Paper IX: Arithmetic Packets, Dyadic Towers, and Local Polynomial Deformation

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6 \@writefile{toc}{\contentsline {section}{What this final edition establishes}{1}{Doc-Start}\protected@file@percent }
7 \@writefile{toc}{\contentsline {section}{\numberline {1}Scope and conventions}{3}{section.1}\protected@file@percent }
8 \newlabel{sec:scope}{{1}{3}{Scope and conventions}{section.1}{}}
9 \newlabel{sec:scope@cref}{{[section][1][]1}{[1][3][]3}{}{}{}}
10 \@writefile{toc}{\contentsline {section}{\numberline {2}Arithmetic packets and coefficient transport}{9}{section.2}\protected@file@percent }
11 \newlabel{sec:foundations}{{2}{9}{Arithmetic packets and coefficient transport}{section.2}{}}
12 \newlabel{sec:foundations@cref}{{[section][2][]2}{[1][9][]9}{}{}{}}
13 \@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Residue selectors and exact reconstruction}{9}{subsection.2.1}\protected@file@percent }
14 \newlabel{thm:residue-reconstruction}{{2.1}{9}{Residue reconstruction}{theorem.2.1}{}}
15 \newlabel{thm:residue-reconstruction@cref}{{[theorem][1][2]2.1}{[1][9][]9}{}{}{}}
16 \@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Formal exponential coordinates and Hurwitz multiplication}{10}{subsection.2.2}\protected@file@percent }
17 \newlabel{thm:egf-coordinates}{{2.2}{10}{Constructed unit-triangular coordinates}{theorem.2.2}{}}
18 \newlabel{thm:egf-coordinates@cref}{{[theorem][2][2]2.2}{[1][10][]10}{}{}{}}
19 \newlabel{thm:hurwitz-product}{{2.3}{10}{Hurwitz product rules}{theorem.2.3}{}}
20 \newlabel{thm:hurwitz-product@cref}{{[theorem][3][2]2.3}{[1][10][]10}{}{}{}}
21 \@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Triangular inversion and a finite precision filtration}{11}{subsection.2.3}\protected@file@percent }
22 \newlabel{thm:triangular-foundation}{{2.4}{11}{Forward substitution and denominator clearing}{theorem.2.4}{}}
23 \newlabel{thm:triangular-foundation@cref}{{[theorem][4][2]2.4}{[1][11][]11}{}{}{}}
24 \newlabel{thm:filtered-foundation}{{2.5}{11}{Filtered finite reconstruction}{theorem.2.5}{}}
25 \newlabel{thm:filtered-foundation@cref}{{[theorem][5][2]2.5}{[1][11][]11}{}{}{}}
26 \@writefile{toc}{\contentsline {subsection}{\numberline {2.4}Finite Mahler coordinates and weighted differences}{12}{subsection.2.4}\protected@file@percent }
27 \newlabel{thm:mahler-foundation}{{2.6}{12}{Finite Newton--Mahler identities}{theorem.2.6}{}}
28 \newlabel{thm:mahler-foundation@cref}{{[theorem][6][2]2.6}{[1][12][]12}{}{}{}}
29 \newlabel{thm:weighted-mahler-foundation}{{2.7}{13}{Weighted difference bound}{theorem.2.7}{}}
30 \newlabel{thm:weighted-mahler-foundation@cref}{{[theorem][7][2]2.7}{[1][12][]13}{}{}{}}
31 \@writefile{toc}{\contentsline {subsection}{\numberline {2.5}Unit modes and the dyadic distinction}{13}{subsection.2.5}\protected@file@percent }
32 \newlabel{thm:unit-modes-foundation}{{2.8}{13}{Differences of unit modes}{theorem.2.8}{}}
33 \newlabel{thm:unit-modes-foundation@cref}{{[theorem][8][2]2.8}{[1][13][]13}{}{}{}}
34 \newlabel{thm:finite-unit-foundation}{{2.9}{13}{Finite unit coordinates}{theorem.2.9}{}}
35 \newlabel{thm:finite-unit-foundation@cref}{{[theorem][9][2]2.9}{[1][13][]13}{}{}{}}
36 \@writefile{toc}{\contentsline {subsection}{\numberline {2.6}Recurrence packets and finite arithmetic synchronization}{14}{subsection.2.6}\protected@file@percent }
37 \newlabel{thm:companion-foundation}{{2.10}{14}{Companion jumps and packet recurrences}{theorem.2.10}{}}
38 \newlabel{thm:companion-foundation@cref}{{[theorem][10][2]2.10}{[1][14][]14}{}{}{}}
39 \newlabel{thm:crt-foundation}{{2.11}{14}{Dyadic--odd packet factorization}{theorem.2.11}{}}
40 \newlabel{thm:crt-foundation@cref}{{[theorem][11][2]2.11}{[1][14][]14}{}{}{}}
41 \@writefile{toc}{\contentsline {subsection}{\numberline {2.7}Polyphase notation retained from the recovered manuscript}{15}{subsection.2.7}\protected@file@percent }
42 \newlabel{subsec:polyphase-exposition}{{2.7}{15}{Polyphase notation retained from the recovered manuscript}{subsection.2.7}{}}
43 \newlabel{subsec:polyphase-exposition@cref}{{[subsection][7][2]2.7}{[1][14][]15}{}{}{}}
44 \@writefile{toc}{\contentsline {section}{\numberline {3}Congruential Euler packets and their arithmetic}{15}{section.3}\protected@file@percent }
45 \newlabel{sec:arithmetic}{{3}{15}{Congruential Euler packets and their arithmetic}{section.3}{}}
46 \newlabel{sec:arithmetic@cref}{{[section][3][]3}{[1][15][]15}{}{}{}}
47 \@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Original coefficients, normalized packets, and precision}{15}{subsection.3.1}\protected@file@percent }
48 \newlabel{def:arith-euler}{{3.1}{15}{Congruential Euler coefficients}{theorem.3.1}{}}
49 \newlabel{def:arith-euler@cref}{{[theorem][1][3]3.1}{[1][15][]15}{}{}{}}
50 \newlabel{eq:arith-euler-egf}{{3.1}{15}{Congruential Euler coefficients}{equation.3.1}{}}
51 \newlabel{eq:arith-euler-egf@cref}{{[equation][1][3]3.1}{[1][15][]15}{}{}{}}
52 \newlabel{eq:arith-kernel}{{3.2}{16}{Original coefficients, normalized packets, and precision}{equation.3.2}{}}
53 \newlabel{eq:arith-kernel@cref}{{[equation][2][3]3.2}{[1][16][]16}{}{}{}}
54 \newlabel{eq:arith-triangular}{{3.3}{16}{Original coefficients, normalized packets, and precision}{equation.3.3}{}}
55 \newlabel{eq:arith-triangular@cref}{{[equation][3][3]3.3}{[1][16][]16}{}{}{}}
