Defeating the Sieve Part 2: The Mod 3 Trap

Last week, I wrote about Defeating the Sieve. I showed how specific prime gaps (like Gap 34) act as “Shields,” locking the sum of consecutive primes into a safe residue class that mathematically guarantees they cannot be divisible by 3, 5, or 7.

I thought I had the complete picture. I was wrong.

To validate the theory, I decided to run a “Boss Fight.” I pointed my Rust tool at Gap 210.

Gap 210 ($2\times 3\times 5\times 7$) is a “Champion Number.” It is divisible by every prime up to 7. Under my original theory, I expected it to be neutral—or perhaps even a “Super Shield” due to its massive neighbors (209 and 211).

Instead, it crashed.

The 100 Billion Stress Test

I spun up a simulation to scan the first 100 Billion ($10^{11}$) primes. This wasn’t just a sampling; it was a census.

When the results came back, Gap 210 wasn’t just underperforming; it was failing spectacularly.

• Gap 14 Success Rate: 16.8% (Predicted: ~17%)
• Gap 210 Success Rate: 6.5%

For context, a random odd number in this range has about an 8.6% chance of being prime. Gap 210 wasn’t just “exposed” to randomness; it was performing worse than random.

How can a number so structurally perfect be so toxic to prime sums?

The “Mod 3 Trap”

I realized my “Shielding Theory” was only half the equation. It explained the Highs (why Gap 34 wins), but not the Lows.

The culprit is the number 3.

Recall the sum formula: $S=2p+g-1$.

The Shield (Gap 14)

Gap 14 is not divisible by 3 ($14\equiv 2(\mathrm{mod}3)$). Because $p$ is prime, it cannot be divisible by 3. But for Gap 14, $p$ also cannot be $1(\mathrm{mod}3)$, or the neighbor $p+14$ would be divisible by 3. This forces $p$ into a specific slot ($2(\mathrm{mod}3)$). Result: The sum is guaranteed safe.

The Trap (Gap 210)

Gap 210 is divisible by 3 ($210\equiv 0(\mathrm{mod}3)$). This breaks the lock. The prime $p$ is free to be $1$ or $2(\mathrm{mod}3)$.

• If $p\equiv 1$, the sum is prime-safe.
• If $p\equiv 2$, the sum is divisible by 3.

It becomes a coin flip. The sum fails 50% of the time.

Why “Exposed” is Worse Than Random

You might think, “Okay, 50% failure isn’t great, but isn’t that just standard randomness?”

No. It is worse.

In the wild, if you pick a random odd number, it has a 33% chance of being divisible by 3. That means 66% of random odd numbers are “safe” from 3.

But for Gap 210, only 50% are safe.

By being divisible by 3, Gap 210 loses the natural “odd number advantage.” It falls into a statistical trap that penalizes it relative to the baseline.

We calculated the exact penalty factor: $\frac{50\%\text{ (Trap Safety)}}{66\%\text{ (Baseline Safety)}}=0.75$

The Unified Field Theory

When we apply this 0.75x Penalty to gaps divisible by 3 (like 6, 30, and 210), the data snaps into focus.

The scatter plot above shows our new prediction model against the actual data from the $10^{11}$ run. The correlation is no longer just “good”; it is mathematically precise.

• Gap 14 rides high because of the Neighbor Shield.
• Gap 210 sits low because of the Mod 3 Trap.
• Gap 30 is the battleground: It has a Shield from its neighbors (29, 31), but it pays the Mod 3 Penalty. The math predicts the winner perfectly.

Conclusion

Gap 34 isn’t just lucky because it has armor. It is lucky because it avoids the trap.

In the architecture of prime sums, there are three forces at play:

1. The Baseline: The raw logarithmic density of primes.
2. The Shield: Neighbors ($g\pm 1$) that block divisibility.
3. The Trap: Factors ($g$) that invite divisibility.

The “randomness” of primes is just a lack of resolution. If you look close enough, the dice are loaded.

View the Updated Code on GitHub