A32_4df86775: audit/snapshots/RQ-D_MAIN_RESULT_1722.tex md5 4df86775aaa4b0e44d1ef733cefaeb16 OK
A34_bba5c599: audit/rqd/bstar_1912/MAIN_RESULT.tex md5 bba5c599262882085752622923179a03 OK
A38_9498b44f: audit/rqd/heavy_2247/MAIN_RESULT.tex md5 9498b44f068b943684ce94e8ff8b473e OK
CUR_212c6f23: audit/snapshots/s0005/RQ-D/MAIN_RESULT.tex md5 212c6f230b7bd8b8ef0c377b21f7c937 OK

===== A32_4df86775 -> A34_bba5c599
  sections in A32_4df86775: ['PREAMBLE', 'Setting', 'sec:tools', 'sec:comp', 'sec:onejump', 'sec:thmB', 'sec:outlook']
  sections in A34_bba5c599: ['PREAMBLE', 'Setting', 'sec:tools', 'sec:comp', 'sec:onejump', 'sec:thmB', 'sec:thmBstar']
  [PREAMBLE] CHANGED: -4 +9 lines
    --- A32_4df86775:PREAMBLE
    +++ A34_bba5c599:PREAMBLE
    @@ -31 +31,6 @@
    -($\sup_K\E|Z_K|=\sqrt{2/e}$) to the complementary far region (Corollary~\ref{cor:c5}). The proof is not a moment
    +($\sup_K\E|Z_K|=\sqrt{2/e}$) to the complementary far region (Corollary~\ref{cor:c5}). In the form needed at finite
    +$q$ (the $q$-free condition ID$^*$ of the audited reduction, with $C_5=g(2\ln\frac54)$) we prove the band lemma with
    +$\theta=0.85$, $A=1.54$: $\E|Z_K|\le\max(g(\lambda^+_{0.85}(K)),C_5)$ for every finitely supported probability $K$ on
    +$[-1.54,1.54]$ with $K([0.85,1.54])\ge4/5$, and the mirror statement (Theorem~\ref{thm:bandstar}). So ID$^*$, and with
    +it Conjecture S for all $q$, reduces to $\E|Z_K|\le C_5$ on the far region $\{K([0.85,1.54])<4/5,\ K([-1.54,-0.85])<4/5\}$
    +(Corollary~\ref{cor:idstar}). The proof is not a moment
    @@ -34,2 +39,2 @@
    -increases $\E\psi(Z_K)$ whenever $\psi''$ is nondecreasing) together with a convexity bound in the jump law. The final
    -inequality on a four-parameter domain is checked in ball arithmetic. Sections~\ref{sec:tools}--\ref{sec:onejump} are
    +increases $\E\psi(Z_K)$ whenever $\psi''$ is nondecreasing) together with convexity bounds in the jump law. The final
    +inequalities on four-parameter domains are checked in ball arithmetic. Sections~\ref{sec:tools}--\ref{sec:onejump} are
    @@ -37 +42 @@
    -(Section~\ref{sec:outlook}, in progress, not claimed).
    +(Section~\ref{sec:thmBstar}).
  [Setting] IDENTICAL (21 lines)
  [sec:tools] IDENTICAL (51 lines)
  [sec:comp] IDENTICAL (95 lines)
  [sec:onejump] CHANGED: -0 +42 lines
    --- A32_4df86775:sec:onejump
    +++ A34_bba5c599:sec:onejump
    @@ -7,0 +8,8 @@
    +(c) (two-point bound) Let $\varphi\le\psi$ on $[j_0,\infty)$, where $\psi$ is $C^1$ on $[j_0,\infty)$ with $\psi'$ absolutely
    +continuous and $\psi''$ nonincreasing a.e.\ (that is, $\psi'$ is concave). Let $\mu>j_0$, $d=\mu-j_0$,
    +$x=\mu+\sigma^2/d$ and $w=d^2/(d^2+\sigma^2)$. Then $\E\varphi(J)\le T_\psi(\sigma^2):=(1-w)\psi(j_0)+w\,\psi(x)$.
    +If moreover $\psi$ is convex and nonincreasing on $\R$, then for all $s^2\ge\sigma^2$, $j_0'\le j_0$,
    +$0\le w'\le d^2/(d^2+s^2)$ and $u'\le d+s^2/d$,
    +\[
    +  \E\varphi(J)\le(1-w')\,\psi(j_0')+w'\,\psi(j_0'+u').
