Flatness-Preserving Residuals

• Flatness-preserving residuals are structural modifications that retain the key flatness property in algebraic, analytic, and geometric models.
• They are applied to preserve stability in control theory, secure convergence in adic completions, and maintain invariance under neural network reparameterizations.
• Implementations span analytic criteria, variational methods, and geometric constructions, underpinning robustness in algebraic geometry, control, and deep learning.

Flatness-preserving residuals are structural, algebraic, or analytical modifications designed to augment models or spaces—often in algebraic geometry, control, or machine learning—while preserving the crucial geometric or module-theoretic property of flatness. These constructions appear in various domains, from analytic criteria for flatness in complex geometry to control theory, commutative algebra, and the design of deep and geometric neural networks. Flatness-preserving residuals ensure that after introducing correction terms, "residual" modifications, or base changes, the essential property of flatness (in the sense appropriate to the context) remains intact, enabling continued application of powerful analytic or algorithmic frameworks.

1. Analytic Criterion for Flatness and Residual Testing

In several analytic and algebraic settings, preserving flatness under residual constructions requires combining local algebraic tests with inductive geometric reductions. The inductive analytic criterion for flatness of a morphism $X \to Y$ of analytic spaces, or more generally of a coherent $\mathcal{O}_X$-module over $Y$, rests on two conditions:

• Codimension-Zero Linear Algebraic Criterion: Locally, for a coherent module $F$ presented as

$$\mathcal{O}_X^p \xrightarrow{\Phi} \mathcal{O}_X^q \to F \to 0,$$

if $r = \operatorname{rank} \Phi(a)$ at $a \in X$, flatness at $a$ is equivalent to the vanishing of all $(r+1)\times(r+1)$ minors of $\Phi$ near $a$. This addresses flatness in "generic" loci.

• Codimension-One Residual Condition: For fiber-dimension reduction, via choosing $g\in A$ and a homomorphism $\psi:A^l \to F$ such that $g(0,x)\ne 0$, $gF\subset \operatorname{im}\psi$, and $\ker\psi\subset \mathfrak{m}A^l$, one performs a block decomposition of $\Phi$. The residual condition demands that $\ker\psi\subset J\cdot A^l$ for an ideal $J$, and that the quotient $F/\operatorname{im}\psi$ is flat over $\mathcal{O}_Y/J$. The application of the Weierstrass preparation theorem then allows inductively reducing the problem to modules over fewer variables, leading to a residual module whose flatness implies that of the original.

The interplay between these two conditions assures that residual constructions—those local or stepwise modifications—do not introduce new torsion or rank jumps, maintaining the flatness of the total family through the induction. This approach is essential for constructing local flatteners, proving the openness of flatness, and justifying the behavior of flatness under analytic base change (Adamus et al., 2011).

2. Flatness Under Completion and Adic Residuals

In commutative algebra, flatness-preserving residuals emerge prominently in the context of module completions and adic systems.

• Adic Flatness Preservation: For a weakly proregular ideal $\mathfrak{a}$ in a ring $A$, an $A$-module $M$ is $\mathfrak{a}$-adically flat if $\operatorname{Tor}^A_i(N, M) = 0$ for all $\mathfrak{a}$-torsion $A$-modules $N$ and $i > 0$. The $\mathfrak{a}$-adic completion $\hat{M}$ of a $\mathfrak{a}$-adically flat module preserves this property provided $\mathfrak{a}$ is weakly proregular. In the noetherian case, $\mathfrak{a}$-adic flatness and flatness coincide for complete modules.
• Residual Approximation by Adic Systems: The structure of an adic system $\{M_k\}$, with $M_k = A/\mathfrak{a}^{k+1}\otimes_A M$, allows studying flatness via residual "slices" $M_k$, each capturing level-wise module-theoretic properties. For finitely generated $\mathfrak{a}$, any such system converges to the completed module, furnishing a residual framework in which flatness can be preserved level by level and passed to the limit (Yekutieli, 2016).

Counterexamples in non-noetherian settings show that even when adic flatness is preserved under completion, full flatness may fail, highlighting the subtleties of flatness preservation in residual settings.

3. Flatness-Preserving Residuals in Geometric PDEs and Variational Problems

In differential geometry, residuals that preserve flatness arise within the framework of geometric PDEs through least squares Lagrangian densities:

• Flatness Deviation Functionals: Defining functionals such as

$I(V) = \int_M |V|^2\,du$

where $|V|^2$ is the norm squared of geometric quantities (connection coefficients, curvature, Ricci tensor, scalar curvature), one measures deviation from flatness.

• Euler-Lagrange Prolongations: The solutions to the Euler-Lagrange equations of these least squares functionals, termed prolongations, yield residuals that approach or maintain flatness. These solutions minimize deviations, ensuring the residuals vanish at flat geometries. The variational approach unifies different notions of geometric flatness—connection, curvature, Ricci, scalar—by focusing on residual vanishing and Lagrangian minimization, where the exact minimizers ("residuals" in the functional sense) correspond to flat metrics or connections (Hirica et al., 2019).

