docs: revise MITC4 shell formulation after review

docs/formulations/mitc4-shell-formulation.md

@@ -5,9 +5,13 @@
 - feature_id: `linear-static-mitc4-shell`
 - source_requirement: `docs/requirements/linear-static-mitc4-shell.md`
 - source_research: `docs/research/linear-static-mitc4-shell-research.md`
+- source_numerical_review: `docs/numerical-reviews/linear-static-mitc4-shell-review.md`
 - status: `ready-for-numerical-review`
 - owner_agent: `formulation-agent`
 - date: `2026-08-11`
+- revision_basis: numerical review commit `0a5aad4`; findings `NR-C01` through
+  `NR-C05` and decisions `NR-D01` through `NR-D02`
+- revision_state: `ready-for-numerical-rereview-not-implementation-planning`
 - product_execution_scope: `small-strain, small-rotation linear static only`
 - future_formulation_scope: `geometrically nonlinear Total Lagrangian residual/tangent; not executable`
 
@@ -37,10 +41,11 @@ Source labels `S4` and `S4R` both select this one FESA formulation by an approve
 semantic mapping. They do not select Abaqus integration or stabilization behavior,
 and this document makes no Abaqus formulation-equivalence claim.
 
-Exact drilling scale/coefficient, drilling-energy warning ratio, smooth-director
+Exact drilling reference family/coefficient, drilling-energy warning ratio,
-angle, drilling-load projection tolerance, and Jacobian/warpage thresholds remain
+smooth-director angle, and geometry thresholds remain Numerical Review decisions.
-Numerical Review decisions. Symbols are retained for them; no implementation default
+The common dimensionless drilling coordinate, geometry-measure inventory,
-is implied.
+drilling-load projection tolerance, and normalized algebraic checks are fixed below
+by the first Numerical Review. No remaining open symbol is an implementation default.
 
 ## 2. Scope and assumptions
 
@@ -192,7 +197,20 @@ using the same four midsurface Gauss locations as the element formulation.
 
 ### 4.2 Smooth shared-node director
 
-For source node `I` and its incident accepted shell elements `E(I)`, define
+For source node `I`, sort its incident accepted shell elements `E(I)` by stable
+source-element identity. Before any averaging, require every unordered incident
+pair to have the same orientation hemisphere:
+
+$$
+c_{pair,I}=\min_{e<f,\ e,f\in E(I)}\mathbf n_e\cdot\mathbf n_f>0.
+$$
+
+For a node with only one incident element this pair test is vacuous. A nonpositive
+pair dot product is an `opposed-incident-normal` geometry error; it is not repaired
+by flipping a candidate or by letting cancellation occur in the average. This
+pairwise rule is deterministic and stronger than testing only the final average.
+
+After the pairwise test, define
 
 $$
 \mathbf s_I=\sum_{e\in E(I)} A_e\mathbf n_e,
@@ -200,8 +218,8 @@ $$
 \mathbf d_I=\frac{\mathbf s_I}{\|\mathbf s_I\|}.
 $$
 
-Incident elements are accumulated in stable source-element order. Before averaging,
+Incident elements are accumulated in the same stable order. After averaging, every
-opposing orientations fail. After averaging, every incident deviation
+incident deviation
 
 $$
 \theta_{eI}=\cos^{-1}\!\left(
@@ -352,12 +370,19 @@ Prescribed global shell DOFs form the Dirichlet data. The only external generali
 forces are aggregated nodal `CLOAD` values. For a nodal moment `M_I`, define
 
 $$
-M_{d,I}=\mathbf d_I\cdot\mathbf M_I.
+\rho_{M,I}=\frac{|\mathbf d_I\cdot\mathbf M_I|}{\|\mathbf M_I\|}.
 $$
 
-If `M_d,I` is nonzero beyond an approved scale-aware projection tolerance, model
+An exactly zero aggregated moment is accepted as a separate zero case and does not
-validation fails as `unsupported-drilling-load`. The tolerance is a Numerical
+form this ratio. Every nonzero moment is accepted only when
-Review/I-O decision and is not replaced by an arbitrary absolute epsilon.
+
+$$
+\boxed{\rho_{M,I}\le 10^{-12}}.
+$$
+
+Otherwise model validation fails as `unsupported-drilling-load`. The denominator is
+never clamped. This is a direction/projection check and is invariant under a
+consistent moment-unit conversion.
 
 Distributed natural-boundary terms may appear in the parent continuum derivation,
 but their existence is not an executable load contract.
@@ -366,31 +391,43 @@ but their existence is not an executable load contract.
 
 ### 7.1 Physical virtual work
 
-Let `V_20` be the admissible five-DOF element trial space and `V_20^0` its homogeneous
+Let `V_24` be the admissible global six-DOF element trial space after essential
-test space. The physical linear equilibrium is
+boundary data are applied, and let `V_24^0` be its homogeneous test space. With
 
 $$
-\delta W_{int}^{phys}-\delta W_{ext}=0
+\mathbf q_{20}=\mathbf T_p\mathbf q_g,
-\qquad\forall\delta\mathbf q_{20}\in V_{20}^0,
+\qquad
+\delta\mathbf q_{20}=\mathbf T_p\delta\mathbf q_g,
+$$
+
+the MITC-projected physical internal virtual work is
+
+$$
+\delta W_{int}^{phys}
+=\int_{\Omega_0}
+(\delta\overline{\boldsymbol\epsilon})^T
+\mathbf C_5
+\overline{\boldsymbol\epsilon}\,dV
+=\delta\mathbf q_{20}^T\mathbf f_{int}^{20}
+=\delta\mathbf q_g^T\mathbf T_p^T\mathbf f_{int}^{20},
 $$
 
 where
 
 $$
-\delta W_{int}^{phys}
+\mathbf f_{int}^{20}
-=\int_{\Omega_0}
+=\int_{\Omega_0}\overline{\mathbf B}^T
-\delta\overline{\boldsymbol\epsilon}^{,T}
+\mathbf C_5\overline{\boldsymbol\epsilon}\,dV.
-\mathbf C_5
-\overline{\boldsymbol\epsilon}\,dV,
 $$
 
