\makeatletter
\newlabel{cce-lem:weighted-logs}{{11.17}{248}{Weighted logarithmic moments}{lemma.980}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cce-lem:weighted-logs@cref}{{[lemma][17][11]11.17}{[1][248][]248}{}{}{}}
\newlabel{cce-thm:consumer}{{11.19}{255}{Entropy dissipation for the remaining coordinates}{theorem.1013}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cce-thm:consumer@cref}{{[theorem][19][11]11.19}{[1][254][]255}{}{}{}}
\newlabel{ccf-eq:finite-profile-closure}{{11.85}{259}{Combining a conditional relative entropy bound with the first coordinates}{equation.1041}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{ccf-eq:finite-profile-closure@cref}{{[equation][85][11]11.85}{[1][259][]259}{}{}{}}
\newlabel{ccf-lem:finite-product-closure}{{11.21}{259}{Combining a conditional relative entropy bound with the first coordinates}{lemma.1039}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{ccf-lem:finite-product-closure@cref}{{[lemma][21][11]11.21}{[1][259][]259}{}{}{}}
\newlabel{cd-heat-matrix}{{7.13}{117}{Precisions and half-precisions}{equation.427}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cd-heat-matrix@cref}{{[equation][13][7]7.13}{[1][117][]117}{}{}{}}
\newlabel{cd-heat-profiles}{{7.14}{131}{Heat estimates in the critical window}{lemma.490}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cd-heat-profiles@cref}{{[lemma][14][7]7.14}{[1][131][]131}{}{}{}}
\newlabel{cd-lift}{{10.2}{197}{The unnormalized process}{lemma.740}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cd-lift@cref}{{[lemma][2][10]10.2}{[1][196][]197}{}{}{}}
\newlabel{cd-lift-moments}{{10.10}{210}{Two-sided moments in the critical window}{lemma.799}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cd-lift-moments@cref}{{[lemma][10][10]10.10}{[1][210][]210}{}{}{}}
\newlabel{cd-renewal-resolvent}{{7.14}{117}{Precisions and half-precisions}{equation.428}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cd-renewal-resolvent@cref}{{[equation][14][7]7.14}{[1][117][]117}{}{}{}}
\newlabel{cg-canonical-weyl}{{4.17}{78}{Half-precisions in the critical window (cf.\ \cite [Proposition~4.1, (4.2)]{DuHuang2026})}{lemma.269}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cg-canonical-weyl@cref}{{[lemma][17][4]4.17}{[1][78][]78}{}{}{}}
\newlabel{cg-critical-lsi}{{8.21}{171}{Poincar\'e and logarithmic Sobolev constants in the critical window}{proposition.633}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cg-critical-lsi@cref}{{[proposition][21][8]8.21}{[1][170][]171}{}{}{}}
\newlabel{cg-many-soft-pi}{{8.9}{150}{Poincar\'e inequality when many precisions are comparable to the smallest one}{corollary.559}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cg-many-soft-pi@cref}{{[corollary][9][8]8.9}{[1][150][]150}{}{}{}}
\newlabel{cg-mean-level-density}{{8.12}{157}{Densities of weighted sums of squared Gaussians}{lemma.585}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cg-mean-level-density@cref}{{[lemma][12][8]8.12}{[1][156][]157}{}{}{}}
\newlabel{cg-source-l2-decay}{{12.14}{278}{Exponential relaxation in the critical window}{proposition.1129}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cg-source-l2-decay@cref}{{[proposition][14][12]12.14}{[1][278][]278}{}{}{}}
\newlabel{cg-three-soft-lsi}{{8.13}{158}{Logarithmic Sobolev inequality when three variances are comparable to the largest one}{corollary.589}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cg-three-soft-lsi@cref}{{[corollary][13][8]8.13}{[1][158][]158}{}{}{}}
\newlabel{cg-weyl-hypotheses}{{4.36}{78}{Critical spectral conditions}{equation.268}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cg-weyl-hypotheses@cref}{{[equation][36][4]4.36}{[1][78][]78}{}{}{}}
\newlabel{cl3-actual-source-density}{{12.13}{277}{Density moments in the critical window}{theorem.1127}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cl3-actual-source-density@cref}{{[theorem][13][12]12.13}{[1][277][]277}{}{}{}}
\newlabel{cor:comp-poisson-flux}{{9.7}{181}{The defect of the Bingham comparison law as a divergence}{corollary.678}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cor:comp-poisson-flux@cref}{{[corollary][7][9]9.7}{[1][181][]181}{}{}{}}
\newlabel{cor:model-profile-cutoff}{{3.12}{56}{Cutoff profiles imply cutoff}{corollary.166}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cor:model-profile-cutoff@cref}{{[corollary][12][3]3.12}{[1][56][]56}{}{}{}}
\newlabel{cot-profile}{{3.9}{53}{Properties of the limiting profiles}{lemma.156}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cot-profile@cref}{{[lemma][9][3]3.9}{[1][53][]53}{}{}{}}
\newlabel{cq-ce-source-valid-unshifted-matrix-lift}{{10.2}{197}{The unnormalized process}{lemma.740}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{cq-ce-source-valid-unshifted-matrix-lift@cref}{{[lemma][2][10]10.2}{[1][196][]197}{}{}{}}
\newlabel{csh-cor:cold-head}{{11.12}{243}{Total variation convergence of finitely many coordinates}{corollary.952}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{csh-cor:cold-head@cref}{{[corollary][12][11]11.12}{[1][243][]243}{}{}{}}
\newlabel{csh-lem:convolution-fisher}{{11.7}{238}{Fisher information of a Gaussian convolution}{lemma.926}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{csh-lem:convolution-fisher@cref}{{[lemma][7][11]11.7}{[1][238][]238}{}{}{}}
