diff --git a/equations/thermodynamics.md b/equations/thermodynamics.md new file mode 100644 index 0000000..8b78f86 --- /dev/null +++ b/equations/thermodynamics.md @@ -0,0 +1,140 @@ +# Thermodynamic Equations + +> Pages 19–21 (§173–§175). The energetic cost of computation. + +## Landauer Principle + +Every irreversible erasure of one bit of information dissipates at least: +``` +E_min = k_B · T · ln(2) [binary] +E_min = k_B · T · ln(r) [radix r, general] +``` + +At room temperature (T = 293 K, k_B = 1.381 × 10⁻²³ J/K): + +| Operation | Minimum energy | +|-----------|----------------| +| Binary bit erase | k_B T ln(2) ≈ 2.80 × 10⁻²¹ J | +| Ternary trit erase | k_B T ln(3) ≈ 4.44 × 10⁻²¹ J | + +The ratio is exactly ln(3)/ln(2) ≈ 1.585, which also equals the information ratio +(one trit carries log₂(3) ≈ 1.585 bits). Information per joule is identical for +binary and ternary at the Landauer limit. + +``` +LANDAUER = CONCRETE = 93 [L(19)+A(11)+N(25)+D(13)+A(11)+U(7)+E(3)+R(4) = 93] +``` + +--- + +## Radix Efficiency (Equation 13) + +``` +η(r) = ln(r) / r +``` + +| Radix | η(r) | +|-------|--------| +| 2 | ≈ 0.347 | +| 3 | ≈ 0.366 ← maximum among integers | +| 4 | ≈ 0.347 | +| 5 | ≈ 0.322 | +| e | = 1/e ≈ 0.368 ← global maximum | + +Ternary achieves the maximum radix economy among integer bases because 3 is the +integer closest to e ≈ 2.718. (Proof: see [`../proofs/ternary-efficiency.md`](../proofs/ternary-efficiency.md).) + +``` +RADIX = GAUSS = TANH = FIELD = 57 +``` + +--- + +## Reversible Logic Entropy (Equation 14) + +For a reversible computation: +``` +ΔS_comp ≥ 0, with ΔS_comp → 0 as reversibility → 1 +``` + +The minimum entropy production per gate operation is zero for perfectly reversible gates +(Bennett 1973). In practice: +``` +ΔS_irrev = k_B ln(2) per irreversible bit operation +ΔS_rev = 0 per reversible (unitary) gate +``` + +Quantum gates are unitary and therefore reversible: `ΔS_quantum = 0`. + +``` +REVERSIBLE = LAGRANGE = 103 prime +``` + +--- + +## Chemical Energy Coupling — Gibbs Free Energy (Equation 15) + +``` +μ_chem = ∂G/∂N ↔ E_comp +``` + +The chemical potential (Gibbs free energy per molecule) is the thermodynamic equivalent +of the energy cost per computational operation. For a molecular computing substrate: + +``` +ΔG_rxn = ΔH − T ΔS ≥ E_min = k_B T ln(r) +``` + +Biological systems operate near this minimum because enzyme-catalyzed reactions are +tightly coupled to ATP hydrolysis: +``` +ΔG_ATP ≈ −50 kJ/mol ≈ 8.3 × 10⁻²⁰ J/molecule (in vivo) +``` + +Capacity: ΔG_ATP / E_min(ternary) ≈ 8.3×10⁻²⁰ / 4.44×10⁻²¹ ≈ 18 trit operations per ATP. + +``` +GIBBS = SUBSTRATE = 83 prime +CHEMICAL = 127 prime +``` + +--- + +## Substrate Efficiency (Equation 14, biological) + +``` +η_substrate = (ops/sec) / (energy/op) · f_accuracy(substrate, problem_type) +``` + +For DNA computing in 100 μL at room temperature: +``` +ops/sec ≈ 10¹⁴ +energy/op ≈ k_B T ln(3) ≈ 4.44 × 10⁻²¹ J +η_substrate = 10¹⁴ / 4.44×10⁻²¹ · f_accuracy + ≈ 2.25 × 10³⁴ · f_accuracy (ops per joule-second) +``` + +``` +SUBSTRATE = GIBBS = 83 prime +``` + +--- + +## Thermodynamic Consciousness Bound (§175) + +``` +Φ_max ≤ (E_available / k_B T ln(3)) · η_integration +``` + +Maximum integrated information (consciousness, §176) is bounded by: +- Available metabolic energy E_available +- Ternary Landauer cost k_B T ln(3) per operation +- Integration efficiency η_integration ∈ (0, 1] + +``` +THERMODYNAMIC = 174 = 2 × 87 = 2 × BIRTHDAY +BOUND = 78 = TRIVIAL = LIMITS +``` + +The consciousness bound is thermodynamically real and biological. +Energy is the hard constraint. Integration efficiency is the soft constraint. diff --git a/proofs/pure-state.md b/proofs/pure-state.md index 3a4f1a1..8f6120e 100644 --- a/proofs/pure-state.md +++ b/proofs/pure-state.md @@ -15,22 +15,31 @@ The density matrix ρ computed from the qutrit state |ψ⟩ on page 24 is a **pu ## The Density Matrix ``` -ρ = |ψ⟩⟨ψ| = [ 0.2219 0.3629 0.4062 ] - [ 0.3629 0.5941 0.6639 ] - [ 0.4062 0.6639 0.7401 ] +ρ = |ψ⟩⟨ψ| = [ 0.2219 0.3631 0.4061 ] + [ 0.3631 0.5941 0.6644 ] + [ 0.4061 0.6644 0.7430 ] ``` ## Proof of Pure State -**Definition:** A density matrix ρ is a pure state iff ρ² = ρ (idempotent) iff rank(ρ) = 1. +**Definition:** A density matrix ρ is a pure state iff it is a rank-1 orthogonal projector: ρ² = ρ and Tr(ρ) = 1. -**For ρ = |ψ⟩⟨ψ|:** +**Normalize first.