-and `overbar` denotes the MITC-projected shear field. In the current product path
+The overbar denotes the common MITC projection used by strain, residual, and
+stiffness. The current product external virtual work is
 
 $$
 \delta W_{ext}=\delta\mathbf q_g^T\mathbf f_e^{CLOAD}.
 $$
 
-### 7.2 Numerical drilling potential
+### 7.2 Numerical drilling potential and complete weak form
 
 Drilling regularization is algebraically separate:
 
@@ -404,6 +441,22 @@ $$
 It does not approximate a shell strain energy and produces no physical generalized
 strain, resultant, or stress.
 
+The complete linear 24-DOF equilibrium statement is therefore
+
+$$
+\boxed{
+\delta\mathbf q_g^T\left[
+\mathbf T_p^T\mathbf f_{int}^{20}
++\mathbf T_d^T\mathbf K_d^l\boldsymbol\gamma
+-\mathbf f_e^{CLOAD}
+\right]=0
+\qquad\forall\delta\mathbf q_g\in V_{24}^0.}
+$$
+
+This is the normative weak form. A physical-only statement in `V_20` is merely the
+restriction of this equation and must not be combined with a 24-DOF external-work
+term as if both variations occupied the same test space.
+
 ## 8. Bilinear discretization and degenerated geometry
 
 ### 8.1 Shape functions
@@ -517,22 +570,88 @@ $$
 - any additional bottom/middle/top recovery evaluation location before recovery is
   committed.
 
+The center `(0,0,0)` is also included in the orientation and warpage inventory.
 Duplicate nodes, self-intersection, degenerate midsurface area, and reversed
-connectivity are separate fail-closed geometry errors.
+connectivity are separate fail-closed geometry errors. No failed location is
+discarded or replaced by a value from another point.
 
 ### 9.3 Scale-aware measures
 
-A dimensionless local determinant measure is
+Let the consecutive midsurface edge inventory be
+
+$$
+\mathcal E=\{(1,2),(2,3),(3,4),(4,1)\},
+\qquad
+\boxed{L_e=\max_{(I,J)\in\mathcal E}\|\mathbf X_J-\mathbf X_I\|}.
+$$
+
+`L_e` must be finite and strictly positive. It is the common element length used by
+geometry checks and the DOF scaling in Section 12.5. At every distinct in-plane
+location in the center, Gauss, tying, and committed recovery inventory, define
+
+$$
+a(\xi,\eta)=\|\mathbf A_\xi\times\mathbf A_\eta\|,
+\qquad
+\boxed{a_g(\xi,\eta)=\frac{a(\xi,\eta)}{L_e^2}},
+$$
+
+$$
+\mathbf n_s(\xi,\eta)=
+\frac{\mathbf A_\xi\times\mathbf A_\eta}{a(\xi,\eta)},
+\qquad
+c_d(\xi,\eta)=\mathbf n_s\cdot\overline{\mathbf d}.
+$$
+
+`a_g` is the normalized surface-collapse/aspect measure. In particular, for
+`A_xi=(1,0,0)` and `A_eta=(0,epsilon,0)` with `L_e=O(1)`, `a_g -> 0` as
+`epsilon -> 0`; the angular measure below alone cannot detect that collapse.
+
+At every full three-dimensional validation point, define the dimensionless
+angular/director determinant measure
 
 $$
 j_s=\frac{J}
 {\|\mathbf G_\xi\|\,\|\mathbf G_\eta\|\,\|\mathbf G_\zeta\|}.
 $$
 
-A valid point requires `J>0` and `j_s>tau_J`; a valid element also satisfies an
+For all positive finite point determinants, define the element-variation measure
-approved distortion/warpage inventory such as `J_min/J_max`. The exact `tau_J` and
+
-warpage bounds are Numerical Review decisions. No `max(1, geometry_scale)` clamp is
+$$
-permitted because it destroys consistent-unit scaling.
+\boxed{r_J=\frac{J_{min}}{J_{max}}},
+\qquad
+J_{min}=\min_{p\in\mathcal P_V}J_p,
+\quad
+J_{max}=\max_{p\in\mathcal P_V}J_p,
+$$
+
+and the surface-normal warpage measure relative to the center normal
+
+$$
+\boxed{\theta_w=
+\max_{p\in\mathcal P_S}
+\cos^{-1}\!\left(\operatorname{clamp}
+(\mathbf n_s(0,0)\cdot\mathbf n_s(p),-1,1)\right)}.
+$$
+
+Here `P_V` contains every volume Gauss point, tying point at `zeta=0`, center, and
+every committed bottom/middle/top recovery point; `P_S` contains their distinct
+in-plane projections. A valid element must satisfy, without denominator clamping,
+
+$$
+\boxed{
+J_p>0,\quad
+j_{s,p}>\tau_{ang},\quad
+a_{g,p}>\tau_{area},\quad
+c_{d,p}>\tau_{dir},\quad
+r_J>\tau_{var},\quad
+\theta_w<\theta_{warp}.}
+$$
+
+The measures and their location inventory are fixed by this formulation revision.
+The positive dimensionless thresholds remain `needs-numerical-calibration`; they
+must separate valid distortion/warp sweeps from collapsed negative sequences before
+Numerical Review may approve them. No `max(1, geometry_scale)`, zero denominator,
+failed-point omission, or pointwise orientation repair is permitted.
 
