diff --git a/zero_distillable_entanglement_v2.txt b/zero_distillable_entanglement_v2.txt
index 87ee821..10af3da 100644
--- a/zero_distillable_entanglement_v2.txt
+++ b/zero_distillable_entanglement_v2.txt
@@ -70,7 +70,7 @@ If \(\{e_0,e_1\}\) is the standard basis of \(\C^2\), the vector
 
 When the state is not known---because someone flipped a coin and
 prepared \(|e_0\rangle\) or \(|e_1\rangle\) so that we only know a
-probability---the right object is a *density operator*. This is a
+probability---the right object is a \emph{density operator}. This is a
 positive semidefinite matrix \(\rho\) with \(\tr\rho=1\). If the state
 is pure, prepared from a known unit vector \(|\psi\rangle\), then
 \(\rho=|\psi\rangle\langle\psi|\). Measuring an observable \(A\) yields
@@ -83,7 +83,7 @@ Bob share a quantum system that lives on a tensor product
 \(\C^a\otimes\C^b\). The joint density operator \(\rho\) records the
 statistics of all joint measurements one could perform there.
 
-A state is *separable* if it can be written as a finite sum
+A state is \emph{separable} if it can be written as a finite sum
 \[
 \rho=\sum_{r} p_r\; |\psi_r\rangle\langle\psi_r|_A
                   \otimes |\phi_r\rangle\langle\phi_r|_B,
@@ -93,7 +93,7 @@ variable chooses a label \(r\), and each laboratory prepares a state
 depending only on \(r\). Separable states therefore carry only classical
 correlations.
 
-A state is *entangled* if it is not separable. Entanglement is the extra
+A state is \emph{entangled} if it is not separable. Entanglement is the extra
 correlation that lets Bell-inequality experiments beat any classical
 model of the same statistics. It is not the same as classical randomness
 shared through a hidden variable.
@@ -105,10 +105,10 @@ authenticated public channel. Question: can they produce identical secret
 random bits?
 
 The adversary is an eavesdropper Eve. The most pessimistic and most
-natural model is: Eve started with a *purification* of \(\rho\), a pure
+natural model is: Eve started with a \emph{purification} of \(\rho\), a pure
 state \(|\Psi\rangle\langle\Psi|\) on \(\C^a\otimes\C^b\otimes\C^E\)
 whose restriction to \(\C^a\otimes\C^b\) is \(\rho\). She also sees the
-full public transcript. An *ideal secret bit* is one for which the joint
+full public transcript. An \emph{ideal secret bit} is one for which the joint
 state of Alice, Bob, and Eve has the form
 \[
 \gamma = \frac12\sum_{z\in\{0,1\}} |z,z\rangle\langle z,z| \otimes \sigma,
@@ -125,7 +125,7 @@ traditional "trace distance" and has the advantage that
 \(0\le\norm{\tau-\gamma}_1\le2\). If it is close to zero, Eve learns
 essentially nothing.
 
-The *distillable secret key* \(K_D(\rho)\) is the supremum over secure
+The \emph{distillable secret key} \(K_D(\rho)\) is the supremum over secure
 protocols of the asymptotic rate of bits produced per copy of \(\rho\),
 in the limit of many copies. Ordinary entropic protocols (privacy
 amplification) show that some states have \(K_D>0\).
@@ -134,12 +134,12 @@ amplification) show that some states have \(K_D>0\).
 Eve's purification is a summation over product states, so she already
 knows everything about each branch and no announcement can surprise her.
 
-The converse is *a priori* not obvious. If \(\rho\) is entangled, does it
+The converse is \emph{a priori} not obvious. If \(\rho\) is entangled, does it
 have \(K_D>0\)? By the mid-2000s this was a standard open problem. It
 appears as Problem 24 in Kr\"uger and Werner's 2005 collection, in the
 form: "Secret key from all entangled states?", attributed to
 P.~Horodecki. The intuition was: entanglement is a resource; surely
-*some* useful amount of it must survive public discussion.
+\emph{some} useful amount of it must survive public discussion.
 
