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It is not a proof of Beal, a complete solution of the signature $(3,5,7)$, or an independent acceptance of every upstream arithmetic dependency.","\\section{Three powers and one common prime}\\label{sec:statement}\n\\begin{conjecture}[Beal]\\label{conj:beal}\nIf positive integers $A,B,C$ and integers $x,y,z>2$ satisfy\n\\begin{equation}\\label{eq:beal}\nA^x+B^y=C^z,\n\\end{equation}\nthen $A,B,C$ have a common prime divisor.\n\\end{conjecture}\nThe exponents are allowed to differ. That is the essential extra freedom beyond the equal exponent Fermat equation. The common prime belongs to all three bases, not merely to the product of two of them. The historical statement and its generalized Fermat context are discussed in Mauldin's account and its authenticated published translation \\cite{mauldin1997,mauldin1998translation}.","\\begin{definition}\nA positive solution of \\eqref{eq:beal} is \\emph{primitive} when $\\gcd(A,B,C)=1$. Its ordered exponent triple $(x,y,z)$ is its signature. The first two entries may be exchanged together with the first two bases.\n\\end{definition}\nPrimitivity becomes stronger on an actual solution than it is on an arbitrary integer triple.\n\\begin{lemma}\\label{lem:pairwise}\nFor a solution of \\eqref{eq:beal}, every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases.\n\\end{lemma}\n\\begin{proof}\nIf a prime $p$ divides $A$ and $B$, the equation gives $p\\mid C^z$, hence $p\\mid C$. If $p$ divides $A$ and $C$, it gives $p\\mid B^y$, hence $p\\mid B$. The third case is the same. Thus a prime shared by any pair would divide the gcd of all three.\n\\e","\\begin{definition}\nA positive solution of \\eqref{eq:beal} is \\emph{primitive} when $\\gcd(A,B,C)=1$. Its ordered exponent triple $(x,y,z)$ is its signature. The first two entries may be exchanged together with the first two bases.\n\\end{definition}\nPrimitivity becomes stronger on an actual solution than it is on an arbitrary integer triple.\n\\begin{lemma}\\label{lem:pairwise}\nFor a solution of \\eqref{eq:beal}, every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases.\n\\end{lemma}\n\\begin{proof}\nIf a prime $p$ divides $A$ and $B$, the equation gives $p\\mid C^z$, hence $p\\mid C$. If $p$ divides $A$ and $C$, it gives $p\\mid B^y$, hence $p\\mid B$. The third case is the same. Thus a prime shared by any pair would divide the gcd of all three.\n\\end{proof}\nBeal is therefore equivalent","first two entries may be exchanged together with the first two bases.\n\\end{definition}\nPrimitivity becomes stronger on an actual solution than it is on an arbitrary integer triple.\n\\begin{lemma}\\label{lem:pairwise}\nFor a solution of \\eqref{eq:beal}, every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases.\n\\end{lemma}\n\\begin{proof}\nIf a prime $p$ divides $A$ and $B$, the equation gives $p\\mid C^z$, hence $p\\mid C$. If $p$ divides $A$ and $C$, it gives $p\\mid B^y$, hence $p\\mid B$. The third case is the same. Thus a prime shared by any pair would divide the gcd of all three.\n\\end{proof}\nBeal is therefore equivalent to the absence of primitive positive solutions for all signatures with entries greater than two. This reformulation is the one used throughout the research program. It makes the support of each base disjoint from the other two supports, which is useful when assigning local valuation sectors and conductor conditions.","tries may be exchanged together with the first two bases.\n\\end{definition}\nPrimitivity becomes stronger on an actual solution than it is on an arbitrary integer triple.\n\\begin{lemma}\\label{lem:pairwise}\nFor a solution of \\eqref{eq:beal}, every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases.\n\\end{lemma}\n\\begin{proof}\nIf a prime $p$ divides $A$ and $B$, the equation gives $p\\mid C^z$, hence $p\\mid C$. If $p$ divides $A$ and $C$, it gives $p\\mid B^y$, hence $p\\mid B$. The third case is the same. Thus a prime shared by any pair would divide the gcd of all three.\n\\end{proof}\nBeal is therefore equivalent to the absence of primitive positive solutions for all signatures with entries greater than two. This reformulation is the one used throughout the research program. It makes the support of each base disjoint from the other two supports, which is useful when assigning local valuation sectors and conductor conditions.","y be exchanged together with the first two bases.\n\\end{definition}\nPrimitivity becomes stronger on an actual solution than it is on an arbitrary integer triple.\n\\begin{lemma}\\label{lem:pairwise}\nFor a solution of \\eqref{eq:beal}, every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases.\n\\end{lemma}\n\\begin{proof}\nIf a prime $p$ divides $A$ and $B$, the equation gives $p\\mid C^z$, hence $p\\mid C$. If $p$ divides $A$ and $C$, it gives $p\\mid B^y$, hence $p\\mid B$. The third case is the same. Thus a prime shared by any pair would divide the gcd of all three.\n\\end{proof}\nBeal is therefore equivalent to the absence of primitive positive solutions for all signatures with entries greater than two. This reformulation is the one used throughout the research program. It makes the support of each base disjoint from the other two supports, which is useful when assigning local valuation sectors and conductor conditions.","with the first two bases.\n\\end{definition}\nPrimitivity becomes stronger on an actual solution than it is on an arbitrary integer triple.\n\\begin{lemma}\\label{lem:pairwise}\nFor a solution of \\eqref{eq:beal}, every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases.\n\\end{lemma}\n\\begin{proof}\nIf a prime $p$ divides $A$ and $B$, the equation gives $p\\mid C^z$, hence $p\\mid C$. If $p$ divides $A$ and $C$, it gives $p\\mid B^y$, hence $p\\mid B$. The third case is the same. Thus a prime shared by any pair would divide the gcd of all three.\n\\end{proof}\nBeal is therefore equivalent to the absence of primitive positive solutions for all signatures with entries greater than two. This reformulation is the one used throughout the research program. It makes the support of each base disjoint from the other two supports, which is useful when assigning local valuation sectors and conductor conditions.","bases.\n\\end{definition}\nPrimitivity becomes stronger on an actual solution than it is on an arbitrary integer triple.\n\\begin{lemma}\\label{lem:pairwise}\nFor a solution of \\eqref{eq:beal}, every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases.\n\\end{lemma}\n\\begin{proof}\nIf a prime $p$ divides $A$ and $B$, the equation gives $p\\mid C^z$, hence $p\\mid C$. If $p$ divides $A$ and $C$, it gives $p\\mid B^y$, hence $p\\mid B$. The third case is the same. Thus a prime shared by any pair would divide the gcd of all three.\n\\end{proof}\nBeal is therefore equivalent to the absence of primitive positive solutions for all signatures with entries greater than two. This reformulation is the one used throughout the research program. It makes the support of each base disjoint from the other two supports, which is useful when assigning local valuation sectors and conductor conditions.","finition}\nPrimitivity becomes stronger on an actual solution than it is on an arbitrary integer triple.\n\\begin{lemma}\\label{lem:pairwise}\nFor a solution of \\eqref{eq:beal}, every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases.\n\\end{lemma}\n\\begin{proof}\nIf a prime $p$ divides $A$ and $B$, the equation gives $p\\mid C^z$, hence $p\\mid C$. If $p$ divides $A$ and $C$, it gives $p\\mid B^y$, hence $p\\mid B$. The third case is the same. Thus a prime shared by any pair would divide the gcd of all three.\n\\end{proof}\nBeal is therefore equivalent to the absence of primitive positive solutions for all signatures with entries greater than two. This reformulation is the one used throughout the research program. It makes the support of each base disjoint from the other two supports, which is useful when assigning local valuation sectors and conductor conditions.","imitivity becomes stronger on an actual solution than it is on an arbitrary integer triple.\n\\begin{lemma}\\label{lem:pairwise}\nFor a solution of \\eqref{eq:beal}, every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases.\n\\end{lemma}\n\\begin{proof}\nIf a prime $p$ divides $A$ and $B$, the equation gives $p\\mid C^z$, hence $p\\mid C$. If $p$ divides $A$ and $C$, it gives $p\\mid B^y$, hence $p\\mid B$. The third case is the same. Thus a prime shared by any pair would divide the gcd of all three.\n\\end{proof}\nBeal is therefore equivalent to the absence of primitive positive solutions for all signatures with entries greater than two. This reformulation is the one used throughout the research program. It makes the support of each base disjoint from the other two supports, which is useful when assigning local valuation sectors and conductor conditions.","y becomes stronger on an actual solution than it is on an arbitrary integer triple.\n\\begin{lemma}\\label{lem:pairwise}\nFor a solution of \\eqref{eq:beal}, every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases.