EMERGENT MEASUREMENT maths — every equation

The maths the project actually runs on.

Everything below is a real equation appearing in the code — the standard quantum mechanics we measure against, the reference collapse models the founders implement, the seven scorers that grade each candidate, and the search machinery that picks what to try next. Each card carries the equation in proper notation, who came up with it, what it says, and where it lives in the code.

For the wave-form intuition behind these symbols, see the Learn page. For why we're searching for any of this in the first place, see The Big Idea.

section 1

The classical baseline

The two equations every candidate is measured against. If a candidate breaks either one in a regime where experiment confirms them, it's ruled out before scoring even finishes.

classical · 1926

The Schrödinger equation

Erwin Schrödinger · 1926
\[ i\hbar\,\frac{\partial\psi}{\partial t} \;=\; \hat{H}\,\psi \;=\; \Bigl(-\frac{\hbar^{2}}{2m}\,\nabla^{2} \,+\, V(\vec{x})\Bigr)\psi \]

What it says

The wavefunction \(\psi\) evolves smoothly in time under the Hamiltonian \(\hat{H}\). Kinetic-plus-potential, just like classical mechanics — but acting on a complex-valued field over configuration space. Linear, deterministic, unitary; never collapses on its own.

In this project

The base equation our integrator advances every step. Every candidate is a modification of this — an extra non-linear or stochastic term added to the right-hand side. See src/emergent_measurement/schrodinger/integrator.py.

classical · 1926

The Born rule

Max Born · 1926 (Nobel 1954)
\[ P(\vec{x}) \;=\; |\psi(\vec{x})|^{2}, \qquad \int |\psi|^{2}\,d^{n}x \;=\; 1 \]

What it says

The probability of finding the particle near \(\vec{x}\) is the squared amplitude there. The translation rule from the smooth wavefunction to the discrete clicks of a detector.

In this project

A constraint every candidate must preserve in the single-particle, weak-coupling limit. Scorer M1 runs the double-slit founder on a candidate and checks that the resulting screen pattern still matches \(|\psi|^{2}\). See scoring/m1_born_rule.py.

section 2

The numerical substrate

How we actually advance the equation in time on a grid. Both the pure-state propagator and the density-matrix evolver sit underneath every collapse model in the gallery.

integrator · 1976

Split-step Fourier propagator

Hardin & Tappert · 1973; Feit, Fleck, Steiger · 1982
\[ \psi(t+dt) \;=\; e^{-i\,V\,dt/(2\hbar)}\;\mathcal{F}^{-1}\!\Bigl[\,e^{-i\,\hbar k^{2}\,dt/(2m)}\;\mathcal{F}\bigl[e^{-i\,V\,dt/(2\hbar)}\,\psi(t)\bigr]\Bigr] \]

What it says

Apply the potential operator (diagonal in position space) for a half-step, FFT into momentum space, apply the kinetic operator (diagonal in momentum space) for a full step, FFT back, apply the potential again for the other half-step. Second-order accurate in \(dt\), exact for the linear pieces it splits.

In this project

The propagator under every wavefunction-level experiment. Norm conservation stays at \(10^{-13}\) over \(10^{4}\) steps — well past PLAN's \(10^{-6}\) gate. See schrodinger/integrator.py → SplitStepFourier.step().

substrate · 1976

Lindblad master equation

Lindblad; Gorini, Kossakowski, Sudarshan · 1976
\[ \frac{d\rho}{dt} \;=\; -\frac{i}{\hbar}\,[\hat{H},\,\rho] \;+\; \sum_{k}\Bigl(\hat{L}_{k}\,\rho\,\hat{L}_{k}^{\dagger} \,-\, \tfrac{1}{2}\bigl\{\hat{L}_{k}^{\dagger}\hat{L}_{k},\,\rho\bigr\}\Bigr) \]

What it says

The most general Markovian, completely-positive, trace-preserving evolution of a density matrix \(\rho\). The first term is Schrödinger; the sum is everything else — decoherence, dissipation, and the ensemble-averaged effect of collapse. Each \(\hat{L}_k\) is a jump operator.

