📐 Math — TFT_3Pack Example Suite

Problem 1 — Triadic Sequence Scaling

aₙ = D₃ⁿ τ_r

If \( τ_r \) triples, how does \( aₙ \) change?

Problem 2 — Resonant Integral

I = ∫₀^{τ_r} T_f D₆ dt

Evaluate the integral in terms of \( τ_r \).

Problem 3 — Exponential Resonance Equation

X e^{ΛΘ t} = D₉

Solve for \( t \).

Solution 1 — Sequence Scaling

Tripling \( τ_r \) triples every term: \( aₙ' = 3aₙ \).

Solution 2 — Integral

\( I = T_f D₆ τ_r \).

Solution 3 — Exponential Equation

\( t = \frac{1}{ΛΘ} \ln\left(\frac{D₉}{X}\right) \).

Problem 4 — Multi-Triad Polynomial

P(x) = D₃ x² + D₆ τ_r x + X

Compute P(τ_r) and describe how the linear term changes if τ_r doubles.

Problem 5 — Resonant Differential Equation

dy/dt = D₆ T_f y

Solve with y(0) = y₀.

Problem 6 — Triadic Transform

𝒯{s}(ω) = ∫₀^∞ e^{-D₃ t} e^{-i ω τ_r t} dt

Combine exponentials and evaluate the integral.

Problem 7 — Resonant Fixed Point

f(x) = X √x - D₉

Solve f(x) = x.

Problem 8 — Triadic Matrix Resonance

M = [ D₃   τ_r ]
    [  0    D₆ ]
        

Compute det(M) and eigenvalues.

Diagram 1 — Triadic Sequence

D₃ → power n → × τ_r → aₙ
        

Diagram 2 — Resonant Integral

T_f D₆ → integrate 0→τ_r → I
        

Diagram 3 — Exponential Resonance

X × e^{ΛΘ t} = D₉ → solve for t