From 2f4fd8d9eeda6ad5f16653282c2cb4d55c9178a4 Mon Sep 17 00:00:00 2001 From: allaun Date: Sat, 8 Aug 2026 03:34:11 -0500 Subject: [PATCH] docs(p28): post-submission lint polish (chktex W2/W24) on the audited release line Whitespace-only polish of solution.tex: 46 chktex warnings fixed - W2 (43x): 'word \eqref{...}' -> 'word~\eqref{...}' and continuation-line joins so references stay glued to their prose - W24 (3x): \label glued to the \begin{...} line - W8 kept (5x, all in the DOI identifier 10.1090/S0025-5718-1965-0194620-7: single hyphens are correct there, not prose dashes) Verified: chktex W2+W24 = 0 (total 128, all W3/W25 brace suggestions + W8 DOI false positives); pdflatex 3-pass clean build (0 errors, 0 warnings, 0 overfull, 17 pages, 0 '??'); token-level PDF text diff vs the shipped e85d7bf9 PDF shows only glyph-extraction/wrap artifacts (content identical, whitespace-only tex diff). 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b/experiments/ramanujan_28/submission/solution.tex @@ -323,14 +323,10 @@ The constant and generic coefficient equations are \frac{(A(j-1)+B(j-1))\chi_{j-1}}{\psi_{j-1}}\right]=1 \quad(j\ge1). \end{equation} -Clearing the displayed nonzero factors turns -\eqref{eq:ore0}--\eqref{eq:tail-coefficients} into polynomial +Clearing the displayed nonzero factors turns~\eqref{eq:ore0}--\eqref{eq:tail-coefficients} into polynomial equalities with every coefficient zero. The independent script \texttt{p28\_standalone\_equations.py} expands precisely these equalities -using only rational addition and multiplication. Equation -\eqref{eq:tail-coefficients} gives the first row of -\eqref{eq:tail-contiguity}; applying -\eqref{eq:ore0}--\eqref{eq:ore3} successively gives the other three +using only rational addition and multiplication. Equation~\eqref{eq:tail-coefficients} gives the first row of~\eqref{eq:tail-contiguity}; applying~\eqref{eq:ore0}--\eqref{eq:ore3} successively gives the other three Euler-jet rows. \end{proof} @@ -377,9 +373,8 @@ z(\theta+\tfrac16)(\theta+\tfrac12)(\theta+\tfrac56)\right]y=0. \] Substituting \(z=-x/(1-x)\), using \(\theta=(1-x)x\partial_x\), and multiplying by \(72(1-x)\) -expands to \eqref{eq:transformed-ode}. -Using the displayed decomposition of \(C\), equations -\eqref{eq:ascension}--\eqref{eq:transformed-ode} give +expands to~\eqref{eq:transformed-ode}. +Using the displayed decomposition of \(C\), equations~\eqref{eq:ascension}--\eqref{eq:transformed-ode} give \[ Ck_0=-\frac54. \] @@ -417,8 +412,7 @@ direct substitution in the authoritative matrix gives \end{equation} Thus \(J_N(0)\) has rank one. -The first nonconstant coefficient of the \(\F43\) in -\eqref{eq:tail} equals +The first nonconstant coefficient of the \(\F43\) in~\eqref{eq:tail} equals \[ c_N=\frac{u(3u-2)(3u+2)}{144(u-1)^2}=a_N^{-1}. \] @@ -430,8 +424,8 @@ Since \(z=-x+O(x^2)\), +O(x) \right],\qquad \eta_N\ne0. \end{equation} -The leading vector in \eqref{eq:tail-direction} is precisely the image -direction in \eqref{eq:rank-one}. +The leading vector in~\eqref{eq:tail-direction} is precisely the image +direction in~\eqref{eq:rank-one}. The other expansion required below is equally direct. Since \[ F_{N+1}=\kappa_{N+1}z^{N+2}(1+O(z)),\qquad @@ -443,13 +437,12 @@ k_{N+1}=x^{N+2}\widetilde\eta_N \left(\e_1+xs_N+O(x^2)\right), \qquad\widetilde\eta_N\ne0. \end{equation} -Moreover, \eqref{eq:rank-one} gives +Moreover,~\eqref{eq:rank-one} gives \[ J_N(0)\e_1=u^3(a_N,-1,-1,-1)^T\ne0. \] -\begin{lemma}[DVR step, including the extra first-column zero] -\label{lem:dvr} +\begin{lemma}[DVR step, including the extra first-column zero]\label{lem:dvr} Let \(R_0=\Q[[x]]\), \(H=\diag(x,1,1,1)\), and suppose \[ E=x^NLH,\qquad L\in\operatorname{Mat}_4(R_0),\qquad Ek=0. @@ -514,12 +507,9 @@ Consequently the constant term in the first column of \(fC\) is \(-5/4\), cancelling the first component of every row of \((5/4)\mathcal P\). Hence \(\mathcal E_0=L_0H\). -Apply Lemma~\ref{lem:dvr} inductively, using -\eqref{eq:annihilation}, \eqref{eq:rank-one}, -\eqref{eq:tail-direction}, \eqref{eq:next-tail-direction}, the displayed +Apply Lemma~\ref{lem:dvr} inductively, using~\eqref{eq:annihilation},~\eqref{eq:rank-one},~\eqref{eq:tail-direction},~\eqref{eq:next-tail-direction}, the displayed nonzero first column of \(J_N(0)\), and \(M_Nk_{N+1}=k_N\). -The stronger last-row assertion follows from -\eqref{eq:row-relation}. +The stronger last-row assertion follows from~\eqref{eq:row-relation}. \end{proof} \section{The terminating denominator} @@ -536,8 +526,7 @@ and define Since \(x=-z/(1-z)\), this is also the first component of \((-z)^nCG_N\). -\begin{proposition}[Exact terminating denominator] -\label{prop:terminating} +\begin{proposition}[Exact terminating denominator]\label{prop:terminating} For every \(N\ge0\), \begin{equation}\label{eq:qhat} \frac{\widehat Q_N(z)}{\alpha_n} @@ -619,8 +608,7 @@ integer \(n\) are nonzero for \(n\ge1\); no value is obtained by dividing at \(z=0\), because the verifier cross-multiplies first and the reconstructed base functions have removable limits there. The mandatory sparse-polynomial verifier checks every -reconstruction equation, the closing factorization, and -\eqref{eq:full-gauge}--\eqref{eq:matrix-scalar-bridge} by cross +reconstruction equation, the closing factorization, and~\eqref{eq:full-gauge}--\eqref{eq:matrix-scalar-bridge} by cross multiplication. For any reconstructed horizontal row these identities give \begin{equation}\label{eq:terminating-step} p_{n+1} @@ -642,7 +630,7 @@ P(n,t)={}&-5n-76n^2+1404n^3+4360n^4+4320n^5+1440n^6\\ &+(-51+659n+3086n^2+4500n^3+2232n^4)t^2\\ &+(-72-432n-864n^2-576n^3)t^3. \end{align*} -Thus \eqref{eq:terminating-step} is the scalar form of the displayed +Thus~\eqref{eq:terminating-step} is the scalar form of the displayed matrix recurrence; no differential-equation uniqueness is used below. Let @@ -698,13 +686,11 @@ Moreover, \[ \deg((d_0(\theta)+zd_1(\theta))p_n)\le n+1. \] -Equations \eqref{eq:term-constant}--\eqref{eq:term-top} therefore prove, -coefficient by coefficient, that the right side of -\eqref{eq:terminating-step} is +Equations~\eqref{eq:term-constant}--\eqref{eq:term-top} therefore prove, +coefficient by coefficient, that the right side of~\eqref{eq:terminating-step} is \(\alpha_{n+1}\sum_{k=0}^{n+1}h_{n+1,k}z^k\). -Starting from \eqref{eq:base-horizontal-row}, -\eqref{eq:full-gauge} propagates