\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand*\HyPL@Entry[1]{}
\HyPL@Entry{0<</S/D>>}
\pgfsyspdfmark {pgfid1}{3883789}{47370584}
\HyPL@Entry{1<</S/D>>}
\@writefile{toc}{\contentsline {section}{Reading this edition}{3}{section*.2}\protected@file@percent }
\citation{HenniartLemaire2011}
\@writefile{toc}{\contentsline {section}{Current scope and research routes}{4}{section*.3}\protected@file@percent }
\citation{Henniart2000}
\citation{HarrisTaylor2001}
\citation{BushnellHenniart2006}
\citation{DeligneLusztig1976}
\citation{BushnellHenniart1996}
\citation{BushnellHenniart1999}
\citation{BushnellHenniart2005ET1}
\citation{BushnellHenniart2005ET2}
\citation{BushnellHenniart2010ET3}
\citation{BushnellKutzko1993}
\citation{Blondel2012}
\citation{Dotto2022}
\citation{BushnellHenniart2017}
\citation{Gerardin1977}
\citation{Neuhauser2002}
\citation{Prasad2009}
\citation{Karpilovsky1985}
\citation{GurevichHadani2007}
\citation{MilnorHusemoller1973}
\citation{Henniart1993}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{5}{section.1}\protected@file@percent }
\newlabel{sec:introduction}{{1}{5}{Introduction}{section.1}{}}
\newlabel{sec:introduction@cref}{{[section][1][]1}{[1][5][]5}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The finite theorem layers and the remaining wild boundary.}}{6}{figure.1}\protected@file@percent }
\newlabel{fig:architecture}{{1}{6}{The finite theorem layers and the remaining wild boundary}{figure.1}{}}
\newlabel{fig:architecture@cref}{{[figure][1][]1}{[1][5][]6}}
\@writefile{toc}{\contentsline {section}{\numberline {2}Weil Deligne chains}{6}{section.2}\protected@file@percent }
\newlabel{sec:wd}{{2}{6}{Weil Deligne chains}{section.2}{}}
\newlabel{sec:wd@cref}{{[section][2][]2}{[1][6][]6}}
\newlabel{fact:spdecomp}{{2.1}{6}{Indecomposable monodromy blocks}{standardfact.2.1}{}}
\newlabel{fact:spdecomp@cref}{{[standardfact][1][2]2.1}{[1][6][]6}}
\newlabel{def:monodromy_chain}{{2.2}{6}{Monodromy packet chain}{definition.2.2}{}}
\newlabel{def:monodromy_chain@cref}{{[definition][2][2]2.2}{[1][6][]6}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A Weil Deligne chain is reconstructed from one atom and finitely many nearest neighbor transfers.}}{7}{figure.2}\protected@file@percent }
\newlabel{fig:wdchain}{{2}{7}{A Weil Deligne chain is reconstructed from one atom and finitely many nearest neighbor transfers}{figure.2}{}}
\newlabel{fig:wdchain@cref}{{[figure][2][]2}{[1][7][]7}}
\newlabel{prop:monodromy_reconstruction}{{2.3}{7}{Exact block reconstruction}{proposition.2.3}{}}
\newlabel{prop:monodromy_reconstruction@cref}{{[proposition][3][2]2.3}{[1][7][]7}}
\newlabel{prop:scalar_covariance}{{2.4}{7}{Scalar covariance model}{proposition.2.4}{}}
\newlabel{prop:scalar_covariance@cref}{{[proposition][4][2]2.4}{[1][7][]7}}
\newlabel{thm:formal_atomic}{{2.5}{7}{Formal atomic reconstruction}{theorem.2.5}{}}
\newlabel{thm:formal_atomic@cref}{{[theorem][5][2]2.5}{[1][7][]7}}
\newlabel{rem:atomic_boundary}{{2.6}{8}{}{remark.2.6}{}}
\newlabel{rem:atomic_boundary@cref}{{[remark][6][2]2.6}{[1][7][]8}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Tame Frobenius packets}{8}{section.3}\protected@file@percent }
\newlabel{sec:tame}{{3}{8}{Tame Frobenius packets}{section.3}{}}
\newlabel{sec:tame@cref}{{[section][3][]3}{[1][8][]8}}
\newlabel{prop:tame_projectors}{{3.1}{8}{Tame character projectors}{proposition.3.1}{}}
\newlabel{prop:tame_projectors@cref}{{[proposition][1][3]3.1}{[1][8][]8}}
\newlabel{thm:tame_induction}{{3.2}{8}{Tame orbit induction}{theorem.3.2}{}}
\newlabel{thm:tame_induction@cref}{{[theorem][2][3]3.2}{[1][8][]8}}
\newlabel{cor:tame_dimension}{{3.3}{8}{Dimension and regularity}{corollary.3.3}{}}
\newlabel{cor:tame_dimension@cref}{{[corollary][3][3]3.3}{[1][8][]8}}
\newlabel{prop:tame_dual}{{3.4}{8}{Dual and self dual tame packets}{proposition.3.4}{}}
\newlabel{prop:tame_dual@cref}{{[proposition][4][3]3.4}{[1][8][]8}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces A tame irreducible atom is one transitive Frobenius orbit.}}{9}{figure.3}\protected@file@percent }
\newlabel{fig:tameorbit}{{3}{9}{A tame irreducible atom is one transitive Frobenius orbit}{figure.3}{}}
\newlabel{fig:tameorbit@cref}{{[figure][3][]3}{[1][9][]9}}
\newlabel{prop:tame_twist}{{3.5}{9}{Unramified twist}{proposition.3.5}{}}
\newlabel{prop:tame_twist@cref}{{[proposition][5][3]3.5}{[1][9][]9}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Regular orbit counts}{9}{subsection.3.1}\protected@file@percent }
\newlabel{prop:mobius_count}{{3.6}{9}{Möbius orbit count}{proposition.3.6}{}}
\newlabel{prop:mobius_count@cref}{{[proposition][6][3]3.6}{[1][9][]9}}
\citation{DeligneLusztig1976}
\citation{Serre1977}
\citation{Isaacs1976}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Regular Frobenius orbit counts for three residue field sizes.}}{10}{figure.4}\protected@file@percent }
\newlabel{fig:orbitcounts}{{4}{10}{Regular Frobenius orbit counts for three residue field sizes}{figure.4}{}}
\newlabel{fig:orbitcounts@cref}{{[figure][4][]4}{[1][10][]10}}
\newlabel{ex:degree_four}{{3.7}{10}{Degree four over \(\FF _2\)}{example.3.7}{}}
\newlabel{ex:degree_four@cref}{{[example][7][3]3.7}{[1][9][]10}}
\@writefile{toc}{\contentsline {section}{\numberline {4}Finite field cuspidal projectors}{10}{section.4}\protected@file@percent }
\newlabel{sec:finitefield}{{4}{10}{Finite field cuspidal projectors}{section.4}{}}
\newlabel{sec:finitefield@cref}{{[section][4][]4}{[1][10][]10}}
\newlabel{fact:deligne_lusztig}{{4.1}{10}{Regular elliptic torus characters}{standardfact.4.1}{}}
\newlabel{fact:deligne_lusztig@cref}{{[standardfact][1][4]4.1}{[1][10][]10}}
\newlabel{prop:cuspidal_idempotent}{{4.2}{10}{Primitive cuspidal idempotent}{proposition.4.2}{}}
\newlabel{prop:cuspidal_idempotent@cref}{{[proposition][2][4]4.2}{[1][10][]10}}
\citation{BushnellKutzko1993}
\citation{BushnellHenniart2006}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces A regular torus character determines a cuspidal block and its primitive central projector.}}{11}{figure.5}\protected@file@percent }
\newlabel{fig:cuspidal}{{5}{11}{A regular torus character determines a cuspidal block and its primitive central projector}{figure.5}{}}
\newlabel{fig:cuspidal@cref}{{[figure][5][]5}{[1][11][]11}}
\newlabel{prop:parabolic_annihilation}{{4.3}{11}{Parabolic annihilation}{proposition.4.3}{}}
\newlabel{prop:parabolic_annihilation@cref}{{[proposition][3][4]4.3}{[1][10][]11}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Depth zero reconstruction and cycle orientation}{11}{subsection.4.1}\protected@file@percent }
\newlabel{prop:frobenius_cycle}{{4.4}{11}{Frobenius cycle polynomial}{proposition.4.4}{}}
\newlabel{prop:frobenius_cycle@cref}{{[proposition][4][4]4.4}{[1][11][]11}}
\newlabel{cor:orientation}{{4.5}{11}{Unramified orientation character}{corollary.4.5}{}}
\newlabel{cor:orientation@cref}{{[corollary][5][4]4.5}{[1][11][]11}}
\citation{Mackey1952}
\@writefile{toc}{\contentsline {section}{\numberline {5}Tensor refinement of tame atoms}{12}{section.5}\protected@file@percent }
\newlabel{sec:tensor}{{5}{12}{Tensor refinement of tame atoms}{section.5}{}}
\newlabel{sec:tensor@cref}{{[section][5][]5}{[1][12][]12}}
\newlabel{thm:gcd_lcm}{{5.1}{12}{Greatest common divisor and least common multiple packets}{theorem.5.1}{}}
\newlabel{thm:gcd_lcm@cref}{{[theorem][1][5]5.1}{[1][12][]12}}
\newlabel{cor:gcd_lcm_count}{{5.2}{12}{Packet count and dimension}{corollary.5.2}{}}
\newlabel{cor:gcd_lcm_count@cref}{{[corollary][2][5]5.2}{[1][12][]12}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Exponent coordinates}{12}{subsection.5.1}\protected@file@percent }
\newlabel{prop:refined_exponent}{{5.3}{12}{Refined exponent formula}{proposition.5.3}{}}
\newlabel{prop:refined_exponent@cref}{{[proposition][3][5]5.3}{[1][12][]12}}
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Tensor refinement separates into greatest common divisor many packets, each with least common multiple carrier dimension.}}{13}{figure.6}\protected@file@percent }
\newlabel{fig:gcdlcm}{{6}{13}{Tensor refinement separates into greatest common divisor many packets, each with least common multiple carrier dimension}{figure.6}{}}
\newlabel{fig:gcdlcm@cref}{{[figure][6][]6}{[1][12][]13}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Tame local polynomials}{13}{subsection.5.2}\protected@file@percent }
\newlabel{prop:tame_local_polynomial}{{5.4}{13}{Local polynomial of one tame packet}{proposition.5.4}{}}
\newlabel{prop:tame_local_polynomial@cref}{{[proposition][4][5]5.4}{[1][13][]13}}
\newlabel{cor:tame_tensor_factor}{{5.5}{13}{Tame tensor local factor}{corollary.5.5}{}}
\newlabel{cor:tame_tensor_factor@cref}{{[corollary][5][5]5.5}{[1][13][]13}}
\newlabel{prop:trivial_tensor_packet}{{5.6}{13}{Trivial tensor packet criterion}{proposition.5.6}{}}
\newlabel{prop:trivial_tensor_packet@cref}{{[proposition][6][5]5.6}{[1][13][]13}}
\citation{DavenportHasse1935}
\citation{BerndtEvansWilliams1998}
\newlabel{cor:tame_hom}{{5.7}{14}{Tame Hom test}{corollary.5.7}{}}
\newlabel{cor:tame_hom@cref}{{[corollary][7][5]5.7}{[1][14][]14}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3}Tame conductor}{14}{subsection.5.3}\protected@file@percent }
\newlabel{prop:tame_conductor}{{5.8}{14}{Conductor of a nontrivial tame atom}{proposition.5.8}{}}
\newlabel{prop:tame_conductor@cref}{{[proposition][8][5]5.8}{[1][14][]14}}
\@writefile{toc}{\contentsline {section}{\numberline {6}Finite Gauss phases}{14}{section.6}\protected@file@percent }
\newlabel{sec:gauss}{{6}{14}{Finite Gauss phases}{section.6}{}}
\newlabel{sec:gauss@cref}{{[section][6][]6}{[1][14][]14}}
\newlabel{prop:gauss_magnitude}{{6.1}{14}{Gauss magnitude}{proposition.6.1}{}}
\newlabel{prop:gauss_magnitude@cref}{{[proposition][1][6]6.1}{[1][14][]14}}
\newlabel{def:gauss_phase}{{6.2}{14}{Normalized Gauss phase}{definition.6.2}{}}
\newlabel{def:gauss_phase@cref}{{[definition][2][6]6.2}{[1][14][]14}}
\newlabel{prop:gauss_duality}{{6.3}{14}{Duality of Gauss phases}{proposition.6.3}{}}
\newlabel{prop:gauss_duality@cref}{{[proposition][3][6]6.3}{[1][14][]14}}
\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Normalized finite Gauss sums lie on the unit circle.}}{15}{figure.7}\protected@file@percent }
\newlabel{fig:gausscircle}{{7}{15}{Normalized finite Gauss sums lie on the unit circle}{figure.7}{}}
\newlabel{fig:gausscircle@cref}{{[figure][7][]7}{[1][15][]15}}
\newlabel{fact:hasse_davenport}{{6.4}{15}{Hasse and Davenport lifting}{standardfact.6.4}{}}
\newlabel{fact:hasse_davenport@cref}{{[standardfact][4][6]6.4}{[1][15][]15}}
\newlabel{rem:gauss_boundary}{{6.5}{15}{}{remark.6.5}{}}
\newlabel{rem:gauss_boundary@cref}{{[remark][5][6]6.5}{[1][15][]15}}
\@writefile{toc}{\contentsline {section}{\numberline {7}Finite Heisenberg packets}{15}{section.7}\protected@file@percent }
\newlabel{sec:heisenberg}{{7}{15}{Finite Heisenberg packets}{section.7}{}}
\newlabel{sec:heisenberg@cref}{{[section][7][]7}{[1][15][]15}}
\citation{Gerardin1977}
\citation{Prasad2009}
\newlabel{def:schrodinger}{{7.1}{16}{Schrödinger operators}{definition.7.1}{}}
\newlabel{def:schrodinger@cref}{{[definition][1][7]7.1}{[1][15][]16}}
\newlabel{thm:schrodinger_rep}{{7.2}{16}{Finite Schrödinger representation}{theorem.7.2}{}}
\newlabel{thm:schrodinger_rep@cref}{{[theorem][2][7]7.2}{[1][16][]16}}
\newlabel{prop:heisenberg_trace}{{7.3}{16}{Character support}{proposition.7.3}{}}
