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).

NOTE: this is the parallel audited line (492c8ab provenance), NOT the
authority manuscript (completed-submissions 1cbb5982, hand-maintained,
must not be regenerated).

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
+Clearing the displayed nonzero factors turns~\eqref{eq:ore0}--\eqref{eq:tail-coefficients} into polynomial
-\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
+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
-\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}.
+expands to~\eqref{eq:transformed-ode}.
-Using the displayed decomposition of \(C\), equations
+Using the displayed decomposition of \(C\), equations~\eqref{eq:ascension}--\eqref{eq:transformed-ode} give
-\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
+The first nonconstant coefficient of the \(\F43\) in~\eqref{eq:tail} equals
-\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
+The leading vector in~\eqref{eq:tail-direction} is precisely the image
-direction in \eqref{eq:rank-one}.
+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]
+\begin{lemma}[DVR step, including the extra first-column zero]\label{lem:dvr}
-\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
+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
-\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
+The stronger last-row assertion follows from~\eqref{eq:row-relation}.
-\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]
+\begin{proposition}[Exact terminating denominator]\label{prop:terminating}
-\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
+reconstruction equation, the closing factorization, and~\eqref{eq:full-gauge}--\eqref{eq:matrix-scalar-bridge} by cross
-\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,
+Equations~\eqref{eq:term-constant}--\eqref{eq:term-top} therefore prove,
-coefficient by coefficient, that the right side of
+coefficient by coefficient, that the right side of~\eqref{eq:terminating-step} is
-\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},
+Starting from~\eqref{eq:base-horizontal-row},~\eqref{eq:full-gauge} propagates the horizontal form at every step.
-\eqref{eq:full-gauge} propagates the horizontal form at every step.
+This proves~\eqref{eq:qhat} and~\eqref{eq:normalization} for the actual
-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
+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
-\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
+\(A_1G_N\e_1=S q_N\), and using~\eqref{eq:error-vanish}, we get
-\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]
+\begin{lemma}[Explicit dominant-product dichotomy]\label{lem:dominant-product}
-\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
+\(\xi_m=\Lambda(a)L_m\).  Equations~\eqref{eq:block-bounds}--\eqref{eq:block-separation} ensure
-\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
+Multiplying by \(P^{-1}\) proves~\eqref{eq:dominant-asymptotic} with
-\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
+If \(\Lambda(C)=0\), the first coordinate of~\eqref{eq:exceptional-decay} would contradict~\eqref{eq:q-lower}.
-\eqref{eq:exceptional-decay} would contradict \eqref{eq:q-lower}.
 Therefore
 \[
 \Lambda(A_1)=S\Lambda(C)\ne0.
 \]
-If \(\Lambda(A_0)=0\), then
+If \(\Lambda(A_0)=0\), then~\eqref{eq:exceptional-decay},~\eqref{eq:dominant-asymptotic}, and
-\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:
 \[