The stack is an RH-observable

Lay the same fractal grid on the unit square at many scales at once - scale n puts its cell boundaries at x = k/n - drop the opacity and add the layers up. The result is a moire, and a bright point is one that many scales agree on. The question this page answers is what the bright points are, and the answer is not decorative: the lit nodes are the Farey fractions, the amount of new structure each scale contributes is Euler's totient phi(n), and how evenly those nodes spread is - by a pair of theorems from 1924 - literally equivalent to the Riemann hypothesis.

Every claim below carries a tag. Proved means derived here from definitions. Verified means recomputed from scratch, or checked against the published literature. Nothing on this page is a conjecture.

Where the lines land

Stack the scales n = 1..N. A point a/b in lowest terms receives a grid line from exactly the scales that are multiples of b, so over 1..N its brightness is floor(N/b). Proved, and Verified by direct simulation at N = 30: building the stack node by node and comparing every node's hit count against floor(30/b) gives no mismatch anywhere.

Brightness therefore falls as one over the denominator, which is the Stern-Brocot ordering of the rationals. The top of the table at N = 30:

The lit nodes are also exactly the lattice points visible from the origin, since a/b is in lowest terms precisely when gcd(a,b) = 1. That is the "lighthouse" reading of the picture. Proved. The density of visible points is 6/pi^2 - the same constant, and the same base-blindness, discussed in what base 3 hides, where it is measured as 0.608042 on a 3000 x 3000 grid. Verified, by recounting that grid.

Primes are the maximally novel scales

The nodes scale n introduces for the first time are the fractions a/n with gcd(a,n) = 1, since any a/n that reduces was already lit by the smaller scale it reduces to. There are exactly phi(n) of them. Proved, and Verified by set difference over the stack for n = 2..30:

n       2  3  4  5  6  7  8  9 10 11 12 13 14 15 16
new     1  2  2  4  2  6  4  6  4 10  4 12  6  8  8

n      17 18 19 20 21 22 23 24 25 26 27 28 29 30
new    16  6 18  8 12 10 22  8 20 12 18 12 28  8

Every count equals phi(n), and the running maxima 1, 2, 4, 6, 10, 12, 16, 18, 22, 28 occur at n = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. The reason is one line: phi(n) = n - 1 if and only if n is prime, because every one of 1..n-1 is coprime to n exactly when n has no smaller factor. Proved, and Verified by testing the equivalence against trial division for all n up to 200.

So primality is readable off the picture. Stack 1..n-1, then add scale n, and count what appeared: n - 1 new nodes means n is prime, fewer means composite. Proved (it is the previous claim restated). A composite scale mostly re-lights nodes its own divisors already drew - scale 30 adds only 8 new lines, the rest of its grid falling on lines from 1, 2, 3, 5, 6, 10 and 15.

Franel and Landau, 1924

Over scales 1..Q the stack lights exactly the reduced fractions of denominator at most Q: the Farey sequence F_Q. Its size in (0,1] is m = sum_{k<=Q} phi(k). Proved, and Verified by generating F_Q through the next-term recurrence and comparing its length with the totient sum at Q = 10, 30, 60.

Write rho_1 < ... < rho_m for those nodes and delta_j = rho_j - j/m for how far each one sits from perfect equidistribution. Then:

• Franel (1924) proved that sum_j delta_j^2 = O(Q^(-1+eps)) for every eps > 0 is

equivalent to the Riemann hypothesis.

• Landau (1924), in a note published immediately after Franel's, proved the same for

sum_j |delta_j| = O(Q^(1/2+eps)).

Verified against the literature: both statements, with the original 1924 citations to the Göttingen Nachrichten, are the standard Franel-Landau formulation, and are reproduced in Edwards, Riemann's Zeta Function, chapter 12.

Put the two halves together. The nodes whose discrepancy Franel and Landau are talking about are the nodes the stack draws - not an analogue of them, the same set. So the question "how evenly are the bright points spread?" is not related to the Riemann hypothesis; at this level of precision it is the Riemann hypothesis. Proved, given the identification above, which is what the first two sections establish.

The meter reads what RH predicts

Both sums are computable. Generating F_Q exactly and measuring, with S2 = sum delta_j^2 and S1 = sum |delta_j|:

Verified by recomputation. S2*Q flattens near 0.656 and the local exponent walks to -1, which is the Franel condition; S1 stays under its Q^(1/2) envelope and its own local exponent runs between 0.27 and 0.43, under the Landau threshold of 0.5. The node count matches sum phi(k) exactly at every rung, which is the control that says the object being measured really is the stack's node set.

The honest cap

An observable is not a handle. What the last two sections establish is that this picture renders a genuinely RH-equivalent object, which is a real upgrade over the vaguer "fractals and zeta both have self-similar structure" gestures. What it does not do is supply any route to a proof. The Riemann hypothesis is already checked numerically far beyond any range this or any other meter can reach, so the table above can only ever illustrate the expected behaviour - it is consistent with RH, it is not evidence for it, and no amount of extra Q changes that. The research notes score the link quality at 6 out of 10 and the tractability at 0, and both numbers deserve to be stated together: the connection is exact, and it is untouchable.