56 \newlabel{prop:arith-denominator}{{3.2}{16}{A denominator bound for every address}{theorem.3.2}{}}
57 \newlabel{prop:arith-denominator@cref}{{[theorem][2][3]3.2}{[1][16][]16}{}{}{}}
58 \newlabel{ex:arith-bernoulli}{{3.3}{16}{The Bernoulli boundary address}{theorem.3.3}{}}
59 \newlabel{ex:arith-bernoulli@cref}{{[theorem][3][3]3.3}{[1][16][]16}{}{}{}}
60 \newlabel{def:arith-precision}{{3.4}{17}{Differences and valuation conventions}{theorem.3.4}{}}
61 \newlabel{def:arith-precision@cref}{{[theorem][4][3]3.4}{[1][17][]17}{}{}{}}
62 \@writefile{toc}{\contentsline {subsection}{\numberline {3.2}The standard address window and its common reduction}{17}{subsection.3.2}\protected@file@percent }
63 \newlabel{prop:arith-lucas-window}{{3.5}{17}{Lucas reduction and address independence}{theorem.3.5}{}}
64 \newlabel{prop:arith-lucas-window@cref}{{[theorem][5][3]3.5}{[1][17][]17}{}{}{}}
65 \@writefile{toc}{\contentsline {subsection}{\numberline {3.3}The full Nishibuchi statement and its starting bound}{18}{subsection.3.3}\protected@file@percent }
66 \newlabel{thm:arith-nishibuchi}{{3.6}{18}{Nishibuchi congruence}{theorem.3.6}{}}
67 \newlabel{thm:arith-nishibuchi@cref}{{[theorem][6][3]3.6}{[1][17][]18}{}{}{}}
68 \newlabel{eq:arith-nishibuchi}{{3.4}{18}{Nishibuchi congruence}{equation.3.4}{}}
69 \newlabel{eq:arith-nishibuchi@cref}{{[equation][4][3]3.4}{[1][17][]18}{}{}{}}
70 \newlabel{thm:arith-divisor-base}{{3.7}{18}{All-address differences on the divisor branch}{theorem.3.7}{}}
71 \newlabel{thm:arith-divisor-base@cref}{{[theorem][7][3]3.7}{[1][18][]18}{}{}{}}
72 \newlabel{eq:arith-divisor-integral}{{3.5}{18}{All-address differences on the divisor branch}{equation.3.5}{}}
73 \newlabel{eq:arith-divisor-integral@cref}{{[equation][5][3]3.5}{[1][18][]18}{}{}{}}
74 \newlabel{eq:arith-divisor-bound}{{3.6}{18}{All-address differences on the divisor branch}{equation.3.6}{}}
75 \newlabel{eq:arith-divisor-bound@cref}{{[equation][6][3]3.6}{[1][18][]18}{}{}{}}
76 \newlabel{eq:arith-divisor-normalized}{{3.7}{18}{All-address differences on the divisor branch}{equation.3.7}{}}
77 \newlabel{eq:arith-divisor-normalized@cref}{{[equation][7][3]3.7}{[1][18][]18}{}{}{}}
78 \newlabel{ex:arith-normalization-loss}{{3.8}{20}{Why the factorial loss is necessary}{theorem.3.8}{}}
79 \newlabel{ex:arith-normalization-loss@cref}{{[theorem][8][3]3.8}{[1][20][]20}{}{}{}}
80 \newlabel{cor:arith-window-period}{{3.9}{20}{A common period in the standard window}{theorem.3.9}{}}
81 \newlabel{cor:arith-window-period@cref}{{[theorem][9][3]3.9}{[1][20][]20}{}{}{}}
82 \newlabel{lem:arith-newton-period}{{3.10}{21}{Prime-power period from finite differences}{theorem.3.10}{}}
83 \newlabel{lem:arith-newton-period@cref}{{[theorem][10][3]3.10}{[1][21][]21}{}{}{}}
84 \@writefile{toc}{\contentsline {subsection}{\numberline {3.4}All-row dyadic differences and displacement amplification}{21}{subsection.3.4}\protected@file@percent }
85 \newlabel{thm:arith-dyadic-differences}{{3.11}{22}{All-row dyadic higher differences}{theorem.3.11}{}}
86 \newlabel{thm:arith-dyadic-differences@cref}{{[theorem][11][3]3.11}{[1][21][]22}{}{}{}}
87 \newlabel{eq:arith-allrow-c}{{3.9}{22}{All-row dyadic higher differences}{equation.3.9}{}}
88 \newlabel{eq:arith-allrow-c@cref}{{[equation][9][3]3.9}{[1][21][]22}{}{}{}}
89 \newlabel{eq:arith-allrow-u}{{3.10}{22}{All-row dyadic higher differences}{equation.3.10}{}}
90 \newlabel{eq:arith-allrow-u@cref}{{[equation][10][3]3.10}{[1][21][]22}{}{}{}}
91 \newlabel{eq:arith-charged-product}{{3.11}{22}{All-row dyadic differences and displacement amplification}{equation.3.11}{}}
92 \newlabel{eq:arith-charged-product@cref}{{[equation][11][3]3.11}{[1][22][]22}{}{}{}}
93 \newlabel{cor:arith-amplified}{{3.12}{23}{Step-sensitive divisor-branch bound}{theorem.3.12}{}}
94 \newlabel{cor:arith-amplified@cref}{{[theorem][12][3]3.12}{[1][23][]23}{}{}{}}
95 \newlabel{cor:arith-allrow-period}{{3.13}{23}{Dyadic periods from row zero}{theorem.3.13}{}}
96 \newlabel{cor:arith-allrow-period@cref}{{[theorem][13][3]3.13}{[1][23][]23}{}{}{}}
97 \@writefile{toc}{\contentsline {subsection}{\numberline {3.5}Exact displacement valuations and least packet periods}{23}{subsection.3.5}\protected@file@percent }
98 \newlabel{lem:arith-dominance}{{3.14}{23}{Finite Newton dominance}{theorem.3.14}{}}
99 \newlabel{lem:arith-dominance@cref}{{[theorem][14][3]3.14}{[1][23][]23}{}{}{}}
100 \newlabel{thm:arith-exact}{{3.15}{24}{Exact four-packet displacement law}{theorem.3.15}{}}
101 \newlabel{thm:arith-exact@cref}{{[theorem][15][3]3.15}{[1][24][]24}{}{}{}}
102 \newlabel{cor:arith-least-period}{{3.16}{25}{All positive periods and their least element}{theorem.3.16}{}}
103 \newlabel{cor:arith-least-period@cref}{{[theorem][16][3]3.16}{[1][25][]25}{}{}{}}
104 \@writefile{toc}{\contentsline {subsection}{\numberline {3.6}Classical odd-length coordinates and their congruence ladder}{25}{subsection.3.6}\protected@file@percent }
105 \newlabel{def:arith-odd-source}{{3.17}{25}{Odd-length source and inverse coefficients}{theorem.3.17}{}}
106 \newlabel{def:arith-odd-source@cref}{{[theorem][17][3]3.17}{[1][25][]25}{}{}{}}
107 \newlabel{eq:arith-signed-zigzag}{{3.12}{26}{Classical odd-length coordinates and their congruence ladder}{equation.3.12}{}}
108 \newlabel{eq:arith-signed-zigzag@cref}{{[equation][12][3]3.12}{[1][26][]26}{}{}{}}
