Complex Reality (2D) — Advanced Notes (α & γ, v2)

The law stays simple. What changes is what we keep in view. We retain the angle we previously dropped and read the pair.

Premise

Right-hand side (Expectation) is unconscious; the lawful readout on the left is the log of the aperture:

The firewall stands: you don’t touch A or $||E||$ , you only witness $S$ .

Core objects (2D geometry)

Prediction (real/x axis): $P$
Ideal (imag/y axis): $I$
Steering angle (who’s mixing): $alpha = atan2(I,P)$
Aperture (unitless): $r=A/||E||$ with felt readout $S=ln r$

γ (coherence gain) — diagnostic, not a dial

γ names a context-driven lock that amplifies the ideal axis when scenes phase-align with an idea. You don’t set γ; you infer it.

Elliptical norm (clean γ model)

Let γ act only on $I$ when forming the magnitude:

Two angles (keep roles distinct)

Steering $alpha=atan2(I,P)$ — mix diagnostic (γ-independent).
Metric $beta=atan2(gamma I,P)$ — how γ bites (depends on γ).

Telemetry identities (teach these)

Ideal edge ( $beta → 90°$ ): unit drop per log-gain, $ΔS≈-Δ ln γ$ . Predictive edge: negligible effect.

Asymptotic regimes

Predictive edge ( $alpha→0°$ ): $||E||→|P|$ (I, γ irrelevant).
Ideal edge ( $alpha→90°$ ): $||E||→γ|I|$ , so $ΔS=-Δ ln γ$ .
Silence ( $||E||→0+$ ): $r→∞, S→+∞$ (limit; no report).
Possession ( $||E||→∞$ ): $r→0, S→−∞$ (large |I| with $alpha≈90°$ or high γ).

Auto vs. Manual (binary modes)

Auto (hands off): autoguide enacts the arriving pair $(r, α)$ . At that moment, $P$ is your best whole-sky proxy.
Manual (hands on): you choose the sample.
- Search (roaming): $α$ wobbles; Expectation drifts slowly and noisily.
- Fixation (camping): across sessions $α$ marches toward the niche’s axis; $||E||$ inflates for that niche → $r$ shrinks (tunnel).

Independence and reconstruction

Lawful independence: angle does not determine aperture, and aperture does not determine angle.

Reconstruction (state assumptions):

Neutral (γ≈1): $||E||0 = A/r$ ; then $P=||E||0 cos α$ , $I=||E||0 sin α$ .
With coherence (γ known): $||E||0 = (A/r)/sqrt(cos^2 α + γ^2 sin^2 α)$ Then $P=||E||0 cos α, I=||E||0 sin α$ .

Inference recipes

1) Neutral calibration (baseline shot)

Capture $I≈0$ so $α≈0°$ ; then $r0 = A/|P|$ . For a later shot $(r,α)$ :

2) Two-angle method (no baseline)

With comparable $A$ and unchanged underlying mix across contexts, take shots $(r1,α1), (r2,α2)$ ; let $q=(r1/r2)^2$ . Eliminating $||E||0$ yields:

γ^2 = [cos^2 α2 − q cos^2 α1] / [q sin^2 α1 − sin^2 α2]

3) Quick read: ΔS from Δγ

Behavioral signatures

Auto after Manual: skewed pairs relax toward baseline without intervention.
Manual–Search: $α$ wobbles; $r$ drifts slowly; little stable overfit.
Manual–Fixation: $α$ aligns to the niche axis; $r$ contracts; $S$ trends negative (tunnel).

Worked anchor

Baseline: A = $5, $P=6$ , $I=5.29$ ⇒ $||E||0≈8$ , $r=0.625$ , $S≈-0.470$ , $α≈41°$ .
Extreme ideal (γ=1): keep A = $5, $P=6$ , raise $I→1000$ ⇒ $||E||≈1000$ , $r≈0.005$ , $S≈-5.30$ , $α≈89.66°$ .
Coherent ideal (γ doubles): $ΔS=-ln 2$ with $P,I$ fixed — “tightening without more content.”

Design notes (stay 2D this semester)

Keep $α$ in the spine: always read “who’s steering” alongside “how tight.”
Treat $γ$ as telemetry: recognize fixation history, rapid $S$ contraction, and angle marching toward the ideal axis.
Keep Auto/Manual binary; leave “which idea” identity rings for later courses.

Common pitfalls to retire

“Angle tells me the aperture.” No—independent in the law.
“Tighter means truer.” No—tighter means larger Expectation for that niche; truth is the whole sky.
“The eyepiece shows Actual.” No—the eyepiece shows a sample.

One-line takeaway

Two numbers cross $(r, α)$ . Witness feels $ln r$ . Participant samples via $(r, α)$ . Auto lets the world teach the denominator; Manual teaches it what you hold. $α$ tells who’s steering; $γ$ explains why the same idea can shrink $r$ even when magnitudes are modest—readouts to read, not knobs to turn.

Notation: $A$ actual (scalar, units possible), $P$ predicted (real), $I$ ideal (imag), $E$ expectation (complex), $r$ aperture, $S$ felt readout, $α$ steering angle, $β$ metric angle, $γ$ coherence gain.

Complex Reality (2D) — Advanced Notes (α & γ, v2)

Complex Reality (2D) — Advanced Notes (α & γ, v2)

Premise

Core objects (2D geometry)