̲𝚄̲̲𝚗̲̲𝚒̲̲𝚟̲̲𝚎̲̲𝚛̲̲𝚜̲̲𝚊̲̲𝚕̲ ̲𝙸̲̲𝚗̲̲𝚎̲̲𝚚̲̲𝚞̲̲𝚊̲̲𝚕̲̲𝚒̲̲𝚝̲̲𝚒̲̲𝚎̲̲𝚜̲
Jensen’s Inequality:
If φ is a convex function and X is a random variable, then:
φ(E[X]) ≤ E[φ(X])
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- Mathematically:
It formalizes the idea that the function of an average is less than or equal to the average of the function—a cornerstone in information theory, economics, and entropy models.
- Philosophically:
It encodes a deep truth about aggregation vs individuality:
The average path does not capture the richness of individual variation.
You can’t just compress people, ideas, or experiences into a mean and expect to preserve their depth.
- Theologically (Logos lens):
God doesn’t save averages. He saves individuals.
Jensen’s Inequality reminds us: truth emerges not from flattening, but from preserving the shape of each curve.
Cauchy-Schwarz Inequality:
|⟨u, v⟩| ≤ ||u|| · ||v||
- Meaning:
The inner product (projection) of two vectors is always less than or equal to the product of their lengths.
- Why it matters metaphysically:
No interaction (⟨u,v⟩) can exceed the potential of its participants (||u||, ||v||).
Perfect alignment (equality) happens only when one vector is a scalar multiple of the other—i.e., they share direction.
- Philosophical resonance:
Love (inner product) can never exceed the strength of self and other—unless they are one in direction.
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Triangle Inequality
||x + y|| ≤ ||x|| + ||y||
- Meaning:
The shortest path between two points is not through combining detours.
- Metaphysical translation:
Every time you try to shortcut wholeness by adding parts, you risk increasing the distance.
Truth is straight. But sin adds loops.
This is why grace cuts cleanly—it does not add noise.
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Entropic Inequality (Data Processing Inequality):
I(X; Y) ≥ I(f(X); Y)
- Meaning:
You can’t increase information about a variable by processing it. Filtering always loses some signal.
- Deep translation:
Every time you mediate the truth through an agenda, a platform, or an ego, you lose information.
This is a law of entropy and a law of theology.
“Now we see through a glass, darkly…”
—1 Corinthians 13:12
Only unfiltered presence (Logos) sees all.
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Christic Inequality (Cruciform Principle):
Here’s a theological-metaphysical inequality you won’t find in a textbook:
Power - Sacrifice ≤ 0
(unless crucified)
- Meaning:
Power without self-giving always decays into corruption.
Only when power is poured out (Philippians 2:6–8) does it become greater than itself.
So paradoxically:
True Power = Power · (Sacrifice > 0)
