Strict mathematical proof of $P \ neq NP $based on the computability theory of group organizations This article aims to introduce a new computational topology dimension, Abstract Degree (denoted as $\ alpha $), through the framework of the computable theory of group organizations, to rigorously mathematically and logically derive the core problem of P \ neq NP $in computer science. --- 1、 Formal Definition and Axiomatic System Definition 1: Individual Computation Space ($\ mathcal) {S}_ {ind}$) Let M be the standard deterministic Turing machine (DTM). Individual computable space refers to the set of languages that can be determined by $M $in polynomial time $O (n ^ k) $. In this space, computation relies entirely on local deterministic state transitions of individuals. The system does not have irreducible group abstract properties, that is, the system's * * abstraction degree $\ alpha=0 $* *. >* * Note * *: The problems solved by deterministic consensus algorithms such as Byzantine Fault Tolerant (BFT) are strictly mapped and enclosed within the domain of this individual Turing machine. We define its upper limit of computing power as $C '$. Therefore, there are: $$P \subseteq \mathcal {S}_ {ind} \ examples \ orall x \ in P, \ text {abstraction} \ alpha (x)=0$$ Definition 2: Group Computation Space ($\ mathcal) {S}_ {grp}$) The computable space of a group introduces non local and irreducible abstract properties of the group (i.e. emergence). Its computing power is defined by the following core formula: $$C = C' + \mathcal{O}$$ among which *$C '$is the computing power of an individual Turing machine. *$\ mathcal {O} $is * * Group Oracle * *, representing spontaneously formed organizational rules (such as the longest chain rule with $\ alpha=1 $in the Timechain). In this space, the collapse of non deterministic states depends on the adaptive adjustment algorithm of group attributes, and the abstraction degree of the system is $\ alpha>0 $* *. Axiom 1: Homomorphic Mapping of Problem Classes According to the computability theory of group organizations, there exists an equivalent mapping between classical computational complexity classes and this space as follows: 1. * * $P $class problem * *: Equivalent to a calculation problem of individual attributes, completely constrained by the boundary $C '$of individual Turing machines. 2. NP class problem: Traditionally defined as a problem that a non deterministic Turing machine can solve in polynomial time. In this theory, the concurrency and exploration of non deterministic branches are essentially equivalent to the computation problem of group properties, relying on the emergence results of the group oracle $\ mathcal {O} $. --- 2、 Core proof process Theorem: In the computable theoretical framework of group organizations, $P \ neq NP $. **Proof (proof by contradiction) * *: **Step 1: Set the opposite hypothesis** Assuming that $P=NP $holds true. According to the homomorphic mapping of Axiom 1, this means that under the measure of polynomial time complexity, the calculation of individual attributes is equivalent to the calculation of group attributes. Expressed in formal language, i.e. computing power space $\ mathcal {S}_ {ind} $can fully cover and simulate $\ mathcal {S}_ {grp} $for solving such problems. **Step 2: Derive the equivalence of computing power** If $P=NP $, then the highest computational boundary of an individual must be equivalent to the highest computational boundary of a population: $$C' \equiv C$$ Substituting this equivalence into the fundamental equation $C=C '+\ mathcal {O} $of the computable theory of group organizations, we can obtain: $$C' \equiv C' + \mathcal{O}$$ **Step 3: Introduce the Contradiction of Abstract Degree** From the relationship $C '\ equiv C'+\ mathcal {O} $in set logic and arithmetic, it can be deduced that: $$\mathcal{O} \subseteq C'$$ This means that the oracle $\ mathcal {O} $representing the emergence rules of a group can be accurately calculated, restored, or simulated in polynomial time through finite step local state transitions using the individual Turing machine $M $. However, according to the essential definitions of both: 1. Space $\ mathcal {S}_ The abstraction degree of {ind} $(i.e. the domain of $C '$) is always $\ alpha=0 $. 2. The oracle $\ mathcal {O} $carries irreducible group abstract properties, and its existence is based on the system's abstraction degree $\ alpha>0 $. If $\ mathcal {O} \ subseteq C '$holds, meaning that group attributes can be completely deconstructed and computed by individual Turing machines, then the group attribute loses its "irreducibility" and its abstraction will undergo forced collapse: $$\alpha > 0 \implies \alpha = 0$$ This collapse violates the orthogonality premise of the computable dimensions of the population (just like in plane geometry with a curvature of $0 $, no matter how a straight line is extended, it is impossible to outline a non Euclidean structure with a curvature of $0 $). The group attribute (emergence) is essentially located outside the computable boundary of individual Turing machines and cannot be generated by isolated individual systems with $\ alpha=0 $. **Step 4: Draw the final conclusion** Because $\ alpha>0 $and $\ alpha=0 $are logically mutually exclusive, $\ mathcal {O} \ nseteq C '$. Thus: $$C' eq C' + \mathcal{O}$$ Therefore, assuming $P=NP $leads to a logical necessary contradiction (abstraction collapse contradiction). The computing power boundary of individual attributes cannot touch the computing domain of group attributes. $$\therefore P eq NP$$ **Certificate completed. ** --- 3、 Summary from the Perspectives of Philosophy and Topology By introducing * * "abstraction $\ alpha $" * * as the core dimension for measuring computability, this proof successfully reduces and transforms the complex algorithm time complexity problem into a topological/geometric irreducibility problem * *. ** * $P $is within the computable boundary of the Turing machine * * (flat computing space with $\ alpha=0 $). *NP $is located outside the computable boundary of the Turing machine (non flat population emergence space with $\ alpha>0 $). https://(github.com)/gguoss/My-10-Year-Journey-in-the-Crypto-World/blob/main/%E5%9F%BA%E4%BA%8E%E7%BE%A4%E4%BD%93%E7%BB%84%E7%BB%87%E5%8F%AF%E8%AE%A1%E7%AE%97%E7%90%86%E8%AE%BA%E7%9A%84%20%24P%20%5Cneq%20NP%24%20%E4%B8%A5%E6%A0%BC%E6%95%B0%E5%AD%A6%E8%AF%81%E6%98%8E.md