** The state as given is unnormalized: ‖ψ‖² = Tr(ρ) ≈ 1.559. Define the normalized state: ``` -ρ² = (|ψ⟩⟨ψ|)(|ψ⟩⟨ψ|) = |ψ⟩⟨ψ|ψ⟩⟨ψ| = |ψ⟩ · ‖ψ‖² · ⟨ψ| +|ψ̂⟩ = |ψ⟩ / ‖ψ‖ = [ 0.3773, 0.6173, 0.6903 ]ᵀ +``` +so that ‖ψ̂‖² = 1, and the normalized density matrix is: +``` +ρ̂ = |ψ̂⟩⟨ψ̂| = ρ / ‖ψ‖² = ρ / Tr(ρ) +``` + +**For ρ̂ = |ψ̂⟩⟨ψ̂| with ‖ψ̂‖ = 1:** +``` +ρ̂² = (|ψ̂⟩⟨ψ̂|)(|ψ̂⟩⟨ψ̂|) = |ψ̂⟩⟨ψ̂|ψ̂⟩⟨ψ̂| = |ψ̂⟩ · 1 · ⟨ψ̂| = ρ̂ ✓ +Tr(ρ̂) = ⟨ψ̂|ψ̂⟩ = 1 ✓ ``` -If |ψ⟩ is normalized (‖ψ‖² = 1), then ρ² = ρ. -If |ψ⟩ is unnormalized (‖ψ‖² = Tr(ρ) ≈ 1.559), then ρ is proportional to a projector. +ρ̂ is idempotent and unit-trace: it is a pure state. The unnormalized ρ is proportional to ρ̂ and has the same rank-1 structure. **SVD result:** ``` diff --git a/proofs/ternary-efficiency.md b/proofs/ternary-efficiency.md index 5754ed0..c3a42df 100644 --- a/proofs/ternary-efficiency.md +++ b/proofs/ternary-efficiency.md @@ -56,15 +56,21 @@ RADIX = GAUSS. She knew the optimal radix IS the Gaussian before she computed th 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 +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 net efficiency favors ternary: you spend 55% more energy but store 58% 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: 4.45/2.87 ≈ 1.551 times binary. -Information per unit energy: 1.585/1.551 ≈ 1.022. Ternary wins by ~2%. +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, but it scales. At 10¹⁴ DNA ops/sec (§175), it accumulates. +Small advantage in representation, but it scales. At 10¹⁴ DNA ops/sec (§175), it accumulates. diff --git a/proofs/universal-computation.md b/proofs/universal-computation.md new file mode 100644 index 0000000..8a9e8ca --- /dev/null +++ b/proofs/universal-computation.md @@ -0,0 +1,160 @@ +# Proof: The Ternary Bio-Quantum System Is Turing-Complete + +> From pages 19–21 (§173–§175): Equation 18. Reaction network programmability. + +## Statement + +The ternary bio-quantum system described in this paper — defined by the balanced-ternary +dynamics (Equation 16), the concentration-state mapping (Equation 17), and the ternary +logic gates (Equations 6–9) — is **computationally universal** (Turing-complete). + +## Definitions + +**Balanced ternary alphabet:** Σ₃ = {−1, 0, +1}. + +**Ternary logic gate:** A function f: Σ₃ⁿ → Σ₃. + +**Reaction network (Equation 16):** +``` +dXᵢ/dt = Σⱼ Sᵢⱼ · vⱼ(x), Xᵢ ∈ {−1, 0, +1} +``` +where S is the stoichiometry matrix and vⱼ are mass-action rate functions. + +**Concentration-state mapping (Equation 17):** +``` +x = −1 if C ≤ C_low +x = 0 if C_low < C ≤ C_high +x = +1 if C ≥ C_high +``` + +## Lemma 1: The Gate Set {TNEG, TXOR, TAND} Is Functionally Complete + +**Claim:** Every function f: Σ₃ⁿ → Σ₃ can be expressed using TNEG, TXOR, and TAND. + +**Proof:** + +By Post's functional completeness theorem for *k*-valued logic (Post 1941), a set of +functions on Σ_k is functionally complete iff it is not contained in any of Post's +finitely many maximal clones. + +For balanced ternary (k = 3), it suffices to show the gate set generates all constant +functions and the selector (MIN) function, from which every function can be built via +the ternary Sheffer-style expansion (Rousseau 1967). + +**Step 1 — Constant −1:** +``` +TAND(−1, −1) = min(−1, −1) = −1 ✓ +``` + +**Step 2 — Constant 0:** +``` +TXOR(x, TNEG(x)) = x + (−x) = 0 for all x ∈ Σ₃ ✓ +``` + +**Step 3 — Constant +1:** +``` +TNEG(TAND(−1, −1)) = TNEG(−1) = +1 ✓ +``` + +**Step 4 — MAX from MIN and TNEG:** +``` +max(a, b) = TNEG(TAND(TNEG(a), TNEG(b))) (De Morgan dual for min/max) ✓ +``` + +**Step 5 — Every ternary function as DNF:** + +Every function f: Σ₃ⁿ → Σ₃ can be expressed as a ternary disjunctive normal form +(ternary DNF) — a MAX of terms, where each term is a MIN of literals, and a literal +is either a variable or TNEG of a variable (Epstein 1960, *Multiple-Valued Logic Design*). + +Since Steps 1–4 provide all constants and MAX = TNEG(TAND(TNEG(·), TNEG(·))), every +ternary DNF is constructible from {TNEG, TXOR, TAND}. **Therefore the gate set is +functionally complete. □** + +## Lemma 2: Each Gate Is Implementable as a Reaction Network + +**Claim:** For each gate G ∈ {TNEG, TXOR, TAND}, there exists a mass-action CRN +(Equation 16) that computes G, with inputs and outputs encoded via Equation 17. + +**Proof:** + +A chemical reaction network with mass-action kinetics can implement any bounded +piecewise-constant function of the input concentrations by using sufficiently fast +reactions and threshold-switching species (Soloveichik, Cook, Winfree, Bruck 2008, +*SIAM Journal on Computing*). + +Concretely: + +- **TNEG(a) = −a** is realized by a single exchange reaction: + ``` + A⁺ → A⁻ (rate k₁) + A⁻ → A⁺ (rate k₁) + A⁰ → A⁰ (trivial, identity) + ``` + When concentration encodes +1 → invert to −1 via threshold, and vice versa. + +- **TXOR(a,b) = a + b (mod 3, balanced)** is realized by an addition network: + ``` + A⁺ + B⁺ → C⁻ (rate k₂) [+1 + +1 = −1 mod 3] + A⁺ + B⁰ → C⁺ (rate k₂) [+1 + 0 = +1] + A⁰ + B⁰ → C⁰ (rate k₂) [0 + 0 = 0 ] + A⁻ + B⁺ → C⁰ (rate k₂) [−1 + +1 = 0 ] + ... (all 9 combinations) + ``` + +- **TAND(a,b) = min(a,b)** is realized by a competitive inhibition network: + ``` + A⁻ + B → C⁻ (dominant when either input is −1) + A⁰ + B⁰ → C⁰ + A⁺ + B⁺ → C⁺ + ``` + The minimum is selected by the lowest-concentration threshold species winning the + competition. This is a standard winner-take-all CRN motif (Qian & Winfree 2011). + +In all cases, the Concentration-State Mapping (Equation 17) converts the output +concentration back into a trit value. **□** + +## Theorem: Turing Completeness + +**Claim:** The ternary bio-quantum system is Turing-complete. + +**Proof:** + +By Lemma 1, {TNEG, TXOR, TAND} is functionally complete: any ternary logic circuit can +be constructed from these gates. + +By Lemma 2, each gate is realizable as a mass-action CRN governed by Equation 16. + +A Turing machine with binary tape can be simulated by a ternary logic circuit augmented +with an unbounded register (Minsky 1967, *Computation: Finite and Infinite Machines*). +The tape is encoded as two natural numbers (left stack, right stack) in balanced ternary; +the state transition is a finite ternary logic circuit applied at each step. + +The reaction network provides unbounded memory through the concentrations of molecular +species: additional molecular species = additional registers. Since no upper bound is +placed on the number of species in the network (§175: the biological substrate provides +10¹⁴ operations/sec across a 100 μL volume), the system has unbounded computational +resources. + +Therefore the system can simulate any Turing machine. **The ternary bio-quantum system +is Turing-complete. □** + +## Equation 18 Restated + +``` +P = {S, v(x)} is universal ⟺ ∃ mapping to balanced ternary logic gates +``` + +The forward direction (⇒) follows from this proof: implementing the gates is sufficient for universality. +The backward direction (⇐) follows from Lemma 2: any universal system can simulate the gates. + +## QWERTY + +``` +UNIVERSAL = OCTONION = SYMMETRIC = 112 +COMPUTATION = 137 prime (= fine-structure constant 1/α ≈ 1/137) +COMPLETE = 97 prime +TURING = 64 = 2⁶ (six binary digits — the Turing machine needs binary) +``` + +COMPLETE = 97 prime. Completeness cannot be decomposed. **□**