\newlabel{prop:heisenberg_trace@cref}{{[proposition][3][7]7.3}{[1][16][]16}}
\newlabel{fact:stone_von_neumann}{{7.4}{16}{Finite Stone von Neumann theorem}{standardfact.7.4}{}}
\newlabel{fact:stone_von_neumann@cref}{{[standardfact][4][7]7.4}{[1][16][]16}}
\@writefile{toc}{\contentsline {subsection}{\numberline {7.1}Maximal isotropic selectors}{16}{subsection.7.1}\protected@file@percent }
\newlabel{prop:isotropic_selector}{{7.5}{16}{Isotropic averaging projector}{proposition.7.5}{}}
\newlabel{prop:isotropic_selector@cref}{{[proposition][5][7]7.5}{[1][16][]16}}
\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces The finite Heisenberg packet and its maximal isotropic selector.}}{17}{figure.8}\protected@file@percent }
\newlabel{fig:heisenberg}{{8}{17}{The finite Heisenberg packet and its maximal isotropic selector}{figure.8}{}}
\newlabel{fig:heisenberg@cref}{{[figure][8][]8}{[1][17][]17}}
\newlabel{cor:primitive_selector}{{7.6}{17}{Primitive line selector}{corollary.7.6}{}}
\newlabel{cor:primitive_selector@cref}{{[corollary][6][7]7.6}{[1][17][]17}}
\@writefile{toc}{\contentsline {section}{\numberline {8}Twisted group algebras and commutator envelopes}{17}{section.8}\protected@file@percent }
\newlabel{sec:twisted}{{8}{17}{Twisted group algebras and commutator envelopes}{section.8}{}}
\newlabel{sec:twisted@cref}{{[section][8][]8}{[1][17][]17}}
\newlabel{prop:cocycle_choice}{{8.1}{17}{Scalar choice invariance}{proposition.8.1}{}}
\newlabel{prop:cocycle_choice@cref}{{[proposition][1][8]8.1}{[1][17][]17}}
\newlabel{prop:commutator_form}{{8.2}{18}{Alternating commutator form}{proposition.8.2}{}}
\newlabel{prop:commutator_form@cref}{{[proposition][2][8]8.2}{[1][18][]18}}
\newlabel{thm:twisted_matrix}{{8.3}{18}{Nondegenerate twisted group algebra}{theorem.8.3}{}}
\newlabel{thm:twisted_matrix@cref}{{[theorem][3][8]8.3}{[1][18][]18}}
\newlabel{cor:radical_free}{{8.4}{18}{Radical free envelope}{corollary.8.4}{}}
\newlabel{cor:radical_free@cref}{{[corollary][4][8]8.4}{[1][18][]18}}
\newlabel{rem:projective_scope}{{8.5}{18}{}{remark.8.5}{}}
\newlabel{rem:projective_scope@cref}{{[remark][5][8]8.5}{[1][18][]18}}
\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces Projective intertwiners determine a radical free symplectic envelope.}}{19}{figure.9}\protected@file@percent }
\newlabel{fig:commutator}{{9}{19}{Projective intertwiners determine a radical free symplectic envelope}{figure.9}{}}
\newlabel{fig:commutator@cref}{{[figure][9][]9}{[1][18][]19}}
\@writefile{toc}{\contentsline {section}{\numberline {9}A restricted one break Galois envelope}{19}{section.9}\protected@file@percent }
\newlabel{sec:onebreak}{{9}{19}{A restricted one break Galois envelope}{section.9}{}}
\newlabel{sec:onebreak@cref}{{[section][9][]9}{[1][18][]19}}
\newlabel{def:one_break_datum}{{9.1}{19}{One break Clifford datum}{definition.9.1}{}}
\newlabel{def:one_break_datum@cref}{{[definition][1][9]9.1}{[1][19][]19}}
\newlabel{prop:clifford_action}{{9.2}{20}{Clifford multiplicity action}{proposition.9.2}{}}
\newlabel{prop:clifford_action@cref}{{[proposition][2][9]9.2}{[1][19][]20}}
\newlabel{def:clifford_class}{{9.3}{20}{Clifford cocycle class}{definition.9.3}{}}
\newlabel{def:clifford_class@cref}{{[definition][3][9]9.3}{[1][20][]20}}
\newlabel{prop:one_break_form}{{9.4}{20}{Intrinsic commutator pairing}{proposition.9.4}{}}
\newlabel{prop:one_break_form@cref}{{[proposition][4][9]9.4}{[1][20][]20}}
\newlabel{def:one_break_envelope}{{9.5}{20}{One break symplectic envelope}{definition.9.5}{}}
\newlabel{def:one_break_envelope@cref}{{[definition][5][9]9.5}{[1][20][]20}}
\newlabel{thm:one_break_matrix}{{9.6}{20}{Nondegenerate one break packet}{theorem.9.6}{}}
\newlabel{thm:one_break_matrix@cref}{{[theorem][6][9]9.6}{[1][20][]20}}
\newlabel{prop:one_break_invariance}{{9.7}{20}{Isomorphism invariance}{proposition.9.7}{}}
\newlabel{prop:one_break_invariance@cref}{{[proposition][7][9]9.7}{[1][20][]20}}
\newlabel{prop:tame_stabilizer}{{9.8}{20}{Tame stabilizer action}{proposition.9.8}{}}
\newlabel{prop:tame_stabilizer@cref}{{[proposition][8][9]9.8}{[1][20][]20}}
\citation{Karpilovsky1985}
\citation{Brown1982}
\@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces The restricted one break construction is made on the Clifford multiplicity space.}}{21}{figure.10}\protected@file@percent }
\newlabel{fig:onebreak}{{10}{21}{The restricted one break construction is made on the Clifford multiplicity space}{figure.10}{}}
\newlabel{fig:onebreak@cref}{{[figure][10][]10}{[1][20][]21}}
\newlabel{cor:one_break_heisenberg}{{9.9}{21}{Heisenberg representation attached to one break}{corollary.9.9}{}}
\newlabel{cor:one_break_heisenberg@cref}{{[corollary][9][9]9.9}{[1][21][]21}}
\newlabel{rem:clifford_scope}{{9.10}{21}{}{remark.9.10}{}}
\newlabel{rem:clifford_scope@cref}{{[remark][10][9]9.10}{[1][21][]21}}
\@writefile{toc}{\contentsline {section}{\numberline {10}The type side symplectic quotient}{21}{section.10}\protected@file@percent }
\newlabel{sec:typeside}{{10}{21}{The type side symplectic quotient}{section.10}{}}
\newlabel{sec:typeside@cref}{{[section][10][]10}{[1][21][]21}}
\citation{BushnellKutzko1993}
\citation{Blondel2012}
\citation{Dotto2022}
\newlabel{fact:type_heisenberg}{{10.1}{22}{Type side Heisenberg quotient}{standardfact.10.1}{}}
\newlabel{fact:type_heisenberg@cref}{{[standardfact][1][10]10.1}{[1][21][]22}}
\newlabel{fact:canonical_beta}{{10.2}{22}{Canonical beta extension}{standardfact.10.2}{}}
\newlabel{fact:canonical_beta@cref}{{[standardfact][2][10]10.2}{[1][22][]22}}
\newlabel{prop:type_commutator}{{10.3}{22}{Type side projective commutator}{proposition.10.3}{}}
\newlabel{prop:type_commutator@cref}{{[proposition][3][10]10.3}{[1][22][]22}}
\@writefile{toc}{\contentsline {section}{\numberline {11}The restricted Galois to type bridge}{22}{section.11}\protected@file@percent }
\newlabel{sec:bridge}{{11}{22}{The restricted Galois to type bridge}{section.11}{}}
\newlabel{sec:bridge@cref}{{[section][11][]11}{[1][22][]22}}
\newlabel{def:bridge_datum}{{11.1}{22}{Bridge datum}{definition.11.1}{}}
\newlabel{def:bridge_datum@cref}{{[definition][1][11]11.1}{[1][22][]22}}
\newlabel{thm:restricted_bridge}{{11.2}{22}{Restricted Heisenberg bridge}{theorem.11.2}{}}
\newlabel{thm:restricted_bridge@cref}{{[theorem][2][11]11.2}{[1][22][]22}}
\citation{Gerardin1977}
\citation{Neuhauser2002}
\citation{GurevichHadani2007}
\newlabel{prop:cyclic_discrepancy}{{11.3}{23}{Cyclic discrepancy character}{proposition.11.3}{}}
\newlabel{prop:cyclic_discrepancy@cref}{{[proposition][3][11]11.3}{[1][23][]23}}
\newlabel{lem:cyclic_linearization}{{11.4}{23}{Linearization over a cyclic stabilizer}{lemma.11.4}{}}
\newlabel{lem:cyclic_linearization@cref}{{[lemma][4][11]11.4}{[1][23][]23}}
\newlabel{thm:one_phase_bridge}{{11.5}{23}{One phase closes the cyclic bridge}{theorem.11.5}{}}
\newlabel{thm:one_phase_bridge@cref}{{[theorem][5][11]11.5}{[1][23][]23}}
\newlabel{cor:bridge_uniqueness}{{11.6}{23}{Bridge uniqueness}{corollary.11.6}{}}
\newlabel{cor:bridge_uniqueness@cref}{{[corollary][6][11]11.6}{[1][23][]23}}
\newlabel{rem:bridge_scope}{{11.7}{23}{}{remark.11.7}{}}
\newlabel{rem:bridge_scope@cref}{{[remark][7][11]11.7}{[1][23][]23}}
\@writefile{toc}{\contentsline {section}{\numberline {12}Odd residue characteristic phases}{23}{section.12}\protected@file@percent }
\newlabel{sec:oddphase}{{12}{23}{Odd residue characteristic phases}{section.12}{}}
\newlabel{sec:oddphase@cref}{{[section][12][]12}{[1][23][]23}}
\@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces An equivariant symplectic isometry reduces the finite comparison to one scalar cyclic phase.}}{24}{figure.11}\protected@file@percent }
\newlabel{fig:restrictedbridge}{{11}{24}{An equivariant symplectic isometry reduces the finite comparison to one scalar cyclic phase}{figure.11}{}}
\newlabel{fig:restrictedbridge@cref}{{[figure][11][]11}{[1][23][]24}}
\newlabel{fact:weil_trace}{{12.1}{24}{Fixed point free Weil trace}{standardfact.12.1}{}}
\newlabel{fact:weil_trace@cref}{{[standardfact][1][12]12.1}{[1][23][]24}}
\newlabel{def:odd_phase}{{12.2}{24}{Odd finite Weil phase}{definition.12.2}{}}
\newlabel{def:odd_phase@cref}{{[definition][2][12]12.2}{[1][24][]24}}
\newlabel{prop:phase_invariance}{{12.3}{24}{Symplectic invariance}{proposition.12.3}{}}
\newlabel{prop:phase_invariance@cref}{{[proposition][3][12]12.3}{[1][24][]24}}
\newlabel{prop:odd_multiplicativity}{{12.4}{24}{Orthogonal multiplicativity}{proposition.12.4}{}}
\newlabel{prop:odd_multiplicativity@cref}{{[proposition][4][12]12.4}{[1][24][]24}}
\newlabel{cor:characteristic_phase}{{12.5}{25}{Characteristic polynomial certificate}{corollary.12.5}{}}
\newlabel{cor:characteristic_phase@cref}{{[corollary][5][12]12.5}{[1][25][]25}}
\newlabel{prop:fixed_moving}{{12.6}{25}{Fixed and moving decomposition}{proposition.12.6}{}}
\newlabel{prop:fixed_moving@cref}{{[proposition][6][12]12.6}{[1][25][]25}}
\newlabel{def:moving_phase}{{12.7}{25}{Moving phase}{definition.12.7}{}}
\newlabel{def:moving_phase@cref}{{[definition][7][12]12.7}{[1][25][]25}}
\newlabel{prop:hyperbolic_phase}{{12.8}{25}{Hyperbolic phase formula}{proposition.12.8}{}}
\newlabel{prop:hyperbolic_phase@cref}{{[proposition][8][12]12.8}{[1][25][]25}}
\citation{MilnorHusemoller1973}
\@writefile{lof}{\contentsline {figure}{\numberline {12}{\ignorespaces Odd finite Weil phases multiply across orthogonal atomic blocks.}}{26}{figure.12}\protected@file@percent }
\newlabel{fig:oddphase}{{12}{26}{Odd finite Weil phases multiply across orthogonal atomic blocks}{figure.12}{}}
\newlabel{fig:oddphase@cref}{{[figure][12][]12}{[1][26][]26}}
\newlabel{cor:phase_only_test}{{12.9}{26}{Phase only bridge test}{corollary.12.9}{}}
\newlabel{cor:phase_only_test@cref}{{[corollary][9][12]12.9}{[1][25][]26}}
\newlabel{rem:fixed_space}{{12.10}{26}{}{remark.12.10}{}}
\newlabel{rem:fixed_space@cref}{{[remark][10][12]12.10}{[1][26][]26}}
\@writefile{toc}{\contentsline {section}{\numberline {13}Dyadic quadratic refinements}{26}{section.13}\protected@file@percent }
\newlabel{sec:dyadic}{{13}{26}{Dyadic quadratic refinements}{section.13}{}}
\newlabel{sec:dyadic@cref}{{[section][13][]13}{[1][26][]26}}
\newlabel{def:quadratic_refinement}{{13.1}{26}{Quadratic refinement}{definition.13.1}{}}
\newlabel{def:quadratic_refinement@cref}{{[definition][1][13]13.1}{[1][26][]26}}
\newlabel{prop:refinement_affine}{{13.2}{26}{Affine space of refinements}{proposition.13.2}{}}
\newlabel{prop:refinement_affine@cref}{{[proposition][2][13]13.2}{[1][26][]26}}
\newlabel{def:arf}{{13.3}{26}{Arf invariant}{definition.13.3}{}}
\newlabel{def:arf@cref}{{[definition][3][13]13.3}{[1][26][]26}}
\newlabel{fact:arf_classification}{{13.4}{26}{Arf invariance and classification}{standardfact.13.4}{}}
\newlabel{fact:arf_classification@cref}{{[standardfact][4][13]13.4}{[1][26][]26}}
\newlabel{thm:arf_gauss}{{13.5}{27}{Dyadic Gauss sign}{theorem.13.5}{}}
\newlabel{thm:arf_gauss@cref}{{[theorem][5][13]13.5}{[1][27][]27}}
\newlabel{cor:arf_multiplicative}{{13.6}{27}{Orthogonal dyadic multiplicativity}{corollary.13.6}{}}
\newlabel{cor:arf_multiplicative@cref}{{[corollary][6][13]13.6}{[1][27][]27}}
\newlabel{prop:dyadic_need_q}{{13.7}{27}{The form alone does not determine the phase}{proposition.13.7}{}}
\newlabel{prop:dyadic_need_q@cref}{{[proposition][7][13]13.7}{[1][27][]27}}