\n\\end{lemma}\n\\begin{proof}\nIf a prime $p$ divides $A$ and $B$, the equation gives $p\\mid C^z$, hence $p\\mid C$. If $p$ divides $A$ and $C$, it gives $p\\mid B^y$, hence $p\\mid B$. The third case is the same. Thus a prime shared by any pair would divide the gcd of all three.\n\\end{proof}\nBeal is therefore equivalent to the absence of primitive positive solutions for all signatures with entries greater than two. This reformulation is the one used throughout the research program. It makes the support of each base disjoint from the other two supports, which is useful when assigning local valuation sectors and conductor conditions.","nger on an actual solution than it is on an arbitrary integer triple.\n\\begin{lemma}\\label{lem:pairwise}\nFor a solution of \\eqref{eq:beal}, every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases.\n\\end{lemma}\n\\begin{proof}\nIf a prime $p$ divides $A$ and $B$, the equation gives $p\\mid C^z$, hence $p\\mid C$. If $p$ divides $A$ and $C$, it gives $p\\mid B^y$, hence $p\\mid B$. The third case is the same. Thus a prime shared by any pair would divide the gcd of all three.\n\\end{proof}\nBeal is therefore equivalent to the absence of primitive positive solutions for all signatures with entries greater than two. This reformulation is the one used throughout the research program. It makes the support of each base disjoint from the other two supports, which is useful when assigning local valuation sectors and conductor conditions.","olution than it is on an arbitrary integer triple.\n\\begin{lemma}\\label{lem:pairwise}\nFor a solution of \\eqref{eq:beal}, every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases.\n\\end{lemma}\n\\begin{proof}\nIf a prime $p$ divides $A$ and $B$, the equation gives $p\\mid C^z$, hence $p\\mid C$. If $p$ divides $A$ and $C$, it gives $p\\mid B^y$, hence $p\\mid B$. The third case is the same. Thus a prime shared by any pair would divide the gcd of all three.\n\\end{proof}\nBeal is therefore equivalent to the absence of primitive positive solutions for all signatures with entries greater than two. This reformulation is the one used throughout the research program. It makes the support of each base disjoint from the other two supports, which is useful when assigning local valuation sectors and conductor conditions.","\\begin{example}\\label{ex:commonprime}\nThe equality $3^3+6^3=3^5$ is a valid power equation. Both terms on the left are positive, and all exponents exceed two. The common prime is $3$, so this equality illustrates the conjecture rather than contradicting it. Its powered terms are $27,216,243$.\n\\end{example}\n\\figplaceholder{01_beal_terms}{The equation in Example~\\ref{ex:commonprime} has a common prime in its bases. Exact equality alone does not supply primitivity.}\nAn ordinary division by $\\gcd(A,B,C)$ is not a valid reduction to a primitive solution. If $A=dA'$, $B=dB'$, $C=dC'$, the three terms carry factors $d^x,d^y,d^z$. Unless the exponents agree, division by one of these powers changes the other coefficients. Later, a weighted scaling will preserve","\\begin{example}\\label{ex:commonprime}\nThe equality $3^3+6^3=3^5$ is a valid power equation. Both terms on the left are positive, and all exponents exceed two. The common prime is $3$, so this equality illustrates the conjecture rather than contradicting it. Its powered terms are $27,216,243$.\n\\end{example}\n\\figplaceholder{01_beal_terms}{The equation in Example~\\ref{ex:commonprime} has a common prime in its bases. Exact equality alone does not supply primitivity.}\nAn ordinary division by $\\gcd(A,B,C)$ is not a valid reduction to a primitive solution. If $A=dA'$, $B=dB'$, $C=dC'$, the three terms carry factors $d^x,d^y,d^z$. Unless the exponents agree, division by one of these powers changes the other coefficients. Later, a weighted scaling will preserve the equation. That operation has a precise valuation criterion and is not the same as dividing every base by $d$.","oth terms on the left are positive, and all exponents exceed two. The common prime is $3$, so this equality illustrates the conjecture rather than contradicting it. Its powered terms are $27,216,243$.\n\\end{example}\n\\figplaceholder{01_beal_terms}{The equation in Example~\\ref{ex:commonprime} has a common prime in its bases. Exact equality alone does not supply primitivity.}\nAn ordinary division by $\\gcd(A,B,C)$ is not a valid reduction to a primitive solution. If $A=dA'$, $B=dB'$, $C=dC'$, the three terms carry factors $d^x,d^y,d^z$. Unless the exponents agree, division by one of these powers changes the other coefficients. Later, a weighted scaling will preserve the equation. That operation has a precise valuation criterion and is not the same as dividing every base by $d$.","he common prime is $3$, so this equality illustrates the conjecture rather than contradicting it. Its powered terms are $27,216,243$.\n\\end{example}\n\\figplaceholder{01_beal_terms}{The equation in Example~\\ref{ex:commonprime} has a common prime in its bases. Exact equality alone does not supply primitivity.}\nAn ordinary division by $\\gcd(A,B,C)$ is not a valid reduction to a primitive solution. If $A=dA'$, $B=dB'$, $C=dC'$, the three terms carry factors $d^x,d^y,d^z$. Unless the exponents agree, division by one of these powers changes the other coefficients. Later, a weighted scaling will preserve the equation. That operation has a precise valuation criterion and is not the same as dividing every base by $d$.","prime is $3$, so this equality illustrates the conjecture rather than contradicting it. Its powered terms are $27,216,243$.\n\\end{example}\n\\figplaceholder{01_beal_terms}{The equation in Example~\\ref{ex:commonprime} has a common prime in its bases. Exact equality alone does not supply primitivity.}\nAn ordinary division by $\\gcd(A,B,C)$ is not a valid reduction to a primitive solution. If $A=dA'$, $B=dB'$, $C=dC'$, the three terms carry factors $d^x,d^y,d^z$. Unless the exponents agree, division by one of these powers changes the other coefficients. Later, a weighted scaling will preserve the equation. That operation has a precise valuation criterion and is not the same as dividing every base by $d$.","$3$, so this equality illustrates the conjecture rather than contradicting it. Its powered terms are $27,216,243$.\n\\end{example}\n\\figplaceholder{01_beal_terms}{The equation in Example~\\ref{ex:commonprime} has a common prime in its bases. Exact equality alone does not supply primitivity.}\nAn ordinary division by $\\gcd(A,B,C)$ is not a valid reduction to a primitive solution. If $A=dA'$, $B=dB'$, $C=dC'$, the three terms carry factors $d^x,d^y,d^z$. Unless the exponents agree, division by one of these powers changes the other coefficients. Later, a weighted scaling will preserve the equation. That operation has a precise valuation criterion and is not the same as dividing every base by $d$.","conjecture rather than contradicting it. Its powered terms are $27,216,243$.\n\\end{example}\n\\figplaceholder{01_beal_terms}{The equation in Example~\\ref{ex:commonprime} has a common prime in its bases. Exact equality alone does not supply primitivity.}\nAn ordinary division by $\\gcd(A,B,C)$ is not a valid reduction to a primitive solution. If $A=dA'$, $B=dB'$, $C=dC'$, the three terms carry factors $d^x,d^y,d^z$. Unless the exponents agree, division by one of these powers changes the other coefficients. Later, a weighted scaling will preserve the equation. That operation has a precise valuation criterion and is not the same as dividing every base by $d$.","dinary division by $\\gcd(A,B,C)$ is not a valid reduction to a primitive solution. If $A=dA'$, $B=dB'$, $C=dC'$, the three terms carry factors $d^x,d^y,d^z$. Unless the exponents agree, division by one of these powers changes the other coefficients. Later, a weighted scaling will preserve the equation. That operation has a precise valuation criterion and is not the same as dividing every base by $d$.","The positivity assumptions also carry information. They imply\n\\begin{equation}\\label{eq:heightbasic}\nA^x<C^z,\\qquad B^y<C^z,\\qquad C\\ge2.\n\\end{equation}\nThe last inequality follows because $A^x+B^y\\ge2$. A base on the left may equal $1$, so a finite classification of powered values need not be a finite classification of exponent labels. The equality $1^x=1$ holds for every $x$. This small point is relevant when comparing a theorem for one fixed signature with a statement uniform in all exponents.","\\section{Which signatures a reduction must retain}\\label{sec:signatures}\nEvery exponent greater than two has a useful small divisor. The divisor need not be the same for the three exponents.\n\\begin{proposition}[Prime or four reduction]\\label{prop:primefour}\nIf a primitive solution of \\eqref{eq:beal} exists, there is a primitive solution whose three exponents each belong to\n\\[\n\\{4\\}\\cup\\{p:p\\text{ is an odd prime}\\}.\n\\]\nConversely, absence of primitive solutions for every ordered signature in this set implies Beal.\n\\end{proposition}\n\\begin{proof}\nLet $n>2$. If $n$ has an odd prime divisor $p$, write $D^n=(D^{n/p})^p$. If it has no odd prime divisor, it is a power of two at least four, so write $D^n=(D^{n/4})^4$. Apply this operation to each term separately. The new bases have exactly the same prime supports as the old bases, so primitivity is preserved. The converse follows by reducing any hypothetical primitive solution in this way.