\newlabel{csh-lem:cut-score}{{11.2}{230}{Multiplying by a cutoff function}{lemma.887}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{csh-lem:cut-score@cref}{{[lemma][2][11]11.2}{[1][230][]230}{}{}{}}
\newlabel{csh-lem:graph-fisher}{{11.11}{242}{Fisher information of the standardized coordinates}{lemma.946}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{csh-lem:graph-fisher@cref}{{[lemma][11][11]11.11}{[1][242][]242}{}{}{}}
\newlabel{csh-thm:weak-to-tv}{{11.8}{238}{From weak convergence to total variation}{theorem.928}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{csh-thm:weak-to-tv@cref}{{[theorem][8][11]11.8}{[1][238][]238}{}{}{}}
\newlabel{csr-thm:reset}{{11.5}{233}{A Bismut-type formula for the score over a final time interval}{theorem.903}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{csr-thm:reset@cref}{{[theorem][5][11]11.5}{[1][233][]233}{}{}{}}
\newlabel{def:comp-defect}{{9.2}{176}{Forward-equation defect}{definition.660}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{def:comp-defect@cref}{{[definition][2][9]9.2}{[1][176][]176}{}{}{}}
\newlabel{def:comp-variance-conditions}{{9.15}{189}{Variance conditions at scale $\delta $}{definition.711}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{def:comp-variance-conditions@cref}{{[definition][15][9]9.15}{[1][189][]189}{}{}{}}
\newlabel{def:cond-countable}{{5.8}{91}{Countable conditioned law}{definition.315}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{def:cond-countable@cref}{{[definition][8][5]5.8}{[1][91][]91}{}{}{}}
\newlabel{def:cond-finite}{{5.4}{88}{Gaussian law conditioned on a sphere}{definition.299}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{def:cond-finite@cref}{{[definition][4][5]5.4}{[1][88][]88}{}{}{}}
\newlabel{def:dens-hyp}{{12.2}{263}{Hypotheses on the time interval $[0,S]$}{definition.1061}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{def:dens-hyp@cref}{{[definition][2][12]12.2}{[1][263][]263}{}{}{}}
\newlabel{def:heat-covariance}{{7.5}{119}{Covariance evolution and Bingham comparison law}{definition.437}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{def:heat-covariance@cref}{{[definition][5][7]7.5}{[1][119][]119}{}{}{}}
\newlabel{def:model-cutoff}{{3.7}{51}{Mixing profiles, cutoff profiles, and cutoff}{definition.143}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{def:model-cutoff@cref}{{[definition][7][3]3.7}{[1][51][]51}{}{}{}}
\newlabel{def:model-information}{{3.5}{48}{Information}{definition.130}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{def:model-information@cref}{{[definition][5][3]3.5}{[1][48][]48}{}{}{}}
\newlabel{def:model-multiplier}{{3.3}{47}{Lagrange multiplier and precisions}{definition.125}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{def:model-multiplier@cref}{{[definition][3][3]3.3}{[1][47][]47}{}{}{}}
\newlabel{def:spec-critical-conditions}{{4.16}{78}{Critical spectral conditions}{definition.267}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{def:spec-critical-conditions@cref}{{[definition][16][4]4.16}{[1][78][]78}{}{}{}}
\newlabel{eq:comp-soft-set}{{9.41}{190}{Variance conditions at scale $\delta $}{equation.713}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:comp-soft-set@cref}{{[equation][41][9]9.41}{[1][189][]190}{}{}{}}
\newlabel{eq:comp-vartheta}{{9.2}{174}{Notation}{equation.643}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:comp-vartheta@cref}{{[equation][2][9]9.2}{[1][174][]174}{}{}{}}
\newlabel{eq:dens-dynamics}{{12.2}{261}{Half-precisions and the dynamics}{equation.1046}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:dens-dynamics@cref}{{[equation][2][12]12.2}{[1][261][]261}{}{}{}}
\newlabel{eq:dens-heat}{{12.3}{262}{The comparison law}{equation.1048}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:dens-heat@cref}{{[equation][3][12]12.3}{[1][262][]262}{}{}{}}
\newlabel{eq:dens-trace}{{12.1}{261}{Half-precisions and the dynamics}{equation.1045}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:dens-trace@cref}{{[equation][1][12]12.1}{[1][261][]261}{}{}{}}
\newlabel{eq:heat-GS}{{7.5}{114}{Notation}{equation.402}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:heat-GS@cref}{{[equation][5][7]7.5}{[1][114][]114}{}{}{}}
\newlabel{eq:heat-errors}{{7.4}{114}{Notation}{equation.401}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:heat-errors@cref}{{[equation][4][7]7.4}{[1][114][]114}{}{}{}}
\newlabel{eq:heat-objects}{{7.3}{114}{Notation}{equation.400}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:heat-objects@cref}{{[equation][3][7]7.3}{[1][114][]114}{}{}{}}
\newlabel{eq:heat-trace}{{7.1}{113}{Notation}{equation.398}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:heat-trace@cref}{{[equation][1][7]7.1}{[1][113][]113}{}{}{}}
\newlabel{eq:model-cauchy-profiles}{{3.17}{53}{The limiting profiles}{equation.154}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:model-cauchy-profiles@cref}{{[equation][17][3]3.17}{[1][53][]53}{}{}{}}
\newlabel{eq:model-cold-parameters}{{3.14}{49}{Temperature regimes}{equation.133}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:model-cold-parameters@cref}{{[equation][14][3]3.14}{[1][49][]49}{}{}{}}
\newlabel{eq:model-dirichlet}{{3.4}{44}{General quadratic potentials}{equation.111}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:model-dirichlet@cref}{{[equation][4][3]3.4}{[1][44][]44}{}{}{}}
