TFPT scientific source excerpt
Original repository path: documents/archive--tfpt-45--source_extracts--03_em_flavor_source/sections/003-subsection-discrete-phase-operator-and-transport-grammar.tex
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Inclusive original lines: 170-208
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000170 | \[
000171 | \tau=0
000172 | \]
000173 | after admissibility restriction. Then
000174 | \[
000175 | Q_y(x,0)=(y^6-z)^2,
000176 | \]
000177 | and the stationarity equation reduces to
000178 | \[
000179 | \frac{1}{1-z}
000180 | +
000181 | \frac{1}{1/729-z}
000182 | +
000183 | \frac{1}{64/729-z}
000184 | =0.
000185 | \]
000186 | Equivalently, for
000187 | \[
000188 | P(z):=
000189 | (z-1)\left(z-\frac{64}{729}\right)\left(z-\frac{1}{729}\right),
000190 | \]
000191 | one has
000192 | \[
000193 | P'(z)=0.
000194 | \]
000195 | The admissible transport pole is therefore the lower critical point of this cubic:
000196 | \[
000197 | z_\star
000198 | =
000199 | \frac{794-7\sqrt{9961}}{2187},
000200 | \qquad
000201 | \delta_{\mathrm{ph}}=z_\star^{1/6}
000202 | =
000203 | \left(\frac{794-7\sqrt{9961}}{2187}\right)^{1/6}
000204 | \approx \num{0.593277280133224}.
000205 | \]
000206 | As an expanded control form, one may equivalently write
000207 | \[
000208 | 1594323\,\delta^{12}-1157652\,\delta^6+47449=0.