{"collection_revision":"2f443f44c190a64c713d6ea71c3d99ce48f78252","source_inputs":{"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex":"5a9d0f9bffc69839a0d68a1598e18a4a161329a7e6da0e7c580c237608657835","papers/paper-xi/current/publication/source/visual_guide/reader_route.tex":"0c2eb21bc7479c4903e6374b0c6f73915e0366865e24df1b19e89888dda2d1a0","papers/paper-xi/current/publication/source/visual_guide/generate_figures.py":"6b6b12f945ff6a4c8a6ee9b3ef191c86c31d4e4ba74bc13e6daf94fdd4efc99b","papers/paper-xi/current/publication/source/visual_guide/figures/figure_data.json":"907889057b53fa590ba336dfcee92bd6ee85e401aa238a5ae7d89ec1d16b6cc8","papers/paper-xi/current/READING_CONCORDANCE.md":"6150041792848fa061b252fc949967ae848792582b16d2df8eadf088876bd3ee"},"source_pdf":"papers/paper-xi/current/publication/pdfs/Paper_XI_Visual_Guide_v1.1.0.pdf","source_pdf_sha256":"feebaffc77ca8da24572a4b67b3ba5550feedf476995526ea5119fab345cd03e","index":"read/paper-xi/atlas/index.html","total_sections":47,"html_sections":[{"number":1,"url":"read/paper-xi/atlas/l01.html","math":["\\GL_n(K)"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":70,"end_line":76},{"number":2,"url":"read/paper-xi/atlas/l02.html","math":["F=\\mathrm{diag}(1,q,q^2,q^3,q^4),\\qquad 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\\gamma(\\chi,\\psi)\\gamma(\\chi^{-1},\\psi)=\\chi(-1).\n","x=ty","q-1","t=1","-1"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":203,"end_line":218},{"number":12,"url":"read/paper-xi/atlas/l12.html","math":["X","T_x f(u)=f(u+x)","X^*","M_\\xi f(u)=\\psi(\\xi(u))f(u)","\\mathcal{H}=\\mathbf{C}^X","T_xM_\\xi=\\psi(\\xi(x))M_\\xi T_x","p","V=X\\oplus X^*","\n b((x,\\xi),(y,\\eta))=\\eta(x)-\\xi(y).\n","H(V)=V\\times\\F_p","\n (v,z)(w,t)=(v+w,z+t+\\tfrac12b(v,w)).\n","\\mathcal H=\\C^X","\n (\\rho(x,\\xi,z)f)(u)\n =\\psi(z+\\xi(u)+\\tfrac12\\xi(x))f(u+x).\n","|X|=\\sqrt{|V|}"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":219,"end_line":235},{"number":13,"url":"read/paper-xi/atlas/l13.html","math":["\n \\Tr\\rho(v,z)=\\begin{cases}|X|\\psi(z),&v=0,\\\\0,&v\\ne0.\\end{cases}\n","L","\\widetilde L","\n P_L=\\frac1{|L|}\\sum_{\\ell\\in\\widetilde L}\\rho(\\ell)\n","|X|/|L|=1","L=X","\n (P_Lf)(u)=|X|^{-1}\\sum_{x\\in X}f(u+x).\n"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":236,"end_line":254},{"number":14,"url":"read/paper-xi/atlas/l14.html","math":["T_aT_b=c(a,b)T_{a+b}","b_c(a,b)","A/\\mathrm{rad}(b_c)","T_aT_b=c(a,b)T_{a+b}","b_c(a,b)=c(a,b)c(b,a)^{-1}","T_a","\\lambda_aT_a","c","\\lambda_a\\lambda_b\\lambda_{a+b}^{-1}","b_c","a","b","\n \\C_c[A]\\simeq M_{\\sqrt{|A|}}(\\C).\n"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":255,"end_line":264},{"number":15,"url":"read/paper-xi/atlas/l15.html","math":["N\\triangleleft G","A=G/N","\\Pi","\\rho","\\rho","G","\\Pi|_N","\n M_\\rho=\\operatorname{Hom}_N(W,X),\\qquad W\\otimes M_\\rho\\simeq X.\n","s:A\\to G","I_a","\\rho","\n \\Pi(s(a))=I_a\\otimes T_a.\n","T_a","M_\\rho","\\dim 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V=2r.\n"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":326,"end_line":338},{"number":20,"url":"read/paper-xi/atlas/l20.html","math":["b(v,\\cdot)","\n q_v(x)=q(x)+b(v,x),\\qquad \\Arf(q_v)=\\Arf(q)+q(v).\n","\n 2^{r-1}(2^r+1),\\qquad 2^{r-1}(2^r-1).\n","r=1","r=3","q"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":339,"end_line":351},{"number":21,"url":"read/paper-xi/atlas/l21.html","math":["A_e:V_u\\to V_v","B_e:W_u\\to W_v","U_r","p_v","\n U_v=B_{p_v}U_rA_{p_v}^{-1}.\n","U_vA_e=B_eU_u","U_rA_\\gamma=B_\\gamma U_r"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":352,"end_line":362},{"number":22,"url":"read/paper-xi/atlas/l22.html","math":["e:u\\to v","\n D_e=U_rA_{p_v}^{-1}A_eA_{p_u}\n      -B_{p_v}^{-1}B_eB_{p_u}U_r.\n","\n U_vA_e-B_eU_u=B_{p_v}D_eA_{p_u}^{-1}.\n","\n \\|U_vA_e-B_eU_u\\|\n \\le\\|B_{p_v}\\|\\,\\|D_e\\|\\,\\|A_{p_u}^{-1}\\|.\n","|E|-|V|+1"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":363,"end_line":381},{"number":23,"url":"read/paper-xi/atlas/l23.html","math":["V=V_{s_1}\\perp V_{s_2}\\perp V_{s_3}","\\omega_V\\simeq\\omega_{s_1}\\otimes\\omega_{s_2}\\otimes\\omega_{s_3}","V=\\perp_sV_s","\n \\omega_V\\simeq\\bigotimes_s\\omega_{V_s},\\qquad\n \\prod_s\\sqrt{|V_s|}=\\sqrt{|V|}.\n"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":382,"end_line":391},{"number":24,"url":"read/paper-xi/atlas/l24.html","math":["E=L(\\alpha)","C\\in M_m(E)","B=\\alpha I_m+\\mathrm{Lift}_E(C)","L","t_L","\n E=L(\\alpha),\\qquad \\alpha^p-\\alpha=t_L^{-q}.\n","C","E","\n B=\\alpha I_m+\\operatorname{Lift}_E(C).\n","E[C]","L[B]"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":392,"end_line":404},{"number":25,"url":"read/paper-xi/atlas/l25.html","math":["p=2,\\ h=(1,3,5)","p=3,\\ h=(1,2,4)","h_1<\\cdots<h_d","q_i=(p+1)h_i","N=p^d","p,p^2,\\ldots,p^d","\n n_j=p^{d-j}q_{d-j+1}.\n","p","p\\nmid h_i","p=2","h_i","n_1","n_2,\\ldots,n_d","n_1"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":405,"end_line":413},{"number":26,"url":"read/paper-xi/atlas/l26.html","math":["p","\\Tr_{E/L}(\\alpha)=0","b","qb\\equiv-1\\pmod p","a=(qb+1)/p","z=t_L^a\\alpha^b","z","f","P_0","\\kappa=-f'(z)/z^{p-1}","u_0=s(P_0)","\n \\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\\qquad\n \\tr_L\\operatorname{Lift}_E(C)\n =\\Tr_{E/L}(\\kappa^{-1}\\tr_EC).