Theoretical Cartography 3: Criteria for Illegal Bridges—Invariant Spectra, Massey Obstructions, and Kan Duality

In theoretical ecology, candidate bridges with the same name or similar structure emergespontaneously in large numbers. If a cartographic framework only registers bridges and lacksthe capacity to reject them, the information content of its map tends to zero. Building on theweighted hypergraph foundation and homological detection tools, this paper establishes a threetier system of illegality criteria for candidate bridges. The first-tier criterion is based on derivedinvariant spectra and role signatures; it is a purely combinatorial, machine-checkable necessarycondition used to identify same-name traps. The second-tier criterion is based on Massey tripleproducts; it characterizes higher obstructions that are pairwise compatible but triply incompatible, and on Borromean-type differential graded algebras it machine-checks and recomputestheir non-vanishing over GF(2). The third-tier criterion is Kan duality: an archive-level legalbridge requires that the left Kan extension and the right Kan extension along the bridge benaturally isomorphic, i.e., transporting an archive does not depend on the direction of transport; its prototype in finite group representation theory is Frobenius reciprocity, confirmed bycharacter-level machine checks over GF(7), and its deeper literature anchor is the ambidexteritytheorem of Hopkins and Lurie. The framework-level status of the third-tier criterion is explicitlymarked: on finite-group-type archives it is a theorem, while on general theoretical ecology it isa framework-level claim accompanied by a falsification path. The unified output format of thecriterion system is a combination of level, obstruction degree, and reversibility clause; illegalverdicts, undecidability, and conditional release are listed separately, and obstruction degreesare not compared across levels. Verdicts are executed on three candidate bridges: the samename bridge between the G2 root system and cellular hierarchical traversal is rejected at thearchive level by the first-tier criterion; the bridge between fractal zeta zeros and an intertextualspectral gap is refused registration because the archives are incomplete; the bridge betweenthe time ladder and the Cayley–Dickson sequence 1,2,4,8 is not rejected by the first tier andis registered as conditionally released, with certificate-registration obligations and second- andthird-tier inspection obligations attached. All machine-checkable assertions, protocols, readings,and certificate indices are given in the appendix.