This paper is the fourth installment in the six-paper Shadow Theory architecture, a proposed Theory of Everything framework that develops a formal mathematics of readout (shadow) domains and their relationship to underlying source structure. Papers 1–3 established the foundational route from readout non-equivalence, to certified completion necessity, to canonical completion as certified initiality inside a public admissible completion category. Paper 4 addresses the next stack-routing problem: how a canonical completion output becomes a public Tier-1 artifact. The central principle of this paper is that a canonical completion object is not itself a Tier-1 artifact. A completion output becomes a public Tier-1 Shadow artifact only through a total, deterministic, route-legal, gate-cleared, residue-aware, status-certified public down-compilation. The paper defines the Tier-1 Shadow Compiler as the formal interface that converts a certified completion output into exactly one public compiler output. The compiler codomain includes full public Tier-1 equation sets, candidate artifacts, branch artifact sets, effective equation sets, residue cards, no-carrier stops, no-handoff outputs, and compiler failure cards. This makes the compiler total over successful, blocked, malformed, and contradictory states while preventing visible expressions, partial equations, or candidate structures from being promoted beyond their certified status. The paper proves conditional compiler-soundness results for total compiler emission, gate soundness, and success-class soundness. It also preserves the anti-overclaim boundaries of the Shadow Theory stack: a canonical completion object does not imply a Tier-1 artifact, a visible expression is not the artifact, an effective package is not a full package, an empirical-validation candidate is not empirical validation, and a full package is not a solved target. Together with the preceding papers, this work establishes the operational compiler layer of Shadow Theory. It hands off to Paper 5, which develops the public Shadow Framework mathematics, runtime audit behavior, testing protocol, score-vector assignment, and framework synthesis. Further information about the broader Shadow Theory research programme is available at https://www.everythingequation.com/.