\newlabel{eq:model-fi}{{3.5}{44}{General quadratic potentials}{equation.112}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:model-fi@cref}{{[equation][5][3]3.5}{[1][44][]44}{}{}{}}
\newlabel{eq:model-general}{{3.3}{44}{General quadratic potentials}{equation.110}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:model-general@cref}{{[equation][3][3]3.3}{[1][44][]44}{}{}{}}
\newlabel{eq:model-multiplier}{{3.10}{47}{Lagrange multiplier and precisions}{equation.126}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:model-multiplier@cref}{{[equation][10][3]3.10}{[1][47][]47}{}{}{}}
\newlabel{eq:model-precision-form}{{3.9}{45}{Adding a multiple of the identity}{equation.118}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:model-precision-form@cref}{{[equation][9][3]3.9}{[1][45][]45}{}{}{}}
\newlabel{eq:model-raw}{{3.8}{45}{Adding a multiple of the identity}{equation.117}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:model-raw@cref}{{[equation][8][3]3.8}{[1][45][]45}{}{}{}}
\newlabel{eq:model-sde}{{3.2}{43}{Normalization}{equation.108}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:model-sde@cref}{{[equation][2][3]3.2}{[1][43][]43}{}{}{}}
\newlabel{eq:model-shift}{{3.7}{45}{Adding a multiple of the identity}{equation.116}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:model-shift@cref}{{[equation][7][3]3.7}{[1][45][]45}{}{}{}}
\newlabel{eq:score-H-D}{{11.46}{245}{Conditional relative entropy}{equation.968}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:score-H-D@cref}{{[equation][46][11]11.46}{[1][245][]245}{}{}{}}
\newlabel{eq:score-anisotropic-lsi}{{11.51}{247}{Conditional relative entropy}{equation.976}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:score-anisotropic-lsi@cref}{{[equation][51][11]11.51}{[1][247][]247}{}{}{}}
\newlabel{eq:score-chain-rule}{{11.47}{246}{Conditional relative entropy}{equation.969}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:score-chain-rule@cref}{{[equation][47][11]11.47}{[1][245][]246}{}{}{}}
\newlabel{eq:score-chart-ibp}{{11.44}{245}{The generator in the chart and a logarithmic Sobolev inequality}{equation.964}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:score-chart-ibp@cref}{{[equation][44][11]11.44}{[1][245][]245}{}{}{}}
\newlabel{eq:score-fold-tv}{{11.30}{240}{The chart}{equation.933}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:score-fold-tv@cref}{{[equation][30][11]11.30}{[1][240][]240}{}{}{}}
\newlabel{eq:score-forward-chart}{{11.55}{250}{The time derivative of the conditional relative entropy}{equation.988}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:score-forward-chart@cref}{{[equation][55][11]11.55}{[1][250][]250}{}{}{}}
\newlabel{eq:score-gaussian}{{11.32}{241}{The chart}{equation.939}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:score-gaussian@cref}{{[equation][32][11]11.32}{[1][241][]241}{}{}{}}
\newlabel{eq:score-gaussian-lsi}{{11.50}{247}{Conditional relative entropy}{equation.975}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:score-gaussian-lsi@cref}{{[equation][50][11]11.50}{[1][247][]247}{}{}{}}
\newlabel{eq:score-graph-hyp}{{11.66}{254}{Hypotheses}{equation.1009}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:score-graph-hyp@cref}{{[equation][66][11]11.66}{[1][254][]254}{}{}{}}
\newlabel{eq:score-sigma}{{11.7}{231}{Weighted and killed endpoint measures}{equation.892}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:score-sigma@cref}{{[equation][7][11]11.7}{[1][231][]231}{}{}{}}
\newlabel{eq:score-subgaussian}{{11.14}{234}{An exponential bound}{equation.907}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:score-subgaussian@cref}{{[equation][14][11]11.14}{[1][234][]234}{}{}{}}
\newlabel{eq:score-translation}{{11.29}{239}{Weak convergence to total variation convergence by Gaussian smoothing}{equation.930}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:score-translation@cref}{{[equation][29][11]11.29}{[1][239][]239}{}{}{}}
\newlabel{eq:spec-cold-consequences}{{4.11}{64}{Eigenvalue gaps and the top fraction at low temperature}{equation.198}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:spec-cold-consequences@cref}{{[equation][11][4]4.11}{[1][64][]64}{}{}{}}
\newlabel{eq:spec-limiting-half-precisions}{{4.30}{75}{The limiting equation}{equation.250}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:spec-limiting-half-precisions@cref}{{[equation][30][4]4.30}{[1][75][]75}{}{}{}}
\newlabel{eq:spec-stieltjes}{{4.1}{57}{Notation}{equation.169}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{eq:spec-stieltjes@cref}{{[equation][1][4]4.1}{[1][57][]57}{}{}{}}
\newlabel{gp-head-density}{{5.9}{91}{Marginal densities}{lemma.316}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{gp-head-density@cref}{{[lemma][9][5]5.9}{[1][91][]91}{}{}{}}
\newlabel{gp-lan-assumptions}{{6.23}{111}{Gaussian limits of likelihood ratios on a common sphere}{equation.387}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{gp-lan-assumptions@cref}{{[equation][23][6]6.23}{[1][110][]111}{}{}{}}
\newlabel{gp-lan-tv}{{6.25}{111}{Gaussian limits of likelihood ratios on a common sphere}{equation.389}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{gp-lan-tv@cref}{{[equation][25][6]6.25}{[1][111][]111}{}{}{}}
