verdict is not a valid current
-formulation verdict: the current numerical-review gate is based on numerical and
-formulation consistency, while downstream artifact administration and comparison
-execution are separate gates.
+### 3.1 Confirmed defects
 
-### 1. Previous finding disposition
+현재 승인된 선형 정적 범위에서 구현계획 전에 Formulation 또는 Research로 돌려보낼
+confirmed mathematical defect는 없다.
 
-| previous item | current disposition | evidence and strict consequence |
+`K20`의 exact-arithmetic 대칭/positive-semidefinite 구조와 20-to-24 congruence는
+일관된다. 다만 실제 구현의 rank, rigid action, patch field와 reference error는 문서
+검토만으로 통과했다고 볼 수 없으며 Section 6의 downstream test evidence가 필요하다.
+
+### 3.2 Previous finding disposition
+
+| previous item | rerun disposition | current independent basis |
 | --- | --- | --- |
-| `NR-C01` Jacobian/geometry inventory | resolved | Formulation Sections 9.2-9.3 enumerate center, stiffness, tying, and recovery locations and require finite bases, nonzero area, and `J>0`; Requirements 014/016 intentionally define no calibrated smooth-angle, distortion, or warp threshold. |
-| `NR-C02` drilling normalization | resolved | Formulation Section 12.2 and Requirements 033-036 now define one exact dimensional rule using only positive physical tangent-rotation diagonals. No calibration decision remains. |
-| `NR-C03` mixed-DOF algebraic scaling | resolved | Formulation Sections 12.3 and 12.5 define physical length scaling separately from the physical/drilling stiffness split and provide normalized rank, symmetry, and rigid-action evidence. |
-| `NR-C04` 20/24-DOF weak-form mismatch | resolved | Formulation Sections 5.2-5.3 and 7.1-7.2 place physical, drilling, and external work in the common global 24-DOF test space using the required transpose maps. |
-| `NR-C05` nonlinear global tangent closure | resolved for current scope | Formulation Section 15 now labels the nonlinear equations non-executable and identifies the missing nonlinear `Phi` map, map Hessian, and objective drilling potential. Those items block only a future nonlinear feature. |
-| `NR-D01` drilling-direction nodal moment | retained/resolved | The exact-zero branch and `rho_M=|d dot M|/||M|| <= 1e-12` rule are consistent in Formulation Section 6.2 and I/O Section 4.3. |
-| `NR-D02` normalized algebraic checks | retained/resolved | Formulation Section 17.1 defines scale-aware symmetry, rigid-action, frame, transformation-energy, residual, and equilibrium checks without a denominator clamp. |
-| `NR-O01` drilling coefficient/plateau | resolved by approved fixed rule | `k_d=10^-3 min(R+)` replaces the former coefficient-family/sweep question. A sweep, plateau, response sensitivity, or condition-number calibration is not an acceptance gate. |
-| `NR-O02` drilling-energy warning | removed from approved scope | Drilling is an internal numerical potential only. No drilling-energy ratio, warning threshold, or drilling-specific output is required. |
-| `NR-O03` smooth-director angle | removed from approved scope | Requirements 014/054 and Formulation Sections 4.2 and 9.3 use exact orientation/finite/nonzero predicates and explicitly remove `NR-O03`. It is not an open numerical decision. |
-| `NR-O04` distortion/warp calibration | removed from approved scope | Requirements 016/054 and Formulation Sections 9.2-9.3 require exact finite/positive validity checks and explicitly remove `NR-O04`. It is not an open numerical decision. |
-| `NR-O05` U/UR tolerance | resolved | Requirements 058-062, Formulation Section 17.5, I/O Section 7.6, and Reference Case Section 5 all use the exact approved B33 component-scale formula. |
+| `NR-C01` Jacobian/geometry inventory | `resolved` | Formulation 9.2-9.3은 center, eight stiffness points, four tying points 및 committed recovery points를 공통 fail-closed inventory로 두고 finite bases, nonzero surface measure와 `J>0`를 요구한다. 승인 범위는 calibrated distortion/warp cutoff를 요구하지 않는다. |
+| `NR-C02` drilling normalization | `resolved` | Formulation 12.2는 `R+`를 오직 8 physical tangent-rotation diagonals의 finite positive 값으로 제한하므로 모든 후보의 단위가 `force*length`로 같다. |
+| `NR-C03` mixed-DOF spectrum scaling | `resolved` | Formulation 12.5의 `(L_e I3,I2)` 및 `(L_e I3,I3)` congruence는 rank/condition evidence에서 translation/rotation 단위 혼합을 제거한다. Raw mixed-unit spectrum은 금지된다. |
+| `NR-C04` 20/24 weak-form mismatch | `resolved` | Formulation 5.2-5.3과 7.1-7.2는 physical, drilling, external work를 모두 `V24`에서 `T_p^T`와 `T_d^T`로 결합한다. |
+| `NR-C05` nonlinear 20-to-24 closure | `resolved for current scope` | Section 15는 physical chart tangent와 conditional global pullback을 분리하고 map-curvature 항을 보존하며, 미정인 global map/objective drill을 future-only blocker로 명시한다. |
+| `NR-D01` drilling-direction moment | `retained and consistent` | Exact-zero moment는 별도 처리하고 nonzero moment에 `rho_M=abs(d dot M)/norm(M)<=1e-12`를 적용한다. Numerical drilling은 거부된 moment를 운반하지 않는다. |
+| `NR-D02` normalized algebraic checks | `retained and consistent` | `1e-12` symmetry/frame/energy와 `1e-10` rigid/residual/equilibrium 기준은 scaled matrices와 unclamped denominators에 적용된다. |
+| `NR-O01` coefficient sweep/plateau | `closed by product decision` | `k_d=1e-3 min(R+)`가 고정 계약이다. Sweep, plateau 및 coefficient optimality는 구현 gate가 아니다. |
+| `NR-O02` drilling-energy ratio | `removed from scope` | Drilling energy는 내부 quadratic identity일 뿐 physical energy나 mandatory output이 아니며 ratio/warning threshold도 요구하지 않는다. |
+| `NR-O03` smooth-director calibration | `removed from scope` | Pairwise positive incident-normal orientation, finite/nonzero averaging 및 duplicate-node fold modeling이 승인된 exact predicate다. 별도 angle calibration은 gate가 아니다. |
+| `NR-O04` distortion/warp calibration | `removed from scope` | Basic topology, finite/nonzero surface measure 및 required-point `J>0`가 승인된 predicate다. Quality sweep이나 cutoff는 gate가 아니다. |
+| `NR-O05` U/UR tolerance | `resolved` | 모든 관련 문서가 `1e-9+1e-6*reference_scale_c`, U blocking, UR warning-only를 동일하게 정의한다. |
 
