Published March 27, 2026 | Version v1
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A Proposed Categorical Framework for Multiscale Physical Combination

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A Proposed Categorical Framework for Multiscale Physical Combination

 
This preprint proposes a unified mathematical formalism to address fragmented notation across scientific disciplines for describing the combination and evolution of physical quantities. Fields from electrical engineering to population dynamics currently use distinct, domain-specific equations to model fundamentally similar systemic behaviors—including strain relief, state transition, and saturating growth.
To bridge these notational gaps, the paper introduces four primary static algebraic operators:
  • The Harmonic Operator (⊥ ): A commutative monoid modeling systemic relief and paths of least resistance.
  • The Morph Operator (⋈t ): A non-commutative operator for path-dependent state transitions and affine blending.
  • The Limit Operator (⊞L ): A commutative monoid modeling bounded throttling and asymptotic saturation.
  • The Addponent Operator (× ): A non-commutative magma for base-preserving generative accretion.
To extend these static combinations across time and dimension, the framework introduces two dynamic modifiers: a Feedback Evolution Operator (⊛γ ) for recursive time evolution toward fixed-point equilibria, and a Scale Operator (⇓r ) for dimensional coarse-graining and spatial averaging.
 
Structured using Applied Category Theory, the paper explicitly calculates mixed associators to demonstrate inherent non-commutativity between operators. Specifically, the formalism proves that scaling a combined system versus combining scaled systems yields mathematically distinct results—providing a rigorous, calculable method for identifying emergent correction terms across physical scales. This framework aims to offer a standardized algebraic syntax for multiscale physical modeling.

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