TFPT scientific source excerpt
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000229 | \begin{definition}[Projected modular free energy and stratified master functional]
000230 | \label{def:projected-modular-free-energy}
000231 | Fix the unique joint discrete datum $d^\star_{\mathrm{disc}}$ of
000232 | \Cref{thm:joint-discrete-admissible-sector}, and let $\Padm$ be the derived full selector of
000233 | \Cref{cor:derived-post-carrier-kernel}. For
000234 | \[
000235 | \xi=(\alpha,\chigeo,\delta_{\mathrm{ph}},\rho_{\mathrm{vac}})
000236 | \]
000237 | on the retained admissible branch, define
000238 | \[
000239 | \begin{aligned}
000240 | \mathcal M_{d^\star_{\mathrm{disc}}}(\xi)
000241 | :={}&-\log Z^{\Padm}_{\mathrm{rel}}[\xi]
000242 | +\Re\log\det_{\zeta,\Padm}(D_\xi^2+\mu^2)\\
000243 | &+\frac12\eta_{\Sigma,\Padm}(D_\xi)
000244 | -\log\det(1-U_{6,\xi})
000245 | +\Gamma_{\mathrm{vac}}(\rho_\xi),
000246 | \end{aligned}
000247 | \]
000248 | where every term is evaluated on the retained admissible sector $\operatorname{Ran}(\Padm)$.
000249 | For a general admissible spectral-bordism state
000250 | \[
000251 | \mathfrak s=(\mathcal B,\xi),
000252 | \qquad
000253 | \mathcal B\in\mathbf{SBord}^{\mathrm{adm}},
000254 | \]
000255 | define the lexicographic barrier
000256 | \[
000257 | \mathbb B_{\mathrm{lex}}(\mathfrak s)
000258 | :=
000259 | \begin{cases}
000260 | 0,& \mathcal B\cong \mathcal B_{\min},\\
000261 | +\infty,& \text{otherwise},
000262 | \end{cases}
000263 | \]
000264 | and the stratified master functional
000265 | \[
000266 | \mathbb M(\mathfrak s)
000267 | :=
000268 | \mathbb B_{\mathrm{lex}}(\mathfrak s)+\mathcal M_{d^\star_{\mathrm{disc}}}(\xi).
000269 | \]
000270 | Thus the continuous retained-branch functional $\mathcal M_{d^\star_{\mathrm{disc}}}$ is the
000271 | restriction of one stratified functional to the unique minimal admissible stratum.
000272 | \end{definition}
000273 | 
000274 | \begin{theorem}[Stratified master functional and continuous closure]
000275 | \label{thm:continuous-closure-from-master-functional}
000276 | On the retained admissible branch, the projected modular free energy factorizes as
000277 | \[
000278 | \mathcal M_{d^\star_{\mathrm{disc}}}(\xi)
000279 | =
000280 | \Gamma_{U(1)}(\alpha)
000281 | +
000282 | \Gamma_{\mathrm{grav}}(\chigeo)
000283 | +
000284 | \Gamma_{\mathrm{tr}}(\delta_{\mathrm{ph}})
000285 | +
000286 | \Gamma_{\mathrm{vac}}(\rho_{\mathrm{vac}})
000287 | +\mathrm{const}.
000288 | \]
000289 | Its admissible critical points are therefore exactly the solutions of
000290 | \[
000291 | \partial_\alpha\Gamma_{U(1)}(\alpha)=0,
000292 | \qquad
000293 | \partial_{\chigeo}\Gamma_{\mathrm{grav}}(\chigeo)=0,
000294 | \qquad
000295 | \partial_\delta\Gamma_{\mathrm{tr}}(\delta_{\mathrm{ph}})=0,
000296 | \qquad
000297 | \delta_\rho\Gamma_{\mathrm{vac}}(\rho_{\mathrm{vac}})=0.
000298 | \]
000299 | Equivalently, the continuous closure laws are
000300 | \[
000301 | F_{U(1)}(\alpha)=0,
000302 | \qquad
000303 | \mathcal R_{\mathrm{grav}}=0,
000304 | \qquad
000305 | P_{\mathrm{tr}}(\delta_{\mathrm{ph}})=0,
000306 | \qquad
000307 | \delta_\rho\Gamma_{\mathrm{vac}}(\rho_{\mathrm{vac}})=0.
000308 | \]
000309 | In particular the continuous closure equations are derived from one primitive variational object
000310 | rather than imposed as independent sector primitives. Equivalently, the stationary points of the
000311 | stratified master functional $\mathbb M$ are exactly the stationary points of
000312 | $\mathcal M_{d^\star_{\mathrm{disc}}}$ on the minimal admissible stratum.
000313 | \end{theorem}
000314 | 
000315 | \begin{proof}
000316 | By construction,
000317 | \[
000318 | \mathbb M(\mathfrak s)=+\infty
000319 | \]
000320 | off the unitary-equivalence class of the minimal admissible bordism $\mathcal B_{\min}$. Hence every
000321 | stationary point of $\mathbb M$ must lie on that minimal stratum, where the barrier vanishes and
000322 | \[
000323 | \mathbb M(\mathfrak s)=\mathcal M_{d^\star_{\mathrm{disc}}}(\xi).
000324 | \]
000325 | 
000326 | Now apply \Cref{cor:derived-post-carrier-kernel}: all continuous terms are evaluated on one and the
000327 | same retained sector $\operatorname{Ran}(\Padm)$. On that branch the family, determinant, transport,
000328 | and vacuum projectors commute, and the physical Hilbert space splits into orthogonal direct summands
000329 | corresponding to the electromagnetic, geometric, transport, and vacuum blocks. For a block-diagonal
000330 | operator, the relative partition function factorizes, the zeta determinant factorizes, and the eta
000331 | invariant is additive. Therefore the logarithms split $\mathcal M_{d^\star_{\mathrm{disc}}}$ into a
000332 | sum of four sector functionals plus a constant.
000333 | 
000334 | The $U(1)$ block is exactly the already-defined electromagnetic effective action, so
000335 | \[
000336 | \partial_\alpha\Gamma_{U(1)}(\alpha)=F_{U(1)}(\alpha).
000337 | \]
000338 | The geometric block is the reduced spectral Einstein functional, hence