109 \newlabel{thm:arith-odd-reconstruction}{{3.18}{26}{Classical odd-length reconstruction}{theorem.3.18}{}}
110 \newlabel{thm:arith-odd-reconstruction@cref}{{[theorem][18][3]3.18}{[1][26][]26}{}{}{}}
111 \newlabel{cor:arith-odd-ladder}{{3.19}{27}{Odd-prime truncation ladder}{theorem.3.19}{}}
112 \newlabel{cor:arith-odd-ladder@cref}{{[theorem][19][3]3.19}{[1][26][]27}{}{}{}}
113 \@writefile{toc}{\contentsline {subsection}{\numberline {3.7}The optimal classical dyadic edge tower}{27}{subsection.3.7}\protected@file@percent }
114 \newlabel{def:arith-truncation}{{3.20}{27}{Dyadic truncation}{theorem.3.20}{}}
115 \newlabel{def:arith-truncation@cref}{{[theorem][20][3]3.20}{[1][27][]27}{}{}{}}
116 \newlabel{eq:arith-tm}{{3.13}{27}{Dyadic truncation}{equation.3.13}{}}
117 \newlabel{eq:arith-tm@cref}{{[equation][13][3]3.13}{[1][27][]27}{}{}{}}
118 \newlabel{lem:arith-reconstruction}{{3.21}{27}{Exact reconstruction and source integrality}{theorem.3.21}{}}
119 \newlabel{lem:arith-reconstruction@cref}{{[theorem][21][3]3.21}{[1][27][]27}{}{}{}}
120 \citation{DLMF24}
121 \newlabel{lem:arith-kappa}{{3.22}{28}{Exact weighted inverse coefficient}{theorem.3.22}{}}
122 \newlabel{lem:arith-kappa@cref}{{[theorem][22][3]3.22}{[1][28][]28}{}{}{}}
123 \newlabel{thm:arith-sharp-tower}{{3.23}{29}{Optimal uniform dyadic tower}{theorem.3.23}{}}
124 \newlabel{thm:arith-sharp-tower@cref}{{[theorem][23][3]3.23}{[1][28][]29}{}{}{}}
125 \newlabel{eq:arith-sharp-tail}{{3.14}{29}{Optimal uniform dyadic tower}{equation.3.14}{}}
126 \newlabel{eq:arith-sharp-tail@cref}{{[equation][14][3]3.14}{[1][28][]29}{}{}{}}
127 \newlabel{prop:arith-row-weight}{{3.24}{29}{Row-dependent summand weights}{theorem.3.24}{}}
128 \newlabel{prop:arith-row-weight@cref}{{[theorem][24][3]3.24}{[1][29][]29}{}{}{}}
129 \@writefile{toc}{\contentsline {subsection}{\numberline {3.8}Isometric continuation of the four dyadic packets}{29}{subsection.3.8}\protected@file@percent }
130 \newlabel{thm:arith-isometry}{{3.25}{30}{Isometric packet bijections}{theorem.3.25}{}}
131 \newlabel{thm:arith-isometry@cref}{{[theorem][25][3]3.25}{[1][29][]30}{}{}{}}
132 \newlabel{cor:arith-affine-image}{{3.26}{30}{Exact image of the original coefficients}{theorem.3.26}{}}
133 \newlabel{cor:arith-affine-image@cref}{{[theorem][26][3]3.26}{[1][30][]30}{}{}{}}
134 \@writefile{toc}{\contentsline {section}{\numberline {4}Threshold-free reciprocal transfer at prime-power section orders}{31}{section.4}\protected@file@percent }
135 \newlabel{sec:prime-power-transfer-final}{{4}{31}{Threshold-free reciprocal transfer at prime-power section orders}{section.4}{}}
136 \newlabel{sec:prime-power-transfer-final@cref}{{[section][4][]4}{[1][31][]31}{}{}{}}
137 \newlabel{eq:final-normalization}{{4.1}{31}{Threshold-free reciprocal transfer at prime-power section orders}{equation.4.1}{}}
138 \newlabel{eq:final-normalization@cref}{{[equation][1][4]4.1}{[1][31][]31}{}{}{}}
139 \newlabel{eq:final-original-normalization}{{4.2}{31}{Threshold-free reciprocal transfer at prime-power section orders}{equation.4.2}{}}
140 \newlabel{eq:final-original-normalization@cref}{{[equation][2][4]4.2}{[1][31][]31}{}{}{}}
141 \@writefile{toc}{\contentsline {subsection}{\numberline {4.1}The Hurwitz lattice and translated differential flatness}{31}{subsection.4.1}\protected@file@percent }
142 \newlabel{eq:hurwitz-reciprocal-recurrence}{{4.3}{31}{The Hurwitz lattice and translated differential flatness}{equation.4.3}{}}
143 \newlabel{eq:hurwitz-reciprocal-recurrence@cref}{{[equation][3][4]4.3}{[1][31][]31}{}{}{}}
144 \newlabel{lem:translated-hurwitz-product}{{4.2}{32}{Translated product formula}{theorem.4.2}{}}
145 \newlabel{lem:translated-hurwitz-product@cref}{{[theorem][2][4]4.2}{[1][31][]32}{}{}{}}
146 \newlabel{eq:translated-product}{{4.4}{32}{Translated product formula}{equation.4.4}{}}
147 \newlabel{eq:translated-product@cref}{{[equation][4][4]4.4}{[1][31][]32}{}{}{}}
148 \newlabel{eq:translated-multinomial}{{4.5}{32}{The Hurwitz lattice and translated differential flatness}{equation.4.5}{}}
149 \newlabel{eq:translated-multinomial@cref}{{[equation][5][4]4.5}{[1][32][]32}{}{}{}}
150 \newlabel{lem:translated-hurwitz-inverse}{{4.3}{32}{Reciprocal transfer}{theorem.4.3}{}}
151 \newlabel{lem:translated-hurwitz-inverse@cref}{{[theorem][3][4]4.3}{[1][32][]32}{}{}{}}
152 \@writefile{toc}{\contentsline {subsection}{\numberline {4.2}The actual Euler source}{32}{subsection.4.2}\protected@file@percent }
153 \newlabel{eq:actual-prime-power-kernel}{{4.6}{32}{The actual Euler source}{equation.4.6}{}}
154 \newlabel{eq:actual-prime-power-kernel@cref}{{[equation][6][4]4.6}{[1][32][]32}{}{}{}}
155 \newlabel{eq:affine-reciprocal-exact}{{4.7}{32}{The actual Euler source}{equation.4.7}{}}
156 \newlabel{eq:affine-reciprocal-exact@cref}{{[equation][7][4]4.7}{[1][32][]32}{}{}{}}
157 \newlabel{eq:discrete-leibniz-final}{{4.8}{33}{The actual Euler source}{equation.4.8}{}}
158 \newlabel{eq:discrete-leibniz-final@cref}{{[equation][8][4]4.8}{[1][32][]33}{}{}{}}
159 \newlabel{thm:all-prime-power-transfer}{{4.4}{33}{All-row prime-power transfer}{theorem.4.4}{}}
160 \newlabel{thm:all-prime-power-transfer@cref}{{[theorem][4][4]4.4}{[1][33][]33}{}{}{}}
161 \newlabel{eq:prime-power-signed-bound}{{4.9}{33}{All-row prime-power transfer}{equation.4.9}{}}
