Catalan Constant: source and evidence

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1 \chapter{Formula and source reference}
2 \label{app:formula-reference}
3 
4 \section{Recurrence and initial values}
5 
6 The normalized rational sequences start with
7 \begin{equation}
8  (u_0,u_1)=(1,7/4),\qquad(v_0,v_1)=(0,13/8).
9  \label{eq:app-initial}
10 \end{equation}
11 For $n\geq1$, set
12 \begin{align}
13  p(n)&=20n^2-8n+1,\notag\\
14  q(n)&=3520n^6+5632n^5+2064n^4\notag\\
15  &\hspace{1.6em}-384n^3-156n^2+16n+7,\notag\\
16  a_n&=(2n+1)^2(2n+2)^2p(n),\notag\\
17  c_n&=(2n-1)^2(2n)^2p(n+1).
18  \label{eq:app-recurrence-coeffs}
19 \end{align}
20 Each sequence $w=u$ or $w=v$ obeys
21 \begin{equation}
22  a_nw_{n+1}=q(n)w_n+c_nw_{n-1}.
23  \label{eq:app-recurrence}
24 \end{equation}
25 The forward recurrence is used only at $n\geq1$. Its coefficients $a_n$ are positive there. The first additional terms are
26 \begin{equation}
27  u_2=\frac{649}{64},\quad u_3=\frac{19471}{256},\qquad
28  v_2=\frac{10699}{1152},\quad v_3=\frac{8025653}{115200}.
29  \label{eq:app-small-sequences}
30 \end{equation}
31 
32 The Wronskian in this edition uses the orientation
33 \begin{equation}
34  W_n=u_{n+1}v_n-u_nv_{n+1}
35  =\frac{(-1)^{n+1}p(n+1)}{2(2n+1)^2(2n+2)^2}.
36  \label{eq:app-Wronskian}
37 \end{equation}
38 Reversing its two terms reverses the sign. The formula applies to $n\geq0$, with the base case supplied by the initial pair and the induction step supplied by \eqref{eq:app-recurrence}.
39 
40 \section{Integer packets and integrals}
41 
42 Set $T_n=16^nu_n$. Its forward recurrence is
43 \begin{equation}
44  a_nT_{n+1}=16q(n)T_n+256c_nT_{n-1},
45  \qquad T_0=1,\quad T_1=28.
46  \label{eq:app-T-recurrence}
47 \end{equation}
48 The initial terms are $1,28,2596,311536$. The universally integral expression is
49 \begin{equation}
50  u_n=\sum_{j=0}^n\binom nj
51  \binom{n-\tfrac12}{j}\binom{n+j-\tfrac12}{j},
52  \label{eq:app-half-binomial}
53 \end{equation}
54 with the external coefficient identity and the written half-binomial denominator argument kept distinct.
55 
56 For $n,m\geq0$, the integer and rational packets are
57 \begin{equation}
58  S_{n,m}=\sum_{i=0}^m\binom miT_{n+i},\qquad
59  H_{n,m}=\sum_{i=0}^m\binom mi16^{n+i}v_{n+i}.
60  \label{eq:app-packets}
61 \end{equation}
62 Their raw linear form and integral filter satisfy
63 \begin{align}
64  F_{n,m}&=8(-1)^n(S_{n,m}G-H_{n,m}),\notag\\
65  \widetilde J_{n,m}
66  &=\sum_{i=0}^m(-16)^i\binom miJ_{n+i}
67  =16^{-n}F_{n,m}.
68  \label{eq:app-filter}
69 \end{align}
70 The corrected unfiltered integral is
71 \begin{equation}
72  J_n=\int_0^1\!\int_0^1
73  \frac{x^{n-1/2}(1-x)^ny^n(1-y)^{n-1/2}}
74  {(1-xy)^{n+1}}\,\dd x\,\dd y
75  =8(-1)^n(u_nG-v_n).
76  \label{eq:app-J}
77 \end{equation}
78 These are fixed normalization identities. Their factors should be checked together when translating a differently normalized source.
79 
80 \section{Clearing and primitive reduction}
81 
82 For a fixed pair, put $U=S_{n,m}$ and write $V=H_{n,m}=N/D$ in reduced form. The exact dictionary is
83 \begin{align}
84  Q&=8L_{2(n+m)}^2,\notag\\
85  A&=(-1)^nQU,\qquad B=(-1)^nQV,\notag\\
86  h&=\gcd(DU,N)=\gcd(U,N),\notag\\
87  g&=\gcd(|A|,|B|)=\frac QD\,h,\notag\\
88  q_{\rm red}&=\frac{DU}{h},\qquad p_{\rm red}=\frac Nh,\notag\\
89  \Lambda_{n,m}&=\frac AgG-\frac Bg
90  =\frac{16^nL_{2(n+m)}^2}{g}\widetilde J_{n,m}.