 ## 10. Linear kinematics and MITC4 shear projection
 
@@ -736,15 +855,36 @@ $$
 \qquad k_{d,I}>0,
 $$
 
-with rotational-stiffness units `force*length`. Write
+with rotational-stiffness units `force*length`. The common physical normalization is
 
 $$
-k_{d,I}=\alpha_d k_{ref,I},
+D_{iso}=\frac{Et^3}{12(1-\nu^2)},
+\qquad
+\boxed{\rho_{d,I}=\frac{k_{d,I}}{D_{iso}}}.
 $$
 
-where both the dimensionless coefficient `alpha_d` and the reference family remain
+`rho_d,I` is dimensionless and is the only common coordinate for comparing drilling
-unapproved until Numerical Review. Dimensionally compatible candidates carried from
+families. For any candidate `c` written as
-research are:
+
+$$
+k_{d,I}^{(c)}=\alpha_d^{(c)}k_{ref,I}^{(c)},
+$$
+
+the candidate-specific conversion is
+
+$$
+\boxed{
+\rho_{d,I}^{(c)}=\alpha_d^{(c)}
+\frac{k_{ref,I}^{(c)}}{D_{iso}},
+\qquad
+\alpha_{d,I}^{eq,(c)}=\rho_{d,I}^{(c)}
+\frac{D_{iso}}{k_{ref,I}^{(c)}}.}
+$$
+
+The second expression is the nodewise equivalent coefficient for a target `rho_d,I`.
+A single actual candidate coefficient may therefore generate a range of `rho_d,I`; that
+entire range is part of the calibration evidence. The dimensionally compatible
+candidate distributions carried from research are:
 
 1. transverse-shear/area transition family
 
@@ -765,15 +905,38 @@ $$
 2. isotropic bending rigidity
 
 $$
-k_{ref}^{(B)}=D_{iso}=\frac{Et^3}{12(1-\nu^2)};
+k_{ref}^{(B)}=D_{iso},
+\qquad
+\rho_{d,I}^{(B)}=\alpha_d^{(B)};
 $$
 
 3. a documented positive statistic formed only from the physical rotational block
-   of `K_20`, whose entries all have `force*length` units.
+   of `K_20`, whose entries all have `force*length` units, converted by the same
+   `k_ref/D_iso` ratio. A raw statistic is not comparable until this conversion is
+   reported.
+
+The first two candidate scales differ sharply in the thin-shell limit:
+
+$$
+\boxed{\displaystyle \lim_{A_{eI}/t^2\to\infty}
+k_{ref,I}^{(A)}/D_{iso}=6(1-\nu)/q}.
+$$
+
+For `nu=0.3` and `q=2.5e-5`, this ratio is `168000`. Consequently the same raw
+coefficient, including `10^-3`, cannot represent the same small drilling stiffness
+for candidates A and B.
 
 The thesis rule `10^-3 min(all K_ii)` is not admissible because it can mix
-translational `force/length` and rotational `force*length` diagonals. The order
+translational `force/length` and rotational `force*length` diagonals. A sweep must
-`10^-3` may only center a sensitivity study; it is not a nominal coefficient.
+instead expand logarithmically in actual `rho_d,I` until it brackets both:
+
+1. a low-side scaled-rank/conditioning or factorization failure; and
+2. a high-side physical `U/N/M/Q` contamination boundary.
+
+A nominal value may be proposed only as the smallest point in a stable plateau, with
+the adjacent lower and higher decades and separate physical/drilling energies
+reported. The reference family, plateau, nominal value, and response/energy bounds
+remain `needs-numerical-calibration`; no common `10^-3` center is retained.
 
 ### 12.3 Stabilized 24-DOF matrix
 
@@ -802,7 +965,9 @@ For a valid free isolated element, the expected physical rank is 14. Embedding i
 24 coordinates creates the six physical rigid modes plus four drilling null modes.
 Four positive independent `k_d,I` values should remove only those drilling modes,
 giving expected stabilized rank 18 and nullity 6. These are verification targets,
-not substitutes for a computed singular-value/rank study.
+not substitutes for the scaled singular-value/rank study defined in Section 12.5.
+The exact-arithmetic rank statement is independent of the calibrated numerical-rank
+threshold.
 
 ### 12.4 Energy split
 
@@ -827,6 +992,51 @@ The ratio `E_drill/E_phys` is reported only when mathematically classifiable. If
 `E_phys` is zero or near zero, the two energies are reported explicitly; no arbitrary
 denominator clamp is used. The warning ratio remains open.
 
