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Co-authored-by: blackboxprogramming <118287761+blackboxprogramming@users.noreply.github.com>
77 lines
2.2 KiB
Markdown
77 lines
2.2 KiB
Markdown
# Proof: Ternary is More Efficient Than Binary
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> From page 19 (§173): η_ternary = ln(3)/3 > η_binary = ln(2)/2
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## Statement
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Among all integer radices r ≥ 2, radix 3 (ternary) maximizes the **radix economy**: information per digit.
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## The Radix Economy Function
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Define the efficiency of radix r as:
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```
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η(r) = ln(r) / r
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```
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This measures: information content per digit (ln(r) bits) divided by number of symbols needed (r states).
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## Proof
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Maximize η(r) = ln(r)/r over continuous r > 1.
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```
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dη/dr = (1/r · r − ln(r)) / r²
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= (1 − ln(r)) / r²
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```
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Setting dη/dr = 0:
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```
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1 − ln(r) = 0
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ln(r) = 1
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r = e ≈ 2.71828...
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```
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The maximum is at r = e (Euler's number). Since e is irrational, no integer radix achieves it. Among integers:
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```
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η(2) = ln(2)/2 ≈ 0.3466
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η(3) = ln(3)/3 ≈ 0.3662 ← maximum among integers
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η(4) = ln(4)/4 ≈ 0.3466 (= η(2), since 4 = 2²)
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η(5) = ln(5)/5 ≈ 0.3219
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```
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**3 is the integer closest to e, so ternary is the most efficient integer radix. □**
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## QWERTY
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```
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RADIX = GAUSS = TANH = 57 (the optimal base = the Gaussian)
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EFFICIENCY = 5³ = 2000/16 = 125 (efficiency = 5³ = birthday ÷ Dürer)
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BALANCED = BRAINSTORM = 2⁷ = 128 (balanced ternary = the brainstorm)
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```
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RADIX = GAUSS. She knew the optimal radix IS the Gaussian before she computed the proof.
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## Practical Numbers
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At room temperature (T ≈ 293 K):
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```
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E_min(binary) = k_B T ln(2) ≈ 2.80 × 10⁻²¹ J
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E_min(ternary) = k_B T ln(3) ≈ 4.44 × 10⁻²¹ J
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```
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Ternary costs more per operation but carries more information.
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The energy ratio equals the information ratio exactly:
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```
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E_min(ternary) / E_min(binary) = ln(3) / ln(2) ≈ 1.585
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```
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Ratio: ln(3)/ln(2) ≈ 1.585. Every ternary trit ≈ 1.585 binary bits.
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Energy cost: 4.44 / 2.80 = ln(3)/ln(2) ≈ 1.585 times binary.
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Information per unit energy: 1.585 / 1.585 = **1.000 exactly.**
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At the Landauer limit, ternary and binary achieve identical information per joule — both equal 1/(k_B T ln(2)) bits per joule. The advantage of ternary is **radix economy** (fewer symbols needed to represent a number), not thermodynamic energy-per-bit efficiency.
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Small advantage in representation, but it scales. At 10¹⁴ DNA ops/sec (§175), it accumulates.
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