-### 2. Required policy classifications
+이전의 `needs-reference-model` 판정에 포함됐던 canonical naming, README,
+`metadata.json`, provenance, expanded portfolio 및 아직 없는 comparison result는 현재
+프로젝트 정책상 formulation verdict의 blocker가 아니다. 현 Reference Model 문서는
+정확한 기존 input/displacement path와 row/tolerance 계약을 제공한다.
 
-#### 2.1 Fixed drilling rule — resolved and implementation-ready
+### 3.3 Open questions
 
-Let `R+` contain only the finite, strictly positive diagonal entries of the physical
-local stiffness `K20` associated with the eight director-tangent rotational DOFs.
-The formulation fixes
+- current_linear_scope: `none blocking`
+- future_geometric_nonlinearity: finite global rotation coordinate, `Phi`와 그 1/2차
+  미분, chart recentering, objective drilling, nodal-moment work 및 nonlinear output/state
+  계약이 미정이다. 이는 별도 future formulation/review가 소유한다.
+- optional_characterization: near-singular positive-J geometry의 conditioning과 original
+  MITC4의 distorted-curved membrane locking을 더 넓게 정량화할 수 있으나 현재 승인된
+  planning/completion gate는 아니다.
+- downstream_results: implementation rank/patch evidence와 S4/S4R comparison 결과는
+  아직 없으며 해당 후속 Agent가 판정한다. 부재 자체는 pre-implementation review의
+  결함이 아니다.
 
-```text
-k_ref = min(R+)
-k_d = 1e-3 * k_ref
-K_drill_local = k_d * I4
-K_drill_24 = T_d^T * K_drill_local * T_d
-```
+## 4. Numerical Risk Assessment
 
