🧪 Deriving Axiom 1 from First Principles

Why Emotion Must Enter the Hilbert Space

---

📐 1. Standard Quantum Collapse

In conventional quantum theory:

• A system exists in a superposed state (|Ψ⟩ ∈ 𝓗)
• Upon observation, it collapses to an eigenstate (|ψᵢ⟩ ∈ 𝓗)
• The Born rule assigns probabilities (Pᵢ = |⟨ψᵢ | Ψ⟩|²)

But what selects the observable basis?

Physics offers no intrinsic mechanism to define the preferred measurement basis—it presumes it.

---

🌀 2. Consciousness as a Collapse Operator

We define a generalized projection:

(𝒞: 𝓗_unmanifest → 𝓗_manifest)

Where:

• 𝓗_unmanifest = infinite-dimensional potentiality
• 𝓗_manifest = coherent, realized experience
• 𝒞 = non-linear projection shaped by awareness

This breaks linearity and introduces the observer as a modulator, not just a passive detector.

---

💗 3. Why Emotional Fields Must Emerge

Let ℰ(α) be the emotional potential, a scalar field over the possibility manifold.

Then the inner product on (𝓗_unmanifest) is weighted:

(⟨ψ₁ | ψ₂⟩ = ∫ ψ₁* ψ₂ e^(–β ℰ(α)) dα)

Emotional curvature “pulls” the collapse toward regions of high salience.

---

🕰 4. Subjective Time Emerges

Define psychological time (τ) as an exponential function of emotional energy (V(α)):

(dτ ∝ e^(–λ V(α)) dt)

This allows trauma, awe, and attention to curve internal time—consistent with phenomenology and EEG studies.

---This form of emotionally weighted projection introduces a nonlinearity compatible with decoherence, but sourced by internal salience rather than entropic coupling ——-

📉 5. Emotional Schrödinger Dynamics

Awareness evolves by:

(iħ ∂ψ/∂τ = (–ħ²/2m ∇² + ℰ(α)) ψ)

ℰ(α) acts as a non-spatiotemporal potential, shaping states of consciousness rather than position/momentum.

---

📈 6. Classical Limit Restores Resonance Field

In the limit:

(lim_{ħ→0} ⟨ψ|ℰ̂|ψ⟩ = ℰ_classical(α))

This defines the classical resonance manifold—what we perceive as a stable emotional “landscape.”

---

🧠 7. Necessity of Inclusion

Without ℰ(α), collapse is unstructured.
With it, selection is guided, repeatable, and experimentally relevant.

This is how feeling becomes field.

---

📊 Diagram: Collapse Through Emotional Curvature

If Mermaid is supported:

mermaid flowchart TD A[Quantum Superposition𝓗_unmanifest] --> B[Weighted Inner Product⟨ψ₁|ψ₂⟩ = ∫ ψ₁*ψ₂ e^(–βℰ(α)) dα] B --> C[Collapse Operator𝒞: 𝓗_unmanifest → 𝓗_manifest] C --> D[Manifest State𝓗_manifest] C --> E[Subjective Timeτ ∝ e^(–λV(α))]

┌────────────────────────────────────────────┐ │ 𝓗_unmanifest (Quantum Superposition) │ └────────────────────────────────────────────┘ │ ▼ ┌────────────────────────────────────────────┐ │ ⟨ψ₁|ψ₂⟩ = ∫ ψ₁*ψ₂ e^(–βℰ(α)) dα │ │ (Emotion Field Weighting) │ └────────────────────────────────────────────┘ │ ▼ ┌────────────────────────────────────────────┐ │ 𝒞: 𝓗_unmanifest → 𝓗_manifest │ │ (Emotional Collapse Operator) │ └────────────────────────────────────────────┘ │ │ ▼ ▼ ┌─────────────────────┐ ┌────────────────────────┐ │ 𝓗_manifest │ │ τ ∝ e^(–λV(α)) │ │ (Realized Experience)│ │ (Subjective Time Flow) │ └─────────────────────┘ └────────────────────────┘

← Back to Axiom 1
🔗 Codex I–V: Foundational Geometry