In this project

The shared form behind CSL, GRW (continuum limit), Diósi-Penrose, and Joos-Zeh environmental decoherence. The DensityMatrixEvolver integrates this directly, and we cross-validated it against the CSL stochastic ensemble. See schrodinger/density.py.

numerical · stability

The stochastic-step stability bound

Measured in this project; standard EM-SDE result
\[ \sqrt{\lambda}\;\sigma_{\text{tail}}\;\sqrt{dt} \;\ll\; 1 \]

What it says

Euler-Maruyama applied to the CSL Itô SDE is stable only when the per-step stochastic kick is small relative to the wavepacket scale. \(\sigma_{\text{tail}}\) is the standard deviation of the localising operator's tail; \(\lambda\) is the collapse rate; \(dt\) is the integrator step. Violate the bound and \(\psi\) blows up to infinity in a few steps.

In this project

A reviewer-side audit gate. Any candidate the AI proposes is checked for the strong-coupling regime first; we use an adaptive \(dt\) when the bound would otherwise be violated. See schrodinger/stochastic.py.

section 3

The reference collapse models — founders

The three textbook recipes the simulator has to reproduce before the AI is allowed to invent anything. Each lives at a known point in the parameter map; each is a starting genome the bandit can perturb.

founder · 1986

GRW — random Gaussian hits

Ghirardi, Rimini, Weber · 1986
\[ \psi(\vec{x}) \;\longrightarrow\; \frac{1}{\sqrt{\mathcal{N}}}\; e^{-(\vec{x}-\vec{r})^{2}/(4\,r_C^{2})}\;\psi(\vec{x}), \qquad \text{at Poisson rate}\;\lambda \] \[ P(\vec{r}) \;=\; \int |\psi(\vec{x})|^{2}\,e^{-(\vec{x}-\vec{r})^{2}/(2\,r_C^{2})}\,d^{3}x \]

What it says

Every particle has a tiny per-second chance \(\lambda\) of being "hit": its wavefunction is multiplied by a narrow Gaussian centred on a random point \(\vec{r}\), then renormalised. The probability of hitting \(\vec{r}\) is itself Born-distributed — so the rule respects the Born rule on average.

In this project

Implemented as F3. Validated: first-hit times follow \(\mathrm{Exp}(\lambda)\) on two moments; ensemble position-basis decoherence rate matches the master-equation prediction. See schrodinger/grw.py.

founder · 1989/90

CSL — continuous stochastic localisation

Pearle 1989; Ghirardi, Pearle, Rimini · 1990
\[ d\psi \;=\; \Bigl[\,-\frac{i}{\hbar}\,\hat{H} \,-\, \tfrac{1}{2}\sum_{k}\bigl(\hat{L}_{k} - \langle\hat{L}_{k}\rangle\bigr)^{2}\,\Bigr]\,\psi\,dt \;+\; \sum_{k}\bigl(\hat{L}_{k} - \langle\hat{L}_{k}\rangle\bigr)\,\psi\;dW_{k} \]

What it says

An Itô stochastic differential equation. The first bracket is the deterministic drift — Schrödinger plus a non-linear "compensator" that keeps the average density matrix on its Lindblad track. The second sum is the stochastic kick: independent Wiener increments \(dW_{k}\) weighted by mean-zero localisation operators \(\hat{L}_{k} - \langle\hat{L}_{k}\rangle\). For mass-coupled CSL, \(\hat{L}_{k}\) is a smoothed position-density operator with width \(r_{C}\).

