Einstein's ⟨r²⟩ = 4Dt recovered from raw random-walk tracks

A particle taking blind, memoryless random steps wanders away from home — but HOW? Einstein's 1905 answer, ⟨r²⟩ = 4Dt in the plane with D fixed by the step statistics alone, plus the deeper claim that the long-time cloud is GAUSSIAN whatever the single steps look like. Can the abstract law, its coefficient, and the central-limit universality behind it all be recovered from simulated tracks that were never told any of it?

▶ Run this simulationRead how it works

The finding

Einstein's ⟨r²⟩ = 4Dt recovered from raw random-walk tracks — D̂ = 1.8794 ± 0.0077 vs σ²/(2Δt) = 1.875 (0.57 SE), with the Gaussian propagator (CLT) and Pearson-walk universality gated, ballistic and Cauchy rivals rejected; honest-module certified: the shipped DiffusionModule.ts is EXECUTED in a 1600-call lockstep (bit-exact through the recollapse), its f64 twin lands on the oracle replica bit-for-bit, and the on-screen D̂'s gap from 1.875 is priced as f32-storage (+1.37e-8) + one cycle's noise (−0.55σ)

Method

Replicate the module's exact microscopic world headless (N=1500 walkers, per-axis Gaussian steps σ=0.25 wu every Δt=1/60 s, same mulberry32 PRNG and Box–Muller call order) and its exact on-screen estimator (sample ⟨r²⟩ every 12th step, stop at the recollapse cap ⟨r²⟩=80, through-origin slope → D̂=slope/4). 24 seeds give D̂ with a cross-seed SE; the log-log exponent of ⟨r²⟩ vs t tests linearity; pooled per-axis excess kurtosis at cycle end tests the Gaussian propagator. Perturbation: σ swept ×4 → D̂ must track σ²/(2Δt) point-by-point with log-log slope 2. Universality: Pearson's 1905 fixed-step-length walk (ℓ=σ√2, single-step per-axis excess kurtosis exactly −1.5) must land on the SAME D with κ→0. Rivals through the same pipeline: ballistic transport (coherent fixed velocities) and heavy-tailed Cauchy steps. The known D is loaded from the reference only to score.

The law it recovers

<r^2> = 4*D*t (d=2), D = sigma^2/(2*dt) = 1.875 wu^2/s

Measurements, controls & cross-checks

Recovered sigma

0.0077

Recovered se units

0.57

Worst seed rel error

0.0445

Diffusive exponent

Mean

0.9991

Worst abs dev from 1

0.0227

Gaussian propagator

Excess kurtosis mean

0.026

Excess kurtosis worst

0.2181

Module replica

Seed

12345

D hat

1.8541439

Rel dev from known

0.0111

Note

byte-identical PRNG stream and estimator to DiffusionModule._stepSim/_updateMeasuredD, float-pinned to 1e-12 — the on-screen D̂ is this same measurement; its 1.1% deviation is one typical cycle's statistical scatter (0.6 single-seed σ), not a bias

Module certificate

Summary

Gates H–M EXECUTE the shipped src/modules/DiffusionModule.ts (sha-pinned, mechanically stripped, run under recorder Babylon/DOM stubs at the reference seed 12345). First certified RNG world: the noise source is a module-internal seeded mulberry32, so the shipped code is exactly as deterministic as the clockwork worlds and the full lockstep pattern applies.

Lockstep

1600 engine calls at fl(1/120); 2·fl(1/120) === fl(1/60) EXACTLY, so one sim step fires every 2nd call and the accumulator returns to EXACTLY 0. Walker state (3000 f32), ⟨r²⟩ history, through-origin fit, and the 24000-float instance buffer match a Float32Array replica bit-for-bit at ALL 1600 calls, THROUGH THE RECOLLAPSE: the cap sample (⟨r²⟩ = 80.67 > 80) never enters the fit, cycle 1 = 53 samples, reset at pinned call 1295, D̂ → NaN → re-earned live in cycle 2 (12 samples at tick end). %6 HUD and chart BOTH LIVE: 267 writes each, payloads template-bit-equal at every write, 50/66 distinct.