-All entries in `R+` have rotational-stiffness dimension `force*length`; translations,
-off-diagonals, nonpositive values, and nonfinite values are excluded. Therefore
-`k_d` has the correct dimension, `K_drill_24` is symmetric positive on the four pure
-drilling coordinates, and the physical and drilling channels are algebraically
-separate. An empty `R+` is a deterministic numerical-validation failure. This is a
-complete algorithm contract, not a calibration placeholder.
-
-The deterministic nodal frames fix the local-coordinate representation, and
-Formulation Section 5.3 supplies the virtual-work/energy congruence used by the
-coordinate-transformation check. No unselected drilling coefficient or family
-remains for Implementation Planning.
-
-#### 2.2 No drilling outputs — resolved and consistent
-
-Formulation Sections 7.2, 12.3, 12.5, 14, and 17.4 keep drilling out of physical
-strain, resultant, stress, and reported physical shell energy. The internal identity
-`E_drill = 0.5 gamma^T K_drill_local gamma` is permissible verification algebra; it
-does not create an external result quantity. Requirements 035/036/046 and I/O
-Sections 6.1/6.4/6.5 consistently require no drilling coefficient, stiffness, ratio,
-or energy dataset. There is no output-contract defect.
-
-#### 2.3 Exact B33 U/UR tolerance — resolved
-
-For each case and component `c`, using only finite Abaqus rows,
-
-```text
-reference_scale_c = max(abs(reference_value_i))
-tolerance_c = 1e-9 + 1e-6 * reference_scale_c
-```
-
-The `1e-9` floor is in the model's user-consistent length unit for `U1/U2/U3` and is
-dimensionless for `UR1/UR2/UR3`. No row is zero-clamped and no row-specific relative
-denominator replaces the component scale. `U1/U2/U3` exceedance is blocking at the
-later Reference Verification gate; `UR1/UR2/UR3` uses the same formula but is
-warning-only. The formula is exact and needs no MITC4-specific calibration.
-
-#### 2.4 Removed and administrative items — nonblocking
-
-`NR-O03`, `NR-O04`, drilling sweeps, drilling-energy criteria, and expanded flat,
-thin/thick, distorted, warped, curved-shell, or mesh-convergence portfolios are not
-part of the approved implementation-planning gate. They may remain future research
-or release evidence but shall not be reintroduced as missing numerical evidence.
-
-Canonical reference naming, a bundle README, `metadata.json`, provenance, an Abaqus
-version, duplicated units/coordinates/model/step/frame descriptions, or a schema
-version are administrative information and cannot block this formulation verdict.
-The absence of optional `metadata.json` is therefore not a defect. If such a file is
-added later, it remains read-only context and cannot override the approved exact
-paths, row mapping, or tolerance.
-
-## Numerical Risk Assessment
-
-| risk | current assessment | required in-scope control |
+| risk label | assessment | required in-scope control |
 | --- | --- | --- |
-| Transverse-shear locking | controlled for the original MITC4 scope, not claimed eliminated for every mesh | Use the exact edge-midpoint covariant shear projection of Sections 10.2-10.4 and pass transverse-shear patch/reference checks. |
-| Curved/distorted-mesh membrane locking | known limitation of the original MITC4 family | Preserve the documented limitation. An expanded convergence portfolio is nonblocking and does not authorize MITC4+. |
-| Volumetric locking | not applicable to the approved plane-stress shell contract | Do not reinterpret `C5` as a full 3D nearly incompressible material law. |
-| Hourglass modes | no reduced-integration/hourglass path is approved | Both source `S4` and `S4R` use the same full `2x2x2` FESA integration and MITC tying path. |
-| Degenerate, inverted, or self-intersecting geometry | fail-closed contract is present | Enforce distinct connectivity, non-self-intersection, finite nonzero surface measure, finite reciprocal bases, and `J>0` at every required location. |
-| Near-singular but still positive geometry | conditioning may degrade because no calibrated quality threshold is in scope | Preserve finite checks and deterministic solver failure diagnostics; do not invent `NR-O04` thresholds. |
-| Opposed or invalid nodal directors | would corrupt frames, signs, and tying | Reject nonfinite/zero candidates, nonpositive incident-normal agreement, and nonfinite/zero averages; use duplicate source nodes for folds. |
-| Drilling singularity | four nonphysical modes would remain without regularization | Apply the exact fixed `R+` rule and verify stabilized nullity six. |
-| Drilling contamination of physics | possible if drill enters the physical operator or recovery | Keep `T_p` and `T_d` separate and verify pure drill has zero physical strain/resultant/stress and no physical energy contribution. |
-| Rigid-mode test contamination | a full spatial rotation vector can contain director-parallel rotation | Construct physical rigid rotation with `u_I=omega x X_I`, `delta d_I=omega x d_I`, and `gamma_I=0` as specified in Section 8.3. |
-| Wrong shear component/factor | would cause patch failure or incorrect shear energy | Keep the `xi-zeta`/`eta-zeta` tying pairs and engineering factor `gamma_ij=2 epsilon_ij` explicit. |
-| Recovery sign/location drift | could hide a correct stiffness behind wrong outputs | Reuse stiffness frames, tying, material, and thickness quadrature; preserve four Gauss identities and bottom/middle/top positions without averaging. |
-| Future nonlinear misuse | current Section 15 does not define a complete global nonlinear element | Keep it non-executable until a separate approved nonlinear formulation closes `Phi`, map curvature, objective drill, load work, and state. |
+| `rigid_body_modes` | Physical `K20`은 six rigid modes와 expected rank 14를 가져야 한다. 24-DOF embedding은 네 drill null coordinates를 더하고 fixed drill block 뒤 expected rank 18/nullity 6이다. | 세 translation과 세 rotation을 명시적으로 구성한다. Rotation mode는 `u_I=omega x X_I`, `theta_I=omega-(omega dot d_I)d_I`, `gamma_I=0`를 사용한다. |
+| `patch_test` | Bilinear membrane/bending field와 MITC tied shear는 required patch states를 표현할 계약을 갖는다. | `E11/E22/G12`, `K11/K22/K12`, `G13/G23`를 독립 시험하고 signs/component order/resultants/stress를 함께 확인한다. |
+| `symmetry` | `B^T C B`, `T_p^T K20 T_p`, `T_d^T(k_d I)T_d`는 exact arithmetic에서 symmetric이다. | Scaled Frobenius check `<=1e-12`; deterministic assembly가 대칭을 깨지 않는지 확인한다. |
+| `positive_definiteness` | Free element는 six-mode semidefinite이고, 충분히 구속된 nonsingular `Kff`는 positive definite가 기대된다. Geometry 또는 supports가 부적절하면 singularity가 정당하다. | Scaled spectrum/rank, non-rigid positive physical energy, constrained solve 및 singular negative cases를 분리한다. |
+| `hourglass` | `1 x 1` reduced integration을 쓰지 않으므로 Abaqus-style hourglass path는 `N/A`다. | Full `2 x 2` midsurface rank test는 유지한다. S4R source label로 reduced rule을 선택하지 않는다. |
+| `shear_locking` | Edge-midpoint MITC projection이 transverse-shear locking을 다루지만 모든 mesh/thickness에서 완전 제거를 주장할 수 없다. | Required shear/bending patch와 declared references를 통과한다. Broader thin/thick convergence는 nonblocking characterization이다. |
+| `membrane_locking` | Original MITC4는 membrane strain을 수정하지 않아 distorted curved meshes에서 알려진 locking 위험이 남는다. | Known limitation을 유지하고 MITC4+ 성능을 주장하지 않는다. Expanded curved/distorted portfolio는 optional이다. |
+| `volumetric_locking` | 승인된 homogeneous plane-stress shell에는 `N/A`다. | `C5`를 3D nearly-incompressible law로 확장하지 않는다. |
+| `distortion` | 양의 `J`를 유지하는 심한 distortion/warpage는 정확도와 rank/conditioning을 악화할 수 있다. | 모든 required location의 exact predicates와 rank/finite-result checks를 시행한다. 승인되지 않은 quality cutoff를 추가하지 않는다. |
+| `singular_jacobian` | Nonfinite/nonpositive `J`, zero surface measure, invalid reciprocal basis는 mapping을 무효화한다. | Center, stiffness, tying 및 recovery inventory 전체를 omission/clamp 없이 fail closed한다. |
+| `conditioning` | Thin shells, near-degenerate positive-J geometry 및 작은 fixed drill scale에서 `Kff` conditioning이 나빠질 수 있다. | Spectrum/condition evidence는 오직 length-scaled matrix를 사용한다. Threshold calibration은 gate가 아니지만 factorization failure는 결정적으로 진단한다. |
+| `convergence` | Current solve는 direct linear solve라 Newton convergence는 `N/A`; spatial convergence와 locking trend는 모델 의존이다. | Free residual/global equilibrium `<=1e-10`과 declared reference cases를 확인한다. Broader mesh sequences는 optional이다. |
+| `drilling_contamination` | Numerical drill이 physical strain/recovery에 들어가면 비물리 결과가 생긴다. | `T_p`/`T_d`를 분리하고 pure drill에서 physical strain/resultant/stress/energy가 zero임을 시험한다. Full residual은 의도대로 stabilized system 전체를 포함한다. |
+| `future_nonlinear_misuse` | Section 15만으로 global nonlinear element를 만들면 nonobjective drill 또는 inconsistent tangent가 된다. | Current plan에서 완전히 제외하고 별도 승인 전 실행하지 않는다. |
 
