08 — Delay Destabilises
Supplied working report; original verdicts not validated.
Review status: Needs review. This is an archival source note, not a passed proof.
README.md
# 08 — Delay Destabilises **Verdict: supported** ## Summary | Quantity | Observed | Theory | Criterion | |---|---|---|---| | Onset of non-decay (k·τ) | **1.5750** | π/2 = 1.5708 | within 2% | | Relative deviation of onset | +0.268% | — | ≤ 2% | | Oscillation period at onset + 0.05 | **3.962 · τ** | 4 · τ | 3.8–4.2 | | Period at the onset point itself | 3.998 · τ | 4 · τ | reported for context | | Logistic variant (r·τ = 1.7) | sustained cycle, std 0.564, 25 crossings | oscillates for r·τ > π/2 | robustness | | Logistic variant (r·τ = 0.2, 0.8) | monotone convergence, std ≈ 0 | stable below π/2 | robustness | | Second-order variant, b = 0 → 0.2 | amplitude ratio 1.02 → 316.7 | any b > 0 amplifies | robustness | Primary system: x′(t) = −k·x(t−τ) with k = 1, Euler integration at dt = τ/2000 (2,000 steps per delay), amplitude ratio R = (max|x| in the final 25τ) / (max|x| in the 20–45τ window) over 120τ runs. Coarse sweep 0.05 ≤ kτ ≤ 3 step 0.05, refined to 0.005 around the crossing. ## Headline numbers - The loop changes from damping (R = 0.50 at kτ = 1.55) to amplification (R = 2.73 at kτ = 1.60) across the theoretical boundary, with the observed onset of non-decay at kτ = 1.5750, which is 0.268% above π/2 (well inside the 2% pre-specified tolerance; the Euler discretisation is expected to shift the boundary by −0.0004, i.e. −0.025%, and the 0.005 measurement grid contributes about ±0.3%). - Near the threshold the oscillation period is 3.96–4.00 · τ, matching the predicted 4τ. - Both robustness variants behave as the claim predicts: the logistic loop develops a bounded sustained cycle once r·τ exceeds π/2 (and only then), and the second-order loop turns from marginal stability into amplification as soon as any delayed coupling is present. ## What this shows and cannot show It shows that in the standard delayed negative-feedback model the destabilising threshold is precisely the predicted fraction of the oscillation period (kτ = π/2, a quarter of the 4τ period), and that delay alone converts damping into amplification. It cannot show that any particular real feedback system (economic, ecological, engineering) sits above or below its own threshold: the result is about the mathematics of the delay equation, and real systems additionally have noise, saturation and distributed rather than discrete delays. ## Limitations 1. **Idealised linear model without saturation.** Above the threshold the linear solution grows without bound; "sustained oscillation" is therefore defined operationally as non-decay (R ≥ 1), exactly the damping-to-amplification transition the claim asserts. The bounded, genuinely sustained cycle is demonstrated in the logistic variant, which confirms the mechanism under saturation. 2. **Discrete, constant delay.** Real loops have distributed or state-dependent delays, and the exact π/2 boundary is specific to a single discrete delay; a different delay kernel would move the threshold, which is a rival explanation for why the exact fraction (one quarter of the period) may not hold in any given empirical system. 3. **Numerical method.** Euler integration is the method the design specifies; its known threshold shift (−π/(4N) ≈ −0.0004 for N = 2,000) is small but nonzero, and the onset is located on a 0.005 grid, so the 1.5750 value carries roughly ±0.005 measurement resolution. 4. **No stochastic forcing.** With noise present, sustained oscillation can appear statistically below the deterministic threshold; this test measures the deterministic boundary only.
hypothesis.md
# 08 — Delay Destabilises
**Source:** Incoprea Research library, Vol. XXXI, "feedback-delay-destabilises" family
(entries published 30 October 2026). Round-V assignment sheet, test 08. The claim is quoted
verbatim from the assignment.
## The posted claim (verbatim)
> "A negative feedback loop destabilises when its delay exceeds a fraction of the oscillation
> period, turning damping into amplification."
**Prediction:** "Oscillation appears above a delay threshold; stability independent of delay
would refute."
## What this test does
Pure NumPy, fully seeded, no network access.
**Primary system.** The linear delayed negative feedback x′(t) = −k·x(t−τ), k = 1, with
constant history x(t ≤ 0) = 0.5, integrated by the Euler method with step dt = τ/2000
(2,000 steps per delay). For kτ from 0.05 to 3.0 (coarse step 0.05, refined to 0.005
around the detected onset), each simulation runs 120τ and records the amplitude ratio
R = (max|x| over the last 25τ of the run) / (max|x| over the 20–45τ window), so R < 1
means damping and R > 1 amplification. The observed onset of non-decay is the first grid
point with R ≥ 1. Theory: x′ = −k·x(t−τ) destabilises at kτ = π/2 ≈ 1.5708, with period
4τ at the boundary (the Euler discretisation with N = 2000 shifts the boundary by about
−π/(4N) ≈ −0.0004, i.e. about −0.03%, negligible against the 2% criterion). The
oscillation period is measured by mean up-crossing intervals over the final 60τ of a run at
onset + 0.05.
**Robustness variant A (logistic).** dx/dt = r·x·(1 − x(t−τ)), r = 1, with
τ ∈ {0.2, 0.8, 1.7}: without delay the logistic map is monotone stable at x = 1; with
rτ > π/2 the delay is expected to induce sustained oscillation. Sustained oscillation is
declared when the standard deviation of x over the final 50τ of a 300τ run exceeds 0.02
and there are at least 4 zero crossings of x − 1.
**Robustness variant B (second order).** x″ = −a·x − b·x(t−τ), a = 1, τ = 1, with
b ∈ {0, 0.02, 0.05, 0.1, 0.2}: b = 0 is a marginally stable (undamped) oscillator;
theory gives Re λ ≈ (b/2)·sin(τ) > 0 for small b, so any positive delay coupling should
turn marginal stability into amplification. The amplitude ratio is measured over 120τ runs.
## Pre-specified criteria for this test
- **supported** — the observed onset of non-decay lies within 2% of kτ = π/2 **and** the
measured period at onset + 0.05 lies within [3.8τ, 4.2τ] (the predicted 4τ).
- **partly supported** — an onset of non-decay exists at or below kτ = 3.0 but lies more
than 2% away from π/2 (the threshold exists but is shifted).
- **not supported as stated** — no onset at any kτ ≤ 3.0: delay never destabilises the loop.
These rules are exhaustive. The logistic and second-order variants are reported as
robustness checks; they do not alter the verdict, which rests on the primary linear system.
## Notes declared in advance
- Above the threshold the linear system grows without bound (it has no nonlinear
saturation); "sustained oscillation" in the primary test is therefore defined as
non-decaying oscillation (R ≥ 1), which is exactly the "damping turned into
amplification" the claim asserts. The logistic variant provides the bounded,
genuinely sustained cycle.
- The onset is measured on a 0.005 grid, so the measurement resolution is about
±0.005/1.5708 ≈ ±0.3% of the theoretical threshold; the 2% criterion is well above
resolution.
- Euler integration with N = 2,000 steps per delay is the method the design specifies;
its known threshold shift is computed and reported alongside the observed onset.