Fix mathematical operator rendering in the tutorial and proof notes

AGENTS.md

@@ -18,6 +18,10 @@ Keep active documents claim-led. Use GitHub-supported inline math and fenced mat
 do not place an unescaped vertical bar in a Markdown table cell. Do not invent sources,
 experimental performance, email addresses, a chosen tutorial or completed priority checks.
 New source claims need exact primary attribution and an honest access-depth record.
+GitHub rejects `\operatorname` in the observed renderer. In current exposition, use
+`\mathop{\mathrm{PV}}\nolimits` (and the corresponding upright operator name) instead;
+follow `\nolimits` with whitespace before a bare letter argument. Preserve the
+byte-identical imported notes and dated evidence.
 
 Use archive/ for obsolete attempts, not as part of the active numerical import path.
 Do not include third-party PDFs, credentials, font files or unrelated scout archives.

research/AMPLITUDE_IDENTIFICATION.md

@@ -22,7 +22,7 @@ H_\alpha(t)=H_0-(u+\alpha t^2)Q,\qquad
 
 Use $`f(t)=(2\pi)^{-1}\int e^{-iEt}\widehat f(E)\,dE`$. Write
 $`g(E)=\langle0|(E-H_0+i0)^{-1}|0\rangle`$ and
-$`\rho(E)=-\operatorname{Im}g(E)/\pi`$ inside the band. Select the nonzero
+$`\rho(E)=-\mathop{\mathrm{Im}}\nolimits g(E)/\pi`$ inside the band. Select the nonzero
 recessive solution at positive infinity of
 
 ```math
@@ -198,7 +198,7 @@ The incident coefficient cannot vanish for a nonzero recessive solution. The
 following current identity also proves this and checks the normalization:
 
 ```math
-j=\operatorname{Im}(\overline c\,c'),\qquad
+j=\mathop{\mathrm{Im}}\nolimits (\overline c\,c'),\qquad
 j'=\frac{\pi}{\alpha}\rho(E)|q(E)|^2,\qquad
 |a_-|^2-|a_+|^2=\frac{\pi}{\alpha}
 \int_0^8\rho(E)|q(E)|^2\,dE.

research/FINITE_VOLUME.md

@@ -106,8 +106,8 @@ $`-e^{\mu(d_M(x)-d_M(y))}`$. Its Hermitian and anti-Hermitian parts have the
 corresponding $`-\cosh`$ and $`i\sinh`$ entries. Symmetric row/column-sum bounds give
 
 ```math
-\operatorname{Re}(W_MH_0W_M^{-1})\ge-a_\mu I,\qquad
-\|\operatorname{Im}(W_MH_0W_M^{-1})\|\le v_\mu.
+\mathop{\mathrm{Re}}\nolimits (W_MH_0W_M^{-1})\ge-a_\mu I,\qquad
+\|\mathop{\mathrm{Im}}\nolimits (W_MH_0W_M^{-1})\|\le v_\mu.
 ```
 
 For $`f_j=(H_{0,j}+\eta_j)^{-1}|0\rangle`$, take the real part of the weighted

research/MODEL_RESIDUAL.md

@@ -106,7 +106,7 @@ criterion. Let $`N\ge2`$ be an integer,
 $`\Omega=N/(N-1)`$, $`g=\tfrac12\sqrt{\Omega^2-1}`$, and use
 
 ```math
-H_{\rm ref}=0\quad\text{on }\operatorname{span}\{|0\rangle,|1\rangle\},\qquad
+H_{\rm ref}=0\quad\text{on }\mathop{\mathrm{span}}\nolimits \{|0\rangle,|1\rangle\},\qquad
 h=\begin{pmatrix}0&0&0\\0&0&g\\0&g&1\end{pmatrix},\qquad
 |s\rangle=(|0\rangle+|1\rangle)/\sqrt2.
 ```

