Particle Life condensation

Is the number the condensation sweep optimizes — the bound fraction that ClusterMetricsModule's union-find reports and ?world=condense crosses at ½ to declare μ_c — a calibrated instrument? Does it read the exactly known combinatorics of the substrate's solvable configuration (the iid uniform initial layout), and does the reported gas→condensed transition survive against a coupling-blind rival on the same machinery?

Units: E[isolated fraction] of the t = 0 bond graph — exact: (1−p)^(N−1), p = V_ball(2)/V_box = 1.1635528e-3, mean degree c = 1.3951

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The finding

Particle Life condensation — the condensation instrument calibrated on the substrate's one exactly solvable configuration: the module's own t = 0 state is 1200 iid uniform points on the 40×18×40 3-torus, so the ClusterMetricsModule bond graph (link R = 2) is EXACTLY Gilbert's random geometric graph — the isolated fraction reads 0.247801 ± 0.000485 vs the exact (1−p)^(N−1) = 0.2476071 (0.08%, 0.40 SE), the full degree histogram sits on Binomial(N−1, p) bin-by-bin (max 1.5 SE over d = 0…6), the bond count matches BOTH exact moments (mean 836.4 ± 1.0 vs 837.06, variance ratio 0.965 vs the exact C(N,2)p(1−p) — bond indicators on a torus are pairwise uncorrelated), and the dimer fraction lands on the exact two-sphere lens quadrature to 0.12% (0.35 SE), with the law tracked across a 42× link-radius swing (log–log slope 0.9995, r 0.99999) and a 2.85× density swing — then the world's open question lands as branch separation on the calibrated ruler: the documented seed-7 ascending μ-sweep reproduces its on-screen apparent μ_c at −0.191 (screen: ≈ −0.18), condenses to bound fraction 0.945 ≫ the measured random floor 0.4645 ± 0.0009, the all-repulsive gas digs BELOW the floor to 0.020 (the instrument reads anti-structure too), and a zero-coupling rival (attract ≡ 0) with identical machinery never leaves the floor (worst excursion 0.016 — a 29× separation): condensation belongs entirely to the species couplings the μ knob shifts; rung 6 (honest module): the SHIPPED module chain is itself certified — a 12240-call engine-protocol lockstep runs ParticleLifeModule + ClusterMetricsModule + ParticleLifeSweepModule bit-exact against an independent replica from the seed-7 boot to the certified screen 'done · apparent μ_c ≈ -0.18' (the documented value, now pinned as executed output), with the shipped instrument reproducing the oracle's 800 reference configurations bit-for-bit (closure, 0 mismatches) and the first substrate-MUTATING driver certified (the sweep's live setAttract edits land at pinned calls)

Method

The generator is ParticleLifeModule + ClusterMetricsModule transcribed verbatim headless (mulberry32 stream: 25 matrix draws then per-particle species/x/y/z, Float32 state; _stepSim's cell list, triangular force profile, R_MIN = 1.1 universal repulsion, FORCE = 50, FRICTION = 0.85, dt = 1/45; _measure's R = 2 cell list + union-find with strict r² < R², MIN_SIZE = 4, extended in-pass to count degrees, bonds and the size-1/2/3 census without changing any bond decision) — no Gilbert formula, no binomial, no quadrature anywhere in the generator; every exact law lives in the scoring path only. Calibration: 800 seeds at t = 0 (headline + degree histogram + bond moments + dimer fraction), 200-seed link-radius sweep R = 0.8→2.8, 200-seed prefix-subsample density sweep M = 300/600/1200. Phenomenon: the sweep module's own protocol mirrored in sim time at seed 7 (settle 11/6 s, measure 3.5 s, carry-over, ascending μ = −0.6→+1.0), condensed-branch check at seeds 8/9, zero-coupling rival scored on its worst excursion over the full 100.5 s run, full-pipeline determinism pin. Rung 6: the shipped modules themselves are executed under the engine protocol in lockstep against an independent replica whose instrument is the oracle's own validated measure() (gates L–P). 22 gates in scripts/condense-derisk.mjs (~18 s); tamper ⇒ exit 1. ?world=condense.

