Pair correlation g(r)

Is the lab's g(r) instrument — the radial distribution function it uses to ask what phase of matter an emergent Particle Life world forms — a calibrated ruler? Does it recover the known structure of the best-understood liquid in statistical mechanics, the hard-sphere fluid, from nothing but raw sampled configurations?

Units: dimensionless g(σ⁺) at η = 0.40 (ρσ³ = 0.7639437), Carnahan–Starling

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The finding

Pair correlation g(r) — the lab's structure ruler calibrated on the hard-sphere fluid AND the shipped instrument itself certified: the SAME toroidal pair-histogram ÷ ideal-gas-shell math the module points at Particle Life, fed raw Metropolis Monte Carlo whose ONLY physics is overlap rejection (no equation of state, no g(r) theory anywhere in the generator), reads the contact value ĝ(σ⁺) = 3.7009 ± 0.0108 at η = 0.40 vs Carnahan–Starling's (1 − η/2)/(1 − η)³ = 3.7037 (0.08%, 0.3 SE) — tracks the CS curve across η = 0.10 → 0.40 (max dev 1.6%, slope 0.9967, r 0.99990), reads exactly 1 on uncorrelated matter, and rejects BOTH exact Percus–Yevick routes in opposite directions (virial −10% below at 34 SE, compressibility +5% above at 17 SE) — resolving the classic PY thermodynamic inconsistency where the near-exact interpolation says it must; the module's coarse-bin scheme is mirrored bin-exactly (0.04%) and its systematic quantified: the on-screen tallest-bin peakG under-reads contact by ~30% and biases the bond length ~7% outward at this bin-width-to-slope ratio — pure bin-width convolution, disclosed. HONEST-MODULE: the shipped PairCorrelationModule.ts + ParticleLifeModule.ts are sha-pinned, type-stripped and EXECUTED headlessly from the seed-7 default boot in a 2400-call fl(1/120) engine-protocol lockstep — histogram, g(r), analysis, status panel and SVG chart all bit/byte-exact vs an independent replica, first run — so the default screen's 'peak g(r) 21.39 · phase-separated · dense droplets + gas' is a certified output of the certified code; and the same shipped estimator driven over the oracle's own seed-101 hard-sphere stream reproduces the validated coarse mirror to 1.2e-15 and labels the reference fluid 'liquid-like · short-range order' — the instrument, executed, reads the best-understood liquid in statistical mechanics AS a liquid

Method

Raw Metropolis MC of 512 hard spheres (σ = 1) in a periodic cube from a simple-cubic start, cell-list overlap test — the ONLY accept/reject rule is 'no two centres closer than σ'; no EOS, no closure, no CS/PY form appears in generation or estimation. The estimator is the module's own math: toroidal pair histogram normalized bin-by-bin by the ideal-gas shell expectation N·ρ·(4π/3)(r₂³−r₁³), here on fine 0.005σ bins over [σ, 1.20σ], with a least-squares quadratic in (r̄_b − σ) over [σ, 1.10σ] (volume-weighted bin radii) extrapolated to contact. 6 seeds × 300 samples at η = 0.40 with half-split drift check; 4-point η-sweep (3 seeds each); ideal-gas Poisson control through the identical machinery; both PY routes as quantitative rivals; and a mirror of PairCorrelationModule's coarse binning (60 bins over 0.95·L/2 from r = 0) accumulated from the same pair stream and gated against the fine truth block-averaged. Honest-module certificate (gates L–P): both shipped modules sha-pinned + mechanically type-stripped + executed under src/core/Engine.ts's own protocol (1 fixedUpdate + 1 render per call at fl(1/120), elapsed by repeated addition) from the seed-7 default boot for 2400 calls against an independently written replica — bit/byte-equality of PL state (899 sim steps, census {0:1501,1:899}), all 34 pair-histogram snapshots, every g(r) recompute, latest{peakR,peakG,shells,phase}, 23 status-panel writes and 40 SVG chart writes; seed-8 twin moves every displayed digit; dt twins prove the budget-3 clamp + overflow reset live; closure drives the executed _snapshot/_computeGr/_analyse over the oracle's seed-101 MC stream against the validated coarse mirror. 16 gates in scripts/paircorr-derisk.mjs (~11 s); tamper ⇒ exit 1 (sha → L only; pair-symmetric PL force tamper → lockstep call 3 with ALL physics gates green; g(r)-normalization tamper → lockstep + closure; known ±0.1 → the 2 scoring gates, both directions, recovery unchanged). ?world=paircorr.

Measurements, controls & cross-checks

Recovered se

0.0108

Worst seed rel error

0.0126

Seeds

6

Half split drift

0.0013

Sweep

Eta

• 0.1
• 0.2
• 0.3
• 0.35
• 0.4

G measured

• 1.3055
• 1.7866
• 2.4976
• 3.0281
• 3.7009

G CS

• 1.30316
• 1.75781
• 2.47813
• 3.00405
• 3.7037

Max rel dev

0.0164

Slope vs CS

0.9967

Pearson r

0.9999

Note

a 2.8× swing in the contact value across dilute gas → dense liquid, on the CS curve point-by-point and strictly monotone — the law holds across the density axis, not at one tuned state point

