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fractals · first study · where quantum turns classical

Witness Map

Every object here is asked the same question: can it be in two places at once, a set gap apart? The answer comes down to a race between two clocks.

Exploration, not peer reviewed, no new result claimed. By Aska Digital with Claude (Anthropic) for Foundation0. The same model file runs this page and the automated checks.

objects on the map
14
surroundings
6
powers of ten spanned
>150
method
order of magnitude
live from witness-map/model.js checked by npm test

00//The two clocks

W = wave clock / witness clock

Two clocks race.

One clock says how fast the wave spreads. The other says how fast the surroundings find out where the object is. Whichever finishes first decides how it behaves.

Wave clock

How long the object's wave needs to spread across the gap. Heavier things spread slower, so their clock runs long.

Witness clock

How long until something in its surroundings (a photon, a gas atom, its own heat glow) carries off a record of which place it's in.

Witness number W = wave clock ÷ witness clock
W below 1: the wave wins, it can act quantum W above 1: witnesses win, it acts classical within 10× of the line nothing witnesses it

01//The map

live from witness-map/model.js

Slide the gap. Watch who crosses.

Each row is one object in its surroundings, placed by its witness number W. Pick a row for the full readout. Share a view with ?gap= (metres), ?env=, ?bh=1 and ?sel= in the address.

250 nm
Each object sits where it's normally found.
Sagittarius A*, 4.3 million Suns. Uses the horizon formula of Danielson, Satishchandran and Wald.
◀ wave winswitnesses wined ▶
10−4011040108010120
W

Giant molecule (~2,000 atoms)

Wave wins · W ≈ 10−2

Its wave crosses the gap about 102 times faster than the surroundings can record which side it's on.

Wave clock25 msto spread across 250 nm
Witness clock900 msuntil a record of its place exists
SurroundingsLab vacuum, room temperature4.2×10−23 kg · 4 nm across

Who's watching, by share of records

Its own heat glow94%
Leftover gas molecules5.6%
Glow of room-temperature walls<0.1%

Seen behaving as a wave in experiments (Vienna, 2019), at gaps of roughly a few hundred nanometres or less.

02//What the map shows

What the map shows.

  • Lab-proven waves sit on the wave side. At the current gap (250 nm), 4 of the 4 objects that have been seen acting as waves in experiments land left of the line. Slide the gap up and the giant molecule crosses first, watched by its own heat glow.
  • Everything bigger is flooded with witnesses at home. The closest call among the rest is the interstellar dust grain, at W ≈ 107 in interstellar space.
  • Silence the witnesses and size matters less. A virus moved into a 4 K vacuum scores W ≈ 10−1 at this gap. That is why physicists have proposed putting a virus in two places at once inside a cold vacuum.
  • Wave dark matter has no witnesses here. One light particle stays on the wave side at every gap: nothing in this model records where it is. A heavy particle is different, but not because it is watched: its wave clock at a 1 kpc gap is about 1030 ages of the universe at 100 GeV, so the superposition never forms. And the galaxy-sized wave is a crowd anyway, its lumps felt by every star (see the crowd panel).
  • A black hole watches, but feebly. Even 10 horizon radii from the Milky Way's black hole, a wave dark matter particle split across a 1 µm gap would wait about 1.7×10170 years for the horizon to record it in the formula. This is a formal extrapolation below the particle's reduced Compton wavelength, ħ/(mc), about 0.06 pc for 10⁻²² eV. It is not a validated prediction for a particle confined to such a small region. The horizon only beats ordinary witnesses for objects that are already classical.
  • Mass still counts, through the wave clock. Heavier objects spread slowly, so they need to stay unwatched for longer. At this gap the Sun's wave clock is 3.7×1043 years, so even a trickle of witnesses settles it.

03//The crowd

The crowd: where the map stops.

not a witness number

Every row above asks about one particle. Wave dark matter in a galaxy is never one particle. Its wave breaks into lumps, and each lump is shared by so many particles that together they act as one ordinary wave, the way countless photons make one radio wave. The crowd keeps its own clocks, and they are measured against the age of the universe, not against a gap.

And the crowd is watched. One particle's pull is far too weak to leave a record, but a lump is a vast number of particles pulling together, and every star it passes is nudged. A nudge of that size is enormous by quantum standards, so the star now holds a record of where the lump was. The wave survives anyway: recording the shape of an ordinary wave does not erase it, any more than measuring a sea wave flattens the sea.

Milky Way halo at the Sun's orbitparticle of 10⁻²² eV, the map's wave dark matter particle
Lump size1,200 light-yearshalf a de Broglie wavelength
Lump mass9.7×104 Suns1.1×1093 particles share it
Lump lifetime710,000 yearsone particle's wave clock at a 390 light-years gap
Stirring time4.7×1012 years340 times the age of the universe
Kick from one lump44 m/son each star it passes
Witnessed?Yes, by gravityevery star it nudges holds a record
Milky Way halo at the Sun's orbitparticle of 3×10⁻¹⁹ eV, the lowest mass dwarf-galaxy stars allow
Lump size0.4 light-yearshalf a de Broglie wavelength
Lump mass3.6×10−6 Sunsabout 1.2 Earths; 1.3×1079 particles share it
Lump lifetime240 yearsone particle's wave clock at a 0.13 light-years gap
Stirring time4.8×1022 years3.5×1012 times the age of the universe
Kick from one lump4.9 µm/son each star it passes
Witnessed?Yes, by gravityevery star it nudges holds a record
Core of a dwarf galaxy like Eridanus IIparticle of 8×10⁻²³ eV, core 0.7 kpc in radius (simulated)
Core radius2,300 light-yearsone standing wave, no lumps inside
Core mass5.1×108 Suns7.1×1096 particles share it
Peak density0.13 Suns per pc³12 times the dark matter density near the Sun
Wave turns over every110 million yearsthe core wanders on this time
Breathes every260 million yearscentral density swings more than twofold
Witnessed?Yes, by gravityits stars are shaken by it

Order of magnitude, like the map. Lump lifetime ħ/(mσ²), lump mass π3/2ρ(ħ/mσ)³ and stirring time from Bar-Or, Fouvry & Tremaine 2019; the core is the ground state of the Schrödinger–Poisson equations, checked against Schive et al. 2014 and 2020; breathing from Veltmaat et al. 2018. Inputs and derivations: README.

