TFPT scientific source excerpt
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000201 | 
000202 | \begin{definition}[Scheme observable category]
000203 | \label{def:obs-category}
000204 | Let $\OschemeTFPT$ denote the category whose objects are triples
000205 | \[
000206 | \mathcal O_{\mathrm{scheme}}=(v,\Sigma,\mathcal D),
000207 | \]
000208 | where $v$ is a scheme-specified representative of a physical observable package, $\Sigma$ its
000209 | declared covariance data, and $\mathcal D$ the induced dependency DAG. Morphisms are smooth
000210 | covariance-preserving reparameterizations respecting $\mathcal D$.
000211 | \end{definition}
000212 | 
000213 | \begin{definition}[Canonical readout hierarchy]
000214 | \label{def:canonical-measurement-orbit}
000215 | Define the functors
000216 | \[
000217 | \Rren:\mathbf{TFPT}^{\mathrm{rig}}\to\GrenTFPT,
000218 | \qquad
000219 | \Mphys:\GrenTFPT\to\OphysTFPT,
000220 | \qquad
000221 | \Mscheme:\OphysTFPT\to\OschemeTFPT/\SchGrp.
000222 | \]
000223 | For every scheme object $s\in\SchGrp$, let
000224 | \[
000225 | \Mscheme^{(s)}\bigl(\Mphys(\Rren(\Tstar))\bigr)\in\OschemeTFPT
000226 | \]
000227 | be the chosen scheme representative. The canonical appendix comparison orbit is
000228 | \[
000229 | [\Mop(\Tstar)]_{\SchGrp}
000230 | :=
000231 | \Mscheme\bigl(\Mphys(\Rren(\Tstar))\bigr)
000232 | \in \OschemeTFPT/\SchGrp,
000233 | \]
000234 | and the appendix comparison map is any chosen representative
000235 | \[
000236 | \Rcmp^{(s)}(\Tstar):=
000237 | \Mscheme^{(s)}\bigl(\Mphys(\Rren(\Tstar))\bigr).
000238 | \]
000239 | \end{definition}
000240 | 
000241 | \begin{definition}[Minimal independent observable basis]
000242 | \label{def:minimal-observable-basis}
000243 | Define the minimal independent basis package
000244 | \[
000245 | \Bind(\Tstar):=
000246 | \bigl(
000247 | C_{\mathrm{em}},
000248 | K_{\mathrm{UV}},
000249 | F_{\mathrm{fl}},
000250 | N_{\nu},
000251 | R_{\mathrm{CS}},
000252 | T_{\mathrm{phys}},
000253 | C_{\mathrm{cos}},
000254 | C_{\mathrm{CP}}
000255 | \bigr),
000256 | \]
000257 | where
000258 | \[
000259 | C_{\mathrm{em}}:=\alpha^{-1}(0),
000260 | \qquad
000261 | K_{\mathrm{UV}}:=\phiz,
000262 | \qquad
000263 | F_{\mathrm{fl}}:=(\lambda_C,\delta_{\mathrm{CKM}}),
000264 | \qquad
000265 | R_{\mathrm{CS}}:=\beta,
000266 | \qquad
000267 | C_{\mathrm{CP}}:=\theta_{\mathrm{eff}},
000268 | \]
000269 | \[
000270 | N_{\nu}:=(\sin^2\theta_{13},\delta_{\mathrm{CP}}^\nu,\sin^2\theta_{23},\Sigma m_\nu,m_{\beta\beta},M_R),
000271 | \]
000272 | \[
000273 | T_{\mathrm{phys}}
000274 | :=
000275 | \bigl(
000276 | G_F^{\mathrm{TFPT}},
000277 | m_W^{\mathrm{pole}},
000278 | m_Z^{\mathrm{pole}},
000279 | m_H^{\mathrm{pole}},
000280 | \{m_f^{\mathrm{pole}}\}_f
000281 | \bigr),
000282 | \]
000283 | \[
000284 | C_{\mathrm{cos}}
000285 | :=
000286 | \bigl(
000287 | \Omega_b,S_\Sigma,\Lambda_{\mathrm{IR}},T_R,f_a,m_a,\nu_a,\eta_B
000288 | \bigr).
000289 | \]
000290 | The retained seed $K_{\mathrm{UV}}=\phiz$ is part of the UV kernel package; it is no longer the
000291 | direct basis package of the flavor, neutrino, determinant-response, or cosmology observables.
000292 | \end{definition}
000293 | 
000294 | \begin{theorem}[Renormalized observable hierarchy from the admissible 1PI action]
000295 | \label{thm:operational-completeness}
000296 | There exist functors
000297 | \[
000298 | \Rren:\mathbf{TFPT}^{\mathrm{rig}}\to\GrenTFPT,
000299 | \qquad
000300 | \Mphys:\GrenTFPT\to\OphysTFPT,
000301 | \qquad
000302 | \Mscheme:\OphysTFPT\to\OschemeTFPT/\SchGrp
000303 | \]
000304 | with the following properties:
000305 | \begin{enumerate}[label=\textbf{(O\arabic*)}, leftmargin=2em, topsep=3pt, itemsep=3pt]
000306 |     \item \textbf{Reality first.}
000307 |     Main-text observables factor through $\Mphys\!\circ\!\Rren$ and therefore belong to
000308 |     $\OphysTFPT$ before any scheme choice is made.