research/PREPARATION_READOUT.md

@@ -74,13 +74,13 @@ binary effect between zero and identity.
 \epsilon_{\rm p}&=\tfrac12\|\rho_{\rm p}-P_b\|_1,\\
 \epsilon_{\rm d}&=\tfrac12\|\rho_{\rm a}-\mathcal U\rho_{\rm p}\mathcal U^\dagger\|_1,\\
 \epsilon_{\rm m}&=\|M-P_b\|,\\
-|\operatorname{Tr}(M\rho_{\rm a})-P_\infty(T,u)|
+|\mathop{\mathrm{Tr}}\nolimits (M\rho_{\rm a})-P_\infty(T,u)|
 &\le\epsilon_{\rm p}+\epsilon_{\rm d}+\epsilon_{\rm m}.
 \end{aligned}
 ```
 
-To prove the inequality, insert $`\operatorname{Tr}(P_b\rho_{\rm a})`$ and
-$`\operatorname{Tr}(P_b\mathcal U\rho_{\rm p}\mathcal U^\dagger)`$ between the
+To prove the inequality, insert $`\mathop{\mathrm{Tr}}\nolimits (P_b\rho_{\rm a})`$ and
+$`\mathop{\mathrm{Tr}}\nolimits (P_b\mathcal U\rho_{\rm p}\mathcal U^\dagger)`$ between the
 two probabilities. The detector term is bounded by its operator norm. For either
 state term, a traceless Hermitian difference has positive and negative parts with
 equal trace $`\|\Delta\|_1/2`$; an effect between zero and identity therefore has

research/REFLECTION_MATCHING.md

@@ -62,12 +62,12 @@ endpoint-basis effects are smaller than $`\mathcal Q_L`$.
 
 ## 2. Physical passivity controls the positive side without an absorber
 
-For the **exact** physical coefficient, $`\operatorname{Im}F_{\rm ex}\le0`$ in
+For the **exact** physical coefficient, $`\mathop{\mathrm{Im}}\nolimits F_{\rm ex}\le0`$ in
 the spectral band. The current
 
 ```math
-j=\operatorname{Im}(\overline y\,y'),\qquad
-j'=-\operatorname{Im}F_{\rm ex}|y|^2
+j=\mathop{\mathrm{Im}}\nolimits (\overline y\,y'),\qquad
+j'=-\mathop{\mathrm{Im}}\nolimits F_{\rm ex}|y|^2
 ```
 
 is nondecreasing there, and is zero in the real evanescent region beyond the upper
@@ -86,19 +86,19 @@ r'=2iqr+\frac{i\Omega}{2q}(1+r)^2.
 ```
 
 The differential identity follows by substitution of $`y''=-q^2y`$.
-Where $`y\ne0`$, passivity says $`\operatorname{Im}(y'/y)\le0`$. Hence
+Where $`y\ne0`$, passivity says $`\mathop{\mathrm{Im}}\nolimits (y'/y)\le0`$. Hence
 
 ```math
-|iq-y'/y-h/2|\ge\operatorname{Re}q-|h|/2.
+|iq-y'/y-h/2|\ge\mathop{\mathrm{Re}}\nolimits q-|h|/2.
 ```
 
-For large $`L`$, $`|h|\le\operatorname{Re}q`$ and
-$`|q|/\operatorname{Re}q\le\sqrt2`$ throughout this interval. Therefore
+For large $`L`$, $`|h|\le\mathop{\mathrm{Re}}\nolimits q`$ and
+$`|q|/\mathop{\mathrm{Re}}\nolimits q\le\sqrt2`$ throughout this interval. Therefore
 $`|r|\le1+4\sqrt2<7`$. At a zero of $`y`$, uniqueness gives $`y'\ne0`$ and
 $`r=-1`$; the denominator is still nonzero. This also includes a Dirichlet terminal
 load. The bound uses no favorable sign of the small lattice correction to $`h`$.
 
-Define $`\mathcal A=\int_R^B-\operatorname{Im}q\,dx`$ and
+Define $`\mathcal A=\int_R^B-\mathop{\mathrm{Im}}\nolimits q\,dx`$ and
 $`\mathcal I_+=\int_R^B|\Omega/q|\,dx`$. Backwards variation of constants gives
 
 ```math
@@ -117,7 +117,7 @@ The last two terms follow from the derivative bounds of Section 1 and are
 exponentially small. Uniformly on this interval,
 