91  \label{eq:app-reduction-dictionary}
92 \end{align}
93 The symbol $q_{\rm red}$ denotes a reduced denominator. It is unrelated to the polynomial $q(n)$ in the recurrence. The rational coefficients of Eskandari's integral normalization are $\widetilde U_{n,m}=8(-1)^n16^{-n}S_{n,m}$ and $\widetilde V_{n,m}=8(-1)^n16^{-n}H_{n,m}$. They must not be mistaken for the fixed-pair aliases $U,V$ above.
94 
95 \section{External inputs used by the piece}
96 
97 The following table is a compact index of the contracts used in this edition. It is not a substitute for their hypotheses, normalization checks, or the original papers. The contracts and acquisition diagnostics accompany the source bundle.
98 
99 \begin{longtable}{@{}p{0.10\linewidth}p{0.27\linewidth}p{0.56\linewidth}@{}}
100 \toprule
101 Contract & Source & Application and boundary\\
102 \midrule
103 \endfirsthead
104 \toprule
105 Contract & Source & Application and boundary\\
106 \midrule
107 \endhead
108 \bottomrule
109 \endfoot
110 LC01 & Zudilin, 2003 & Recurrence, initial pairs, and identification of the limit with $G$. The recurrence is solved forwards only for $n\geq1$.\\[5pt]
111 LC02 & Zudilin, 2003 & Exponential coefficient and error rates. These are asymptotic inputs, not results of the finite run.\\[5pt]
112 LC03 & Zudilin, 2003 & Integral identity with an explicitly audited normalization discrepancy. This edition uses the corrected factor $1/8$.\\[5pt]
113 LC04 & Zudilin, 2002 & Asymptotic denominator bounds. They do not alone prove the exact integrality of every $T_n$.\\[5pt]
114 LC05 & Krattenthaler and Rivoal, 2008 & Strong coefficient arithmetic and the half-binomial formula. The notation bridge is matched at the coefficient level.\\[5pt]
115 LC06 & Eskandari, 2026 & The filter and its coefficient dictionary. The base measure has $1-xy$ outside the square root.\\[5pt]
116 LC07 & Eskandari, 2026 & The bound $M(\alpha)$ and approximation exponent. It is a preprint input below the irrationality threshold.\\[5pt]
117 LC08 & Sun and Wachs, 2026 & Dated irrationality claim and criticism. Neither is imported as an established conclusion here.\\[5pt]
118 LC09 & Zudilin, 2019 & Background disjunction: at least one of six even beta values is irrational. It does not isolate $\beta(2)=G$.\\[5pt]
119 LC10 & Zagier, 1997 & Prime number theorem input and the conversion to $\log L_k=k+o(k)$.\\
120 \end{longtable}
121 
122 The sources corresponding to LC01--LC07 are cited as Zudilin~\cite{Zudilin2003,Zudilin2002}, Krattenthaler and Rivoal~\cite{KR2008}, and Eskandari~\cite{Eskandari2026}. The dated claims in LC08 are Sun~\cite{Sun2026} and Wachs~\cite{Wachs2026}. The background disjunction is Zudilin~\cite{Zudilin2019}. The prime number theorem contract is tied to Zagier's presentation~\cite{Zagier1997}. No complete third-party source paper is redistributed in this bundle.
123 
124 \chapter{Worked small cases and reproduction}
125 \label{app:worked-reproduction}
126 
127 \section{The first recurrence step}
128 
129 At $n=1$, direct substitution in \eqref{eq:app-recurrence-coeffs} gives
130 \begin{equation*}
131  p(1)=13,\quad p(2)=65,\quad q(1)=10699,
132  \quad a_1=1872,\quad c_1=260.
133 \end{equation*}
134 The recurrence therefore yields
135 \begin{align*}
136  u_2&=\frac{10699\cdot(7/4)+260}{1872}
137  =\frac{649}{64},\\
138  v_2&=\frac{10699\cdot(13/8)}{1872}
139  =\frac{10699}{1152}.
140 \end{align*}
141 Multiplying $u_2$ by $16^2$ gives $T_2=2596$. Alternatively, the integer recurrence numerator is
142 \begin{equation*}
143  16\cdot10699\cdot28+256\cdot260\cdot1
144  =4\,859\,712=1872\cdot2596.
