This paper synthesizes a series of results on the (even) Goldbach conjecture and completes two previously unaddressed components: textbf{error analysis} and textbf{fractional sequence encoding (additive sieve)}. The main results are as follows. (1) We systematically present limit-type elementary constructions of the Goldbach conjecture (using both exponential and cosine functions) and their equivalent logical formulations, and extend the method to the Collatz conjecture, the twin prime conjecture, and the prime counting function. (2) For the truncated construction with finite error factor $k$, we derive a rigorous error bound and prove that the criterion holds provided $kn,e^{-4k/n^{2}}ln n$. This determines the textbf{critical exponent for the sufficient condition to be $2$}, and rigorously verifies that the empirically discovered choice $k=10n^{e}$ ($eapprox 2.718>2$) satisfies this sufficient condition for all even $nge 4$, yielding a textbf{limit-free elementary closed-form equivalent proposition} for Goldbach's conjecture. Numerical experiments show that for $nle 120$, the difference between the truncated construction $gamma_{k}(n)$ and the true prime pair count $G(n)$ is less than $10^{-9}$ (see Remark~ef{rem:numerics}).(3) We prove the closed form $A_{j}=j,(d(j)-2)$ of the "additive sieve" sequence ($d$ being the divisor-counting function), so that $A_{j}=0$ if and only if $j$ is prime; we prove that Goldbach's conjecture is equivalent to the existence of zero positions in the symmetric superposition sequence, and give the zero-count identity $#{j}=2G(n)-[,n/2inPP,]$.(4) We completely resolve the textbf{carry problem} in fractional sequence encoding: with block length $dgelceillog_{10} (4n^{3/2}) ceil$, the encoding is carry-free and fully faithful; meanwhile we prove that for fixed $d$ the infinite sequence inevitably carries ($sup_j A_j=infty$), thereby clarifying the precise scope of previous objections to the method's feasibility. (5) We provide conditional proofs for related propositions ($p_1+p_2+3$ and $p_a+p_b+p_c-1$ forms are consequences of the strong Goldbach conjecture), relate the $2p+q$ representation to Lemoine's conjecture with numerical verification up to $10^6$, and correct a limit-order error in the original "irrationality test formula" (the single-limit version actually diverges as $tfrac{sqrt{pi}}{2}k^{3/2}$). It is emphasized: all constructions in this paper are textbf{logical equivalents} of Goldbach's conjecture, not proofs; their value lies in recasting the conjecture into elementary analytic or combinatorial forms with precisely controllable errors.