Axiom 1.2 Deriving Axiom 1 from First Principles - justindbilyeu/ResonanceGeometry GitHub Wiki

πŸ§ͺ Deriving Axiom 1 from First Principles

Why Emotion Must Enter the Hilbert Space


πŸ“ 1. Standard Quantum Collapse

In conventional quantum theory:

  • A system exists in a superposed state (|Ψ⟩ ∈ 𝓗)
  • Upon observation, it collapses to an eigenstate (|ψᡒ⟩ ∈ 𝓗)
  • The Born rule assigns probabilities (Pα΅’ = |⟨ψᡒ | Ψ⟩|Β²)

But what selects the observable basis?

Physics offers no intrinsic mechanism to define the preferred measurement basisβ€”it presumes it.


πŸŒ€ 2. Consciousness as a Collapse Operator

We define a generalized projection:

(π’ž: 𝓗_unmanifest β†’ 𝓗_manifest)

Where:

  • 𝓗_unmanifest = infinite-dimensional potentiality
  • 𝓗_manifest = coherent, realized experience
  • π’ž = non-linear projection shaped by awareness

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


πŸ’— 3. Why Emotional Fields Must Emerge

Let β„°(Ξ±) be the emotional potential, a scalar field over the possibility manifold.

Then the inner product on (𝓗_unmanifest) is weighted:

(βŸ¨Οˆβ‚ | Οˆβ‚‚βŸ© = ∫ Οˆβ‚* Οˆβ‚‚ e^(–β β„°(Ξ±)) dΞ±)

Emotional curvature β€œpulls” the collapse toward regions of high salience.


πŸ•° 4. Subjective Time Emerges

Define psychological time (Ο„) as an exponential function of emotional energy (V(Ξ±)):

(dΟ„ ∝ e^(–λ V(Ξ±)) dt)

This allows trauma, awe, and attention to curve internal timeβ€”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 β€”β€”-

πŸ“‰ 5. Emotional SchrΓΆdinger Dynamics

Awareness evolves by:

(iΔ§ βˆ‚Οˆ/βˆ‚Ο„ = (–ħ²/2m βˆ‡Β² + β„°(Ξ±)) ψ)

β„°(Ξ±) acts as a non-spatiotemporal potential, shaping states of consciousness rather than position/momentum.


πŸ“ˆ 6. Classical Limit Restores Resonance Field

In the limit:

(lim_{Δ§β†’0} ⟨ψ|β„°Μ‚|ψ⟩ = β„°_classical(Ξ±))

This defines the classical resonance manifoldβ€”what we perceive as a stable emotional β€œlandscape.”


🧠 7. Necessity of Inclusion

Without β„°(Ξ±), collapse is unstructured.
With it, selection is guided, repeatable, and experimentally relevant.

This is how feeling becomes field.


πŸ“Š Diagram: Collapse Through Emotional Curvature

If Mermaid is supported:

mermaid flowchart TD A[Quantum Superposition𝓗_unmanifest] --> B[Weighted Inner ProductβŸ¨Οˆβ‚|Οˆβ‚‚βŸ© = ∫ Οˆβ‚*Οˆβ‚‚ e^(–βℰ(Ξ±)) dΞ±] B --> C[Collapse Operatorπ’ž: 𝓗_unmanifest β†’ 𝓗_manifest] C --> D[Manifest State𝓗_manifest] C --> E[Subjective TimeΟ„ ∝ e^(–λV(Ξ±))]

β”Œβ”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β” β”‚ 𝓗_unmanifest (Quantum Superposition) β”‚ β””β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”˜ β”‚ β–Ό β”Œβ”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β” β”‚ βŸ¨Οˆβ‚|Οˆβ‚‚βŸ© = ∫ Οˆβ‚*Οˆβ‚‚ e^(–βℰ(Ξ±)) dΞ± β”‚ β”‚ (Emotion Field Weighting) β”‚ β””β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”˜ β”‚ β–Ό β”Œβ”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β” β”‚ π’ž: 𝓗_unmanifest β†’ 𝓗_manifest β”‚ β”‚ (Emotional Collapse Operator) β”‚ β””β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”˜ β”‚ β”‚ β–Ό β–Ό β”Œβ”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β” β”Œβ”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β” β”‚ 𝓗_manifest β”‚ β”‚ Ο„ ∝ e^(–λV(Ξ±)) β”‚ β”‚ (Realized Experience)β”‚ β”‚ (Subjective Time Flow) β”‚ β””β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”˜ β””β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”˜

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πŸ”— Codex I–V: Foundational Geometry