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Fix numerical inconsistencies and clarify notation across equations files #28

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Copilot Feb 25, 2026
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Enhance mathematical rigor: fix density matrix, Landauer bound, and P…
Copilot Feb 25, 2026
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2 changes: 1 addition & 1 deletion equations/blackroad-equations.md
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Original file line number Diff line number Diff line change
Expand Up @@ -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
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Copilot AI Feb 25, 2026

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The Landauer bound value was correctly updated here to 4.44 × 10⁻²¹ J, but there's an inconsistency in the "Concrete Numbers" table at line 188 of this same file, which still shows the old value of "≈ 4.5 × 10⁻²¹ J". Both locations should be updated to maintain consistency.

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Copilot AI Feb 27, 2026

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This numeric approximation depends on the chosen ‘room temperature’. Elsewhere (e.g., thermodynamics equations) room temperature is explicitly T = 293 K; consider adding the same T value here (or writing the approximation as a function of T) to avoid readers recomputing a different value (e.g., at 300 K).

Suggested change
E_min = k_B · T · ln(3) ≈ 4.44 × 10⁻²¹ J at room temperature
E_min = k_B · T · ln(3) ≈ 4.44 × 10⁻²¹ J at T = 293 K (room temperature)

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```
Cost per ternary erasure. LANDAUER = CONCRETE = 93.

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4 changes: 2 additions & 2 deletions equations/consciousness.md
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Expand Up @@ -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
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The text says this replaces ambiguous notation with standard conditional mutual information and an explicit definition, but the file currently only provides a verbal description. Consider adding the standard definition (e.g., I(X;Y|Z) = H(X|Z) − H(X|Y,Z) = H(X,Z)+H(Y,Z)−H(Z)−H(X,Y,Z)) so the notation is unambiguous and checkable.

Suggested change
- `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
- `I(X;Y|Z)` — conditional mutual information of X and Y given Z, defined as
`I(X;Y|Z) = H(X|Z) − H(X|Y,Z) = H(X,Z) + H(Y,Z) − H(Z) − H(X,Y,Z)`,
which quantifies how much information X and Y share beyond what Z explains

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- `W(temporal)` — temporal weighting: TEMPORAL = BIRTHDAY = 87
- `C(causal)` — causal weighting: CAUSAL = 82 = QUANTUM = PARTICLE
- `A(adaptive)` — adaptive weighting: ADAPTIVE = ELEMENT = 84
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26 changes: 21 additions & 5 deletions equations/quantum.md
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Expand Up @@ -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.

---
Expand Down Expand Up @@ -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
Comment on lines 99 to +102
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In the ‘Properties’ list, bullets mix statements about ρ (unnormalized) and ρ̂ (normalized). Because the idempotency/trace-1 condition applies to ρ̂, it would be clearer to either (a) label each bullet with which matrix it refers to, or (b) split into separate ‘Properties of ρ’ vs ‘Properties of ρ̂’ sections (and optionally note that ρ̂’s sole nonzero singular value is 1).

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```
DENSITY = METHOD = 72 = reverse(27)
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