000309 |     \item \textbf{Unique factorization through the basis.}
000310 |     Every physical readout row and every appendix scheme row factors through exactly one component
000311 |     package of $\Bind(\Tstar)$ via a declared physical or scheme map.
000312 |     \item \textbf{Scheme covariance.}
000313 |     For every scheme morphism $\psi:s\to s'$ one has
000314 |     \[
000315 |     \psi\!\left(\Mscheme^{(s)}(\Mphys(\Rren(\Tstar)))\right)
000316 |     =
000317 |     \Mscheme^{(s')}(\Mphys(\Rren(\Tstar))).
000318 |     \]
000319 |     \item \textbf{No hidden continuous fit parameter.}
000320 |     Any continuous freedom in a readout row is either a declared scheme degree of freedom in
000321 |     $\SchGrp$ or already a theorem-level branch parameter fixed in $\Tstar$.
000322 |     \item \textbf{Complete kill surface.}
000323 |     Every admissible empirical kill test of the manuscript is the pullback of a component test on
000324 |     $\OphysTFPT$ and only afterwards, if needed, of a scheme representative in $\OschemeTFPT$.
000325 | \end{enumerate}
000326 | \end{theorem}
000327 | 
000328 | \begin{proof}
000329 | Define $\Rren$ by sending the rigid object $\Tstar$ to the renormalized 1PI package extracted from
000330 | the exact admissible flow of \Cref{thm:exact-admissible-rg-flow}. Its infrared endpoint is
000331 | \[
000332 | \Gamma_{\mathrm{TFPT}}^{\mathrm{ren}}[\Phi_c]
000333 | =
000334 | \lim_{k\to0}\Gamma_k[\Phi_c],
000335 | \]
000336 | equivalently the Legendre transform of the retained admissible generating functional after removing
000337 | the infrared regulator. By \Cref{thm:admissible-1pi-graph-expansion}, the same object carries the
000338 | admissible 1PI graph expansion around each admissible background. Reflection positivity, lower
000339 | boundedness, and coercivity on the admissible sector ensure the convexity needed for this Legendre
000340 | construction in a neighborhood of the closed branch. Hence the dressed inverse propagators,
000341 | determinant-response kernels, and cosmology saddle system belong to one and the same renormalized
000342 | package $\GrenTFPT$.
000343 | 
000344 | Next define $\Mphys$ by reading physical observables from that 1PI package before any scheme choice:
000345 | pole masses are the zeros of the dressed inverse propagators, mixing data are the branch invariants
000346 | of the corresponding residue matrices, determinant-response observables are read from the exact
000347 | response kernels, and cosmology rows factor through the same FRW / reheating / Boltzmann interface
000348 | block. Final late-time numerical representatives are attached only after the declared comparison
000349 | map. This proves \textbf{(O1)}.
000350 | 
000351 | For \textbf{(O2)}, the factorization through the minimal basis package $\Bind(\Tstar)$ is exactly
000352 | the theorem-level content of the rewritten sector split. The flavor rows factor through
000353 | $F_{\mathrm{fl}}$, neutrino rows through $N_\nu$, determinant-response rows through
000354 | $R_{\mathrm{CS}}$, strong-CP rows through $C_{\mathrm{CP}}$, cosmology rows through
000355 | $C_{\mathrm{cos}}$, and the retained seed $\phiz$ survives only in the UV kernel package
000356 | $K_{\mathrm{UV}}$. No row uses two basis packages at once.
000357 | 
000358 | For \textbf{(O3)}, every scheme morphism in $\SchGrp$ acts only after $\Mphys$ has produced a
000359 | physical observable package. Therefore threshold transfer, finite renormalization, and snapshot
000360 | change act on scheme representatives without altering the underlying physical point, exactly as in
000361 | the displayed covariance identity.
000362 | 
000363 | For \textbf{(O4)}, every continuous parameter entering a readout row is already fixed either by the
000364 | closed branch $\Tstar$ or by the declared scheme object $s\in\SchGrp$. Since $\Rren$ and $\Mphys$
000365 | are defined from the fixed admissible measure and its 1PI action, no extra continuous fit parameter
000366 | can appear in between.
000367 | 
000368 | Finally, \textbf{(O5)} follows because every empirical kill criterion in the manuscript is attached
000369 | to one of the physical rows in $\OphysTFPT$ and only afterwards, if needed, represented by a scheme
000370 | object in $\OschemeTFPT/\SchGrp$. Thus the whole observable hierarchy factors through the same
000371 | admissible 1PI action and then through the declared physical / scheme split.
000372 | \end{proof}