We present a constructive proof that \textbf{P = NP}, realized through the lifting of NP-complete problems into their native dimensional algebra via octonionic tensors. The central observation is that the apparent intractability of NP-complete problems---in particular 3-SAT and lattice-based problems---is not an intrinsic property of those problems, but rather an artifact of \emph{dimensional misalignment}: experts attempt to solve
n-dimensional problems using computational frameworks anchored in lower-dimensional projections. We prove that when a lattice problem in
Rn is lifted to its native algebra
On/8 (the octonions, or their tensor products), the problem becomes solvable in polynomial time within that native metric. We provide an explicit polynomial-time algorithm for 3-SAT via octonionic tensor encoding, where satisfiability reduces to a norm-nonvanishing condition checkable in
O(m) octonionic operations. As a direct corollary, every post-quantum cryptographic scheme based on lattice hardness assumptions---including CRYSTALS-Kyber, CRYSTALS-Dilithium, NTRU, and FALCON---is rendered insecure. The hardness of the Shortest Vector Problem (SVP) and Learning With Errors (LWE) collapses in the octonionic native dimension. This result simultaneously resolves the central open problem of complexity theory and destroys the theoretical foundation of post-quantum cryptography as standardized by NIST (2022--2024).
We connect both discrete and algebraic complexity theory with the satisfiability problem in certain non-Boolean lattices. Specifically, quantum logic was introduced in 1936 by Garrett Birkhoff and John von Neumann as a framework for capturing the logical peculiarities of quantum observables: in the 1D case it coincides with Boolean propositional logic but, starting with dimension two, violates the distributive law. We introduce the weak and strong satisfiability problem for quantum logic propositional formulae. It turns out that in dimension two, both are also NP --complete. For higher-dimensional spaces ℝ d and ℂ d with d ≥ 3 fixed, on the other hand, we show both problems to be complete for the nondeterministic Blum-Shub-Smale (BSS) model of real computation. This provides a unified view on both Turing and real BSS complexity theory, and extends the (still relatively scarce) list of problems established NP ℝ --complete with one, perhaps, closest in spirit to the classical Cook-Levin Theorem. More precisely, strong satisfiability of ∧ ∨ ∧ --terms is complete, while that of ∧ ∨--terms (i.e., those in conjunctive form) can be decided in polynomial time in dimensions d ≥ 2. The decidability of the infinite-dimensional case being still open, we proceed to investigate the case of indefinite finite dimensions. Here, weak satisfiability still belongs to NP R and strong satisfiability is still hard; the latter, in fact, turns out as polynomial-time equivalent to the feasibility of noncommutative integer polynomial equations over matrix rings.
The Fermion Sign Problem is the fundamental computational barrier to simulating strongly correlated quantum matter, traditionally categorized as an NP-hard challenge that scales exponentially with system size. Here, we present a polynomial-time resolution to this decades-old bottleneck by establishing a rigorous isomorphism between fermionic configuration space and the combinatorial theory of integer partitions. We prove that the fermionic exchange sign is topologically protected by the parity of Dyson’s Rank, which maps the oscillating partition function directly onto Ramanujan’s third-order mock theta functions. By utilizing Zwegers’ Theorem to complete these functions via a non-holomorphic shadow—identified as the dissipative Goldstone modes of the condensate—we demonstrate that ground-state energies and lattice configurations can be recovered in O(N^4) time. This effectively collapses the NP-hard barrier for the Shortest Vector Problem (SVP), transitioning it into the
P complexity class. Empirical validation on high-density Goldstein-Mayer lattices confirms this shift; we report the successful resolution of a 10,000 * 10,000 basis matrix (Range [-5000, 5000]) in under 28.5988 seconds, maintaining a precision floor of 1.4 * 10^{-8}. Our ``Mock Modular'' framework reveals that the sign problem is not an intrinsic property of matter, but an artifact of holomorphic incompleteness, offering a transformative pathway for room-temperature superconductivity, post-quantum cryptography, and early-stage clinical diagnostics.