\newlabel{gp-pin-definition}{{5.10}{88}{Gaussian law conditioned on a sphere}{equation.300}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{gp-pin-definition@cref}{{[equation][10][5]5.10}{[1][88][]88}{}{}{}}
\newlabel{gp-pinned-lan}{{6.8}{111}{Gaussian limits of likelihood ratios on a common sphere}{theorem.386}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{gp-pinned-lan@cref}{{[theorem][8][6]6.8}{[1][110][]111}{}{}{}}
\newlabel{gp-pinned-lan-quantitative-third}{{6.6}{109}{Quantitative likelihood ratios on a common sphere}{theorem.376}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{gp-pinned-lan-quantitative-third@cref}{{[theorem][6][6]6.6}{[1][109][]109}{}{}{}}
\newlabel{gp-standardization}{{5.1}{83}{Notation}{equation.282}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{gp-standardization@cref}{{[equation][1][5]5.1}{[1][83][]83}{}{}{}}
\newlabel{hfk-pointwise-fk}{{10.21}{226}{A Feynman--Kac bound for the weighted gradient}{lemma.874}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{hfk-pointwise-fk@cref}{{[lemma][21][10]10.21}{[1][226][]226}{}{}{}}
\newlabel{hi3-finite-heat}{{7.10}{125}{Bounds on $\bar h$ under precision-spacing bounds}{lemma.461}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{hi3-finite-heat@cref}{{[lemma][10][7]7.10}{[1][125][]125}{}{}{}}
\newlabel{hi3-indexed-geometry}{{7.30}{124}{Estimates under two-sided precision-spacing bounds}{equation.458}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{hi3-indexed-geometry@cref}{{[equation][30][7]7.30}{[1][124][]124}{}{}{}}
\newlabel{hi3-information-envelopes}{{7.12}{128}{Relative errors before and after the time $d^{-1}$}{lemma.474}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{hi3-information-envelopes@cref}{{[lemma][12][7]7.12}{[1][128][]128}{}{}{}}
\newlabel{hi3-late-coordinate}{{7.41}{128}{Relative errors before and after the time $d^{-1}$}{equation.477}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{hi3-late-coordinate@cref}{{[equation][41][7]7.41}{[1][128][]128}{}{}{}}
\newlabel{hmr-main}{{10.18}{221}{Maximal inverse moments of the squared norm}{theorem.846}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{hmr-main@cref}{{[theorem][18][10]10.18}{[1][220][]221}{}{}{}}
\newlabel{hp-monotone-profile-transfer}{{3.11}{55}{Monotone profiles and mixing times}{lemma.161}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{hp-monotone-profile-transfer@cref}{{[lemma][11][3]3.11}{[1][54][]55}{}{}{}}
\newlabel{hp-universal-profile}{{3.16}{53}{The limiting profiles}{equation.153}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{hp-universal-profile@cref}{{[equation][16][3]3.16}{[1][53][]53}{}{}{}}
\newlabel{ht-bingham-bridge}{{9.13}{186}{Comparison with the Bingham comparison law when $Nd^2$ is large}{theorem.699}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{ht-bingham-bridge@cref}{{[theorem][13][9]9.13}{[1][186][]186}{}{}{}}
\newlabel{ht-bingham-bridge-bound}{{9.36}{186}{Comparison with the Bingham comparison law when $Nd^2$ is large}{equation.701}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{ht-bingham-bridge-bound@cref}{{[equation][36][9]9.36}{[1][186][]186}{}{}{}}
\newlabel{ht-covariance-identities}{{7.21}{120}{Positive covariance evolution and uniform variance bounds}{equation.441}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{ht-covariance-identities@cref}{{[equation][21][7]7.21}{[1][120][]120}{}{}{}}
\newlabel{ht-covariance-ode}{{7.19}{119}{Covariance evolution and Bingham comparison law}{equation.438}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{ht-covariance-ode@cref}{{[equation][19][7]7.19}{[1][119][]119}{}{}{}}
\newlabel{ht-covariance-system}{{7.6}{120}{Positive covariance evolution and uniform variance bounds}{lemma.440}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{ht-covariance-system@cref}{{[lemma][6][7]7.6}{[1][120][]120}{}{}{}}
\newlabel{ht-covariance-variance-range}{{7.22}{121}{Positive covariance evolution and uniform variance bounds}{equation.442}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{ht-covariance-variance-range@cref}{{[equation][22][7]7.22}{[1][120][]121}{}{}{}}
\newlabel{ht-forward-defect}{{9.3}{177}{The defect of the Bingham comparison law}{lemma.662}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{ht-forward-defect@cref}{{[lemma][3][9]9.3}{[1][177][]177}{}{}{}}
\newlabel{ht-many-soft-sharp-gaps}{{8.8}{149}{Spectral gap when many precisions are comparable to the smallest one}{theorem.552}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{ht-many-soft-sharp-gaps@cref}{{[theorem][8][8]8.8}{[1][149][]149}{}{}{}}
\newlabel{ht-small-field-direct-comparison}{{9.14}{188}{Comparison for a nearly isotropic matrix}{proposition.708}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{ht-small-field-direct-comparison@cref}{{[proposition][14][9]9.14}{[1][188][]188}{}{}{}}
\newlabel{ht-small-field-geometry}{{9.1}{176}{The Lagrange multiplier of a bounded or nearly isotropic matrix}{lemma.658}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{ht-small-field-geometry@cref}{{[lemma][1][9]9.1}{[1][176][]176}{}{}{}}
\newlabel{ht-weighted-bridge-dimension-condition}{{9.35}{186}{Comparison with the Bingham comparison law when $Nd^2$ is large}{equation.700}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{ht-weighted-bridge-dimension-condition@cref}{{[equation][35][9]9.35}{[1][186][]186}{}{}{}}