\n\\end{proof}\nThe proposition retains infinitely many signatures. It does not reduce them all to $(3,5,7)$. A theorem for that signature also handles signa…","the same for the three exponents.\n\\begin{proposition}[Prime or four reduction]\\label{prop:primefour}\nIf a primitive solution of \\eqref{eq:beal} exists, there is a primitive solution whose three exponents each belong to\n\\[\n\\{4\\}\\cup\\{p:p\\text{ is an odd prime}\\}.\n\\]\nConversely, absence of primitive solutions for every ordered signature in this set implies Beal.\n\\end{proposition}\n\\begin{proof}\nLet $n>2$. If $n$ has an odd prime divisor $p$, write $D^n=(D^{n/p})^p$. If it has no odd prime divisor, it is a power of two at least four, so write $D^n=(D^{n/4})^4$. Apply this operation to each term separately. The new bases have exactly the same prime supports as the old bases, so primitivity is preserved. The converse follows by reducing any hypothetical primitive solution in this way.\n\\end{proof}\nThe proposition retains infinitely many signatures. It does not reduce them all to $(3,5,7)$. A theorem for that signature also handles signatures $(3a,5b,7c)$ after absorbing the multipliers into the bases, and the symmetric version with the first two terms exchanged. It does not handle an unrela…","or the three exponents.\n\\begin{proposition}[Prime or four reduction]\\label{prop:primefour}\nIf a primitive solution of \\eqref{eq:beal} exists, there is a primitive solution whose three exponents each belong to\n\\[\n\\{4\\}\\cup\\{p:p\\text{ is an odd prime}\\}.\n\\]\nConversely, absence of primitive solutions for every ordered signature in this set implies Beal.\n\\end{proposition}\n\\begin{proof}\nLet $n>2$. If $n$ has an odd prime divisor $p$, write $D^n=(D^{n/p})^p$. If it has no odd prime divisor, it is a power of two at least four, so write $D^n=(D^{n/4})^4$. Apply this operation to each term separately. The new bases have exactly the same prime supports as the old bases, so primitivity is preserved. The converse follows by reducing any hypothetical primitive solution in this way.\n\\end{proof}\nThe proposition retains infinitely many signatures. It does not reduce them all to $(3,5,7)$. A theorem for that signature also handles signatures $(3a,5b,7c)$ after absorbing the multipliers into the bases, and the symmetric version with the first two terms exchanged. It does not handle an unrelated prime …","n{proposition}[Prime or four reduction]\\label{prop:primefour}\nIf a primitive solution of \\eqref{eq:beal} exists, there is a primitive solution whose three exponents each belong to\n\\[\n\\{4\\}\\cup\\{p:p\\text{ is an odd prime}\\}.\n\\]\nConversely, absence of primitive solutions for every ordered signature in this set implies Beal.\n\\end{proposition}\n\\begin{proof}\nLet $n>2$. If $n$ has an odd prime divisor $p$, write $D^n=(D^{n/p})^p$. If it has no odd prime divisor, it is a power of two at least four, so write $D^n=(D^{n/4})^4$. Apply this operation to each term separately. The new bases have exactly the same prime supports as the old bases, so primitivity is preserved. The converse follows by reducing any hypothetical primitive solution in this way.\n\\end{proof}\nThe proposition retains infinitely many signatures. It does not reduce them all to $(3,5,7)$. A theorem for that signature also handles signatures $(3a,5b,7c)$ after absorbing the multipliers into the bases, and the symmetric version with the first two terms exchanged. It does not handle an unrelated prime signature such as $(5,7,11)$ …","on}[Prime or four reduction]\\label{prop:primefour}\nIf a primitive solution of \\eqref{eq:beal} exists, there is a primitive solution whose three exponents each belong to\n\\[\n\\{4\\}\\cup\\{p:p\\text{ is an odd prime}\\}.\n\\]\nConversely, absence of primitive solutions for every ordered signature in this set implies Beal.\n\\end{proposition}\n\\begin{proof}\nLet $n>2$. If $n$ has an odd prime divisor $p$, write $D^n=(D^{n/p})^p$. If it has no odd prime divisor, it is a power of two at least four, so write $D^n=(D^{n/4})^4$. Apply this operation to each term separately. The new bases have exactly the same prime supports as the old bases, so primitivity is preserved. The converse follows by reducing any hypothetical primitive solution in this way.\n\\end{proof}\nThe proposition retains infinitely many signatures. It does not reduce them all to $(3,5,7)$. A theorem for that signature also handles signatures $(3a,5b,7c)$ after absorbing the multipliers into the bases, and the symmetric version with the first two terms exchanged. It does not handle an unrelated prime signature such as $(5,7,11)$ merely beca…","ists, there is a primitive solution whose three exponents each belong to\n\\[\n\\{4\\}\\cup\\{p:p\\text{ is an odd prime}\\}.\n\\]\nConversely, absence of primitive solutions for every ordered signature in this set implies Beal.\n\\end{proposition}\n\\begin{proof}\nLet $n>2$. If $n$ has an odd prime divisor $p$, write $D^n=(D^{n/p})^p$. If it has no odd prime divisor, it is a power of two at least four, so write $D^n=(D^{n/4})^4$. Apply this operation to each term separately. The new bases have exactly the same prime supports as the old bases, so primitivity is preserved. The converse follows by reducing any hypothetical primitive solution in this way.\n\\end{proof}\nThe proposition retains infinitely many signatures. It does not reduce them all to $(3,5,7)$. A theorem for that signature also handles signatures $(3a,5b,7c)$ after absorbing the multipliers into the bases, and the symmetric version with the first two terms exchanged. It does not handle an unrelated prime signature such as $(5,7,11)$ merely because the same packet language can describe both.","rime divisor, it is a power of two at least four, so write $D^n=(D^{n/4})^4$. Apply this operation to each term separately. The new bases have exactly the same prime supports as the old bases, so primitivity is preserved. The converse follows by reducing any hypothetical primitive solution in this way.\n\\end{proof}\nThe proposition retains infinitely many signatures. It does not reduce them all to $(3,5,7)$. A theorem for that signature also handles signatures $(3a,5b,7c)$ after absorbing the multipliers into the bases, and the symmetric version with the first two terms exchanged. It does not handle an unrelated prime signature such as $(5,7,11)$ merely because the same packet language can describe both.","(D^{n/4})^4$. Apply this operation to each term separately. The new bases have exactly the same prime supports as the old bases, so primitivity is preserved. The converse follows by reducing any hypothetical primitive solution in this way.\n\\end{proof}\nThe proposition retains infinitely many signatures. It does not reduce them all to $(3,5,7)$. A theorem for that signature also handles signatures $(3a,5b,7c)$ after absorbing the multipliers into the bases, and the symmetric version with the first two terms exchanged. It does not handle an unrelated prime signature such as $(5,7,11)$ merely because the same packet language can describe both.","by reducing any hypothetical primitive solution in this way.\n\\end{proof}\nThe proposition retains infinitely many signatures. It does not reduce them all to $(3,5,7)$. A theorem for that signature also handles signatures $(3a,5b,7c)$ after absorbing the multipliers into the bases, and the symmetric version with the first two terms exchanged. It does not handle an unrelated prime signature such as $(5,7,11)$ merely because the same packet language can describe both.","If all three exponents have a common divisor $n>2$, absorbing the remaining factors produces an equal exponent Fermat equation. Fermat's Last Theorem then excludes the primitive solution \\cite{wiles1995,taylor-wiles1995}. This use requires a common divisor of all three exponents. Sharing a divisor in only one pair is weaker. The general prime or four reduction must therefore retain mixed signatures even after the equal exponent cases are removed.","The fixed signature finiteness theorem of Darmon and Granville provides another useful perspective \\cite{darmon-granville1995}. In the hyperbolic range\n\\[\n\\frac1x+\\frac1y+\\frac1z<1,\n\\]\ntheir theorem gives finiteness of primitive solutions for a fixed equation and fixed exponents. For $x,y,z>2$, the only nonhyperbolic possibility is $(3,3,3)$, where the reciprocal sum is one and Fermat already applies. Finiteness is valuable, but it is not absence. It also does not give a uniform finite enumeration over all signatures.","\\subsection{The conditional abc height bound}\nAn abc estimate makes the role of the signature explicit. Let\n\\[\nH=C^z,\\qquad R=\\rad(ABC),\\qquad s=\\frac1x+\\frac1y+\\frac1z.\n\\]\nFor a primitive solution, the powered terms are pairwise coprime and their radical is $R$. Positivity gives $ABC<H^s$. Assume the all epsilon abc estimate\n\\begin{equation}\\label{eq:abc-assumed}\nH\\le K_\\eps R^{1+\\eps}\n\\end{equation}\nfor every $\\eps>0$, with one constant $K_\\eps$ independent of the triple. Then\n\\[\nH<K_\\eps H^{(1+\\eps)s}.\n\\]\nWhenever $(1+\\eps)s<1$, it follows that\n\\begin{equation}\\label{eq:signatureheight}\nH<K_\\eps^{1/(1-(1+\\eps)s)}.