 ```math
--\operatorname{Im}q_0=\frac{\pi}{2L}
+-\mathop{\mathrm{Im}}\nolimits q_0=\frac{\pi}{2L}
 \left[1+O\!\left(\frac{\ln L}{L}\right)\right].
 ```
 
@@ -128,7 +128,7 @@ $`r(R)=O((LR)^{-1})+O(e^{-cL})`$.
 
 For $`F_0`$ itself, the earlier stronger bound
 $`4e^{-2\mathcal A}+(25/2)\mathcal I_+`$ in AUDIT remains valid. The constants
-7 and 32 above simply avoid relying on preservation of $`\operatorname{Im}h_0<0`$
+7 and 32 above simply avoid relying on preservation of $`\mathop{\mathrm{Im}}\nolimits h_0<0`$
 when transferring the estimate to the exact lattice. No old finite bound is changed.
 
 ## 3. The negative tail can be bounded directly from the exact spectral measure
@@ -258,14 +258,14 @@ silently discarded. The transfer integral alone contains artificial boundary ter
 
 ## 5. Both causal contributions survive, with equal leading signs
 
-In distributions, $`V'=\operatorname{PV}(1/x)-i\pi\delta(x)`$. Integration by
+In distributions, $`V'=\mathop{\mathrm{PV}}\nolimits (1/x)-i\pi\delta(x)`$. Integration by
 parts in the preceding expression cancels its explicit boundary term and gives
 
 ```math
 r_c=-\frac1{4L}\int_{-R}^{R}
-\left[\operatorname{PV}\frac1x-i\pi\delta(x)\right]e^{-2ix}dx
+\left[\mathop{\mathrm{PV}}\nolimits \frac1x-i\pi\delta(x)\right]e^{-2ix}dx
 +O(\mathcal Q_L)
-=\frac{i}{4L}[2\operatorname{Si}(2R)+\pi]+O(\mathcal Q_L).
+=\frac{i}{4L}[2\mathop{\mathrm{Si}}\nolimits (2R)+\pi]+O(\mathcal Q_L).
 ```
 
 This identity can also be obtained by separately integrating the ordinary

research/TUTORIAL_BRIDGE.md

@@ -128,7 +128,7 @@ amplitude is
 ```math
 \widetilde r_{L,b}=
 -\frac1{4Lk^2}\int_{-R}^{R}
-\left[\operatorname{PV}\frac1x-i\pi\delta_0(x)\right]e^{-2ikx}\,dx
+\left[\mathop{\mathrm{PV}}\nolimits \frac1x-i\pi\delta_0(x)\right]e^{-2ikx}\,dx
 +o_\delta(L^{-1})
 =\frac{i\pi}{2L(1-b)}+o_\delta(L^{-1}),\qquad R=L^{1/4}.
 ```

research/UNIFORM_MINIMUM.md

@@ -56,7 +56,7 @@ F_b^0=k^2+\frac VL+\frac{V^2}{L(L-V)}.
 
 The subscript $`0`$ on $`F_0`$ here means the exact zero-minimum coefficient.
 For $`|x|<1`$, put $`t=-\ln|x|`$. The measure becomes $`e^{-t}dt`$ and
-$`|L-V|\ge L`$. On $`1\le|x|\le R`$, $`\operatorname{Re}(L-V)\ge L/2`$
+$`|L-V|\ge L`$. On $`1\le|x|\le R`$, $`\mathop{\mathrm{Re}}\nolimits (L-V)\ge L/2`$
 eventually. Thus, without removing any neighborhood of zero,
 
 ```math
@@ -89,30 +89,30 @@ source of uniform transfer control, not a collection of fixed-$`b`$ limits.
 ## 3. Positive outer matching does not assume a sign for the logarithmic derivative
 
 For the physical recessive solution the current satisfies
-$`j'=-\operatorname{Im}F_b|y|^2\ge0`$ in the band, with zero current beyond its
+$`j'=-\mathop{\mathrm{Im}}\nolimits F_b|y|^2\ge0`$ in the band, with zero current beyond its
 upper edge. Subtracting real $`b`$ changes neither sign nor the absence of point
 sources at spectral singularities. Hence $`j\le0`$ as in Gate B.
 