\n","\\tr_EC=0","\\alpha I_m","p"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":414,"end_line":426},{"number":27,"url":"read/paper-xi/atlas/l27.html","math":["\\mathrm{tr}\\delta=0","\\mathrm{tr}\\delta=1","v_F(\\mathrm{tr}\\delta)+s\\geq1","\\psi_F((u-1)\\mathrm{tr}\\delta)=1","t\\OO_F","\\psi_F","t\\OO_F","u\\in U_F^s","(u-1)\\tr\\delta","\n v_F(\\tr\\delta)+s\\ge1\n \\quad\\Longrightarrow\\quad\n \\psi_F((u-1)\\tr\\delta)=1.\n","s=1"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":427,"end_line":437},{"number":28,"url":"read/paper-xi/atlas/l28.html","math":["q","a=(q+1)/2","z=t_L^a\\alpha","\n z^2+t_L^az+t_L=0,\\quad \\Tr_{E/L}\\alpha=1,\n \\quad\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n","\n \\tr_L\\operatorname{Lift}_E(C)=\\Tr_{E/L}(\\alpha\\tr_EC).\n","2^j","1-2\\cdot1=1","0,0"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":438,"end_line":451},{"number":29,"url":"read/paper-xi/atlas/l29.html","math":["u\\in U_F^1","p","\n \\Theta_j(uI)=\\theta_j(u^p)=1.\n","r","m=\\lfloor r/2\\rfloor","2(m+1)-r=1"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":452,"end_line":463},{"number":30,"url":"read/paper-xi/atlas/l30.html","math":["N=p^d","\\GL_N(F)","p","\n \\begin{aligned}\n \\Sw(\\pi)&=p^{d-1}(p+1)h_d,\\\\\n \\dim_{\\F_p}(J^1/H^1)&=N-1,\\quad\\dim\\eta=p^{(N-1)/2},\\\\\n a(\\pi^\\vee\\times\\pi)&=p^{2d}-1+\\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i.\n \\end{aligned}\n","p=2","\n \\begin{aligned}\n \\Sw(\\pi)&=3\\,2^{d-1}h_d,\\\\\n \\dim_{\\F_2}(J^1/H^1)&=N-2,\\quad\\dim\\eta=2^{N/2-1},\\\\\n a(\\pi^\\vee\\times\\pi)&=N^2-1+\\sum_{i=1}^d3\\,4^{i-1}h_i.\n 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E=2.\n","J^1/H^1"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":528,"end_line":540},{"number":36,"url":"read/paper-xi/atlas/l36.html","math":["E_0/F","B=\\operatorname{End}_{E_0}(F^4)","r=3h_1=2m+1","a=3h_2>r","\n P=J\\,U_B^m.\n","J","E_0\\subset K","u","\\lambda=\\theta^u/\\theta","H^1","H_0=\\ker\\lambda","\n P/H_0\\simeq\\F_2^4,\\qquad E=\\Ind_J^P\\eta,\\quad\\dim E=4.\n","\\mu_2"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":541,"end_line":554},{"number":37,"url":"read/paper-xi/atlas/l37.html","math":["\\mathcal{N}=K^\\times J","\\mathcal{M}=K^\\times P","\\Lambda","\\Sigma=\\mathrm{Ind}_{\\mathcal{N}}^{\\mathcal{M}}\\Lambda","\\pi\\simeq\\mathrm{c\\!-\\!Ind}_{\\mathcal{M}}^G\\Sigma","\\Lambda","\\Sigma","K^\\times","P","\n \\mathcal N=K^\\times J,\\quad\\mathcal M=K^\\times P,\n \\qquad\\Sigma=\\Ind_{\\mathcal N}^{\\mathcal M}\\Lambda.\n","\\mathcal N","\\Sigma|_P\\simeq E","\\Sigma","\n \\pi\\simeq\\cInd_{\\mathcal M}^{G}\\Sigma.\n"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":555,"end_line":567},{"number":38,"url":"read/paper-xi/atlas/l38.html","math":["X=\\cInd_J^G\\eta","A_0=\\Lambda(\\varpi_K)","\n (Tf)(g)=A_0^{-1}f(\\varpi_Kg).\n","\n X/(T-1)X\\simeq\\pi.\n","X","J","N","T^N=1","\n e_\\Lambda=\\frac1N\\sum_{i=0}^{N-1}T^i\n","N","\\eta"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":568,"end_line":585},{"number":39,"url":"read/paper-xi/atlas/l39.html","math":["d_\\rho","R(d_\\rho,\\pi)","U_F^1","\\omega","U_F^1","F=\\F_2((t))","N","\\C^\\times","\\omega(t)"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":586,"end_line":594},{"number":40,"url":"read/paper-xi/atlas/l40.html","math":["p^r","\n N/(F^\\times J^1)\\simeq\\mathbb Z/p^r\\mathbb Z.\n","p","C","p","W","\\operatorname{Hom}(W,\\C^\\times)","p"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":595,"end_line":605},{"number":41,"url":"read/paper-xi/atlas/l41.html","math":["C=1","p","N","\\chi(c)I_N","\\widetilde C\\twoheadrightarrow C","C","C"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":606,"end_line":613},{"number":42,"url":"read/paper-xi/atlas/l42.html","math":["J\\subset\\GL_n(F)","v_F\\circ\\det","\\mathbb Z","\n \\nu_a(g)=a^{v_F(\\det g)}\n","J","\\pi","\\pi\\otimes\\nu_a","a^n","a^n\\ne1","8\\to2"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":614,"end_line":623},{"number":43,"url":"read/paper-xi/atlas/l43.html","math":["d_\\rho","R(d_\\rho,\\pi)","\\pi\\simeq\\mathrm{LLC}(\\rho)","\\mathcal A","\\rho\\in\\mathcal A","d_\\rho","\\pi","\\GL_{\\dim\\rho}(F)","R(d_\\rho,\\pi)","R"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":624,"end_line":630},{"number":44,"url":"read/paper-xi/atlas/l44.html","math":["\\pi","R(d_\\rho,\\pi)","\\LLC(\\rho)","\n \\pi\\simeq\\LLC(\\rho).\n","\n \\forall\\pi\\,[R(d,\\pi)\\Rightarrow\\mathrm{H5}(\\pi)]\n \\quad\\text{and}\\quad\n \\exists\\pi\\,[R(d,\\pi)\\land\\mathrm{H5}(\\pi)].\n","d"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":631,"end_line":646},{"number":45,"url":"read/paper-xi/atlas/l45.html","math":["C=1"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":647,"end_line":657},{"number":46,"url":"read/paper-xi/atlas/l46.html","math":["h=(1,3,5)","\n -60,\\quad-18,\\quad-3.