Mode-Dependent Differentials and Stability of Limits in Partial Differential Equations

This work introduces a structural framework for differential calculus and partial differential equations based on the concept of convergence modes.

Classical analysis defines limits and derivatives through asymptotic behavior, but does not explicitly encode the operational structure by which convergence is realized. In practical settings—numerical approximation, physical measurement, and multiscale modeling—limits arise from structured procedures involving discretization, scaling, ordering, and regularization.

In this work, convergence modes are introduced as primary mathematical objects representing such procedures. Derivatives are defined as limits taken along modes, leading to mode-dependent differentials. Stability of these limits with respect to classes of modes becomes the central organizing principle.

Within this framework:
- classical differentiability is reinterpreted as invariance of the differential under admissible variations of convergence modes,
- partial differential equations generate mode-dependent solution clouds rather than single solutions,
- entropy conditions are reinterpreted as restrictions on admissible classes of modes rather than additional constraints on solutions,
- linear PDE emerge as mode-invariant systems,
- nonlinear PDE are characterized as mode-sensitive systems.

The framework provides a unified interpretation linking:
- classical analysis,
- weak and entropy solutions,
- numerical schemes,
- and physical observability.

It naturally extends to:
- multiscale systems,
- stochastic processes,
- compactness theory,
- and mode-dependent numerical analysis.

Conceptually, this work shifts the role of limits from primary objects to derived invariants, and proposes that differential calculus and PDE theory should be understood as theories of stability over spaces of admissible convergence modes.

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This paper constitutes the first part of the research program:

**Mode-Based Analysis**

which aims to develop a systematic theory of analysis, partial differential equations, and applied mathematics grounded in the structure of convergence procedures.

Future work will extend this framework to:
- variational and optimization principles,
- geometric and topological structures,
- categorical formulations of modes and solution spaces.