Flocking (Vicsek)

Is the polar order parameter φ = |Σe^{iθ}|/N — the single number the flock world prints on-screen and sweeps to locate its order–disorder transition — a calibrated instrument? Does it read the exactly known statistics of the Vicsek model's solvable regime, and zero (the correct noise floor) when there is genuinely nothing to read?

Units: N·E[φ²] for iid uniform headings — exact at every N (Rayleigh 1880)

▶ Run this simulationRead how it works

The finding

Flocking (Vicsek) — the lab's polar-order instrument calibrated on the model's one exactly solvable regime: at maximum noise the module's own update rule (alignment sum + full-circle uniform kick) IS Pearson's random flight, and the same φ = |Σe^{iθ}|/N the screen prints reads Rayleigh's exact 1880 moment N·⟨φ²⟩ = 0.9979 ± 0.0099 vs 1 (0.21%, 0.2 SE), the Rayleigh mean √N·⟨φ⟩ = 0.8858 vs √π/2 = 0.8862, and the exact fourth moment 1.980 vs 2 − 1/N = 1.998 — tracks the N^{−1/2} disordered floor across an 8× size sweep (slope −1.0055), still reads order when order exists (φ ≥ 0.94 at η = 0.1), and an interaction-free rival (alignment radius → 0) NEVER leaves the floor at the same noise (max 0.11 vs floor 0.036) — order in a flock is carried by the alignment coupling, and the instrument that will locate η_c is now a calibrated ruler at both ends of its range; honest-module CERTIFIED (first live-RNG-stream cert): the shipped module + sweep, executed sha-pinned, run the whole 19800-call on-screen experiment bit-identically to independent replicas — the displayed η_c ≈ 0.35 is a certified output of the executed code, and the f32-state-vs-f64-oracle trajectory gap is priced at 9.2% (seed-42 pin) while the executed ensemble moment stays 1.4 SE from Rayleigh

Method

The generator is VicsekModule._stepSim transcribed verbatim headless (toroidal alignment sum over radius R = 1.5 with self, uniform kick (rng()−0.5)·2πη, mulberry32 PRNG, N = 600, L = 30, v = 1.2, dt = 1/25) — no Rayleigh/√N formula anywhere in generation or estimation. At η = 1 the kick is uniform on the full circle, so the update reduces identically to iid uniform headings each step: Pearson's random flight, with Rayleigh's exact moments as the oracle. 24 seeds × 420 independent samples for the main moment (successive samples independent because every heading refreshes each step); half-split drift check; 4-point N-sweep at fixed density (75 → 600); ordered-limit check at η = 0.1 (3 seeds); interaction-free rival at the same η; pinned-seed determinism regression. 11 gates in scripts/flock-derisk.mjs (~24 s); tamper ⇒ exit 1. ?world=flock.

Measurements, controls & cross-checks

Recovered se

0.0099

Seeds

24

Worst seed z

2.64

Half split abs

0.0145

Rayleigh mean

Measured sqrtN phi

0.8858

Known sqrt pi over 2

0.8862

Abs dev

0.0004

Fourth moment

Measured N2 phi4

1.9796

Known 2 minus 1 over N

1.9983

Abs dev

0.019

Note

the exponential-tail value E[R⁴] = 2N² − N; a deterministic-resultant rival (R ≡ √N ⇒ moment 1) is rejected at ~22 SE

N sweep

N

• 75
• 150
• 300
• 600

N phi2

• 1.0106
• 0.9926
• 0.9925
• 0.9979

Loglog slope

-1.0055

Note

the N^{−1/2} disordered floor holds point-by-point across an 8× size swing at fixed density — every finite flock's φ measurement sits on this floor, now verified rather than assumed

Ordered limit

Eta

0.1

Phi per seed

• 0.952
• 0.949
• 0.942

Order over floor

26

Note

the instrument still reads order when order exists — calibrated at both ends of its range

Rival rejected

Name

interaction-free flock (alignment radius → 0): θ' = θ + noise, no communication

Max phi

0.1115

Mean phi

0.0397

Floor

0.0362

Note

at the SAME η = 0.1 where real Vicsek reaches φ ≥ 0.94, the rival never leaves the Rayleigh floor — order is carried by the alignment coupling, not by low noise

Determinism pin

Seed

42

Phi2 sum

0.0883764

Rel dev

0

Module certificate

What

Gates L–P EXECUTE the shipped VicsekModule.ts AND VicsekSweepModule.ts (sha-pinned, mechanically type-stripped, headless against recorder stubs) in lockstep with independent hand replicas over the ENTIRE on-screen experiment — the lab's FIRST live-RNG-stream cert: the dynamics slice draws 600 noise kicks per sim step from the seeded mulberry32 (all 15 prior RNG certs froze their draws at init), so per-call bit-equality doubles as a stream-position proof — one draw offset would desync every subsequent heading.

Rng accounting

1 Math.random site = the no-seed fallback (executed proof: seeded init draws 0, unseeded exactly 1, seed === (draw·0xffffffff)>>>0); 5 rng() sites, exactly 1 in the fixedUpdate-onward slice (the per-agent kick); sweep module draws nothing; executed stream-position proof: init consumes exactly 1800 draws, +randomizeHeadings 2400.