-## Consistency Checks
+## 5. Consistency Checks
 
-### 1. DOF order, director sign, and coordinate transforms — pass
+### 5.1 Units, dimensions, DOF order, and constrained/free system — `pass`
 
-- Global order is exactly `[UX,UY,UZ,URX,URY,URZ]` per node.
-- `R_I=[a_I b_I d_I]` is right-handed and orthonormal, with
-  `[alpha,beta,gamma]^T=R_I^T theta_I^g`.
-- The director variation `delta d_I=beta_I a_I-alpha_I b_I` has the correct sign for
-  `theta_I x d_I`.
-- `T_p` is `20x24`; `T_d` is `4x24`. The physical operator receives only `q20`, and
-  the drilling potential receives only `gamma`.
-- The transpose maps in Formulation Sections 5.3 and 7.2 preserve virtual work and
-  energy in the common 24-DOF space.
+- Per-node global order is exactly `[UX,UY,UZ,URX,URY,URZ]`; element order is 24
+  global coordinates and 20 physical coordinates
+  `[uX,uY,uZ,alpha,beta]` per node plus four separately selected `gamma` coordinates.
+- `T_p` is `20 x 24`, `T_d` is `4 x 24`, `K20` is `20 x 20`, and both global
+  stiffness contributions are `24 x 24`.
+- Translation-translation, translation-rotation, and rotation-rotation stiffness
+  blocks have units `force/length`, `force`, and `force*length`; `R+` therefore
+  excludes every translational diagonal.
+- The constrained/free equation is `Kff*df=Ff-Kfc*dc`. Stiffness partition and
+  factorization precede load assembly, and an all-constrained valid `0 x 0 Kff` is
+  not reclassified as singular.
+- `r=K*d-F` fixes the internal-minus-external sign. Constrained entries are the
+  required reaction rows and free entries remain residual evidence.
 