\n\\end{equation}\nThis is a conditional bound, not an unconditional result obtained from the exponent equation alone.","conditional abc height bound}\nAn abc estimate makes the role of the signature explicit. Let\n\\[\nH=C^z,\\qquad R=\\rad(ABC),\\qquad s=\\frac1x+\\frac1y+\\frac1z.\n\\]\nFor a primitive solution, the powered terms are pairwise coprime and their radical is $R$. Positivity gives $ABC<H^s$. Assume the all epsilon abc estimate\n\\begin{equation}\\label{eq:abc-assumed}\nH\\le K_\\eps R^{1+\\eps}\n\\end{equation}\nfor every $\\eps>0$, with one constant $K_\\eps$ independent of the triple. Then\n\\[\nH<K_\\eps H^{(1+\\eps)s}.\n\\]\nWhenever $(1+\\eps)s<1$, it follows that\n\\begin{equation}\\label{eq:signatureheight}\nH<K_\\eps^{1/(1-(1+\\eps)s)}.\n\\end{equation}\nThis is a conditional bound, not an unconditional result obtained from the exponent equation alone.","}\nAn abc estimate makes the role of the signature explicit. Let\n\\[\nH=C^z,\\qquad R=\\rad(ABC),\\qquad s=\\frac1x+\\frac1y+\\frac1z.\n\\]\nFor a primitive solution, the powered terms are pairwise coprime and their radical is $R$. Positivity gives $ABC<H^s$. Assume the all epsilon abc estimate\n\\begin{equation}\\label{eq:abc-assumed}\nH\\le K_\\eps R^{1+\\eps}\n\\end{equation}\nfor every $\\eps>0$, with one constant $K_\\eps$ independent of the triple. Then\n\\[\nH<K_\\eps H^{(1+\\eps)s}.\n\\]\nWhenever $(1+\\eps)s<1$, it follows that\n\\begin{equation}\\label{eq:signatureheight}\nH<K_\\eps^{1/(1-(1+\\eps)s)}.\n\\end{equation}\nThis is a conditional bound, not an unconditional result obtained from the exponent equation alone.","gnature explicit. Let\n\\[\nH=C^z,\\qquad R=\\rad(ABC),\\qquad s=\\frac1x+\\frac1y+\\frac1z.\n\\]\nFor a primitive solution, the powered terms are pairwise coprime and their radical is $R$. Positivity gives $ABC<H^s$. Assume the all epsilon abc estimate\n\\begin{equation}\\label{eq:abc-assumed}\nH\\le K_\\eps R^{1+\\eps}\n\\end{equation}\nfor every $\\eps>0$, with one constant $K_\\eps$ independent of the triple. Then\n\\[\nH<K_\\eps H^{(1+\\eps)s}.\n\\]\nWhenever $(1+\\eps)s<1$, it follows that\n\\begin{equation}\\label{eq:signatureheight}\nH<K_\\eps^{1/(1-(1+\\eps)s)}.\n\\end{equation}\nThis is a conditional bound, not an unconditional result obtained from the exponent equation alone.","R=\\rad(ABC),\\qquad s=\\frac1x+\\frac1y+\\frac1z.\n\\]\nFor a primitive solution, the powered terms are pairwise coprime and their radical is $R$. Positivity gives $ABC<H^s$. Assume the all epsilon abc estimate\n\\begin{equation}\\label{eq:abc-assumed}\nH\\le K_\\eps R^{1+\\eps}\n\\end{equation}\nfor every $\\eps>0$, with one constant $K_\\eps$ independent of the triple. Then\n\\[\nH<K_\\eps H^{(1+\\eps)s}.\n\\]\nWhenever $(1+\\eps)s<1$, it follows that\n\\begin{equation}\\label{eq:signatureheight}\nH<K_\\eps^{1/(1-(1+\\eps)s)}.\n\\end{equation}\nThis is a conditional bound, not an unconditional result obtained from the exponent equation alone.","\\]\nFor a primitive solution, the powered terms are pairwise coprime and their radical is $R$. Positivity gives $ABC<H^s$. Assume the all epsilon abc estimate\n\\begin{equation}\\label{eq:abc-assumed}\nH\\le K_\\eps R^{1+\\eps}\n\\end{equation}\nfor every $\\eps>0$, with one constant $K_\\eps$ independent of the triple. Then\n\\[\nH<K_\\eps H^{(1+\\eps)s}.\n\\]\nWhenever $(1+\\eps)s<1$, it follows that\n\\begin{equation}\\label{eq:signatureheight}\nH<K_\\eps^{1/(1-(1+\\eps)s)}.\n\\end{equation}\nThis is a conditional bound, not an unconditional result obtained from the exponent equation alone.","For $(3,5,7)$, the reciprocal sum is $71/105$. Thus $0<\\eps<34/71$ gives\n\\begin{equation}\\label{eq:357abc}\nH<K_\\eps^{105/(34-71\\eps)},\\qquad\nC<K_\\eps^{15/(34-71\\eps)}.\n\\end{equation}\nAt $\\eps=1/10$, the latter exponent is $150/269$. To turn this into a proof of absence would require the abc input, an explicit usable constant, and a complete verification below the resulting bound.","There is a stronger uniform bound across the Beal range. Apart from $(3,3,3)$, already excluded by Fermat, one exponent is at least four. Thus $s\\le1/3+1/3+1/4=11/12$. Take $\\eps=1/22$ and enlarge its constant to $K_{1/22}\\ge1$. Then $(1+\\eps)s\\le23/24$, and \\eqref{eq:signatureheight} gives\n\\begin{equation}\\label{eq:uniformbealabc}\nH<K_{1/22}^{24}.\n\\end{equation}\nThis conditional constant is independent of the signature. It bounds the powered terms and yields finitely many possible primitive powered triples. It does not show that the finite set is empty. Since $2^z\\le H$, it also bounds $z$ and excludes solutions once the minimum exponent is sufficiently large. The all epsilon hypothesis and the gap between finiteness and absence remain essential \\cite{darmon-granvi","There is a stronger uniform bound across the Beal range. Apart from $(3,3,3)$, already excluded by Fermat, one exponent is at least four. Thus $s\\le1/3+1/3+1/4=11/12$. Take $\\eps=1/22$ and enlarge its constant to $K_{1/22}\\ge1$. Then $(1+\\eps)s\\le23/24$, and \\eqref{eq:signatureheight} gives\n\\begin{equation}\\label{eq:uniformbealabc}\nH<K_{1/22}^{24}.\n\\end{equation}\nThis conditional constant is independent of the signature. It bounds the powered terms and yields finitely many possible primitive powered triples. It does not show that the finite set is empty. Since $2^z\\le H$, it also bounds $z$ and excludes solutions once the minimum exponent is sufficiently large. The all epsilon hypothesis and the gap between finiteness and absence remain essential \\cite{darmon-granville1995}.","Take $\\eps=1/22$ and enlarge its constant to $K_{1/22}\\ge1$. Then $(1+\\eps)s\\le23/24$, and \\eqref{eq:signatureheight} gives\n\\begin{equation}\\label{eq:uniformbealabc}\nH<K_{1/22}^{24}.\n\\end{equation}\nThis conditional constant is independent of the signature. It bounds the powered terms and yields finitely many possible primitive powered triples. It does not show that the finite set is empty. Since $2^z\\le H$, it also bounds $z$ and excludes solutions once the minimum exponent is sufficiently large. The all epsilon hypothesis and the gap between finiteness and absence remain essential \\cite{darmon-granville1995}.","e its constant to $K_{1/22}\\ge1$. Then $(1+\\eps)s\\le23/24$, and \\eqref{eq:signatureheight} gives\n\\begin{equation}\\label{eq:uniformbealabc}\nH<K_{1/22}^{24}.\n\\end{equation}\nThis conditional constant is independent of the signature. It bounds the powered terms and yields finitely many possible primitive powered triples. It does not show that the finite set is empty. Since $2^z\\le H$, it also bounds $z$ and excludes solutions once the minimum exponent is sufficiently large. The all epsilon hypothesis and the gap between finiteness and absence remain essential \\cite{darmon-granville1995}.","Unit bases can be disposed of separately by the classical theorem on consecutive perfect powers. If $A=1$ and $B>1$, then $C^z-B^y=1$. Mih\\u{a}ilescu's theorem gives the only pair $3^2-2^3=1$, whose upper exponent is two and violates the Beal range \\cite{mihailescu2004,mihailescu2005,bilu2005}. The same argument handles $B=1$. If both are one, $C^z=2$ is impossible. Hence every actual Beal solution has all bases greater than one. After this accepted deep input, a powered height bound also bounds every exponent label. This is a use of a classical theorem, not a new proof of it.","\\subsection{Why a generic radical ceiling fails}\nThe exponent equation gives $R<H^{71/105}$ for the signature $(3,5,7)$. It is tempting to reverse this relation by postulating $H\\le R^{105/71}$ for every primitive sum. That universal assertion is false. The exact equality\n\\[\n2+109\\cdot3^{10}=23^5=6\\,436\\,343\n\\]\nhas radical $2\\cdot3\\cdot109\\cdot23=15\\,042$, and\n\\[\n(23^5)^{71}>15\\,042^{105}.\n\\]\nIts logarithmic quality is about $1.629911684$. It is not a $(3,5,7)$ counterexample. It shows that an estimate strong enough to close the desired signature must retain its power conditions or c","\\subsection{Why a generic radical ceiling fails}\nThe exponent equation gives $R<H^{71/105}$ for the signature $(3,5,7)$. It is tempting to reverse this relation by postulating $H\\le R^{105/71}$ for every primitive sum. That universal assertion is false. The exact equality\n\\[\n2+109\\cdot3^{10}=23^5=6\\,436\\,343\n\\]\nhas radical $2\\cdot3\\cdot109\\cdot23=15\\,042$, and\n\\[\n(23^5)^{71}>15\\,042^{105}.\n\\]\nIts logarithmic quality is about $1.629911684$. It is not a $(3,5,7)$ counterexample. It shows that an estimate strong enough to close the desired signature must retain its power conditions or contain the constants and exc","\\subsection{Why a generic radical ceiling fails}\nThe exponent equation gives $R<H^{71/105}$ for the signature $(3,5,7)$. It is tempting to reverse this relation by postulating $H\\le R^{105/71}$ for every primitive sum. That universal assertion is false. The exact equality\n\\[\n2+109\\cdot3^{10}=23^5=6\\,436\\,343\n\\]\nhas radical $2\\cdot3\\cdot109\\cdot23=15\\,042$, and\n\\[\n(23^5)^{71}>15\\,042^{105}.