Constraint Rigidity, A Combinatorial Framework for Controlled Hamiltonian Stability

A recurring engineering problem in quantum many body systems is to anticipate when an intended interaction pattern admits a controlled low energy sector that remains stable within a specified, constraint defined family of local deformations. Existing gap stability results are powerful but model dependent and typically do not yield a simple graph level diagnostic that can be applied before any diagonalisation. This work proposes a combinatorial notion of constraint rigidity for Hamiltonians specified by an interaction graph. The central construction is a rigidity style linearisation of how local basis degrees of freedom and edge level interaction constraints trade off, producing a Maxwell type count and a sparsity condition that plays the role of a quantum analogue of minimal rigidity for a specified constraint model. The framework yields graph level necessary conditions for eliminating constraint preserving local mechanisms under that model, proves a local spectral invariance statement for constraint preserving drifts in a controlled family, and motivates a conjectural link between gap preservation along calibrated paths and generic full rank of an associated “Hamiltonian rigidity matrix”. The result is a concrete interface between classical rigidity theory and Hamiltonian engineering that isolates when stability can plausibly be attributable to interaction structure under an explicit constraint model, rather than to fine tuned couplings.