-### 2. Shape functions, geometry, and B operator — pass
+### 5.2 Local/global transforms, congruence, and energy — `pass`
 
-- The bilinear shape functions satisfy partition of unity, nodal interpolation, and
-  derivative-sum identities.
-- The degenerated geometry uses a dimensionless unit director and separate thickness
-  factor `t*zeta/2`, avoiding thickness double-counting.
-- The direct covariant strain column is the symmetric gradient written in covariant
-  bases. The two transverse covariant shear components alone are replaced by the
-  canonical MITC4 edge-midpoint interpolation.
-- Reconstructing with reciprocal bases before local projection preserves the tensor
-  meaning. Engineering shear factors are applied once in the local five-component
-  vector.
-- The same projected `B_bar` is used in strain, residual, stiffness, and recovery;
-  no direct/tied shear mismatch remains.
+- `(a_I,b_I,d_I)` and `(e1,e2,e3)` are deterministic right-handed orthonormal frames.
+  The least-aligned-axis nodal rule avoids a fixed-axis parallel singularity.
+- `[alpha,beta,gamma]^T=R_I^T theta_I^g` gives
+  `delta d=beta*a-alpha*b=theta x d` with the required sign.
+- `q20=T_p qg` and `gamma=T_d qg` preserve virtual work. Congruence gives
+  `Kphys24=T_p^T K20 T_p` and `Kdrill24=T_d^T(k_d I4)T_d`; the corresponding local and
+  global quadratic energies are identical.
+- A physical rigid rotation uses only the tangent projection of `omega`, so `gamma=0`
+  and drilling does not destroy the six physical rigid modes.
 
-### 3. Constitutive matrix and dimensional consistency — pass
+### 5.3 Kinematic operators and MITC tying — `pass`
 
-- For finite `E>0` and `-1<nu<0.5`, the plane-stress block and
-  `kappa_s G I2`, with `kappa_s=5/6`, are symmetric positive definite.
-- `B_bar^T C5 B_bar J dxi deta dzeta` has stiffness-consistent dimensions because
-  the geometry Jacobian contains the through-thickness scale.
-- Membrane/shear strain is dimensionless, curvature is `1/length`, `N/Q` is
-  `force/length`, `M` is `force`, stress is `force/length^2`, and physical energy is
+- Bilinear `N_I` satisfies partition, Kronecker and derivative-sum identities.
+- Membrane and bending content comes from the direct covariant small-strain operator.
+  Only `epsilon_xi-zeta` and `epsilon_eta-zeta` are replaced.
+- `epsilon_xi-zeta` is tied at `(0,-1,0)` and `(0,+1,0)` and interpolated in `eta`;
+  `epsilon_eta-zeta` is tied at `(-1,0,0)` and `(+1,0,0)` and interpolated in `xi`.
+  Each interpolation reproduces its own edge value and is constant along the edge
+  direction, matching the original MITC4 construction.
+- The assumed covariant tensor is reconstructed through reciprocal bases, projected
+  into the stored local Cartesian frame, and converted once to engineering shear
+  `gamma_ij=2 epsilon_ij`. The same projected `B_bar` drives strain, internal force,
+  stiffness and recovery.
+
+### 5.4 Constitutive and section matrices — `pass`
+
+- `Cps=E/(1-nu^2)[[1,nu,0],[nu,1,0],[0,0,(1-nu)/2]]` uses engineering `G12`; its
+  shear coefficient is exactly `G=E/[2(1+nu)]`.
+- `C5=diag(Cps,(5/6)G I2)` is symmetric positive definite for `E>0` and
+  `-1<nu<0.5`. `sigma33=0` and absent thickness stretch remain assumptions.
+- `A=t Cps`, `B=0`, `D=t^3 Cps/12`, and `As=(5/6)Gt I2` have consistent dimensions.
+  Membrane/shear strains are dimensionless, curvature is `1/length`, `N/Q` is
+  `force/length`, `M` is `force`, stress is `force/length^2`, and energy is
   `force*length`.
-- The drilling reference uses rotational stiffness only, so no translation/rotation
-  unit mixing occurs.
 
-### 4. Integration, residual, stiffness, and modes — pass
+### 5.5 Jacobian, derivative transform, and integration — `pass`
 
-- Stiffness uses deterministic two-point Gauss quadrature in each of `xi`, `eta`, and
-  `zeta`, with points `+-1/sqrt(3)` and unit weights.
-- Both source types select this one rule; FESA does not emulate Abaqus S4/S4R internal
-  integration or hourglass behavior.
-- `K20 = integral(B_bar^T C5 B_bar dV)` is symmetric positive semidefinite. The
-  expected physical rank is 14: 20 physical coordinates minus six rigid modes.
-- `Kphys24=T_p^T K20 T_p` adds four drilling null coordinates. The fixed positive
-  drilling block removes those four, leaving exactly six physical rigid modes.
-- `f_int=K_e q_g` and `r=K d-F` use a consistent linear sign. Partitioning uses
-  `Kff df=Ff-Kfc dc`, including the valid `0x0 Kff` all-constrained case.
+- The three-dimensional degenerated mapping uses
+  `X=sum(N X_I)+(t*zeta/2)sum(N d_I)` with a unit nodal director and separate
+  thickness, preventing nodal thickness double counting.
+- `J=det[G_xi,G_eta,G_zeta]` and finite covariant/reciprocal bases are checked at all
+  eight stiffness points, all four midsurface tying points, center, and every
+  committed bottom/middle/top recovery evaluation. Failed points are not skipped,
+  averaged, clamped or repaired.
+- Direct natural derivatives are converted covariantly and then to the local
+  Cartesian tensor through contravariant bases; no flat-element derivative shortcut
+  is substituted for curved/warped accepted geometry.
+- Both S4 and S4R use the common in-plane `2 x 2` points
+  `+-1/sqrt(3)` with unit weights and two identical thickness points, for eight
+  volume evaluations. Tied shear is evaluated at `zeta=0` and reused at both
+  thickness points while the remaining mapping and direct components use the actual
+  thickness point.
 