162 \newlabel{eq:prime-power-signed-bound@cref}{{[equation][9][4]4.9}{[1][33][]33}{}{}{}}
163 \newlabel{eq:prime-power-original-twist}{{4.10}{33}{All-row prime-power transfer}{equation.4.10}{}}
164 \newlabel{eq:prime-power-original-twist@cref}{{[equation][10][4]4.10}{[1][33][]33}{}{}{}}
165 \newlabel{cor:all-dyadic-final}{{4.5}{33}{Threshold-free dyadic congruence}{theorem.4.5}{}}
166 \newlabel{cor:all-dyadic-final@cref}{{[theorem][5][4]4.5}{[1][33][]33}{}{}{}}
167 \newlabel{eq:all-dyadic-final-bound}{{4.11}{33}{Threshold-free dyadic congruence}{equation.4.11}{}}
168 \newlabel{eq:all-dyadic-final-bound@cref}{{[equation][11][4]4.11}{[1][33][]33}{}{}{}}
169 \citation{gessel}
170 \@writefile{toc}{\contentsline {section}{\numberline {5}Factorial precision at section order four}{34}{section.5}\protected@file@percent }
171 \newlabel{sec:factorial-four-final}{{5}{34}{Factorial precision at section order four}{section.5}{}}
172 \newlabel{sec:factorial-four-final@cref}{{[section][5][]5}{[1][34][]34}{}{}{}}
173 \newlabel{thm:four-factorial-bound}{{5.1}{34}{Four-address factorial bound}{theorem.5.1}{}}
174 \newlabel{thm:four-factorial-bound@cref}{{[theorem][1][5]5.1}{[1][34][]34}{}{}{}}
175 \newlabel{eq:factorial-four-lower}{{5.1}{34}{Four-address factorial bound}{equation.5.1}{}}
176 \newlabel{eq:factorial-four-lower@cref}{{[equation][1][5]5.1}{[1][34][]34}{}{}{}}
177 \newlabel{eq:factorial-four-j2-filtration}{{5.2}{34}{Four-address factorial bound}{equation.5.2}{}}
178 \newlabel{eq:factorial-four-j2-filtration@cref}{{[equation][2][5]5.2}{[1][34][]34}{}{}{}}
179 \@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Euler moments and an integral divided-power algebra}{34}{subsection.5.1}\protected@file@percent }
180 \newlabel{eq:classical-frobenius-exact}{{5.3}{34}{Euler moments and an integral divided-power algebra}{equation.5.3}{}}
181 \newlabel{eq:classical-frobenius-exact@cref}{{[equation][3][5]5.3}{[1][34][]34}{}{}{}}
182 \newlabel{eq:euler-pd-odd-moments}{{5.4}{34}{Euler moments and an integral divided-power algebra}{equation.5.4}{}}
183 \newlabel{eq:euler-pd-odd-moments@cref}{{[equation][4][5]5.4}{[1][34][]34}{}{}{}}
184 \newlabel{eq:four-euler-factorization}{{5.5}{35}{Euler moments and an integral divided-power algebra}{equation.5.5}{}}
185 \newlabel{eq:four-euler-factorization@cref}{{[equation][5][5]5.5}{[1][34][]35}{}{}{}}
186 \newlabel{eq:four-euler-H}{{5.6}{35}{Euler moments and an integral divided-power algebra}{equation.5.6}{}}
187 \newlabel{eq:four-euler-H@cref}{{[equation][6][5]5.6}{[1][35][]35}{}{}{}}
188 \newlabel{eq:components-divided-powers}{{5.7}{35}{Euler moments and an integral divided-power algebra}{equation.5.7}{}}
189 \newlabel{eq:components-divided-powers@cref}{{[equation][7][5]5.7}{[1][35][]35}{}{}{}}
190 \newlabel{eq:euler-W-factorial}{{5.8}{35}{Euler moments and an integral divided-power algebra}{equation.5.8}{}}
191 \newlabel{eq:euler-W-factorial@cref}{{[equation][8][5]5.8}{[1][35][]35}{}{}{}}
192 \newlabel{eq:base-row-finite-expansion}{{5.9}{35}{Euler moments and an integral divided-power algebra}{equation.5.9}{}}
193 \newlabel{eq:base-row-finite-expansion@cref}{{[equation][9][5]5.9}{[1][35][]35}{}{}{}}
194 \@writefile{toc}{\contentsline {subsection}{\numberline {5.2}A finite step-expansion lemma}{36}{subsection.5.2}\protected@file@percent }
195 \newlabel{eq:finite-step-expansion}{{5.10}{36}{A finite step-expansion lemma}{equation.5.10}{}}
196 \newlabel{eq:finite-step-expansion@cref}{{[equation][10][5]5.10}{[1][35][]36}{}{}{}}
197 \@writefile{toc}{\contentsline {subsection}{\numberline {5.3}An elementary Bernoulli denominator bound}{36}{subsection.5.3}\protected@file@percent }
198 \newlabel{eq:bernoulli-stirling-denominator}{{5.11}{36}{An elementary Bernoulli denominator bound}{equation.5.11}{}}
199 \newlabel{eq:bernoulli-stirling-denominator@cref}{{[equation][11][5]5.11}{[1][36][]36}{}{}{}}
200 \newlabel{eq:elementary-bernoulli-bound}{{5.12}{36}{An elementary Bernoulli denominator bound}{equation.5.12}{}}
201 \newlabel{eq:elementary-bernoulli-bound@cref}{{[equation][12][5]5.12}{[1][36][]36}{}{}{}}
202 \@writefile{toc}{\contentsline {subsection}{\numberline {5.4}Address one and the scaled-binomial transfer}{36}{subsection.5.4}\protected@file@percent }
203 \newlabel{eq:address-one-tangent-factor}{{5.13}{36}{Address one and the scaled-binomial transfer}{equation.5.13}{}}
204 \newlabel{eq:address-one-tangent-factor@cref}{{[equation][13][5]5.13}{[1][36][]36}{}{}{}}
205 \newlabel{eq:address-one-packets}{{5.14}{36}{Address one and the scaled-binomial transfer}{equation.5.14}{}}
206 \newlabel{eq:address-one-packets@cref}{{[equation][14][5]5.14}{[1][36][]36}{}{}{}}
207 \newlabel{eq:address-one-exact-transfer}{{5.15}{37}{Address one and the scaled-binomial transfer}{equation.5.15}{}}
208 \newlabel{eq:address-one-exact-transfer@cref}{{[equation][15][5]5.15}{[1][36][]37}{}{}{}}
209 \newlabel{eq:scaled-binomial-coefficients}{{5.16}{37}{Address one and the scaled-binomial transfer}{equation.5.16}{}}
210 \newlabel{eq:scaled-binomial-coefficients@cref}{{[equation][16][5]5.16}{[1][37][]37}{}{}{}}
211 \newlabel{eq:scaled-binomial-degree-bound}{{5.17}{37}{Address one and the scaled-binomial transfer}{equation.5.17}{}}
212 \newlabel{eq:scaled-binomial-degree-bound@cref}{{[equation][17][5]5.17}{[1][37][]37}{}{}{}}