\n","0,0,1","p=3","h=(1,2,4)","144,24,4"],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":658,"end_line":674},{"number":47,"url":"read/paper-xi/atlas/l47.html","math":[],"source":"papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex","start_line":675,"end_line":691}],"opening_math":["K","F=\\F_p((t))","K=F[\\beta]"],"figures":{"01_three_objects":{"receipt":{"name":"01_three_objects","source":"papers/paper-xi/current/publication/source/visual_guide/figures/01_three_objects.pdf","source_sha256":"5cde89302743e9e5a4ea4a6e87171c96aa7e9d31a35b6994938b98e32aaed209","image":"assets/atlas/paper-xi/01_three_objects.png","image_sha256":"5f77af7445ad80cb23086ad0eb4fe92269af0658f176f8a5bdfd6a71bfd849d1","dimensions":[1919,990],"native_text":"Weil atoms Monodromy chains\nFinite packet comparison Actual arithmetic realization\nConjectural realization\nThe finite comparison does not create the arithmetic witness."},"alt":"Weil atoms, monodromy chains, finite comparison, and arithmetic realization","description":"Four boxes distinguish Weil atoms, monodromy chains, finite packet comparison, and actual arithmetic realization. The horizontal arrow from finite comparison to arithmetic realization is marked conjectural. Finite comparison does not create the arithmetic witness.","math_labels":[],"tables":[]},"02_monodromy":{"receipt":{"name":"02_monodromy","source":"papers/paper-xi/current/publication/source/visual_guide/figures/02_monodromy.pdf","source_sha256":"a899a43b2047c8e7d887758ed6abd112ff0a52b56ff58a30c501ee5d0863bc91","image":"assets/atlas/paper-xi/02_monodromy.png","image_sha256":"be82e31353d32360576179d94e0faee62bacef1436927fcdd2949967324a134b","dimensions":[1919,947],"native_text":"V0\nq\n0\nV1\nq\n1\nV2\nq\n2\nV3\nq\n3\nV4\nq\n4\nF = diag(1, q, q\n2\n, q\n3\n, q\n4\n), FNF\n−1 = q\n−1N\nN(Vj) ⊆ Vj − 1; N\n5 = 0"},"alt":"A five-term monodromy chain","description":"Five spaces run from V zero to V four. The arrows for N point toward the preceding space. A path of five transfers cannot exist, giving the illustrated nilpotence statement.","math_labels":[{"line":62,"text":"$F=\\mathrm{diag}(1,q,q^2,q^3,q^4),\\qquad FNF^{-1}=q^{-1}N$"},{"line":63,"text":"$N(V_j)\\subseteq V_{j-1};\\qquad N^5=0$"}],"tables":[]},"03_tame_orbit":{"receipt":{"name":"03_tame_orbit","source":"papers/paper-xi/current/publication/source/visual_guide/figures/03_tame_orbit.pdf","source_sha256":"1ed8c2d13c619b61f92134044f068809b555ba1fc6a10a433ec9e6b9890191ac","image":"assets/atlas/paper-xi/03_tame_orbit.png","image_sha256":"ad1677a3bb0ca2048ba0c9b83af4c301c9df66774fd4b36e761dc621a51ba688","dimensions":[1309,1098],"native_text":"1\n2\n4\n8\n× 2\nmod 15"},"alt":"Multiplication by two around a tame orbit modulo fifteen","description":"The four labels 1, 2, 4 and 8 form a cycle under multiplication by two modulo fifteen. The arrows encode the fixed geometric Frobenius convention.","math_labels":[],"tables":[]},"04_orbit_counts":{"receipt":{"name":"04_orbit_counts","source":"papers/paper-xi/current/publication/source/visual_guide/figures/04_orbit_counts.pdf","source_sha256":"bbe5a2a7218a9274cbf7c9dfbaa62d927ef63f28c5e8d6998427a10eb9c1e54d","image":"assets/atlas/paper-xi/04_orbit_counts.png","image_sha256":"ae45f166b990f286ef9c19155701bd3cfd581c06a33a2dac413474b19a38cdbc","dimensions":[1916,939],"native_text":"1\n2\n3\n4\n5\n6\n7\n8\nExact Frobenius orbit length n\n1\n0\n0\n1\n0\n1\n1\n0\n2\n1\n0\n3\n1\n0\n4\nNumber of regular orbits\nq=2\nq=3\nq=5"},"alt":"Exact regular Frobenius orbit counts","description":"Three series give regular orbit counts for residue sizes two, three and five, at orbit lengths one through eight. The vertical axis is logarithmic. The following table retains the recorded integers.","math_labels":[],"tables":[{"caption":"Recorded exact regular orbit counts","headers":["Orbit length n","q = 2","q = 3","q = 5"],"rows":[[1,1,2,4],[2,1,3,10],[3,2,8,40],[4,3,18,150],[5,6,48,624],[6,9,116,2580],[7,18,312,11160],[8,30,810,48750]]}]},"05_tensor_grid":{"receipt":{"name":"05_tensor_grid","source":"papers/paper-xi/current/publication/source/visual_guide/figures/05_tensor_grid.pdf","source_sha256":"8d955a3557d0329ebeb91f965ea40bd60978064224a806fe2b2f5938633f6ea2","image":"assets/atlas/paper-xi/05_tensor_grid.png","image_sha256":"05d86cffaf29267c50ae55385a10121990a3de0de1ebf1e2e739a3e849ace4ce","dimensions":[1486,1047],"native_text":"0\n1\n2\n3\n4\n5\nZ/6Z\n0123\nZ/4Z\n0\n1\n0\n1\n0\n1\n1\n0\n1\n0\n1\n0\n0\n1\n0\n1\n0\n1\n1\n0\n1\n0\n1\n0"},"alt":"Two diagonal orbits on a four by six grid","description":"Rows are coordinates modulo four and columns coordinates modulo six. Each cell is labelled zero or one. Simultaneous translation preserves that label and partitions the grid into two orbits of twelve cells.","math_labels":[],"tables":[{"caption":"Recorded diagonal orbit labels","headers":["Coordinate mod 4","Column 0","Column 1","Column 2","Column 3","Column 4","Column 5"],"rows":[[0,0,1,0,1,0,1],[1,1,0,1,0,1,0],[2,0,1,0,1,0,1],[3,1,0,1,0,1,0]]}]},"06_cycle_scalar":{"receipt":{"name":"06_cycle_scalar","source":"papers/paper-xi/current/publication/source/visual_guide/figures/06_cycle_scalar.pdf","source_sha256":"65f7b9c3f76299ca1c7528a89bc1a10869179f3dd9eef098a46bbddbc6679386","image":"assets/atlas/paper-xi/06_cycle_scalar.png","image_sha256":"f2f68a9fa57789276d1d3173c3692c8d008f10dee71c641f15e41fec0841e828","dimensions":[1919,882],"native_text":"e0 e1 e2 e3\nClosing edge a: characteristic polynomial X\n4 − a"},"alt":"A cyclic chain with a distinguished closing scalar","description":"A four-step cycle has ordinary transfer edges and one closing edge marked a. Its characteristic polynomial is X to the fourth minus a. The closing scalar retains information beyond the inertia orbit.","math_labels":[{"line":87,"text":"Closing edge $a$:  characteristic polynomial $X^4-a$"}],"tables":[]},"07_heisenberg":{"receipt":{"name":"07_heisenberg","source":"papers/paper-xi/current/publication/source/visual_guide/figures/07_heisenberg.pdf","source_sha256":"8ed92590e26fe40f4d420a70a92308482fae7e90f2e801a440876c0ed12d50c6","image":"assets/atlas/paper-xi/07_heisenberg.png","image_sha256":"1431af674c6ce76088a9448736b281058c6a56a5d1db0110eaa263b0e2d10053","dimensions":[1919,1098],"native_text":"X translations\nTxf(u) = f(u + x)\nX\n* multipliers\nMξf(u) = ψ(ξ(u))f(u)\n = C\nX\nTxMξ = ψ(ξ(x))MξTx\nNoncommuting operators generate the full matrix algebra."