Self-Dual Trace Confinement: A Six Birds Structural Law for Formed Closures Under Involutive Self-Duality

Abstract When must an operator-valued obstruction associated with an involutive system vanish? This paper answers that question inside the Six Birds framework by naming Self-Dual Trace Confinement (SDTC) as the recognition source for formed closures with genuine involutive self-duality. The mechanism is the duality-confinement membrane theorem: for an involutive object ledger with separating anti-invariant readout, a sequence of positive domination records AX ≼ Bn with tr Bn → 0 forces the anti-invariant ledger AX to vanish and confines visible mass to the fixed locus of the duality. The supporting apparatus develops the involutive object ledger, the anti-invariant ledger and trace identity, direct qualitative and quantitative confinement, completed-domination bridges, Douglas factorization in typed-cone form, exhaustive moving ledgers, and optimized scalar trace budgets. Operator-theoretic realization over trace-class operators is an open extension, as is a derivation of the AM–GM step inside a typed real-arithmetic structure. The development is formalized in Lean as a mathlib-free mechanization at the typed positive-cone level. Cross-substrate generalizations such as CPT symmetry, gauge invariance, quantum self-adjointness, and function-field RH are presented as structural predictions, not theorems; the paper develops the abstract apparatus layer and names the hypotheses any substrate-specific application must supply.