\n\\]\nIts logarithmic quality is about $1.629911684$. It is not a $(3,5,7)$ counterexample. It shows that an estimate strong enough to close the desired signature must retain its power conditions or contain the constants and exceptions that a genuine abc type estimate allows.","cal ceiling fails}\nThe exponent equation gives $R<H^{71/105}$ for the signature $(3,5,7)$. It is tempting to reverse this relation by postulating $H\\le R^{105/71}$ for every primitive sum. That universal assertion is false. The exact equality\n\\[\n2+109\\cdot3^{10}=23^5=6\\,436\\,343\n\\]\nhas radical $2\\cdot3\\cdot109\\cdot23=15\\,042$, and\n\\[\n(23^5)^{71}>15\\,042^{105}.\n\\]\nIts logarithmic quality is about $1.629911684$. It is not a $(3,5,7)$ counterexample. It shows that an estimate strong enough to close the desired signature must retain its power conditions or contain the constants and exceptions that a genuine abc type estimate allows.","nent equation gives $R<H^{71/105}$ for the signature $(3,5,7)$. It is tempting to reverse this relation by postulating $H\\le R^{105/71}$ for every primitive sum. That universal assertion is false. The exact equality\n\\[\n2+109\\cdot3^{10}=23^5=6\\,436\\,343\n\\]\nhas radical $2\\cdot3\\cdot109\\cdot23=15\\,042$, and\n\\[\n(23^5)^{71}>15\\,042^{105}.\n\\]\nIts logarithmic quality is about $1.629911684$. It is not a $(3,5,7)$ counterexample. It shows that an estimate strong enough to close the desired signature must retain its power conditions or contain the constants and exceptions that a genuine abc type estimate allows.","\\section{A rational parametrization and its integral defect}\\label{sec:scaling}\nFor the signature $(3,5,7)$, rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact.\n\\begin{proposition}[Universal power identity]\\label{prop:universal}\nIn any commutative ring, put $s=1-t$ and\n\\[\nA_0=t^{12}s^{28},\\qquad B_0=t^7s^{17},\\qquad C_0=t^5s^{12}.\n\\]\nThen $A_0^3+B_0^5=C_0^7$. Weighted scaling\n\\begin{equation}\\label{eq:weightedscale}\n(A,B,C)=(\\lambda^{35}A_0,\\lambda^{21}B_0,\\lambda^{15}C_0)\n\\end{equation}\npreserves this equality.\n\\end{proposition}\n\\begin{proof}\nThe two terms on the left are $t^{36}s^{84}$ and $t^{35}s^{85}$. Their sum is $t^{35}s^{84}(t+s)=t^{35}s^{84}$, which is $C_0^7$. The three scaling weights multiply the po","\\section{A rational parametrization and its integral defect}\\label{sec:scaling}\nFor the signature $(3,5,7)$, rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact.\n\\begin{proposition}[Universal power identity]\\label{prop:universal}\nIn any commutative ring, put $s=1-t$ and\n\\[\nA_0=t^{12}s^{28},\\qquad B_0=t^7s^{17},\\qquad C_0=t^5s^{12}.\n\\]\nThen $A_0^3+B_0^5=C_0^7$. Weighted scaling\n\\begin{equation}\\label{eq:weightedscale}\n(A,B,C)=(\\lambda^{35}A_0,\\lambda^{21}B_0,\\lambda^{15}C_0)\n\\end{equation}\npreserves this equality.\n\\end{proposition}\n\\begin{proof}\nThe two terms on the left are $t^{36}s^{84}$ and $t^{35}s^{85}$. Their sum is $t^{35}s^{84}(t+s)=t^{35}s^{84}$, which is $C_0^7$. The three scaling weights multiply the powered terms by the same factor $\\lambda^{105}$.\n\\end{proof}\nOver a field, if $t$ and $1-t$ are nonzero, all three bases are nonzero and $A_0^3/C_0^7=t$. Thus every regular parameter is locally populated. This works modulo arbitrary prime power","\\section{A rational parametrization and its integral defect}\\label{sec:scaling}\nFor the signature $(3,5,7)$, rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact.\n\\begin{proposition}[Universal power identity]\\label{prop:universal}\nIn any commutative ring, put $s=1-t$ and\n\\[\nA_0=t^{12}s^{28},\\qquad B_0=t^7s^{17},\\qquad C_0=t^5s^{12}.\n\\]\nThen $A_0^3+B_0^5=C_0^7$. Weighted scaling\n\\begin{equation}\\label{eq:weightedscale}\n(A,B,C)=(\\lambda^{35}A_0,\\lambda^{21}B_0,\\lambda^{15}C_0)\n\\end{equation}\npreserves this equality.\n\\end{proposition}\n\\begin{proof}\nThe two terms on the left are $t^{36}s^{84}$ and $t^{35}s^{85}$. Their sum is $t^{35}s^{84}(t+s)=t^{35}s^{84}$, which is $C_0^7$. The three scaling weights multiply the powered terms by the same factor $\\lambda^{105}$.\n\\end{proof}\nOver a field, if $t$ and $1-t$ are nonzero, all three bases are nonzero and $A_0^3/C_0^7=t$. Thus every regular parameter is locally populated. This works modulo arbitrary prime powers whenever $t$ and $1-t$ are units. A finite field …","rization and its integral defect}\\label{sec:scaling}\nFor the signature $(3,5,7)$, rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact.\n\\begin{proposition}[Universal power identity]\\label{prop:universal}\nIn any commutative ring, put $s=1-t$ and\n\\[\nA_0=t^{12}s^{28},\\qquad B_0=t^7s^{17},\\qquad C_0=t^5s^{12}.\n\\]\nThen $A_0^3+B_0^5=C_0^7$. Weighted scaling\n\\begin{equation}\\label{eq:weightedscale}\n(A,B,C)=(\\lambda^{35}A_0,\\lambda^{21}B_0,\\lambda^{15}C_0)\n\\end{equation}\npreserves this equality.\n\\end{proposition}\n\\begin{proof}\nThe two terms on the left are $t^{36}s^{84}$ and $t^{35}s^{85}$. Their sum is $t^{35}s^{84}(t+s)=t^{35}s^{84}$, which is $C_0^7$. The three scaling weights multiply the powered terms by the same factor $\\lambda^{105}$.\n\\end{proof}\nOver a field, if $t$ and $1-t$ are nonzero, all three bases are nonzero and $A_0^3/C_0^7=t$. Thus every regular parameter is locally populated. This works modulo arbitrary prime powers whenever $t$ and $1-t$ are units. A finite field test that asks only whether…","For the signature $(3,5,7)$, rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact.\n\\begin{proposition}[Universal power identity]\\label{prop:universal}\nIn any commutative ring, put $s=1-t$ and\n\\[\nA_0=t^{12}s^{28},\\qquad B_0=t^7s^{17},\\qquad C_0=t^5s^{12}.\n\\]\nThen $A_0^3+B_0^5=C_0^7$. Weighted scaling\n\\begin{equation}\\label{eq:weightedscale}\n(A,B,C)=(\\lambda^{35}A_0,\\lambda^{21}B_0,\\lambda^{15}C_0)\n\\end{equation}\npreserves this equality.\n\\end{proposition}\n\\begin{proof}\nThe two terms on the left are $t^{36}s^{84}$ and $t^{35}s^{85}$. Their sum is $t^{35}s^{84}(t+s)=t^{35}s^{84}$, which is $C_0^7$. The three scaling weights multiply the powered terms by the same factor $\\lambda^{105}$.\n\\end{proof}\nOver a field, if $t$ and $1-t$ are nonzero, all three bases are nonzero and $A_0^3/C_0^7=t$. Thus every regular parameter is locally populated. This works modulo arbitrary prime powers whenever $t$ and $1-t$ are units. A finite field test that asks only whether such a nonzero local solution exists cannot remove a…","Universal power identity]\\label{prop:universal}\nIn any commutative ring, put $s=1-t$ and\n\\[\nA_0=t^{12}s^{28},\\qquad B_0=t^7s^{17},\\qquad C_0=t^5s^{12}.\n\\]\nThen $A_0^3+B_0^5=C_0^7$. Weighted scaling\n\\begin{equation}\\label{eq:weightedscale}\n(A,B,C)=(\\lambda^{35}A_0,\\lambda^{21}B_0,\\lambda^{15}C_0)\n\\end{equation}\npreserves this equality.\n\\end{proposition}\n\\begin{proof}\nThe two terms on the left are $t^{36}s^{84}$ and $t^{35}s^{85}$. Their sum is $t^{35}s^{84}(t+s)=t^{35}s^{84}$, which is $C_0^7$. The three scaling weights multiply the powered terms by the same factor $\\lambda^{105}$.\n\\end{proof}\nOver a field, if $t$ and $1-t$ are nonzero, all three bases are nonzero and $A_0^3/C_0^7=t$. Thus every regular parameter is locally populated. This works modulo arbitrary prime powers whenever $t$ and $1-t$ are units. A finite field test that asks only whether such a nonzero local solution exists cannot remove any of these parameter values.","ntity]\\label{prop:universal}\nIn any commutative ring, put $s=1-t$ and\n\\[\nA_0=t^{12}s^{28},\\qquad B_0=t^7s^{17},\\qquad C_0=t^5s^{12}.\n\\]\nThen $A_0^3+B_0^5=C_0^7$. Weighted scaling\n\\begin{equation}\\label{eq:weightedscale}\n(A,B,C)=(\\lambda^{35}A_0,\\lambda^{21}B_0,\\lambda^{15}C_0)\n\\end{equation}\npreserves this equality.\n\\end{proposition}\n\\begin{proof}\nThe two terms on the left are $t^{36}s^{84}$ and $t^{35}s^{85}$. Their sum is $t^{35}s^{84}(t+s)=t^{35}s^{84}$, which is $C_0^7$. The three scaling weights multiply the powered terms by the same factor $\\lambda^{105}$.\n\\end{proof}\nOver a field, if $t$ and $1-t$ are nonzero, all three bases are nonzero and $A_0^3/C_0^7=t$. Thus every regular parameter is locally populated. This works modulo arbitrary prime powers whenever $t$ and $1-t$ are units. A finite field test that asks only whether such a nonzero local solution exists cannot remove any of these parameter values.","In any commutative ring, put $s=1-t$ and\n\\[\nA_0=t^{12}s^{28},\\qquad B_0=t^7s^{17},\\qquad C_0=t^5s^{12}.\n\\]\nThen $A_0^3+B_0^5=C_0^7$. Weighted scaling\n\\begin{equation}\\label{eq:weightedscale}\n(A,B,C)=(\\lambda^{35}A_0,\\lambda^{21}B_0,\\lambda^{15}C_0)\n\\end{equation}\npreserves this equality.\n\\end{proposition}\n\\begin{proof}\nThe two terms on the left are $t^{36}s^{84}$ and $t^{35}s^{85}$. Their sum is $t^{35}s^{84}(t+s)=t^{35}s^{84}$, which is $C_0^7$. The three scaling weights multiply the powered terms by the same factor $\\lambda^{105}$.\n\\end{proof}\nOver a field, if $t$ and $1-t$ are nonzero, all three bases are nonzero and $A_0^3/C_0^7=t$. Thus every regular parameter is locally populated. This works modulo arbitrary prime powers whenever $t$ and $1-t$ are units. A finite field test that asks only whether such a nonzero local solution exists cannot remove any of these parameter values.","A_0=t^{12}s^{28},\\qquad B_0=t^7s^{17},\\qquad C_0=t^5s^{12}.