},"alt":"Translations and multipliers generate a Heisenberg packet","description":"Translation and multiplication operators feed a common representation space. Their noncommutation is the central scalar relation; together they generate the full matrix algebra in the supplied finite model.","math_labels":[{"line":90,"text":"$X$ translations"},{"line":90,"text":"$T_x f(u)=f(u+x)$"},{"line":91,"text":"$X^*$ multipliers"},{"line":91,"text":"$M_\\xi f(u)=\\psi(\\xi(u))f(u)$"},{"line":92,"text":"$\\mathcal{H}=\\mathbf{C}^X$"},{"line":92,"text":"$T_xM_\\xi=\\psi(\\xi(x))M_\\xi T_x$"}],"tables":[]},"08_radical":{"receipt":{"name":"08_radical","source":"papers/paper-xi/current/publication/source/visual_guide/figures/08_radical.pdf","source_sha256":"a20c5ce067bacec0c8b2a981e2dd56ce2d7cd698e025bd679d32776e17ddb308","image":"assets/atlas/paper-xi/08_radical.png","image_sha256":"7af2cc1dfbb9506a8c719d25a87753a785deabf76ed56282c12ac1253efea782","dimensions":[1919,882],"native_text":"Projective action\nTaTb = c(a, b)Ta + b\nScalar commutator\nbc(a, b)\nSymplectic quotient\nA/rad(bc)\nRadical directions commute with every projective generator."},"alt":"Radical directions and a nondegenerate quotient","description":"A projective commutator space passes to its quotient by the radical. Radical directions commute with every generator; the remaining quotient carries the nondegenerate alternating form.","math_labels":[{"line":96,"text":"$T_aT_b=c(a,b)T_{a+b}$"},{"line":97,"text":"$b_c(a,b)$"},{"line":98,"text":"$A/\\mathrm{rad}(b_c)$"}],"tables":[]},"09_bridge":{"receipt":{"name":"09_bridge","source":"papers/paper-xi/current/publication/source/visual_guide/figures/09_bridge.pdf","source_sha256":"cde612f114e68598ef6e99cc8cfb8af27bf3af323db37c97ab7cfdb8667ae717","image":"assets/atlas/paper-xi/09_bridge.png","image_sha256":"371eea64d49bc5039b624a855c4c966f7e9daf18bf91f47d3e0b78d56d1fb9e3","dimensions":[1919,1098],"native_text":"(Vρ, bρ) (Vθ, hθ)\nωρ ωθ\nι: equivariant isometry\nU: unique up to scalar\nA cyclic generator phase removes the discrepancy character."},"alt":"Two supplied finite models compared by a bridge","description":"The two representations are related by a supplied intertwiner. Their remaining discrepancy is a scalar character, which a cyclic generator phase can remove under the stated assumptions.","math_labels":[{"line":102,"text":"$(V_\\rho,b_\\rho)$"},{"line":102,"text":"$(V_\\theta,h_\\theta)$"},{"line":103,"text":"$\\omega_\\rho$"},{"line":103,"text":"$\\omega_\\theta$"},{"line":104,"text":"$\\iota$: equivariant isometry"},{"line":106,"text":"$U$: unique up to scalar"}],"tables":[]},"10_quadratic_planes":{"receipt":{"name":"10_quadratic_planes","source":"papers/paper-xi/current/publication/source/visual_guide/figures/10_quadratic_planes.pdf","source_sha256":"7a818195bb6bc2914d00135156398543e4b10dfb61bacd3c4852f96bec9bd3d4","image":"assets/atlas/paper-xi/10_quadratic_planes.png","image_sha256":"b369fe1f49084046f1a40a94cdd2d43ac5a22d76fbb439a08f9473dad44d5a63","dimensions":[1724,871],"native_text":"0 1\nx\n0\n1\ny\n0 0\n0 1\nq0(x, y) = xy\n0 1\nx\n0\n1\ny\n0 1\n1 1\nq1(x, y) = xy + x + y"},"alt":"Two quadratic refinements of the same symplectic plane","description":"Two four-point diagrams share the underlying alternating plane but assign different quadratic values. The retained point table records the two refinements and distinguishes their signs.","math_labels":[{"line":111,"text":"$q_0(x,y)=xy$"},{"line":111,"text":"$q_1(x,y)=xy+x+y$"}],"tables":[{"caption":"Recorded quadratic values","headers":["Point (x, y)","q0","q1"],"rows":[["(0, 0)",0,0],["(1, 0)",0,1],["(0, 1)",0,1],["(1, 1)",1,1]]}]},"11_arf_counts":{"receipt":{"name":"11_arf_counts","source":"papers/paper-xi/current/publication/source/visual_guide/figures/11_arf_counts.pdf","source_sha256":"b43d69045ee704af2ba77f0f9d964424352f069759b8541cbe7d47e58f8740dd","image":"assets/atlas/paper-xi/11_arf_counts.png","image_sha256":"46d4f17f1be3ca90be2b2ee1d647c44c7dad405f8426740defe9a0455709f3b7","dimensions":[1917,918],"native_text":"1\n2\n3\n4\n5\nHalf dimension r\n1\n0\n0\n1\n0\n1\n1\n0\n2\nNumber of refinements\nArf 0\nArf 1"},"alt":"Counts of refinements of Arf invariant zero and one","description":"Paired bars show the two Arf counts in dimensions two through ten, indexed by half dimension one through five. The vertical scale is logarithmic; the table retains each exact count.","math_labels":[],"tables":[{"caption":"Recorded counts by Arf invariant","headers":["Half dimension r","Arf 0","Arf 1"],"rows":[[1,3,1],[2,10,6],[3,36,28],[4,136,120],[5,528,496]]}]},"12_holonomy":{"receipt":{"name":"12_holonomy","source":"papers/paper-xi/current/publication/source/visual_guide/figures/12_holonomy.pdf","source_sha256":"fadd1f588439167d8e47dec3a5f1963301676246386a21d4e08a4b9b39566143","image":"assets/atlas/paper-xi/12_holonomy.png","image_sha256":"c85fcc2e46fd017246817e0c29bf6e18f4ef2c75f3d3be8525e8a7ff31545473","dimensions":[1919,1163],"native_text":"Solid tree: fix gauges. Dashed chords: test cycles.