    +\]
    @@ -12,0 +21,16 @@
    +
    +(c) The law $P^*=(1-w)\delta_{j_0}+w\delta_x$ has mean $j_0+w(x-j_0)=j_0+\frac{d^2}{d^2+\sigma^2}\cdot\frac{d^2+\sigma^2}{d}=\mu$
    +and variance $(1-w)d^2+w\sigma^4/d^2=\sigma^2$. Let $q$ be the quadratic with $q(j_0)=\psi(j_0)$, $q(x)=\psi(x)$,
    +$q'(x)=\psi'(x)$ (if $\sigma^2=0$ then $x=\mu$, $w=1$, $J=\mu$ a.s.\ and there is nothing to prove), and $k=q-\psi$. Then
    +$k(j_0)=k(x)=k'(x)=0$ and $k'=q'-\psi'$ is convex. By Rolle $k'(\xi)=0$ for some $\xi\in(j_0,x)$; a convex function
    +vanishing at the two points $\xi<x$ is $\le0$ on $[\xi,x]$ and $\ge0$ on $[j_0,\xi]\cup[x,\infty)$. So $k$ is nondecreasing on
    +$[j_0,\xi]$, nonincreasing on $[\xi,x]$ and nondecreasing on $[x,\infty)$; with $k(j_0)=k(x)=0$ this gives $k\ge0$, i.e.\
    +$\psi\le q$ on $[j_0,\infty)$. As $\E q(J)$ depends only on the mean and variance of $J$,
    +$\E\varphi(J)\le\E\psi(J)\le\E q(J)=\E_{P^*}q=\E_{P^*}\psi=T_\psi(\sigma^2)$.
    +
    +For the second statement: if $\sigma^2<s^2$, the two-point law $P^{**}$ on $\{j_0,x(s^2)\}$ with mean $\mu$ arises from
    +$P^*$ by splitting the mass at $x(\sigma^2)$ between $j_0$ and $x(s^2)>x(\sigma^2)$ with conditional mean $x(\sigma^2)$,
    +so $T_\psi(\sigma^2)\le T_\psi(s^2)$ for convex $\psi$ (Jensen). Write $\tilde w=d^2/(d^2+s^2)$ and $\tilde u=d+s^2/d$, so
    +$T_\psi(s^2)=\psi(j_0)-\tilde w\,(\psi(j_0)-\psi(j_0+\tilde u))$. The bracket is $\ge0$ as $\psi$ is nonincreasing,
    +so $T_\psi(s^2)\le(1-w')\psi(j_0)+w'\psi(j_0+\tilde u)$ for $w'\le\tilde w$, and
    +$\psi(j_0)\le\psi(j_0')$, $\psi(j_0+\tilde u)\le\psi(j_0'+u')$ because $j_0'\le j_0$ and $j_0'+u'\le j_0+\tilde u$.
    @@ -62,0 +87,18 @@
    +\emph{The $\varphi$-form of these choices} (used in Section~\ref{sec:thmBstar}). Put $\varphi(y)=2\E(-W-y)_+$, so that
    +$O'\ge\E\varphi(J-m)$ is required; $\varphi$ is convex and nonincreasing, and by Lemmas~\ref{lem:ell} and~\ref{lem:cost}
    +\[
    +  \varphi(y)\le\chi^C_h(y),\quad \varphi(y)\le\sqrt{y^2+\rho^2}-y,\quad
    +  \varphi(y)=-2y+2\E(W+y)_+\le-2y+\chi^B_h(-y)
    +\]
    +for all $y\in\R$ and $h>0$.
