Where in the maths is the collapse rule hiding?

Schrödinger's equation has a missing piece.

The equation that powers all of chemistry, electronics, and materials science — Schrödinger's — describes a wavefunction evolving smoothly. It does not describe a wavefunction that ever collapses. Yet detectors click, screens flash, cats are either alive or dead.

Either standard quantum mechanics is the whole story and the collapse is a story we tell ourselves (Many Worlds, Bohmian, QBism), or there's a tiny extra term in the equation we haven't found yet. This project bets on the second option.

How tiny? Tiny enough that a single electron in a lab still does the textbook double-slit. Big enough that the 10²³ atoms in a pointer settle on a definite reading in microseconds.

The simplest version has two numbers.

If the missing term is a random kick toward localisation — the wavefunction gets nudged toward "being somewhere definite" — then in its simplest form it has just two free parameters:

How often the kicks happen — call it λ, the collapse rate. Bigger λ = more frequent kicks.

How localised each kick is — call it r_C, the localisation length. Smaller r_C = sharper kicks.

Every choice of (λ, r_C) is a different theory of nature. The map on the right plots them. Somewhere on it — if collapse is real — is the right answer.

This is the parameter space of the CSL family. Other families (gravity-driven, multi-basis, non-Markovian) have different axes — but the same logic. The structure of the problem is "where on this map?".

Ghirardi, Rimini, Weber put the first pin in the map.

Three Italian physicists — Ghirardi, Rimini, Weber — wrote down the first concrete candidate in 1986. Each particle has a per-second probability λ of getting "hit": its wavefunction suddenly multiplies by a narrow Gaussian centred on a randomly chosen point.

For a single electron, λ is so small the hit never happens during a lab experiment — interference is preserved. For a macroscopic object, the hits on different particles are amplified by entanglement: hit one atom of the pointer, the whole pointer's wavefunction collapses with it. Big things collapse fast for free.

Their pin: λ ≈ 10⁻¹⁶ per second per particle, r_C ≈ 10⁻⁷ metres. That's the GRW point on the map.

At GRW's parameters, a pointer with 10²³ atoms collapses in about a microsecond. An electron alone collapses in 100 million years. Both are needed for the model to work.

Pearle and Ghirardi smoothed the kicks into a continuous wind.

GRW's discrete hits are awkward — they violate energy conservation in spikes and treat identical particles oddly. Philip Pearle (1989) and then Ghirardi-Pearle-Rimini (1990) replaced the hits with a continuous random nudge — a stochastic background field that constantly localises the wavefunction a little.

Continuous Spontaneous Localisation — CSL — lives on the same map. Same two numbers. Different mathematical form (an Itô stochastic differential equation), same physical picture.

Adler (2007) revisited the CSL parameters and argued for a higher rate than GRW — around λ ≈ 10⁻⁸ — so that latent-image formation on photographic plates triggers collapse fast enough to be "the measurement". That's a second pin on the map.

GRW and CSL are predictively close: at common scales they give similar collapse rates. They're two reference points, not two rival camps.

If λ is too small, nothing collapses — and that's testable.

If the kicks are too rare, even macroscopic objects stay in superposition longer than experiments allow. We can see interference fringes from molecules up to thousands of atoms in mass. We do not see them from grains of dust. That gap gives a lower bound on λ.

Bassi, Lochan, Satin, Singh and Ulbricht in their 2013 review (Models of wavefunction collapse, Rev. Mod. Phys.) compiled all these limits. The bottom-left of the map — small λ, small r_C — is the "doesn't actually collapse" zone.

That cuts off the bottom of the map. The hunt shrinks.

If λ is too large, matter heats up — and we'd notice.

Every random kick on a charged particle is a tiny shake — and shaking charged particles radiates. Stronger collapse rates produce more spontaneous X-ray emission from ordinary matter. Underground germanium detectors, shielded from cosmic rays, can measure exactly this background.

Adler & collaborators (2018) looked. Spontaneous X-rays from germanium and silicon are far below what a strong-collapse CSL would predict. That carved out the top-right corner of the map: large λ at any r_C is gone.

Donadi et al. (2021), looking at the same germanium data, ruled out the simplest version of Diósi-Penrose (gravity-driven collapse) for any cutoff length below about 10⁻¹⁰ metres. Whole structural family of theories — gone.

This is hard ground. These bounds aren't speculative — they are direct measurements. Any candidate the AI proposes has to clear them.

Two more sides get squeezed in.

Bose et al. (2017) proposed using milligram masses in a quantum interferometer to bound gravity-mediated collapse rates. The proposal alone constrains a chunk of the gravitational-coupling axis.

Fein et al. (2019) pushed matter-wave interference up to 25,000 atomic mass units — molecules of 2,000 atoms. Seeing interference at that scale means collapse can't happen faster than a certain rate at those masses. Another chunk of the (λ, r_C) plane shaded out.

Each result narrows the map. What's left is a slim corridor of parameter space — and a much larger landscape of structurally different candidates that nobody has written down yet.

These six bounds are already loaded into the project's bounds.yaml — every candidate the AI proposes gets checked against them before it's allowed to run.