\nr\nu\nv\nw"},"alt":"A spanning tree and two cycle-testing chords","description":"Vertices r, u, v and w are connected by solid tree edges and dashed extra edges. The tree fixes vertex gauges; each extra edge closes a cycle whose residual must be checked.","math_labels":[],"tables":[]},"13_break_tensor":{"receipt":{"name":"13_break_tensor","source":"papers/paper-xi/current/publication/source/visual_guide/figures/13_break_tensor.pdf","source_sha256":"5dfd137d221ebdad5125e3dde110f67d948aee2779b98445a672cea004f029b6","image":"assets/atlas/paper-xi/13_break_tensor.png","image_sha256":"c3313fbcd4a63cfb7536ff3f8f4e9ee0b20272684ce0da2fa2be41eb8a10c421","dimensions":[1919,990],"native_text":"Vs1\nωs1\nVs2\nωs2\nVs3\nωs3\nV = Vs1 ⟂ Vs2 ⟂ Vs3\nωV ≃ ωs1 ⊗ ωs2 ⊗ ωs3"},"alt":"An orthogonal sum becomes a tensor factorization","description":"Three break spaces and their representations appear above an orthogonal decomposition and its tensor-product representation. The source gives the hypotheses for that factorization.","math_labels":[{"line":136,"text":"$V=V_{s_1}\\perp V_{s_2}\\perp V_{s_3}$"},{"line":137,"text":"$\\omega_V\\simeq\\omega_{s_1}\\otimes\\omega_{s_2}\\otimes\\omega_{s_3}$"}],"tables":[]},"14_recursive_parameter":{"receipt":{"name":"14_recursive_parameter","source":"papers/paper-xi/current/publication/source/visual_guide/figures/14_recursive_parameter.pdf","source_sha256":"345803510284f134285081b7dd4e609944a00291b221d52992dddd985d239ae5","image":"assets/atlas/paper-xi/14_recursive_parameter.png","image_sha256":"e127c94c25835d621ddd930bf3ea811448a8f954c0e813751dd057f0d3265d80","dimensions":[1919,1098],"native_text":"Native parameter field\nE = L(α)\nDerived parameter\nC ∈ Mm(E)\nActual ambient matrix\nB = αIm + LiftE(C)\nThe derived field and the final field are not identified."},"alt":"Native field, derived parameter, and ambient matrix","description":"A native parameter field and a matrix over it feed an actual ambient matrix obtained by adding the scalar parameter to a lifted matrix. The diagram keeps the derived field separate from the final parameter field.","math_labels":[{"line":140,"text":"$E=L(\\alpha)$"},{"line":141,"text":"$C\\in M_m(E)$"},{"line":142,"text":"$B=\\alpha I_m+\\mathrm{Lift}_E(C)$"}],"tables":[]},"15_degree_levels":{"receipt":{"name":"15_degree_levels","source":"papers/paper-xi/current/publication/source/visual_guide/figures/15_degree_levels.pdf","source_sha256":"f162b6e04fe55817fffafb73044b944c05ab493e5cb3451dfaaa64546d99ac38","image":"assets/atlas/paper-xi/15_degree_levels.png","image_sha256":"9bc749c252c61cf5b79a4520d856392c6545ce4e5b3ca79a830b14293db43436","dimensions":[1910,979],"native_text":"2 4 8\nApproximation-field degree\n10\n20\n30\n40\n50\n60\nCritical level\n60\n18\n3\np = 2, h = (1, 3, 5)\n3 9 27\nApproximation-field degree\n0\n20\n40\n60\n80\n100\n120\n140\nCritical level\n144\n24\n4\np = 3, h = (1, 2, 4)"},"alt":"Two recorded approximation-degree and critical-level profiles","description":"The dyadic example uses degrees two, four and eight with levels sixty, eighteen and three. The odd example uses degrees three, nine and twenty-seven with levels one hundred forty-four, twenty-four and four. These are finite illustrative profiles.","math_labels":[{"line":148,"text":"$p=2,\\ h=(1,3,5)$"},{"line":148,"text":"$p=3,\\ h=(1,2,4)$"}],"tables":[{"caption":"Recorded critical levels","headers":["Stage","Dyadic level","Odd p = 3 level"],"rows":[[1,60,144],[2,18,24],[3,3,4]]}]},"16_scalar_depth":{"receipt":{"name":"16_scalar_depth","source":"papers/paper-xi/current/publication/source/visual_guide/figures/16_scalar_depth.pdf","source_sha256":"30427871a8356e9a370102489f7262f461628eecb5cebf9bf23370e222b14e71","image":"assets/atlas/paper-xi/16_scalar_depth.png","image_sha256":"02e93eeaf05812c2ce3faff0941f457baf0ef5afe294c391be371c928e656e1c","dimensions":[1919,1034],"native_text":"trδ = 0\nodd trace-zero tower\ntrδ = 1\ndyadic native tower\nvF(trδ) + s ≥ 1\nψF((u − 1)trδ) = 1"},"alt":"Odd trace zero and dyadic trace one satisfy a depth condition","description":"The odd tower has native trace zero; the dyadic tower has native trace one. Both feed a depth inequality ensuring the displayed additive character is trivial on the required scalar perturbations.","math_labels":[{"line":154,"text":"$\\mathrm{tr}\\delta=0$"},{"line":155,"text":"$\\mathrm{tr}\\delta=1$"},{"line":156,"text":"$v_F(\\mathrm{tr}\\delta)+s\\geq1$"},{"line":156,"text":"$\\psi_F((u-1)\\mathrm{tr}\\delta)=1$"}],"tables":[]},"17_native_ambient_trace":{"receipt":{"name":"17_native_ambient_trace","source":"papers/paper-xi/current/publication/source/visual_guide/figures/17_native_ambient_trace.pdf","source_sha256":"0aad83ced1ec86c32cc4e483a7825f7ffab3cfe48662028e2feb110ac7dec545","image":"assets/atlas/paper-xi/17_native_ambient_trace.png","image_sha256":"2942855c44a5e61a633670c0f18e558834e02031b846708c1f4b4131b1720622","dimensions":[1919,969],"native_text":"Native rank 2: trace 1\nIn rank 8: trace 0\nNative rank 4: trace 1\nIn rank 8: trace 0\nNative rank 8: trace 1\nIn rank 8: trace 1\nNative trace and amplified ambient trace have different roles."