Theoretical Cartography 2: Homological Detection of Missing Bridges—Component Criteria, Mayer–Vietoris Counting, and Obstruction Stratification

Theoretical space can be formalized as an axiom-annotated weighted hypergraph.This paper transforms the structural question “whether there exists a direct channelbetween two subtheories that does not pass through the axiomatic root layer” into acomputable decision theory. Let the flag complex decompose as K = K1 ∪ K2, withshared layer L = K1∩K2 containing the root layer. This paper establishes a two-levelcriterion. The order-0 criterion uses the number of connected components after deletingthe shared layer as its reading: under a closed cover decomposition, an order-0 missingpair exists within a connected component C of K if and only if c0(C−L) ≥ 2; when Kis connected, the number of missing component pairs is C(c0(K −L),2). The order-1criterion takes the Mayer–Vietoris exact sequence as its framework: when H1(L) = 0,β1(K) = β1(K1)+β1(K2)+dimkerφ, where dimkerφ = c0(L)−c0(K1)−c0(K2)+c0(K); more generally, under a closed cover decomposition one always has β1(K) =β1(K1) + β1(K2) − rankj + dimkerφ, where j : H1(L) → H1(K1) ⊕ H1(K2) is theinclusion-induced map. The nonzero classes of the quotient space H1(K)/im(H1(K1)⊕H1(K2)) are defined as the order-1 missing bridge classes of this decomposition; thisquotient space is isomorphic to kerφ, so the number of order-1 missing bridge classesequals dimkerφ. The closed cover condition is a premise of the Mayer–Vietoris exactsequence: if K ̸ = K1 ∪ K2, cross-side simplices can make the counting identity fail.This paper proves that closed cover repair is linear relative to the size of the explicitflag complex, so the closed cover condition does not constitute a restriction on theinput. Both criteria are polynomial-time computable over GF(2), and deterministicmachine verification is completed on three explicit configurations: a regression checkof the running example, a testbed with controllable ground-truth implantation, anda cycle–arc standard example. Missing structures of order 2 and above are given aprogrammatic formulation within the framework of obstruction theory and are strictlydistinguished from the proven criteria. Four open problems are registered at the endof the paper.