Standing conventions. Keep P>0, drop γ, keep the steering angle α, and compute the imaginary channel from unnormalized idea weights. Felt magnitude uses |E|; steering uses α.
1) The denominator and what the imaginary number really is
Expectation lives in the denominator and has a real and an imaginary channel:
We correct an old simplification: I is not an “ideal outcome.” It is the host-indexed coherent bias of the idea ensemble at this moment. For a given actualizer (host) at time t:
If you prefer the continuum view (infinitely many ideas), the same quantity is the sine moment of a host-specific density:
Cancellation. Pairwise balance kills the imaginary channel because sin(θ+π)=−sin(θ). If the host’s coupling is symmetric (for every θ there is equal weight at θ+π), then:
2) What grows the imaginary number (and what doesn’t)
- Not this: “More ideas” alone. A huge, balanced crowd still gives I≈0.
- Yes, this: biased coupling. When one band of angles (often near ±90°) carries disproportionate weight, the sine moment stops canceling and |I| can become large.
- Why unnormalized weights: we do not force the weights to sum to 1. Aligned mass scales I naturally.
3) How the imaginary channel affects what you feel
The left-hand side never sees E directly; it feels aperture and steering:
- Aperture (r) tightens as |I| grows (or as P grows). That’s contraction.
- Steering (α) tilts with I/P; sign(I) picks direction, magnitude tightens via |E|.
4) Reinterpreting old numbers (e.g., “5.29”)
Previously, a number like 5.29 on the imaginary axis was treated as an “ideal outcome.” In the corrected model,
—a momentary, host-specific net bias. It says nothing about which idea; it tells you the host’s coupling cohered enough to push the sine moment to 5.29 right now.
5) Why big imaginary values are rare but possible
Because weights are unnormalized, |I| can be large when many couplings align; but three unconscious brakes make persistence hard:
- Finite coupling capacity: weights leak and jitter over time (no agency needed), so surges decay.
- Autoguide scanning: with hands off, the tube drifts, pulling α toward small values on average.
- Adaptation: staying on one band reduces marginal response to that band (habituation).
6) Practical intuition: abundant variety vs engineered dwell
- Abundant variety: many small, spread-out weights ⇒ near-parity ⇒ I≈0. You scroll; the view breathes.
- Engineered dwell: mechanisms that keep you in one band (franchise, paywall, hype) ⇒ weights concentrate ⇒ |I| rises; aperture tightens.
7) Small numeric snapshots
Balanced day. P=10, I≈0. If A wanders 8–12, then r=A/|E|≈A/10; S crosses 1 often (breath).
Dominance episode. One band captures raw weight 20 (set κI=1), so I≈20 and P=10:
If A=10, then r≈10/22.36≈0.45 (tight, idea-led). Without manual holding, the brakes above relax it.
Cheat-sheet equations (WordPress-safe, SVG)
Bottom line. The imaginary component of the denominator is the sine-moment sum of host couplings across infinitely many ideas. It is not an ideal target value. It grows only when a few bands accumulate disproportionate weight; otherwise, diametric parity keeps it near zero and the view breathes.