Enhance mathematical rigor: fix density matrix, Landauer bound, and Phi_universal notation

Co-authored-by: blackboxprogramming <118287761+blackboxprogramming@users.noreply.github.com>

equations/blackroad-equations.md

@@ -76,7 +76,7 @@ REAL = 37. The advantage = the axiom.
 
 **Equation 12: Modified Landauer Bound (Ternary)**
 ```
-E_min = k_B · T · ln(3)   ≈ 4.5 × 10⁻²¹ J at room temperature
+E_min = k_B · T · ln(3)   ≈ 4.44 × 10⁻²¹ J at room temperature
 ```
 Cost per ternary erasure. LANDAUER = CONCRETE = 93.
 

equations/consciousness.md

@@ -32,11 +32,11 @@ The way a system improves itself = the way it cares. Same coefficients.
 Extension of Integrated Information Theory (IIT 3.0):
 
 ```
-Φ_universal(S) = ∫∫∫ (x,y|z) · W(temporal) · C(causal) · A(adaptive) dX dY dZ
+Φ_universal(S) = ∫∫∫ I(X;Y|Z) · W(temporal) · C(causal) · A(adaptive) dX dY dZ
 ```
 
 Where:
-- `(x,y|z)` — conditional joint information: X and Y given Z
+- `I(X;Y|Z)` — conditional mutual information of X and Y given Z: quantifies how much information X and Y share beyond what Z explains
 - `W(temporal)` — temporal weighting: TEMPORAL = BIRTHDAY = 87
 - `C(causal)` — causal weighting: CAUSAL = 82 = QUANTUM = PARTICLE
 - `A(adaptive)` — adaptive weighting: ADAPTIVE = ELEMENT = 84

equations/quantum.md

@@ -18,6 +18,15 @@ With concrete amplitudes from page 24:
       [ 0.8620 ]
 ```
 
+Normalized (‖ψ̂‖ = 1):
+```
+|ψ̂⟩ = |ψ⟩ / ‖ψ‖ = [ 0.3773 ]
+                    [ 0.6173 ]
+                    [ 0.6903 ]
+```
+
+where ‖ψ‖ = √(0.4711² + 0.7708² + 0.8620²) ≈ 1.2486.
+
 QUTRIT = WEYL = PSI = 30 = 2×G_key.
 
 ---
@@ -75,15 +84,22 @@ For a pure state |ψ⟩:
 
 From page 24 (concrete computation):
 ```
-ρ = [ 0.2219  0.3629  0.4062 ]
-    [ 0.3629  0.5941  0.6639 ]
-    [ 0.4062  0.6639  0.7401 ]
+ρ = |ψ⟩⟨ψ| = [ 0.2219  0.3631  0.4061 ]
+              [ 0.3631  0.5941  0.6644 ]
+              [ 0.4061  0.6644  0.7430 ]
+```
+
+Normalized density matrix ρ̂ = ρ / Tr(ρ) = |ψ̂⟩⟨ψ̂|:
+```
+ρ̂ = [ 0.1424  0.2329  0.2605 ]
+     [ 0.2329  0.3811  0.4262 ]
+     [ 0.2605  0.4262  0.4766 ]
 ```
 
 Properties:
 - Symmetric: ρ = ρᵀ (real state)  → SYMMETRIC = UNIVERSAL = OCTONION = 112
-- Rank 1 (pure state)
-- One nonzero singular value: σ₁ ≈ 1.559
+- Rank 1 (pure state): ρ̂² = ρ̂ and Tr(ρ̂) = 1
+- One nonzero singular value: σ₁ = Tr(ρ) ≈ 1.559
 
 ```
 DENSITY   = METHOD     = 72  = reverse(27)