+### 12.5 DOF scaling for rank and conditioning evidence
+
+Raw shell stiffness matrices mix translational and rotational units and therefore
+must not be used for singular-value, eigenvalue, rank, or condition-number
+acceptance. Define diagnostic coordinate scalings
+
+$$
+\mathbf q_{20}=\mathbf S_{20}\widehat{\mathbf q}_{20},\qquad
+\mathbf S_{20}=\operatorname{blockdiag}_{I=1}^{4}(L_e\mathbf I_3,\mathbf I_2),
+$$
+
+$$
+\mathbf q_g=\mathbf S_{24}\widehat{\mathbf q}_g,\qquad
+\mathbf S_{24}=\operatorname{blockdiag}_{I=1}^{4}(L_e\mathbf I_3,\mathbf I_3).
+$$
+
+The corresponding scaled element matrices are
+
+$$
+\boxed{\widehat{\mathbf K}_{20}=\mathbf S_{20}^T\mathbf K_{20}\mathbf S_{20},
+\qquad \widehat{\mathbf K}_e=\mathbf S_{24}^T\mathbf K_e^{24}\mathbf S_{24}}.
+$$
+
+Every entry of these matrices has units `force*length`. Rigid and other diagnostic
+vectors are transformed by `q_hat=S^{-1}q`. Scaling is used only for algebraic
+evidence; it does not modify the physical assembly, prescribed values, or solve.
+
+For a global shell model, use the deterministic model length
+
+$$
+L_m=\max_{e\in\mathcal E_{active}}L_e
+$$
+
+and construct the analogous full-model scaling with `(L_m I_3,I_3)` per six-DOF
+node. Restrict it to the stable free-DOF order as `S_f` and define
+
+$$
+\widehat{\mathbf K}_{ff}=\mathbf S_f^T\mathbf K_{ff}\mathbf S_f.
+$$
+
+Global condition and numerical-rank evidence uses `K_hat_ff`; a valid `0 x 0 Kff`
+case is classified separately and is not reported as singular. Numerical
+rank/condition thresholds remain calibration decisions, but no raw mixed-unit
+matrix may be used to choose them.
+
 ## 13. Numerical integration
 
 ### 13.1 Stiffness rule
@@ -1019,15 +1229,17 @@ locations. This formulation does not define nodal averaging or extrapolation.
 ## 15. Future-only geometrically nonlinear residual and tangent
 
 > **Non-executable boundary:** This section records the requested future formulation.
-> It must not be routed through the current linear-static procedure or state.
+> It must not be routed through the current linear-static procedure or state. The
+> equations through Section 15.3 live only in a 20-coordinate physical director
+> chart; they are not a complete global 24-DOF nonlinear element.
 
 ### 15.1 Total Lagrangian kinematics
 
-Let the physical nonlinear coordinate vector retain three translations and two
+Let `q_20` be the physical nonlinear coordinate vector with three translations and
-director parameters per node. In the reference configuration,
+two director parameters per node. In the reference configuration,
 
 $$
-\mathbf x(\xi,\eta,\zeta;\mathbf q)
+\mathbf x(\xi,\eta,\zeta;\mathbf q_{20})
 =\sum_I N_I(\mathbf X_I+\mathbf u_I)
 +\frac{t\zeta}{2}\sum_I N_I\mathbf d_I(\boldsymbol\phi_I),
 $$
@@ -1050,13 +1262,13 @@ $$
 \mathbf E=\frac12(\mathbf F^T\mathbf F-\mathbf I).
 $$
 
-For physical coordinates `q_A` and `q_B`, define
+For physical chart coordinates `q_20,A` and `q_20,B`, define
 
 $$
-\mathbf h_A=\frac{\partial\mathbf x}{\partial q_A},
+\mathbf h_A=\frac{\partial\mathbf x}{\partial q_{20,A}},
 \qquad
 \mathbf h_{AB}=\frac{\partial^2\mathbf x}
-{\partial q_A\partial q_B}.
+{\partial q_{20,A}\partial q_{20,B}}.
 $$
 
 The covariant Green-Lagrange components and their exact first and second derivatives
@@ -1124,8 +1336,8 @@ Use second Piola-Kirchhoff stress `S` work-conjugate to `E`. In engineering-vect
 notation, define
 
 $$
-\mathbf B_A(\mathbf q)=
+\mathbf B_A(\mathbf q_{20})=
-\frac{\partial\overline{\mathbf e}}{\partial q_A}.
+\frac{\partial\overline{\mathbf e}}{\partial q_{20,A}}.
 $$
 
 For the geometrically nonlinear but materially linear candidate,
@@ -1145,17 +1357,14 @@ Then the physical internal residual is
 
 $$
 \boxed{
-r_A^{phys}(\mathbf q)
+r_A^{phys}(\mathbf q_{20})
 =\int_{\Omega_0}\mathbf B_A^T\mathbf s\,dV}.
 $$
 
-The total residual is
+Collecting these entries gives the physical chart residual
-
+`r_20^phys(q_20) in R^20`. A total 24-DOF residual is deliberately not written at
-$$
+this point: drilling and external load can be combined with the physical residual
-\mathbf r(\mathbf q)=
+only after the nonlinear 20-to-24 coordinate map in Section 15.4 is selected.
-\mathbf r^{phys}(\mathbf q)
-+\mathbf r^{drill}(\mathbf q)-\mathbf f_{ext}(\mathbf q).
-$$
 
 ### 15.3 Consistent material and geometric tangent
 
@@ -1166,10 +1375,10 @@ $$
 {\partial\overline{\mathbf e}},
 \qquad
 \mathbf G_{AB}=\frac{\partial^2\overline{\mathbf e}}
-{\partial q_A\partial q_B}.
+{\partial q_{20,A}\partial q_{20,B}}.
 $$
 
-The consistent physical tangent is
+The consistent `20 x 20` physical-chart tangent is
 
 $$
 \boxed{
@@ -1189,56 +1398,94 @@ K_{AB}^{geo}=\int_{\Omega_0}
 $$
 
 `K_geo` contains the stress-dependent second variation of Green-Lagrange strain,
-including consistent finite-director derivatives. The future total tangent is
+including consistent finite-director derivatives. These equations define
+`K_20^phys=K_20^mat+K_20^geo`; they do not yet define congruence to a global
+24-DOF tangent. For a constant generalized dead-load vector in this selected
+20-coordinate chart, its chart load tangent is zero. Follower pressure and other
+configuration-dependent loads are outside scope. Finite-rotation nodal-moment work
+must be separately defined before assuming its load tangent is zero.
+
+### 15.4 Unresolved 20-to-24 mapping and drilling boundary
+
+Let a future finite global six-DOF coordinate vector be `q_g` and suppose an
+approved nonlinear map exists:
 
 $$
-\boxed{
+\mathbf q_{20}=\boldsymbol\Phi(\mathbf q_g),\qquad
-\mathbf K_T=
+\mathbf A(\mathbf q_g)=\partial\boldsymbol\Phi/\partial\mathbf q_g.
-\mathbf K^{mat}+\mathbf K^{geo}+\mathbf K^{drill}-\mathbf K^{load}}.
 $$
 