 \paragraph{What this paper proves.} It does not. There is a specific
 entangled \(\rho\) on \(\C^{10}\otimes\C^{10}\) with \(K_D(\rho)=0\). In
@@ -264,7 +264,7 @@ details.
 
 \paragraph{Why this is surprising.} For two decades, evidence pointed the
 other way. Horodecki, Horodecki, Horodecki, and Oppenheim~\cite{%
-keyBoundEntanglement} proved that *some* PPT entangled (bound)
+keyBoundEntanglement} proved that \emph{some} PPT entangled (bound)
 states have positive secret-key cost, via the private-state paradigm of
 Refs.~\cite{keyBoundEntanglement,keyPrivateStates}. Devetak and Winter~%
 \cite{DevetakWinter05} gave one-way positive-key criteria. On the collapse
@@ -527,7 +527,7 @@ Represent \(\omega=(\mathcal L\otimes\mathcal M)(\chi\chi^*)\) with
 \(\mathcal L,\mathcal M\) input-transpose CP. Pull the \(X_i,Y_j\) back
 by adjoints to positive matrices on the input spaces. A congruence
 regularizes the two pullback pairs so they commute with sum equal to a
-projection. Then a direct block bound and a *reverse* block bound (via
+projection. Then a direct block bound and a \emph{reverse} block bound (via
 the input transpose) sandwich the off-diagonal term. Splitting the input
 vector according to which of the two commuting effects is larger, the
 off-diagonal contributions obey a spectral weighting that gives the four
@@ -569,7 +569,7 @@ applying Lemma~\ref{lem:positive-block} yields
 \norm{Z(u,z)}_1\le\sqrt{p_{00}(u)p_{11}(z)}.
 \end{equation}
 
-For the reverse estimate, the *global input transpose*
+For the reverse estimate, the \emph{global input transpose}
 \(T_{rs}=T_r\otimes T_s\) makes \(C\circ T_{rs}\) CP. Apply the same
 block argument to this map on the ordered pair \((\overline z,\overline u)\)
 and the same output filters. Since
@@ -580,7 +580,7 @@ unchanged but the two diagonal weights swap:
 \begin{equation}\label{eq:reversed-block-bound}
 \norm{Z(u,z)}_1\le\sqrt{p_{00}(z)p_{11}(u)}.
 \end{equation}
-It is the transpose on *both* input factors that supplies this reverse
+It is the transpose on \emph{both} input factors that supplies this reverse
 estimate.
 
 \paragraph{Splitting the input vector.}
@@ -621,7 +621,7 @@ at each complete public transcript.
 \section{Complete public transcripts}
 \label{sec:transcript}
 
-At each public transcript, a protocol induces *the same two pairs* of
+At each public transcript, a protocol induces \emph{the same two pairs} of
 positive local matrices on Alice's and Bob's input spaces, for all four
 possible bit values. These matrices are not just the final measurements;
 they are the aggregated effects of the whole protocol on its input. This
@@ -854,7 +854,10 @@ V_j(t)=\sum_i\tr_{A_0}W_{ij}(t),\quad
 \sum_i\tr U_i(t)=\sum_j\tr V_j(t)=1.
 \]
 The conditional POVM statistics and their factorized form give
-\(\tr[(E_r\otimes G_s)W_{ij}(t)]=\tr[E_rU_i(t)]\tr[G_sV_j(t)]\) a.e.
+\[
+\tr[(E_r\otimes G_s)W_{ij}(t)]=\tr[E_rU_i(t)]\tr[G_sV_j(t)]
+\]
+almost everywhere.
 Since \(E_r\otimes G_s\) span the Hermitian matrices on
 \(A_0\otimes B_0\), we conclude
 \begin{equation}\label{eq:probe-product}
@@ -1368,13 +1371,12 @@ symmetry gives \(M(x)^*M(x)\). Since
 the rank identity. Finally \(M_0^{\T}M_0=36I_4\).
 \end{proof}
 
