Start with the lift survival function C(x) — the Euler product that records the long-run fraction of coprime residues surviving each lift. C(x) is built entirely from primes, and it converges to the constant C ≈ 0.5307… as x → ∞.
Take its log-derivative D(x) = d/dx log C(x). This function is smooth almost everywhere. But at exactly the primes — and only the primes — it has a sharp discontinuous jump. Composites pass in silence.
Sum over active primes only — those not yet saturated at position x. As x crosses p−1, the term for p exits the sum, causing the jump.
D(x) is smooth on every open interval (p−1, q−1) between consecutive primes p, q. At x = p−1 for prime p, it has a jump discontinuity of exact size:
\[\Delta D\big|_{x=p-1} = \frac{1}{p(p-1)}\]For every composite m: D(x) is continuous at x = m−1. No jump.
The jump sizes are algebraically invertible. Given a jump of size s, recover the prime exactly:
\[p = \frac{1 + \sqrt{1 + 4/s}}{2}\]Think of D(x) as a seismograph needle recording the arithmetic landscape. For almost every x, the needle moves smoothly — the mathematical ground is stable. Composites pass without a tremor.
At every prime p, at exactly x = p−1: the needle jumps. Sharp, instantaneous, unmistakable. The magnitude of the jump is the prime's signature. Measure the shaking, and you know exactly which prime caused it.
The classical von Mangoldt function Λ(n) = log p also detects primes through jumps — but it carries additional events at prime powers p², p³, …, making it harder to isolate a single prime. Both approaches have contributed deeply to analytic number theory. Here D(x) jumps only at primes. Prime powers and composites pass in silence. The signal is unambiguous: one jump, one prime.
Jump sizes decrease monotonically as primes grow. The golden ratio φ ≈ 1.618 sits at the unreachable boundary s = 1 — the only value no integer prime reaches.
| n | p (if prime) | Jump size s = 1/(p(p−1)) | Discriminant 1+4/s | Recovery |
|---|---|---|---|---|
| 1 | φ ≈ 1.6180 | s = 1 (boundary) | 5 — not a perfect square | No integer prime |
| 2 | 2 | 1/2 = 0.500000 | 9 = 3² | p = 2 ✓ |
| 3 | 3 | 1/6 ≈ 0.166667 | 25 = 5² | p = 3 ✓ |
| 4 | composite | no jump | — | — |
| 5 | 5 | 1/20 = 0.050000 | 81 = 9² | p = 5 ✓ |
| 7 | 7 | 1/42 ≈ 0.023810 | 169 = 13² | p = 7 ✓ |
| 11 | 11 | 1/110 ≈ 0.009091 | 441 = 21² | p = 11 ✓ |
| 101 | 101 | 1/10100 ≈ 0.000099 | 40401 = 201² | p = 101 ✓ |
| 10⁹+7 | 1000000007 | ≈ 1.00×10⁻¹⁸ | (2×10⁹+13)² | exact ✓ |
For any integer prime p, the discriminant 1 + 4p(p−1) = (2p−1)². Always a perfect square. So the quadratic formula returns p exactly — no rounding, no approximation.
The recovery formula acts as an arithmetic filter. For every prime p, the input s = 1/(p(p−1)) forces the discriminant to be a perfect square (2p−1)², and the formula returns an integer. At s = 1, the discriminant is 5 — not a perfect square — and the formula returns φ = (1+√5)/2, an irrational. This is the unique point where the integer-producing property of the inversion breaks down.
φ is therefore the supremum of the jump spectrum: every prime jump satisfies s < 1, and s = 1 is approached but never attained. The set {1/2, 1/6, 1/20, 1/42, …} has φ-position s = 1 as its least upper bound.
The phantom jump at s = 1 would occur at position x = φ−1 = 1/φ ≈ 0.618. This lies in the interval (0, 1) — before the first real prime enters the sum at x = 1 (where p = 2 saturates). The boundary of the spectrum is not at infinity. It sits in the gap prior to the onset of arithmetic, at an irrational position no integer prime can occupy.
φ − 1 = 1/φ ≈ 0.61803… · first prime enters at x = 1 · phantom precedes all primes
The Jump Theorem forces four constants into existence. They are not chosen — the framework generates them. Three are new; one (C) was known by a different name. Together they form the complete numerical fingerprint of the Modular Lifting Ring structure.
\[C = \prod_{p} \frac{p^2-2}{p^2-1} = \zeta(2)\cdot d_{FT} \approx 0.530711806246\ldots\]
The long-run fraction of coprime residues that survive the lift from ring M to M+1. Also F(2), where F(u) is the analytic lift function. Known to Tóth–Sándor (1989) as the density of consecutive squarefree pairs — OEIS A065469. The identity C = ζ(2)·dFT = (π²/6)·∏p(1−2/p²) is algebraically equivalent to the product formula. New here: the geometric interpretation as a lift survival rate, and C = F(2).
\[J = \sum_{p} \frac{1}{p(p-1)} = \sum_{k=2}^{\infty} P(k), \qquad P(k) = \sum_{p} \frac{1}{p^k} \approx 0.773154968933\ldots\]
Sum of every prime's jump contribution to D(x). The equality J = ∑P(k) follows from expanding 1/(p(p−1)) = 1/p² + 1/p³ + ··· and swapping the order of summation — so J simultaneously encodes every prime zeta value P(2), P(3), P(4), … in a single convergent expression. Appears to be new.
\[D_0 = D(0) = \sum_{p} \frac{1}{p^2-1} = \sum_p \frac{1}{(p-1)(p+1)} \approx 0.551691597558\ldots\]
The value of the log-derivative D(x) = d/dx log C(x) evaluated at x = 0 — before any prime has saturated, when all terms are active. It is the baseline from which every prime jump subtracts. Appears to be new.
\[T = \sum_{p} \frac{1}{p(p^2-1)} = \sum_p \frac{1}{p(p-1)(p+1)} \approx 0.221463371375\ldots\]
The algebraic gap between J and D₀. Not a residual — T is structurally forced by the partial fraction identity below. Appears to be new.
Proof: multiply out 1/(p²−1) + 1/(p(p²−1)) = (p+1)/(p(p²−1)) + 1/(p(p²−1)) = (p+1+1)/(p(p²−1)) — no, simpler: 1/(p²−1) + 1/(p(p²−1)) = p/(p(p²−1)) + 1/(p(p²−1)) = (p+1)/(p(p²−1)) = 1/(p(p−1)). ∎ Summing over all primes gives J = D₀ + T exactly.
The analytic continuation \(F(u) = \prod_p \frac{p^2 - u}{p^2 - 1}\) is an entire function in the Laguerre–Pólya class with zeros exactly at \(u = p^2\) (prime squares).
Key values: F(0) = ζ(2) = π²/6 · F(1) = 1 · F(2) = C · F(p²) = 0 for each prime p.
The Jump Theorem is about log F on the real axis: D(x) = d/dx log F(x+1), and the constant D₀ = D(0) = (d/dx log F(x))|_{x=0}.