-### 5. Recovery and external result meaning — pass
+### 5.6 Internal force, residual, stiffness, and future tangent — `pass`
 
-- Nodal reactions are constrained entries of the assembled full residual; free
-  entries remain equilibrium evidence.
-- Generalized strains are thickness moments with order
-  `[E11,E22,G12,K11,K22,K12,G13,G23]`.
-- Resultants use `[N11,N22,N12,M11,M22,M12,Q13,Q23]` and the stated centered-layer
-  `A/D/A_s` cross-check.
-- Bottom/middle/top `[S11,S22,S12]` are direct section-position evaluations. `S33=0`
-  is documented but not emitted, and `S13/S23` point stress is outside the output
-  contract.
-- Physical shell energy excludes numerical drilling stabilization, matching the I/O
-  schema.
+- Current `K20=integral(B_bar^T C5 B_bar dV)` and `f_int20=K20 q20` are mutually
+  consistent and symmetric positive semidefinite in exact arithmetic.
+- The complete current weak form is in `V24` and adds the numerical drilling
+  gradient before subtracting the global nodal `CLOAD` vector.
+- No geometric stiffness or nonlinear state enters the current product path.
+  Future Section 15 correctly separates `Kmat` and the stress-dependent `Kgeo` and,
+  conditionally on a future `Phi`, includes both `A^T K20 A` and the
+  residual-weighted map-Hessian term.
 
-### 6. Architecture and deterministic lifecycle — pass for planning
+### 5.7 Fixed drilling contract — `pass`
 
-The formulation and I/O handoff match ADR-007/008/009/016/017 and the architecture:
-stable element-local computation, deterministic COO/reduction, stiffness assembly
-and partition before load assembly, factorization before substitution, full residual
-recovery, stable row identity, and failure-atomic HDF5 commit. These are planning and
-later implementation-test obligations, not unresolved equations.
+- `R+` contains only finite strictly positive diagonals of the eight physical
+  tangent-rotation coordinates. `k_ref=min(R+)`, `k_d=1e-3 k_ref`, and
+  `Kd_local=k_d I4` are dimensionally consistent and deterministic; empty `R+`
+  fails validation.
+- `T_d^T(k_d I4)T_d` is symmetric and positive on the four pure drilling
+  coordinates. It must remove those four nonphysical null modes without changing
+  the physical rank/null modes.
+- Drilling is excluded from generalized strain/resultant/stress and reported
+  physical strain energy. A director-parallel applied nodal moment is rejected as
+  `unsupported-drilling-load`; no numerical drill load channel exists.
 
-## Verification Readiness
+### 5.8 Recovery, signs, locations, units, and external comparison — `pass`
 
-### Required element and algebraic tests
+- Nodal `[U1,U2,U3,UR1,UR2,UR3]` and full-residual
+  `[RF1,RF2,RF3,RM1,RM2,RM3]` are global and source-node ordered.
+- Four midsurface Gauss rows recover
+  `[E11,E22,G12,K11,K22,K12,G13,G23]` and
+  `[N11,N22,N12,M11,M22,M12,Q13,Q23]` in the stored local frame. The definition
+  `e_m(z)=epsilon0+z*kappa` fixes curvature, moment, and bottom/top stress signs.
+- `[S11,S22,S12]` is evaluated directly at `zeta=-1,0,+1`; `S33=0` is documented but
+  not emitted, and `S13/S23` point stress is not synthesized. Different natural or
+  section locations are never averaged.
+- Reference comparison first rejects missing, extra, duplicate, nonfinite or
+  identity-mismatched rows. For each case/component,
+  `reference_scale_c=max(abs(finite Abaqus values))` and
+  `tolerance_c=1e-9+1e-6*reference_scale_c`; no zero clamp or row denominator is
+  introduced. U1/U2/U3 is blocking and UR1/UR2/UR3 is warning-only.
+- Source S4 and S4R select the same FESA MITC4 kernel/quadrature/recovery path while
+  preserving source type. This is an input mapping, not an Abaqus formulation,
+  integration, stabilization or recovery equivalence claim.
 
-Implementation Planning shall trace RED/GREEN/VERIFY tests for:
+## 6. Verification Readiness
 
-1. shape-function identities and deterministic right-handed nodal/integration frames;
-2. valid and invalid geometry at every center, Gauss, tying, and recovery location;
-3. `T_p`/`T_d` dimensions, virtual-work equality, and transformation-energy equality;
-4. direct versus tied shear component construction and engineering-shear factors;
-5. constitutive symmetry/positive definiteness and exact `2x2x2` quadrature;
-6. normalized symmetry at `1e-12`, rigid action at `1e-10`, physical rank 14, and
-   stabilized rank 18/nullity six;
-7. exact `R+`, `k_ref`, `k_d`, and `K_drill_local` construction, including empty-`R+`
-   failure and exclusion of translational diagonals;
-8. pure drilling: positive drill action, zero physical strain/resultant/stress, and
-   no drilling-specific output;
-9. deterministic assembly/recovery ordering and thread-count repeatability;
-10. partition/effective-RHS/full-residual reaction behavior, including all-constrained
-    `0x0 Kff` handling.
+### 6.1 Downstream unit and invariant tests
 
