[proofplan] The proof is a direct linearity argument. A parity-blind divisor-sum functional in $\mathcal C_D$ depends, by definition, on a finite sequence only through the local divisor counts $|S_d|$ for squarefree $d \leq D$ dividing the primorial-type product $P(z)$. Since $A$ and $B$ have identical values for exactly those local counts, every coefficient-weighted linear combination of them has the same value on $A$ and on $B$. The parity conclusion follows because the permitted functionals have no access to any information beyond these local counts. [/proofplan] [step:Expand an arbitrary ordinary sieve functional in terms of its divisor counts] Fix $\mathcal L \in \mathcal C_D$. The quantity $P(z)$ is the product of all primes $p$ with $p \leq z$, as defined in the theorem statement. Define the finite divisor set $\mathcal D_{D,z}$ to be the set of squarefree positive integers $d$ such that $d \leq D$ and $d \mid P(z)$. The assumptions $z \geq 2$ and $D \geq 1$ are part of the ordinary sieve setup and ensure that this is the standard finite level-$D$ family of squarefree divisors drawn from primes up to $z$; the linearity argument below uses only that $\mathcal D_{D,z}$ is exactly the finite indexing set specified in the definition of $\mathcal C_D$. For a finite sequence $S=(s_1,\dots,s_N)$ of positive integers and a positive integer $d$, define \begin{align*} S_d := \{j \in \{1,\dots,N\}: d \mid s_j\}. \end{align*} Thus $|S_d|$ counts the terms of $S$ divisible by $d$, counted with their multiplicities in the sequence. By the definition of the class $\mathcal C_D$, and not by any additional external theorem, there are complex coefficients $\lambda_d$, indexed by $d \in \mathcal D_{D,z}$, such that for every finite sequence $S$ of positive integers, \begin{align*} \mathcal L(S) = \sum_{d \in \mathcal D_{D,z}} \lambda_d |S_d|. \end{align*} Applying this formula first to $S=A$ and then to $S=B$ gives \begin{align*} \mathcal L(A) = \sum_{d \in \mathcal D_{D,z}} \lambda_d |A_d|. \end{align*} Similarly, \begin{align*} \mathcal L(B) = \sum_{d \in \mathcal D_{D,z}} \lambda_d |B_d|. \end{align*} [guided] We fix one permitted functional $\mathcal L \in \mathcal C_D$ because the desired conclusion must hold for every such functional. First define the objects that the functional is allowed to inspect. The quantity $P(z)$ is the product of all primes $p$ with $p \leq z$, as specified in the theorem statement. Define $\mathcal D_{D,z}$ to be the set of squarefree positive integers $d$ such that $d \leq D$ and $d \mid P(z)$. The hypotheses $z \geq 2$ and $D \geq 1$ place us in the ordinary sieve convention and fix the standard finite level-$D$ collection of squarefree divisors built from primes up to $z$; for the equality argument, the relevant point is that $\mathcal D_{D,z}$ is exactly the finite list of divisor counts available to the functional. For a finite sequence $S=(s_1,\dots,s_N)$ of positive integers and a positive integer $d$, define \begin{align*} S_d := \{j \in \{1,\dots,N\}: d \mid s_j\}. \end{align*} The number $|S_d|$ therefore counts, with sequence multiplicity, how many terms of $S$ are divisible by $d$. The definition of a parity-blind divisor-sum functional in $\mathcal C_D$ is precisely that $\mathcal L$ is a finite linear combination of these divisor-counting data $|S_d|$, where $d$ ranges only over $\mathcal D_{D,z}$. No additional representation theorem is being invoked here. Thus there exist complex coefficients $\lambda_d$ such that \begin{align*} \mathcal L(S) = \sum_{d \in \mathcal D_{D,z}} \lambda_d |S_d| \end{align*} for every finite sequence $S$ of positive integers. The important point