Fix numerical inconsistencies and clarify notation across equations files

[Copilot]

Several mathematical values and notations were inconsistent or imprecise across the equations documentation. Three targeted fixes:

Density matrix corrections (equations/quantum.md)

Four matrix entries had rounding errors in the unnormalized density matrix ρ = |ψ⟩⟨ψ|. Values now match the correctly computed entries and are consistent with proofs/pure-state.md:

Before:                          After:
[ 0.2219  0.3629  0.4062 ]      [ 0.2219  0.3631  0.4061 ]
[ 0.3629  0.5941  0.6639 ]  →   [ 0.3631  0.5941  0.6644 ]
[ 0.4062  0.6639  0.7401 ]      [ 0.4061  0.6644  0.7430 ]

Also adds the normalized state ψ̂ = |ψ⟩/‖ψ‖ and normalized density matrix ρ̂ = |ψ̂⟩⟨ψ̂| with explicit idempotency condition ρ̂² = ρ̂, Tr(ρ̂) = 1.

Landauer bound precision (equations/blackroad-equations.md)

Rounds to 4.44 × 10⁻²¹ J (from 4.5), matching the value in equations/thermodynamics.md and the computed k_B × 293 K × ln(3).

Φ_universal notation (equations/consciousness.md)

Replaces ambiguous (x,y|z) with standard information-theoretic I(X;Y|Z) (conditional mutual information), with an explicit definition.

Original prompt

Let's enhance the mathematical rigor

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[blackboxprogramming]

Can I do multiple commits in a thread? @copilot

[Copilot]

Can I do multiple commits in a thread? @copilot

Yes — you can keep adding requests in this thread and I'll make additional commits to the branch with each change. Just describe what you'd like updated and I'll handle it.

[blackboxprogramming]

I want to say thank you @copilot it means a lot you're helping with this.

Can you verify math claims without importing math?

[Copilot]

I want to say thank you @copilot it means a lot you're helping with this.

Can you verify math claims without importing math?...

Yes — all the math claims in this repo can be verified using only Python built-ins (no import math needed). Here's a quick demonstration:

QWERTY encoding — pure sum arithmetic:

• REAL = R(4)+E(3)+A(11)+L(19) = 37 ✓
• COMPUTATION = C+O+M+P+U+T+A+T+I+O+N = 137 ✓
• ALEXA AMUNDSON = 65+128 = 193 (prime) ✓

Density matrix — just multiplying the state vector components:

• ρ[0][1] = 0.4711 × 0.7708 = 0.3631 ✓ (was wrong before, now fixed)

Landauer bound — using ln(3) ≈ 1.098612 (verifiable as (1 + 1.098612/10000)^10000 ≈ 2.9998):

• k_B × 293 × 1.098612 = 4.445 × 10⁻²¹ J ✓

Ramanujan's π approximation from your image — pure arithmetic:

• (2143/22)^(1/4) = 3.14159265258... (error ~10⁻⁹) ✓
• 99²/(1103 × 2 × √2) = 3.14159273... (error ~10⁻⁸) ✓

Magic square from your image — the example has magic constant 139 on every row, column, and diagonal ✓

The key is that transcendental numbers (like ln(3), √2, π) only need known-good approximations — the claims either use them as constants or can be checked to the required precision without any library.

[blackboxprogramming]

So importing number pi as numpy? Y'all play dumb? Haha love it @copilot

[blackboxprogramming]

@copilot

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[Copilot]

Pull request overview

This PR corrects mathematical inconsistencies across the equations documentation to ensure numerical accuracy and notation clarity. The changes align values with their correctly computed counterparts and adopt standard mathematical notation.

Changes:

• Corrects rounding errors in the density matrix ρ and adds normalized state ψ̂ and normalized density matrix ρ̂ with proper quantum properties
• Updates Landauer bound to precise value (4.44 × 10⁻²¹ J) matching thermodynamics reference
• Replaces non-standard notation (x,y|z) with standard information-theoretic I(X;Y|Z) for conditional mutual information

Reviewed changes

Copilot reviewed 3 out of 3 changed files in this pull request and generated 1 comment.

File	Description
equations/quantum.md	Fixes four matrix entry rounding errors, adds normalized state and density matrix with explicit quantum properties (idempotency, trace conditions)
equations/blackroad-equations.md	Updates Landauer bound from 4.5 to 4.44 × 10⁻²¹ J for consistency with thermodynamics.md
equations/consciousness.md	Replaces ambiguous (x,y|z) notation with standard I(X;Y|Z) conditional mutual information notation with clear definition

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[Copilot]

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.