In this project

Implemented as F4. Direct ρ-evolution matches the stochastic ensemble to ensemble-noise — two independent code paths agree to machine precision. Subject to the stability bound above. See schrodinger/csl.py.

founder · 1989/1996

Diósi-Penrose — gravity-driven collapse

Lajos Diósi · 1989; Roger Penrose · 1996
\[ \Gamma_{\text{DP}}(\rho_A,\rho_B) \;=\; \frac{1}{\hbar}\,\bigl[\,U(\rho_A)\,+\,U(\rho_B)\,-\,2\,U(\rho_{AB})\,\bigr] \] \[ \Gamma_{\text{sat}} \;=\; \frac{G\,m^{2}}{\sigma\,\hbar}, \qquad \lambda_{\text{eff}}^{\text{small-}d} \;=\; \frac{G\,m^{2}}{\sigma^{3}\,\hbar} \]

What it says

The decoherence rate between two mass distributions \(\rho_A\) and \(\rho_B\) equals their gravitational self-energy difference, divided by \(\hbar\). \(U(\rho)\) is the Newtonian self-energy of a Gaussian-smeared density (regularisation length \(\sigma\)). In the saturated regime (well-separated peaks) the rate is constant \(G m^{2}/(\sigma\hbar)\); for small separation \(d\), it looks like CSL with effective rate \(G m^{2}/(\sigma^{3}\hbar)\).

In this project

Implemented as F5. Donadi 2021's underground germanium-detector bound on spontaneous X-rays forces \(\sigma \gtrsim 10^{-10}\,\text{m}\) — encoded in data/bounds.yaml. See schrodinger/diosi_penrose.py.

section 4

The seven judges — how a candidate gets graded

Each candidate gets seven scores in \([0, 1]\). Three of them (M3, M4, M6) are hard gates — a fail kills the candidate. Four (M1, M2, M5, M7) are soft scorers — they shape the bandit's reward signal. M3 and M4 are the most intellectually demanding.

scorer · M4

CHSH inequality — preserving Bell violations

Clauser, Horne, Shimony, Holt · 1969
\[ S \;=\; \bigl|\,E(a,b) \,-\, E(a,b') \,+\, E(a',b) \,+\, E(a',b')\,\bigr|, \qquad E(a,b) \;=\; \langle\,\hat{\sigma}_a \otimes \hat{\sigma}_b\,\rangle \] \[ S_{\text{classical}}^{\max} \;=\; 2, \qquad S_{\text{QM}}^{\max} \;=\; 2\sqrt{2}\;\;\text{(Tsirelson)} \]

What it says

For an entangled pair, the CHSH combination of correlation functions is bounded by 2 for any local hidden-variable theory. Quantum mechanics saturates at \(2\sqrt{2}\) for the singlet at optimal angles. A collapse model that kills entanglement before measurement would pull \(S\) below \(2\sqrt{2}\).

In this project

chsh() with optimal-angle defaults — Tsirelson saturation is locked at machine precision for the bare singlet. M4 scores a candidate by how close its stochastically-averaged \(|S|\) stays to \(2\sqrt{2}\). See scoring/m4_chsh.py.

scorer · M3

No-signalling — locality of marginals

Ghirardi, Rimini, Weber · 1980; Gisin · 1989
\[ \rho_{A}^{(U_{B})} \;\equiv\; \mathrm{Tr}_{B}\!\bigl[\,(I_{A}\otimes U_{B})\,\rho\,(I_{A}\otimes U_{B}^{\dagger})\,\bigr] \;=\; \rho_{A} \qquad \forall\, U_{B} \]

What it says

Tracing out Bob's side of a bipartite state must give the same reduced state \(\rho_A\) for Alice, no matter which local unitary \(U_B\) Bob applies — otherwise Bob could signal Alice faster than light by choosing his measurement. Standard QM passes this automatically; non-linear modifications of Schrödinger don't, unless built carefully.

In this project

Hard constraint. M3 computes the Frobenius distance between \(\rho_A\) and \(\rho_A^{(U_B)}\) over a sweep of \(U_B\) and fails the candidate if it exceeds \(10^{-12}\). A wired-up CNOT acts as the negative control. See scoring/m3_no_signalling.py.

scorer · M2 + M5

Decoherence-time scaling with mass

GRW 1986; CSL 1990; standard scaling
\[ \log \tau_{\text{coll}}(m) \;=\; -\alpha\,\log m \,+\, c \] \[ \alpha_{\text{CSL}} \;=\; 1, \qquad \alpha_{\text{mass-power}} \;=\; \alpha\;\;\text{(searched)} \]

What it says

For any objective-collapse model, the collapse timescale \(\tau_{\text{coll}}\) must shrink as mass grows — so a single atom stays quantum (M2: \(\tau\) very long) and a macroscopic object goes classical (M5: \(\tau \le 1\,\text{s}\)). CSL gives \(\alpha = 1\); the AI explores other powers. \(R^{2}\) of the log-log fit + sign check + tolerance window on \(\alpha\) make the M2 score; a hard threshold at macroscopic \(N\) makes M5.

