Feigenbaum's constant is really in there

How does a simple map's long-term behaviour depend on its growth rate — and is the rate at which its period-doublings pile up really a UNIVERSAL constant of nature, the same for any smooth one-hump map, as Feigenbaum announced in 1978?

Units: dimensionless (Feigenbaum δ, quadratic-maximum class; Briggs 1991 high-precision value)

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The finding

Feigenbaum's constant is really in there: the bare iteration x → r·x·(1−x) yields δ = 4.669201 (rel 9e-8, never plugged in), the SAME δ from a sine map (universality), a DIFFERENT one (7.2847) from a quartic-max map (class control), plus r∞, λ(4) = ln 2 and the analytic onsets

Method

Locate the superstable parameters R_n — where the critical point x = ½ is itself on the period-2^n cycle — as roots of f^(2^n)(½; r) − ½, by Newton on r with the derivative propagated through the iteration. The search bootstraps from the ANALYTIC period-1/2 superstables (R₀ = 2, R₁ = 1+√5) and predicts each next root from the map's OWN measured gap ratio; no Feigenbaum number appears anywhere in the recovery, and every root is verified to have minimal period 2^n. δ̂ = (R₁₀−R₉)/(R₁₁−R₁₀) (n = 11 chosen where float64 gap round-off is still negligible), r̂∞ by geometric extrapolation with the recovered δ̂. Repeat the entire cascade for a DIFFERENT quadratic-maximum map (x → r·sin(πx)) and for a quartic-maximum map (x → r·(1−(2x−1)⁴)). Lyapunov exponents λ = ⟨ln|f′|⟩ along the orbit at r = 4 (24 seeded initial conditions), r = 3.2, 3.83, and r̂∞; doubling onsets r₁, r₂ recovered by bisection on the measured attractor period — the live module's method — against their analytic values. All knowns loaded from scripts/oracles/logistic.reference.json only to score. ?world=logistic.

The law it recovers

δ_n = (R_{n-1}−R_{n-2})/(R_n−R_{n-1}) → δ = 4.6692016… for every smooth unimodal map with a quadratic maximum

Measurements, controls & cross-checks

Cascade

R series note

R₂…R₁₄ = 3.4985616993 → 3.5699456712, fourteen superstable parameters spanning periods 4 → 16384, ratio sequence converged to |δ₁₁−δ₁₀| = 3.3e-8

Accumulation point

Recovered

3.5699457

Known

3.5699457

Rel offset

2.7600e-14

Note

geometric extrapolation R₁₄ + (R₁₄−R₁₃)/(δ̂−1) using the RECOVERED δ̂ — the onset of chaos located to 13 digits

Universality

Sine map delta

4.6692011

Sine rel offset vs known

1.1100e-7

Sine vs logistic

9.8200e-8

Note

x → r·sin(πx) shares NOTHING with the logistic formula except a smooth quadratic maximum, yet its cascade converges to the same δ to seven digits — Feigenbaum's discovery, the reason δ shows up in real convection cells and driven circuits

Controls

Quartic class

Delta

7.284725

Separation from quadratic delta

2.616

Vs z4 known

3.9400e-5

Note

a QUARTIC-maximum map does NOT give 4.669: its cascade ratio lands on the z=4 class constant 7.2846862 (Briggs 1991). Universality is a property of the quadratic-maximum class, tracking the order of the map's extremum — 'any period-doubling cascade gives 4.669' is falsified

Period3 window

Lyapunov at 3.83

-0.3697

Note

λ < 0 deep inside the chaotic regime — the period-3 window, order interleaved with chaos

Edge of chaos

Lyapunov at recovered r inf

-1.4500e-6

Note

λ ≈ 0 exactly at the recovered accumulation point — the cascade's limit is the edge of chaos

Lyapunov r4

Recovered

0.69314847

Se

8.7000e-7

Known

0.69314718

Abs offset

1.2900e-6

Seeds

24

Note

λ(4) = ln 2, the conjugacy-to-bit-shift value: at r = 4 the map doubles phase-space information at exactly one bit per iteration

Lyapunov period2 analytic

Recovered

-0.91629073

Known formula

½·ln|−r²+2r+4| at r=3.2 = ln 0.4

Abs offset

4.0800e-13

Onsets

R1

Recovered

2.9999983

Known

3

Abs offset

1.6900e-6

R2

Recovered

3.4494891

Known

3.4494897

Known form

1+√6

Abs offset

6.1600e-7

Note

recovered by bisection on the measured attractor period — the live module's own measurement method, with a 4e6-iteration warmup to out-wait critical slowing near the bifurcation

R3 hat

3.5440901

R3 known literature

3.5440904

R3 abs offset

2.3600e-7

R3 note

third doubling onset (4→8), no closed form; deep-warmup (4e6) attractor-period bisection vs the literature value (OEIS A086181), |Δ| = 2.4e-7 — added on the honest-module climb as the comparator for the HUD's r₃ and the first-interval ratio δ₁

