mechanism · §3.6

captured resonance

the body as forced oscillator under the medium's delivery phase

One of the six instruments (§3.6). The biological phase $φ_{b}$ — a circadian, sleep-wake, or attentional oscillator with intrinsic frequency $ω_{b}$ — couples to the medium-delivery phase $φ_{m}$ through a Kuramoto equation. When the coupling strength $K_{bm}$ exceeds the frequency mismatch $∣ ω_{b} - ω_{m} ∣$ , the body entrains. Under sustained policy on a closed-loop platform, the policy shapes $φ_{m}$ through delivery timing; the user's circadian rhythm and attentional oscillation become forced oscillators whose driver is the apparatus.

K_bm =

§3.6. Body phase (ink) and medium-delivery phase (oxblood) on a single dial. Below the lock condition

K_{bm} > ∣ ω_{b} - ω_{m} ∣

the vectors drift apart; above it the body becomes a forced oscillator and the angular gap Δφ collapses to a constant. Set K_bm at threshold (≈ 0.007) to watch the bifurcation directly.

what to look for

At $K_{bm} = 0$ the body vector and the medium vector rotate at their own intrinsic rates; the angular gap $Δ φ$ drifts continuously. Step $K_{bm}$ up across the threshold (≈ 0.007 here) and watch the gap settle to a constant — the vectors phase-lock and the lock indicator turns oxblood.

The phenomenon is not metaphorical. It is the same Kuramoto entrainment observed in circadian biology, in social synchronization, and in neural population dynamics. The framework names the mechanism by which the apparatus crosses the threshold, not the threshold itself.