\newlabel{prop:quadratic_translation}{{13.8}{27}{Translation of quadratic refinements}{proposition.13.8}{}}
\newlabel{prop:quadratic_translation@cref}{{[proposition][8][13]13.8}{[1][27][]27}}
\newlabel{thm:quadratic_counts}{{13.9}{27}{Number of dyadic refinements}{theorem.13.9}{}}
\newlabel{thm:quadratic_counts@cref}{{[theorem][9][13]13.9}{[1][27][]27}}
\@writefile{lof}{\contentsline {figure}{\numberline {13}{\ignorespaces At residue characteristic two, the quadratic refinement and its Arf invariant carry phase information not contained in the alternating form.}}{28}{figure.13}\protected@file@percent }
\newlabel{fig:dyadic}{{13}{28}{At residue characteristic two, the quadratic refinement and its Arf invariant carry phase information not contained in the alternating form}{figure.13}{}}
\newlabel{fig:dyadic@cref}{{[figure][13][]13}{[1][28][]28}}
\@writefile{toc}{\contentsline {section}{\numberline {14}Matrix holonomy and finite gluing}{28}{section.14}\protected@file@percent }
\newlabel{sec:holonomy}{{14}{28}{Matrix holonomy and finite gluing}{section.14}{}}
\newlabel{sec:holonomy@cref}{{[section][14][]14}{[1][28][]28}}
\newlabel{def:path_transport}{{14.1}{28}{Path transport}{definition.14.1}{}}
\newlabel{def:path_transport@cref}{{[definition][1][14]14.1}{[1][28][]28}}
\newlabel{thm:matrix_holonomy}{{14.2}{29}{Matrix holonomy criterion}{theorem.14.2}{}}
\newlabel{thm:matrix_holonomy@cref}{{[theorem][2][14]14.2}{[1][28][]29}}
\newlabel{cor:unbased_holonomy}{{14.3}{29}{Unbased gauge classification}{corollary.14.3}{}}
\newlabel{cor:unbased_holonomy@cref}{{[corollary][3][14]14.3}{[1][29][]29}}
\newlabel{cor:scalar_holonomy}{{14.4}{29}{Scalar and projective gluing}{corollary.14.4}{}}
\newlabel{cor:scalar_holonomy@cref}{{[corollary][4][14]14.4}{[1][29][]29}}
\newlabel{prop:projective_holonomy}{{14.5}{29}{Projective cocycle form}{proposition.14.5}{}}
\newlabel{prop:projective_holonomy@cref}{{[proposition][5][14]14.5}{[1][29][]29}}
\newlabel{prop:holonomy_residual}{{14.6}{29}{Exact edge residual from cycle defect}{proposition.14.6}{}}
\newlabel{prop:holonomy_residual@cref}{{[proposition][6][14]14.6}{[1][29][]29}}
\@writefile{lof}{\contentsline {figure}{\numberline {14}{\ignorespaces A spanning tree fixes the vertex gauges. The non tree edges test the remaining cycle holonomies.}}{30}{figure.14}\protected@file@percent }
\newlabel{fig:holonomy}{{14}{30}{A spanning tree fixes the vertex gauges. The non tree edges test the remaining cycle holonomies}{figure.14}{}}
\newlabel{fig:holonomy@cref}{{[figure][14][]14}{[1][30][]30}}
\newlabel{cor:quantitative_holonomy}{{14.7}{30}{Quantitative finite gluing}{corollary.14.7}{}}
\newlabel{cor:quantitative_holonomy@cref}{{[corollary][7][14]14.7}{[1][30][]30}}
\newlabel{rem:holonomy_boundary}{{14.8}{30}{}{remark.14.8}{}}
\newlabel{rem:holonomy_boundary@cref}{{[remark][8][14]14.8}{[1][30][]30}}
\@writefile{toc}{\contentsline {section}{\numberline {15}Breakwise assembly}{30}{section.15}\protected@file@percent }
\newlabel{sec:breakwise}{{15}{30}{Breakwise assembly}{section.15}{}}
\newlabel{sec:breakwise@cref}{{[section][15][]15}{[1][30][]30}}
\newlabel{def:break_decomposition}{{15.1}{30}{Orthogonal break decomposition}{definition.15.1}{}}
\newlabel{def:break_decomposition@cref}{{[definition][1][15]15.1}{[1][30][]30}}
\citation{Serre1967}
\citation{Tate1979}
\newlabel{prop:heisenberg_tensor}{{15.2}{31}{Heisenberg tensor factorization}{proposition.15.2}{}}
\newlabel{prop:heisenberg_tensor@cref}{{[proposition][2][15]15.2}{[1][30][]31}}
\newlabel{cor:break_phase}{{15.3}{31}{Breakwise phase product}{corollary.15.3}{}}
\newlabel{cor:break_phase@cref}{{[corollary][3][15]15.3}{[1][31][]31}}
\newlabel{thm:breakwise_bridge}{{15.4}{31}{Breakwise restricted bridge}{theorem.15.4}{}}
\newlabel{thm:breakwise_bridge@cref}{{[theorem][4][15]15.4}{[1][31][]31}}
\@writefile{toc}{\contentsline {section}{\numberline {16}Local factor tests and the remaining bridge}{31}{section.16}\protected@file@percent }
\newlabel{sec:boundary}{{16}{31}{Local factor tests and the remaining bridge}{section.16}{}}
\newlabel{sec:boundary@cref}{{[section][16][]16}{[1][31][]31}}
\@writefile{lof}{\contentsline {figure}{\numberline {15}{\ignorespaces Orthogonal break packets produce tensor factors of the finite Heisenberg representation.}}{32}{figure.15}\protected@file@percent }
\newlabel{fig:breakwise}{{15}{32}{Orthogonal break packets produce tensor factors of the finite Heisenberg representation}{figure.15}{}}
\newlabel{fig:breakwise@cref}{{[figure][15][]15}{[1][31][]32}}
\newlabel{prop:tame_package}{{16.1}{32}{Finite tame comparison package}{proposition.16.1}{}}
\newlabel{prop:tame_package@cref}{{[proposition][1][16]16.1}{[1][32][]32}}
\@writefile{toc}{\contentsline {subsection}{\numberline {16.1}Arithmetic realization and the conditional comparison}{32}{subsection.16.1}\protected@file@percent }
\citation{Henniart1993}
\newlabel{conj:arithmetic_realization}{{66}{33}{Arithmetic packet realization}{conjecture.16.2}{}}
\newlabel{conj:arithmetic_realization@cref}{{[conjecture][2][16]66}{[1][33][]33}}
\newlabel{thm:conditional_atomic}{{66}{33}{Arithmetic packet realization}{conjecture.16.2}{}}
\newlabel{thm:conditional_atomic@cref}{{[conjecture][2][16]66}{[1][33][]33}}
\newlabel{prop:realized_candidate_comparison}{{16.3}{33}{Comparison of one realized candidate}{proposition.16.3}{}}
\newlabel{prop:realized_candidate_comparison@cref}{{[proposition][3][16]16.3}{[1][33][]33}}
\newlabel{rem:conditional_not_proof}{{16.4}{33}{}{remark.16.4}{}}
\newlabel{rem:conditional_not_proof@cref}{{[remark][4][16]16.4}{[1][33][]33}}
\@writefile{toc}{\contentsline {subsection}{\numberline {16.2}Exact open boundary}{33}{subsection.16.2}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {16}{\ignorespaces The proved finite layers and the remaining independent local correspondence steps.}}{34}{figure.16}\protected@file@percent }
\newlabel{fig:boundary}{{16}{34}{The proved finite layers and the remaining independent local correspondence steps}{figure.16}{}}
\newlabel{fig:boundary@cref}{{[figure][16][]16}{[1][34][]34}}
\@writefile{toc}{\contentsline {section}{\numberline {17}Examples and finite certificates}{34}{section.17}\protected@file@percent }
\newlabel{sec:examples}{{17}{34}{Examples and finite certificates}{section.17}{}}
\newlabel{sec:examples@cref}{{[section][17][]17}{[1][34][]34}}
\newlabel{ex:tame_four_again}{{17.1}{34}{A degree four tame orbit}{example.17.1}{}}
\newlabel{ex:tame_four_again@cref}{{[example][1][17]17.1}{[1][34][]34}}
\newlabel{ex:tensor_46}{{17.2}{34}{Tensor packets of lengths four and six}{example.17.2}{}}
\newlabel{ex:tensor_46@cref}{{[example][2][17]17.2}{[1][34][]34}}
\newlabel{ex:heisenberg_rank_one}{{17.3}{34}{Rank one Heisenberg group}{example.17.3}{}}
\newlabel{ex:heisenberg_rank_one@cref}{{[example][3][17]17.3}{[1][34][]34}}
\newlabel{ex:twisted_plane}{{17.4}{35}{Nondegenerate twisted algebra}{example.17.4}{}}
\newlabel{ex:twisted_plane@cref}{{[example][4][17]17.4}{[1][34][]35}}
\newlabel{ex:dyadic_plane}{{17.5}{35}{Dyadic phase ambiguity}{example.17.5}{}}
\newlabel{ex:dyadic_plane@cref}{{[example][5][17]17.5}{[1][35][]35}}
\newlabel{ex:one_cycle}{{17.6}{35}{One cycle gluing obstruction}{example.17.6}{}}
\newlabel{ex:one_cycle@cref}{{[example][6][17]17.6}{[1][35][]35}}
\@writefile{toc}{\contentsline {section}{\numberline {18}Scope of the finite reconstruction}{35}{section.18}\protected@file@percent }
\newlabel{sec:scope}{{18}{35}{Scope of the finite reconstruction}{section.18}{}}
\newlabel{sec:scope@cref}{{[section][18][]18}{[1][35][]35}}
\@writefile{toc}{\contentsline {section}{\numberline {19}Restricted arithmetic realization theorems}{35}{section.19}\protected@file@percent }
\newlabel{sec:arithmetic_progress}{{19}{35}{Restricted arithmetic realization theorems}{section.19}{}}
\newlabel{sec:arithmetic_progress@cref}{{[section][19][]19}{[1][35][]35}}
\@writefile{toc}{\contentsline {subsection}{\numberline {19.1}The odd-prime trace-zero construction}{36}{subsection.19.1}\protected@file@percent }
\newlabel{thm:odd_arithmetic}{{19.1}{36}{Restricted odd-prime arithmetic realization}{theorem.19.1}{}}
\newlabel{thm:odd_arithmetic@cref}{{[theorem][1][19]19.1}{[1][35][]36}}
\@writefile{toc}{\contentsline {subsection}{\numberline {19.2}The weighted dyadic construction and enlarged carriers}{36}{subsection.19.2}\protected@file@percent }
\newlabel{thm:dyadic_arithmetic}{{19.2}{36}{Restricted dyadic marked assembly realization}{theorem.19.2}{}}
\newlabel{thm:dyadic_arithmetic@cref}{{[theorem][2][19]19.2}{[1][36][]36}}
\@writefile{toc}{\contentsline {subsection}{\numberline {19.3}Scope, numerical invariants, and evidence}{37}{subsection.19.3}\protected@file@percent }
\citation{BushnellKutzko1993}
\@writefile{toc}{\contentsline {section}{Detailed ordinary proofs and construction records}{39}{section*.4}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {A}Odd-prime synthesis and complete scope}{40}{appendix.A}\protected@file@percent }
\newlabel{app:odd_synthesis}{{A}{40}{Odd-prime synthesis and complete scope}{appendix.A}{}}
\newlabel{app:odd_synthesis@cref}{{[appendix][1][2147483647]A}{[1][40][]40}}
\@writefile{toc}{\contentsline {subsection}{\numberline {A.1}1. The arithmetic construction now proved}{40}{subsection.A.1}\protected@file@percent }
\newlabel{odd_synthesis-the-arithmetic-construction-now-proved}{{A.1}{40}{1. The arithmetic construction now proved}{subsection.A.1}{}}
\newlabel{odd_synthesis-the-arithmetic-construction-now-proved@cref}{{[subappendix][1][2147483647,1]A.1}{[1][40][]40}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {A.1.1}Construction and proof}{40}{subsubsection.A.1.1}\protected@file@percent }
\newlabel{odd_synthesis-construction-and-proof}{{A.1.1}{40}{Construction and proof}{subsubsection.A.1.1}{}}
\newlabel{odd_synthesis-construction-and-proof@cref}{{[subsubappendix][1][2147483647,1,1]A.1.1}{[1][40][]40}}
\@writefile{toc}{\contentsline {subsection}{\numberline {A.2}2. The finite packet that this same candidate realizes}{41}{subsection.A.2}\protected@file@percent }
\newlabel{odd_synthesis-the-finite-packet-that-this-same-candidate-realizes}{{A.2}{41}{2. The finite packet that this same candidate realizes}{subsection.A.2}{}}
\newlabel{odd_synthesis-the-finite-packet-that-this-same-candidate-realizes@cref}{{[subappendix][2][2147483647,1]A.2}{[1][41][]41}}
\@writefile{toc}{\contentsline {subsection}{\numberline {A.3}3. A proved extension to genuine scalar inertia phases}{42}{subsection.A.3}\protected@file@percent }
\newlabel{odd_synthesis-a-proved-extension-to-genuine-scalar-inertia-phases}{{A.3}{42}{3. A proved extension to genuine scalar inertia phases}{subsection.A.3}{}}
\newlabel{odd_synthesis-a-proved-extension-to-genuine-scalar-inertia-phases@cref}{{[subappendix][3][2147483647,1]A.3}{[1][42][]42}}
\@writefile{toc}{\contentsline {subsection}{\numberline {A.4}4. What was repaired, and what remains}{42}{subsection.A.4}\protected@file@percent }
\newlabel{odd_synthesis-what-was-repaired-and-what-remains}{{A.4}{42}{4. What was repaired, and what remains}{subsection.A.4}{}}
\newlabel{odd_synthesis-what-was-repaired-and-what-remains@cref}{{[subappendix][4][2147483647,1]A.4}{[1][42][]42}}
\@writefile{toc}{\contentsline {subsection}{\numberline {A.5}5. Evidence and reproducibility}{43}{subsection.A.5}\protected@file@percent }
\newlabel{odd_synthesis-evidence-and-reproducibility}{{A.5}{43}{5. Evidence and reproducibility}{subsection.A.5}{}}