-\begin{proposition}[Twenty rank-loss directions]
+\begin{proposition}[Twenty directions]
 \label{prop:twenty-directions}
 The projective directions \([x]\in\mathbb P^3(\C)\) with
 \(\rank M(x)<4\) consist of exactly twenty distinct points, each with
 unique representative \(x=(1,t_1,t_2,t_3)\). Every nonzero homogeneous
-quadratic in \(\C[x_0,x_1,x_2,x_3]\) vanishes on at most nine of these
-representatives.
+quadratic in \(\C[x_0,x_1,x_2,x_3]\) vanishes on at most nine of them.
 \end{proposition}
 
 The equation \(M(x)y=0\) with \(x,y\ne0\) relates to bilinear kernel
@@ -2077,7 +2079,7 @@ the six selected projections would violate Lemma~%
 
 \begin{theorem}[Combinatorial construction]
 \label{thm:old-construction}
-Define data by \eqref{eq:combinatorial-exact-data}:
+Take the explicit data:
 \(p=257\), \(k=1028\), \(b=\binom{1027}{513}\),
 \(a_0=\sum_{j=0}^{256}\binom{1028}{j}\), \(Q=b^3\),
 \(K=1086373952\), \(\kappa=3a_0b^2<Q/K\), \(L=\binom Q8\),
@@ -2141,8 +2143,8 @@ the standard library and integer arithmetic. Its inputs are the matrices
       \(M_i^{\T}M_j=M_j^{\T}M_i\) for all \(0\le i,j<4\);
 \item the integer matrices \(D\in\Z^{15\times20}\) and
       \(H\in\Z^{15\times15}\) in the decomposition
-      \((p_\rho(t))_\rho=Dv(t)+Hw(t)\) of \eqref{eq:DH}, where the
-      monomial order is \eqref{eq:monomial-order};
+      \((p_\rho(t))_\rho=Dv(t)+Hw(t)\), where the
+      monomial order is total degree first, then lexicographic;
 \item for each of \(p=41,131,139\): the residue of \(\det H\); the
       twenty-one characteristic coefficients of the \(\F_p\)-reduction
       of \(f(z)=\det(zI_{20}-N)\); and the gcd degrees
@@ -2163,8 +2165,7 @@ vanish identically (equal matrix entries), while swapping either also
 leaves the same value; so strict loops cover every equality. The array
 \texttt{R} is \([D\mid H]\); \texttt{T} is the reduction of \(N\)
 modulo the current prime. The program expands each determinant over the
-\(4!=24\) permutations, collects coefficients in the order
-\eqref{eq:monomial-order}, and performs Gaussian elimination on \(H\)
+\(4!=24\) permutations, collects coefficients in that order, and performs Gaussian elimination on \(H\)
 modulo each prime.
 
 For \(f(z)=z^{20}+c_1z^{19}+\cdots+c_{20}\), Newton's identities
@@ -2212,7 +2213,7 @@ Python standard library. It checks:
 (1) primality of \(257\) by trial division by the primes up to
      \(\sqrt{257}\), namely \(2,3,5,7,11,13\);
 (2) the intermediate binomial and power inequalities used to derive
-     \(\delta<2^{-65}\) in \eqref{eq:combinatorial-analytic-bound};
+     \(\delta<2^{-65}\);
 (3) \(3a_0K<b\) and \(3a_0b^2\ge3\).
 Its recorded output \texttt{combinatorics\_exact\_checks.json} stores
 the exact integers \(a_0\) (under key \texttt{A}) and \(b\), confirms
@@ -2255,22 +2256,10 @@ objects are specified and analyzed by the finite formulas in the text.
 
 \begin{thebibliography}{99}
 
-\bibitem{ABS1991}
-N.~Alon, L.~Babai, and H.~Suzuki,
-``Multilinear polynomials and Frankl--Ray-Chaudhuri--Wilson type
-intersection theorems,'' \emph{J.\ Combin.\ Theory Ser.\ A} \textbf{58}
-(1991), 165--180.
-
 \bibitem{AnLee2026}
 J.~An and S.~Lee, ``Beyond the Positive Partial Transpose Squared
 Conjecture: The Qutrit Case,'' arXiv:2607.15947 [quant-ph] (2026).
 