In this project

Meta-scorers — they run a candidate at several masses and fit. Standard QM fails M2 by construction (\(\tau\) is infinite for all \(m\)). See scoring/m2_decoherence.py and scoring/m5_classical.py.

scorer · M6

Spontaneous-emission heating

Adler 2007; Donadi et al. · 2021
\[ \dot{E}_{\text{heat}} \;\propto\; \frac{\lambda\,\hbar^{2}}{m\,r_C^{2}}, \qquad \dot{N}_{\gamma}(\omega) \;\propto\; \frac{\lambda\,e^{2}}{m^{2}\,r_C^{2}\,\omega} \]

What it says

Random kicks on a charged particle radiate. The rate of spontaneous X-ray emission per unit frequency, summed over all the protons in a sample, gives a count-rate that underground germanium detectors can measure. The bound from Donadi 2021 cuts off the top-right of the parameter map.

In this project

M6 computes the predicted spontaneous-emission count-rate for the candidate's \(\lambda, r_C\) and checks it against the Adler 2018 / Donadi 2021 entries in data/bounds.yaml. A fail kills the candidate. See scoring/m6_emission.py.

section 5

How the AI picks what to try next

Once a candidate has seven scores, the bandit decides which family of recipes to explore next. The maths is borrowed from the sibling gravity project — same Thompson sampler, same fitness aggregation.

project

The fitness score — seven judges, weighted

Built for this project
\[ \text{score} \;=\; \prod_{i\in\mathcal{G}}\,\mathbf{1}[\,s_{i}\,\ge\,\theta_{i}\,] \;\cdot\; \sum_{j\in\mathcal{S}} w_{j}\,s_{j} \] \[ \mathcal{G} \;=\; \{M_{3}, M_{4}, M_{6}\}, \qquad \mathcal{S} \;=\; \{M_{1}, M_{2}, M_{5}, M_{7}\} \]

What it says

The score factors into two pieces. The first is a product of hard gates: each gate-scorer \(s_i\) must exceed its threshold \(\theta_i\); fail any and the score is zero. The second is a soft sum of the remaining scorers weighted by importance. Together they implement "must pass no-signalling, Bell, and heating; optimise everything else".

In this project

See scoring/aggregator.py. The aggregator also captures audit-only diagnostics (s_audit) that don't enter the score but flag candidates worth a human look.

algorithm · 1933

Thompson-sampling bandit

William R. Thompson · 1933
\[ \theta_{i} \;\sim\; \text{Beta}(\alpha_{i},\,\beta_{i}), \qquad \text{pick family} \;=\; \mathop{\mathrm{argmax}}_{i}\,\theta_{i} \] \[ \alpha_{i} \;\to\; \alpha_{i} + 1 \;\;\text{if score}>0;\qquad \beta_{i} \;\to\; \beta_{i} + 1 \;\;\text{otherwise.} \]

What it says

For each candidate family (CSL-like, mass-power, curvature, entropy, memory, multi-basis…), keep a Beta posterior over its success rate. Sample one draw per family, pick the one with the highest draw. Auto-balances exploring untried families (wide posterior, occasional high draws) with exploiting winning ones (high mean, consistently high draws). Provably near-optimal for many bandit setups.

In this project

Same module as the gravity project, repointed at the measurement families. See search/strategy.py → BanditStrategy.

That's every equation that does meaningful work in the project. If you want the wave-form intuition that goes with these symbols, head to the Learn page. If you want why we're testing any of this in the first place, the Big Idea page lays out the chain of reasoning — and the map of which combinations are already ruled out.

And once the search starts trying recipes against this scoring, the Watch page is where it'll happen.