-For a constant generalized dead-load vector in the selected coordinate chart,
+Here `A` is a `20 x 24` mapping Jacobian. Conditional on that map, virtual work gives
-`K_load=0`. Follower pressure and other configuration-dependent loads are outside
-scope. Finite-rotation nodal-moment work must be separately defined before assuming
-its load tangent is zero.
-
-### 15.4 Future drilling tangent boundary
-
-If the linear initial-frame penalty is formally retained,
 
 $$
-\Pi_d=\frac12\boldsymbol\gamma(\mathbf q)^T
+\mathbf r_g^{phys}=\mathbf A^T\mathbf r_{20}^{phys}.
-\mathbf K_d^l(\mathbf q)\boldsymbol\gamma(\mathbf q),
+$$
+
+Its consistent global physical tangent must contain both the congruence and the
+coordinate-map curvature term:
+
+$$
+\boxed{\mathbf K_g^{phys}=\mathbf A^T\mathbf K_{20}^{phys}\mathbf A+\sum_{a=1}^{20}r_{20,a}^{phys}(\partial^2\Phi_a/\partial\mathbf q_g^2)}.
+$$
+
+The candidate two-parameter director chart in Section 15.1 does not by itself
+define `Phi`: a finite three-component global rotation coordinate, its
+director-parallel gauge, chart update/recentering, and the first and second
+derivatives of the complete map are still unspecified.
+
+Likewise, a future drilling potential would have to be objective and expressed in
+the same global coordinates:
+
+$$
+\Pi_d=\frac12\boldsymbol\gamma(\mathbf q_g)^T
+\mathbf K_d^l(\mathbf q_g)\boldsymbol\gamma(\mathbf q_g),
 \qquad
-\mathbf r^{drill}=\frac{\partial\Pi_d}{\partial\mathbf q},
+\mathbf r_g^{drill}=\frac{\partial\Pi_d}{\partial\mathbf q_g},
 \qquad
-\mathbf K^{drill}=\frac{\partial^2\Pi_d}{\partial\mathbf q^2}.
+\mathbf K_g^{drill}=\frac{\partial^2\Pi_d}{\partial\mathbf q_g^2}.
 $$
 
-However, a constant initial-frame diagonal penalty is not automatically objective
+A constant initial-frame diagonal penalty is not automatically objective under
-under large rotation. The nonlinear frame update, drilling coordinate, scale update,
+large rotation. The nonlinear frame, drilling coordinate, scale update, and all
-and their first/second derivatives are unresolved future decisions. They do not
+their first/second derivatives remain unresolved. Until those choices and
-block the current linear kernel but do block geometrically nonlinear implementation.
+configuration-dependent load work are approved, neither a complete global residual
+nor a complete global 24-DOF tangent exists. This does not block the current linear
+kernel but blocks geometrically nonlinear implementation.
 
 ### 15.5 Newton equation and linear limit
 
-At iteration `k`,
+Only after Section 15.4 is closed may the global residual and tangent be defined as
 
 $$
-\mathbf K_T(\mathbf q^{(k)})\Delta\mathbf q
+\mathbf r_g=\mathbf A^T\mathbf r_{20}^{phys}+\mathbf r_g^{drill}-\mathbf f_{ext},
-=-\mathbf r(\mathbf q^{(k)}),
+\qquad \mathbf K_T=\mathbf K_g^{phys}+\mathbf K_g^{drill}-\mathbf K_g^{load}.
-\qquad
-\mathbf q^{(k+1)}=\mathbf q^{(k)}+\Delta\mathbf q.
 $$
 
-At the undeformed, stress-free state with the small-rotation director derivative,
+The conditional Newton equation at iteration `k` is
 
 $$
-\mathbf K^{geo}=\mathbf0,
+\mathbf K_T(\mathbf q_g^{(k)})\Delta\mathbf q_g=-\mathbf r_g(\mathbf q_g^{(k)}),
+\qquad \mathbf q_g^{(k+1)}=\mathbf q_g^{(k)}+\Delta\mathbf q_g.
 $$
 
-and the physical material tangent reduces to `K_phys^24` in Section 12. With the
+At the undeformed, stress-free state, recovering the current linear formulation
-approved linear drill block, the complete limit is `K_e^24`.
+requires all of the following identities:
+
+$$
+\mathbf A_0=\mathbf T_p,\qquad \mathbf r_{20}^{phys}=\mathbf0,
+\qquad \mathbf K_{20}^{geo}=\mathbf0.
+$$
+
+Then the coordinate-map curvature term vanishes and
+
+$$
+\mathbf K_g^{phys}=\mathbf T_p^T\mathbf K_{20}\mathbf T_p=\mathbf K_{phys}^{24}.
+$$
+
+If the future map also recovers `gamma=T_d q_g` and the approved constant linear
+drill block in this limit, the complete tangent reduces to `K_e^24`. This is a
+conditional consistency requirement, not evidence that a nonlinear `Phi` or
+objective drilling potential has already been selected.
 
