PrimeMind · Structural atlas · the terrain

Terrain Lab

The bounce game, with the line pushing back. The line has a shape: a hill at every prime (or at any set you choose, placed by a rule you choose). Runners roll on it — uphill they slow, at a hill taller than their energy they turn back — so each runner's rhythm is set by the whole terrain, not by its own number. The rule that turns runners into standing waves is simple: a pitch is an energy whose round trip fits a whole number of wavelengths. The lab reads the pitches off that rule, checks them against the exact wave computation, and scores them against the zeta zeros with a fake-prime terrain as the control. Bending the line into a third dimension enters as a specific extra term; the complex dimension enters as a twist.

The essentials — the line and its hills

numbers that make hills

distance rule p(n) — where the hill for n stands

distance scale (multiplies p(n))

hill height (whole units — every hill the same)

hill width

line half-length U

ends

pitches to compute

custom list

fold: mirror every hill at −p(n) fake-prime terrain as control

Optional — each of these is off unless you turn it on

hill heights by a weight rule instead of whole units

background terrain b(u), an expression in u — or a named bowl

bend into a 3rd dimension: curvature rule κ(u) — adds the dip −κ²/4

twist (the complex dimension): phase added per round trip, in turns

Working backwards — fit the free dials to the first M zeros, then check whether the rest follow

fit to the first M zeros (shape only — ratios to the first)

dials to free

search effort (evaluations)

Batch — run many setups at once, rank them against chance, export the table

distance rules to try (one per line)

bowls to try (one per line; 0 = none; logwell allowed)

hill heights to try (space-separated)

twists to try (turns, space-separated; twist affects the round-trip rule only)

pitches per run (fewer = faster)

fake-prime twins per setup (more = a real null distribution)

Every combination of the lists is built (with the current numbers, distance scale, width, fold, ends, and weights setting), struck, and scored the same way as a single run: the density-matched match to the zeros, the same for its fake-prime twin, and the spacing statistics. Each setup is also built from several fake-prime twins, so the table shows the fakes' mean, spread and best — a null distribution, not one draw — and ranks by z, the number of fake spreads the prime terrain sits above the fake mean. A single fake twin can be unlucky; six or more can't be, much. A row turns amber only when z is above 3 and the prime terrain beats every fake twin; and even then the next step is a larger K and more twins, not a celebration. With only a dozen hills the chance spread is wide (about ±5 points around 30%), so small sets need many twins.

The line — runners at the first pitches, rolling on the terrain (each runner drawn at its own energy level; it turns where the terrain reaches that level)

speed

The pitches — violet: zeros · cyan: this terrain (exact wave) · green: this terrain (round-trip rule) · grey: fake-prime terrain

compare up to height

Spacing statistics

What each essential does. The distance rule and scale place the hills — "distance" in your sense. The hill height is one number; every hill is the same size unless you switch on weights. The ends are the two dots: hard walls turn a runner around instantly, free ends let the wave sit at full swing there. The fold mirrors the line about its middle. The round-trip rule. For an energy E, the runner moves where the terrain is below E, with speed √(E − h); its round trip fits when ∫√(E − h(u)) du over that region equals a whole number of half-turns (plus the twist, if on). Those energies are the green ladder. The cyan ladder is the exact wave computation of the same terrain; they agree except at the lowest few, which is the known limit of the runner picture. The score uses the exact ladder. The bend. A line that curves into a third dimension and comes back is, for a wave travelling along it, the same as a straight line with an extra dip wherever it bends: the dip is −κ²/4, κ being how sharply the line curves there. That is a theorem about thin curved channels, so "bending the line" is not a new kind of number — it is a shape you draw with the curvature rule, and the lab converts it to terrain. The twist. Waves carry a phase as well as a height — that is where the complex numbers live — and a twist adds a fixed phase to every round trip. On a line with two walls that phase can be relabelled away and changes nothing, so with hard walls or free ends the twist only enters the green round-trip ladder. Choose loop ends and it becomes real: the two dots are the same point, the wave goes round and meets itself, and the twist decides which pitches fit — a ring threaded by a field. In turns: 0 is a plain loop, 0.5 flips the sign each time round, and whole turns are the same as none. It is lossless. Weights. Off by default, as you asked; switch them on to test whether hill sizes matter. Scoring. Ladders are rescaled to equal counts below the chosen height; chance is about 30%; only a cyan score far above the grey means anything.