145 \end{equation*}
146 The exact division has no remainder. This calculation illustrates the program's division check. The all-index integrality proof uses the separate half-binomial argument.
147 
148 The same small case checks the Wronskian orientation:
149 \begin{align*}
150  W_0&=\frac74\cdot0-1\cdot\frac{13}{8}
151  =-\frac{13}{8},\\
152  W_1&=\frac{649}{64}\cdot\frac{13}{8}
153  -\frac74\cdot\frac{10699}{1152}
154  =\frac{65}{288}.
155 \end{align*}
156 Both agree with \eqref{eq:app-Wronskian}. They show why the sign is $(-1)^{n+1}$ for this orientation.
157 
158 \section{A raw packet and its valuation}
159 
160 For $(n,m)=(1,1)$,
161 \begin{equation*}
162  S_{1,1}=T_1+T_2=28+2596=2624=2^6\cdot41.
163 \end{equation*}
164 Its exact valuation is $v_2(S_{1,1})=6$. Pascal evolution gives the same result from the initial row: $S_{1,1}=S_{1,0}+S_{2,0}$.
165 
166 If this calculation is carried out modulo $2^4$, its residue is zero. The residue therefore certifies only $v_2(S_{1,1})\geq4$. It does not identify the exact valuation as $4$. At modulus $2^8$, the residue is $64$, whose valuation is $6<8$, so the modular valuation criterion certifies the exact value $6$. This concrete example explains both the unresolved branch and the sufficient precision branch in the scan.
167 
168 The associated rational packet is
169 \begin{equation*}
170  H_{1,1}=16v_1+16^2v_2
171  =26+\frac{21398}{9}
172  =\frac{21632}{9}.
173 \end{equation*}
174 Therefore the raw form is
175 \begin{equation*}
176  F_{1,1}=-8\left(2624G-\frac{21632}{9}\right)
177  =\frac{173056}{9}-20992G.
178 \end{equation*}
179 A certified interval for $G$ makes this value positive. This is the smallest counterexample in the finite audit to the proposed sign rule $\operatorname{sign}F_{n,m}=(-1)^m$.
180 
181 \section{Minimal and redundant clearing in the same example}
182 
183 Continue with $(n,m)=(1,1)$. The reduced rational packet has $N=21632$ and $D=9$. The minimal pair is
184 \begin{equation*}
185  (DS_{1,1},N)=(23616,21632).
186 \end{equation*}
187 Its gcd is $h=64$. In fact $23616=64\cdot369$ and $21632=64\cdot338$, and $369$ and $338$ are coprime. Thus the reduced approximation is
188 \begin{equation*}
189  \frac{H_{1,1}}{S_{1,1}}=\frac{338}{369}.
190 \end{equation*}
191 
192 The universal factor uses $L_4=12$, hence $Q=8L_4^2=1152$. Its redundant scale is $Q/D=128$. Consequently
193 \begin{equation*}
194  g=128\cdot64=8192.
195 \end{equation*}
196 The signed cleared pair is $(-3\,022\,848,-2\,768\,896)$. Dividing by $8192$ gives $(-369,-338)$, so the primitive form is
197 \begin{equation}
198  \Lambda_{1,1}=338-369G.
199  \label{eq:app-small-primitive}
200 \end{equation}
201 The integral identity gives the same expression. Since $\widetilde J_{1,1}=F_{1,1}/16$ and $16L_4^2/g=9/32$,
202 \begin{equation*}
203  \frac{16L_4^2}{g}\widetilde J_{1,1}
204  =\frac9{512}F_{1,1}=338-369G.
205 \end{equation*}
206 This checks the factor $8$, the sign $(-1)^n$, and the denominator index in one exact example.
207 
208 For comparison, $(n,m)=(1,0)$ has $S=28,H=26,D=1,h=2,Q=32,g=64$. Its primitive form is $13-14G=J_1$. At $(0,0)$, the convention $H=0,D=1$ gives $S=1,Q=8,g=8$ and primitive form $G=J_0/8$. These boundary cases require no valuation of the integer zero.