    +Admissible $O'$ (with $J\in[\theta,A]$, mean $\mu_1$, variance $\sigma_J^2$, $y=J-m$):
    +(O1) the chord of $\varphi$ itself (Lemma~\ref{lem:onejump}(a)), with the endpoint values $\varphi(\theta-m)$, $\varphi(A-m)$
    +replaced by any of the upper bounds above;
    +(O2) Lemma~\ref{lem:onejump}(b) with $\Gamma(y)=-2y+\chi^B_h(-y)$, $\kappa=1/h$ (Lemma~\ref{lem:cost}(iii));
    +(O3) Lemma~\ref{lem:onejump}(c) with $j_0=\theta-m$ and $\psi=\chi^C_h$, whose second derivative
    +$\PP(W_C>y-h)/h$ is nonincreasing (Lemma~\ref{lem:cost}(iii)), $\psi$ being convex and nonincreasing;
    +(O4) if $\theta\ge m$: Lemma~\ref{lem:onejump}(c) with $j_0=\theta-m\ge0$ and $\psi(y)=\sqrt{y^2+r^2}-y$ for any
    +$r^2\ge\rho^2$, whose second derivative $r^2/(y^2+r^2)^{3/2}$ is nonincreasing on $[0,\infty)$.
    +In (O3) and (O4) the mean of $J-m$ exceeds $j_0$ unless $J\equiv\theta$, in which case $\E\varphi(J-m)=\varphi(\theta-m)$.
    +Upper bounds for $\sigma_J^2$: $(\Lambda V-m^2)/\Lambda^2$ (exact) and $(\mu_1-\theta)(A-\mu_1)$ (Bhatia--Davis).
    +
  [sec:thmB] IDENTICAL (102 lines)
  [sec:outlook] REMOVED in A34_bba5c599
  [sec:thmBstar] NEW in A34_bba5c599 (246 lines)

===== A34_bba5c599 -> A38_9498b44f
  sections in A34_bba5c599: ['PREAMBLE', 'Setting', 'sec:tools', 'sec:comp', 'sec:onejump', 'sec:thmB', 'sec:thmBstar']
  sections in A38_9498b44f: ['PREAMBLE', 'Setting', 'sec:tools', 'sec:comp', 'sec:onejump', 'sec:thmB', 'sec:thmBstar', 'sec:heavy']
  [PREAMBLE] CHANGED: -3 +8 lines
    --- A34_bba5c599:PREAMBLE
    +++ A38_9498b44f:PREAMBLE
    @@ -19 +19,2 @@
    -\title{RQ-D: flip formula, a comparison lemma for stop-loss terms,\\ and the band lemma near the Dirac curve\\
    +\title{RQ-D: flip formula, a comparison lemma for stop-loss terms,\\ the band lemma near the Dirac curve,\\
    +and the heavy part of the far region\\
    @@ -36 +37 @@
    -(Corollary~\ref{cor:idstar}). The proof is not a moment
    +(Corollary~\ref{cor:idstar}). The proof of the band lemmas is not a moment
    @@ -42 +43,5 @@
    -(Section~\ref{sec:thmBstar}).
    +(Section~\ref{sec:thmBstar}). On the heavy part of that far region, $\{\frac12\le K([0.85,1.54])<\frac45\}$ and its
    +mirror, we prove $\E|Z_K|<C_5$ (Theorem~\ref{thm:heavy}). The positive edge is replaced by its chord law (a convex-order
    +increase) and fixed on boxes of the two chord rates. The rest enters through a certified degree-16 moment bound
    +(polynomial majorant, Lagrangian relaxation on cumulant cells). So ID$^*$ and C5 reduce to the light part
    +$\{K([0.85,1.54])<\frac12,\ K([-1.54,-0.85])<\frac12\}$ (Corollary~\ref{cor:light}).
  [Setting] IDENTICAL (21 lines)
  [sec:tools] IDENTICAL (51 lines)
  [sec:comp] IDENTICAL (95 lines)
  [sec:onejump] IDENTICAL (104 lines)
  [sec:thmB] IDENTICAL (102 lines)
  [sec:thmBstar] CHANGED: -1 +0 lines
    --- A34_bba5c599:sec:thmBstar
    +++ A38_9498b44f:sec:thmBstar
    @@ -246 +245,0 @@
    -\end{document}
  [sec:heavy] NEW in A38_9498b44f (345 lines)

===== A38_9498b44f -> CUR_212c6f23
  sections in A38_9498b44f: ['PREAMBLE', 'Setting', 'sec:tools', 'sec:comp', 'sec:onejump', 'sec:thmB', 'sec:thmBstar', 'sec:heavy']
  sections in CUR_212c6f23: ['PREAMBLE', 'Setting', 'sec:tools', 'sec:comp', 'sec:onejump', 'sec:thmB', 'sec:thmBstar', 'sec:heavy']
  [PREAMBLE] IDENTICAL (49 lines)
  [Setting] IDENTICAL (21 lines)
  [sec:tools] IDENTICAL (51 lines)
  [sec:comp] IDENTICAL (95 lines)
  [sec:onejump] IDENTICAL (104 lines)
  [sec:thmB] IDENTICAL (102 lines)
  [sec:thmBstar] IDENTICAL (245 lines)
  [sec:heavy] CHANGED: -3 +36 lines
    --- A38_9498b44f:sec:heavy
    +++ CUR_212c6f23:sec:heavy
    @@ -309,0 +310,12 @@
    +\paragraph{Certificate files.} Each line of \texttt{certs\_v1\_r*.jsonl} is a JSON object for one leaf box.