This framework underpins modern geometric analysis and informs approaches in geometric flows, conservation laws, and general relativity.

4. Reparameterization-Invariant Flatness in Neural Networks

In high-dimensional machine learning, flatness-preserving residuals are linked to invariance of flatness measures under neural network reparameterizations:

• Relative Flatness Measures: Classical flatness measures (Hessian-based curvature, spectral norm, or trace) are not invariant under layer-wise or neuron-wise rescalings. Relative flatness metrics

$$\kappa_\phi(w) = \| \operatorname{vec}(w_l) \|^2 \cdot \lambda_{\max}^{H,l}(w_l)$$

and its variants incorporate the scaling of weights, ensuring invariance under rescaling transformations that leave the function unchanged. This adjustment preserves the "flatness" property in the presence of residual or skip connections (as in ResNets), which frequently induce reparameterizations (Petzka et al., 2019, Petzka et al., 2020).

• Soft Rank and Generalization Gap: Flatness measured by the soft rank of the Hessian,

$$\text{rank}_\lambda(H(\theta^*)) = \operatorname{Tr}( H(\theta^*)( H(\theta^*)+\lambda I )^{-1} ),$$

tightly predicts the expected generalization gap for calibrated models. Preservation of flatness under residual reparameterizations ensures that generalization properties are properly assessed and that overfitting is avoided (Shoham et al., 21 Jun 2025).

Flatness-preserving regularization strategies, such as FAM (Relative Flatness Aware Minimization), leverage these principles to structure optimization landscapes so that minima remain robust under parameter perturbations, improving generalization in both vision and LLMs (Adilova et al., 2023).

5. Structural Residuals Preserving Differential Flatness in Control

In control theory, especially for differentially flat systems:

• Flatness-Compatible Residuals in Pure-Feedback Systems: When learning correction terms (residual dynamics) for a nominal, differentially flat, pure-feedback system, generic corrections may destroy flatness, impeding inversion and flat-output-based planning. By constraining residuals to a lower-triangular structure—each correction $\Delta_i$ depending only on $(x_1, ..., x_i)$—the flat outputs and associated diffeomorphism of the augmented system are preserved. An explicit recursive procedure recovers the new flatness map, enabling flatness-based control with learned corrections (Yang et al., 6 Apr 2025).

This preserves the system's core structure post-learning, allowing computationally efficient planning and trajectory tracking with real-world disturbances included.

6. Flatness Preservation in Commutative Algebra and Algebraic Geometry

In algebraic contexts, flatness-preserving residuals are exemplified by conditions on ring extensions and modules:

• Stable Prime Extension Property: A homomorphism $R \to S$ is flat if, for reduced $R$ with finitely many minimal primes per maximal ideal, all prime ideals extend to primes or units after arbitrary base changes. This structural property ensures that fiberwise, flatness persists under residual algebraic modifications, and counterexamples with infinite minimal primes underline the necessity of the finiteness condition (Hochster et al., 2020).
• Intersection Flatness: When tensoring with a module or algebra $S$ commutes with the intersection of submodules or ideals, flatness and extension properties are preserved in graded and residual settings, allowing one to reduce flatness checks to closed fibers or minimal residual cases.

7. Geometric and Manifold-Preserving Residuals in Modern Architectures

In geometric learning, particularly for hyperbolic neural networks:

• Lorentzian Residual Connections: LResNet constructs residual connections in hyperbolic space using the weighted Lorentzian centroid, ensuring that the combined output remains on the hyperbolic manifold and preserves geodesic structure. Operations such as

$$x \oplus_\ell f(x) = \frac{w_x x + w_y f(x)}{\sqrt{-K}\| w_x x + w_y f(x) \|_\ell}$$

are commutative, numerically stable, and maintain the essential geometric "flatness" (absence of distortion), even after multiple residual layers. This property is critical for modeling hierarchical or tree-structured data and avoids the flattening artifacts present in tangent-space or parallel transport-based approaches (He et al., 19 Dec 2024).

Such geometric flatness-preserving residuals ensure data relationships, class semantics, and hierarchical relations are robustly represented in learned embeddings, supporting high expressivity and transferability across tasks.

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Flatness-preserving residuals are thus a unifying paradigm wherein residual modifications—be they algebraic corrections, learned dynamics, architectural alterations, or variational prolongations—are structured to preserve the essential feature of flatness, enabling robust transfer of analytic, algebraic, geometric, or computational properties across modifications and base changes. This concept is fundamental to both the theoretical understanding and practical implementation of stable, expressive, and predictive models in geometry, algebra, control, and machine learning.