We construct a mathematical object, the Coulon Manifold M, as a topological category whose points are transactions, paths are sequences of commitments, and submanifolds are blocks.

The manifold is characterized by its guardable regions G, cavities C, and a viscosity relation ~ that governs flow. We establish an injectivity condition (no overlapping paths) which, when violated, generates homology classes representing consensus failures.

We prove that bijectivity between nouns (p) and noun phrases (np) in the engineering notebook corresponds to a compact submanifold with vanishing first homology. We define the Coulon Number n² = m, its generalization n^n = m, and the Coulon Region S(t) as a manageable state.

Using minimal lambda calculus and Lisp, we implement functionals that compute coverage, detect cavities, and map energy. We demonstrate counterexamples, prove five theorems, and conclude with implications for distributed commitment systems.

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ABSTRACT

We construct a mathematical object, the Coulon Manifold M, as a topological category whose points are transactions, paths are sequences of commitments, and submanifolds are blocks. 

The manifold is characterized by its guardable regions G, cavities C, and a viscosity relation ~ that governs flow. We establish an injectivity condition (no overlapping paths) which, when violated, generates homology classes representing consensus failures. 

We prove that bijectivity between nouns (p) and noun phrases (np) in the engineering notebook corresponds to a compact submanifold with vanishing first homology. We define the Coulon Number n² = m, its generalization n^n = m, and the Coulon Region S(t) as a manageable state. 

Using minimal lambda calculus and Lisp, we implement functionals that compute coverage, detect cavities, and map energy. We demonstrate counterexamples, prove five theorems, and conclude with implications for distributed commitment systems.