# Threshold calibration — measured evidence (all numbers from real runs)

System: x' = 0.21 x(t-tau)/(1+x(t-tau)^9.8) - 0.105 x, with tau free.
Task: locate tau* where the dominant nontrivial Floquet multiplier is -1.

## The bifurcation point (measured)

Continuation of the periodic orbit downward in tau, recomputing the dominant
multiplier at each step, gives a cleanly monotone branch:

| tau | period T | mu |
|---|---|---|
| 22.70 | 57.6894 | -2.676218 |
| 20.30 | 52.6475 | -2.318055 |
| 17.90 | 47.8077 | -1.897613 |
| 16.30 | 44.8243 | -1.602322 |
| 12.70 | 37.5134 | -0.984631 |

Secant refinement on mu(tau) + 1 then converges in 3 iterations:

    tau* = 12.795726635219
    T*   = 37.732804942870
    mu   = -1.0000000000161      (|mu + 1| = 1.6e-11)
    orbit residual (L-inf, spectral) = 8.8e-14
    amplitude 0.615, fundamentality gap 0.333, spectral tail 3.0e-17

Total oracle wall time: 1089 s (18 min), inside the 3 h task budget.

## Thresholds

| check | full credit | partial |
|---|---|---|
| tau* in [10, 20], period > tau | required | required |
| positivity, amplitude >= 0.4 | required | required |
| spectral tail (top 1/8 modes) | <= 1e-10 | <= 1e-10 |
| fundamentality gap | >= 0.1 | >= 0.1 |
| DDE residual (L-inf) | <= 1e-8 | <= 1e-8 |
| bifurcation condition abs(mu + 1) | <= 1e-4 | <= 1e-2 |

## Why abs(mu + 1) is capped at 1e-4 and not tighter

The threshold is set by the GRADER's own certified accuracy, not by wishful
precision. At the bifurcation orbit the grader's two-resolution Richardson
estimator shows a spread of 2.7e-6 between resolutions, and its
trivial-multiplier canary (the exact eigenvalue mu = 1, which must be
reproduced) sits at 2.3e-5. Demanding better than ~1e-4 would ask for
precision the grader cannot itself vouch for. The oracle clears the
threshold by seven orders of magnitude (1.6e-11), so there is ample headroom
on the accepted side.

Because dmu/dtau ~ -0.168 along this branch, |mu + 1| <= 1e-4 corresponds to
locating tau* to about 6e-4 absolute -- so the agent must genuinely run a
parameter continuation and a root-solve, not merely find one orbit.

## Second deliverable: branch switching, and the trap it guards

The agent must also produce the PERIOD-DOUBLED orbit born at the
bifurcation, at the fixed evidence delay tau = 14.0. This forces an actual
branch switch rather than a scalar root-solve. Reference (measured):
T2 = 83.42643132, residual 4.0e-14, fundamentality gap 0.191, amplitude
0.768.

The check that matters is fundamentality, and the reason is a genuine trap
we hit while building this: the period-1 orbit written at double period is
an exact solution of the delay equation at the same tau. Measured, that
decoy has residual 4.3e-14 -- it passes the residual check with room to
spare -- and multiplier mu_1^2 rather than mu_1. Its fundamentality gap is
4.4e-16, so ONLY that check separates it from a genuine doubled orbit. Our
own first continuation attempt slid onto exactly this branch (fundamentality
gap collapsing to 0.000 while the residual stayed at 1e-14), which is why
the reference solver re-seeds from forward simulation at every step instead
of Newton-continuing.

## Self-certification

No hidden reference value is used. "The dominant nontrivial Floquet
multiplier equals -1" is an intrinsic property of the submitted (tau*, orbit)
pair, and the grader recomputes it from that pair alone: it rebuilds the
monodromy operator of the linearised delay equation on the history space and
extracts its spectrum. Any correct bifurcation point passes; a wrong tau*
fails regardless of how it was obtained.

## Determinism

No data files; fixed grading resolutions (nh = 280/400, steps_per_delay =
700/1400); pure float64 numpy on the submitted arrays; re-runs are bitwise
identical. Grader wall time ~90 s.