    +\texttt{box} $=[l_a,i_a,l_b,i_b]$ is the box $[i_a,i_a+1]\cdot\frac{28}{25}2^{-l_a}\times[i_b,i_b+1]\cdot\frac{17}{50}2^{-l_b}$
    +(here $l_a=l_b=4+d$ at box level $d$), and \texttt{status} is \texttt{empty} or \texttt{ok}. An \texttt{ok} box has a list
    +\texttt{certs} with one entry per cell. In an entry, \texttt{cell} $=[\ell,i,j]$ is the cell $(i,j)$ of the
    +$2^\ell\times2^\ell$ grid on $R_B$, and \texttt{status} is \texttt{ok} or \texttt{empty\_farkas}. An \texttt{ok} entry holds
    +\texttt{coef} (the 17 coefficients of $\varphi$, ascending), \texttt{Lt} ($L$), \texttt{rmult} (for each $j=3,\dots,16$ the
    +pair $(s_3,s_4)$ of (a) for $L_j$, then the pair for $U_j$) and \texttt{mult}, the multipliers of (c) in the order of the
    +display there: \texttt{cell} for the four cell rows, \texttt{mcc} for the four McCormick rows and \texttt{cheb} for the two
    +H\"older rows of a tuple $\alpha$, keyed by $\alpha$. An \texttt{empty\_farkas} entry holds \texttt{farkas} $=(s_3,s_4)$ of
    +(b). Every number is a double, read as an exact dyadic rational. The other keys (\texttt{t}, \texttt{worst},
    +\texttt{ncell}, \texttt{bound}, \texttt{lp}) are the generator's bookkeeping, and the verifier does not read them.
    +
    @@ -340,3 +352,24 @@
    -The float supremum over the heavy region of the bound with \emph{exact} moments of $W$, i.e.\ with no cell or McCormick
    -loss, is about $0.8387$ (\texttt{scripts/plus\_scan.py}, \texttt{outputs/far\_extra/plus\_scan/}, 7 seeds). The box,
    -cell and relaxation losses together raise the maximum from about $0.8387$ to the certified $0.8549$.
    +
    +\paragraph{The design's value (not part of the proof).} For $K$ in the heavy region, with chord rates $(a_+,b_+)$, let
    +$J(K)$ be the supremum of $\E h(Y)$ over all random variables $Y$ whose moments up to degree 16 are those of $Z_{W_0}$,
    +where $W_0=K|_{[-A,\tau)}$ and $h(r)=\E|\tau(N_{a_+}-a_+)+A(N_{b_+}-b_+)+r|$. This is the value of the design itself,
    +without boxes, cells or relaxations. One $Y$ serves all values of the counts, as one $\varphi$ does in
    +Lemma~\ref{lem:cellcert}. For the box $B$ and the rest $W$ of $K$ in the proof of
    +Lemma~\ref{lem:box}(ii), the bound $\beta+\varepsilon$ of the cell that contains $(\kappa_3(W),\kappa_4(W))$ is at least
    +$J(K)$. Indeed, let $Y'$ be the sum of $Y$, an independent copy of $Z_{W_1-W_0}$ and an
    +independent $\sigma G$. Then $Y'$ has the moments of $Z_W$ up to degree 16, so
    +$\E h_B(Y')\le\E\varphi(Y')+\varepsilon\le\beta+\varepsilon$ by Lemma~\ref{lem:maj} and (c). Also
    +$C_B+Z_{W_1-W_0}\overset d=\tau(N_{a_+}-a_+)+A(N_{b_+}-b_+)$, so $\E h_B(Y')=\E h(Y+\sigma G)\ge\E h(Y)$ by Jensen.