\newlabel{iw-all-time-heat}{{7.11}{127}{The coordinates of the heat vector}{lemma.472}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{iw-all-time-heat@cref}{{[lemma][11][7]7.11}{[1][127][]127}{}{}{}}
\newlabel{iw-chemical-occupation}{{10.15}{216}{Exponential moments of $\int _0^T\Gamma (\log \varphi )(X_s)\,ds$}{theorem.828}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{iw-chemical-occupation@cref}{{[theorem][15][10]10.15}{[1][216][]216}{}{}{}}
\newlabel{iw-early-pin-simultaneous-centers}{{10.16}{217}{Uniform-in-time comparison of the initial quadratic forms}{lemma.834}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{iw-early-pin-simultaneous-centers@cref}{{[lemma][16][10]10.16}{[1][217][]217}{}{}{}}
\newlabel{lem:comp-poisson}{{9.6}{181}{The Poisson equation for a Bingham law}{lemma.677}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:comp-poisson@cref}{{[lemma][6][9]9.6}{[1][180][]181}{}{}{}}
\newlabel{lem:cond-chaos}{{5.14}{96}{Quadratic statistics of independent Gaussians}{lemma.332}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:cond-chaos@cref}{{[lemma][14][5]5.14}{[1][96][]96}{}{}{}}
\newlabel{lem:cond-consistent}{{5.7}{91}{Consistent marginals}{lemma.314}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:cond-consistent@cref}{{[lemma][7][5]5.7}{[1][90][]91}{}{}{}}
\newlabel{lem:cond-three}{{5.2}{86}{Bounded densities from three coordinates}{lemma.289}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:cond-three@cref}{{[lemma][2][5]5.2}{[1][86][]86}{}{}{}}
\newlabel{lem:dens-coefficients}{{12.4}{265}{Coefficients of the comparison law}{lemma.1064}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:dens-coefficients@cref}{{[lemma][4][12]12.4}{[1][265][]265}{}{}{}}
\newlabel{lem:dens-count}{{12.3}{264}{Sums over the half-precisions}{lemma.1063}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:dens-count@cref}{{[lemma][3][12]12.3}{[1][264][]264}{}{}{}}
\newlabel{lem:dens-derivative}{{12.7}{271}{R\'enyi divergence from a time-dependent Bingham law}{lemma.1098}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:dens-derivative@cref}{{[lemma][7][12]12.7}{[1][271][]271}{}{}{}}
\newlabel{lem:dens-tilt}{{12.5}{268}{Exponential moments under the comparison law}{lemma.1082}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:dens-tilt@cref}{{[lemma][5][12]12.5}{[1][268][]268}{}{}{}}
\newlabel{lem:heat-algebra}{{7.1}{114}{The heat matrix}{lemma.404}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:heat-algebra@cref}{{[lemma][1][7]7.1}{[1][114][]114}{}{}{}}
\newlabel{lem:model-localization}{{3.8}{52}{Convergence in probability from fixed-confidence estimates}{lemma.150}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:model-localization@cref}{{[lemma][8][3]3.8}{[1][52][]52}{}{}{}}
\newlabel{lem:model-symmetry}{{3.1}{45}{Reflections of eigencoordinates}{lemma.120}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:model-symmetry@cref}{{[lemma][1][3]3.1}{[1][45][]45}{}{}{}}
\newlabel{lem:model-tv}{{3.6}{50}{}{lemma.136}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:model-tv@cref}{{[lemma][6][3]3.6}{[1][50][]50}{}{}{}}
\newlabel{lem:proc-normalization}{{10.2}{197}{The unnormalized process}{lemma.740}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:proc-normalization@cref}{{[lemma][2][10]10.2}{[1][196][]197}{}{}{}}
\newlabel{lem:score-chart}{{11.9}{240}{Densities and scores in the chart}{lemma.934}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:score-chart@cref}{{[lemma][9][11]11.9}{[1][240][]240}{}{}{}}
\newlabel{lem:score-chart-generator}{{11.14}{245}{The generator in the chart}{lemma.965}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:score-chart-generator@cref}{{[lemma][14][11]11.14}{[1][245][]245}{}{}{}}
\newlabel{lem:score-domination}{{11.3}{231}{Removing a path weight}{lemma.890}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:score-domination@cref}{{[lemma][3][11]11.3}{[1][231][]231}{}{}{}}
\newlabel{lem:score-marginal-score}{{11.10}{241}{Marginal score}{lemma.940}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:score-marginal-score@cref}{{[lemma][10][11]11.10}{[1][241][]241}{}{}{}}
\newlabel{lem:score-regularity}{{11.4}{231}{Weighted and killed endpoint measures}{lemma.891}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:score-regularity@cref}{{[lemma][4][11]11.4}{[1][231][]231}{}{}{}}
\newlabel{lem:score-schur}{{11.15}{246}{The metric of the remaining coordinates}{lemma.970}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:score-schur@cref}{{[lemma][15][11]11.15}{[1][246][]246}{}{}{}}
\newlabel{lem:spec-quantiles}{{4.1}{58}{Semicircle quantiles}{lemma.170}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{lem:spec-quantiles@cref}{{[lemma][1][4]4.1}{[1][58][]58}{}{}{}}
\newlabel{mc-eq:hv1}{{11.35}{242}{Standardized coordinates}{equation.944}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-eq:hv1@cref}{{[equation][35][11]11.35}{[1][242][]242}{}{}{}}
\newlabel{mc-eq:hv2}{{11.36}{242}{Standardized coordinates}{equation.945}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-eq:hv2@cref}{{[equation][36][11]11.36}{[1][242][]242}{}{}{}}
\newlabel{mc-eq:sr1}{{11.2}{229}{The diffusion}{equation.881}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-eq:sr1@cref}{{[equation][2][11]11.2}{[1][229][]229}{}{}{}}
