# 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`: | node | brightness | `floor(30/b)` | |---|---|---| | `0`, `1` | 30 | 30 | | `1/2` | 15 | 15 | | `1/3`, `2/3` | 10 | 10 | | `1/4`, `3/4` | 7 | 7 | | `1/5` ... | 6 | 6 | 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](bases.md), 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|`: | `Q` | nodes | `S2*Q` | `S1/sqrt(Q)` | local exponent of `S2` | |---|---|---|---|---| | 125 | 4796 | 0.5395 | 0.2040 | - | | 250 | 19024 | 0.5848 | 0.1942 | -0.884 | | 500 | 76116 | 0.6241 | 0.1852 | -0.906 | | 1000 | 304192 | 0.6387 | 0.1634 | -0.967 | | 2000 | 1216588 | 0.6560 | 0.1512 | -0.961 | | 4000 | 4863602 | 0.6538 | 0.1314 | -1.005 | | 8000 | 19455782 | 0.6564 | 0.1123 | -0.994 | **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.