209 
210 \section{Current bounds and precision}
211 
212 \begin{longtable}{@{}p{0.28\linewidth}p{0.31\linewidth}p{0.34\linewidth}@{}}
213 \toprule
214 Calculation & Stated domain or precision & What is decided\\
215 \midrule
216 \endfirsthead
217 \toprule
218 Calculation & Stated domain or precision & What is decided\\
219 \midrule
220 \endhead
221 \bottomrule
222 \endfoot
223 Wronskian run & $0\leq n\leq200$ & Exact rational equalities at all recorded indices.\\[5pt]
224 Integer $T$ run & $0\leq n\leq10000$ & Every forward exact division has zero remainder.\\[5pt]
225 Large Pascal scan & $1\leq n\leq2000$, $0\leq m\leq4n$, modulus $2^{128}$ & All $8\,006\,000$ valuations are exact, maximum $31$.\\[5pt]
226 Direct grid & $1\leq n\leq100$, $0\leq m\leq4n$ & All $20\,300$ direct sums agree with the modular scan.\\[5pt]
227 Independent recurrence & $0\leq n\leq500$ & Rational sequence agreement and stated clearing checks.\\[5pt]
228 Half-binomial check & $0\leq n\leq50$, $0\leq j\leq n$ & All $1\,326$ summands and sequence sums agree.\\[5pt]
229 Certified $G$ interval & Euler terms $j=0,\ldots,1000$ & Exact rational endpoints with a tail of at most $2^{-1001}$.\\[5pt]
230 Raw filter signs & $1\leq n\leq20$, $0\leq m\leq4n$ & All $860$ forms are nonzero, and $138$ disagree with the sign guess.\\[5pt]
231 Cleared gcds & The five pairs in Table~\ref{tab:content-five} & Exact factorization and decomposition at those pairs.\\[5pt]
232 Precision negative control & $1\leq n\leq3$, $0\leq m\leq4n$, modulus $2^4$ & Eight zero residues are unresolved, and the direct maximum is $6$.\\
233 \end{longtable}
234 
235 The Euler interval uses $a_k=(2k+1)^{-2}$ and the positive difference convention $\Delta a_k=a_k-a_{k+1}$. The identity
236 \begin{equation*}
237  \Delta^ja_0=\int_0^1(1-x^2)^j(-\log x)\,\dd x
238 \end{equation*}
239 gives $0\leq\Delta^ja_0\leq1$. The positive Euler series for $G$ is $\sum_{j\geq0}2^{-j-1}\Delta^ja_0$. Keeping the first $1001$ terms therefore gives a lower endpoint and an upper endpoint obtained by adding $2^{-1001}$. All retained terms are rational. The analytic tail bound is a written argument. The computed interval is its finite arithmetic application.
240 
241 \section{Run the public checks}
242 
243 The following commands are issued from the root of the unpacked source bundle. They use fresh output paths and do not overwrite the preserved evidence. Python 3.12.14 was used for the recorded computation runs. The public arithmetic programs use the Python standard library.
244 
245 \begin{quote}\small\ttfamily\raggedright
246 python3 computations/catalan\_packet\_audit.py\ \textbackslash\\
247 \hspace*{1em}--scan-n 2000 --precision-bits 128\ \textbackslash\\
248 \hspace*{1em}--output fresh\_default\_results.json
249 \end{quote}
250 
251 \begin{quote}\small\ttfamily\raggedright
252 python3 computations/independent\_audit\_final.py\ \textbackslash\\
253 \hspace*{1em}--output-dir fresh\_independent
254 \end{quote}
255 
256 \begin{quote}\small\ttfamily\raggedright
257 python3 computations/witness\_and\_precision\_audit.py\ \textbackslash\\
258 \hspace*{1em}--output-dir fresh\_witness
259 \end{quote}
260 
261 \begin{quote}\small\ttfamily\raggedright
262 python3 computations/half\_binomial\_audit.py\ \textbackslash\\
263 \hspace*{1em}--output fresh\_half\_binomial\_results.json
264 \end{quote}
265 
266 The first command should reproduce the historical default output exactly. The other commands supply independent constructions, direct witnesses, the precision negative control, and the half-binomial normalization check. A successful exit is not an irrationality certificate. Read the reported ranges, exactness flags, and mismatch records.
267 
268 To rebuild the two publication PDFs from their supplied figures, run \texttt{python3 build.py}. To regenerate the figures as well, run \texttt{python3 build.py --figures}. The latter requires the plotting dependencies recorded in the source bundle, and rebuilding PDFs also requires a compatible LaTeX installation. Figure generation does not participate in the exact arithmetic decisions. A regenerated plot may differ in rendering metadata while carrying the same mathematical data.
269 
270 The source bundle records the actual child command arrays, input and output hashes, timestamps, exit statuses, and resource limits for the preserved runs. Fresh runs create new evidence and should be compared with those records. Floating logarithms may differ in their final displayed digit across platforms. Exact integer and rational strings, valuation decisions, and certified signs are the load-bearing comparisons.
271 

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