Title: A deterministic sieve framework for Twin Primes and Goldbach partitions via minimal interval thresholds
Introduction
I am sharing a deterministic, non-probabilistic framework that uses a modified Eratosthenes sieve to address the Twin Prime and Goldbach conjectures. Unlike standard probabilistic models or heuristic density estimates, this approach relies strictly on the combinatorial shifting properties of residue classes to find a guaranteed lower bound for the macro-interval size $L$ required to contain a set number of surviving pairs.
A well-known constraint is that the maximum length of consecutive integers coprime to a single prime $p$ is strictly capped at $p-1$. Rather than trying to extend consecutive prime sequences, this framework analyzes the exact shifting thresholds where "blocked cells" (multiples of primes) overlap or separate, ensuring a zero-free survival zone.
1. Deterministic Sieve Expansion for a Single Progression
Let $N \in \mathbb{Z}^+$ and $P_z = {p \in \text{Prime} \mid p \le z}$. Suppose we demand that at least $N$ elements must survive after sifting out the multiples of a prime $p$.
By tracking the exact geometric shift of blocked periods (represented mechanically as pushing overlapping configurations to the right), the framework deduces that to deterministically guarantee at least $N$ unblocked elements, the total window size $L$ must expand by a strict arithmetic ratio:
$$L \ge \frac{p+1}{p-1} \cdot N$$
Crucially, this is treated as a hard geometric threshold for existence, not an expected value.
2. Simultaneous Bounded Sieve for Linked Progressions
Evaluating twin primes and Goldbach partitions requires analyzing two linked arithmetic progressions simultaneously under a combined sieve constraint.
To ensure that at least $N$ pairs deterministically survive the simultaneous exclusion by a prime $p$, the local combinatorial configurations require the window $L$ to scale as:
$$L \ge \frac{p+2}{p-2} \cdot N$$
Compounding these structural restrictions across all primes $p \in P_z$ establishes a definitive product threshold for the total required macro-interval width $L$:
$$L \ge \prod_{p \le z} \left( \frac{p+2}{p-2} \right) \cdot N$$
To analyze the behavior of this product for large scales, the framework utilizes an analytical bound under a Riemann Hypothesis framework from Kevin Broughan's Equivalents of the Riemann Hypothesis (2017). For a sieved limit $z \ge 10^5$:
$$\prod_{p \le z} \left( \frac{p+2}{p-2} \right) \le e^{C \ln z} \left( 1 + \frac{1}{2\ln^2 z} \right)$$
This product functions as a rigid combinatorial scaling factor, dictating how large our search scope must be to guarantee the preservation of unblocked slots.
3. Application to the Conjectures
A. Twin Prime Formulation
We construct two shifted intervals of length $L(x)$:
- $A = {1, 2, 3, \dots, x}$
- $B = {3, 4, 5, \dots, x+2}$
As $x \to \infty$, the combinatorial scaling shows that the necessary interval bounds diverge slower than the available space. Because the configurations are governed by these exact arithmetic ratios, the number of sieved pairs $(p, p+2)$ is forced to diverge, implying an infinite number of twin primes.
B. Goldbach's Conjecture
For any given even integer $x$, we map two symmetric intervals:
- $A = {1, 2, 3, \dots, x-1}$
- $B = {x-1, x-2, x-3, \dots, 1}$ (such that $A_k + B_k = x$)
Applying the deterministic product threshold over all primes up to $\sqrt{x}$, the framework concludes that the remaining structure guarantees the number of valid prime pairs $\rho \ge 1$ for all large even $x$.
Questions for MathOverflow:
Validity of the Shifting Principle ($L \ge \frac{p+2}{p-2} N$):
In standard sieve theory, Chinese Remainder Theorem structures are usually treated via densities ($1 - \frac{2}{p}$). The author’s framework asserts that by viewing the sieve as a rigid system of overlapping periodic blocks, the ratio $\frac{p+2}{p-2}$ acts as a strict structural threshold to force a non-empty intersection of size $N$. Has a similar deterministic "pushed cell" boundary logic been studied in combinatorial number theory, and is it free from the parity barrier?Error terms in a Multi-Prime Deterministic Cascade:
When we multiply these thresholds across multiple primes ($\prod \frac{p+2}{p-2}$), the framework assumes the local worst-case alignment conditions can be scaled multiplicatively without generating an external remainder term error. Does this purely structural composition hold rigorously when combining multiple distinct prime periods, or does the lack of absolute coprimality between different shifting configurations introduce a hidden remainder term that breaks the determinism?
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