 ## 16. Algorithm pseudocode
 
@@ -1247,12 +1494,14 @@ approved linear drill block, the complete limit is `K_e^24`.
 ```text
 for each shell element in stable source order:
     validate four distinct source nodes and source-order geometry
+    compute L_e from the four consecutive midsurface edges
     compute center positive normal candidate n_e
     compute 2x2 surface area A_e
 
 for each shell source node in stable source order:
     gather incident candidates in stable element order
-    reject opposed, degenerate, or too-sharp incident normals
+    reject any nonpositive pairwise incident-normal dot product before averaging
+    reject degenerate or too-sharp incident normals
     d_I = normalize(sum(A_e * n_e))
     select least-aligned global axis with deterministic tie break
     construct right-handed (a_I, b_I, d_I)
@@ -1265,13 +1514,15 @@ input: X_I, d_I, a_I, b_I, E, nu, t, q_g, nodal CLOAD share
 build T, T_p, T_d
 initialize K20[20,20] = 0
 
+evaluate the complete center/Gauss/tying/recovery geometry inventory
+validate pointwise J, j_s, a_g, c_d and aggregate r_J, theta_w
 evaluate and validate four midsurface tying locations
 for each 2x2 midsurface Gauss location in fixed order:
     construct and validate local frame (e1,e2,e3)
     compute the four direct covariant tying shear B columns at zeta=0
     for each thickness Gauss point in fixed order:
         build degenerated geometry, J, reciprocal bases, H derivatives
-        validate finite positive scale-aware Jacobian
+        reuse the validated point geometry and reciprocal bases
         form direct covariant B columns
         replace only xi-zeta and eta-zeta shear by MITC tying interpolation
         transform to local engineering B_bar[5,20]
@@ -1279,9 +1530,10 @@ for each 2x2 midsurface Gauss location in fixed order:
 
 check K20 finite and symmetric within approved normalized tolerance
 Kphys24 = T_p^T * K20 * T_p
-construct candidate-aware positive Kd_local without closing open coefficient choices
+construct positive Kd_local and report every k_d,I through rho_d,I = k_d,I/D_iso
 Kdrill24 = T_d^T * Kd_local * T_d
 Ke24 = Kphys24 + Kdrill24
+form S20, S24, Khat20, and Khat_e for rank/conditioning evidence only
 fint24 = Ke24 * q_g
 residual24 = fint24 - f_CLOAD
 return matrices, residual, transforms, frames, and separate energy operators
@@ -1291,9 +1543,11 @@ return matrices, residual, transforms, frames, and separate energy operators
 
 ```text
 assemble all Ke24 contributions with stable element-local COO ordering
+form model-length DOF scaling and Khat_ff for global rank evidence only
 partition full K into Kff, Kfc, Kcf, Kcc in stable free/constrained order
 factorize Kff before load assembly
 assemble and deterministically aggregate nodal CLOAD
+accept an exact-zero nodal moment separately; otherwise require rho_M <= 1e-12
 solve Kff * df = Ff - Kfc * dc
 reconstruct full displacement d
 compute full residual r = K*d - F
@@ -1317,11 +1571,13 @@ compute E_physical and E_drill separately; never clamp a near-zero denominator
 ### 16.5 Future nonlinear tangent check
 
 ```text
-given a future approved finite-director map and nonlinear drill potential:
+given a future approved 20-coordinate finite-director chart:
-    evaluate residual r(q)
+    evaluate physical residual r20(q20) and consistent K20(q20)
-    evaluate analytic consistent K_T(q)
+    verify its directional derivative
+given an approved global Phi map, objective drill potential, and load work:
+    evaluate global residual rg(qg) and complete consistent K_T(qg)
     for several normalized perturbation directions p and decreasing h:
-        compare K_T*p with [r(q+h*p)-r(q-h*p)]/(2*h)
+        compare K_T*p with [rg(qg+h*p)-rg(qg-h*p)]/(2*h)
     require the error to decrease in the expected truncation range
 ```
 
@@ -1331,18 +1587,42 @@ given a future approved finite-director map and nonlinear drill potential:
 
 - Shape-function partition of unity, Kronecker delta, and derivative sums.
 - Nodal and integration frames orthonormal and right-handed.
-- Finite positive Jacobian at every Gauss, tying, and committed recovery point.
+- The complete `J/j_s/a_g/c_d/r_J/theta_w` inventory at center, Gauss, tying,
-- `K20`, `Kphys24`, `Kdrill24`, and `Ke24` symmetry.
+  and committed recovery points.
+- Scaled `K20`, `Kphys24`, `Kdrill24`, and `Ke24` symmetry and spectrum.
 - Transformation work/energy invariance.
-- Physical rigid modes satisfy normalized `Kphys24 r=0`.
+- Physical rigid modes satisfy normalized scaled stiffness action.
 - Stabilized free-element nullity is exactly six; accepted non-rigid physical modes
   have positive physical energy.
 - Pure drill vectors have zero physical energy and positive drilling energy.
 - Consistent force/length unit rescaling leaves dimensionless decisions unchanged.
 