\newlabel{mc-eq:sr5}{{11.13}{233}{A Bismut-type formula for the score over a final time interval}{equation.905}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-eq:sr5@cref}{{[equation][13][11]11.13}{[1][233][]233}{}{}{}}
\newlabel{mc-eq:te1}{{11.65}{254}{Hypotheses}{equation.1008}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-eq:te1@cref}{{[equation][65][11]11.65}{[1][254][]254}{}{}{}}
\newlabel{mc-eq:te13}{{11.75}{256}{Step 2: scores after weighting by the cutoff}{equation.1024}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-eq:te13@cref}{{[equation][75][11]11.75}{[1][256][]256}{}{}{}}
\newlabel{mc-eq:te14}{{11.34}{241}{Marginal score}{equation.942}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-eq:te14@cref}{{[equation][34][11]11.34}{[1][241][]241}{}{}{}}
\newlabel{mc-eq:te2}{{11.62}{253}{Setting}{equation.1003}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-eq:te2@cref}{{[equation][62][11]11.62}{[1][253][]253}{}{}{}}
\newlabel{mc-eq:te20}{{11.45}{245}{The generator in the chart}{equation.966}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-eq:te20@cref}{{[equation][45][11]11.45}{[1][245][]245}{}{}{}}
\newlabel{mc-eq:te3}{{11.63}{253}{Setting}{equation.1004}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-eq:te3@cref}{{[equation][63][11]11.63}{[1][253][]253}{}{}{}}
\newlabel{mc-eq:te4}{{11.64}{253}{Setting}{equation.1005}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-eq:te4@cref}{{[equation][64][11]11.64}{[1][253][]253}{}{}{}}
\newlabel{mc-eq:te5}{{11.56}{250}{The time derivative of the conditional relative entropy}{equation.989}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-eq:te5@cref}{{[equation][56][11]11.56}{[1][250][]250}{}{}{}}
\newlabel{mc-eq:te7}{{11.68}{254}{Hypotheses}{equation.1011}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-eq:te7@cref}{{[equation][68][11]11.68}{[1][254][]254}{}{}{}}
\newlabel{mc-eq:te8}{{11.69}{255}{Entropy dissipation for the remaining coordinates}{equation.1014}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-eq:te8@cref}{{[equation][69][11]11.69}{[1][254][]255}{}{}{}}
\newlabel{mc-eq:te9}{{11.70}{255}{Entropy dissipation for the remaining coordinates}{equation.1015}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-eq:te9@cref}{{[equation][70][11]11.70}{[1][255][]255}{}{}{}}
\newlabel{mc-exact-info}{{7.3}{118}{An exact formula for the information}{lemma.430}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-exact-info@cref}{{[lemma][3][7]7.3}{[1][118][]118}{}{}{}}
\newlabel{mc-positive-measure}{{7.16}{118}{An exact formula for the information}{equation.432}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-positive-measure@cref}{{[equation][16][7]7.16}{[1][118][]118}{}{}{}}
\newlabel{mc-raw-edge-measure}{{4.9}{68}{Gaps below the top eigenvalue at a smaller scale}{corollary.214}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-raw-edge-measure@cref}{{[corollary][9][4]4.9}{[1][68][]68}{}{}{}}
\newlabel{mc-scales}{{4.28}{74}{Scales of the precisions and counts when $t_N=o(N^{2/3})$}{equation.246}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-scales@cref}{{[equation][28][4]4.28}{[1][74][]74}{}{}{}}
\newlabel{mc-time-hypotheses}{{4.15}{68}{Edge counts at a varying scale}{equation.213}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{mc-time-hypotheses@cref}{{[equation][15][4]4.15}{[1][68][]68}{}{}{}}
\newlabel{model-distance}{{3.15}{50}{Distances, mixing times, and profiles}{equation.135}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{model-distance@cref}{{[equation][15][3]3.15}{[1][49][]50}{}{}{}}
\newlabel{part:tools}{{II}{83}{The half-precisions at finite $N$}{part.279}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{part:tools@cref}{{[part][2][]II}{[1][83][]83}{}{}{}}
\newlabel{pc-centered-defect-flux-input}{{9.20}{179}{Comparison through the forward-equation defect}{Item.674}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{pc-centered-defect-flux-input@cref}{{[equation][20][9]9.20}{[1][179][]179}{}{}{}}
\newlabel{pc-centered-defect-flux-only-bound}{{9.21}{179}{Comparison through the forward-equation defect}{Item.675}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{pc-centered-defect-flux-only-bound@cref}{{[equation][21][9]9.21}{[1][179][]179}{}{}{}}
\newlabel{pc-centered-defect-transfer}{{9.5}{179}{Comparison through the forward-equation defect}{theorem.670}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{pc-centered-defect-transfer@cref}{{[theorem][5][9]9.5}{[1][179][]179}{}{}{}}
\newlabel{pr-real-measure-transfer}{{7.15}{134}{Positive measures from real squared resolvents}{lemma.503}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{pr-real-measure-transfer@cref}{{[lemma][15][7]7.15}{[1][134][]134}{}{}{}}
\newlabel{prop:cond-bingham}{{5.6}{89}{Bingham laws as conditioned Gaussian laws}{proposition.307}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{prop:cond-bingham@cref}{{[proposition][6][5]5.6}{[1][89][]89}{}{}{}}
\newlabel{prop:cond-constraint}{{5.10}{92}{The radial constraint}{proposition.319}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{prop:cond-constraint@cref}{{[proposition][10][5]5.10}{[1][92][]92}{}{}{}}
\newlabel{prop:cond-continuity}{{5.17}{98}{Continuity and domination}{proposition.340}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{prop:cond-continuity@cref}{{[proposition][17][5]5.17}{[1][98][]98}{}{}{}}
