Why P ≠ NP Flows from the Addition–Scaling Path: Prime-Generator Isomorphism, Symmetry-Free Dual Witnesses, and the Quantization–Uncertainty Valley Beyond OpenAI's ETH Refutation

We investigate the algebraic and spectral mechanisms governing subexponential counting algorithms for k-SAT following OpenAI's recent refutation of the Exponential Time Hypothesis (T(n) = 2O(n (log log n)2 / √(log n))). Any Q-point polynomial-interpolation decoupling scheme partitions variable-clause incidences across an auxiliary state space of dimension D(Q, R) and decomposes the incidence query vector as ζ = Mζ + (I − M)ζ into a degree-(R−1) polynomial bulk term Mζ (READ) and a residual coupling (I − M)ζ (RESIDUE) of rank c = rank(I − M). On the 2Q-element fibers of the swap group Σ = ⟨Π1, …, ΠQ⟩ ≅ {±1}Q, this decomposition is governed by the interplay between two non-overlapping operators: The Additive Disagreement Counter: W0 = (1/2) Σs=1Q (I − Πs) (with Binomial spectrum Bin(Q, p) of Gaussian width Θ(√Q)), and The Multiplicative Room-Transfer Parity Involution: V0 = Πs=1Q Πs = (−1)W0. Balancing the Taylor readout cost of degree ⌊3nR / Q⌋ against the residual rank cost under the integer copy-multiplicity constraint h = max(1, ⌈6n / D⌉) yields the Master Bottom-Up Exponent Law: E(n, Q, R) ≈ n (R / Q) log2(2eQ / R) + max(n, D(Q, R)) exp(−R2 / (8Q)) log2 D(Q, R). Crucially, we prove that P ≠ NP is not an isolated complexity-theoretic condition, but a direct mathematical consequence of the non-overlap between Addition (ADD) and Scaling (SCALE)—the exact same root bifurcation that generates the prime numbers as unreachable empty slots on the integer lattice. Key Contributions and Certified Results (Theorems I–VI) Theorem I (Master Bottom-Up Exponent Law and V-Shaped Quantization–Uncertainty Valley): Proves that for 1 < Q ≤ Q* = Θ(log n), integer copy multiplicity h = ⌈6n / D(Q)⌉ ≥ 1 cancels the auxiliary dimension D(Q) = r2Q, driving the runtime exponent down the Gaussian square-root slope R/Q ~ 1/√Q to OpenAI's 2Θ̃(n / √(log n)) valley minimum at Q* = Θ(log n). For Q > Q*, discrete incidence quantization locks h = 1, halting the cancellation of D(Q) = r2Q, forcing R = Θ(Q), and rebounding T(n) back toward 2Ω(n). Theorem II (Telescoping Circuit Decomposition, Fundamental-Theorem-of-Algebra Locality Lock, and Antipode Bijection): Proves that any circuit C of size S and depth d telescopes into a polynomial READ term plus a cumulative RESIDUE of rank Ctot ≤ Σg=1S rank(I − Mg). By the Fundamental Theorem of Algebra, any degree-(R−1) decoupling matrix M vanishes identically whenever the cross-room mismatch count satisfies |Ex,y| = dHam(x̄, y) ≥ R, locking the minimal auxiliary subspace dimension on every 2Q-element swap fiber to Dmin(Q, R) = Σj=0R−1 C(Q, j) ≥ C(Q, R−1). Theorem III (Rolle Sign-Alternation Lock on Adjacent Pairs and Conjugate Computational Uncertainty Hyperbola): Proves via Rolle's Theorem and the Intermediate Value Theorem that any real polynomial f(z) of degree ≤ R − 1 agreeing with (−1)j on an arbitrary (possibly non-contiguous) subset S(f) ⊆ {0, 1, …, Q} can cover both elements of at most ⌊R/2⌋ adjacent even–odd pairs {2k, 2k+1}. This enforces the universal closed-form lower bound mindeg(f) ≤ R−1 Pr[f(W) ≠ (−1)W] ≥ 1/2 − Θ(R / √Q) and locks every admissible READ → RESIDUE operator above the Conjugate Computational Uncertainty Hyperbola: Δdeg · Δdim = (R / Q) · (log2 D / log2(Q / R)) ≥ R2 / Q ≥ Ω(1). Theorem IV (Exact Algebraic Isomorphism Between Reality's Prime Generator and the #3-SAT Swap Algebra): Proves that under the Q-fold Walsh–Hadamard transform HQ, the #3-SAT swap algebra ℚ[Π1, …, ΠQ] on any 2Q-element fiber is canonically isomorphic to the square-free divisor algebra on the first Q primes p1, …, pQ, mapping HQ W0 HQT = diag(ω(n)) (Erdős–Kac prime-factor counter) and HQ V0 HQT = diag(μ(n)) (Möbius empty-slot detector). Under HQ, any degree-(R−1) READ operator maps to a truncated Brun–Selberg prime sieve of degree R − 1, and the degree threshold R ≥ Ω(√Q) is the exact algebraic counterpart of Selberg's Parity Barrier (1949). Theorem V (Symmetry-Free Delsarte–Krawtchouk Positive-Semidefinite Dual Witness for Arbitrary Matrices M): Constructs an explicit positive-semidefinite spectral witness Wdual(W0) = Σj0=0Q−R αj0 Pj0,R(W0) ⪰ 0 with ||Wdual||op ≤ 1 whose entries vanish identically on all allowed support pairs |Ex,y| ≤ R − 1. Without any symmetry or commutativity assumptions on M ∈ ℂ2Q × 2Q, Tr(Wdual M) ≡ 0, proving the universal nuclear-norm and rank lower bound via unimodal sliding-window bijection: rank(I − M) · ||I − M||op ≥ ||I − M||* ≥ Tr(Wdual) ≡ Σj ∉ I* C(Q, j) = 2Q min|I|=R Pr[W ∉ I]. Theorem VI & Corollary 7.1 (Unconditional Amortized Extraction on Three Concrete Circuit Classes & Classical Barrier Audit): Applies Wdual(W0) to establish unconditional extraction lower bounds for depth-d AC0[⊕] circuits (S ≥ 2Ω(Q1/(2d))), bounded-norm algebraic/threshold decoupling layers (rank(I − M) ≥ (2Q / K) min|I|=R Pr[W ∉ I]), and read-k oblivious branching programs (ks ≥ Ω(Q)), together with a complete audit of the Relativization, Algebrization, and Natural Proofs barriers. Author Contact:Oleksandr Khropak (Kyiv, Ukraine) — oleksandr.hropak@gmail.com