| Layerwise decomposition of the tangent Gram matrix | statement | lemma K.21 | arxiv |
| Finite-witness lower bound for the smallest eigenvalue | statement | proposition D.2 | arxiv |
| Row regularity of finite-order strong-form linear constraints | statement | proposition C.8 | arxiv |
| Compatible targets and exact compression | statement | lemma H.3 | arxiv |
| Positivity criterion at depth | conjecture | conjecture K.26 | arxiv |
| No exponent yields a nonzero deterministic polynomial kernel | statement | corollary K.13 | arxiv |
| Depth-$H$ network | definition | definition K.20 | arxiv |
| Taylor remainder bound | statement | lemma I.7 | arxiv |
| Positive-definiteness certificate | definition | definition 2.7 | arxiv |
| Empty-interior hidden support breaks the exact positivity criterion | example | example E.5 | arxiv |
| Gradient descent | definition | definition 2.4 | arxiv |
| Finite-width positive definiteness at initialization | statement | proposition C.11 | arxiv |
| Independence of activation-derivative polynomial functions | statement | lemma D.1 | arxiv |
| Convergence of gradient descent under abstract row regularity | statement | theorem I.2 | arxiv |
| Exact direct-sum criterion for the NTK | statement | corollary D.5 | arxiv |
| NTK existence and exact nullspace for weak-form constraints | statement | theorem D.10 | arxiv |
| Initial error bound | statement | lemma H.1 | arxiv |
| Automatic regularity for bounded functional rows | statement | lemma C.4 | arxiv |
| NTK analysis for finite linear constraints | statement | theorem C.1 | arxiv |
| Trichotomy in the width exponent | statement | theorem K.12 | arxiv |
| Weak-form NTK: existence, nullspace, and rank | statement | proposition 3.2 | arxiv |
| Two obstructions to geometric decay | statement | proposition H.5 | arxiv |
| Spectral one-step descent | statement | lemma I.8 | arxiv |
| Target projection for an architecture linear in its output weights | statement | lemma G.3 | arxiv |
| Depth and differential-order dependence of the constants | conjecture | conjecture K.24 | arxiv |
| Classwise GD transfer | statement | corollary 4.3 | arxiv |
| Limits of the two empirical objects | statement | lemma K.4 | arxiv |
| Dichotomy: affine rows concentrate, nonlinear rows do not | statement | corollary K.7 | arxiv |
| Depth-$H$ substitution into the two-layer argument | statement | proposition K.23 | arxiv |
| Lower bound on the NTK eigenvalue along training | statement | proposition I.4 | arxiv |
| Exact mixed nonlocal residual--boundary certificate | example | example E.11 | arxiv |
| A coupled system with a co-located dependency | example | example H.6 | arxiv |
| NTK existence and exact nullspace for nonlocal integral constraints | statement | theorem D.14 | arxiv |
| Sampled linear constraints | definition | definition 2.2 | arxiv |
| Nonlocal residuals with nonlocal or Dirichlet boundary data | statement | corollary D.15 | arxiv |
| Automatic regularity for standard weak \(H^1\) rows | statement | corollary C.5 | arxiv |
| Depth-$H$ row regularity, condition $(\mathsf G^H_\tau)$ | definition | definition K.22 | arxiv |
| Joint weak-residual and boundary criterion | statement | corollary D.11 | arxiv |
| Cubic ReLU degenerates for a fourth-order strong residual | example | example B.1 | arxiv |
| Product row against a linear row | statement | corollary K.8 | arxiv |
| Structural spaces and compatibility | definition | definition H.2 | arxiv |
| Nonlocal NTK: existence, nullspace, and rank | statement | proposition 3.3 | arxiv |
| Exact pointwise coefficient certificate | example | example E.9 | arxiv |
| Exact composite tangent Gram identity | statement | lemma K.3 | arxiv |
| Composite rows over a base family of linear observations | definition | definition K.2 | arxiv |
| Two-dimensional weak Poisson problem with tested or pointwise boundary data | statement | corollary D.13 | arxiv |
| Exact mixed weak residual--boundary certificate | example | example E.10 | arxiv |
| Compatible duplicate rows need only an active-subspace gap | example | example B.4 | arxiv |
| A compactly supported $C^\infty$ activation breaks the weak criterion | example | example E.4 | arxiv |
| Explicit nondegeneracy for a convective row | example | example K.9 | arxiv |
| The uncentered branch | statement | proposition K.15 | arxiv |
| Prediction difference between consecutive iterates | statement | proposition I.3 | arxiv |
| Taylor-coefficient and evaluation rank characterization | statement | proposition D.6 | arxiv |
| Rank conditions from monomials and moments | statement | corollary D.7 | arxiv |
| A sharp interface lies outside the strong-form class but gives a bounded weak row | example | example B.5 | arxiv |
| Convergence when all errors lie in a fixed subspace | statement | corollary H.4 | arxiv |
| Coupled linear advection--diffusion system | example | example F.1 | arxiv |
| Continuum weak Poisson problem with a discontinuous coefficient | statement | corollary D.12 | arxiv |
| Singular NTK with distinct constraints and zero nullspace projection | example | example E.8 | arxiv |
| Width exponent | definition | definition K.1 | arxiv |
| Linear independence across all parameter contributions | statement | lemma G.1 | arxiv |
| One-step parameter motion | statement | lemma I.5 | arxiv |
| Automatic regularity for weak \(H^k\) rows in any dimension | statement | corollary C.6 | arxiv |
| Parameter displacement from initialization | statement | lemma I.6 | arxiv |
| A fully numerical finite-width GD certificate | example | example D.3 | arxiv |
| Two-layer network | definition | definition 2.1 | arxiv |
| Loss and tangent Gram matrix | definition | definition 2.3 | arxiv |
| Initialization law and moment existence | example | example B.6 | arxiv |
| Smoothness without analyticity collapses the weak feature span | example | example E.3 | arxiv |
| Abstract row-regularity condition | definition | definition C.2 | arxiv |
| Pointwise NTK: existence, nullspace, and rank | statement | proposition 3.1 | arxiv |
| Exact limiting-NTK nullspace for a network with hidden biases | statement | proposition G.2 | arxiv |
| Abstract NTK analysis | statement | theorem C.10 | arxiv |
| Positive definiteness during training | statement | proposition C.14 | arxiv |
| Positive definiteness from independent continuous parameter-derivative functions | statement | lemma D.4 | arxiv |
| Ridge analyticity and density in Sobolev spaces, any dimension | statement | lemma D.8 | arxiv |
| Coupled vector rows require joint rank | example | example B.3 | arxiv |
| The linear certificate controls both base matrices | statement | lemma K.6 | arxiv |
| Local row regularity | definition | definition 4.1 | arxiv |
| A shared PDE--boundary point can be singular or benign | example | example B.2 | arxiv |
| Finite-width GD for fixed linear observations | statement | theorem 4.2 | arxiv |
| Random limit of the tangent Gram matrix for composite rows | statement | proposition K.5 | arxiv |
| The ridge-density clause of $\mathsf W_{\mathrm{pd}}$ is load-bearing | example | example E.1 | arxiv |
| Positive definiteness for a Poisson PINN | example | example E.7 | arxiv |
| Automatic regularity of finite-measure rows | statement | proposition C.7 | arxiv |
| Deleting $\gamma>0$ from $\mathsf N_{\mathrm{pd}}$: an even activation collapses two distinct Dirac rows | example | example E.2 | arxiv |
| A heavy-tailed initialization law destroys the deterministic limit | example | example E.6 | arxiv |
