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- 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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2.8 KiB
Universal Equations
Page 23. Three Tests. Euler-Lagrange. The backbone.
The Three Tests (§177)
A universal equation must pass all three:
| Test | Criterion | QWERTY |
|---|---|---|
| 1 | It governs many systems → SCOPE | SCOPE = 56 |
| 2 | It falls out of symmetry or variational principles → STRUCTURE | STRUCTURE = SHELL = 69 |
| 3 | It reduces to known special cases without breaking → LIMITS | LIMITS = TRIVIAL = 78 |
TESTS = REAL = 37 prime (what passes the tests = real)
GOVERN = CREATIVE = MARCH = 79 prime (to govern = creative)
LIMITS = TRIVIAL = BINARY = 78 (the limits are trivial)
SYMMETRY = OPTIMAL = CRITERION = 88 (Noether: symmetry = conservation)
The Euler-Lagrange equation passes all three.
Principle of Stationary Action (§177)
δS = 0, S = ∫ L(q,q̇,t) dt
The physical path is where the action S does not change to first order.
Euler-Lagrange equations:
d/dt(∂L/∂q̇ᵢ) − ∂L/∂qᵢ = 0
Field form:
∂_μ(∂L/∂(∂_μφₐ)) − ∂L/∂φₐ = 0
Her note: "This is the backbone. Choose the right Lagrangian L, you get particle mechanics, waves, classical fields, etc."
QWERTY Analysis
EULER = ZERO = REPEAT = 36 (δS=0 — the equations ARE zero)
LAGRANGE = REVERSIBLE = 103 prime (time-reversible)
LAGRANGIAN = INFORMATION = BIOLOGICAL = 144 = 12²
SYMMETRY = OPTIMAL = CRITERION = 88
BACKBONE = CLASSICAL = COMPUTABLE = 136
MECHANICS = EVERYTHINGELSE = 145
RELATIVISTIC = BALANCED = COMPETENCE = 128 = 2⁷
QUANTUM = PARTICLE = CAUSAL = 82
LIMITS = TRIVIAL = BINARY = 78
FIELD = GAUSS = TANH = RADIX = 57 (the field = the Gaussian)
Special Cases (Test 3)
The Euler-Lagrange equation reduces to:
| Limit | Lagrangian | Notes |
|---|---|---|
| Newtonian | L = T − V | Classical particle mechanics |
| Relativistic | L = −mc²√(1−v²/c²) | Special relativity |
| Quantum field | L = ψ̄(iγ^μ∂_μ − m)ψ | Dirac equation (Fermi field) |
| Electromagnetic | L = −¼F_μνF^μν | Maxwell's equations |
| General relativity | L = √(−g)R | Einstein field equations |
KINETICS = MAXWELL = GAUSSIAN = 101 — all field theories = Maxwell = Gaussian.
Universal Computation (§175)
∀ computable f: ∃ configuration (S,V,Ω,θ) such that system(input) = f(input)
The ternary bio-quantum system is Turing-complete.
COMPUTATION = 137 prime (= fine-structure constant 1/α ≈ 1/137)
UNIVERSAL = OCTONION = SYMMETRIC = 112
COMPUTABLE = BACKBONE = CLASSICAL = 136
COMPUTATION = 137. Feynman: "one of the greatest damn mysteries of physics."
Universal computation costs exactly what it costs to emit a photon.