-For a nonzero scaled stiffness, use the formulation's normalized metrics without a
-fallback denominator. Exactly zero constructed energy cases are classified by their
-separate rigid/null action tests rather than clamped to pass.
+Implementation Planning shall convert the following to `RED -> GREEN -> VERIFY`:
 
-### Required patch and sign tests
+1. Shape identities; nodal/integration frame orthonormality, handedness and axis
+   tie-break determinism.
+2. `T_p`/`T_d` dimensions, orthogonal channel selection, virtual-work equality and
+   nonzero transformation-energy equality.
+3. Hand-calculated direct membrane/bending columns, all four covariant tying values,
+   interpolation weights and engineering-shear factors.
+4. `Cps/C5/A/D/As` coefficients, symmetry, positive definiteness, dimensions and
+   force/length unit-rescaling invariance.
+5. Common `2 x 2 x 2` point/weight order and an independent analytical or
+   higher-order flat-element stiffness/recovery cross-check.
+6. Required-location geometry validation: valid planar/rotated/warped cases and
+   duplicate, bow-tie/self-intersecting, zero-area, reversed, nonfinite,
+   nonpositive-J and opposed-normal negative cases.
+7. Scaled symmetry `<=1e-12`, physical rigid action `<=1e-10`, expected physical
+   rank 14, stabilized rank 18/nullity six, and positive non-rigid physical energy.
+8. Exact `R+` membership, exclusion of translations, fixed coefficient, empty-`R+`
+   failure, pure drill action and zero physical recovery/energy.
+9. Stable COO/reduction, source/result/diagnostic order and thread-count
+   repeatability.
+10. `Kff/Kfc` effective RHS, nonzero prescribed values, full-residual reaction,
+    singular-support negative case, and valid all-constrained `0 x 0 Kff` case.
+11. Exact-zero and accepted/rejected `rho_M` moment projections, including proof
+    that rejected drilling moments never reach stabilization.
+12. Mandatory HDF5 locations/components/units, physical-only energy, nonfinite
+    recovery failure and atomic finalization.
 
-- constant membrane strain/stress and `N` sign;
-- pure bending about both local axes, curvature/moment order, and bottom/top stress sign;
-- constant transverse shear and `Q13/Q23` order;
-- pure twist and `K12/M12` convention;
-- zero physical recovery from a pure drilling vector.
+### 6.2 Required patch and sign tests
 
-### Declared reference readiness
+- independent constant `E11`, `E22`, and `G12` membrane fields with `N` and
+  middle-stress signs;
+- pure `K11` and `K22` bending with `M` order and bottom/top stress reversal;
+- pure `K12` twist with `M12` sign;
+- constant `G13` and `G23` transverse shear with `Q13/Q23` order;
+- six physical rigid states and four pure drilling states;
+- source-type-only S4/S4R variants producing identical FESA numeric rows and
+  different preserved source metadata.
 
-The lightweight inventory identifies these read-only required pairs:
+### 6.3 Reference and physics handoff readiness
 
-- `reference/shell/shell.inp` and
-  `reference/shell/shell displacements.csv` for source `S4`;
-- `reference/shellR/shellR.inp` and
-  `reference/shellR/shellR displacements.csv` for source `S4R`.
+The declared read-only pairs are:
 
-Read-only inspection confirmed that all four declared files exist, their SHA-256
-values match the Reference Case inventory, and each required displacement CSV has
-49 data rows. This is inventory evidence only. No FESA output exists yet in this
-review, and no reference-comparison pass/fail decision was made.
+- `reference/shell/shell.inp` with
+  `reference/shell/shell displacements.csv` for S4;
+- `reference/shellR/shellR.inp` with
+  `reference/shellR/shellR displacements.csv` for S4R.
 
-The later comparator must require exact normalized source-row/component sets,
-finite/unique values, U blocking, UR warning-only, and the approved mixed tolerance.
-The two Abaqus cases are not expected to equal one another, while identical supported
-FESA models labeled S4 or S4R must take the same internal numerical path.
+The Reference Model and I/O documents define deterministic HDF5-to-CSV identity,
+precheck and tolerance sufficiently for later comparison. This review did not assert
+that `results.h5` exists or that any row passes. Reference Verification owns numeric
+U/UR outcome; Physics Evaluation owns force/moment balance, displacement direction,
+symmetry, result signs, recovered-resultant consistency and physical plausibility.
 