-\bibitem{BIRS2012}
-M.~B.~Ruskai, M.~Junge, D.~Kribs, P.~Hayden, and A.~Winter
-(organizers), \emph{Operator structures in quantum information theory},
-Banff International Research Station workshop report 12w5084 (2012),
-Section~3, Problem~G.
-
 \bibitem{Choi1975}
 M.-D.~Choi, ``Completely positive linear maps on complex matrices,''
 \emph{Linear Algebra Appl.} \textbf{10} (1975), 285--290.
@@ -2294,12 +2283,6 @@ channel,'' \emph{J.\ Phys.\ A} \textbf{54} (2021), 255301.
 P.~Frankl and V.~R\"odl, ``Forbidden intersections,''
 \emph{Trans.\ Amer.\ Math.\ Soc.} \textbf{300} (1987), 259--286.
 
-\bibitem{FuchsVdG1999}
-C.~A.~Fuchs and J.~van~de~Graaf,
-``Cryptographic distinguishability measures for quantum-mechanical
-states,'' \emph{IEEE Trans.\ Inf.\ Theory} \textbf{45} (1999),
-1216--1227.
-
 \bibitem{HansenEtAl2012}
 L.~O.~Hansen, A.~Hauge, J.~Myrheim, and P.~{\O}.~Sollid,
 ``Low-rank positive-partial-transpose states and their relation to
@@ -2347,12 +2330,6 @@ theorems,'' \emph{Phys.\ Rev.\ Lett.} \textbf{65} (1990), 3373--3376.
 S.-J.~Park, ``Every PPT channel has finite entanglement breaking
 index,'' arXiv:2608.13551 [quant-ph] (2026).
 
-\bibitem{Parthasarathy2004}
-K.~R.~Parthasarathy, ``On the maximal dimension of a completely
-entangled subspace for finite level quantum systems,''
-\emph{Proc.\ Indian Acad.\ Sci.\ (Math.\ Sci.)} \textbf{114} (2004),
-365--374.
-
 \bibitem{Peres1990}
 A.~Peres, ``Incompatible results of quantum measurements,''
 \emph{Phys.\ Lett.\ A} \textbf{151} (1990), 107--108.
@@ -2361,21 +2338,6 @@ A.~Peres, ``Incompatible results of quantum measurements,''
 A.~Peres, ``Two simple proofs of the Kochen--Specker theorem,''
 \emph{J.\ Phys.\ A} \textbf{24} (1991), L175--L178.
 
-\bibitem{Pittenger2003}
-A.~O.~Pittenger, ``Unextendible product bases and the construction of
-inseparable states,'' \emph{Linear Algebra Appl.} \textbf{359} (2003),
-235--248.
-
-\bibitem{PowersStormer1970}
-R.~T.~Powers and E.~St{\o}rmer, ``Free states of the canonical
-anticommutation relations,''
-\emph{Comm.\ Math.\ Phys.} \textbf{16} (1970), 1--33.
-
-\bibitem{RennerKonig05}
-R.~Renner and R.~K\"onig, ``Universally composable privacy
-amplification against quantum adversaries,''
-\emph{Theory of Cryptography (TCC) 2005}, LNCS \textbf{3378}, 407--425.
-
 \bibitem{SchliemannLossMacDonald2001}
 J.~Schliemann, D.~Loss, and A.~H.~MacDonald, ``Double-occupancy
 errors, adiabaticity, and entanglement of spin qubits in quantum dots,''