-Requirements currently prescribe normalized targets `1e-12` for frame, symmetry,
+For any nonzero scaled stiffness under test, the approved normalized checks are
-and transformation-energy checks and `1e-10` for rigid action, residual, and global
+the following. Here `r_hat` denotes a constructed scaled rigid-mode vector, not the
-equilibrium unless Numerical Review documents replacements.
+assembled residual.
+
+$$
+e_{sym}=\|\widehat{\mathbf K}-\widehat{\mathbf K}^T\|_F/\|\widehat{\mathbf K}\|_F\le10^{-12},
+$$
+
+$$
+e_{rigid}=\|\widehat{\mathbf K}\widehat{\mathbf r}\|_2/(\|\widehat{\mathbf K}\|_2\|\widehat{\mathbf r}\|_2)\le10^{-10},
+$$
+
+$$
+e_{frame}=\|\mathbf R^T\mathbf R-\mathbf I\|_F\le10^{-12}.
+$$
+
+For a nonzero transformation-energy case,
+
+$$
+e_T=|E_g-E_l|/(|E_g|+|E_l|)\le10^{-12}.
+$$
+
+A zero matrix/vector denominator is invalid test construction and is not clamped to
+pass. When both transformation energies are exactly zero, the rigid or pure-null
+case is classified by its separate stiffness-action test. Linear-system residual
+and global-equilibrium evidence retain the normalized `1e-10` requirement.
 
 ### 17.2 Patch and sign checks
 
@@ -1369,16 +1649,20 @@ Independently verify:
 
 ### 17.4 Drilling sensitivity
 
-For every candidate reference scale, sweep dimensionless coefficients around the
+For every candidate reference scale, convert candidate coefficients to the actual
-research order of magnitude and record:
+nodewise `rho_d,I` inventory. Expand a logarithmic sweep until both the low-side
+rank/conditioning failure and high-side physical-response contamination boundary
+are observed, and record:
 
-- free-element rank and constrained system conditioning;
+- free-element scaled rank and scaled constrained-system conditioning;
 - global `U` and physical `N/M/Q` sensitivity;
 - `E_phys` and `E_drill` without denominator clamping;
 - invariance under consistent unit conversion, thickness ratios, and mesh sizes.
 
-The study must select the reference family, nominal coefficient/range, and warning
+Candidate runs are compared by overlapping actual `rho_d,I` ranges, never by equal
-criterion before Implementation Planning.
+raw `alpha_d`. The study must select the reference family, the smallest stable
+plateau value, its adjacent-decade sensitivity, and the energy warning criterion
+before Implementation Planning.
 
 ### 17.5 Reference-comparison boundary
 
@@ -1393,8 +1677,10 @@ not expected to be numerically identical on finite meshes.
 
 Before any geometrically nonlinear implementation, verify residual directional
 derivatives, tangent symmetry for conservative loading, objectivity under large
-rigid motion, zero stress-free geometric stiffness, and convergence of Newton
+rigid motion, the first/second derivatives of `Phi`, the coordinate-map curvature
-iterations. These checks are future-only and do not authorize a nonlinear procedure.
+term, objective drilling, zero stress-free geometric stiffness, and convergence of
+Newton iterations. These checks are future-only and do not authorize a nonlinear
+procedure.
 
 ## 18. Numerical risks
 
@@ -1404,17 +1690,18 @@ iterations. These checks are future-only and do not authorize a nonlinear proced
 | Membrane locking on distorted curved meshes | slow or nonuniform convergence | distortion/curvature sweeps; document original MITC4 limitation |
 | Volumetric locking | N/A for the approved plane-stress shell contract | do not infer a three-dimensional incompressible formulation |
 | Wrong tying pair or engineering-shear factor | swapped/incorrect shear and loss of patch consistency | component-level tying and patch tests |
-| Reversed or degenerate Jacobian | invalid basis, sign, or energy | fail at every required point using scale-aware measures |
+| Reversed or degenerate Jacobian | invalid basis, sign, or energy | common location inventory with `J/j_s/a_g/c_d/r_J/theta_w` |
 | Discontinuous shared director | artificial coupling or undefined frame | reject; require duplicate nodes at folds |
 | Fixed-axis tangent singularity | nondeterministic rotation transform | least-aligned global-axis construction |
 | Drilling coefficient too small | rank/conditioning failure | rank and conditioning sensitivity sweep |
 | Drilling coefficient too large | contaminated displacement/resultant | physical-output and separate-energy sensitivity sweep |
-| Mixed-unit diagonal scale | unit-dependent stiffness | prohibit `min(all Kii)`; use rotationally compatible scale |
+| Mixed-unit drilling or spectrum scale | unit-dependent stabilization/rank | use `rho_d,I` and `S^T K S`; prohibit raw mixed-unit comparison |
 | Misconstructed rigid test | false drill energy in a physical mode | use rigid translation plus tangent director change with `gamma=0` |
 | Recovery/stiffness mismatch | inconsistent energy and section output | same frames, tying, material, and thickness rule |
 | Location averaging | hidden sign/identity error | preserve exact location rows; no nodal extrapolation |
 | Treating S4R as reduced FESA integration | divergent kernel and unsupported hourglass behavior | one documented rule for S4/S4R |
 | Nonlinear director Hessian omission | inconsistent future tangent | exact first/second derivative of approved rotation map |
+| Missing nonlinear 20-to-24 map Hessian | incomplete global tangent | require `Phi`, `A`, and coordinate-map curvature term |
 | Nonobjective nonlinear drill penalty | artificial large-rotation energy | future review of rotating frame and drill potential |
 