the horizontal form at every step. -This proves \eqref{eq:qhat} and \eqref{eq:normalization} for the actual +Starting from~\eqref{eq:base-horizontal-row},~\eqref{eq:full-gauge} propagates the horizontal form at every step. +This proves~\eqref{eq:qhat} and~\eqref{eq:normalization} for the actual row \(CG_N\), not merely for a scalar surrogate. The mandatory standalone checker independently expands the cleared identities and rejects any nonzero coefficient. @@ -722,15 +708,14 @@ Moreover, \end{corollary} \begin{proof} -For \(0\le k\le n\), the coefficient of \(z^k\) in -\eqref{eq:qhat} has sign \((-1)^k\). Since \(z_0<0\), every summand is +For \(0\le k\le n\), the coefficient of \(z^k\) in~\eqref{eq:qhat} has sign \((-1)^k\). Since \(z_0<0\), every summand is nonnegative. Also \[ \frac{576n^2(2n+1)^2}{(6n+1)(6n+5)}-29n^2 =\frac{n^2(1260n^2+1260n+431)} {(6n+1)(6n+5)}>0. \] -Iterating \eqref{eq:normalization} proves the lower bound. +Iterating~\eqref{eq:normalization} proves the lower bound. \end{proof} \section{From formal contact to convergence at @@ -802,7 +787,7 @@ We now verify the analytic hypothesis behind the next step. On so the series defining \(y\) and its first three Euler derivatives are holomorphic on a neighborhood of the closed disk. Because \(f=O(x)\), the simple pole of \(C\) cancels in \(fC\), so \(\mathcal E_0\) is -holomorphic there. Inspection of \eqref{eq:deformed-matrix} shows that +holomorphic there. Inspection of~\eqref{eq:deformed-matrix} shows that each \(M_m\) is holomorphic off \(x=0\) in this disk and has at most a simple pole at \(0\). Hence \(\mathcal R_{N,r}=x^n(\mathcal E_N)_{r,1}\), with \(n=N+1\), is @@ -828,7 +813,7 @@ Corollary~\ref{cor:positivity} yields Q_N(x_0)\ge 18\cdot29^N(N!)^2(1-x_0)^n. \] -Combining this with \eqref{eq:cauchy}, we obtain +Combining this with~\eqref{eq:cauchy}, we obtain \begin{equation}\label{eq:geometric-error} \left|\frac{E_{N,r}(x_0)}{q_N(x_0)}\right| =\left|\frac{\mathcal R_{N,r}(x_0)}{Q_N(x_0)}\right| @@ -850,20 +835,19 @@ Therefore \section{Identification of the first-column limit} -By \eqref{eq:seed-identities} and the definition of \(\Phi\), +By~\eqref{eq:seed-identities} and the definition of \(\Phi\), \[ A_0-\Phi A_1 =-A(\mathcal E_0)_{0,*}-B(\mathcal E_0)_{1,*}. \] Multiplying by \(G_N\e_1\), dividing by -\(A_1G_N\e_1=S q_N\), and using -\eqref{eq:error-vanish}, we get +\(A_1G_N\e_1=S q_N\), and using~\eqref{eq:error-vanish}, we get \[ \lim_{N\to\infty} \frac{A_0G_N\e_1}{A_1G_N\e_1} =\Phi(x_0). \] -Equation \eqref{eq:CM-value} therefore proves +Equation~\eqref{eq:CM-value} therefore proves \begin{equation}\label{eq:first-column} \boxed{ \lim_{N\to\infty}\frac{P_{N,1}}{Q_{N,1}} @@ -927,8 +911,7 @@ Q_R(1)=-(64R^3-105R^2+274R-233)<0,\qquad Consequently the unique exterior zero is a simple real number \(\rho>1\). -\begin{lemma}[Explicit dominant-product dichotomy] -\label{lem:dominant-product} +\begin{lemma}[Explicit dominant-product dichotomy]\label{lem:dominant-product} Fix \(\tau\) with \[ \max_{\lambda\ne\rho}|\lambda|<\tau<1, @@ -997,7 +980,7 @@ For stable columns \(\|h\|\le1\), set \[ \Psi_m(h)=\frac{E_mh-c_m}{a_m-b_mh}. \] -Equations \eqref{eq:block-bounds}--\eqref{eq:block-separation} give +Equations~\eqref{eq:block-bounds}--\eqref{eq:block-separation} give \[ \|\Psi_m(h)\| \le\frac{d_*+\epsilon}{a_*-\epsilon}<1. @@ -1009,7 +992,7 @@ For two such columns, +\frac{(E_mk-c_m)b_m(h-k)} {(a_m-b_mh)(a_m-b_mk)}, \] -so \eqref{eq:graph-contraction-constant} gives +so~\eqref{eq:graph-contraction-constant} gives \[ \|\Psi_m(h)-\Psi_m(k)\|\le q\|h-k\|. \] @@ -1042,7 +1025,7 @@ U_m(a)=Z_m(a)P=(\alpha_m,\beta_m),\qquad \xi_m=\alpha_m-\beta_mh_m,\qquad d_m=a_m-b_mh_{m+1}. \] -Using \eqref{eq:graph-invariance} in +Using~\eqref{eq:graph-invariance} in \(U_{m+1}=U_mT_m\) gives the exact scalar equation \[ \xi_{m+1}=d_m\xi_m. @@ -1053,8 +1036,7 @@ Define the composed seed functional and product L_m=\prod_{\ell=m_0}^{m-1}d_\ell. \] Both are now explicit, \(\Lambda\) is linear, and -\(\xi_m=\Lambda(a)L_m\). Equations -\eqref{eq:block-bounds}--\eqref{eq:block-separation} ensure +\(\xi_m=\Lambda(a)L_m\). Equations~\eqref{eq:block-bounds}--\eqref{eq:block-separation} ensure \(d_m\ne0\). If \(\Lambda(a)=0\), then \(\alpha_m=\beta_mh_m\) and @@ -1063,7 +1045,7 @@ If \(\Lambda(a)=0\), then \(\alpha_m=\beta_mh_m\) and \|\beta_{m+1}\|<(d_*+\epsilon)\|\beta_m\| <\tau\|\beta_m\|, \] -which proves \eqref{eq:exceptional-decay}. +which proves~\eqref{eq:exceptional-decay}. If \(\Lambda(a)\ne0\), put \(r_m=\beta_m/\xi_m\). Exact substitution gives @@ -1089,8 +1071,7 @@ Since \(\alpha_m/\xi_m=1+r_mh_m\to1\), \[ U_m(a)=\Lambda(a)L_m\bigl((1,0)+o_a(1)\bigr). \] -Multiplying by \(P^{-1}\) proves -\eqref{eq:dominant-asymptotic} with +Multiplying by \(P^{-1}\) proves~\eqref{eq:dominant-asymptotic} with \[ w=(1,0)P^{-1},\qquad w\mathcal S=\rho w. \] @@ -1119,7 +1100,7 @@ inverse rescaling absorbed into \(L_m\). Since \(R>7\) and \(\rho>1\), every displayed grouping is positive. Thus \(w_j(\rho)>0\) for \(j=1,2,3,4\); below we abbreviate \(w_j=w_j(\rho)\). -Undoing the balancing in \eqref{eq:dominant-asymptotic} gives, whenever +Undoing the balancing in~\eqref{eq:dominant-asymptotic} gives, whenever \(\Lambda(a)\ne0\), \begin{equation}\label{eq:all-column-asymptotic} aG_m\e_j=(m!)^2m^{-(j-1)} @@ -1131,19 +1112,17 @@ The positivity estimate and \(Q_m=x_0^{m+1}q_m\) give \frac{q_m(x_0)}{(m!)^2} \ge18(R-1)\,[29(R-1)]^m. \end{equation} -If \(\Lambda(C)=0\), the first coordinate of -\eqref{eq:exceptional-decay} would contradict \eqref{eq:q-lower}. +If \(\Lambda(C)=0\), the first coordinate of~\eqref{eq:exceptional-decay} would contradict~\eqref{eq:q-lower}. Therefore \[ \Lambda(A_1)=S\Lambda(C)\ne0. \] -If \(\Lambda(A_0)=0\), then -\eqref{eq:exceptional-decay}, \eqref{eq:dominant-asymptotic}, and +If \(\Lambda(A_0)=0\), then~\eqref{eq:exceptional-decay},~\eqref{eq:dominant-asymptotic}, and \(|d_m|>1\) for large \(m\) would make the first-column quotient tend -to zero, contradicting \eqref{eq:first-column}. Hence +to zero, contradicting~\eqref{eq:first-column}. Hence \(\Lambda(A_0)\ne0\) as well. -Because \(w_j>0\), equation \eqref{eq:all-column-asymptotic} first proves +Because \(w_j>0\), equation~\eqref{eq:all-column-asymptotic} first proves that every \(Q_{m,j}\) is nonzero for all sufficiently large \(m\), and only then permits division: \[