Measurements, controls & cross-checks

Recovered se

0.000485

Seeds

800

Worst seed z

3.15

Degree histogram

Bins

d = 0…6 vs exact Binomial(1199, p)

Max abs z

1.47

Mean abs z

0.61

Bond moments

Mean

836.42

Mean se

1

Exact mean

837.0599

Mean z

0.64

Var ratio

0.965

Note

variance gated against the EXACT C(N,2)p(1−p) = 836.09 — on a torus, bond indicators sharing a vertex are pairwise uncorrelated (P(both) = p² by conditioning on the shared point), so the bond count carries no Poisson approximation

Dimer fraction

Recovered

0.16776

Se

0.00058

Exact quadrature

0.1675519

Z

0.35

Note

exact 1-D Simpson quadrature over the two-sphere lens volume U(r) = 2V_ball − (π/12)(4R+r)(2R−r)² — no Monte Carlo in the scorer

Radius sweep

R

• 0.8
• 1.2
• 1.6
• 2
• 2.4
• 2.8

Loglog slope

0.9995

Pearson r

0.99999

Max abs z

2.03

Exact swing

42.3

Density sweep

M

• 300
• 600
• 1200

Max abs z

2.03

Exact swing

2.85

Random floor

Clustered fraction

0.46445

Se

0.00088

Note

the on-screen order parameter (components ≥ 4) at the random configuration — measured and disclosed; its exact expectation is a k ≥ 4 cluster-integral series with no closed form, but the machinery producing it is pinned by the four exact laws plus the partition identity f1+f2+f3+f(≥4) = 1 (holds to 7e-16)

Phenomenon

Seed7 mu c

-0.191

Onscreen mu c

-0.18

Condensed bound

0.945

Gas bound

0.02

Branch seeds

8

0.965

9

0.994

Note

the sweep module's protocol mirrored in sim time reproduces the documented on-screen apparent μ_c within 0.011 — well inside the a-priori bracket −0.18 ± one grid interval; μ_c itself stays an honest OPEN question (single-seed, protocol-dependent, ascending sweep carries the known Δμ ≈ 0.04 hysteresis)

Rival rejected

Name

zero-coupling limit: identical layout, PRNG, integrator, sweep schedule and instrument, but attract ≡ 0 (only the universal r < 1.1 repulsion; the μ knob shifts couplings that no longer exist)

Max abs dev from floor

0.0164

Condensed displacement

0.48

Separation

29.3

Note

over every sample of the full 100.5 s run the rival never leaves the random floor — the condensation the screen shows is carried entirely by the species couplings, not by the instrument or the overdamped churn

Determinism pin

Seed

42

Mu

0.5

Steps

200

Clustered fraction

0.95

Rel dev

0

Module certificate

Scope

gates L–P: the SHIPPED ParticleLifeModule.ts + ClusterMetricsModule.ts + ParticleLifeSweepModule.ts (the whole ?world=condense measurement chain minus the generic Hud) sha-pinned, type-stripped and EXECUTED headlessly in the engine's own protocol (all fixedUpdates in registration order → elapsed += fl(1/120) by repeated addition → all renders, src/core/Engine.ts), from the seed-7 default boot to the on-screen final status — the FIRST certificate of a substrate-MUTATING driver: the sweep edits the live interaction matrix through setAttract at executed state-machine transitions

Lockstep

12240 calls bit-exact vs an independent replica on the first full run: PL trajectory (4589 sim steps, census {0:7651, 1:4589} of the 45 Hz accumulator), the live-mutated matrix, the sweep state machine (10 measure windows opened at calls 1320…11589 and committed at 1741…12010, done at 12010, final draw 12017 — every fl(1/120)-sum crossing executed, never computed), 408 cluster samples at exact 30-call cadence (history capped 240), all 10 {μ, giant, bound} points, 408 cluster-panel + 1503 status + 1503 SVG chart writes byte-identical, thin-instance buffers probed every 600 calls, final elapsed 102.00000000002724 (repeated-addition clock)

Certified screen

final status 'done · apparent μ_c ≈ -0.18' — the shipped interpolator over the shipped instrument's live readings lands ON the documented value; final cluster panel 'count 27 · mean 42 · largest 223 · in-clust 95%' pinned; seed-8 twin reads μ_c ≈ -0.17, last bound 0.958 vs 0.948 — every displayed digit moves

Closure

shipped code × oracle configurations: the EXECUTED shipped _measure over all 800 of the oracle's calibration configs reproduces the oracle's own validated instrument with 0 mismatches (bit-for-bit in clusters/meanSize/largest/clusteredFraction) and its ensemble mean is Object.is-identical to the calibrated floor 0.46445 — the shipped instrument inherits the four exact-law gates; the shipped _criticalMu executed on the oracle's sim-time mirror points returns -0.19, in the documented bracket

