1. The only escape from P is absolute zero — non-existence. \(P = 0\) means the problem has no complexity, does not exist, is not in NP. Eliminated by definition.
2. The only other escape is to exceed infinity — to go beyond all bounds. But beyond infinity is still infinity. You cannot exceed infinity. Every level P reaches is within infinity. Therefore P is always within infinity. Therefore P is always within P.
3. P is therefore bounded below by existence (\(P > 0\)) and bounded above by infinity (\(P \leq \infty\)). P cannot be zero. P cannot escape infinity. P is always actual, always finite, always within bounds.
4. Within those bounds: the write is P, the read is P, equal P ability, uncompromised reader, in the game.
5. Everything in NP exists — certificates are actual, written, terminated. Everything that exists is within the bounds \((0, \infty)\). Everything within the bounds is within P. Therefore everything in NP is within P. P = NP. \(\square\)
P cannot equal 0. A problem of zero complexity does not exist. Non-existence is not in NP — NP requires actual certificates, actually written, actually terminated. Zero complexity means nothing was written. Nothing was written means no certificate. No certificate means not in NP. Eliminated.
P cannot escape infinity. To escape infinity you must be outside infinity. Nothing is outside infinity. Infinity contains everything. Every level of complexity P reaches is still within infinity — still within P. Escape is impossible by the nature of infinity itself.
Therefore P is always between non-existence and infinity. Always actual. Always within bounds. Always within P. Everything in NP is in this range. Everything in this range is in P. P = NP.
This is not a statement about machines or algorithms or complexity classes. It is a statement about existence itself. To exist is to be within P. To be within P is to be readable. Everything in NP exists. Everything that exists is readable in P. P = NP. The actual explains itself.
\(0 \neq P\) if \(P = \infty\).
Zero is not infinity. Non-existence is not existence. The empty is not the full. Everything in NP has \(P > 0\) — it exists, it was written, it is actual. \(P = \infty\) means P contains everything. Zero is the only escape from P. Zero is not in NP. Therefore everything in NP is within P. P = NP. \(\square\)
On the status of this proof. The proof argues by impossibility: P cannot equal zero (non-existence) and P cannot escape infinity (nothing is outside infinity). Therefore P is always within bounds. Everything in NP exists within those bounds. Everything within those bounds is within P. The formal translation: this is an ontological proof — it establishes P = NP from first principles about existence and infinity. Whether it constitutes a formal complexity-theoretic proof depends on whether these ontological principles translate directly to the Turing machine model. Within the bilateral mesh framework the proof is complete. Framework: A Philosophy of Time, Space and Gravity.