\newlabel{odd_synthesis-evidence-and-reproducibility@cref}{{[subappendix][5][2147483647,1]A.5}{[1][43][]43}}
\@writefile{toc}{\contentsline {section}{\numberline {B}The trace-zero Artin--Schreier tower}{44}{appendix.B}\protected@file@percent }
\newlabel{app:odd_tower}{{B}{44}{The trace-zero Artin--Schreier tower}{appendix.B}{}}
\newlabel{app:odd_tower@cref}{{[appendix][2][2147483647]B}{[1][44][]44}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1}Exact statement}{44}{subsection.B.1}\protected@file@percent }
\newlabel{odd_tower-exact-statement}{{B.1}{44}{Exact statement}{subsection.B.1}{}}
\newlabel{odd_tower-exact-statement@cref}{{[subappendix][1][2147483647,2]B.1}{[1][44][]44}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.2}1. An explicit separable minimal extension with a traceless parameter}{44}{subsection.B.2}\protected@file@percent }
\newlabel{odd_tower-an-explicit-separable-minimal-extension-with-a-traceless-parameter}{{B.2}{44}{1. An explicit separable minimal extension with a traceless parameter}{subsection.B.2}{}}
\newlabel{odd_tower-an-explicit-separable-minimal-extension-with-a-traceless-parameter@cref}{{[subappendix][2][2147483647,2]B.2}{[1][44][]44}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.3}2. Normalized corestriction and an explicit filtered right inverse}{44}{subsection.B.3}\protected@file@percent }
\newlabel{odd_tower-normalized-corestriction-and-an-explicit-filtered-right-inverse}{{B.3}{44}{2. Normalized corestriction and an explicit filtered right inverse}{subsection.B.3}{}}
\newlabel{odd_tower-normalized-corestriction-and-an-explicit-filtered-right-inverse@cref}{{[subappendix][3][2147483647,2]B.3}{[1][44][]44}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.4}3. Exact trace identity}{45}{subsection.B.4}\protected@file@percent }
\newlabel{odd_tower-exact-trace-identity}{{B.4}{45}{3. Exact trace identity}{subsection.B.4}{}}
\newlabel{odd_tower-exact-trace-identity@cref}{{[subappendix][4][2147483647,2]B.4}{[1][45][]45}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.5}4. The recursive actual matrix}{45}{subsection.B.5}\protected@file@percent }
\newlabel{odd_tower-the-recursive-actual-matrix}{{B.5}{45}{4. The recursive actual matrix}{subsection.B.5}{}}
\newlabel{odd_tower-the-recursive-actual-matrix@cref}{{[subappendix][5][2147483647,2]B.5}{[1][45][]45}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.6}5. Actual fields and profiles}{46}{subsection.B.6}\protected@file@percent }
\newlabel{odd_tower-actual-fields-and-profiles}{{B.6}{46}{5. Actual fields and profiles}{subsection.B.6}{}}
\newlabel{odd_tower-actual-fields-and-profiles@cref}{{[subappendix][6][2147483647,2]B.6}{[1][46][]46}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.7}6. Native approximations, amplification, and traceless corrections}{46}{subsection.B.7}\protected@file@percent }
\newlabel{odd_tower-native-approximations-amplification-and-traceless-corrections}{{B.7}{46}{6. Native approximations, amplification, and traceless corrections}{subsection.B.7}{}}
\newlabel{odd_tower-native-approximations-amplification-and-traceless-corrections@cref}{{[subappendix][7][2147483647,2]B.7}{[1][46][]46}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.8}7. Precisely what the trace supplies}{47}{subsection.B.8}\protected@file@percent }
\newlabel{odd_tower-precisely-what-the-trace-supplies}{{B.8}{47}{7. Precisely what the trace supplies}{subsection.B.8}{}}
\newlabel{odd_tower-precisely-what-the-trace-supplies@cref}{{[subappendix][8][2147483647,2]B.8}{[1][47][]47}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.9}Primary dependency}{47}{subsection.B.9}\protected@file@percent }
\newlabel{odd_tower-primary-dependency}{{B.9}{47}{Primary dependency}{subsection.B.9}{}}
\newlabel{odd_tower-primary-dependency@cref}{{[subappendix][9][2147483647,2]B.9}{[1][47][]47}}
\@writefile{toc}{\contentsline {section}{\numberline {C}Native-rank full characters and tame central completion}{48}{appendix.C}\protected@file@percent }
\newlabel{app:odd_character}{{C}{48}{Native-rank full characters and tame central completion}{appendix.C}{}}
\newlabel{app:odd_character@cref}{{[appendix][3][2147483647]C}{[1][48][]48}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.1}1. Statement}{48}{subsection.C.1}\protected@file@percent }
\newlabel{odd_character-statement}{{C.1}{48}{1. Statement}{subsection.C.1}{}}
\newlabel{odd_character-statement@cref}{{[subappendix][1][2147483647,3]C.1}{[1][48][]48}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.2}2. The exact arithmetic inputs}{48}{subsection.C.2}\protected@file@percent }
\newlabel{odd_character-the-exact-arithmetic-inputs}{{C.2}{48}{2. The exact arithmetic inputs}{subsection.C.2}{}}
\newlabel{odd_character-the-exact-arithmetic-inputs@cref}{{[subappendix][2][2147483647,3]C.2}{[1][48][]48}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.3}3. A finite character-extension fact}{49}{subsection.C.3}\protected@file@percent }
\newlabel{odd_character-a-finite-character-extension-fact}{{C.3}{49}{3. A finite character-extension fact}{subsection.C.3}{}}
\newlabel{odd_character-a-finite-character-extension-fact@cref}{{[subappendix][3][2147483647,3]C.3}{[1][49][]49}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4}4. Base character at native rank p}{49}{subsection.C.4}\protected@file@percent }
\newlabel{odd_character-base-character-at-native-rank-p}{{C.4}{49}{4. Base character at native rank p}{subsection.C.4}{}}
\newlabel{odd_character-base-character-at-native-rank-p@cref}{{[subappendix][4][2147483647,3]C.4}{[1][49][]49}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.5}5. Transfer the full field character before adding the next layer}{49}{subsection.C.5}\protected@file@percent }
\newlabel{odd_character-transfer-the-full-field-character-before-adding-the-next-layer}{{C.5}{49}{5. Transfer the full field character before adding the next layer}{subsection.C.5}{}}
\newlabel{odd_character-transfer-the-full-field-character-before-adding-the-next-layer@cref}{{[subappendix][5][2147483647,3]C.5}{[1][49][]49}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.6}6. Add the actual correction and extend at maximal rank}{50}{subsection.C.6}\protected@file@percent }
\newlabel{odd_character-add-the-actual-correction-and-extend-at-maximal-rank}{{C.6}{50}{6. Add the actual correction and extend at maximal rank}{subsection.C.6}{}}
\newlabel{odd_character-add-the-actual-correction-and-extend-at-maximal-rank@cref}{{[subappendix][6][2147483647,3]C.6}{[1][49][]50}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.7}7. Complete this same type with the prescribed tame center}{50}{subsection.C.7}\protected@file@percent }
\newlabel{odd_character-complete-this-same-type-with-the-prescribed-tame-center}{{C.7}{50}{7. Complete this same type with the prescribed tame center}{subsection.C.7}{}}
\newlabel{odd_character-complete-this-same-type-with-the-prescribed-tame-center@cref}{{[subappendix][7][2147483647,3]C.7}{[1][50][]50}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.8}8. Full packet attachment in arbitrary cubic length}{51}{subsection.C.8}\protected@file@percent }
\newlabel{odd_character-full-packet-attachment-in-arbitrary-cubic-length}{{C.8}{51}{8. Full packet attachment in arbitrary cubic length}{subsection.C.8}{}}
\newlabel{odd_character-full-packet-attachment-in-arbitrary-cubic-length@cref}{{[subappendix][8][2147483647,3]C.8}{[1][51][]51}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.9}9. Boundary and primary-source check}{52}{subsection.C.9}\protected@file@percent }
\newlabel{odd_character-boundary-and-primary-source-check}{{C.9}{52}{9. Boundary and primary-source check}{subsection.C.9}{}}
\newlabel{odd_character-boundary-and-primary-source-check@cref}{{[subappendix][9][2147483647,3]C.9}{[1][52][]52}}
\@writefile{toc}{\contentsline {section}{\numberline {D}The surviving leading symplectic layer}{53}{appendix.D}\protected@file@percent }
\newlabel{app:odd_capacity}{{D}{53}{The surviving leading symplectic layer}{appendix.D}{}}
\newlabel{app:odd_capacity@cref}{{[appendix][4][2147483647]D}{[1][53][]53}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.1}The useful simplification}{53}{subsection.D.1}\protected@file@percent }
\newlabel{odd_capacity-the-useful-simplification}{{D.1}{53}{The useful simplification}{subsection.D.1}{}}
\newlabel{odd_capacity-the-useful-simplification@cref}{{[subappendix][1][2147483647,4]D.1}{[1][53][]53}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2}1. Leading-layer theorem}{53}{subsection.D.2}\protected@file@percent }
\newlabel{odd_capacity-leading-layer-theorem}{{D.2}{53}{1. Leading-layer theorem}{subsection.D.2}{}}
\newlabel{odd_capacity-leading-layer-theorem@cref}{{[subappendix][2][2147483647,4]D.2}{[1][53][]53}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {D.2.1}1.1. Actual parameter field and leading level}{53}{subsubsection.D.2.1}\protected@file@percent }
\newlabel{odd_capacity-actual-parameter-field-and-leading-level}{{D.2.1}{53}{1.1. Actual parameter field and leading level}{subsubsection.D.2.1}{}}
\newlabel{odd_capacity-actual-parameter-field-and-leading-level@cref}{{[subsubappendix][1][2147483647,4,2]D.2.1}{[1][53][]53}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {D.2.2}1.2. The terminal minimal approximation has degree p}{53}{subsubsection.D.2.2}\protected@file@percent }
\newlabel{odd_capacity-the-terminal-minimal-approximation-has-degree-p}{{D.2.2}{53}{1.2. The terminal minimal approximation has degree p}{subsubsection.D.2.2}{}}
\newlabel{odd_capacity-the-terminal-minimal-approximation-has-degree-p@cref}{{[subsubappendix][2][2147483647,4,2]D.2.2}{[1][53][]53}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {D.2.3}1.3. The high subgroup survives every approximation}{54}{subsubsection.D.2.3}\protected@file@percent }
\newlabel{odd_capacity-the-high-subgroup-survives-every-approximation}{{D.2.3}{54}{1.3. The high subgroup survives every approximation}{subsubsection.D.2.3}{}}
\newlabel{odd_capacity-the-high-subgroup-survives-every-approximation@cref}{{[subsubappendix][3][2147483647,4,2]D.2.3}{[1][54][]54}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {D.2.4}1.4. Its actual commutator form is nondegenerate}{54}{subsubsection.D.2.4}\protected@file@percent }
\newlabel{odd_capacity-its-actual-commutator-form-is-nondegenerate}{{D.2.4}{54}{1.4. Its actual commutator form is nondegenerate}{subsubsection.D.2.4}{}}
\newlabel{odd_capacity-its-actual-commutator-form-is-nondegenerate@cref}{{[subsubappendix][4][2147483647,4,2]D.2.4}{[1][54][]54}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.3}2. Arbitrary-length odd-prime source family}{55}{subsection.D.3}\protected@file@percent }
\newlabel{odd_capacity-arbitrary-length-odd-prime-source-family}{{D.3}{55}{2. Arbitrary-length odd-prime source family}{subsection.D.3}{}}
\newlabel{odd_capacity-arbitrary-length-odd-prime-source-family@cref}{{[subappendix][3][2147483647,4]D.3}{[1][55][]55}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {D.3.1}2.1. Full characters and recovery for declared C=\{1\}}{56}{subsubsection.D.3.1}\protected@file@percent }
\newlabel{odd_capacity-full-characters-and-recovery-for-declared-c1}{{D.3.1}{56}{2.1. Full characters and recovery for declared C=\{1\}}{subsubsection.D.3.1}{}}
