simulation-theory/proofs/ternary-efficiency.md

Proof: Ternary is More Efficient Than Binary

From page 19 (§173): η_ternary = ln(3)/3 > η_binary = ln(2)/2

Statement

Among all integer radices r ≥ 2, radix 3 (ternary) maximizes the radix economy: information per digit.

The Radix Economy Function

Define the efficiency of radix r as:

η(r) = ln(r) / r

This measures: information content per digit (ln(r) bits) divided by number of symbols needed (r states).

Proof

Maximize η(r) = ln(r)/r over continuous r > 1.

dη/dr = (1/r · r − ln(r)) / r²
       = (1 − ln(r)) / r²

Setting dη/dr = 0:

1 − ln(r) = 0
ln(r) = 1
r = e ≈ 2.71828...

The maximum is at r = e (Euler's number). Since e is irrational, no integer radix achieves it. Among integers:

η(2) = ln(2)/2 ≈ 0.3466
η(3) = ln(3)/3 ≈ 0.3662   ← maximum among integers
η(4) = ln(4)/4 ≈ 0.3466   (= η(2), since 4 = 2²)
η(5) = ln(5)/5 ≈ 0.3219

3 is the integer closest to e, so ternary is the most efficient integer radix. □

QWERTY

RADIX    = GAUSS = TANH = 57   (the optimal base = the Gaussian)
EFFICIENCY = 5³  = 2000/16 = 125   (efficiency = 5³ = birthday ÷ Dürer)
BALANCED = BRAINSTORM = 2⁷  = 128   (balanced ternary = the brainstorm)

RADIX = GAUSS. She knew the optimal radix IS the Gaussian before she computed the proof.

Practical Numbers

At room temperature (T ≈ 293 K):

E_min(binary)  = k_B T ln(2) ≈ 2.80 × 10⁻²¹ J
E_min(ternary) = k_B T ln(3) ≈ 4.44 × 10⁻²¹ J

Ternary costs more per operation but carries more information.
The energy ratio equals the information ratio exactly:

E_min(ternary) / E_min(binary) = ln(3) / ln(2) ≈ 1.585

Ratio: ln(3)/ln(2) ≈ 1.585. Every ternary trit ≈ 1.585 binary bits.
Energy cost: 4.44 / 2.80 = ln(3)/ln(2) ≈ 1.585 times binary.
Information per unit energy: 1.585 / 1.585 = 1.000 exactly.

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.

Small advantage in representation, but it scales. At 10¹⁴ DNA ops/sec (§175), it accumulates.