For problem A and candidate solution x, the verification geodesic V(x) is the shortest path on MA from problem statement to verified solution. Its length |V(x)| defines verification complexity.
Definition 2.2 (Generation Geodesic):
The generation geodesic G(A) is the shortest path on MA from problem statement to any verified solution. Its length |G(A)| defines solution complexity.
Definition 2.3 (Homoiconic Embedding):
A diffeomorphism φ: MA → M'A that preserves computational semantics while altering geodesic distances. This represents finding the "right" problem representation.
Theorem 3.2 (NP as Search Geometry):
A problem A is in NP iff there exists a manifold representation where |V(x)| is polynomial, but no such guarantee exists for |G(A)| in all representations.
Proof: Follows directly from definitions: NP verification is efficient (short V(x)), while solution finding may require exponential search (long G(A)) in the naive geometry.
Corollary 3.5 (Creativity Bound):
Proving P = NP is equivalent to demonstrating a universal mathematical principle for constructing geodesic-shortcut embeddings.
Definition 2.1 (Verification Geodesic):
For problem A and candidate solution x, the verification geodesic V(x) is the shortest path on MA from problem statement to verified solution. Its length |V(x)| defines verification complexity.
Definition 2.2 (Generation Geodesic):
The generation geodesic G(A) is the shortest path on MA from problem statement to any verified solution. Its length |G(A)| defines solution complexity.
Definition 2.3 (Homoiconic Embedding):
A diffeomorphism φ: MA → M'A that preserves computational semantics while altering geodesic distances. This represents finding the "right" problem representation.
Theorem 3.2 (NP as Search Geometry):
A problem A is in NP iff there exists a manifold representation where |V(x)| is polynomial, but no such guarantee exists for |G(A)| in all representations.
Proof: Follows directly from definitions: NP verification is efficient (short V(x)), while solution finding may require exponential search (long G(A)) in the naive geometry.
Corollary 3.5 (Creativity Bound):
Proving P = NP is equivalent to demonstrating a universal mathematical principle for constructing geodesic-shortcut embeddings.