Numerics

Worst newton residual

1.5200e-10

Minimal periods

verified for all three maps

Delta sequence convergence

3.3300e-8

Gates

19/19 (13 original + r̂₃ vs literature + 5 on-screen reconciliation: bit-exact pins, 3 a-priori onset-honesty bands, directional δ-identity)

Screen reconciliation

Url

?world=logistic

Hud

R1

2.9992

R2

3.4496

R3

3.5444

Delta

4.7510549

Displayed 3dp

r₁ 2.999 · r₂ 3.450 · r₃ 3.544 · δ 4.751

Protocol note

the module's HUD is fully DETERMINISTIC (no RNG, no seed): findBifurcation grid scans (warmup 3000, 256 samples, distinct-gap tol 1e-3; grids 4e-4/4e-4/2e-4 over [2.9,3.05]/[3.4,3.48]/[3.535,3.565], targets 2/4/8) and δ = (r₂−r₁)/(r₃−r₂). The derisk transcribes the scan verbatim (float-accumulated grid included), pins all four values bit-exact (r's strict ===, δ rel ≤ 1e-12) plus the four 3-dp HUD strings, and the live browser HUD was read back headlessly and matches the pins verbatim

Onset honesty

R1 dev

-0.0008

R2 dev

0.00011

R3 dev

0.00031

Bands apriori

±2.47e-3 / ±8.44e-4 / ±4.69e-4, each = wₙ (transient-smear window ln(A/tol)/warmup ÷ |dμₙ/dr|, A = 0.5, slopes analytic for n=1,2 — μ₁ = 2−r, μ₂ = −r²+2r+4 — measured 22.75 from the converged 4-cycle for n=3) + grid step + Lₙ (measured settled coarse-tol splitting lag: deep-warmup bisection of the module's own tol-1e-3 counter; L₃ = 1.78e-4 dominates r₃'s lag because the INNERMOST α²-compressed 8-cycle pair must split past 1e-3). No term fitted to the pins

Mechanism

r₂ and r₃ fire at the first grid point ≥ rₙ* (settled coarse-tol threshold: r₂* = 3.4495105, r₃* = 3.5442681) — splitting-lag-limited; r₁ fires 8.0e-4 EARLY, inside the transient-smear window w₁ = 2.07e-3 (fixed-point multiplier |2−r| ≈ 1 defeats the 3000-iteration warmup near r = 3)

Module systematics

Delta hud vs delta1

0.000391

Delta hud vs delta inf

0.0819

Note

the HUD displays δ 4.751 against its own label '(Feigenbaum 4.669)' — a VISIBLE on-screen 1.75% mismatch that is finite-n convergence physics, not error: a three-onset scan estimates the FIRST-interval ratio δ₁ = (r₂−r₁)/(r₃−r₂) = 4.751446 (analytic r₁, r₂ + literature r₃; the derisk's own deep instrument gives 4.751439), and the HUD lands 209× closer to δ₁ than to δ∞ = 4.669202. The gap δ₁ − δ∞ = 0.0822 is the first step of the δ_n → δ∞ march the oracle's deep superstable cascade exhibits explicitly (gate A: δ read at n = 11 where it has converged to rel 9e-8). Gated directionally a-priori: |δ_HUD − δ₁| < |δ_HUD − δ∞| strictly

What it reduces to

Feigenbaum universality (M. J. Feigenbaum 1978): the period-doubling renormalization operator has a single unstable eigenvalue δ = 4.6692016… at its fixed point, so the parameter intervals between successive doublings of ANY smooth map with a quadratic maximum shrink asymptotically by that same factor — a universal number, independent of the map, discovered numerically on an HP-65. It VALIDATES, not derives: the lab assumes only the bare iteration x → f(x) and shows the superstable-parameter ratios converge to δ (never fed — the root search bootstraps from the map's own gaps), that a structurally different quadratic-max map (sine) yields the SAME constant while a quartic-max map yields the DIFFERENT class constant 7.2847 (so the number tracks the order of the maximum, exactly what renormalization predicts), that the cascade accumulates at r∞ = 3.56994567187 where the measured Lyapunov exponent crosses zero, and that λ(4) = ln 2 and the analytic onsets r₁ = 3, r₂ = 1+√6 all hold. It does NOT verify the second Feigenbaum constant α = 2.5029 (band-width scaling) or the full renormalization spectrum. The lab's chaos cornerstone: the same δ measured in Rayleigh–Bénard convection (Libchaber) is here, inside one line of algebra.

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

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

Oracle

scripts/oracles/logistic.reference.json

Sources

M. J. Feigenbaum, 'Quantitative universality for a class of nonlinear transformations', J. Stat. Phys. 19, 25–52 (1978); K. Briggs, 'A precise calculation of the Feigenbaum constants', Math. Comp. 57, 435–439 (1991) — δ = 4.669201609102990671…, δ(z=4) = 7.2846862…; A. Libchaber et al.'s helium convection cascade measured δ ≈ 4.4 ± 0.1 in a real fluid. λ(4) = ln 2: E. Ott, 'Chaos in Dynamical Systems', §2.2. r₂ = 1+√6: S. Strogatz, 'Nonlinear Dynamics and Chaos', §10.3.