From 8f64e18c58cf8220970e4a3f906e0ce129274edb Mon Sep 17 00:00:00 2001 From: Ruge Lin Date: Thu, 8 Oct 2026 00:44:58 +0800 Subject: [PATCH] Fix unsupported operator macros in current mathematical exposition --- AGENTS.md | 4 ++++ research/AMPLITUDE_IDENTIFICATION.md | 4 ++-- research/FINITE_VOLUME.md | 4 ++-- research/MODEL_RESIDUAL.md | 2 +- research/PREPARATION_READOUT.md | 6 +++--- research/REFLECTION_MATCHING.md | 26 +++++++++++++------------- research/TUTORIAL_BRIDGE.md | 2 +- research/UNIFORM_MINIMUM.md | 22 +++++++++++----------- 8 files changed, 37 insertions(+), 33 deletions(-) diff --git a/AGENTS.md b/AGENTS.md index ad161e8..12311cf 100644 --- a/AGENTS.md +++ b/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. diff --git a/research/AMPLITUDE_IDENTIFICATION.md b/research/AMPLITUDE_IDENTIFICATION.md index 00d5ca1..f22ab9a 100644 --- a/research/AMPLITUDE_IDENTIFICATION.md +++ b/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. diff --git a/research/FINITE_VOLUME.md b/research/FINITE_VOLUME.md index db4f371..cdde040 100644 --- a/research/FINITE_VOLUME.md +++ b/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 diff --git a/research/MODEL_RESIDUAL.md b/research/MODEL_RESIDUAL.md index 7569f9e..bc5d6ed 100644 --- a/research/MODEL_RESIDUAL.md +++ b/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. ``` diff --git a/research/PREPARATION_READOUT.md b/research/PREPARATION_READOUT.md index 92605d4..17523a7 100644 --- a/research/PREPARATION_READOUT.md +++ b/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 diff --git a/research/REFLECTION_MATCHING.md b/research/REFLECTION_MATCHING.md index 2f1d036..1c7589e 100644 --- a/research/REFLECTION_MATCHING.md +++ b/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 diff --git a/research/TUTORIAL_BRIDGE.md b/research/TUTORIAL_BRIDGE.md index 94d7214..42ad91b 100644 --- a/research/TUTORIAL_BRIDGE.md +++ b/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}. ``` diff --git a/research/UNIFORM_MINIMUM.md b/research/UNIFORM_MINIMUM.md index 03f0a22..52358f2 100644 --- a/research/UNIFORM_MINIMUM.md +++ b/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 $`O_\delta(L^{-5/4})`$.