Clock systematic

module protocol clock (engine-dt windows, 4 Hz sampling at 120 calls/s; each measure window's first consumed sample can be one ≤0.25 s-old settle reading — a code behavior the certificate executes faithfully) vs the oracle's sim-time mirror (0.5 s sampling): max |Δbound| 0.0371 across the 10 μ points and |Δμ_c| 0.011 (screen -0.18 vs mirror -0.191), both gated a-priori at 0.1 (#156: measured and gated, not assumed away)

Tampers

sha → L only; pair-symmetric PL repulsion −1→−0.9 → ALL 17 physics gates green, lockstep call 3 (px@0) — the oracle reads the physics, the certificate reads the code; instrument MIN_SIZE 4→3 → lockstep call 1 (first sample) AND closure (800/800 mismatches) — the dual catch; known ±0.1 both directions → the scoring gates only, recovery unchanged 0.247801; reference restored byte-identical (cmp)

What it reduces to

Random-geometric-graph combinatorics on the initial layout (E. N. Gilbert, 'Random plane networks', 1961; M. Penrose, Random Geometric Graphs, 2003; nearest-neighbour ancestry Hertz 1909 / Chandrasekhar 1943): with N points iid uniform on a torus, the number of neighbours of any point within R is exactly Binomial(N−1, V_ball/V_box) — no asymptotics, exact at N = 1200 — fixing the isolated fraction, the degree histogram, both bond-count moments, and (by exact 1-D lens quadrature) the dimer fraction. Non-circular: the generator codes ONLY the modules' seeding/force/measure rules transcribed verbatim (the reduction of the fresh layout to a binomial point process is a property of that seeding code, not a formula in it), the estimator is the module's own union-find, and every exact law is computed exclusively in the scoring path. INSTRUMENT-CALIBRATION scope, honestly stated (the paircorr/flock/schelling pattern): this validates the bound-fraction ruler on the exactly solvable configuration and falsifies the zero-coupling rival; the condensation point μ_c itself has NO citable closed form (it depends on the seed's matrix, the box, the settle protocol and the ascending carry-over) and REMAINS the open question the world investigates — now asked with a calibrated instrument, with the oracle's protocol-faithful mirror reproducing the on-screen μ_c ≈ −0.18 at −0.191. Module systematics disclosed: the on-screen clusteredFraction thresholds at component size ≥ 4, so its random floor (0.4645 ± 0.0009) has no closed form and is self-measured (the k ≤ 3 census that pins it is exact); the module samples via render (4 Hz under the certified 120-calls/s engine protocol, ~2 Hz at 60 fps wall-clock) while the oracle samples every 0.5 s of sim time — the certificate now MEASURES this clock systematic instead of assuming it away: max |Δbound| 0.0371 across the 10 μ points and |Δμ_c| 0.011 (screen -0.18 vs mirror -0.191), both gated a-priori at 0.1; the sweep's window bookkeeping also lets each measure window's first consumed sample be one ≤0.25 s-old settle reading (disclosed code behavior, executed faithfully in the lockstep, diluted among ~15 samples per window); Float32 position quantization perturbs bond decisions at relative 1e-7, far below every statistical gate.

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- condense (scripts/condense-derisk.mjs — 22 gates: isolated fraction (SE + rel + worst-seed z + reference self-consistency), 7-bin exact-binomial degree histogram, bond mean + exact-variance fingerprint, dimer quadrature, partition identity, 6-point radius sweep (z + log–log slope + r across 42×), 3-point density sweep, gas-below-floor, condensed-above-floor, μ_c bracket, cross-seed condensed branch, rival on floor + 4× separation, determinism pin, plus the honest-module certificate L–P (source pins + strip + answer-freedom census, executed init, 12240-call lockstep, display reconciliation with seed-motion twin, closure over the oracle's 800 configs + shipped-interpolator-on-mirror + screen-vs-mirror clock gates + dt twins); ~18 s, exit non-zero on any miss; tamper self-tests CONDENSE_TAMPER=sha|rulepl|rulecm)

Oracle

scripts/oracles/condense.reference.json

Sources

E. N. Gilbert, 'Random plane networks', J. SIAM 9, 533–543 (1961); M. D. Penrose, Random Geometric Graphs, Oxford UP (2003); P. Hertz, Math. Ann. 67, 387 (1909); S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943)