},"alt":"Native trace one can amplify to ambient trace zero","description":"Native ranks two, four and eight each have trace one. In ambient rank eight their traces are zero, zero and one. Native trace and amplified ambient trace have different roles.","math_labels":[],"tables":[]},"18_capacity":{"receipt":{"name":"18_capacity","source":"papers/paper-xi/current/publication/source/visual_guide/figures/18_capacity.pdf","source_sha256":"3c3572a483e79c5265ef8a7c0c6fe599caa7c988d20a56a76d11014c056e7ecd","image":"assets/atlas/paper-xi/18_capacity.png","image_sha256":"1b900c5922f08c3fbe640cdc5cec4a75f73c64d11c7e016f9ecf40d4dee9ec9d","dimensions":[1916,1025],"native_text":"d=1\nN=2\nd=2\nN=4\nd=3\nN=8\nd=4\nN=16\nd=5\nN=32\n05\n10\n15\n20\n25\n30\nSymplectic dimension over F₂\nO\nrigin\nal q\nu\no\ntie\nn\nt\nN\n−\n2\nW\nh\nole\ns\no\nu\nr\nc\ne\n2\nd"},"alt":"Original quotient and whole-source symplectic dimensions","description":"Paired bars compare original quotient dimension N minus two with whole-source dimension twice d, for N equal to two to the d and d from one through five. The data table separately records representation carrier dimensions.","math_labels":[{"line":168,"text":"Original quotient $N-2$"},{"line":169,"text":"Whole source $2d$"}],"tables":[{"caption":"Recorded quotient and carrier dimensions","headers":["d","Ambient rank","Host symplectic dimension","Source symplectic dimension","Host representation dimension","Source representation dimension"],"rows":[[1,2,0,2,1,2],[2,4,2,4,2,4],[3,8,6,6,8,8],[4,16,14,8,128,16],[5,32,30,10,32768,32]]}]},"21_marked_extension":{"receipt":{"name":"21_marked_extension","source":"papers/paper-xi/current/publication/source/visual_guide/figures/21_marked_extension.pdf","source_sha256":"88c62a90652122b99b575124bf66f100b5d07c80cb3eaf0baa40525972c487fa","image":"assets/atlas/paper-xi/21_marked_extension.png","image_sha256":"19c579d0176260c4f13a17cf30ae32aba6f8e71417eafd477de9ffb5404c60c5","dimensions":[1919,1012],"native_text":"Marked source extension\nfinite subgroup + lifts + squares\nMarked image in scalar pushout\nF0(a)\n2 = F0(a\n2\n)\nThe whole finite subgroup travels through the map.\nA separately fixed native marking needs its own comparison."},"alt":"The whole marked finite subgroup travels through the map","description":"An arrow carries a marked source extension, including lifts and squares, into a scalar pushout. The square identity is preserved. An independently prescribed native marking needs its own comparison.","math_labels":[{"line":188,"text":"$F_0(a)^2=F_0(a^2)$"}],"tables":[]},"19_enlarged_carrier":{"receipt":{"name":"19_enlarged_carrier","source":"papers/paper-xi/current/publication/source/visual_guide/figures/19_enlarged_carrier.pdf","source_sha256":"25fe0884a19cba6e0f391f1e515b3919d642641eb934c2607a24ea74456e0df2","image":"assets/atlas/paper-xi/19_enlarged_carrier.png","image_sha256":"7f10a5b29662b69c0aad5abea96e48387ef2040be2b69c9dd6ac4795783b2cc4","dimensions":[1919,1098],"native_text":"Original compact type\n(J, η)\nEnlarged compact group\n(P, E = Ind\nP\nJ η)\nSame supercuspidal candidate\nπ = c−Ind\nG\nK\n× JΛ\n[P : J] = 2"},"alt":"An enlarged compact carrier recovers the same candidate","description":"The original compact type maps to an enlarged compact group of index two. Both lead to the same supercuspidal candidate. The enlargement changes the assembly carrier while retaining the inducing representation.","math_labels":[{"line":173,"text":"$(J,\\eta)$"},{"line":174,"text":"$(P,E=\\mathrm{Ind}_J^P\\eta)$"},{"line":175,"text":"$\\pi=\\mathrm{c\\!-\\!Ind}_{K^\\times J}^G\\Lambda$"},{"line":176,"text":"$[P:J]=2$"}],"tables":[]},"20_induction_stages":{"receipt":{"name":"20_induction_stages","source":"papers/paper-xi/current/publication/source/visual_guide/figures/20_induction_stages.pdf","source_sha256":"49f97a4a0ac85da14906edf56a93500806b1635059b2bf8171c0ccf7da498d55","image":"assets/atlas/paper-xi/20_induction_stages.png","image_sha256":"f9b3015b66d54f386a94f3c9f713992c766f031b791ac19bdda5196f942709d9","dimensions":[1919,1055],"native_text":" = K\n× J  = K\n× P\nΛ Σ = Ind\n\n Λ\nπ ≃ c−Ind\nG\nΣ"},"alt":"Induction in stages between two noncompact groups","description":"The upper row enlarges K times J to K times P. The lower row induces Lambda to Sigma. Compact induction of Sigma recovers the displayed same candidate.","math_labels":[{"line":180,"text":"$\\mathcal{N}=K^\\times J$"},{"line":181,"text":"$\\mathcal{M}=K^\\times P$"},{"line":182,"text":"$\\Lambda$"},{"line":182,"text":"$\\Sigma=\\mathrm{Ind}_{\\mathcal{N}}^{\\mathcal{M}}\\Lambda$"},{"line":184,"text":"$\\pi\\simeq\\mathrm{c\\!