Control ideal gas

G contact

0.9942

Flat max abs dev

0.0307

Note

the identical histogram + normalization + extrapolation fed genuinely uncorrelated Poisson configurations reads 1 and stays flat — the 3.70-fold contact enhancement is carried by the sampled ensemble's excluded-volume correlations, not by the instrument

Rivals rejected

PY virial

Value

3.3333

Measured above rel

0.11

Se separation

34

PY compressibility

Value

3.8889

Measured below rel

0.048

Se separation

17.4

Note

the exact Percus–Yevick solution's two mutually inconsistent thermodynamic routes bracket the truth ±5–10% at η = 0.40; the measurement rejects both in opposite directions and lands on the near-exact CS interpolation — sub-percent structure measurement resolves a closure-level theory dispute

Module mirror

Block avg max dev

0.0004

Coarse peak g

2.573

Coarse peak r

1.074

Disclosed systematic

the module's 60-bin-from-zero scheme, applied to this reference fluid, under-reads the contact value by ~30% (2.573 vs 3.701) and biases the first-peak position ~7% outward — entirely explained (to 0.04%) as the fine truth convolved with the bin width against the steep contact slope dg/dr ≈ −19/σ. The mechanism transfers to the module's Particle Life box (its Δr is set by that box); the exact percentages scale with Δr·|dg/dr|. The module's phase LABELS (gas/liquid/crystal thresholds) key off this smeared peakG, so they are conservative near thresholds

Module certificate

Lockstep

2400 calls at fl(1/120) from the seed-7 default boot of ?world=paircorr (ParticleLifeModule + PairCorrelationModule, the engine's fixedUpdate/render/elapsed protocol), bit/byte-exact vs an independent replica on the FIRST full run: PL positions/velocities/matrix, sim-step census {0:1501,1:899} (fl(1/45) incommensurate accumulator), settle crossing at call 1080, 34 snapshots at exact 40-call cadence, every g(r) recompute, 23 panel + 40 chart writes (23/23/24 distinct), 5 thin-instance buffers probed every 600 calls, final SVG sha-pinned

Certified screen

Seed

7

PeakR m

0.64125

PeakG

21.391761

Shells

2

Phase

phase-separated · dense droplets + gas

Note

the default screen the visitor sees, pinned bit-exact — consistent with the finding's long-standing statement that the default seed shows phase separation, and honestly labelled by the module's own ≫3 peak heuristic

Seed motion

Seed

8

PeakR m

0.21375

PeakG

197.36802

Note

every displayed digit moves with the physics (anti-grid-lock twin): 0.21/197.37 vs 0.64/21.39

Closure

Executed on

the oracle's own seed-101 hard-sphere MC stream (300 configs)

Mirror worst rel

1.2000e-15

Executed peak g

2.5648549

Executed peak r sigma

1.0738499

Executed phase

liquid-like · short-range order

Note

the SHIPPED estimator code reproduces the oracle's validated coarse mirror to float-associativity precision and its phase heuristic, executed, reads the hard-sphere reference fluid as a liquid — the instrument's phase language is now itself calibrated on a known liquid

Clock note

the certificate models the engine protocol at one fixed step per render (the same certified protocol as the active/flock/kuramoto certs); in a live browser the render cadence is the display's, so snapshot TIMES differ while the estimator math, settle gate and per-snapshot histogram are the certified code paths

What it reduces to

The radial distribution function of statistical mechanics and the hard-sphere contact-value theorem Z = 1 + 4η·g(σ): recovering g_CS(σ) = (1 − η/2)/(1 − η)³ (Carnahan & Starling 1969) from raw overlap-rejection MC. Non-circular: the generator codes only 'reject overlaps', the estimator codes only the ideal-gas shell expectation, and the CS/PY closed forms are loaded from the reference exclusively to score; the certificate's lockstep never sees the known value. Scope honestly stated: the oracle validates the INSTRUMENT on a citable reference fluid and quantifies its binning systematic; the certificate proves the SHIPPED code — estimator, substrate, screen — is bit-exactly the thing so validated (executed headlessly, no browser needed after all), including driving the shipped estimator over the oracle's own configurations. What remains an investigation is the Particle Life g(r) VALUE itself (peak 21.39, phase-separated at seed 7): an emergent world has no citable known value to score it against — the certified claim is that this number is the true output of a calibrated, disclosed-bias instrument reading the true substrate.

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- paircorr (scripts/paircorr-derisk.mjs — 16 gates: contact scalar (mean/worst-seed/half-split), 5-point CS sweep (point + slope + r + monotone), ideal-gas control (contact + flatness), PY-virial and PY-compressibility rivals, module-binning mirror; plus the honest-module certificate L–P (sha pins + strip + RNG/answer-freedom census + executed fallback, executed init, 2400-call lockstep, display reconciliation + seed-motion twin, closure on the oracle's stream + dt twins); ~11 s, exit non-zero on any miss)

Oracle

scripts/oracles/paircorr.reference.json

Sources

N. F. Carnahan & K. E. Starling, J. Chem. Phys. 51, 635 (1969); M. S. Wertheim, Phys. Rev. Lett. 10, 321 (1963); E. Thiele, J. Chem. Phys. 39, 474 (1963); J. A. Barker & D. Henderson, Rev. Mod. Phys. 48, 587 (1976); J. Kolafa, S. Labík & A. Malijevský, Phys. Chem. Chem. Phys. 6, 2335 (2004); N. Metropolis et al., J. Chem. Phys. 21, 1087 (1953)