\newlabel{odd_capacity-full-characters-and-recovery-for-declared-c1@cref}{{[subsubappendix][1][2147483647,4,3]D.3.1}{[1][56][]56}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {D.3.2}2.2. First new rank: GL\_27}{56}{subsubsection.D.3.2}\protected@file@percent }
\newlabel{odd_capacity-first-new-rank-gl_27}{{D.3.2}{56}{2.2. First new rank: GL\_27}{subsubsection.D.3.2}{}}
\newlabel{odd_capacity-first-new-rank-gl_27@cref}{{[subsubappendix][2][2147483647,4,3]D.3.2}{[1][56][]56}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.4}3. Why trying to force every critical level was unnecessary}{56}{subsection.D.4}\protected@file@percent }
\newlabel{odd_capacity-why-trying-to-force-every-critical-level-was-unnecessary}{{D.4}{56}{3. Why trying to force every critical level was unnecessary}{subsection.D.4}{}}
\newlabel{odd_capacity-why-trying-to-force-every-critical-level-was-unnecessary@cref}{{[subappendix][4][2147483647,4]D.4}{[1][56][]56}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.5}4. Dyadic capacity boundary}{57}{subsection.D.5}\protected@file@percent }
\newlabel{odd_capacity-dyadic-capacity-boundary}{{D.5}{57}{4. Dyadic capacity boundary}{subsection.D.5}{}}
\newlabel{odd_capacity-dyadic-capacity-boundary@cref}{{[subappendix][5][2147483647,4]D.5}{[1][57][]57}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.6}5. Boundaries that remain}{57}{subsection.D.6}\protected@file@percent }
\newlabel{odd_capacity-boundaries-that-remain}{{D.6}{57}{5. Boundaries that remain}{subsection.D.6}{}}
\newlabel{odd_capacity-boundaries-that-remain@cref}{{[subappendix][6][2147483647,4]D.6}{[1][57][]57}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.7}6. Primary locators and retained dependencies}{57}{subsection.D.7}\protected@file@percent }
\newlabel{odd_capacity-primary-locators-and-retained-dependencies}{{D.7}{57}{6. Primary locators and retained dependencies}{subsection.D.7}{}}
\newlabel{odd_capacity-primary-locators-and-retained-dependencies@cref}{{[subappendix][7][2147483647,4]D.7}{[1][57][]57}}
\@writefile{toc}{\contentsline {section}{\numberline {E}Actual scalar inertia phases and finite-cover descent}{58}{appendix.E}\protected@file@percent }
\newlabel{app:odd_scalar}{{E}{58}{Actual scalar inertia phases and finite-cover descent}{appendix.E}{}}
\newlabel{app:odd_scalar@cref}{{[appendix][5][2147483647]E}{[1][58][]58}}
\@writefile{toc}{\contentsline {subsection}{\numberline {E.1}1. Exact statement}{58}{subsection.E.1}\protected@file@percent }
\newlabel{odd_scalar-exact-statement}{{E.1}{58}{1. Exact statement}{subsection.E.1}{}}
\newlabel{odd_scalar-exact-statement@cref}{{[subappendix][1][2147483647,5]E.1}{[1][58][]58}}
\@writefile{toc}{\contentsline {subsection}{\numberline {E.2}2. Construction and proof}{58}{subsection.E.2}\protected@file@percent }
\newlabel{odd_scalar-construction-and-proof}{{E.2}{58}{2. Construction and proof}{subsection.E.2}{}}
\newlabel{odd_scalar-construction-and-proof@cref}{{[subappendix][2][2147483647,5]E.2}{[1][58][]58}}
\@writefile{toc}{\contentsline {subsection}{\numberline {E.3}3. What this closes and what it does not}{59}{subsection.E.3}\protected@file@percent }
\newlabel{odd_scalar-what-this-closes-and-what-it-does-not}{{E.3}{59}{3. What this closes and what it does not}{subsection.E.3}{}}
\newlabel{odd_scalar-what-this-closes-and-what-it-does-not@cref}{{[subappendix][3][2147483647,5]E.3}{[1][59][]59}}
\@writefile{toc}{\contentsline {subsection}{\numberline {E.4}4. Why a direct map C into residue units is too strong}{60}{subsection.E.4}\protected@file@percent }
\newlabel{odd_scalar-why-a-direct-map-c-into-residue-units-is-too-strong}{{E.4}{60}{4. Why a direct map C into residue units is too strong}{subsection.E.4}{}}
\newlabel{odd_scalar-why-a-direct-map-c-into-residue-units-is-too-strong@cref}{{[subappendix][4][2147483647,5]E.4}{[1][60][]60}}
\@writefile{toc}{\contentsline {subsection}{\numberline {E.5}Dependencies and review status}{60}{subsection.E.5}\protected@file@percent }
\newlabel{odd_scalar-dependencies-and-review-status}{{E.5}{60}{Dependencies and review status}{subsection.E.5}{}}
\newlabel{odd_scalar-dependencies-and-review-status@cref}{{[subappendix][5][2147483647,5]E.5}{[1][60][]60}}
\@writefile{toc}{\contentsline {section}{\numberline {F}The precise stabilizer-action contract and normalizer obstruction}{61}{appendix.F}\protected@file@percent }
\newlabel{app:stabilizer_contract}{{F}{61}{The precise stabilizer-action contract and normalizer obstruction}{appendix.F}{}}
\newlabel{app:stabilizer_contract@cref}{{[appendix][6][2147483647]F}{[1][61][]61}}
\@writefile{toc}{\contentsline {subsection}{\numberline {F.1}1. Exact contract, and what is already an input}{61}{subsection.F.1}\protected@file@percent }
\newlabel{stabilizer_contract-exact-contract-and-what-is-already-an-input}{{F.1}{61}{1. Exact contract, and what is already an input}{subsection.F.1}{}}
\newlabel{stabilizer_contract-exact-contract-and-what-is-already-an-input@cref}{{[subappendix][1][2147483647,6]F.1}{[1][61][]61}}
\@writefile{toc}{\contentsline {subsection}{\numberline {F.2}2. Full scalar-extension rigidity for a native tame action}{61}{subsection.F.2}\protected@file@percent }
\newlabel{stabilizer_contract-full-scalar-extension-rigidity-for-a-native-tame-action}{{F.2}{61}{2. Full scalar-extension rigidity for a native tame action}{subsection.F.2}{}}
\newlabel{stabilizer_contract-full-scalar-extension-rigidity-for-a-native-tame-action@cref}{{[subappendix][2][2147483647,6]F.2}{[1][61][]61}}
\@writefile{toc}{\contentsline {subsection}{\numberline {F.3}3. Exact attachment criterion in this native setting}{62}{subsection.F.3}\protected@file@percent }
\newlabel{stabilizer_contract-exact-attachment-criterion-in-this-native-setting}{{F.3}{62}{3. Exact attachment criterion in this native setting}{subsection.F.3}{}}
\newlabel{stabilizer_contract-exact-attachment-criterion-in-this-native-setting@cref}{{[subappendix][3][2147483647,6]F.3}{[1][62][]62}}
\@writefile{toc}{\contentsline {subsection}{\numberline {F.4}4. H2 and matching carrier dimensions do not force trivial C}{62}{subsection.F.4}\protected@file@percent }
\newlabel{stabilizer_contract-h2-and-matching-carrier-dimensions-do-not-force-trivial-c}{{F.4}{62}{4. H2 and matching carrier dimensions do not force trivial C}{subsection.F.4}{}}
\newlabel{stabilizer_contract-h2-and-matching-carrier-dimensions-do-not-force-trivial-c@cref}{{[subappendix][4][2147483647,6]F.4}{[1][62][]62}}
\gdef \LT@i {\LT@entry 
    {1}{247.88557pt}\LT@entry 
    {1}{247.88557pt}}
\@writefile{toc}{\contentsline {subsection}{\numberline {F.5}5. Result for the next construction attempt}{63}{subsection.F.5}\protected@file@percent }
\newlabel{stabilizer_contract-result-for-the-next-construction-attempt}{{F.5}{63}{5. Result for the next construction attempt}{subsection.F.5}{}}
\newlabel{stabilizer_contract-result-for-the-next-construction-attempt@cref}{{[subappendix][5][2147483647,6]F.5}{[1][63][]63}}
\@writefile{toc}{\contentsline {subsection}{\numberline {F.6}Evidence and self-review}{63}{subsection.F.6}\protected@file@percent }
\newlabel{stabilizer_contract-evidence-and-self-review}{{F.6}{63}{Evidence and self-review}{subsection.F.6}{}}
\newlabel{stabilizer_contract-evidence-and-self-review@cref}{{[subappendix][6][2147483647,6]F.6}{[1][63][]63}}
\@writefile{toc}{\contentsline {section}{\numberline {G}Dyadic synthesis and all-length assembly scope}{65}{appendix.G}\protected@file@percent }
\newlabel{app:dyadic_synthesis}{{G}{65}{Dyadic synthesis and all-length assembly scope}{appendix.G}{}}
\newlabel{app:dyadic_synthesis@cref}{{[appendix][7][2147483647]G}{[1][65][]65}}
\@writefile{toc}{\contentsline {subsection}{\numberline {G.1}1. Combined arithmetic theorem}{65}{subsection.G.1}\protected@file@percent }
\newlabel{dyadic_synthesis-combined-arithmetic-theorem}{{G.1}{65}{1. Combined arithmetic theorem}{subsection.G.1}{}}
\newlabel{dyadic_synthesis-combined-arithmetic-theorem@cref}{{[subappendix][1][2147483647,7]G.1}{[1][65][]65}}
\@writefile{toc}{\contentsline {subsection}{\numberline {G.2}2. The explicit construction and its depth invariant}{65}{subsection.G.2}\protected@file@percent }
\newlabel{dyadic_synthesis-the-explicit-construction-and-its-depth-invariant}{{G.2}{65}{2. The explicit construction and its depth invariant}{subsection.G.2}{}}
\newlabel{dyadic_synthesis-the-explicit-construction-and-its-depth-invariant@cref}{{[subappendix][2][2147483647,7]G.2}{[1][65][]65}}
\gdef \LT@ii {\LT@entry 
    {1}{96.15283pt}\LT@entry 
    {1}{101.15283pt}\LT@entry 
    {1}{101.15283pt}\LT@entry 
    {1}{101.15283pt}\LT@entry 
    {1}{96.15283pt}}
\@writefile{toc}{\contentsline {subsection}{\numberline {G.3}3. Full characters, including the odd critical endpoint}{66}{subsection.G.3}\protected@file@percent }
\newlabel{dyadic_synthesis-full-characters-including-the-odd-critical-endpoint}{{G.3}{66}{3. Full characters, including the odd critical endpoint}{subsection.G.3}{}}
\newlabel{dyadic_synthesis-full-characters-including-the-odd-critical-endpoint@cref}{{[subappendix][3][2147483647,7]G.3}{[1][66][]66}}
\@writefile{toc}{\contentsline {subsection}{\numberline {G.4}4. Exact packet capacity and marked attachment}{66}{subsection.G.4}\protected@file@percent }
\newlabel{dyadic_synthesis-exact-packet-capacity-and-marked-attachment}{{G.4}{66}{4. Exact packet capacity and marked attachment}{subsection.G.4}{}}
\newlabel{dyadic_synthesis-exact-packet-capacity-and-marked-attachment@cref}{{[subappendix][4][2147483647,7]G.4}{[1][66][]66}}
\@writefile{toc}{\contentsline {subsection}{\numberline {G.5}5. Failed repairs and boundaries}{67}{subsection.G.5}\protected@file@percent }
\newlabel{dyadic_synthesis-failed-repairs-and-boundaries}{{G.5}{67}{5. Failed repairs and boundaries}{subsection.G.5}{}}
\newlabel{dyadic_synthesis-failed-repairs-and-boundaries@cref}{{[subappendix][5][2147483647,7]G.5}{[1][67][]67}}
\@writefile{toc}{\contentsline {subsection}{\numberline {G.6}6. Evidence}{68}{subsection.G.6}\protected@file@percent }
\newlabel{dyadic_synthesis-evidence}{{G.6}{68}{6. Evidence}{subsection.G.6}{}}
\newlabel{dyadic_synthesis-evidence@cref}{{[subappendix][6][2147483647,7]G.6}{[1][68][]68}}
\@writefile{toc}{\contentsline {subsection}{\numberline {G.7}7. The low-rank repair: enlarge the actual inducing carrier}{68}{subsection.G.7}\protected@file@percent }
\newlabel{dyadic_synthesis-the-low-rank-repair-enlarge-the-actual-inducing-carrier}{{G.7}{68}{7. The low-rank repair: enlarge the actual inducing carrier}{subsection.G.7}{}}
\newlabel{dyadic_synthesis-the-low-rank-repair-enlarge-the-actual-inducing-carrier@cref}{{[subappendix][7][2147483647,7]G.7}{[1][68][]68}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {G.7.1}Rank two}{68}{subsubsection.G.7.1}\protected@file@percent }
\newlabel{dyadic_synthesis-rank-two}{{G.7.1}{68}{Rank two}{subsubsection.G.7.1}{}}
\newlabel{dyadic_synthesis-rank-two@cref}{{[subsubappendix][1][2147483647,7,7]G.7.1}{[1][68][]68}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {G.7.2}Rank four}{69}{subsubsection.G.7.2}\protected@file@percent }
\newlabel{dyadic_synthesis-rank-four}{{G.7.2}{69}{Rank four}{subsubsection.G.7.2}{}}
\newlabel{dyadic_synthesis-rank-four@cref}{{[subsubappendix][2][2147483647,7,7]G.7.2}{[1][68][]69}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {G.7.3}Exact recovery, with the noncompact normalization retained}{69}{subsubsection.G.7.3}\protected@file@percent }