\n\\]\nThen $A_0^3+B_0^5=C_0^7$. Weighted scaling\n\\begin{equation}\\label{eq:weightedscale}\n(A,B,C)=(\\lambda^{35}A_0,\\lambda^{21}B_0,\\lambda^{15}C_0)\n\\end{equation}\npreserves this equality.\n\\end{proposition}\n\\begin{proof}\nThe two terms on the left are $t^{36}s^{84}$ and $t^{35}s^{85}$. Their sum is $t^{35}s^{84}(t+s)=t^{35}s^{84}$, which is $C_0^7$. The three scaling weights multiply the powered terms by the same factor $\\lambda^{105}$.\n\\end{proof}\nOver a field, if $t$ and $1-t$ are nonzero, all three bases are nonzero and $A_0^3/C_0^7=t$. Thus every regular parameter is locally populated. This works modulo arbitrary prime powers whenever $t$ and $1-t$ are units. A finite field test that asks only whether such a nonzero local solution exists cannot remove any of these parameter values.","=C_0^7$. Weighted scaling\n\\begin{equation}\\label{eq:weightedscale}\n(A,B,C)=(\\lambda^{35}A_0,\\lambda^{21}B_0,\\lambda^{15}C_0)\n\\end{equation}\npreserves this equality.\n\\end{proposition}\n\\begin{proof}\nThe two terms on the left are $t^{36}s^{84}$ and $t^{35}s^{85}$. Their sum is $t^{35}s^{84}(t+s)=t^{35}s^{84}$, which is $C_0^7$. The three scaling weights multiply the powered terms by the same factor $\\lambda^{105}$.\n\\end{proof}\nOver a field, if $t$ and $1-t$ are nonzero, all three bases are nonzero and $A_0^3/C_0^7=t$. Thus every regular parameter is locally populated. This works modulo arbitrary prime powers whenever $t$ and $1-t$ are units. A finite field test that asks only whether such a nonzero local solution exists cannot remove any of these parameter values.","el{eq:weightedscale}\n(A,B,C)=(\\lambda^{35}A_0,\\lambda^{21}B_0,\\lambda^{15}C_0)\n\\end{equation}\npreserves this equality.\n\\end{proposition}\n\\begin{proof}\nThe two terms on the left are $t^{36}s^{84}$ and $t^{35}s^{85}$. Their sum is $t^{35}s^{84}(t+s)=t^{35}s^{84}$, which is $C_0^7$. The three scaling weights multiply the powered terms by the same factor $\\lambda^{105}$.\n\\end{proof}\nOver a field, if $t$ and $1-t$ are nonzero, all three bases are nonzero and $A_0^3/C_0^7=t$. Thus every regular parameter is locally populated. This works modulo arbitrary prime powers whenever $t$ and $1-t$ are units. A finite field test that asks only whether such a nonzero local solution exists cannot remove any of these parameter values.","ightedscale}\n(A,B,C)=(\\lambda^{35}A_0,\\lambda^{21}B_0,\\lambda^{15}C_0)\n\\end{equation}\npreserves this equality.\n\\end{proposition}\n\\begin{proof}\nThe two terms on the left are $t^{36}s^{84}$ and $t^{35}s^{85}$. Their sum is $t^{35}s^{84}(t+s)=t^{35}s^{84}$, which is $C_0^7$. The three scaling weights multiply the powered terms by the same factor $\\lambda^{105}$.\n\\end{proof}\nOver a field, if $t$ and $1-t$ are nonzero, all three bases are nonzero and $A_0^3/C_0^7=t$. Thus every regular parameter is locally populated. This works modulo arbitrary prime powers whenever $t$ and $1-t$ are units. A finite field test that asks only whether such a nonzero local solution exists cannot remove any of these parameter values.","_0,\\lambda^{15}C_0)\n\\end{equation}\npreserves this equality.\n\\end{proposition}\n\\begin{proof}\nThe two terms on the left are $t^{36}s^{84}$ and $t^{35}s^{85}$. Their sum is $t^{35}s^{84}(t+s)=t^{35}s^{84}$, which is $C_0^7$. The three scaling weights multiply the powered terms by the same factor $\\lambda^{105}$.\n\\end{proof}\nOver a field, if $t$ and $1-t$ are nonzero, all three bases are nonzero and $A_0^3/C_0^7=t$. Thus every regular parameter is locally populated. This works modulo arbitrary prime powers whenever $t$ and $1-t$ are units. A finite field test that asks only whether such a nonzero local solution exists cannot remove any of these parameter values.","are $t^{36}s^{84}$ and $t^{35}s^{85}$. Their sum is $t^{35}s^{84}(t+s)=t^{35}s^{84}$, which is $C_0^7$. The three scaling weights multiply the powered terms by the same factor $\\lambda^{105}$.\n\\end{proof}\nOver a field, if $t$ and $1-t$ are nonzero, all three bases are nonzero and $A_0^3/C_0^7=t$. Thus every regular parameter is locally populated. This works modulo arbitrary prime powers whenever $t$ and $1-t$ are units. A finite field test that asks only whether such a nonzero local solution exists cannot remove any of these parameter values.","36}s^{84}$ and $t^{35}s^{85}$. Their sum is $t^{35}s^{84}(t+s)=t^{35}s^{84}$, which is $C_0^7$. The three scaling weights multiply the powered terms by the same factor $\\lambda^{105}$.\n\\end{proof}\nOver a field, if $t$ and $1-t$ are nonzero, all three bases are nonzero and $A_0^3/C_0^7=t$. Thus every regular parameter is locally populated. This works modulo arbitrary prime powers whenever $t$ and $1-t$ are units. A finite field test that asks only whether such a nonzero local solution exists cannot remove any of these parameter values.","For a nonzero rational solution, define $t=A^3/C^7$ and $\\lambda=C^8/(AB^4)$. Substitution recovers \\eqref{eq:weightedscale}. If the original bases are positive, then $0<t<1$ and $\\lambda>0$. The rational parametrization therefore captures the entire positive rational problem. Its missing condition is exactly primitive integrality.","\\begin{theorem}[Primitive scaling criterion]\\label{thm:scaling}\nFix a rational $0<t<1$. There is a positive rational $\\lambda$ for which \\eqref{eq:weightedscale} gives primitive positive integers if and only if, for every rational prime $\\ell$,\n\\begin{align*}\nv_\\ell(t)>0&\\quad\\Longrightarrow\\quad3\\mid v_\\ell(t),","\\begin{theorem}[Primitive scaling criterion]\\label{thm:scaling}\nFix a rational $0<t<1$. There is a positive rational $\\lambda$ for which \\eqref{eq:weightedscale} gives primitive positive integers if and only if, for every rational prime $\\ell$,\n\\begin{align*}\nv_\\ell(t)>0&\\quad\\Longrightarrow\\quad3\\mid v_\\ell(t),\\\\\nv_\\ell(1-t)>0&\\quad\\Longrightarrow\\quad5\\mid v_\\ell(1-t),\\\\\nv_\\ell(t)<0&\\quad\\Longrightarrow\\quad7\\mid -v_\\ell(t).\n\\end{align*}\nWhen it exists, $\\lambda$ is unique.\n\\end{theorem}\n\\begin{proof}\nWrite $m=v_\\ell(t)$, $n=v_\\ell(1-t)$ and $h=v_\\ell(\\lambda)$. Because $t+(1-t)=1$, the pair $(m,n)$ belongs to one of four sectors:\n\\[\n(m>0,n=0),\\quad(m=0,n>0),\\quad(m=n<0),\\quad(m=n=0).\n\\]\nThe output valuations are\n\\begin{align*}\na_\\ell&=35h+12m+28n,","v_\\ell(t)<0&\\quad\\Longrightarrow\\quad7\\mid -v_\\ell(t).\n\\end{align*}\nWhen it exists, $\\lambda$ is unique.\n\\end{theorem}\n\\begin{proof}\nWrite $m=v_\\ell(t)$, $n=v_\\ell(1-t)$ and $h=v_\\ell(\\lambda)$. Because $t+(1-t)=1$, the pair $(m,n)$ belongs to one of four sectors:\n\\[\n(m>0,n=0),\\quad(m=0,n>0),\\quad(m=n<0),\\quad(m=n=0).\n\\]\nThe output valuations are\n\\begin{align*}\na_\\ell&=35h+12m+28n,","v_\\ell(t)<0&\\quad\\Longrightarrow\\quad7\\mid -v_\\ell(t).\n\\end{align*}\nWhen it exists, $\\lambda$ is unique.\n\\end{theorem}\n\\begin{proof}\nWrite $m=v_\\ell(t)$, $n=v_\\ell(1-t)$ and $h=v_\\ell(\\lambda)$. Because $t+(1-t)=1$, the pair $(m,n)$ belongs to one of four sectors:\n\\[\n(m>0,n=0),\\quad(m=0,n>0),\\quad(m=n<0),\\quad(m=n=0).\n\\]\nThe output valuations are\n\\begin{align*}\na_\\ell&=35h+12m+28n,\\\\\nb_\\ell&=21h+7m+17n,\\\\\nc_\\ell&=15h+5m+12n.\n\\end{align*}\nThey must be nonnegative, and their minimum must be zero. Equivalently, the minimum of $3a_\\ell,5b_\\ell,7c_\\ell$ must be zero.","c_\\ell&=15h+5m+12n.\n\\end{align*}\nThey must be nonnegative, and their minimum must be zero. Equivalently, the minimum of $3a_\\ell,5b_\\ell,7c_\\ell$ must be zero.","In the first sector these weighted valuations are $105h+36m,105h+35m,105h+35m$. Therefore $h=-m/3$, and the output is $(m/3,0,0)$. In the second sector they are $105h+84n,105h+85n,105h+84n$. This forces $h=-4n/5$ and output $(0,n/5,0)$. In the third sector write $m=n=-r$ with $r>0$. The weighted valuations are $105h-120r,105h-120r,105h-119r$. Thus $h=8r/7$ and the output is $(0,0,r/7)$. In the final sector $h=0$.","t sector these weighted valuations are $105h+36m,105h+35m,105h+35m$. Therefore $h=-m/3$, and the output is $(m/3,0,0)$. In the second sector they are $105h+84n,105h+85n,105h+84n$. This forces $h=-4n/5$ and output $(0,n/5,0)$. In the third sector write $m=n=-r$ with $r>0$. The weighted valuations are $105h-120r,105h-120r,105h-119r$. Thus $h=8r/7$ and the output is $(0,0,r/7)$. In the final sector $h=0$.","An integral $h$ exists in these four cases precisely under the stated divisibility conditions. Only finitely many primes occur in $t$ or $1-t$, so $\\lambda=\\prod_\\ell\\ell^{h_\\ell}$ is a well defined positive rational number. The three output supports are disjoint, proving integrality and primitivity. Every $h_\\ell$ was forced, so two positive choices of $\\lambda$ would have ratio a positive rational with zero valuation at every prime. That ratio is one.\n\\end{proof}\n\\figplaceholder{03_valuation_packets}{The three nontrivial valuation sectors retain different divisibility conditions. A residue projector can preserve their labels without making the conditions hold.}\nIf $t=m/n$ is in lowest terms with $0<m<n$","An integral $h$ exists in these four cases precisely under the stated divisibility conditions. Only finitely many primes occur in $t$ or $1-t$, so $\\lambda=\\prod_\\ell\\ell^{h_\\ell}$ is a well defined positive rational number. The three output supports are disjoint, proving integrality and primitivity. Every $h_\\ell$ was forced, so two positive choices of $\\lambda$ would have ratio a positive rational with zero valuation at every prime. That ratio is one.