213 \newlabel{eq:scaled-binomial-difference-bound}{{5.18}{37}{Address one and the scaled-binomial transfer}{equation.5.18}{}}
214 \newlabel{eq:scaled-binomial-difference-bound@cref}{{[equation][18][5]5.18}{[1][37][]37}{}{}{}}
215 \newlabel{eq:address-one-extra-bit}{{5.19}{37}{Address one and the scaled-binomial transfer}{equation.5.19}{}}
216 \newlabel{eq:address-one-extra-bit@cref}{{[equation][19][5]5.19}{[1][37][]37}{}{}{}}
217 \@writefile{toc}{\contentsline {subsection}{\numberline {5.5}Central factorials and integral Bernoulli divided powers}{38}{subsection.5.5}\protected@file@percent }
218 \newlabel{lem:bernoulli-central-pd}{{5.2}{38}{Bernoulli divided-power moments}{theorem.5.2}{}}
219 \newlabel{lem:bernoulli-central-pd@cref}{{[theorem][2][5]5.2}{[1][37][]38}{}{}{}}
220 \newlabel{eq:bernoulli-central-moment-bound}{{5.20}{38}{Bernoulli divided-power moments}{equation.5.20}{}}
221 \newlabel{eq:bernoulli-central-moment-bound@cref}{{[equation][20][5]5.20}{[1][37][]38}{}{}{}}
222 \newlabel{eq:central-factorial-substitution}{{5.21}{38}{Central factorials and integral Bernoulli divided powers}{equation.5.21}{}}
223 \newlabel{eq:central-factorial-substitution@cref}{{[equation][21][5]5.21}{[1][38][]38}{}{}{}}
224 \newlabel{eq:central-factorial-evaluation}{{5.22}{38}{Central factorials and integral Bernoulli divided powers}{equation.5.22}{}}
225 \newlabel{eq:central-factorial-evaluation@cref}{{[equation][22][5]5.22}{[1][38][]38}{}{}{}}
226 \newlabel{eq:central-factorial-recurrence}{{5.23}{38}{Central factorials and integral Bernoulli divided powers}{equation.5.23}{}}
227 \newlabel{eq:central-factorial-recurrence@cref}{{[equation][23][5]5.23}{[1][38][]38}{}{}{}}
228 \newlabel{eq:central-factorial-term-bound}{{5.24}{38}{Central factorials and integral Bernoulli divided powers}{equation.5.24}{}}
229 \newlabel{eq:central-factorial-term-bound@cref}{{[equation][24][5]5.24}{[1][38][]38}{}{}{}}
230 \@writefile{toc}{\contentsline {subsection}{\numberline {5.6}Address two and its stronger filtration}{39}{subsection.5.6}\protected@file@percent }
231 \newlabel{eq:address-two-factorization}{{5.25}{39}{Address two and its stronger filtration}{equation.5.25}{}}
232 \newlabel{eq:address-two-factorization@cref}{{[equation][25][5]5.25}{[1][39][]39}{}{}{}}
233 \newlabel{eq:address-two-H}{{5.26}{39}{Address two and its stronger filtration}{equation.5.26}{}}
234 \newlabel{eq:address-two-H@cref}{{[equation][26][5]5.26}{[1][39][]39}{}{}{}}
235 \newlabel{eq:address-two-basic-factorial}{{5.27}{39}{Address two and its stronger filtration}{equation.5.27}{}}
236 \newlabel{eq:address-two-basic-factorial@cref}{{[equation][27][5]5.27}{[1][39][]39}{}{}{}}
237 \newlabel{eq:address-two-initial-newton}{{5.28}{39}{Address two and its stronger filtration}{equation.5.28}{}}
238 \newlabel{eq:address-two-initial-newton@cref}{{[equation][28][5]5.28}{[1][39][]39}{}{}{}}
239 \newlabel{eq:shifted-newton-finite}{{5.29}{39}{Address two and its stronger filtration}{equation.5.29}{}}
240 \newlabel{eq:shifted-newton-finite@cref}{{[equation][29][5]5.29}{[1][39][]39}{}{}{}}
241 \newlabel{eq:address-two-filtration-algebra}{{5.30}{39}{Address two and its stronger filtration}{equation.5.30}{}}
242 \newlabel{eq:address-two-filtration-algebra@cref}{{[equation][30][5]5.30}{[1][39][]39}{}{}{}}
243 \newlabel{eq:address-two-filtration-pd}{{5.31}{39}{Address two and its stronger filtration}{equation.5.31}{}}
244 \newlabel{eq:address-two-filtration-pd@cref}{{[equation][31][5]5.31}{[1][39][]39}{}{}{}}
245 \@writefile{toc}{\contentsline {subsection}{\numberline {5.7}Address three and cancellation of the apparent pole}{40}{subsection.5.7}\protected@file@percent }
246 \newlabel{eq:address-three-cotangent-factor}{{5.32}{40}{Address three and cancellation of the apparent pole}{equation.5.32}{}}
247 \newlabel{eq:address-three-cotangent-factor@cref}{{[equation][32][5]5.32}{[1][40][]40}{}{}{}}
248 \newlabel{eq:address-three-packets}{{5.33}{40}{Address three and cancellation of the apparent pole}{equation.5.33}{}}
249 \newlabel{eq:address-three-packets@cref}{{[equation][33][5]5.33}{[1][40][]40}{}{}{}}
250 \newlabel{eq:address-three-exact-transfer}{{5.34}{40}{Address three and cancellation of the apparent pole}{equation.5.34}{}}
251 \newlabel{eq:address-three-exact-transfer@cref}{{[equation][34][5]5.34}{[1][40][]40}{}{}{}}
252 \newlabel{eq:address-three-extra-bit}{{5.35}{40}{Address three and cancellation of the apparent pole}{equation.5.35}{}}
253 \newlabel{eq:address-three-extra-bit@cref}{{[equation][35][5]5.35}{[1][40][]40}{}{}{}}
254 \@writefile{toc}{\contentsline {section}{\numberline {6}Sharp higher-difference families}{40}{section.6}\protected@file@percent }
255 \newlabel{sec:sharp-higher-final}{{6}{40}{Sharp higher-difference families}{section.6}{}}
256 \newlabel{sec:sharp-higher-final@cref}{{[section][6][]6}{[1][40][]40}{}{}{}}
257 \newlabel{thm:zero-one-exact-mixed}{{6.1}{40}{Exact factorial precision at addresses zero and one}{theorem.6.1}{}}
258 \newlabel{thm:zero-one-exact-mixed@cref}{{[theorem][1][6]6.1}{[1][40][]40}{}{}{}}
259 \newlabel{eq:zero-one-homogeneous-exact}{{6.1}{40}{Exact factorial precision at addresses zero and one}{equation.6.1}{}}
260 \newlabel{eq:zero-one-homogeneous-exact@cref}{{[equation][1][6]6.1}{[1][40][]40}{}{}{}}
261 \newlabel{eq:zero-parity-decomposition}{{6.2}{41}{Sharp higher-difference families}{equation.6.2}{}}