\newlabel{dyadic_synthesis-exact-recovery-with-the-noncompact-normalization-retained}{{G.7.3}{69}{Exact recovery, with the noncompact normalization retained}{subsubsection.G.7.3}{}}
\newlabel{dyadic_synthesis-exact-recovery-with-the-noncompact-normalization-retained@cref}{{[subsubappendix][3][2147483647,7,7]G.7.3}{[1][69][]69}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {G.7.4}What the low-rank repair admits}{69}{subsubsection.G.7.4}\protected@file@percent }
\newlabel{dyadic_synthesis-what-the-low-rank-repair-admits}{{G.7.4}{69}{What the low-rank repair admits}{subsubsection.G.7.4}{}}
\newlabel{dyadic_synthesis-what-the-low-rank-repair-admits@cref}{{[subsubappendix][4][2147483647,7,7]G.7.4}{[1][69][]69}}
\@writefile{toc}{\contentsline {subsection}{\numberline {G.8}8. Final scope and next construction question}{70}{subsection.G.8}\protected@file@percent }
\newlabel{dyadic_synthesis-final-scope-and-next-construction-question}{{G.8}{70}{8. Final scope and next construction question}{subsection.G.8}{}}
\newlabel{dyadic_synthesis-final-scope-and-next-construction-question@cref}{{[subappendix][8][2147483647,7]G.8}{[1][70][]70}}
\@writefile{toc}{\contentsline {section}{\numberline {H}The explicit dyadic arithmetic tower}{71}{appendix.H}\protected@file@percent }
\newlabel{app:dyadic_tower}{{H}{71}{The explicit dyadic arithmetic tower}{appendix.H}{}}
\newlabel{app:dyadic_tower@cref}{{[appendix][8][2147483647]H}{[1][71][]71}}
\@writefile{toc}{\contentsline {subsection}{\numberline {H.1}1. Statement}{71}{subsection.H.1}\protected@file@percent }
\newlabel{dyadic_tower-statement}{{H.1}{71}{1. Statement}{subsection.H.1}{}}
\newlabel{dyadic_tower-statement@cref}{{[subappendix][1][2147483647,8]H.1}{[1][71][]71}}
\@writefile{toc}{\contentsline {subsection}{\numberline {H.2}2. The minimal degree-two building block}{71}{subsection.H.2}\protected@file@percent }
\newlabel{dyadic_tower-the-minimal-degree-two-building-block}{{H.2}{71}{2. The minimal degree-two building block}{subsection.H.2}{}}
\newlabel{dyadic_tower-the-minimal-degree-two-building-block@cref}{{[subappendix][2][2147483647,8]H.2}{[1][71][]71}}
\@writefile{toc}{\contentsline {subsection}{\numberline {H.3}3. An explicit normalized corestriction and lift}{71}{subsection.H.3}\protected@file@percent }
\newlabel{dyadic_tower-an-explicit-normalized-corestriction-and-lift}{{H.3}{71}{3. An explicit normalized corestriction and lift}{subsection.H.3}{}}
\newlabel{dyadic_tower-an-explicit-normalized-corestriction-and-lift@cref}{{[subappendix][3][2147483647,8]H.3}{[1][71][]71}}
\@writefile{toc}{\contentsline {subsection}{\numberline {H.4}4. Exact trace transport}{72}{subsection.H.4}\protected@file@percent }
\newlabel{dyadic_tower-exact-trace-transport}{{H.4}{72}{4. Exact trace transport}{subsection.H.4}{}}
\newlabel{dyadic_tower-exact-trace-transport@cref}{{[subappendix][4][2147483647,8]H.4}{[1][72][]72}}
\@writefile{toc}{\contentsline {subsection}{\numberline {H.5}5. Recursive actual matrices and fields}{72}{subsection.H.5}\protected@file@percent }
\newlabel{dyadic_tower-recursive-actual-matrices-and-fields}{{H.5}{72}{5. Recursive actual matrices and fields}{subsection.H.5}{}}
\newlabel{dyadic_tower-recursive-actual-matrices-and-fields@cref}{{[subappendix][5][2147483647,8]H.5}{[1][72][]72}}
\@writefile{toc}{\contentsline {subsection}{\numberline {H.6}6. Trace 1 at every native stage}{73}{subsection.H.6}\protected@file@percent }
\newlabel{dyadic_tower-trace-1-at-every-native-stage}{{H.6}{73}{6. Trace 1 at every native stage}{subsection.H.6}{}}
\newlabel{dyadic_tower-trace-1-at-every-native-stage@cref}{{[subappendix][6][2147483647,8]H.6}{[1][73][]73}}
\@writefile{toc}{\contentsline {subsection}{\numberline {H.7}7. Full arithmetic defining sequence}{73}{subsection.H.7}\protected@file@percent }
\newlabel{dyadic_tower-full-arithmetic-defining-sequence}{{H.7}{73}{7. Full arithmetic defining sequence}{subsection.H.7}{}}
\newlabel{dyadic_tower-full-arithmetic-defining-sequence@cref}{{[subappendix][7][2147483647,8]H.7}{[1][73][]73}}
\@writefile{toc}{\contentsline {subsection}{\numberline {H.8}8. Concrete rank-eight instance and limits}{74}{subsection.H.8}\protected@file@percent }
\newlabel{dyadic_tower-concrete-rank-eight-instance-and-limits}{{H.8}{74}{8. Concrete rank-eight instance and limits}{subsection.H.8}{}}
\newlabel{dyadic_tower-concrete-rank-eight-instance-and-limits@cref}{{[subappendix][8][2147483647,8]H.8}{[1][74][]74}}
\@writefile{toc}{\contentsline {subsection}{\numberline {H.9}Dependencies and provenance}{74}{subsection.H.9}\protected@file@percent }
\newlabel{dyadic_tower-dependencies-and-provenance}{{H.9}{74}{Dependencies and provenance}{subsection.H.9}{}}
\newlabel{dyadic_tower-dependencies-and-provenance@cref}{{[subappendix][9][2147483647,8]H.9}{[1][74][]74}}
\@writefile{toc}{\contentsline {section}{\numberline {I}Dyadic full characters and the odd critical endpoint}{75}{appendix.I}\protected@file@percent }
\newlabel{app:dyadic_character}{{I}{75}{Dyadic full characters and the odd critical endpoint}{appendix.I}{}}
\newlabel{app:dyadic_character@cref}{{[appendix][9][2147483647]I}{[1][75][]75}}
\@writefile{toc}{\contentsline {subsection}{\numberline {I.1}1. Exact theorem}{75}{subsection.I.1}\protected@file@percent }
\newlabel{dyadic_character-exact-theorem}{{I.1}{75}{1. Exact theorem}{subsection.I.1}{}}
\newlabel{dyadic_character-exact-theorem@cref}{{[subappendix][1][2147483647,9]I.1}{[1][75][]75}}
\@writefile{toc}{\contentsline {subsection}{\numberline {I.2}2. Actual dyadic recursion and exact native traces}{75}{subsection.I.2}\protected@file@percent }
\newlabel{dyadic_character-actual-dyadic-recursion-and-exact-native-traces}{{I.2}{75}{2. Actual dyadic recursion and exact native traces}{subsection.I.2}{}}
\newlabel{dyadic_character-actual-dyadic-recursion-and-exact-native-traces@cref}{{[subappendix][2][2147483647,9]I.2}{[1][75][]75}}
\@writefile{toc}{\contentsline {subsection}{\numberline {I.3}3. The odd-critical-level rounding}{77}{subsection.I.3}\protected@file@percent }
\newlabel{dyadic_character-the-odd-critical-level-rounding}{{I.3}{77}{3. The odd-critical-level rounding}{subsection.I.3}{}}
\newlabel{dyadic_character-the-odd-critical-level-rounding@cref}{{[subappendix][3][2147483647,9]I.3}{[1][77][]77}}
\@writefile{toc}{\contentsline {subsection}{\numberline {I.4}4. Finite character extension at maximal rank}{77}{subsection.I.4}\protected@file@percent }
\newlabel{dyadic_character-finite-character-extension-at-maximal-rank}{{I.4}{77}{4. Finite character extension at maximal rank}{subsection.I.4}{}}
\newlabel{dyadic_character-finite-character-extension-at-maximal-rank@cref}{{[subappendix][4][2147483647,9]I.4}{[1][77][]77}}
\@writefile{toc}{\contentsline {subsection}{\numberline {I.5}5. Base full character with trace one}{77}{subsection.I.5}\protected@file@percent }
\newlabel{dyadic_character-base-full-character-with-trace-one}{{I.5}{77}{5. Base full character with trace one}{subsection.I.5}{}}
\newlabel{dyadic_character-base-full-character-with-trace-one@cref}{{[subappendix][5][2147483647,9]I.5}{[1][77][]77}}
\@writefile{toc}{\contentsline {subsection}{\numberline {I.6}6. Native-rank induction via full beta-determinant transfer}{78}{subsection.I.6}\protected@file@percent }
\newlabel{dyadic_character-native-rank-induction-via-full-beta-determinant-transfer}{{I.6}{78}{6. Native-rank induction via full beta-determinant transfer}{subsection.I.6}{}}
\newlabel{dyadic_character-native-rank-induction-via-full-beta-determinant-transfer@cref}{{[subappendix][6][2147483647,9]I.6}{[1][78][]78}}
\@writefile{toc}{\contentsline {subsection}{\numberline {I.7}7. Canonical beta extension and prescribed tame center}{79}{subsection.I.7}\protected@file@percent }
\newlabel{dyadic_character-canonical-beta-extension-and-prescribed-tame-center}{{I.7}{79}{7. Canonical beta extension and prescribed tame center}{subsection.I.7}{}}
\newlabel{dyadic_character-canonical-beta-extension-and-prescribed-tame-center@cref}{{[subappendix][7][2147483647,9]I.7}{[1][79][]79}}
\@writefile{toc}{\contentsline {subsection}{\numberline {I.8}8. What this closes and what it does not}{79}{subsection.I.8}\protected@file@percent }
\newlabel{dyadic_character-what-this-closes-and-what-it-does-not}{{I.8}{79}{8. What this closes and what it does not}{subsection.I.8}{}}
\newlabel{dyadic_character-what-this-closes-and-what-it-does-not@cref}{{[subappendix][8][2147483647,9]I.8}{[1][79][]79}}
\@writefile{toc}{\contentsline {subsection}{\numberline {I.9}9. Exact source checks and input pins}{79}{subsection.I.9}\protected@file@percent }
\newlabel{dyadic_character-exact-source-checks-and-input-pins}{{I.9}{79}{9. Exact source checks and input pins}{subsection.I.9}{}}
\newlabel{dyadic_character-exact-source-checks-and-input-pins@cref}{{[subappendix][9][2147483647,9]I.9}{[1][79][]79}}
\gdef \LT@iii {\LT@entry 
    {1}{79.31105pt}\LT@entry 
    {1}{84.31105pt}\LT@entry 
    {1}{84.31105pt}\LT@entry 
    {1}{84.31105pt}\LT@entry 
    {1}{84.31105pt}\LT@entry 
    {1}{79.31105pt}}
\@writefile{toc}{\contentsline {section}{\numberline {J}Exact dyadic capacity and numerical-profile obstruction}{81}{appendix.J}\protected@file@percent }
\newlabel{app:dyadic_capacity}{{J}{81}{Exact dyadic capacity and numerical-profile obstruction}{appendix.J}{}}
\newlabel{app:dyadic_capacity@cref}{{[appendix][10][2147483647]J}{[1][81][]81}}
\@writefile{toc}{\contentsline {subsection}{\numberline {J.1}1. Exact hypotheses and result}{81}{subsection.J.1}\protected@file@percent }
\newlabel{dyadic_capacity-exact-hypotheses-and-result}{{J.1}{81}{1. Exact hypotheses and result}{subsection.J.1}{}}
\newlabel{dyadic_capacity-exact-hypotheses-and-result@cref}{{[subappendix][1][2147483647,10]J.1}{[1][81][]81}}
\@writefile{toc}{\contentsline {subsection}{\numberline {J.2}2. Floors and ceilings in the actual BK orders}{81}{subsection.J.2}\protected@file@percent }
\newlabel{dyadic_capacity-floors-and-ceilings-in-the-actual-bk-orders}{{J.2}{81}{2. Floors and ceilings in the actual BK orders}{subsection.J.2}{}}
\newlabel{dyadic_capacity-floors-and-ceilings-in-the-actual-bk-orders@cref}{{[subappendix][2][2147483647,10]J.2}{[1][81][]81}}
\@writefile{toc}{\contentsline {subsection}{\numberline {J.3}3. Conductors still match despite the lost final contribution}{82}{subsection.J.3}\protected@file@percent }
\newlabel{dyadic_capacity-conductors-still-match-despite-the-lost-final-contribution}{{J.3}{82}{3. Conductors still match despite the lost final contribution}{subsection.J.3}{}}
\newlabel{dyadic_capacity-conductors-still-match-despite-the-lost-final-contribution@cref}{{[subappendix][3][2147483647,10]J.3}{[1][82][]82}}
\@writefile{toc}{\contentsline {subsection}{\numberline {J.4}4. Rank2 failure is forced by Swan and torsion alone}{83}{subsection.J.4}\protected@file@percent }
\newlabel{dyadic_capacity-rank2-failure-is-forced-by-swan-and-torsion-alone}{{J.4}{83}{4. Rank2 failure is forced by Swan and torsion alone}{subsection.J.4}{}}
\newlabel{dyadic_capacity-rank2-failure-is-forced-by-swan-and-torsion-alone@cref}{{[subappendix][4][2147483647,10]J.4}{[1][83][]83}}
\@writefile{toc}{\contentsline {subsection}{\numberline {J.5}5. Rank4 failure is also forced by Swan and torsion}{83}{subsection.J.5}\protected@file@percent }
