This work presents a systematic and rigorous extension of the entire apparatus of Meta-Operational Mathematics to the domain of computer operations and their compositional inverses. Every computer program– arithmetic, logic, bitwise, control flow, data movement, compression, encryption, compilation, optimization– is regarded as an operation acting on a discrete base space of finite bit strings. A meta-operation is a mapping that takes one or more operations as input and produces another operation as output. The fundamental philosophical principle– that operations upon operations are the natural generalization of operations upon data– implies that every elementary computer operation lifts canonically to a meta-operation, and that meta-operations can be composed, added pointwise, multiplied, differentiated, randomized, exponentiated, logarithmized, and subjected to bornological limits arbitrarily many times (integer, fractional, real, complex, or even infinite). A strict four-level hierarchical framework is developed: Level 0 (data elements, finite bit strings), Level 1 (computer operations, e.g. ADD, XOR, COMPRESS), Level 2 (meta-operations, e.g. COMPILE, OPTIMIZE, DCE), Level 3 (meta-meta-operations, e.g. a compiler of compilers). Within this framework, twelve axioms are established that capture the essential features of computer science: discreteness, side effects, non-global inverses with recovery fidelity η, compilation/decompilation as an antipode pair, concurrency via a non-commutative parallel composition, and bornological convergence for infinite processes. The category of computer meta-operations is shown to carry an endomorphism operad structure, which is further endowed with a Hopf operad structure. The antipode is given by decompilation (inverse of compilation), decryption (inverse of encryption), or, in general, any compositional inverse with recovery fidelity η. Bornological convergence is introduced to handle infinite sums, infinite compositions, fractionally iterated operations, and the limit behaviour of operating systems and recursive programs. All classical abstract data types (stacks, queues, trees, graphs, hash tables, finite automata) are shown to belong to the meta-operational universe, and their fundamental identities become equations of meta-operations. Program analysis (constant propagation, dead code elimination, loop unrolling), compiler optimisations, Hoare logic, dataflow analysis, and cryptographic protocols (encryption, hashing, digital signatures) are reformulated as meta-operational equalities or bornological fixed-point problems. Every conjecture and open problem from the original research programme has either been proved as a theorem within the body of this work or is precisely reduced to a well-known hard problem. All proofs are given in full detail (every theorem is accompanied by a step-by-step proof of at least four steps; central theorems have ≥ 12 steps). Numerical verification tables and algorithm schemas are provided. The work is self-contained, assuming only basic computability theory, elementary category theory, and undergraduate programming language semantics.
Liu S (Wed,) studied this question.