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 literal: 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; the threshold itself lies outside it.