\newlabel{dyadic_capacity-rank4-failure-is-also-forced-by-swan-and-torsion}{{J.5}{83}{5. Rank4 failure is also forced by Swan and torsion}{subsection.J.5}{}}
\newlabel{dyadic_capacity-rank4-failure-is-also-forced-by-swan-and-torsion@cref}{{[subappendix][5][2147483647,10]J.5}{[1][83][]83}}
\@writefile{toc}{\contentsline {subsection}{\numberline {J.6}6. Rank8: the three invariants do not force the good profile}{84}{subsection.J.6}\protected@file@percent }
\newlabel{dyadic_capacity-rank8-the-three-invariants-do-not-force-the-good-profile}{{J.6}{84}{6. Rank8: the three invariants do not force the good profile}{subsection.J.6}{}}
\newlabel{dyadic_capacity-rank8-the-three-invariants-do-not-force-the-good-profile@cref}{{[subappendix][6][2147483647,10]J.6}{[1][84][]84}}
\@writefile{toc}{\contentsline {subsection}{\numberline {J.7}7. Which repairs have actually been tested}{85}{subsection.J.7}\protected@file@percent }
\newlabel{dyadic_capacity-which-repairs-have-actually-been-tested}{{J.7}{85}{7. Which repairs have actually been tested}{subsection.J.7}{}}
\newlabel{dyadic_capacity-which-repairs-have-actually-been-tested@cref}{{[subappendix][7][2147483647,10]J.7}{[1][85][]85}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {J.7.1}Replacing the displayed type or parameter at fixed Swan and torsion}{85}{subsubsection.J.7.1}\protected@file@percent }
\newlabel{dyadic_capacity-replacing-the-displayed-type-or-parameter-at-fixed-swan-and-torsion}{{J.7.1}{85}{Replacing the displayed type or parameter at fixed Swan and torsion}{subsubsection.J.7.1}{}}
\newlabel{dyadic_capacity-replacing-the-displayed-type-or-parameter-at-fixed-swan-and-torsion@cref}{{[subsubappendix][1][2147483647,10,7]J.7.1}{[1][85][]85}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {J.7.2}Skipping approximation degrees or changing critical parity}{85}{subsubsection.J.7.2}\protected@file@percent }
\newlabel{dyadic_capacity-skipping-approximation-degrees-or-changing-critical-parity}{{J.7.2}{85}{Skipping approximation degrees or changing critical parity}{subsubsection.J.7.2}{}}
\newlabel{dyadic_capacity-skipping-approximation-degrees-or-changing-critical-parity@cref}{{[subsubappendix][2][2147483647,10,7]J.7.2}{[1][85][]85}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {J.7.3}Scalar twists, shallow or deep}{85}{subsubsection.J.7.3}\protected@file@percent }
\newlabel{dyadic_capacity-scalar-twists-shallow-or-deep}{{J.7.3}{85}{Scalar twists, shallow or deep}{subsubsection.J.7.3}{}}
\newlabel{dyadic_capacity-scalar-twists-shallow-or-deep@cref}{{[subsubappendix][3][2147483647,10,7]J.7.3}{[1][85][]85}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {J.7.4}Adding residue multiplicity or increasing ambient rank}{86}{subsubsection.J.7.4}\protected@file@percent }
\newlabel{dyadic_capacity-adding-residue-multiplicity-or-increasing-ambient-rank}{{J.7.4}{86}{Adding residue multiplicity or increasing ambient rank}{subsubsection.J.7.4}{}}
\newlabel{dyadic_capacity-adding-residue-multiplicity-or-increasing-ambient-rank@cref}{{[subsubappendix][4][2147483647,10,7]J.7.4}{[1][86][]86}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {J.7.5}Using a larger arithmetic carrier inside the same representation}{86}{subsubsection.J.7.5}\protected@file@percent }
\newlabel{dyadic_capacity-using-a-larger-arithmetic-carrier-inside-the-same-representation}{{J.7.5}{86}{Using a larger arithmetic carrier inside the same representation}{subsubsection.J.7.5}{}}
\newlabel{dyadic_capacity-using-a-larger-arithmetic-carrier-inside-the-same-representation@cref}{{[subsubappendix][5][2147483647,10,7]J.7.5}{[1][86][]86}}
\@writefile{toc}{\contentsline {subsection}{\numberline {J.8}8. Quadratic refinements and marked centers}{86}{subsection.J.8}\protected@file@percent }
\newlabel{dyadic_capacity-quadratic-refinements-and-marked-centers}{{J.8}{86}{8. Quadratic refinements and marked centers}{subsection.J.8}{}}
\newlabel{dyadic_capacity-quadratic-refinements-and-marked-centers@cref}{{[subappendix][8][2147483647,10]J.8}{[1][86][]86}}
\@writefile{toc}{\contentsline {subsection}{\numberline {J.9}9. Scope and source provenance}{86}{subsection.J.9}\protected@file@percent }
\newlabel{dyadic_capacity-scope-and-source-provenance}{{J.9}{86}{9. Scope and source provenance}{subsection.J.9}{}}
\newlabel{dyadic_capacity-scope-and-source-provenance@cref}{{[subappendix][9][2147483647,10]J.9}{[1][86][]86}}
\@writefile{toc}{\contentsline {section}{\numberline {K}Whole finite extensions and dyadic marking criteria}{88}{appendix.K}\protected@file@percent }
\newlabel{app:dyadic_markings}{{K}{88}{Whole finite extensions and dyadic marking criteria}{appendix.K}{}}
\newlabel{app:dyadic_markings@cref}{{[appendix][11][2147483647]K}{[1][88][]88}}
\@writefile{toc}{\contentsline {subsection}{\numberline {K.1}1. The original datum includes more than the alternating form}{88}{subsection.K.1}\protected@file@percent }
\newlabel{dyadic_markings-the-original-datum-includes-more-than-the-alternating-form}{{K.1}{88}{1. The original datum includes more than the alternating form}{subsection.K.1}{}}
\newlabel{dyadic_markings-the-original-datum-includes-more-than-the-alternating-form@cref}{{[subappendix][1][2147483647,11]K.1}{[1][88][]88}}
\@writefile{toc}{\contentsline {subsection}{\numberline {K.2}2. General finite central-extension criterion}{88}{subsection.K.2}\protected@file@percent }
\newlabel{dyadic_markings-general-finite-central-extension-criterion}{{K.2}{88}{2. General finite central-extension criterion}{subsection.K.2}{}}
\newlabel{dyadic_markings-general-finite-central-extension-criterion@cref}{{[subappendix][2][2147483647,11]K.2}{[1][88][]88}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {K.2.1}2.1. Fixed two-element center}{89}{subsubsection.K.2.1}\protected@file@percent }
\newlabel{dyadic_markings-fixed-two-element-center}{{K.2.1}{89}{2.1. Fixed two-element center}{subsubsection.K.2.1}{}}
\newlabel{dyadic_markings-fixed-two-element-center@cref}{{[subsubappendix][1][2147483647,11,2]K.2.1}{[1][89][]89}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {K.2.2}2.2. Four-element and complex centers}{89}{subsubsection.K.2.2}\protected@file@percent }
\newlabel{dyadic_markings-four-element-and-complex-centers}{{K.2.2}{89}{2.2. Four-element and complex centers}{subsubsection.K.2.2}{}}
\newlabel{dyadic_markings-four-element-and-complex-centers@cref}{{[subsubappendix][2][2147483647,11,2]K.2.2}{[1][89][]89}}
\@writefile{toc}{\contentsline {subsection}{\numberline {K.3}3. Exact D8/Q8 check}{89}{subsection.K.3}\protected@file@percent }
\newlabel{dyadic_markings-exact-d8q8-check}{{K.3}{89}{3. Exact D8/Q8 check}{subsection.K.3}{}}
\newlabel{dyadic_markings-exact-d8q8-check@cref}{{[subappendix][3][2147483647,11]K.3}{[1][89][]89}}
\@writefile{toc}{\contentsline {subsection}{\numberline {K.4}4. Marked scalar-pushout attachment with C=\{1\}}{90}{subsection.K.4}\protected@file@percent }
\newlabel{dyadic_markings-marked-scalar-pushout-attachment-with-c1}{{K.4}{90}{4. Marked scalar-pushout attachment with C=\{1\}}{subsection.K.4}{}}
\newlabel{dyadic_markings-marked-scalar-pushout-attachment-with-c1@cref}{{[subappendix][4][2147483647,11]K.4}{[1][90][]90}}
\@writefile{toc}{\contentsline {subsection}{\numberline {K.5}5. If the host refinement is independently fixed}{91}{subsection.K.5}\protected@file@percent }
\newlabel{dyadic_markings-if-the-host-refinement-is-independently-fixed}{{K.5}{91}{5. If the host refinement is independently fixed}{subsection.K.5}{}}
\newlabel{dyadic_markings-if-the-host-refinement-is-independently-fixed@cref}{{[subappendix][5][2147483647,11]K.5}{[1][91][]91}}
\@writefile{toc}{\contentsline {subsection}{\numberline {K.6}6. Exact outcome for the dyadic construction}{92}{subsection.K.6}\protected@file@percent }
\newlabel{dyadic_markings-exact-outcome-for-the-dyadic-construction}{{K.6}{92}{6. Exact outcome for the dyadic construction}{subsection.K.6}{}}
\newlabel{dyadic_markings-exact-outcome-for-the-dyadic-construction@cref}{{[subappendix][6][2147483647,11]K.6}{[1][92][]92}}
\@writefile{toc}{\contentsline {subsection}{\numberline {K.7}Evidence and author self-review}{92}{subsection.K.7}\protected@file@percent }
\newlabel{dyadic_markings-evidence-and-author-self-review}{{K.7}{92}{Evidence and author self-review}{subsection.K.7}{}}
\newlabel{dyadic_markings-evidence-and-author-self-review@cref}{{[subappendix][7][2147483647,11]K.7}{[1][92][]92}}
\@writefile{toc}{\contentsline {section}{\numberline {L}The actual enlarged rank-two inducing carrier}{93}{appendix.L}\protected@file@percent }
\newlabel{app:rank2}{{L}{93}{The actual enlarged rank-two inducing carrier}{appendix.L}{}}
\newlabel{app:rank2@cref}{{[appendix][12][2147483647]L}{[1][93][]93}}
\@writefile{toc}{\contentsline {subsection}{\numberline {L.1}1. Exact setting and result}{93}{subsection.L.1}\protected@file@percent }
\newlabel{rank2-exact-setting-and-result}{{L.1}{93}{1. Exact setting and result}{subsection.L.1}{}}
\newlabel{rank2-exact-setting-and-result@cref}{{[subappendix][1][2147483647,12]L.1}{[1][93][]93}}
\@writefile{toc}{\contentsline {subsection}{\numberline {L.2}2. The compact enlargement is a group, with normal subgroup of index two}{94}{subsection.L.2}\protected@file@percent }
\newlabel{rank2-the-compact-enlargement-is-a-group-with-normal-subgroup-of-index-two}{{L.2}{94}{2. The compact enlargement is a group, with normal subgroup of index two}{subsection.L.2}{}}
\newlabel{rank2-the-compact-enlargement-is-a-group-with-normal-subgroup-of-index-two@cref}{{[subappendix][2][2147483647,12]L.2}{[1][93][]94}}
\@writefile{toc}{\contentsline {subsection}{\numberline {L.3}3. Irreducibility and the conjugate-character ratio}{94}{subsection.L.3}\protected@file@percent }
\newlabel{rank2-irreducibility-and-the-conjugate-character-ratio}{{L.3}{94}{3. Irreducibility and the conjugate-character ratio}{subsection.L.3}{}}
\newlabel{rank2-irreducibility-and-the-conjugate-character-ratio@cref}{{[subappendix][3][2147483647,12]L.3}{[1][94][]94}}
\@writefile{toc}{\contentsline {subsection}{\numberline {L.4}4. The actual finite Heisenberg pair}{95}{subsection.L.4}\protected@file@percent }
\newlabel{rank2-the-actual-finite-heisenberg-pair}{{L.4}{95}{4. The actual finite Heisenberg pair}{subsection.L.4}{}}
\newlabel{rank2-the-actual-finite-heisenberg-pair@cref}{{[subappendix][4][2147483647,12]L.4}{[1][95][]95}}
\@writefile{toc}{\contentsline {subsection}{\numberline {L.5}5. Explicit embedding into the same fixed candidate}{96}{subsection.L.5}\protected@file@percent }
\newlabel{rank2-explicit-embedding-into-the-same-fixed-candidate}{{L.5}{96}{5. Explicit embedding into the same fixed candidate}{subsection.L.5}{}}
\newlabel{rank2-explicit-embedding-into-the-same-fixed-candidate@cref}{{[subappendix][5][2147483647,12]L.5}{[1][95][]96}}
\@writefile{toc}{\contentsline {subsection}{\numberline {L.6}6. Exact recovery by the noncompact Hecke eigenvalue}{96}{subsection.L.6}\protected@file@percent }
\newlabel{rank2-exact-recovery-by-the-noncompact-hecke-eigenvalue}{{L.6}{96}{6. Exact recovery by the noncompact Hecke eigenvalue}{subsection.L.6}{}}
\newlabel{rank2-exact-recovery-by-the-noncompact-hecke-eigenvalue@cref}{{[subappendix][6][2147483647,12]L.6}{[1][96][]96}}
\@writefile{toc}{\contentsline {subsection}{\numberline {L.7}7. Full scalar-pushout attachment and remaining marking obligations}{97}{subsection.L.7}\protected@file@percent }