262 \newlabel{eq:zero-parity-decomposition@cref}{{[equation][2][6]6.2}{[1][41][]41}{}{}{}}
263 \newlabel{eq:zero-parity-leading-pd}{{6.3}{41}{Sharp higher-difference families}{equation.6.3}{}}
264 \newlabel{eq:zero-parity-leading-pd@cref}{{[equation][3][6]6.3}{[1][41][]41}{}{}{}}
265 \newlabel{eq:zero-one-mixed-excess}{{6.4}{41}{Sharp higher-difference families}{equation.6.4}{}}
266 \newlabel{eq:zero-one-mixed-excess@cref}{{[equation][4][6]6.4}{[1][41][]41}{}{}{}}
267 \newlabel{thm:two-exact-power-orders}{{6.3}{42}{Exact address-two power orders}{theorem.6.3}{}}
268 \newlabel{thm:two-exact-power-orders@cref}{{[theorem][3][6]6.3}{[1][41][]42}{}{}{}}
269 \newlabel{eq:two-power-orders-exact}{{6.5}{42}{Exact address-two power orders}{equation.6.5}{}}
270 \newlabel{eq:two-power-orders-exact@cref}{{[equation][5][6]6.5}{[1][41][]42}{}{}{}}
271 \newlabel{eq:two-power-leading-normalized}{{6.6}{42}{Sharp higher-difference families}{equation.6.6}{}}
272 \newlabel{eq:two-power-leading-normalized@cref}{{[equation][6][6]6.6}{[1][42][]42}{}{}{}}
273 \newlabel{eq:two-power-Q-evaluation}{{6.7}{42}{Sharp higher-difference families}{equation.6.7}{}}
274 \newlabel{eq:two-power-Q-evaluation@cref}{{[equation][7][6]6.7}{[1][42][]42}{}{}{}}
275 \@writefile{toc}{\contentsline {section}{\numberline {7}Exact first differences, least periods, and dyadic images}{43}{section.7}\protected@file@percent }
276 \newlabel{sec:sharp-periods-final}{{7}{43}{Exact first differences, least periods, and dyadic images}{section.7}{}}
277 \newlabel{sec:sharp-periods-final@cref}{{[section][7][]7}{[1][43][]43}{}{}{}}
278 \newlabel{lem:newton-exact-isometry-criterion}{{7.1}{43}{A first-difference criterion}{theorem.7.1}{}}
279 \newlabel{lem:newton-exact-isometry-criterion@cref}{{[theorem][1][7]7.1}{[1][43][]43}{}{}{}}
280 \newlabel{eq:newton-isometry-criterion}{{7.1}{43}{A first-difference criterion}{equation.7.1}{}}
281 \newlabel{eq:newton-isometry-criterion@cref}{{[equation][1][7]7.1}{[1][43][]43}{}{}{}}
282 \newlabel{eq:newton-isometry-conclusion}{{7.2}{43}{A first-difference criterion}{equation.7.2}{}}
283 \newlabel{eq:newton-isometry-conclusion@cref}{{[equation][2][7]7.2}{[1][43][]43}{}{}{}}
284 \newlabel{thm:rank-four-exact-periods}{{7.2}{43}{Exact four-address first differences}{theorem.7.2}{}}
285 \newlabel{thm:rank-four-exact-periods@cref}{{[theorem][2][7]7.2}{[1][43][]43}{}{}{}}
286 \newlabel{eq:rank-four-normalized-isometry}{{7.3}{43}{Exact four-address first differences}{equation.7.3}{}}
287 \newlabel{eq:rank-four-normalized-isometry@cref}{{[equation][3][7]7.3}{[1][43][]43}{}{}{}}
288 \newlabel{eq:rank-four-original-isometry}{{7.4}{43}{Exact four-address first differences}{equation.7.4}{}}
289 \newlabel{eq:rank-four-original-isometry@cref}{{[equation][4][7]7.4}{[1][43][]43}{}{}{}}
290 \newlabel{eq:rank-four-least-period-final}{{7.5}{44}{Exact four-address first differences}{equation.7.5}{}}
291 \newlabel{eq:rank-four-least-period-final@cref}{{[equation][5][7]7.5}{[1][43][]44}{}{}{}}
292 \newlabel{eq:exact-reciprocal-row-recurrence}{{7.6}{44}{Exact first differences, least periods, and dyadic images}{equation.7.6}{}}
293 \newlabel{eq:exact-reciprocal-row-recurrence@cref}{{[equation][6][7]7.6}{[1][43][]44}{}{}{}}
294 \newlabel{cor:four-exact-images}{{7.3}{44}{Exact image and residue enumeration}{theorem.7.3}{}}
295 \newlabel{cor:four-exact-images@cref}{{[theorem][3][7]7.3}{[1][44][]44}{}{}{}}
296 \newlabel{eq:four-interpolation-images-final}{{7.7}{44}{Exact image and residue enumeration}{equation.7.7}{}}
297 \newlabel{eq:four-interpolation-images-final@cref}{{[equation][7][7]7.7}{[1][44][]44}{}{}{}}
298 \@writefile{toc}{\contentsline {subsection}{\numberline {7.1}An infinite exact family at all dyadic ranks}{45}{subsection.7.1}\protected@file@percent }
299 \newlabel{thm:lower-half-all-dyadic-isometry}{{7.4}{45}{Lower-half exact laws}{theorem.7.4}{}}
300 \newlabel{thm:lower-half-all-dyadic-isometry@cref}{{[theorem][4][7]7.4}{[1][45][]45}{}{}{}}
301 \newlabel{eq:lower-half-all-dyadic-exact}{{7.8}{45}{Lower-half exact laws}{equation.7.8}{}}
302 \newlabel{eq:lower-half-all-dyadic-exact@cref}{{[equation][8][7]7.8}{[1][45][]45}{}{}{}}
303 \newlabel{eq:lower-half-initial-newton}{{7.9}{45}{An infinite exact family at all dyadic ranks}{equation.7.9}{}}
304 \newlabel{eq:lower-half-initial-newton@cref}{{[equation][9][7]7.9}{[1][45][]45}{}{}{}}
305 \@writefile{toc}{\contentsline {subsection}{\numberline {7.2}Complete exact laws at ranks two and eight}{45}{subsection.7.2}\protected@file@percent }
306 \newlabel{prop:two-four-eight-isometry-table}{{7.5}{45}{Small-rank complete isometry table}{theorem.7.5}{}}
307 \newlabel{prop:two-four-eight-isometry-table@cref}{{[theorem][5][7]7.5}{[1][45][]45}{}{}{}}
308 \@writefile{toc}{\contentsline {section}{\numberline {8}Analytic interpolation and exact Taylor radii}{46}{section.8}\protected@file@percent }
309 \newlabel{sec:analytic-radii-final}{{8}{46}{Analytic interpolation and exact Taylor radii}{section.8}{}}
310 \newlabel{sec:analytic-radii-final@cref}{{[section][8][]8}{[1][46][]46}{}{}{}}
311 \newlabel{thm:four-taylor-radii}{{8.1}{47}{Taylor coefficient bounds and radii}{theorem.8.1}{}}
312 \newlabel{thm:four-taylor-radii@cref}{{[theorem][1][8]8.1}{[1][46][]47}{}{}{}}
313 \newlabel{eq:zero-one-exact-taylor-radius}{{8.1}{47}{Taylor coefficient bounds and radii}{equation.8.1}{}}
