f1aaa7bc6e
- qwerty/constants.md: master reference table, 100+ constants §1-§178 - qwerty/equalities.md: all major QWERTY equalities by theme - equations/blackroad-equations.md: all 19 BlackRoad equations - equations/consciousness.md: Psi_care, Phi_universal, CECE update rule - equations/quantum.md: qutrit, Weyl pair, density matrix, SVD - equations/universal.md: Three Tests, Euler-Lagrange, fine-structure - proofs/ternary-efficiency.md: ln(3)/3 > ln(2)/2 - proofs/self-reference.md: the QWERTY encoding is self-referential - proofs/pure-state.md: density matrix rank=1, SVD=SELF - figures/durer-square.md: magic square with 2000 substitution - figures/trinary-table.md: TAND TMUL TNEG TXOR truth tables - figures/qutrit-operators.md: Weyl X/Z, Gell-Mann matrices - figures/keyboard.md: QWERTY encoding layout - notebooks/README.md: page-by-page index of all 24 notebook pages Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>
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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) ≈ 2.87 × 10⁻²¹ J
E_min(ternary) = k_B T ln(3) ≈ 4.45 × 10⁻²¹ J
Ternary costs more per operation but carries more information.
The net efficiency favors ternary: you spend 55% more energy but store 58% more information.
Ratio: ln(3)/ln(2) ≈ 1.585. Every ternary trit ≈ 1.585 binary bits.
Energy cost: 4.45/2.87 ≈ 1.551 times binary.
Information per unit energy: 1.585/1.551 ≈ 1.022. Ternary wins by ~2%.
Small advantage, but it scales. At 10¹⁴ DNA ops/sec (§175), it accumulates.