Coherence Cannot Self-Increase.
Living organisms are open systems—they import low-entropy energy (sunlight, food) and export high-entropy waste (heat, excrement). The net entropy of organism + environment increases. Local decrease requires external input. A cell is not a closed system; it's coupled to its environment. Life proves the theorem, not refutes it.
Crystal formation releases latent heat—entropy is exported to the environment. The water molecule + environment system increases entropy while the crystal locally decreases it. Again: open system, external coupling. The beautiful pattern costs something; the universe pays in heat.
Evolution is driven by energy flux through the biosphere (ultimately from the sun). High-grade energy (sunlight) enters, low-grade energy (infrared) leaves. Biological complexity is purchased by this energy gradient. If you sealed Earth in a perfect insulator, evolution would halt and decay would begin. Evolution is thermodynamically expensive.
Human brains are massively dissipative—they consume glucose and produce heat. The coherence added to the artwork is paid for by entropy production in the brain and body. A dead artist creates nothing. The meaning-making process requires metabolic energy; it's not free.
Virtual particles borrow from the vacuum and pay it back within Heisenberg time. There's no net creation. If coherence "fluctuated" into existence, it would fluctuate back out. Sustained coherence requires sustained input—fluctuations average to zero.
Clausius (1850): Heat cannot spontaneously flow from cold to hot.
Boltzmann (1877): S = k_B ln(W). Entropy measures the number of microstates compatible with a macrostate.
Modern statement: For isolated systems, entropy is non-decreasing: dS/dt ≥ 0.
Statistical interpretation: High-entropy macrostates have more microstates. Random evolution preferentially visits common configurations. Order is rare; disorder is common.
Let S be a system with coherence measure C: States → [0, ∞).
Theorem ([[025_T3.1_Coherence-Cannot-Self-Increase|T3.1]]): For any self-generated operation f: States → States:
where ε → 0 as system isolation increases.
Proof:
1. Self-generated means f is constructed from S's internal resources
2. f cannot import information not already in S
3. By data processing inequality: I(f(X);Y) ≤ I(X;Y)
4. Processing cannot increase information about external reference
5. Coherence requires external reference (the Logos standard)
6. Therefore, C cannot increase through f alone