314 \newlabel{eq:zero-one-exact-taylor-radius@cref}{{[equation][1][8]8.1}{[1][46][]47}{}{}{}}
315 \newlabel{eq:two-exact-taylor-radius}{{8.2}{47}{Taylor coefficient bounds and radii}{equation.8.2}{}}
316 \newlabel{eq:two-exact-taylor-radius@cref}{{[equation][2][8]8.2}{[1][47][]47}{}{}{}}
317 \newlabel{eq:newton-taylor-stirling}{{8.3}{47}{Analytic interpolation and exact Taylor radii}{equation.8.3}{}}
318 \newlabel{eq:newton-taylor-stirling@cref}{{[equation][3][8]8.3}{[1][47][]47}{}{}{}}
319 \newlabel{eq:taylor-translation-coefficient}{{8.4}{47}{Analytic interpolation and exact Taylor radii}{equation.8.4}{}}
320 \newlabel{eq:taylor-translation-coefficient@cref}{{[equation][4][8]8.4}{[1][47][]47}{}{}{}}
321 \@writefile{toc}{\contentsline {section}{\numberline {9}The final congruence in the original Euler convention}{48}{section.9}\protected@file@percent }
322 \newlabel{sec:final-original-congruence}{{9}{48}{The final congruence in the original Euler convention}{section.9}{}}
323 \newlabel{sec:final-original-congruence@cref}{{[section][9][]9}{[1][47][]48}{}{}{}}
324 \newlabel{cor:final-euler-congruence}{{9.1}{48}{Final common-step congruence}{theorem.9.1}{}}
325 \newlabel{cor:final-euler-congruence@cref}{{[theorem][1][9]9.1}{[1][48][]48}{}{}{}}
326 \newlabel{eq:boxed-final-euler-congruence}{{9.1}{48}{Final common-step congruence}{equation.9.1}{}}
327 \newlabel{eq:boxed-final-euler-congruence@cref}{{[equation][1][9]9.1}{[1][48][]48}{}{}{}}
328 \@writefile{toc}{\contentsline {subsection}{\numberline {9.1}Proof coverage and remaining questions}{48}{subsection.9.1}\protected@file@percent }
329 \@writefile{toc}{\contentsline {section}{\numberline {10}Finite obstructions, compatible lifts, and deformation}{50}{section.10}\protected@file@percent }
330 \newlabel{sec:lifting}{{10}{50}{Finite obstructions, compatible lifts, and deformation}{section.10}{}}
331 \newlabel{sec:lifting@cref}{{[section][10][]10}{[1][50][]50}{}{}{}}
332 \@writefile{toc}{\contentsline {subsection}{\numberline {10.1}The polynomial correction equation}{50}{subsection.10.1}\protected@file@percent }
333 \newlabel{lem:polynomial-taylor}{{10.1}{50}{Polynomial Taylor remainder}{theorem.10.1}{}}
334 \newlabel{lem:polynomial-taylor@cref}{{[theorem][1][10]10.1}{[1][50][]50}{}{}{}}
335 \newlabel{eq:polynomial-taylor}{{10.1}{50}{Polynomial Taylor remainder}{equation.10.1}{}}
336 \newlabel{eq:polynomial-taylor@cref}{{[equation][1][10]10.1}{[1][50][]50}{}{}{}}
337 \newlabel{thm:finite-polynomial-obstruction}{{10.2}{51}{Finite polynomial obstruction}{theorem.10.2}{}}
338 \newlabel{thm:finite-polynomial-obstruction@cref}{{[theorem][2][10]10.2}{[1][50][]51}{}{}{}}
339 \newlabel{eq:finite-obstruction}{{10.2}{51}{Finite polynomial obstruction}{equation.10.2}{}}
340 \newlabel{eq:finite-obstruction@cref}{{[equation][2][10]10.2}{[1][50][]51}{}{}{}}
341 \newlabel{eq:digit-correction}{{10.3}{51}{The polynomial correction equation}{equation.10.3}{}}
342 \newlabel{eq:digit-correction@cref}{{[equation][3][10]10.3}{[1][51][]51}{}{}{}}
343 \newlabel{prop:affine-obstruction}{{10.3}{51}{Affine depth zero}{theorem.10.3}{}}
344 \newlabel{prop:affine-obstruction@cref}{{[theorem][3][10]10.3}{[1][51][]51}{}{}{}}
345 \newlabel{prop:representative-independence}{{10.4}{51}{Independence of the integer representative}{theorem.10.4}{}}
346 \newlabel{prop:representative-independence@cref}{{[theorem][4][10]10.4}{[1][51][]51}{}{}{}}
347 \@writefile{toc}{\contentsline {subsection}{\numberline {10.2}One correction shared by two systems}{52}{subsection.10.2}\protected@file@percent }
348 \newlabel{eq:common-quotient}{{10.4}{52}{One correction shared by two systems}{equation.10.4}{}}
349 \newlabel{eq:common-quotient@cref}{{[equation][4][10]10.4}{[1][52][]52}{}{}{}}
350 \newlabel{thm:common-correction}{{10.5}{52}{Common correction and fiber product}{theorem.10.5}{}}
351 \newlabel{thm:common-correction@cref}{{[theorem][5][10]10.5}{[1][52][]52}{}{}{}}
352 \newlabel{eq:common-correction}{{10.5}{52}{Common correction and fiber product}{equation.10.5}{}}
353 \newlabel{eq:common-correction@cref}{{[equation][5][10]10.5}{[1][52][]52}{}{}{}}
354 \newlabel{eq:joint-fiber-product}{{10.6}{52}{Common correction and fiber product}{equation.10.6}{}}
355 \newlabel{eq:joint-fiber-product@cref}{{[equation][6][10]10.6}{[1][52][]52}{}{}{}}
356 \newlabel{cor:shared-polynomial}{{10.6}{52}{Simultaneous polynomial lifting}{theorem.10.6}{}}
357 \newlabel{cor:shared-polynomial@cref}{{[theorem][6][10]10.6}{[1][52][]52}{}{}{}}
358 \@writefile{toc}{\contentsline {subsection}{\numberline {10.3}Correction torsors and exact residue counts}{53}{subsection.10.3}\protected@file@percent }
359 \newlabel{thm:lift-torsor-count}{{10.7}{53}{Torsor and lift cardinalities}{theorem.10.7}{}}
360 \newlabel{thm:lift-torsor-count@cref}{{[theorem][7][10]10.7}{[1][53][]53}{}{}{}}
361 \newlabel{eq:lift-count}{{10.7}{53}{Torsor and lift cardinalities}{equation.10.7}{}}
362 \newlabel{eq:lift-count@cref}{{[equation][7][10]10.7}{[1][53][]53}{}{}{}}
363 \@writefile{toc}{\contentsline {subsection}{\numberline {10.4}Smooth compatible towers and exact roots}{54}{subsection.10.4}\protected@file@percent }
364 \newlabel{thm:smooth-tower}{{10.8}{54}{Compatible polynomial tower}{theorem.10.8}{}}
365 \newlabel{thm:smooth-tower@cref}{{[theorem][8][10]10.8}{[1][54][]54}{}{}{}}
