Trisduction: Formal Proof of Geometric Orthogonal Certainty Claim by Self-Grounding Architecture

Four structures of the Trisduction architecture are gathered into one self-grounding chain and proven to one discipline: the actuation floor RA, the triaxial count, the sufficient lock, and the twelve-gate roster. The chain is a single self-similar object, a floor that grounds itself four ways, verifies itself by its own cascade, and returns as a smaller posterior copy that grounds again. Beneath the architecture sits its pure-algebra completion. The architecture carries three seals under an explicit independence law, linguistic, geometric, and mathematical, each forbidden to stand on the others' instruments. This paper closes the mathematical seal alone, with the linguistic and geometric instrument sets removed entirely and demoted to corroboration. An Order-Sensitivity Completion theorem shows that non-commutativity of audit composition alone forces the quaternions by the Frobenius classification, fixing exactly three orthogonal axes over one scalar ground and demoting the axis-plurality floor from premise to corollary, so the premise ledger shrinks by one. The twelve-gate roster is recovered three ways inside the integers of the quaternions and corroborated twice from geometry. The truth function stands in closed form, det(R) equals the squared scalar part of the composed triad, with derived bounds that retire the classical composition inequalities from the warrant set, a Weyl-closed invariant catalog, conjugation frame-invariance, explicit precedence, and the Return Law. The reflective register is then run, not asserted: the foundational reflection splits each claim into a decidable bridge and a located residence, the orientation-blindness theorem is executed at machine precision with the discarded sign conserved as a made-zero, and the imprint test separates a genuine determination from a Platonic Ghost. The falsifiable core is two reference batteries totaling twenty-five recorded checks, identity residues at machine precision and parameter-free integer counts, twenty-four, twelve and twelve, and the class equation 1, 3, 4, 4, with seven falsifier clauses stating the exact null results that break each claim. Each link is theorem-grade on grounded premise; the union of the four pillars into one chain is the structural road, named structural and never a new theorem. The word counts three, the geometry closes four, and the algebra is the receipt.