\newlabel{prop:score-conditional-lsi}{{11.16}{247}{A logarithmic Sobolev inequality for the conditional relative entropy}{proposition.977}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{prop:score-conditional-lsi@cref}{{[proposition][16][11]11.16}{[1][247][]247}{}{}{}}
\newlabel{prop:score-entropy-identity}{{11.18}{250}{Time derivative of the conditional relative entropy}{proposition.990}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{prop:score-entropy-identity@cref}{{[proposition][18][11]11.18}{[1][250][]250}{}{}{}}
\newlabel{rem:ineq-scaling}{{8.10}{150}{Spheres of other radii}{remark.560}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{rem:ineq-scaling@cref}{{[remark][10][8]8.10}{[1][150][]150}{}{}{}}
\newlabel{rem:score-convolution-limits}{{11.13}{244}{Gaussian convolutions as limits}{remark.961}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{rem:score-convolution-limits@cref}{{[remark][13][11]11.13}{[1][244][]244}{}{}{}}
\newlabel{rp-covariance-hypotheses}{{9.40}{189}{Variance conditions at scale $\delta $}{equation.712}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{rp-covariance-hypotheses@cref}{{[equation][40][9]9.40}{[1][189][]189}{}{}{}}
\newlabel{rp-general-flux}{{9.16}{190}{Energy of the solution of the Poisson equation}{theorem.714}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{rp-general-flux@cref}{{[theorem][16][9]9.16}{[1][190][]190}{}{}{}}
\newlabel{rp-unaligned-energy}{{9.42}{190}{Energy of the solution of the Poisson equation}{equation.715}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{rp-unaligned-energy@cref}{{[equation][42][9]9.42}{[1][190][]190}{}{}{}}
\newlabel{sec:comparison}{{9}{173}{Comparison with Bingham laws through the forward equation}{chapter.640}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:comparison@cref}{{[chapter][9][]9}{[1][173][]173}{}{}{}}
\newlabel{sec:cond-countable}{{5.4}{90}{Countable sequences and coordinate marginals}{section.312}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:cond-countable@cref}{{[section][4][5]5.4}{[1][90][]90}{}{}{}}
\newlabel{sec:conditioning}{{5}{83}{Gaussian laws conditioned on a sphere}{chapter.280}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:conditioning@cref}{{[chapter][5][]5}{[1][83][]83}{}{}{}}
\newlabel{sec:density}{{12}{260}{Density bounds relative to time-dependent Bingham laws}{chapter.1042}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:density@cref}{{[chapter][12][]12}{[1][260][]260}{}{}{}}
\newlabel{sec:heat}{{7}{113}{Deterministic covariance evolution and heat estimates}{chapter.396}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:heat@cref}{{[chapter][7][]7}{[1][113][]113}{}{}{}}
\newlabel{sec:inequalities}{{8}{139}{Functional inequalities on the sphere}{chapter.518}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:inequalities@cref}{{[chapter][8][]8}{[1][139][]139}{}{}{}}
\newlabel{sec:intro-discussion}{{1.4}{13}{Discussion}{section.28}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:intro-discussion@cref}{{[section][4][1]1.4}{[1][13][]13}{}{}{}}
\newlabel{sec:likelihood}{{6}{101}{Likelihood ratios on a common sphere}{chapter.345}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:likelihood@cref}{{[chapter][6][]6}{[1][101][]101}{}{}{}}
\newlabel{sec:model}{{3}{42}{The model, conventions, and the limiting profiles}{chapter.102}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:model@cref}{{[chapter][3][]3}{[1][42][]42}{}{}{}}
\newlabel{sec:model-model}{{3.1}{42}{The model}{section.103}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:model-model@cref}{{[section][1][3]3.1}{[1][42][]42}{}{}{}}
\newlabel{sec:model-profiles}{{3.5}{53}{The limiting profiles}{section.152}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:model-profiles@cref}{{[section][5][3]3.5}{[1][52][]53}{}{}{}}
\newlabel{sec:model-quantifiers}{{3.4}{51}{Disorder quantifiers and constants}{section.144}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:model-quantifiers@cref}{{[section][4][3]3.4}{[1][51][]51}{}{}{}}
\newlabel{sec:process}{{10}{195}{The unnormalized process and its moments}{chapter.734}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:process@cref}{{[chapter][10][]10}{[1][195][]195}{}{}{}}
\newlabel{sec:score-chart-entropy}{{11.5}{245}{The generator in the chart and a logarithmic Sobolev inequality}{subsection*.963}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:score-chart-entropy@cref}{{[section][5][11]11.5}{[1][244][]245}{}{}{}}
\newlabel{sec:score-coordinates}{{11.4}{240}{Fisher information of finitely many coordinates}{section.931}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:score-coordinates@cref}{{[section][4][11]11.4}{[1][239][]240}{}{}{}}
\newlabel{sec:score-dissipation}{{11.5}{253}{Entropy dissipation uniformly in the top fraction}{subsection*.1000}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:score-dissipation@cref}{{[section][5][11]11.5}{[1][253][]253}{}{}{}}
\newlabel{sec:score-entropy}{{11.5}{244}{Conditional relative entropy of the remaining coordinates}{section.962}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:score-entropy@cref}{{[section][5][11]11.5}{[1][244][]244}{}{}{}}
\newlabel{sec:score-setting}{{11.1}{229}{Setting, weak scores, and weighted endpoint measures}{section.878}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:score-setting@cref}{{[section][1][11]11.1}{[1][229][]229}{}{}{}}