\n\\end{proof}\n\\figplaceholder{03_valuation_packets}{The three nontrivial valuation sectors retain different divisibility conditions. A residue projector can preserve their labels without making the conditions hold.}\nIf $t=m/n$ is in lowest terms with $0<m<n$, the theorem says that $m$ must be a cube, $n-m$ a fifth power, and $n$ a seventh power. The parametrization has not","An integral $h$ exists in these four cases precisely under the stated divisibility conditions. Only finitely many primes occur in $t$ or $1-t$, so $\\lambda=\\prod_\\ell\\ell^{h_\\ell}$ is a well defined positive rational number. The three output supports are disjoint, proving integrality and primitivity. Every $h_\\ell$ was forced, so two positive choices of $\\lambda$ would have ratio a positive rational with zero valuation at every prime. That ratio is one.\n\\end{proof}\n\\figplaceholder{03_valuation_packets}{The three nontrivial valuation sectors retain different divisibility conditions. A residue projector can preserve their labels without making the conditions hold.}\nIf $t=m/n$ is in lowest terms with $0<m<n$, the theorem says that $m$ must be a cube, $n-m$ a fifth power, and $n$ a seventh power. The parametrization has not escaped t","An integral $h$ exists in these four cases precisely under the stated divisibility conditions. Only finitely many primes occur in $t$ or $1-t$, so $\\lambda=\\prod_\\ell\\ell^{h_\\ell}$ is a well defined positive rational number. The three output supports are disjoint, proving integrality and primitivity. Every $h_\\ell$ was forced, so two positive choices of $\\lambda$ would have ratio a positive rational with zero valuation at every prime. That ratio is one.\n\\end{proof}\n\\figplaceholder{03_valuation_packets}{The three nontrivial valuation sectors retain different divisibility conditions. A residue projector can preserve their labels without making the conditions hold.}\nIf $t=m/n$ is in lowest terms with $0<m<n$, the theorem says that $m$ must be a cube, $n-m$ a fifth power, and $n$ a seventh power. The parametrization has not escaped the original power equation. It has iso","An integral $h$ exists in these four cases precisely under the stated divisibility conditions. Only finitely many primes occur in $t$ or $1-t$, so $\\lambda=\\prod_\\ell\\ell^{h_\\ell}$ is a well defined positive rational number. The three output supports are disjoint, proving integrality and primitivity. Every $h_\\ell$ was forced, so two positive choices of $\\lambda$ would have ratio a positive rational with zero valuation at every prime. That ratio is one.\n\\end{proof}\n\\figplaceholder{03_valuation_packets}{The three nontrivial valuation sectors retain different divisibility conditions. A residue projector can preserve their labels without making the conditions hold.}\nIf $t=m/n$ is in lowest terms with $0<m<n$, the theorem says that $m$ must be a cube, $n-m$ a fifth power, and $n$ a seventh power. The parametrization has not escaped the original power equation. It has isolated its defect in three valuation sectors.","egrality and primitivity. Every $h_\\ell$ was forced, so two positive choices of $\\lambda$ would have ratio a positive rational with zero valuation at every prime. That ratio is one.\n\\end{proof}\n\\figplaceholder{03_valuation_packets}{The three nontrivial valuation sectors retain different divisibility conditions. A residue projector can preserve their labels without making the conditions hold.}\nIf $t=m/n$ is in lowest terms with $0<m<n$, the theorem says that $m$ must be a cube, $n-m$ a fifth power, and $n$ a seventh power. The parametrization has not escaped the original power equation. It has isolated its defect in three valuation sectors.","$h_\\ell$ was forced, so two positive choices of $\\lambda$ would have ratio a positive rational with zero valuation at every prime. That ratio is one.\n\\end{proof}\n\\figplaceholder{03_valuation_packets}{The three nontrivial valuation sectors retain different divisibility conditions. A residue projector can preserve their labels without making the conditions hold.}\nIf $t=m/n$ is in lowest terms with $0<m<n$, the theorem says that $m$ must be a cube, $n-m$ a fifth power, and $n$ a seventh power. The parametrization has not escaped the original power equation. It has isolated its defect in three valuation sectors.","itive choices of $\\lambda$ would have ratio a positive rational with zero valuation at every prime. That ratio is one.\n\\end{proof}\n\\figplaceholder{03_valuation_packets}{The three nontrivial valuation sectors retain different divisibility conditions. A residue projector can preserve their labels without making the conditions hold.}\nIf $t=m/n$ is in lowest terms with $0<m<n$, the theorem says that $m$ must be a cube, $n-m$ a fifth power, and $n$ a seventh power. The parametrization has not escaped the original power equation. It has isolated its defect in three valuation sectors.","ambda$ would have ratio a positive rational with zero valuation at every prime. That ratio is one.\n\\end{proof}\n\\figplaceholder{03_valuation_packets}{The three nontrivial valuation sectors retain different divisibility conditions. A residue projector can preserve their labels without making the conditions hold.}\nIf $t=m/n$ is in lowest terms with $0<m<n$, the theorem says that $m$ must be a cube, $n-m$ a fifth power, and $n$ a seventh power. The parametrization has not escaped the original power equation. It has isolated its defect in three valuation sectors.","positive rational with zero valuation at every prime. That ratio is one.\n\\end{proof}\n\\figplaceholder{03_valuation_packets}{The three nontrivial valuation sectors retain different divisibility conditions. A residue projector can preserve their labels without making the conditions hold.}\nIf $t=m/n$ is in lowest terms with $0<m<n$, the theorem says that $m$ must be a cube, $n-m$ a fifth power, and $n$ a seventh power. The parametrization has not escaped the original power equation. It has isolated its defect in three valuation sectors.","\\begin{example}\nAt $t=1/2$, the denominator has valuation one at two, which fails the required divisibility by seven. No positive rational scaling produces a primitive integer triple. The choice $\\lambda=4$ nevertheless gives\n\\[\n(A,B,C)=(2^{30},2^{18},2^{13}),\n\\]\nand the power equation holds. Its common divisor $2^{13}$ makes it a negative control against interpreting rational parametrization as a primitive solution.\n\\end{example}\nThis is where the Komorebi packet viewpoint has concrete force. Each sector carries an exa","\\begin{example}\nAt $t=1/2$, the denominator has valuation one at two, which fails the required divisibility by seven. No positive rational scaling produces a primitive integer triple. The choice $\\lambda=4$ nevertheless gives\n\\[\n(A,B,C)=(2^{30},2^{18},2^{13}),\n\\]\nand the power equation holds. Its common divisor $2^{13}$ makes it a negative control against interpreting rational parametrization as a primitive solution.\n\\end{example}\nThis is where the Komorebi packet viewpoint has concrete force. Each sector carries an exact linear valuation rule, a required divisibility residue, and a reconstruction weight. Keeping all three records prevents information loss. The records alone do not prove that no","\\begin{example}\nAt $t=1/2$, the denominator has valuation one at two, which fails the required divisibility by seven. No positive rational scaling produces a primitive integer triple. The choice $\\lambda=4$ nevertheless gives\n\\[\n(A,B,C)=(2^{30},2^{18},2^{13}),\n\\]\nand the power equation holds. Its common divisor $2^{13}$ makes it a negative control against interpreting rational parametrization as a primitive solution.\n\\end{example}\nThis is where the Komorebi packet viewpoint has concrete force. Each sector carries an exact linear valuation rule, a required divisibility residue, and a reconstruction weight. Keeping all three records prevents information loss. The records alone do not prove that no rational $t$ satisfies every condition. A global closure","\\begin{example}\nAt $t=1/2$, the denominator has valuation one at two, which fails the required divisibility by seven. No positive rational scaling produces a primitive integer triple. The choice $\\lambda=4$ nevertheless gives\n\\[\n(A,B,C)=(2^{30},2^{18},2^{13}),\n\\]\nand the power equation holds. Its common divisor $2^{13}$ makes it a negative control against interpreting rational parametrization as a primitive solution.\n\\end{example}\nThis is where the Komorebi packet viewpoint has concrete force. Each sector carries an exact linear valuation rule, a required divisibility residue, and a reconstruction weight. Keeping all three records prevents information loss. The records alone do not prove that no rational $t$ satisfies every condition. A global closure still requires a relation linking the sectors, rather than","^{13}$ makes it a negative control against interpreting rational parametrization as a primitive solution.