-\\!Ind}_{\\mathcal{M}}^G\\Sigma$"}],"tables":[]},"22_completion":{"receipt":{"name":"22_completion","source":"papers/paper-xi/current/publication/source/visual_guide/figures/22_completion.pdf","source_sha256":"1e2a3afead8dd90dc29d744ee38d3e7b469ab29b088ed25533d0983f7b260da7","image":"assets/atlas/paper-xi/22_completion.png","image_sha256":"31b2662e7aacadf8ff2b69df8ad4261daec1dfda4ada5e5c1a435f43d88dd0ac","dimensions":[1919,1055],"native_text":"Finite compact packet\ndρ\nNoncompact completion\ncentral character + root\nOne actual candidate\nR(dρ, π)"},"alt":"Finite packet and noncompact completion determine one candidate","description":"A finite compact packet and a noncompact central-character and root choice feed one actual candidate with an arithmetic recovery witness. Completion is additional construction data.","math_labels":[{"line":193,"text":"$d_\\rho$"},{"line":195,"text":"$R(d_\\rho,\\pi)$"}],"tables":[]},"23_conjecture_boundary":{"receipt":{"name":"23_conjecture_boundary","source":"papers/paper-xi/current/publication/source/visual_guide/figures/23_conjecture_boundary.pdf","source_sha256":"4c486795a1ec4d9838022f0d590953a925ced4d2446664752df86e42d3a1e9d6","image":"assets/atlas/paper-xi/23_conjecture_boundary.png","image_sha256":"7f3aa13d2a1b0bfa282428dedf7ab956583f0db5b588fd36cac14154251464eb","dimensions":[1919,1120],"native_text":"H1–H4 finite datum\ndρ\nArithmetic witness\nR(dρ, π)\nH5 for this same candidate\ntested pair factors\nConditional comparison\nπ ≃ LLC(ρ)\nConjecture 66"},"alt":"Conjectural existence and conditional same-candidate comparison","description":"The upper arrow from supplied H1 through H4 data to an arithmetic witness is labelled Conjecture 66. The witness and H5 for that same candidate feed the lower conditional comparison. The diagram does not provide the missing general existence witness.","math_labels":[{"line":199,"text":"$d_\\rho$"},{"line":200,"text":"$R(d_\\rho,\\pi)$"},{"line":202,"text":"$\\pi\\simeq\\mathrm{LLC}(\\rho)$"}],"tables":[]}},"macros":{"F":"\\mathbb F","C":"\\mathbb C","OO":"\\mathcal O","GL":"\\mathrm{GL}","Sp":"\\operatorname{Sp}","tr":"\\operatorname{tr}","Tr":"\\operatorname{Tr}","Ind":"\\operatorname{Ind}","cInd":"\\operatorname{c\\!\\!-\\!\\!Ind}","Sw":"\\operatorname{Sw}","Arf":"\\operatorname{Arf}","rad":"\\operatorname{rad}","LLC":"\\operatorname{LLC}"},"manuscript_references":{"1":[{"label":"Section 2","title":"Weil Deligne chains","url":"read/paper-xi/weil-deligne-chains.html#L232"},{"label":"Section 11","title":"The restricted Galois to type bridge","url":"read/paper-xi/restricted-bridge.html#L1192"},{"label":"Section 16","title":"Local factor tests and the remaining bridge","url":"read/paper-xi/local-factors-and-conjecture.html#L1814"}],"2":[{"label":"Section 2","title":"Weil Deligne chains","url":"read/paper-xi/weil-deligne-chains.html#L232"}],"3":[{"label":"Section 3","title":"Tame Frobenius packets","url":"read/paper-xi/tame-packets.html#L315"}],"4":[{"label":"Section 3","title":"Tame Frobenius packets","url":"read/paper-xi/tame-packets.html#L315"}],"5":[{"label":"Section 3","title":"Tame Frobenius packets","url":"read/paper-xi/tame-packets.html#L315"}],"6":[{"label":"Section 4","title":"Finite field cuspidal projectors","url":"read/paper-xi/cuspidal-projectors.html#L452"}],"7":[{"label":"Section 5","title":"Tensor refinement of tame atoms","url":"read/paper-xi/tensor-refinement.html#L552"}],"8":[{"label":"Section 5","title":"Tensor refinement of tame atoms","url":"read/paper-xi/tensor-refinement.html#L552"}],"9":[{"label":"Section 3","title":"Tame Frobenius packets","url":"read/paper-xi/tame-packets.html#L315"},{"label":"Section 4","title":"Finite field cuspidal projectors","url":"read/paper-xi/cuspidal-projectors.html#L452"}],"10":[{"label":"Section 5","title":"Tensor refinement of tame atoms","url":"read/paper-xi/tensor-refinement.html#L552"}],"11":[{"label":"Section 6","title":"Finite Gauss phases","url":"read/paper-xi/gauss-phases.html#L713"}],"12":[{"label":"Section 7","title":"Finite Heisenberg packets","url":"read/paper-xi/heisenberg-packets.html#L787"}],"13":[{"label":"Section 7","title":"Finite Heisenberg packets","url":"read/paper-xi/heisenberg-packets.html#L787"}],"14":[{"label":"Section 8","title":"Twisted group algebras and commutator envelopes","url":"read/paper-xi/commutator-envelopes.html#L916"}],"15":[{"label":"Section 9","title":"A restricted one break Galois envelope","url":"read/paper-xi/one-break-envelope.html#L1010"}],"16":[{"label":"Section 10","title":"The type side symplectic quotient","url":"read/paper-xi/type-side-quotient.html#L1151"}],"17":[{"label":"Section 11","title":"The restricted Galois to type bridge","url":"read/paper-xi/restricted-bridge.html#L1192"}],"18":[{"label":"Section 12","title":"Odd residue characteristic phases","url":"read/paper-xi/odd-phases.html#L1289"}],"19":[{"label":"Section 13","title":"Dyadic quadratic refinements","url":"read/paper-xi/dyadic-refinements.html#L1468"}],"20":[{"label":"Section 13","title":"Dyadic quadratic refinements","url":"read/paper-xi/dyadic-refinements.html#L1468"}],"21":[{"label":"Section 14","title":"Matrix holonomy and finite gluing","url":"read/paper-xi/matrix-holonomy.html#L1609"}],"22":[{"label":"Section 14","title":"Matrix holonomy and finite gluing","url":"read/paper-xi/matrix-holonomy.html#L1609"}],"23":[{"label":"Section 15","title":"Breakwise assembly","url":"read/paper-xi/breakwise-assembly.html#L1750"}],"24":[{"label":"Appendix B","title":"The trace-zero Artin--Schreier tower","url":"read/paper-xi/appendix-odd-tower.html#L2"},{"label":"Appendix H","title":"The explicit dyadic arithmetic tower","url":"read/paper-xi/appendix-dyadic-tower.html#L2"}],"25":[{"label":"Appendix A","title":"Odd-prime synthesis and complete