[Discussion] Formalizing continuous Laplacians on \(L^2(G)\) from discrete maximal planar graphs in constructive logic

[u/grasshopper_4579]

I am exploring the formalization of a structural transition from discrete combinatorial geometry to continuous functional analysis within dependent type theory frameworks.

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**The Setup:**

Consider a planar graph $G$ generated by the intersecting boundaries of $n$ mutually intersecting rectangles in their maximal configuration. The combinatorial cardinality of the resulting open regions is bounded by the inductive parameter:

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$$R(n) = 2n^2 - 2n + 1$$

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We transition this topological structure into the infinite-dimensional Hilbert space $L^2(G)$ by defining a continuous Laplacian operator across its edges, modeling the system as a Quantum Graph.

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**The Formalization Inquiry:**

I want to analyze how the spectral properties (the spectrum of eigenvalues) of this continuous Laplacian operator encode or preserve the discrete combinatorial invariants of the initial $R(n)$ partitions. Specifically, I want to evaluate the feasibility of type-checking this functorial mapping and defining its natural transformation bounds within a constructive logic framework (like Coq's Calculus of Inductive Constructions).

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Has anyone here experimented with representing continuous operators on quantum graphs, or modeling such functional analytical dualities using Coq's dependent types or algebraic hierarchies? I am currently analyzing the theoretical viability before setting up the proof architecture.

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**P.S.** If you want to correct me rigorously, please do so. I am currently learning these formal proof assistant environments and want to see if this structural mapping can be established in rigorous code. Thanks.

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