366 \newlabel{thm:smooth-padic-root}{{10.9}{54}{Completion and smooth branching}{theorem.10.9}{}}
367 \newlabel{thm:smooth-padic-root@cref}{{[theorem][9][10]10.9}{[1][54][]54}{}{}{}}
368 \newlabel{rem:singular-tower-boundary}{{10.10}{55}{}{theorem.10.10}{}}
369 \newlabel{rem:singular-tower-boundary@cref}{{[theorem][10][10]10.10}{[1][55][]55}{}{}{}}
370 \@writefile{toc}{\contentsline {subsection}{\numberline {10.5}Coefficient stability and continuous root charts}{55}{subsection.10.5}\protected@file@percent }
371 \newlabel{prop:smooth-coefficient-stability}{{10.11}{55}{Smooth coefficient stability}{theorem.10.11}{}}
372 \newlabel{prop:smooth-coefficient-stability@cref}{{[theorem][11][10]10.11}{[1][55][]55}{}{}{}}
373 \newlabel{thm:continuous-parameter-chart}{{10.12}{55}{Continuous charts with a selected Jacobian minor}{theorem.10.12}{}}
374 \newlabel{thm:continuous-parameter-chart@cref}{{[theorem][12][10]10.12}{[1][55][]55}{}{}{}}
375 \newlabel{eq:chart-congruence}{{10.8}{55}{Continuous charts with a selected Jacobian minor}{equation.10.8}{}}
376 \newlabel{eq:chart-congruence@cref}{{[equation][8][10]10.8}{[1][55][]55}{}{}{}}
377 \newlabel{eq:polynomial-ball-image}{{10.9}{56}{Coefficient stability and continuous root charts}{equation.10.9}{}}
378 \newlabel{eq:polynomial-ball-image@cref}{{[equation][9][10]10.9}{[1][56][]56}{}{}{}}
379 \@writefile{toc}{\contentsline {subsection}{\numberline {10.6}Prescribed parameters and their obstruction}{56}{subsection.10.6}\protected@file@percent }
380 \newlabel{thm:prescribed-obstruction}{{10.13}{57}{Prescribed-parameter obstruction and count}{theorem.10.13}{}}
381 \newlabel{thm:prescribed-obstruction@cref}{{[theorem][13][10]10.13}{[1][56][]57}{}{}{}}
382 \newlabel{eq:prescribed-count}{{10.10}{57}{Prescribed-parameter obstruction and count}{equation.10.10}{}}
383 \newlabel{eq:prescribed-count@cref}{{[equation][10][10]10.10}{[1][57][]57}{}{}{}}
384 \newlabel{thm:prescribed-integer-tower}{{10.14}{57}{Every compatible integer parameter path}{theorem.10.14}{}}
385 \newlabel{thm:prescribed-integer-tower@cref}{{[theorem][14][10]10.14}{[1][57][]57}{}{}{}}
386 \newlabel{thm:prescribed-padic-tower}{{10.15}{58}{Arbitrary $p$-adic parameter endpoints}{theorem.10.15}{}}
387 \newlabel{thm:prescribed-padic-tower@cref}{{[theorem][15][10]10.15}{[1][57][]58}{}{}{}}
388 \@writefile{toc}{\contentsline {subsection}{\numberline {10.7}Weighted lifting for a nonunit Jacobian}{58}{subsection.10.7}\protected@file@percent }
389 \newlabel{thm:weighted-lifting}{{10.16}{58}{Weighted nonlinear lifting}{theorem.10.16}{}}
390 \newlabel{thm:weighted-lifting@cref}{{[theorem][16][10]10.16}{[1][58][]58}{}{}{}}
391 \newlabel{eq:weighted-hypotheses}{{10.11}{58}{Weighted nonlinear lifting}{equation.10.11}{}}
392 \newlabel{eq:weighted-hypotheses@cref}{{[equation][11][10]10.11}{[1][58][]58}{}{}{}}
393 \newlabel{eq:weighted-displacement}{{10.12}{58}{Weighted nonlinear lifting}{equation.10.12}{}}
394 \newlabel{eq:weighted-displacement@cref}{{[equation][12][10]10.12}{[1][58][]58}{}{}{}}
395 \newlabel{eq:weighted-fixed-point}{{10.13}{59}{Weighted lifting for a nonunit Jacobian}{equation.10.13}{}}
396 \newlabel{eq:weighted-fixed-point@cref}{{[equation][13][10]10.13}{[1][59][]59}{}{}{}}
397 \newlabel{ex:weighted-boundary}{{10.17}{59}{Strict threshold and displacement boundary}{theorem.10.17}{}}
398 \newlabel{ex:weighted-boundary@cref}{{[theorem][17][10]10.17}{[1][59][]59}{}{}{}}
399 \@writefile{toc}{\contentsline {subsection}{\numberline {10.8}First-order coefficient and parameter motion}{59}{subsection.10.8}\protected@file@percent }
400 \newlabel{thm:coefficient-motion}{{10.18}{59}{Weighted coefficient deformation}{theorem.10.18}{}}
401 \newlabel{thm:coefficient-motion@cref}{{[theorem][18][10]10.18}{[1][59][]59}{}{}{}}
402 \newlabel{eq:coefficient-first-order}{{10.14}{59}{Weighted coefficient deformation}{equation.10.14}{}}
403 \newlabel{eq:coefficient-first-order@cref}{{[equation][14][10]10.14}{[1][59][]59}{}{}{}}
404 \newlabel{thm:joint-deformation}{{10.19}{60}{Joint coefficient and parameter deformation}{theorem.10.19}{}}
405 \newlabel{thm:joint-deformation@cref}{{[theorem][19][10]10.19}{[1][60][]60}{}{}{}}
406 \newlabel{eq:joint-first-order}{{10.15}{60}{Joint coefficient and parameter deformation}{equation.10.15}{}}
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1154 \bibcite{Nishibuchi}{1}
1155 \bibcite{gessel}{2}
1156 \bibcite{DLMF24}{3}
1157 \bibcite{Mathlib}{4}
1158 \bibcite{PaperIXArchive}{5}
1159 \bibcite{Amice1964}{6}
1160 \bibcite{BarberoCerrutiMurru2018}{7}
1161 \bibcite{CarlitzLevine1960}{8}
1162 \bibcite{Chen2004}{9}
1163 \bibcite{Comtet1974}{10}
1164 \bibcite{EverestVanDerPoortenShparlinskiWard2003}{11}
1165 \bibcite{Gouvea1997}{12}
1166 \bibcite{GrahamKnuthPatashnik1994}{13}
1167 \bibcite{IrelandRosen1990}{14}
1168 \bibcite{Keigher1997}{15}
1169 \bibcite{Koblitz1984}{16}
1170 \bibcite{LidlNiederreiter1997}{17}
1171 \bibcite{Lucas1878a}{18}
1172 \bibcite{Mahler1958}{19}
1173 \bibcite{Robert2000}{20}
1174 \bibcite{Roman1984}{21}
1175 \bibcite{Serre1979}{22}
1176 \bibcite{Stanley2012}{23}
1177 \bibcite{Sun2008}{24}
1178 \bibcite{Washington1997}{25}
1179 \bibcite{Young1999}{26}
1180 \bibcite{Young2001}{27}
1181 \gdef \@abspage@last{232}

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