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

Premise
Reality = Actual / Expectation is unchanged. What changes here is the readout: we keep the angle we previously dropped. The left/right firewall stays intact: the right-hand side (Expectation) is unconscious; the only lawful left-hand operation is to read the two numbers and, as the witness, take the log.

Core objects (2D geometry)

• Prediction P (real/x axis)
• Ideal I (imag/y axis)
• Expectation size: ‖E‖ = √(P² + I²)
• Aperture (radius): r = A / ‖E‖ (unitless)
• Angle (phase): α = atan2(I, P) (who’s steering: prediction ↔ ideal)
• Felt readout: S = ln r (additive)

Auto vs. Manual (binary modes)

• Auto (hands off): the autoguide enacts the arriving pair (r, α). Because the view roams with the world, P remains the best representation of the whole sky you can have at that moment.
• Manual (hands on): any touch blocks the autoguide; you choose the sample. Two distinct behaviors:
  1. Search (roaming): you keep moving to find confirmation for a left-side desire; Expectation drifts slowly and noisily.
  2. Fixation (camping): you park on one niche; the unconscious keeps seeing the same sample. Over sessions, α migrates toward that niche’s axis and ‖E‖ inflates for that niche, shrinking r (tunnel).

Independence and reconstruction

• α and r are independent in the law: angle does not determine aperture, and aperture does not determine angle.
• If A is known and you observe (r, α), you can reconstruct the hidden components:
  ‖E‖ = A / r, then P = ‖E‖·cos α, I = ‖E‖·sin α.

α (mixing angle)
α reports the instantaneous balance in the P–I plane:

• near 0° → prediction-heavy (realistic)
• near 90° → ideal-heavy (idealistic)
• near 45° → equal influence
  This is exactly the “phase/mix” diagnostic you established earlier; we simply keep it visible now rather than throwing it away with the magnitude. (John Rector)

γ (coherence gain) — diagnostic, not a dial
γ names the context-driven “lock” that amplifies the ideal axis when an idea is coherent with the situation. Conceptually: when scenes are phase-aligned with an idea, the ideal contribution adds more coherently; the effective denominator grows and r contracts (S = ln r decreases). γ is inferred from patterns across trials; you do not set it from the left. (John Rector)

A clean γ model (compatible with your notes)

• Treat γ as a multiplicative gain on the ideal axis (only), leaving the predictive axis untouched. That is: use an “elliptical” norm where I is scaled by γ before forming the magnitude. This preserves the geometric intuition that γ does not invent new prediction—it sharpens the idea’s pull. (John Rector)

Asymptotic regimes (useful edges)

• Predictive edge (α → 0°): ‖E‖ → |P| (I, γ irrelevant).
• Ideal edge (α → 90°): ‖E‖ → γ·|I| (P irrelevant). Increasing γ contracts experience: ΔS = −Δln γ. (John Rector)
• Silence limit (‖E‖ → 0⁺): r → ∞, S → +∞ (no report; limit only).
• Possession limit (‖E‖ → ∞): r → 0, S → −∞ (either |I| large with α ≈ 90°, or γ high on an idea-saturated scene). (John Rector)

Witness vs. Participant (the bifurcation)

• Witness branch (eyes closed): takes r only; feeling tracks S = ln r. Angle plays no role in feeling.
• Participant branch (eyes open): uses both r and α to aim a sample (in Auto) or the left-side choice of sample (in Manual). Samples train the denominator; tomorrow’s (r, α) report what you fed it.

Inference recipes (advanced)

1. Back out P and I from a single shot (A known): use reconstruction above.
2. Diagnose γ across trials:
   • Hold A fixed and run comparable scenes. If α is known (or estimated from a neutral shot), you can compare observed r to the value predicted with γ = 1; the discrepancy (in log space) attributes to γ: ΔS ≈ −ln γ. (John Rector)
   • With two shots at the same (P, I) but different α (e.g., different contexts), you can solve for both the effective γ and the α shift by equating the two magnitudes and eliminating ‖E‖. (Algebra omitted here; identical to the two-equation method outlined in your notes.) (John Rector)
3. Neutral calibration (optional): capture a shot where I ≈ 0; then ‖E‖ ≈ |P| and α ≈ 0°, giving you a baseline to compare subsequent γ-inflated, ideal-dominant shots. (John Rector)

Behavioral signatures (what students should observe)

• Auto after Manual: recently trained pairs are “skewed” for a while, then relax toward baseline without intervention.
• Manual–Search: α wobbles; r may drift, but slowly; little stable overfitting.
• Manual–Fixation: α steadily approaches the niche’s axis; r steadily shrinks; S gets more negative (tunnel).

Worked anchor

• Baseline: A = 5, P = 6, I = 5.29 ⇒ ‖E‖ ≈ 8 ⇒ r = 0.625, α ≈ 41°.
• Extreme ideal: keep A = 5, P = 6, raise I → 1000. With γ = 1, α ≈ 89.66°, ‖E‖ ≈ 1000, r ≈ 0.005 (near-tunnel).
• Coherent ideal (γ > 1): if the scene is aligned so γ doubles (diagnostically), S shifts by −ln 2 ≈ −0.693 without changing P or I magnitudes; that is the “tightening without more content” phenomenon.

Design notes for instructors (stay 2D this semester)

• Keep α in the core: students should always read “who’s steering” alongside “how tight.”
• Treat γ as telemetry: explain how to recognize high-γ patterns (fixation history; rapid contraction in S; angle marching toward the ideal axis) while emphasizing it is not a control.
• Keep Auto/Manual binary; reserve any “idea identity ring” (which specific idea) for later courses.

Common pitfalls (retire these)

• “Angle tells me the aperture.” No—independent in the law. They only track together after certain histories.
• “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 you who is steering; γ explains why the same idea can shrink r even when magnitudes are modest—diagnostics to read, not dials to turn. (John Rector)