04//Method and checks

tolerance: a factor of a few

How it’s computed.

Everything runs live in this page from first principles. Treat the numbers as order-of-magnitude: the method is good to within a factor of a few, and the map spans more than 150 powers of ten.

wave clock = m·Δx² / ħ   ·   witness rate Γ = Σ (hits per second) × (information per hit)   ·   W = Γ × wave clock
  • Hits per second. Photons from the surroundings (a full thermal spectrum at each source's temperature), gas atoms, solar wind, and the object's own heat glow. Cross-sections are geometric for objects bigger than the wavelength and fall off as (size/wavelength)⁴ for smaller ones. Electrons use the standard electron–photon value. Molecules glow through one fitted number, set so a heated fullerene loses its interference at the measured temperature.
  • Information per hit. A hit with wavelength λ can only tell the two places apart if the gap is comparable to λ. The fraction is 1 − exp(−(2πΔx/λ)²/3) for a scattered photon and 1 − sin(2πΔx/λ)/(2πΔx/λ) for one the object gives off: tiny for gaps much smaller than λ, one full record for larger gaps.
  • Black hole horizon. Γ = G⁶M⁵m²Δx⁴ / (ħc¹⁰b¹⁰) for distance b, and an electric version for the electron. Valid only when the gap is much smaller than the distance.
  • The crowd. For particle mass m, density ρ and speed spread σ: lumps about πħ/(mσ) across, living ħ/(mσ²), each pulling like a body of mass π3/2ρ(ħ/mσ)³. The stirring time is how long their tugs take to change a star's speed by as much as the halo's own spread of speeds.

Checks against published numbers

  • Light and gas scattering match the closed-form decoherence formulas to within 2%. They match 11 of the 15 entries of the Joos–Zeh table and 7 of the 8 entries of Schlosshauer's table to within a factor of 5. The tables give powers of ten only, and where they disagree with the model they also disagree with each other.
  • The horizon formula reproduces both worked examples in Danielson, Satishchandran and Wald (2022): Earth split by its own radius beside a Sun-mass black hole (paper: ~10 µs, here: ~22 µs) and an electron split by 1 m (paper: ~10⁴³ years, here: ~4×10⁴² years).
  • Heated C₇₀, the molecule of the 2004 Vienna heating experiment, loses half its fringe visibility here at about 2,475 K, against about 2,500 K measured. The glow is fitted to that point, and the fit gets the light itself wrong: too little of it is visible.
  • The crowd's lump mass and stirring time reproduce the worked numbers of Bar-Or, Fouvry and Tremaine (2019) to 2%. The core, solved here from first principles, matches the density and mass formulas of Schive, Chiueh and Broadhurst (2014) to about 2% and their core wave period (2020) to 1%.
  • All of these run as automated checks (npm test). Mistakes and disagreements found along the way are in CORRECTIONS.md.

What it leaves out

  • Objects are treated as sitting still. Fast motion, internal structure and spin are ignored.
  • The wave clock assumes speeds far below light speed and is unreliable for gaps below ħ/(mc), the reduced Compton wavelength. The particle horizon example also assumes a source confined to a small region, which is not justified at those scales.
  • W is per particle. Wave dark matter's galaxy-wide wave is a crowd effect, enormous numbers of particles sharing one wave. The crowd panel above gives its clocks; it has no witness number.
  • Gravity's only witness here is a black hole horizon. Gravitational nudges from passing stars are treated as zero.
  • Light an object absorbs from its surroundings is not counted. Molecules have no electronic gap, and their glow below 2,000 K is unchecked.

Code: Foundation-0/fractals · open questions

Sources: Joos & Zeh 1985 · Danielson, Satishchandran & Wald 2022 · Dürr, Nonn & Rempe 1998 · Bild et al. 2023 · Riedel & Zurek 2010 · Hackermüller et al. 2004 · Hornberger et al. 2005 · Schlosshauer 2019 · Hui, Ostriker, Tremaine & Witten 2017 · Bar-Or, Fouvry & Tremaine 2019 · Allali & Hertzberg 2020

05//The studies

One question, asked at every scale.

fractals asks where reality repeats itself across scales, and where the repetition stops. Each study is a small model, checked against published numbers, with its mistakes kept on the record.

  • The Witness Map (this page): where quantum turns classical, object by object.
  • Gravitational atom beside hydrogen: the same wave equation at two scales, drawn at matched size.
  • Quantum paths: where a recorded path's fractal dimension 2 starts and stops.
  • The critical point: a magnet whose spins point only up or down repeats itself across scales only when tuned to its critical point; away from it the repetition stops at the correlation length.
  • Turbulence: from the size where a flow is stirred down to a smallest size, the same energy flux passes through every scale; the bursts do not scale exactly, and what sets that smallest size depends on the medium: viscosity in air and water, the ions' own motion in the solar wind.
  • Cosmic homogeneity: the scale where galaxy clustering stops looking fractal. Model and checks written; issue 17 tracks publication.
  • Open questions: halos within halos. The issue list is the programme.