    +The same holds if $\E Y^{16}$ is only bounded above by that of $Z_{W_0}$, as in the LPs below, because the tail test of
    +Lemma~\ref{lem:maj}(iii) gives $c_{16}\ge0$.
    +
    +Float searches over the heavy region (\texttt{scripts/plus\_scan.py}, \texttt{outputs/far\_extra/plus\_scan/}, 15 runs,
    +8 of them stopped early; $J$ by a discretised moment LP) found values of $J$ up to $0.8390$, at $V_+$ close to $\frac45$,
    +while the certified bounds go up to $0.8549$. RQ-C's list of adversarial laws (RQ-C notes, 22:12 UTC) has five laws in
    +the heavy region. At them $J$ is between $0.814$ and $0.836$, and $\E|Z_K|\le\E h_B(Z_W)\le\beta+\varepsilon$ holds for
    +the cell of $K$, with $\beta+\varepsilon-J$ between $0.018$ and $0.027$ (\texttt{scripts/plus\_check\_laws.py},
    +\texttt{outputs/plus\_cert/check\_rqc\_laws.txt}). The LP measures behind these values of $J$ violate the scaled moment
    +rows by at most $2.2\cdot10^{-10}$ beyond the LP tolerance, and the values change by at most $10^{-5}$ between the
    +tolerances $10^{-11}$ and $10^{-7}$ (\texttt{scripts/plus\_lp\_tol.py}, \texttt{outputs/plus\_cert/plus\_lp\_tol.txt}). The
    +best law of the searches, recomputed from its printed parameters with the same check, gives $0.83893$
    +(\texttt{scripts/plus\_scan\_recheck.py}, \texttt{outputs/far\_extra/plus\_scan/recheck\_best.txt}).

===== A34_bba5c599 -> CUR_212c6f23
  sections in A34_bba5c599: ['PREAMBLE', 'Setting', 'sec:tools', 'sec:comp', 'sec:onejump', 'sec:thmB', 'sec:thmBstar']
  sections in CUR_212c6f23: ['PREAMBLE', 'Setting', 'sec:tools', 'sec:comp', 'sec:onejump', 'sec:thmB', 'sec:thmBstar', 'sec:heavy']
  [PREAMBLE] CHANGED: -3 +8 lines
    --- A34_bba5c599:PREAMBLE
    +++ CUR_212c6f23:PREAMBLE
    @@ -19 +19,2 @@
    -\title{RQ-D: flip formula, a comparison lemma for stop-loss terms,\\ and the band lemma near the Dirac curve\\
    +\title{RQ-D: flip formula, a comparison lemma for stop-loss terms,\\ the band lemma near the Dirac curve,\\
    +and the heavy part of the far region\\
    @@ -36 +37 @@
    -(Corollary~\ref{cor:idstar}). The proof is not a moment
    +(Corollary~\ref{cor:idstar}). The proof of the band lemmas is not a moment
    @@ -42 +43,5 @@
    -(Section~\ref{sec:thmBstar}).
    +(Section~\ref{sec:thmBstar}). On the heavy part of that far region, $\{\frac12\le K([0.85,1.54])<\frac45\}$ and its
    +mirror, we prove $\E|Z_K|<C_5$ (Theorem~\ref{thm:heavy}). The positive edge is replaced by its chord law (a convex-order
    +increase) and fixed on boxes of the two chord rates. The rest enters through a certified degree-16 moment bound
    +(polynomial majorant, Lagrangian relaxation on cumulant cells). So ID$^*$ and C5 reduce to the light part
    +$\{K([0.85,1.54])<\frac12,\ K([-1.54,-0.85])<\frac12\}$ (Corollary~\ref{cor:light}).
  [Setting] IDENTICAL (21 lines)
  [sec:tools] IDENTICAL (51 lines)
  [sec:comp] IDENTICAL (95 lines)
  [sec:onejump] IDENTICAL (104 lines)
  [sec:thmB] IDENTICAL (102 lines)
  [sec:thmBstar] CHANGED: -1 +0 lines
    --- A34_bba5c599:sec:thmBstar
    +++ CUR_212c6f23:sec:thmBstar
    @@ -246 +245,0 @@
    -\end{document}
  [sec:heavy] NEW in CUR_212c6f23 (378 lines)

DONE