is that this formula contains no other information about $S$. It does not inspect the prime factorization of individual terms except through divisibility by the allowed squarefree divisors $d$ of $P(z)$. Substituting $S=A$ gives \begin{align*} \mathcal L(A) = \sum_{d \in \mathcal D_{D,z}} \lambda_d |A_d|. \end{align*} Substituting $S=B$ gives \begin{align*} \mathcal L(B) = \sum_{d \in \mathcal D_{D,z}} \lambda_d |B_d|. \end{align*} Both sums use the same divisor set $\mathcal D_{D,z}$ and the same coefficients, because the functional $\mathcal L$ is fixed. [/guided] [/step] [step:Use the matching local data to cancel every term] Subtracting the two displayed formulas gives \begin{align*} \mathcal L(A) - \mathcal L(B) = \sum_{d \in \mathcal D_{D,z}} \lambda_d \bigl(|A_d| - |B_d|\bigr). \end{align*} For every divisor $d \in \mathcal D_{D,z}$, the hypothesis gives $|A_d| = |B_d|$. Hence each summand is zero: \begin{align*} \lambda_d \bigl(|A_d| - |B_d|\bigr) = 0. \end{align*} Therefore \begin{align*} \mathcal L(A) - \mathcal L(B) = 0. \end{align*} Thus $\mathcal L(A)=\mathcal L(B)$. Since $\mathcal L \in \mathcal C_D$ was arbitrary, the equality holds for every $\mathcal L \in \mathcal C_D$. [guided] The two formulas from the previous step have the same coefficients and the same indexing set, so the only possible difference between $\mathcal L(A)$ and $\mathcal L(B)$ comes from the individual differences $|A_d|-|B_d|$. Subtract the formula for $\mathcal L(B)$ from the formula for $\mathcal L(A)$: \begin{align*} \mathcal L(A) - \mathcal L(B) = \sum_{d \in \mathcal D_{D,z}} \lambda_d \bigl(|A_d| - |B_d|\bigr). \end{align*} Now fix an arbitrary divisor $d \in \mathcal D_{D,z}$. By definition of $\mathcal D_{D,z}$, this $d$ is squarefree, satisfies $d \leq D$, and divides $P(z)$. Therefore the theorem's hypothesis applies to this $d$, giving $|A_d|=|B_d|$. Hence \begin{align*} \lambda_d \bigl(|A_d| - |B_d|\bigr) = 0. \end{align*} Because this holds for every $d \in \mathcal D_{D,z}$, every summand in the finite sum is zero. Therefore \begin{align*} \mathcal L(A) - \mathcal L(B) = 0. \end{align*} Thus $\mathcal L(A)=\mathcal L(B)$. The functional $\mathcal L \in \mathcal C_D$ was chosen arbitrarily, so the same equality holds for every parity-blind divisor-sum functional at level $D$. [/guided] [/step] [step:Interpret the identity as the parity obstruction] The preceding identity shows that all information available to the class $\mathcal C_D$ is contained in the family of local counts \begin{align*} \bigl(|S_d|\bigr)_{d \in \mathcal D_{D,z}}. \end{align*} Let $\mathbb N_0 := \{0,1,2,\dots\}$ denote the set of nonnegative integers, and let $\operatorname{Seq}(\mathbb N)$ denote the set of finite sequences of positive integers. Define the local-data map \begin{align*} \pi_{D,z}: \operatorname{Seq}(\mathbb N) \to \mathbb N_0^{\mathcal D_{D,z}} \end{align*} by sending each finite sequence $S \in \operatorname{Seq}(\mathbb N)$ to \begin{align*} \pi_{D,z}(S) = \bigl(|S_d|\bigr)_{d \in \mathcal D_{D,z}}. \end{align*} The hypothesis says exactly that $\pi_{D,z}(A)=\pi_{D,z}(B)$. Conversely, the values of all functionals in $\mathcal C_D$ determine this vector of local counts: for each $e \in \mathcal D_{D,z}$, define the coordinate-count functional $\mathcal L_e: \operatorname{Seq}(\mathbb N) \to \mathbb C$ by $\mathcal L_e(S)=|S_e|$; this belongs to $\mathcal C_D$ by taking $\lambda_e=1$ and $\lambda_d=0$ for all $d \neq e$. Thus two sequences have the same values under every functional in $\mathcal C_D$ if and only if they have the same local-data vector. Therefore, for every set $Y$ and every well-defined rule \begin{align*} \Phi: \pi_{D,z}(\operatorname{Seq}(\mathbb N)) \to Y \end{align*} whose input is the data supplied by all functionals in $\mathcal C_D$, one has \begin{align*} \Phi(\pi_{D,z}(A)) = \Phi(\pi_{D,z}(B)). \end{align*} This is the formal meaning of saying that conclusions derived only from all ordinary divisor-sum functional values cannot distinguish $A$ from $B$. Define the arithmetic function $\Omega: \mathbb N \to \mathbb N_0$ by letting $\Omega(n)$ be the number of prime factors of the positive integer $n$, counted with multiplicity. In particular, if one such sequence is supported on integers $n$ with $\Omega(n)$ odd and another is supported on integers $n$ with $\Omega(n)$ even, while their allowed divisor counts agree, then ordinary divisor-sum weights at level $D$ give identical outputs on the two models. The obstruction is therefore intrinsic to the local divisor-count data supplied to the sieve weights, not to a special choice of the coefficients $\lambda_d$. [guided] The equality $\mathcal L(A)=\mathcal L(B)$ for every $\mathcal L \in \mathcal C_D$ has a precise information-theoretic interpretation. Let $\mathbb N_0 := \{0,1,2,\dots\}$ denote the set of nonnegative integers, and let $\operatorname{Seq}(\mathbb N)$ denote the set of finite sequences of positive integers. Define the local-data map \begin{align*} \pi_{D,z}: \operatorname{Seq}(\mathbb N) \to \mathbb N_0^{\mathcal D_{D,z}} \end{align*} by sending each finite sequence $S \in \operatorname{Seq}(\mathbb N)$ to \begin{align*} \pi_{D,z}(S) = \bigl(|S_d|\bigr)_{d \in \mathcal D_{D,z}}. \end{align*} The hypothesis of the theorem is exactly the statement that $\pi_{D,z}(A)=\pi_{D,z}(B)$. We also need the converse information claim: the full collection of functional values determines this same local-data vector. For each $e \in \mathcal D_{D,z}$, define the coordinate-count functional $\mathcal L_e: \operatorname{Seq}(\mathbb N) \to \mathbb C$ by $\mathcal L_e(S)=|S_e|$. This functional belongs to $\mathcal C_D$ because it is represented by the coefficients $\lambda_e=1$ and $\lambda_d=0$ for every $d \in \mathcal D_{D,z}$ with $d \neq e$. Hence knowing all values $\mathcal L(S)$ with $\mathcal L \in \mathcal C_D$ includes, in particular, knowing every coordinate $|S_e|$ of $\pi_{D,z}(S)$. Conversely, every $\mathcal L \in \mathcal C_D$ is a linear combination of these coordinates. Thus equality of all functional values is equivalent to equality of the local-data vectors. Now let $Y$ be any set, and let \begin{align*} \Phi: \pi_{D,z}(\operatorname{Seq}(\mathbb N)) \to Y \end{align*} be any rule whose input is exactly the information supplied by all ordinary divisor-sum functionals. Since the two inputs agree, applying the same map $\Phi$ to both inputs gives \begin{align*} \Phi(\pi_{D,z}(A)) = \Phi(\pi_{D,z}(B)). \end{align*} This formalizes the phrase "cannot distinguish": any invariant, rule, or conclusion that depends only on the values of all functionals in $\mathcal C_D$ must assign the same output to $A$ and $B$. Finally define the arithmetic function $\Omega: \mathbb N \to \mathbb N_0$ by letting $\Omega(n)$ be the number of prime factors of the positive integer $n$, counted with multiplicity. If one model is supported on integers $n$ with $\Omega(n)$ odd and the other is supported on integers $n$ with $\Omega(n)$ even, but the local data $\pi_{D,z}$ agree, then every ordinary divisor-sum weight at level $D$ has the same value on both models. Thus the parity obstruction comes from the restricted information available to the sieve weights, namely the local divisor counts, rather than from any accidental choice of coefficients. [/guided] [/step]