Zero gap twin

An f64 twin of the lockstep replica (same stream, same op order, only the store width differs) lands cycle-1 D̂ = 1.854143894440838 === the oracle's independently-coded G3 runCycle pin BIT-FOR-BIT — two implementations of the pipeline agree to the last bit, and the shipped f32 value is EARNED (≠ twin ≠ 1.875 as doubles, every ⟨r²⟩ sample ≠ 4Dt).

Display priced

shown D̂ 1.854144 − label 1.875 = −2.086e-2 splits exactly into an f32-storage stage +1.366e-8 (the module walks in Float32Array; per-sample gap ≤ 1.13e-8 under the DERIVED deterministic rounding ceiling 4nP²ε₃₂/msd = 2.39e-4, no free constant) + a finite-ensemble noise stage −2.086e-2 located at z = −0.55 inside G2's own 24-seed scatter (σ_seed = 0.0377). HUD strings pinned ('D 1.854 / 1.875 · 1.1%'); the '/ 1.875' label double-equals the known value but is COMPUTED from σ²/(2Δt) module statics (disclosed). Chart: the cycle-1 polyline's worst pixel gap from the 4Dt guide is 3.14 px on the 128-px axis — a visible, honest single-cycle fluctuation, not smoothed away.

Answer free

The executed measurement path (_stepSim walk + ⟨r²⟩ sampling + _updateMeasuredD + _reset) contains no dPred/slopePred/1.875/7.5/theory; the law feeds only the disclosed HUD label and the chart's 4Dt guide + axis scale (census dPred ×7, slopePred ×3). A planted theory shortcut is caught, proving the scanner live.

Tampers

known_value ×1.013 → only G2 fails, recovery unchanged; sha tamper → only gate H; post-strip 1-ulp SIGMA tamper → only H's statics/getter pins fail while the entire 1600-call trajectory stays bit-exact — the f32 walker stores ABSORB 1-ulp double tampers (4th confirmation of the f32-absorption phenomenon), so pinned doubles are the only working ulp tripwire.

Perturbation

Sigma values

• 0.125
• 0.25
• 0.5

Loglog slope

2.0105

Worst point rel

0.0098

Universality

Pearson over gauss d ratio

0.99648

Pearson single step kurtosis

-1.5

Pearson cycle end kurtosis

-0.0177

Controls

Ballistic

Exponent

2

Line residual

0.253

D rel error

20.68

Note

rejected — coherent motion gives ⟨r²⟩∝t², residual ~26× the diffusive case, 'D̂' off by 2068%

Cauchy

Cycle end kurtosis

1540

Note

rejected — infinite-variance steps never Gaussianize; the kurtosis gate that diffusion passes at |κ|<0.1 reads 1.5e+3 here, so the CLT claim is falsifiable, not automatic

What it reduces to

Einstein's mean-square-displacement law (Ann. Phys. 17, 549, 1905): ⟨x²⟩=2Dt per axis, ⟨r²⟩=4Dt in d=2, with D=σ²/(2Δt) from the microscopic step variance — plus the CLT universality Pearson's random-walk problem (Nature 72, 294, 1905) and Rayleigh's reply made classic: fixed-length steps, uniform steps, any finite-variance steps diffuse identically. Non-circular in the estimator sense: the generator takes raw random steps and the recovery fits measured ⟨r²⟩ vs t through the origin; neither D, the diffusion equation, nor the Gaussian propagator appears in the recovery path (the microscopic identity D=σ²/(2Δt) is computed only to SCORE). The linear law itself IS a short mathematical consequence of independent increments — the validation content is that the full pipeline (finite N, finite cycle, correlated samples, the module's exact estimator) lands on it with quantified uncertainty, that the Gaussian limit emerges from non-Gaussian microsteps, and that the same pipeline rejects ballistic and heavy-tailed rivals. Companion to ?world=perrin, which runs this law in SI units through Stokes–Einstein to recover Avogadro's number; this oracle validates the abstract law and the universality Perrin's analysis takes for granted.

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- diffusion (scripts/diffusion-derisk.mjs)

Oracle

scripts/oracles/diffusion.reference.json

Sources

A. Einstein, Ann. Phys. 17, 549 (1905); K. Pearson, Nature 72, 294 (1905) and Lord Rayleigh, Nature 72, 318 (1905); M. von Smoluchowski, Ann. Phys. 21, 756 (1906).