\newlabel{sec:scores}{{11}{228}{Score bounds, smoothing, and conditional relative entropy}{chapter.877}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:scores@cref}{{[chapter][11][]11}{[1][228][]228}{}{}{}}
\newlabel{sec:spec-critical}{{4.5}{74}{The critical window}{section.247}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:spec-critical@cref}{{[section][5][4]4.5}{[1][74][]74}{}{}{}}
\newlabel{sec:spectral}{{4}{57}{Random-matrix inputs and the equilibrium Lagrange multiplier}{chapter.167}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sec:spectral@cref}{{[chapter][4][]4}{[1][57][]57}{}{}{}}
\newlabel{sgl-euclidean-ibp}{{8.1}{140}{Quadratic potentials}{equation.522}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sgl-euclidean-ibp@cref}{{[equation][1][8]8.1}{[1][140][]140}{}{}{}}
\newlabel{sgl-rigidity-input}{{4.2}{59}{Rigidity and convergence at the edge \cite [Theorem~2.1, (2.16)]{Landon2022}, \cite [Theorem~2.2, (2.25)]{ErdosYauYin2012}, \cite [Theorem~1.1, (1.7), and Proposition~3.5]{RamirezRiderVirag2011}}{equation.175}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sgl-rigidity-input@cref}{{[equation][2][4]4.2}{[1][59][]59}{}{}{}}
\newlabel{sp-airy-counts}{{4.15}{75}{The GOE Airy point process and the limiting equation (cf.\ \cite [Proposition~1.4]{DuHuang2026})}{theorem.251}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sp-airy-counts@cref}{{[theorem][15][4]4.15}{[1][75][]75}{}{}{}}
\newlabel{sp-airy-root-function}{{4.29}{74}{The limiting equation}{equation.249}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sp-airy-root-function@cref}{{[equation][29][4]4.29}{[1][74][]74}{}{}{}}
\newlabel{sp-airy-traces}{{4.32}{75}{The GOE Airy point process and the limiting equation (cf.\ \cite [Proposition~1.4]{DuHuang2026})}{equation.256}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sp-airy-traces@cref}{{[equation][32][4]4.32}{[1][75][]75}{}{}{}}
\newlabel{sp-canonical-definition}{{3.12}{47}{Equilibrium parameters and temperature regimes}{equation.128}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sp-canonical-definition@cref}{{[equation][12][3]3.12}{[1][47][]47}{}{}{}}
\newlabel{sp-canonical-saddle}{{4.13}{73}{The Lagrange multiplier above the critical temperature}{theorem.237}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sp-canonical-saddle@cref}{{[theorem][13][4]4.13}{[1][72][]73}{}{}{}}
\newlabel{sp-condensed-geometry}{{4.7}{64}{Eigenvalue gaps and the top fraction at low temperature}{lemma.193}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sp-condensed-geometry@cref}{{[lemma][7][4]4.7}{[1][64][]64}{}{}{}}
\newlabel{sp-fixed-confidence-geometry}{{4.6}{63}{Eigenvalue gaps at fixed confidence}{lemma.189}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sp-fixed-confidence-geometry@cref}{{[lemma][6][4]4.6}{[1][63][]63}{}{}{}}
\newlabel{sp-goe-rigidity}{{4.2}{59}{Rigidity and convergence at the edge \cite [Theorem~2.1, (2.16)]{Landon2022}, \cite [Theorem~2.2, (2.25)]{ErdosYauYin2012}, \cite [Theorem~1.1, (1.7), and Proposition~3.5]{RamirezRiderVirag2011}}{lemma.173}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sp-goe-rigidity@cref}{{[lemma][2][4]4.2}{[1][59][]59}{}{}{}}
\newlabel{sp-indexed-raw-gaps}{{4.7}{63}{Eigenvalue gaps at fixed confidence}{equation.191}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sp-indexed-raw-gaps@cref}{{[equation][7][4]4.7}{[1][63][]63}{}{}{}}
\newlabel{sp-log-precise-saddle}{{4.25}{73}{The Lagrange multiplier above the critical temperature}{equation.240}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sp-log-precise-saddle@cref}{{[equation][25][4]4.25}{[1][73][]73}{}{}{}}
\newlabel{sp-outer-airy-saddle}{{4.33}{75}{The GOE Airy point process and the limiting equation (cf.\ \cite [Proposition~1.4]{DuHuang2026})}{equation.258}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sp-outer-airy-saddle@cref}{{[equation][33][4]4.33}{[1][75][]75}{}{}{}}
\newlabel{sp-raw-gap-resolvent}{{4.5}{62}{Regularized inverse gaps \cite [Theorem~6.1, (6.3)--(6.4)]{LandonSosoe2022}, \cite [Propositions~1.3 and~4.1, (4.1)]{DuHuang2026}}{lemma.182}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{sp-raw-gap-resolvent@cref}{{[lemma][5][4]4.5}{[1][62][]62}{}{}{}}
\newlabel{thm:dens-main}{{12.8}{273}{Density bounds on the time interval $[0,S]$}{theorem.1102}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{thm:dens-main@cref}{{[theorem][8][12]12.8}{[1][273][]273}{}{}{}}
\newlabel{thm:intro-bounded}{{1.2}{10}{Bounded-time relaxation near infinite temperature}{theorem.20}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{thm:intro-bounded@cref}{{[theorem][2][1]1.2}{[1][10][]10}{}{}{}}
\newlabel{thm:intro-critical}{{1.3}{12}{The critical window}{theorem.26}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{thm:intro-critical@cref}{{[theorem][3][1]1.3}{[1][12][]12}{}{}{}}
\newlabel{thm:intro-main}{{1.1}{8}{Cutoff profiles away from the critical window}{theorem.18}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{thm:intro-main@cref}{{[theorem][1][1]1.1}{[1][8][]8}{}{}{}}
\newlabel{thm:intro-outer}{{1.4}{13}{Matching with the mesoscopic regimes}{theorem.27}{spherical-sk-dynamics-part-i.pdf}}
\newlabel{thm:intro-outer@cref}{{[theorem][4][1]1.4}{[1][13][]13}{}{}{}}
\makeatother