Title. Analytic validity of mapping the regularized sum of natural numbers to the sum of primes via a scalar product projection?

Title

Analytic validity of mapping the regularized sum of natural numbers to the sum of primes via a scalar product projection

Body

In a recent preprint by Ezadiin Redwaan titled "A Scalar Product Approach to Strong Goldbach, Twin Primes, Polignac Conjectures And Geometric Unification of Regularized Prime Manifolds," the author introduces a global boundary identity for an infinite-dimensional prime vector space:

$$\begin{equation} -\frac{n}{12} \cdot \cos(\theta_i) = \sum_{\mathbb{P}_{\text{odd}} \in \mathcal{P}_{\text{k}}}^{\infty}\mathbb{P}_{\text{odd}} \end{equation}$$

Context & Derivation Claims:

• Vector Setup: The author defines an infinite-dimensional inner product space using sequences of prime components and ones, constructing vectors $\mathbf{A} = (1, p_1, 1, p_2, \dots)$ and $\mathbf{B} = (p_1, 1, p_2, 1, \dots)$ such that their standard dot product evaluates to arithmetic sums of the form $a+b$.
• Global Left-Hand Side: By imposing a coordinate invariance constraint over the entire space, the squared Euclidean norm expands into a nested triangular matrix, yielding a global scale expression $2n \cdot (1 + 2 + 3 + 4 + \dots)$. The parameter $n$ is treated as a global metric tensor constant. The divergent series of natural numbers is regularized via Riemann zeta functional reflection to $\zeta(-1) = -1/12$.
• The Phase Projection: The variable $\theta_i$ is a fixed global geometric phase angle constrained strictly to the obtuse interval $\frac{\pi}{2} < \theta_i < \pi$, meaning $\cos(\theta_i) < 0$ structurally cancels the negative sign of the regularized integer baseline.
• Right-Hand Side: This represents the total structural capacity of the prime manifold $\mathcal{P}_k$, treated as a regularized transfinite sum over all odd primes.

The Logic of the Proof: The paper attempts a non-constructive proof by contradiction based on a boundary singularity. It argues that if a specific even integer cannot be partitioned into two primes (violating Goldbach), or if a specific prime gap is finite (violating Polignac), the global phase angle is forced to its limits: $\theta_i \to \frac{\pi}{2}$ or $\theta_i \to \pi$. This triggers an angular collapse ($\cos(\pi/2) = 0$), forcing the global integer side to zero while the right side remains a positive infinite sum of primes, creating a fatal topological contradiction.

My Questions:

1. Analytic Continuation Incompatibility: Standard Ramanujan summation ($\sum n = -1/12$) stems from the smooth analytic continuation of the Riemann Zeta Function $\zeta(s)$. However, the Prime Zeta Function $P(s) = \sum p^{-s}$ possesses a dense line of singularities (a natural boundary) along $\text{Re}(s) = 0$. From a rigorous analytic number theory standpoint, can a smooth global projection ($\cos \theta_i$) validly bridge two Dirichlet series with such fundamentally incompatible analytic structures?
2. Trivializing the Parity Barrier: Sieve theory is restricted by the parity barrier. Does mapping these sets to an infinite-dimensional vector space ($\mathbf{A} \cdot \mathbf{B} = a+b$) and imposing a global regularized boundary condition actually circumvent the parity problem, or does it merely obscure the arithmetic obstruction behind a non-constructive geometric definition?
3. Rigorous Regularization of $\sum p$: Is there any established framework in mathematical physics, conformal field theory, or asymptotic analysis where the infinite sum of primes can be regularized or assigned a finite global value that mathematically pairs with $\zeta(-1)$?