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An Ising model on a chain of corner-sharing triangles, with one base spin and one apex spin per cell. If the apex interacts only with the two base corners, summing each apex independently gives a two-by-two transfer matrix on base spins. For symmetric Ising spins and energy $Js{s}^{\prime }+\bar{J}\bar{s}(s+s^{\prime })$, its entries are $2e^{-\beta Js{s}^{\prime }}\cosh [\beta \bar{J}(s+s^{\prime })]$.

Transfer dynamic programming advances through repeated layers while storing a state on their interface. If $v_j(s)$ counts partial configurations with boundary state $s$, transitions give $v_{j+1}(t)={\sum }_s{T}_{ts}{v}_j(s)$. With identical layers this is repeated multiplication by a transfer matrix.

For binary strings with no consecutive ones, store whether the last digit is zero or one. Appending a zero is allowed from either state; appending a one is allowed only from the zero state. The resulting two-state recurrence gives Fibonacci numbers. The state must retain all information that a later layer can detect, including connectivity when local occupancies alone do not suffice.

For $V(x)=g\delta (x-x_0)$, put $q=\sqrt{2mE}/\hslash >0$ and $\gamma =mg/({\hslash }^{2}q)$. Relate the coefficients of $A{e}^{iqx}+B{e}^{-iqx}$ immediately to the right and left by $v_R=D(x_0)v_L$. Continuity of the wavefunction and integration of the Schrödinger equation across the delta potential give ${\psi }^{\prime }(x_0^{+})-{\psi }^{\prime }(x_0^{-})=2mg\psi (x_0)/{\hslash }^2$. Solving these two matching equations givesFor scatterers at $x_1<\cdots <x_N$, the full transfer matrix is $M=D(x_N)\cdots D(x_1)$: the leftmost scatterer acts first on the coefficient column. With these globally phased plane waves, the propagation phases are already in each $D(x_j)$. For incidence only from the left, $t=1/M_{22}$ and $r=-M_{21}/M_{22}$ follow from $(t,0{)}^T=M(1,r{)}^T$ and $\det M=1$.

For a real-frequency lossless reciprocal periodic line, a cell transfer matrix has determinant one. If half its trace lies outside $[-1,1]$, the Bloch multipliers are real reciprocal numbers rather than unit-modulus propagation factors. The physical interior field decays and carries no net power to infinity. A real reference line is then totally reflected by a semi-infinite structure; a finite structure can still tunnel.