Conclusion

The residual seed theory becomes precise once vertical and horizontal generation are separated. Positive-depth perturbations can be complete within one formal component. They cannot change the residual root. The universal arithmetic seed is therefore a forest rather than one branch:

Projectors add three useful structures to this forest. They split tangent and obstruction complexes into exact subproblems, they measure delayed dyadic visibility through weights, and they turn finite carrier occupancy into an integer trace. Directed transport adds a fourth: it converts the informal question “how many seeds are needed?” into the source-component count of a condensation graph.

The system-level boundary is now narrow enough to state without ambiguity. One must construct a common algebraic spine, populate every residual companion forest, lift those companions through all depths, and prove positive occupancy of the required global packets. The present paper proves the formal reconstruction, counting, transport, compactness, carrier, and finite-projector theorems used in that reduction. It does not assert the missing universal existence statements.

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