Projector Lab v2 with residue phases
The primes as a projector. Each prime is thrown from the center onto a surface that surrounds it. Where it lands is spun by the stretching flow — the flow turns the prime p at rate log p — and how hard it lands is set by an exponent σ, your "dimensional shift": the hit for p has weight p−σ. On a ring this is not a metaphor; it is the Euler product, and its sum is the logarithm of zeta. On a sphere, disc, or torus it is a sandbox where the hits set off ripples that add and subtract.
Projector settings exact — this is Σ p−σ e−it log p
Left. The ring is the surface; the center is the projector. Prime p lands at angle t · log p: at t = 0 every prime lands at the same spot, and as the flow runs, each prime's hit slides around the ring at its own rate — 2 slowest, big primes fastest. Its weight is p−σ. Adding the hits head-to-tail as vectors gives the amber path; its endpoint is the resultant. That resultant is, exactly, the partial sum of log ζ(σ + it) over the primes you allowed. Right. The real part of the resultant as t sweeps, with the zeros marked. At σ = ½ the resultant's real part is log|ζ| (green, computed independently), which plunges to −∞ at every zero: the primes' hits, spun to those particular t, pile up pointing backwards. That is a zero, seen from the primes' side — not a cancellation of waves but a conspiracy of hits all landing on the far side of the ring. With finitely many primes the amber curve only dips; add more and it dips deeper. Phase by residue. Choose a character and every prime's hit is turned by that prime's residue mod q — a multiplicative rule, since the phase of a product is the product of the phases. The projector then stops aiming at ζ and aims at the Dirichlet L-function of that character; the violet ticks on the right switch to its zeros, and the dips land on them. That is the sweep result of the previous round, made visible: the primes sorted by residue have their own drum, and a multiplicative phase is the only kind the flow respects. Your dimensional shift. Slide σ above 1 and the hits fade fast enough that the resultant settles; below 1 it never does. The critical line σ = ½ is the shift at which the projection is balanced on the edge between settling and wandering.
Watch the primes arrive exact
Same projector, but now the primes arrive in order — 2, then 3, then 5 — and the running resultant is traced as a path. Two behaviours, and σ decides between them. Above σ = 1 the hits shrink fast enough that the path homes in on a point: the projection converges, and the point is log ζ(σ + it). At or below σ = 1 the hits don't shrink fast enough and the path wanders forever, never settling, always adding structure — which is what you described as a system that keeps building complexity without gaining or losing. On the critical line that wandering is exact: the Euler product does not converge there. The zeros are the values of t at which the wandering path drifts off toward negative infinity. This is the honest form of "put the primes in and get the zeros out": at σ = ½ the primes, and only the primes, point at the zeros — but only in the limit, and only as a divergence, which is exactly why no finite projection ever proves anything about all of them.
Ripples on a surface sandbox — an illustration, not an identity
Each prime lands somewhere on the surface and sets off a ripple — a standing wave of the surface's pitch, centred on the hit, with the prime's weight. The surface adds all the ripples; colour is the total, cyan positive, amber negative, and the sand-coloured lines are where it's zero. The placement rules, the surface, the surface's pitch and the decay length are all knobs, because this tab is for trying scenarios, not for stating identities: there is no theorem here that the pattern means anything about the zeros. What it is good for is seeing the same three things the exact tabs show, in two dimensions: how the flow spins the primes at different rates, how σ decides whether the far primes matter, and how a pile-up of hits on one side of the surface is what an interference "event" looks like. If you find a placement rule under which something on this surface lines up with the zeros, that is a conjecture to test in tab 1, not a result.