CODA-MUR: Certified Representation Transport, Composition, and Irreversible Forgetting

CODA-MUR: Certified Representation Transport, Composition, and Irreversible Forgetting A Commuting Square Is Not Target Standing: Certified Representation Transport and Irreversible Forgetting In machine learning, compilation, and abstract interpretation, representations are constantly compressed, projected, or quantized: High-dimensional vectors are mapped into lower-dimensional embeddings. Infinite integers are collapsed into parity bits (n mod 2). Verified high-level source code is compiled into optimized binary. When an observation commutes across a move, (q(F(x)) = F̄(p(x))), it is easy to mistake that compatibility for an operational license. It is not. A commuting square is a transport certificate, not target standing. CODA-MUR formalizes the algebra of moving pictures across observation boundaries while enforcing strict claim hygiene: Look After Move vs. Move After Look: A transport certificate χ_F asserts that observing after a transformation yields the same result as transforming the observation. Transports paste associatively: chaining certified moves yields a certified composite move. Collisions Travel: If two states appear identical under observation (p(x) = p(y)), they remain identical after transport. Downstream Processing Cannot Split a Collision: No downstream function or post-processing filter can un-collide states once their distinctions have been merged. Irreversible Forgetting: Non-injective collapse cannot be inverted. Once two distinct values map to a single point, no left inverse recovering every source value exists. The Authority Firewall: Compatibility with a target regime’s observation space does not grant authority within that regime. Operational standing requires reaching a declared target terminal and independently satisfying seven evidence-derived halt guards. Mechanized with zero unproved axioms (0 sorry, 0 admit) in Lean 4 and paired with an exact rational Python verification engine (featuring a closed 5 x 5 morphism composition monoid and strict loss ledgers), CODA-MUR provides the formal foundation for moving data without laundering unauthorized claims.