scope","url":"read/paper-xi/appendix-odd-synthesis.html#L2"},{"label":"Appendix G","title":"Dyadic synthesis and all-length assembly scope","url":"read/paper-xi/appendix-dyadic-synthesis.html#L2"}],"26":[{"label":"Appendix B","title":"The trace-zero Artin--Schreier tower","url":"read/paper-xi/appendix-odd-tower.html#L2"},{"label":"Appendix C","title":"Native-rank full characters and tame central completion","url":"read/paper-xi/appendix-odd-character.html#L2"}],"27":[{"label":"Appendix H","title":"The explicit dyadic arithmetic tower","url":"read/paper-xi/appendix-dyadic-tower.html#L2"},{"label":"Appendix I","title":"Dyadic full characters and the odd critical endpoint","url":"read/paper-xi/appendix-dyadic-character.html#L2"}],"28":[{"label":"Appendix H","title":"The explicit dyadic arithmetic tower","url":"read/paper-xi/appendix-dyadic-tower.html#L2"}],"29":[{"label":"Appendix C","title":"Native-rank full characters and tame central completion","url":"read/paper-xi/appendix-odd-character.html#L2"},{"label":"Appendix I","title":"Dyadic full characters and the odd critical endpoint","url":"read/paper-xi/appendix-dyadic-character.html#L2"}],"30":[{"label":"Appendix A","title":"Odd-prime synthesis and complete scope","url":"read/paper-xi/appendix-odd-synthesis.html#L2"},{"label":"Appendix G","title":"Dyadic synthesis and all-length assembly scope","url":"read/paper-xi/appendix-dyadic-synthesis.html#L2"},{"label":"Appendix J","title":"Exact dyadic capacity and numerical-profile obstruction","url":"read/paper-xi/appendix-dyadic-capacity.html#L2"}],"31":[{"label":"Appendix J","title":"Exact dyadic capacity and numerical-profile obstruction","url":"read/paper-xi/appendix-dyadic-capacity.html#L2"}],"32":[{"label":"Appendix A","title":"Odd-prime synthesis and complete scope","url":"read/paper-xi/appendix-odd-synthesis.html#L2"},{"label":"Appendix G","title":"Dyadic synthesis and all-length assembly scope","url":"read/paper-xi/appendix-dyadic-synthesis.html#L2"},{"label":"Appendix K","title":"Whole finite extensions and dyadic marking criteria","url":"read/paper-xi/appendix-dyadic-markings.html#L2"}],"33":[{"label":"Appendix K","title":"Whole finite extensions and dyadic marking criteria","url":"read/paper-xi/appendix-dyadic-markings.html#L2"}],"34":[{"label":"Appendix K","title":"Whole finite extensions and dyadic marking criteria","url":"read/paper-xi/appendix-dyadic-markings.html#L2"}],"35":[{"label":"Appendix L","title":"The actual enlarged rank-two inducing carrier","url":"read/paper-xi/appendix-rank2.html#L2"},{"label":"Appendix M","title":"The actual enlarged rank-four inducing carrier","url":"read/paper-xi/appendix-rank4.html#L2"}],"36":[{"label":"Appendix M","title":"The actual enlarged rank-four inducing carrier","url":"read/paper-xi/appendix-rank4.html#L2"}],"37":[{"label":"Appendix N","title":"Rank-four normalization and induction in stages","url":"read/paper-xi/appendix-rank4-stages.html#L2"},{"label":"Appendix O","title":"Exact enlarged-carrier recovery and original-contract comparison","url":"read/paper-xi/appendix-lowrank-contract.html#L2"}],"38":[{"label":"Appendix O","title":"Exact enlarged-carrier recovery and original-contract comparison","url":"read/paper-xi/appendix-lowrank-contract.html#L2"}],"39":[{"label":"Appendix C","title":"Native-rank full characters and tame central completion","url":"read/paper-xi/appendix-odd-character.html#L2"},{"label":"Appendix I","title":"Dyadic full characters and the odd critical endpoint","url":"read/paper-xi/appendix-dyadic-character.html#L2"}],"40":[{"label":"Appendix F","title":"The precise stabilizer-action contract and normalizer obstruction","url":"read/paper-xi/appendix-stabilizer-contract.html#L2"}],"41":[{"label":"Appendix E","title":"Actual scalar inertia phases and finite-cover descent","url":"read/paper-xi/appendix-odd-scalar.html#L2"}],"42":[{"label":"Appendix J","title":"Exact dyadic capacity and numerical-profile obstruction","url":"read/paper-xi/appendix-dyadic-capacity.html#L2"},{"label":"Section 16","title":"Local factor tests and the remaining bridge","url":"read/paper-xi/local-factors-and-conjecture.html#L1814"}],"43":[{"label":"Section 16","title":"Local factor tests and the remaining bridge","url":"read/paper-xi/local-factors-and-conjecture.html#L1814"}],"44":[{"label":"Section 16","title":"Local factor tests and the remaining bridge","url":"read/paper-xi/local-factors-and-conjecture.html#L1814"}],"45":[{"label":"Section 19","title":"Restricted arithmetic realization theorems","url":"read/paper-xi/arithmetic-realization.html#L2003"},{"label":"Appendix A","title":"Odd-prime synthesis and complete scope","url":"read/paper-xi/appendix-odd-synthesis.html#L2"},{"label":"Appendix G","title":"Dyadic synthesis and all-length assembly scope","url":"read/paper-xi/appendix-dyadic-synthesis.html#L2"}],"46":[{"label":"Section 17","title":"Examples and finite certificates","url":"read/paper-xi/finite-examples.html#L1935"}],"47":[{"label":"Section 18","title":"Scope of the finite reconstruction","url":"read/paper-xi/finite-scope.html#L1983"},{"label":"Section 19","title":"Restricted arithmetic realization theorems","url":"read/paper-xi/arithmetic-realization.html#L2003"}]},"new_proof_review":false,"new_lean_execution":false,"new_mathematical_example_execution":false,"new_tex_build":false}