Lockstep

19800 calls at fl(1/120) — the full default boot (Vicsek + sweep, the engine's registration order, 1:1 render cadence) from disordered start through the 11-point φ(η) sweep to the displayed 'done · η_c ≈ 0.35': acc/t/η/order, all headings/positions (Float32), the 9600-float instance buffer, and the sweep's state machine (i, phaseT, measuring, phiSum, phiN, points) bit-exact vs the replicas at EVERY call; all 11 measured points and the η_c string bit-identical to pins; seed frozen; done at call 19572/19572.

Accumulator class

incommensurate-subtractive, 2nd sighting (conway's fl(1/9) was the 1st): fl(1/25) is no float multiple of fl(1/120) — no exact zero, census {0:15676, 1:4124}, max acc 0.03999999999999999 pinned one grain under SIM_DT; budget-4 and acc-overflow-reset branches proven DEAD by execution.

Display

HUD byte-equal at all 3960 %5 writes (η .toFixed(2), φ .toFixed(3), FLOCKING/disordered both exercised; 1549 distinct — LIVE), sweep status at all 2448 %8 writes (23 distinct) and the full SVG chart byte-equal at every write (12 distinct); bufferUpdated 4125 / boundsRefresh 4126 / frames 19800 pinned.

Closure

the EXECUTED shipped _stepSim at η = 1 over 6 oracle seeds (same discard-3/420-sample protocol) recovers N⟨φ²⟩ = 1.0288 vs Rayleigh 1 (2.88% = 1.4 SE at 2520 samples, ≤ 7% gate); DISCLOSED f32 SYSTEMATIC: the module stores state in Float32Array while the oracle generator is Float64 — same seed and identical draw stream decorrelate at trajectory level (executed seed-42 Σφ² = 0.08022698401226544 vs f64 pin 0.088376400094688917: 9.2% gap, chaos amplifies f32 rounding) while the ensemble moment stays on the exact law: the φ instrument is calibrated for statistics, not trajectories, and both pins now live in the reference.

Eta c status

the displayed η_c ≈ 0.35 (seed 7, angular noise, N = 600, ρ = 2/3, upward-sweep protocol) is now a CERTIFIED measurement output of the executed shipped code — not an oracle-validated constant: finite-size Vicsek η_c remains genuinely non-citable (Chaté et al. 2008), which is exactly the open question the world investigates.

Tampers

sha ⇒ only L; kick-amplitude halved post-strip (the noise-term analog of a rule flip) ⇒ N at call 5165 — the FIRST sim step after η leaves 0 (at η = 0 a noise-amplitude tamper is physically invisible) — plus O, and P at ~440 SE (N⟨φ²⟩ = 9.78: a half-circle kick leaves residual alignment); known_value 1.0→1.1 by hand ⇒ moment gates fail at 10.3 SE with the recovery byte-unchanged (0.9979), restored by hand. All three exit 1.

What it reduces to

Pearson's random walk / Rayleigh's random flight (Rayleigh 1880; Pearson & Rayleigh, Nature 1905): for iid uniform phases E|Σe^{iθ}|² = N exactly, E|Σe^{iθ}|⁴ = 2N² − N exactly, and E|Σe^{iθ}| → √(πN)/2. Non-circular: the generator codes ONLY the module's alignment-plus-noise update (the reduction to iid uniform headings at η = 1 is a theorem about that update, not code), the estimator is the module's own |Σe^{iθ}|/N, and Rayleigh's moments are loaded from the reference exclusively to score. INSTRUMENT-CALIBRATION scope, honestly stated (the paircorr pattern): this validates the φ ruler at both ends — the exactly-solvable disordered floor and the deep ordered branch — and falsifies the no-interaction rival; the transition point η_c itself remains an OPEN measured quantity (the on-screen sweep's η_c ≈ where φ falls through ½) because finite-size Vicsek η_c is genuinely non-citable (Chaté et al. 2008: it depends strongly on N, density, and noise type) — that is the open question the world exists to investigate, now asked with a calibrated instrument. HONEST-MODULE (gates L–P): the SHIPPED VicsekModule.ts + VicsekSweepModule.ts, executed sha-pinned and type-stripped, are bit-identical to independent replicas over the entire 19800-call on-screen experiment — every HUD string, the SVG transition chart, and the displayed η_c ≈ 0.35 are certified outputs of the executed code (first live-RNG-stream cert), with the module's one systematic vs the oracle priced and disclosed: Float32 state decorrelates individual chaotic trajectories from the Float64 generator (seed-42 Σφ² gap 9.2%) while the executed ensemble moment sits 1.4 SE from Rayleigh's exact value.

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- flock (scripts/flock-derisk.mjs — 16 gates: exact second moment (mean/worst-seed z/half-split), Rayleigh mean, exact fourth moment, 4-point N-sweep (points + slope), ordered limit, order-over-floor separation, interaction-free rival, determinism pin, plus honest-module certification L–P (sha-pin + strip + live-RNG-stream accounting, executed init, 19800-call full-experiment lockstep of BOTH shipped modules, byte-equal display incl. the SVG chart and the η_c ≈ 0.35 headline, executed closure at η = 1 with the f32-vs-f64 trajectory gap priced at 9.2%); ~30 s, exit non-zero on any miss)

Oracle

scripts/oracles/flock.reference.json

Sources

Lord Rayleigh, Phil. Mag. 10, 73 (1880); K. Pearson, Nature 72, 294 (1905) and Rayleigh's reply, Nature 72, 318 (1905); T. Vicsek et al., Phys. Rev. Lett. 75, 1226 (1995); H. Chaté, F. Ginelli, G. Grégoire & F. Raynaud, Phys. Rev. E 77, 046113 (2008)