Features of Ei:
Automatic inversion: Ei detects all operators and the structure of the expression on its own.
Type-independent: works with numbers, strings, arrays, boolean expressions, sets, and even algorithms.
Deterministic: each operation receives a formally defined mirrored counterpart or a MAX result if no direct analog exists.
Structure-preserving: nesting, parentheses, and functions are automatically maintained.
Formal idea: Let E be any expression composed of operations and operands. The low-level inversion Ei is defined as an operator I acting on the structure of the expression: I(E) = E[I(O1), I(O2), ..., I(On)] where I(Oi) = mirrored operation Oi or MAX if the operation is unique and has no direct counterpart.
Examples of inversions: Boolean operators: NOT → TRUE, AND → OR, OR → AND, XOR → XNOR Sets: UNION → INTERSECT, INTERSECT → UNION, BACKSLASH → symmetric difference Numbers: + → *, > → <, MAX → MIN Arrays and strings: append → prepend, sort → reverse sort, concat → reverse concat
Example of Ei in action: Input: NOT A AND B OR C Ei result: TRUE A OR B AND C Structure is preserved, operators are automatically inverted, operand types don’t matter.
Applications of Ei:
Boolean algebra and combinatorics: analyzing symmetries, generating alternative logic circuits
Set theory: building mirrored expressions, finding antimorphisms
Linear algebra: mirrored matrix operations, alternative computation algorithms
Algebraic structures: analyzing symmetries in groups, rings, fields
Information theory and cryptography: alternative encryption schemes, pattern analysis
Automated testing and AI: generating mirrored algorithm variants for code stability checks
Ei is a universal inversion formula that works without user intervention, automatically determines mirrored operations, preserves expression structure, and can be applied in mathematics, programming, cryptography, combinatorial design, AI, and algorithm testing.