Collatz Final Gate (v5.6) — Proof-completion reduction via an EB-only closure packet

# Summary (v5. 6) ## OverviewThis release advances a **certification-to-proof pipeline** for the Collatz conjecture by formalizing an **EB-only closure packet** interface at a base instance \ ( (k_, L) \). The manuscript separates the program into two layers: - **Internal layer (deterministic mathematics): ** once a closure packet is supplied, the operator/absorption implications (Dirichlet gap engine, annealed contraction, and the Gate B closure step) are proved internally as a strict implication chain. - **External layer (finite certificates only): ** the only computational role is to produce **auditable witness objects** and logs at a fixed \ ( (k_, L) \) that mechanically imply the operator inequalities used by the internal layer. The central accounting rule is the budget-to-lift mechanism: \₋₈₅ₓ (k_, L): =₁₋ (k_, L) +₂₎₌₏ (k_, L), ₄₅₅: =_-₋₈₅ₓ. **Level-2A** numerical claim requires *threshold enforcement* (see below), e. g. \ (₄₅₅>0\) and \ (₁₀₃ _0\), \ (₁₀₃_\) ) that passes mechanical audit. ## KeywordsCollatz conjecture; certification-to-proof reduction; closure packet; twisted transfer operators; spectral gap; Dirichlet gap; absorbing sets; conductance/flux bounds; carry leakage budgets; auditable witnesses; reproducible certification; mechanical audit. ========================= Author: Lee Byoungwoo leeclinic@protonmail. com