-### Nonblocking evidence
+### 6.4 Missing evidence classification
 
-The following cannot change this formulation verdict:
+- blocking_for_current_formulation: `none`
+- required_after_implementation: invariant, patch, MSVC build/CTest, declared
+  reference comparison and physics evidence above
+- nonblocking_optional: drilling coefficient sweep/energy ratio, `NR-O03`, `NR-O04`,
+  canonical naming, README/metadata/provenance, expanded benchmark portfolio and
+  broader mesh convergence studies
+- future_only: nonlinear directional-derivative/objectivity/Newton evidence after
+  its missing formulation decisions are separately approved
 
-- absent README, `metadata.json`, provenance, canonical name, schema-version record,
-  or duplicated bundle descriptions;
-- no coefficient sweep, drilling-energy ratio, smooth-angle calibration, or
-  distortion/warp threshold sweep;
-- no expanded flat/thin/thick/distorted/warped/curved/convergence benchmark portfolio;
-- no implementation result, build/test result, Abaqus run, or completed comparison
-  at this pre-implementation gate.
-
-## Required Revisions
+## 7. Required Revisions
 
 ### Formulation Agent
 
 - None for the approved current linear-static implementation scope.
-- Keep Formulation Section 15 non-executable until a separately approved nonlinear
-  feature closes its global coordinate map, consistent tangent, objective drilling,
-  and load-work decisions.
+- Do not promote Section 15 to executable status until a separate formulation closes
+  the nonlinear global coordinate map, objective drilling and load-work contracts.
 
 ### Research Agent
 
 - None before current Implementation Planning.
-- Broader original-MITC4 locking and convergence studies remain optional future
-  characterization and must not silently widen the implementation gate.
-
-### Reference Model Agent
-
-- None for this formulation verdict. Preserve the four declared artifacts read-only.
-- Optional administrative metadata, if later added by an authorized phase, does not
-  replace the approved exact paths, matching, and tolerance contract.
-
-## Downstream Handoff
-
-### Implementation Planning Agent
-
-Implementation Planning is authorized. The plan shall:
-
-- cover the deterministic director/frame preprocessing, `24 -> 20 + 4` transforms,
-  covariant MITC tying, full `2x2x2` integration, fixed drilling split, and recovery;
-- trace every approved must-requirement to TDD tests, including the invariant, patch,
-  fixed-drilling, failure, schema, row-matching, and U/UR decision behaviors above;
-- preserve one internal `FESA-MITC4` numerical path for source S4 and S4R while keeping
-  source metadata distinct;
-- keep drilling out of physical recovery and HDF5 results;
-- exclude future nonlinear execution, calibration sweeps, removed `NR-O03/NR-O04`,
-  expanded portfolios, and administrative reference requirements.
-
-This handoff authorizes planning only. It does not authorize Harness execution,
-production-code changes, reference-artifact mutation, or a claim of implementation
-completion.
+- Optional locking/convergence characterization must remain clearly outside the
+  approved implementation gate and must not imply MITC4+ or Abaqus equivalence.
 
 ### I/O Definition Agent
 
-The current I/O contract is numerically consistent with the formulation. Planning
-shall preserve its exact source identity, load projection, output units/locations,
-physical-energy meaning, reference row mapping, and U/UR decision rule.
+- None for the current numerical verdict. Preserve exact physical/full-residual
+  distinction, source identity, locations, units and U-versus-UR decision rule.
 
-### Reference Verification and Physics Evaluation Agents
+### Reference Model Agent
 
-These remain downstream of implementation and build/test. Reference Verification
-will decide U/UR comparison outcomes; Physics Evaluation will independently assess
-equilibrium, signs, symmetry, and physical plausibility. Neither result is asserted
-by this review.
+- None for the current numerical verdict. Preserve the four declared files read-only
+  and do not add administrative or portfolio gates.
 
-## Review Evidence
+## 8. Downstream Handoff
 
-This review used the repository policy/design files, the approved requirements,
-research, formulation, I/O definition and reference-case inventory listed in
-Metadata, plus read-only inspection of the four declared artifacts. Local FEM wiki
-material cross-checked MITC4 kinematics, edge-midpoint assumed shear and known locking
-risks; the approved repository documents remain the feature source of truth.
+### Implementation Planning Agent
+
+Implementation Planning is authorized and shall:
+
+- trace the required tests in Section 6 to the approved requirement IDs before
+  production work;
+- keep `24 global -> 20 physical + 4 drilling` transforms, covariant MITC tying,
+  common `2 x 2 x 2` integration, fixed drilling and physical recovery as explicit
+  independent test seams;
+- preserve stiffness assembly/partition/factorization-before-load, stable reduction,
+  full-residual reaction and failure-atomic HDF5 lifecycle;
+- keep future nonlinear execution, coefficient calibration, drilling output,
+  `NR-O03/NR-O04`, reference-artifact mutation and Abaqus-equivalence claims outside
+  the plan.
+
+This handoff authorizes planning only. It does not authorize Harness execution,
+production implementation, reference artifact changes, or completion claims.
+
+### Reference Verification Agent
+
+- Compare authoritative FESA HDF5 rows directly with the matching declared Abaqus
+  displacement CSV after exact row-set precheck.
+- Let only U1/U2/U3 affect pass/fail; report every UR1/UR2/UR3 warning without
+  changing the verdict.
+
+### Physics Evaluation Agent
+
+- After reference verification, independently evaluate force and global moment
+  balance, free residual, reaction sign, displacement direction, symmetry, positive
+  physical energy and consistency of local resultants/stresses.
+
+### Coordinator and Release Agents
+
+- Record the Numerical Review gate as passed for planning at HEAD `a058ef7`.
+- Do not infer implementation or release completion. Build/test, reference,
+  physics-sanity and release-readiness gates remain pending.