 ## 19. Evidence basis and applicability
@@ -1461,41 +1748,65 @@ research brief remain the project source of truth.
 | Requirement group | Formulation coverage | Remaining owner |
 | --- | --- | --- |
 | `001-004`, `030`, `037` | linear-static boundary; S4/S4R one FESA path, source identity distinct | I/O, planning |
-| `005`, `031-038` | 24 global DOFs; 20 physical plus four drill coordinates; transform, rank, energy | Numerical Review for drill choices |
+| `005`, `031-038` | 24 global DOFs; 20 physical plus four drill coordinates; `rho_d,I`, scaled rank, energy | Numerical Review for drill calibration |
 | `006-010` | isotropic plane stress, one centered constant-thickness layer | I/O validation |
-| `011-016` | connectivity-positive normals, deterministic area weighting, frames, Jacobian checks | Numerical Review for thresholds |
+| `011-016` | pairwise normals, deterministic averaging/frames, complete geometry measures | Numerical Review for calibrated thresholds |
-| `017-020` | global BC/CLOAD virtual work; parallel moment rejected; distributed loads excluded | I/O projection diagnostic |
+| `017-020` | global 24-DOF virtual work; `rho_M <= 1e-12`; distributed loads excluded | I/O diagnostic/schema |
 | `024-029` | deterministic element buffers, partitioned linear lifecycle, full-residual reaction | planning |
 | `039-048` | nodal/global and shell/local recovery inventory, units, identities, energy split | I/O schema |
-| `049-057` | invariants, patches, locking, geometry, curved shells, drilling, equilibrium | Numerical Review/reference/physics |
+| `049-057` | scaled normalized invariants, patches, locking, geometry, curved shells, drilling, equilibrium | Numerical Review/reference/physics |
 | `058-064` | U blocking, UR warning-only comparison boundary | Numerical Review/reference model |
 | `065-072` | reference immutability and displacement-only evidence boundary acknowledged | reference model |
 
+### 20.1 Numerical Review revision traceability
+
+The first review findings map to this revision as follows.
+
+`NR-C01` maps to Sections 9.2-9.3, 16.2, and 17.1/17.3. The geometry measures
+are fixed while their thresholds still require calibration.
+
+`NR-C02` maps to Sections 12.2 and 17.4. The common `rho_d,I` coordinate and
+candidate conversions are fixed while the stable plateau remains open.
+
+`NR-C03` maps to Sections 12.3/12.5 and 17.1. Element/global DOF scaling is fixed
+while the numerical-rank threshold remains open.
+
+`NR-C04` maps to Sections 7.1-7.2, which fix the global 24-DOF weak form and
+transpose notation. `NR-C05` maps to Sections 15.1-15.5 and 17.6, which expose the
+physical-chart/global-map boundary and required map Hessian term.
+
+`NR-D01` maps to Sections 6.2 and 16.3, which fix the exact-zero case and
+`rho_M <= 1e-12`. `NR-D02` maps to Sections 12.5 and 17.1, which fix the approved
+normalized algebraic metrics.
+
 ## 21. Open issues and downstream handoff
 
 ### 21.1 Blocking Numerical Review decisions
 
 1. Select the dimensionally compatible drilling reference-scale family.
-2. Select `alpha_d`, its sensitivity range, conditioning/rank acceptance, and the
+2. Select the nodewise `rho_d,I` stable plateau, nominal value, scaled
-   physical-output contamination bound.
+   conditioning/rank acceptance, and physical-output contamination bound.
 3. Define classification and warning behavior for `E_drill/E_phys`, including the
    zero/near-zero physical-energy case.
 4. Approve `theta_smooth` after curved-mesh resolution sweeps; `20 degrees` is only
    the initial candidate.
-5. Approve dimensionless Jacobian, distortion, and warpage measures and thresholds.
+5. Calibrate thresholds for the fixed `J/j_s/a_g/c_d/r_J/theta_w` geometry
-6. Approve the scale-aware tolerance used to reject a director-parallel nodal
+   inventory using valid distortion/warp and collapsed negative sequences.
-   moment.
+6. Jointly with Reference Model, approve the U mixed tolerance and nonblocking UR
-7. Re-derive/check the exact `B_bar` component signs, tying-point inventory,
+   large-error warning threshold.
-   engineering-shear factors, rank, and bottom/top stress sign independently.
 
-Until these items are approved, the formulation is ready for Numerical Review but
+The first review already approved `rho_M <= 1e-12`, the normalized algebraic checks,
-not for Implementation Planning.
+the MITC tying/component signs, constitutive law, quadrature, and recovery signs.
+This revision is ready for Numerical Review rerun but not for Implementation
+Planning until the six remaining evidence-backed decisions are closed.
 
 ### 21.2 I/O Definition handoff
 
 - Preserve source `S4`/`S4R` separately from internal `FESA-MITC4`.
 - Define exact keyword subset, section/material resolution, and fail-closed
-  diagnostics for director, folds, Jacobians, unsupported loads, and recovery.
+  diagnostics for director, folds, Jacobians, unsupported loads, and recovery;
+  encode exact-zero nodal moment separately and enforce `rho_M <= 1e-12` without
+  a denominator clamp.
 - Define exact HDF5 row schemas for global `U/UR`, `RF/RM`; four midsurface
   generalized-strain/resultant locations; bottom/middle/top stress positions; full
   residual/equilibrium; and separate energies.
@@ -1522,8 +1833,9 @@ not for Implementation Planning.
 ### 21.5 Future nonlinear formulation handoff
 
 Before geometric-nonlinear implementation, separately approve the finite-director
-chart and update law, objective drilling potential and scale update, exact
+chart and update law, the complete `Phi: R24 -> R20` map and its first/second
-first/second director derivatives, nonlinear constitutive boundary, nodal-moment
+derivatives, objective drilling potential and scale update, exact first/second
-work/load tangent, state ownership, convergence controls, and nonlinear output
+director derivatives, nonlinear constitutive boundary, nodal-moment work/load
-contract. Section 15 supplies the residual/tangent structure but deliberately does
+tangent, state ownership, convergence controls, and nonlinear output contract.
-not close those product decisions.
+Section 15 supplies the physical-chart residual/tangent and the conditional global
+mapping identity but deliberately does not close those product decisions.