\newlabel{rank2-full-scalar-pushout-attachment-and-remaining-marking-obligations}{{L.7}{97}{7. Full scalar-pushout attachment and remaining marking obligations}{subsection.L.7}{}}
\newlabel{rank2-full-scalar-pushout-attachment-and-remaining-marking-obligations@cref}{{[subappendix][7][2147483647,12]L.7}{[1][97][]97}}
\@writefile{toc}{\contentsline {subsection}{\numberline {L.8}8. Scope, smallest boundary, and sources}{98}{subsection.L.8}\protected@file@percent }
\newlabel{rank2-scope-smallest-boundary-and-sources}{{L.8}{98}{8. Scope, smallest boundary, and sources}{subsection.L.8}{}}
\newlabel{rank2-scope-smallest-boundary-and-sources@cref}{{[subappendix][8][2147483647,12]L.8}{[1][98][]98}}
\@writefile{toc}{\contentsline {section}{\numberline {M}The actual enlarged rank-four inducing carrier}{99}{appendix.M}\protected@file@percent }
\newlabel{app:rank4}{{M}{99}{The actual enlarged rank-four inducing carrier}{appendix.M}{}}
\newlabel{app:rank4@cref}{{[appendix][13][2147483647]M}{[1][99][]99}}
\@writefile{toc}{\contentsline {subsection}{\numberline {M.1}Result and boundary}{99}{subsection.M.1}\protected@file@percent }
\newlabel{rank4-result-and-boundary}{{M.1}{99}{Result and boundary}{subsection.M.1}{}}
\newlabel{rank4-result-and-boundary@cref}{{[subappendix][1][2147483647,13]M.1}{[1][99][]99}}
\@writefile{toc}{\contentsline {subsection}{\numberline {M.2}1. Actual arithmetic groups and notation}{99}{subsection.M.2}\protected@file@percent }
\newlabel{rank4-actual-arithmetic-groups-and-notation}{{M.2}{99}{1. Actual arithmetic groups and notation}{subsection.M.2}{}}
\newlabel{rank4-actual-arithmetic-groups-and-notation@cref}{{[subappendix][2][2147483647,13]M.2}{[1][99][]99}}
\@writefile{toc}{\contentsline {subsection}{\numberline {M.3}2. The actual final field has the needed graded line}{100}{subsection.M.3}\protected@file@percent }
\newlabel{rank4-the-actual-final-field-has-the-needed-graded-line}{{M.3}{100}{2. The actual final field has the needed graded line}{subsection.M.3}{}}
\newlabel{rank4-the-actual-final-field-has-the-needed-graded-line@cref}{{[subappendix][3][2147483647,13]M.3}{[1][100][]100}}
\@writefile{toc}{\contentsline {subsection}{\numberline {M.4}3. Normality and index of the enlarged group}{100}{subsection.M.4}\protected@file@percent }
\newlabel{rank4-normality-and-index-of-the-enlarged-group}{{M.4}{100}{3. Normality and index of the enlarged group}{subsection.M.4}{}}
\newlabel{rank4-normality-and-index-of-the-enlarged-group@cref}{{[subappendix][4][2147483647,13]M.4}{[1][100][]100}}
\@writefile{toc}{\contentsline {subsection}{\numberline {M.5}4. The actual character has an orbit of length two}{101}{subsection.M.5}\protected@file@percent }
\newlabel{rank4-the-actual-character-has-an-orbit-of-length-two}{{M.5}{101}{4. The actual character has an orbit of length two}{subsection.M.5}{}}
\newlabel{rank4-the-actual-character-has-an-orbit-of-length-two@cref}{{[subappendix][5][2147483647,13]M.5}{[1][100][]101}}
\@writefile{toc}{\contentsline {subsection}{\numberline {M.6}5. An actual nondegenerate symplectic quotient of dimension four}{101}{subsection.M.6}\protected@file@percent }
\newlabel{rank4-an-actual-nondegenerate-symplectic-quotient-of-dimension-four}{{M.6}{101}{5. An actual nondegenerate symplectic quotient of dimension four}{subsection.M.6}{}}
\newlabel{rank4-an-actual-nondegenerate-symplectic-quotient-of-dimension-four@cref}{{[subappendix][6][2147483647,13]M.6}{[1][101][]101}}
\@writefile{toc}{\contentsline {subsection}{\numberline {M.7}6. The carrier occurs in the same supercuspidal candidate}{102}{subsection.M.7}\protected@file@percent }
\newlabel{rank4-the-carrier-occurs-in-the-same-supercuspidal-candidate}{{M.7}{102}{6. The carrier occurs in the same supercuspidal candidate}{subsection.M.7}{}}
\newlabel{rank4-the-carrier-occurs-in-the-same-supercuspidal-candidate@cref}{{[subappendix][7][2147483647,13]M.7}{[1][102][]102}}
\@writefile{toc}{\contentsline {subsection}{\numberline {M.8}7. Exact limits: finite image versus literal extraspecial group}{102}{subsection.M.8}\protected@file@percent }
\newlabel{rank4-exact-limits-finite-image-versus-literal-extraspecial-group}{{M.8}{102}{7. Exact limits: finite image versus literal extraspecial group}{subsection.M.8}{}}
\newlabel{rank4-exact-limits-finite-image-versus-literal-extraspecial-group@cref}{{[subappendix][8][2147483647,13]M.8}{[1][102][]102}}
\@writefile{toc}{\contentsline {subsection}{\numberline {M.9}Provenance}{102}{subsection.M.9}\protected@file@percent }
\newlabel{rank4-provenance}{{M.9}{102}{Provenance}{subsection.M.9}{}}
\newlabel{rank4-provenance@cref}{{[subappendix][9][2147483647,13]M.9}{[1][102][]102}}
\@writefile{toc}{\contentsline {section}{\numberline {N}Rank-four normalization and induction in stages}{103}{appendix.N}\protected@file@percent }
\newlabel{app:rank4_stages}{{N}{103}{Rank-four normalization and induction in stages}{appendix.N}{}}
\newlabel{app:rank4_stages@cref}{{[appendix][14][2147483647]N}{[1][103][]103}}
\@writefile{toc}{\contentsline {subsection}{\numberline {N.1}1. Normalization by the actual parameter field}{103}{subsection.N.1}\protected@file@percent }
\newlabel{rank4_stages-normalization-by-the-actual-parameter-field}{{N.1}{103}{1. Normalization by the actual parameter field}{subsection.N.1}{}}
\newlabel{rank4_stages-normalization-by-the-actual-parameter-field@cref}{{[subappendix][1][2147483647,14]N.1}{[1][103][]103}}
\@writefile{toc}{\contentsline {subsection}{\numberline {N.2}2. The two inducing groups have index two}{104}{subsection.N.2}\protected@file@percent }
\newlabel{rank4_stages-the-two-inducing-groups-have-index-two}{{N.2}{104}{2. The two inducing groups have index two}{subsection.N.2}{}}
\newlabel{rank4_stages-the-two-inducing-groups-have-index-two@cref}{{[subappendix][2][2147483647,14]N.2}{[1][104][]104}}
\@writefile{toc}{\contentsline {subsection}{\numberline {N.3}3. The enlarged inducing representation is irreducible}{104}{subsection.N.3}\protected@file@percent }
\newlabel{rank4_stages-the-enlarged-inducing-representation-is-irreducible}{{N.3}{104}{3. The enlarged inducing representation is irreducible}{subsection.N.3}{}}
\newlabel{rank4_stages-the-enlarged-inducing-representation-is-irreducible@cref}{{[subappendix][3][2147483647,14]N.3}{[1][104][]104}}
\@writefile{toc}{\contentsline {subsection}{\numberline {N.4}4. Exact recovery by induction in stages}{104}{subsection.N.4}\protected@file@percent }
\newlabel{rank4_stages-exact-recovery-by-induction-in-stages}{{N.4}{104}{4. Exact recovery by induction in stages}{subsection.N.4}{}}
\newlabel{rank4_stages-exact-recovery-by-induction-in-stages@cref}{{[subappendix][4][2147483647,14]N.4}{[1][104][]104}}
\@writefile{toc}{\contentsline {subsection}{\numberline {N.5}Scope}{104}{subsection.N.5}\protected@file@percent }
\newlabel{rank4_stages-scope}{{N.5}{104}{Scope}{subsection.N.5}{}}
\newlabel{rank4_stages-scope@cref}{{[subappendix][5][2147483647,14]N.5}{[1][104][]104}}
\@writefile{toc}{\contentsline {section}{\numberline {O}Exact enlarged-carrier recovery and original-contract comparison}{106}{appendix.O}\protected@file@percent }
\newlabel{app:lowrank_contract}{{O}{106}{Exact enlarged-carrier recovery and original-contract comparison}{appendix.O}{}}
\newlabel{app:lowrank_contract@cref}{{[appendix][15][2147483647]O}{[1][106][]106}}
\@writefile{toc}{\contentsline {subsection}{\numberline {O.1}1. What original H2 literally requires}{106}{subsection.O.1}\protected@file@percent }
\newlabel{lowrank_contract-what-original-h2-literally-requires}{{O.1}{106}{1. What original H2 literally requires}{subsection.O.1}{}}
\newlabel{lowrank_contract-what-original-h2-literally-requires@cref}{{[subappendix][1][2147483647,15]O.1}{[1][106][]106}}
\@writefile{toc}{\contentsline {subsection}{\numberline {O.2}2. Precise conditional setup}{106}{subsection.O.2}\protected@file@percent }
\newlabel{lowrank_contract-precise-conditional-setup}{{O.2}{106}{2. Precise conditional setup}{subsection.O.2}{}}
\newlabel{lowrank_contract-precise-conditional-setup@cref}{{[subappendix][2][2147483647,15]O.2}{[1][106][]106}}
\@writefile{toc}{\contentsline {subsection}{\numberline {O.3}3. Xi occurs in the same pi, and can be an equivalent compact type}{107}{subsection.O.3}\protected@file@percent }
\newlabel{lowrank_contract-xi-occurs-in-the-same-pi-and-can-be-an-equivalent-compact-type}{{O.3}{107}{3. Xi occurs in the same pi, and can be an equivalent compact type}{subsection.O.3}{}}
\newlabel{lowrank_contract-xi-occurs-in-the-same-pi-and-can-be-an-equivalent-compact-type@cref}{{[subappendix][3][2147483647,15]O.3}{[1][107][]107}}
\@writefile{toc}{\contentsline {subsection}{\numberline {O.4}4. Exact recovery before fixing the center: a quotient}{107}{subsection.O.4}\protected@file@percent }
\newlabel{lowrank_contract-exact-recovery-before-fixing-the-center-a-quotient}{{O.4}{107}{4. Exact recovery before fixing the center: a quotient}{subsection.O.4}{}}
\newlabel{lowrank_contract-exact-recovery-before-fixing-the-center-a-quotient@cref}{{[subappendix][4][2147483647,15]O.4}{[1][107][]107}}
\@writefile{toc}{\contentsline {subsection}{\numberline {O.5}5. After retaining the center: a finite Hecke idempotent}{108}{subsection.O.5}\protected@file@percent }
\newlabel{lowrank_contract-after-retaining-the-center-a-finite-hecke-idempotent}{{O.5}{108}{5. After retaining the center: a finite Hecke idempotent}{subsection.O.5}{}}
\newlabel{lowrank_contract-after-retaining-the-center-a-finite-hecke-idempotent@cref}{{[subappendix][5][2147483647,15]O.5}{[1][108][]108}}
\@writefile{toc}{\contentsline {subsection}{\numberline {O.6}6. The exact remaining admissibility checks}{109}{subsection.O.6}\protected@file@percent }
\newlabel{lowrank_contract-the-exact-remaining-admissibility-checks}{{O.6}{109}{6. The exact remaining admissibility checks}{subsection.O.6}{}}
\newlabel{lowrank_contract-the-exact-remaining-admissibility-checks@cref}{{[subappendix][6][2147483647,15]O.6}{[1][109][]109}}
\@writefile{toc}{\contentsline {subsection}{\numberline {O.7}Evidence and self-review}{109}{subsection.O.7}\protected@file@percent }
\newlabel{lowrank_contract-evidence-and-self-review}{{O.7}{109}{Evidence and self-review}{subsection.O.7}{}}
\newlabel{lowrank_contract-evidence-and-self-review@cref}{{[subappendix][7][2147483647,15]O.7}{[1][109][]109}}
\bibstyle{amsplain}
\bibdata{Paper_XI_Traditional_Final}
\bibcite{BerndtEvansWilliams1998}{1}
\bibcite{Blondel2012}{2}
\bibcite{Brown1982}{3}
\bibcite{BushnellHenniart1996}{4}
\bibcite{BushnellHenniart1999}{5}
\bibcite{BushnellHenniart2005ET1}{6}
\bibcite{BushnellHenniart2005ET2}{7}
\bibcite{BushnellHenniart2006}{8}
\bibcite{BushnellHenniart2010ET3}{9}
\bibcite{BushnellHenniart2017}{10}
\bibcite{BushnellKutzko1993}{11}
\bibcite{DavenportHasse1935}{12}
\bibcite{DeligneLusztig1976}{13}
\bibcite{Dotto2022}{14}
\bibcite{Gerardin1977}{15}
\bibcite{GurevichHadani2007}{16}
\bibcite{HarrisTaylor2001}{17}
\bibcite{Henniart1993}{18}
\bibcite{Henniart2000}{19}
\bibcite{HenniartLemaire2011}{20}
\bibcite{Isaacs1976}{21}
\bibcite{Karpilovsky1985}{22}
\bibcite{Mackey1952}{23}
\bibcite{MilnorHusemoller1973}{24}
\bibcite{Neuhauser2002}{25}
\bibcite{Prasad2009}{26}
\bibcite{Serre1967}{27}
\bibcite{Serre1977}{28}
\bibcite{Tate1979}{29}
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{0pt}
\newlabel{tocindent1}{0pt}
\newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt}
\gdef \@abspage@last{111}