Strip Correlations

A point of strip $g$ is $(r,\,r+g)$, and $\gcd(r,\,r+g)=\gcd(r,g)$ — so strip $g$'s coprimality is the event "$r$ is coprime to $g$", with density $\varphi(g)/g=\prod_{p\mid g}(1-\tfrac1p)$. Take two strips $g_1,g_2$ and ask for the joint event: the same $r$ coprime to both. By periodicity mod $\mathrm{lcm}(g_1,g_2)$ (CRT), the joint density is

$$J(g_1,g_2)\;=\;\prod_{p\,\mid\,g_1 g_2}\Bigl(1-\frac1p\Bigr).$$

Independence would predict the product $\frac{\varphi(g_1)}{g_1}\cdot\frac{\varphi(g_2)}{g_2}$ — which counts every shared prime twice. The ratio is the correlation factor:

$$\rho(g_1,g_2)\;=\;\frac{J(g_1,g_2)}{\frac{\varphi(g_1)}{g_1}\frac{\varphi(g_2)}{g_2}} \;=\;\prod_{p\,\mid\,\gcd(g_1,g_2)}\frac{p}{p-1}\;\;\ge\;1.$$

So the correlation is always non-negative, depends only on $\gcd(g_1,g_2)$, and equals $1$ exactly when the strips are coprime to each other. Two strips sharing a factor of 2 are already $2\times$ more correlated than independence predicts; sharing $2\cdot3$ makes it $3\times$. At $g_1=g_2=g$ the factor is $g/\varphi(g)$ — the strip is perfectly correlated with itself, as it must be.

Lattice triangle (dim) · strip g₁ teal · strip g₂ violet · optional strip g₃ gold · optional strip g₄ coral · filled = coprime on that strip · a gold column spans (r,r+g₁)…(r,r+g_k) where r is coprime to every active strip — the joint event. Columns are what independence would get wrong. Each g-slider's range adapts to the current G (a strip can never reach or exceed G), and 🔒 pins a strip to exactly G/2, following G as it moves.

Because $\rho$ depends only on $\gcd(g_1,g_2)$, the map of all strip pairs inherits the full structure of the gcd function — bright rays along common multiples, brightest on the diagonal where a strip meets itself. Click any cell to load that pair into the live view above.

ρ(g₁,g₂) for g₁,g₂ = 1…96, log color scale · dark = ρ=1 (coprime strips, truly independent) · bright red-gold = strong shared-prime correlation · current pair marked. This picture is exact — no sampling.

Page 1's constant is $C(n)=\zeta(2)\prod_p\bigl(1-\min(n{+}1,p)/p^2\bigr)$. The block event behind it — $r$ coprime to all of $M, M{+}1,\dots,M{+}n$ — is a family of $n{+}1$ overlapping strip-type conditions. If those conditions were independent, each prime would contribute $(1-1/p^2)^{\,n+1}$, i.e. the block density would be $\zeta(2)^{-(n+1)}$-shaped. What actually happens, prime by prime:

Large primes, $p > n{+}1$: the block of $n{+}1$ consecutive integers occupies $n{+}1$ distinct residues mod $p$, so the pair $(r,M)$ fails for $\min(n{+}1,p)=n{+}1$ of the $p^2$ residue pairs — giving $1-(n{+}1)/p^2$, which is what near-independence looks like to first order: $(1-1/p^2)^{n+1}\approx 1-(n{+}1)/p^2$. The strips barely feel each other.

Small primes, $p \le n{+}1$: the block covers every residue class mod $p$ — the conditions collapse into one: "$p \nmid r$". That is total correlation, and it is exactly where $\min(n{+}1,p)$ saturates at $p$, contributing $1-p/p^2 = 1-\tfrac1p$ — the same factor as a $\varphi(g)/g$ strip. The saturation points $n=p-1$ that Page 1's sat−/sat+ buttons jump between are precisely the depths where one more prime crosses from the near-independent regime into the fully-correlated one.

So the two regimes of $\min(n{+}1,p)$ are not a formula quirk — they are the independence-to-correlation transition of this page, happening once per prime as $n$ grows.

The identities on this page — $\gcd(r,r{+}g)=\gcd(r,g)$, the joint density over $p \mid g_1g_2$, and the correlation factor $\rho=\prod_{p\mid\gcd} p/(p{-}1)$ — are elementary and classical in method (periodicity and CRT). No novelty is claimed for them. What this page contributes is the reading: the diagonal-strip decomposition of Pages 1–2 fails to be independent in an exactly computable way, that failure is gcd-structured, and the $\min(n{+}1,p)$ kernel of $C(n)$ is the same phenomenon seen prime-by-prime. Measured columns are finite-$G$ counts; the |Δ| shown is the finite-size error, empirically $O(1/G)$, consistent with Page 1's reports.