 On $`[R,B]`$, $`F_b=k^2+O(\ln L/L)+O(e_*B)`$, uniformly. For the exact square
 root $`q=\sqrt{F_b}`$ choose positive real part and nonpositive imaginary part;
 set $`h=q'/q`$, $`\Omega=h^2/4-h'/2`$. For $`L\ge L_\delta`$,
-$`\operatorname{Re}q\ge\sqrt\delta/2`$, $`|q|/\operatorname{Re}q\le\sqrt2`$,
-and $`|h|\le\operatorname{Re}q`$. Where $`y\ne0`$, $`m=y'/y`$ obeys
+$`\mathop{\mathrm{Re}}\nolimits q\ge\sqrt\delta/2`$, $`|q|/\mathop{\mathrm{Re}}\nolimits q\le\sqrt2`$,
+and $`|h|\le\mathop{\mathrm{Re}}\nolimits q`$. Where $`y\ne0`$, $`m=y'/y`$ obeys
 
 ```math
-|iq-m-h/2|\ge\operatorname{Re}q-|h|/2\ge\tfrac12\operatorname{Re}q.
+|iq-m-h/2|\ge\mathop{\mathrm{Re}}\nolimits q-|h|/2\ge\tfrac12\mathop{\mathrm{Re}}\nolimits q.
 ```
 
 The nonsingular chart
 $`r=[y'+(h/2+iq)y]/[(iq-h/2)y-y']`$ consequently has $`|r|<7`$; at a zero
 of $`y`$ it equals $`-1`$. Neither the denominator nor this conclusion uses
-$`\operatorname{Im}h\le0`$. Indeed that sign can reverse after subtracting $`b`$.
+$`\mathop{\mathrm{Im}}\nolimits h\le0`$. Indeed that sign can reverse after subtracting $`b`$.
 The exact Riccati identity is still
 
 ```math
 r'=2iqr+\frac{i\Omega}{2q}(1+r)^2,\qquad
 |r(R)|\le7e^{-2\mathcal A}+32\mathcal I_+,
-\quad \mathcal A=\int_R^B-\operatorname{Im}q\,dx,\quad
+\quad \mathcal A=\int_R^B-\mathop{\mathrm{Im}}\nolimits q\,dx,\quad
 \mathcal I_+=\int_R^B|\Omega/q|\,dx.
 ```
 
@@ -128,7 +128,7 @@ h^0=\frac{L}{2xD(L-bD)},\qquad
 All inverse powers of $`F_b`$ are controlled by the fixed margin. Gate B's
 $`C^2`$ replacement bounds give
 $`\mathcal I_+=O_\delta((LR)^{-1}+e_*\ln B+e_*^2B)`$.
-Also $`-\operatorname{Im}\sqrt{F_b^0}=\pi/(2Lk)[1+O_\delta(\ln L/L)]`$.
+Also $`-\mathop{\mathrm{Im}}\nolimits \sqrt{F_b^0}=\pi/(2Lk)[1+O_\delta(\ln L/L)]`$.
 The integrated exact-root correction is $`O_\delta(e_*B^2)`$, so
 $`\mathcal A=\pi L/(2k)[1+o_\delta(1)]\ge\pi L/4`$ eventually. Thus the
 physical right-end data have reflection $`O_\delta((LR)^{-1})+O(e^{-cL})`$,
@@ -195,13 +195,13 @@ then expanding the incident denominator, gives
 ```math
 r_c=-\frac{i}{2kL}\int_{-R}^RV e^{-2ikx}dx
 -\frac{V(R)e^{-2ikR}-V(-R)e^{2ikR}}{4Lk^2}+O_\delta(\mathcal Q_L)
-=\frac{i[2\operatorname{Si}(2kR)+\pi]}{4Lk^2}+O_\delta(\mathcal Q_L),
+=\frac{i[2\mathop{\mathrm{Si}}\nolimits (2kR)+\pi]}{4Lk^2}+O_\delta(\mathcal Q_L),
 ```
 
 where $`\mathcal Q_L=(LR)^{-1}+R^2(1+\ln R)^2/L^2+e^{-cL}`$.
 The identity uses both terms of
-$`V'=\operatorname{PV}(1/x)-i\pi\delta(x)`$. The bound
-$`|\operatorname{Si}(2kR)-\pi/2|\le1/(kR)`$ is uniform for
+$`V'=\mathop{\mathrm{PV}}\nolimits (1/x)-i\pi\delta(x)`$. The bound
+$`|\mathop{\mathrm{Si}}\nolimits (2kR)-\pi/2|\le1/(kR)`$ is uniform for
 $`k\ge\sqrt\delta`$. Sections 3–4 give the claimed auxiliary amplitude in
 Section 1, with remainder $`O_\delta(L^{-5/4})`$.