\n\\end{example}\nThis is where the Komorebi packet viewpoint has concrete force. Each sector carries an exact linear valuation rule, a required divisibility residue, and a reconstruction weight. Keeping all three records prevents information loss. The records alone do not prove that no rational $t$ satisfies every condition. A global closure still requires a relation linking the sectors, rather than additional names for the same local data.","\\section{The complete local squareclass test}\\label{sec:strata}\nWrite a hypothetical primitive solution as\n\\[\nA=au^2,\\qquad B=bv^2,\\qquad C=cw^2,\n\\]\nwhere $a,b,c$ are positive pairwise coprime squarefree integers. The equation becomes\n\\begin{equation}\\label{eq:strata}\na^3u^6+b^5v^{10}=c^7w^{14}.\n\\end{equation}\nAt a prime $p$, a local solution is called primitive when at most one base is divisible by $p$. Roots and bases are required to be nonzero in $\\Z_p$. The roots may depend on $p$. This last freedom is exactly why local existence can be weaker than an integer solution.","tion{The complete local squareclass test}\\label{sec:strata}\nWrite a hypothetical primitive solution as\n\\[\nA=au^2,\\qquad B=bv^2,\\qquad C=cw^2,\n\\]\nwhere $a,b,c$ are positive pairwise coprime squarefree integers. The equation becomes\n\\begin{equation}\\label{eq:strata}\na^3u^6+b^5v^{10}=c^7w^{14}.\n\\end{equation}\nAt a prime $p$, a local solution is called primitive when at most one base is divisible by $p$. Roots and bases are required to be nonzero in $\\Z_p$. The roots may depend on $p$. This last freedom is exactly why local existence can be weaker than an integer solution.",":strata}\nWrite a hypothetical primitive solution as\n\\[\nA=au^2,\\qquad B=bv^2,\\qquad C=cw^2,\n\\]\nwhere $a,b,c$ are positive pairwise coprime squarefree integers. The equation becomes\n\\begin{equation}\\label{eq:strata}\na^3u^6+b^5v^{10}=c^7w^{14}.\n\\end{equation}\nAt a prime $p$, a local solution is called primitive when at most one base is divisible by $p$. Roots and bases are required to be nonzero in $\\Z_p$. The roots may depend on $p$. This last freedom is exactly why local existence can be weaker than an integer solution.","imitive solution as\n\\[\nA=au^2,\\qquad B=bv^2,\\qquad C=cw^2,\n\\]\nwhere $a,b,c$ are positive pairwise coprime squarefree integers. The equation becomes\n\\begin{equation}\\label{eq:strata}\na^3u^6+b^5v^{10}=c^7w^{14}.\n\\end{equation}\nAt a prime $p$, a local solution is called primitive when at most one base is divisible by $p$. Roots and bases are required to be nonzero in $\\Z_p$. The roots may depend on $p$. This last freedom is exactly why local existence can be weaker than an integer solution.","\\begin{theorem}[Local coefficient criterion]\\label{thm:localstrata}\nFor fixed $a,b,c$ as above, primitive local solvability of \\eqref{eq:strata} is determined by the following conditions.\n\\begin{enumerate}\n\\item At two, at least one row in the table must be admissible. An even coefficient forces its own base to be the even base.\n\\item At an odd prime dividing $a,b,c$, respectively, the condition is\n\\[\n\\left(\\frac{bc}{p}\\right)=1,\\qquad\n\\left(\\frac{ac}{p}\\right)=1,\\qquad\n\\left(\\frac{-ab}{p}\\right)=1.\n\\]\n\\item At every odd prime not dividing $abc$, a primitive local solution exists.\n\\end{enumerate}\n\\end{theorem}\n\\begin{center}\n\\begin{tabular}{lll}\\toprule\nEven base&Required odd coefficients&Relation modulo eight","in{theorem}[Local coefficient criterion]\\label{thm:localstrata}\nFor fixed $a,b,c$ as above, primitive local solvability of \\eqref{eq:strata} is determined by the following conditions.\n\\begin{enumerate}\n\\item At two, at least one row in the table must be admissible. An even coefficient forces its own base to be the even base.\n\\item At an odd prime dividing $a,b,c$, respectively, the condition is\n\\[\n\\left(\\frac{bc}{p}\\right)=1,\\qquad\n\\left(\\frac{ac}{p}\\right)=1,\\qquad\n\\left(\\frac{-ab}{p}\\right)=1.\n\\]\n\\item At every odd prime not dividing $abc$, a primitive local solution exists.\n\\end{enumerate}\n\\end{theorem}\n\\begin{center}\n\\begin{tabular}{lll}\\toprule\nEven base&Required odd coefficients&Relation modulo eight","determined by the following conditions.\n\\begin{enumerate}\n\\item At two, at least one row in the table must be admissible. An even coefficient forces its own base to be the even base.\n\\item At an odd prime dividing $a,b,c$, respectively, the condition is\n\\[\n\\left(\\frac{bc}{p}\\right)=1,\\qquad\n\\left(\\frac{ac}{p}\\right)=1,\\qquad\n\\left(\\frac{-ab}{p}\\right)=1.\n\\]\n\\item At every odd prime not dividing $abc$, a primitive local solution exists.\n\\end{enumerate}\n\\end{theorem}\n\\begin{center}\n\\begin{tabular}{lll}\\toprule\nEven base&Required odd coefficients&Relation modulo eight","midrule\n$A$&$b,c$&$b\\equiv c$","$B$&$a,c$&$a\\equiv c$","$C$&$a,b$&$a+b\\equiv0$","\\subsection{Odd prime lifting}\nFor the polynomial on the left of \\eqref{eq:strata} minus its right side, the partial derivatives are\n\\[\n6a^3u^5,\\qquad10b^5v^9,\\qquad-14c^7w^{13}.\n\\]\nIn a primitive residue solution, at least two bases are units. At an odd prime, at most one of $6,10,14$ is divisible by $p$. Thus at least one unit variable has unit derivative. Lift the other variables to nonzero integers, using $p$ for a zero residue if necessary. Successive correction in the selected variable by multiples of $p^n$ kills one new digit at each stage. The derivative remains a unit, so the corrections converge to a nonzero root with the original primitive pattern. Every primitive residue solution therefore lifts to $\\Z_p$, including at $3,5,7$.","d prime lifting}\nFor the polynomial on the left of \\eqref{eq:strata} minus its right side, the partial derivatives are\n\\[\n6a^3u^5,\\qquad10b^5v^9,\\qquad-14c^7w^{13}.\n\\]\nIn a primitive residue solution, at least two bases are units. At an odd prime, at most one of $6,10,14$ is divisible by $p$. Thus at least one unit variable has unit derivative. Lift the other variables to nonzero integers, using $p$ for a zero residue if necessary. Successive correction in the selected variable by multiples of $p^n$ kills one new digit at each stage. The derivative remains a unit, so the corrections converge to a nonzero root with the original primitive pattern. Every primitive residue solution therefore lifts to $\\Z_p$, including at $3,5,7$.","al derivatives are\n\\[\n6a^3u^5,\\qquad10b^5v^9,\\qquad-14c^7w^{13}.\n\\]\nIn a primitive residue solution, at least two bases are units. At an odd prime, at most one of $6,10,14$ is divisible by $p$. Thus at least one unit variable has unit derivative. Lift the other variables to nonzero integers, using $p$ for a zero residue if necessary. Successive correction in the selected variable by multiples of $p^n$ kills one new digit at each stage. The derivative remains a unit, so the corrections converge to a nonzero root with the original primitive pattern. Every primitive residue solution therefore lifts to $\\Z_p$, including at $3,5,7$.","one unit variable has unit derivative. Lift the other variables to nonzero integers, using $p$ for a zero residue if necessary. Successive correction in the selected variable by multiples of $p^n$ kills one new digit at each stage. The derivative remains a unit, so the corrections converge to a nonzero root with the original primitive pattern. Every primitive residue solution therefore lifts to $\\Z_p$, including at $3,5,7$.","s unit derivative. Lift the other variables to nonzero integers, using $p$ for a zero residue if necessary. Successive correction in the selected variable by multiples of $p^n$ kills one new digit at each stage. The derivative remains a unit, so the corrections converge to a nonzero root with the original primitive pattern. Every primitive residue solution therefore lifts to $\\Z_p$, including at $3,5,7$.","If $p\\mid a$, primitivity forces the other two bases to be units. Reduction gives $b^5v^{10}=c^7w^{14}$. The quadratic character of this equality proves the necessary symbol condition. Conversely, $v^{10}w^{-14}$ ranges over the squares of $\\F_p^\\times$, because $\\gcd(10,14,p-1)=2$. The symbol condition lets us choose the two unit roots, then lift. The other two cases use $\\gcd(6,14,p-1)=2$ and $\\gcd(6,10,p-1)=2$. Moving a term across the equality gives the minus sign when $p\\mid c$.","es $b^5v^{10}=c^7w^{14}$. The quadratic character of this equality proves the necessary symbol condition. Conversely, $v^{10}w^{-14}$ ranges over the squares of $\\F_p^\\times$, because $\\gcd(10,14,p-1)=2$. The symbol condition lets us choose the two unit roots, then lift. The other two cases use $\\gcd(6,14,p-1)=2$ and $\\gcd(6,10,p-1)=2$. Moving a term across the equality gives the minus sign when $p\\mid c$.","\\subsection{Odd primes away from the coefficients}\nLet $\\alpha,\\beta,\\gamma$ be the quadratic characters of $a,b,c$, and let $e=(-1/p)$. A primitive residue solution with one zero base exists if\n\\[\n\\beta=\\gamma,\\qquad\\alpha=\\gamma,\\qquad\\alpha=e\\beta,\n\\]\nrespectively. The same cyclic group argument proves this, and the lifting construction replaces the zero root by a nonzero multiple of $p$.","If $p\\equiv1\\pmod4$, two of the three signs agree, so one of these conditions holds. If $p\\equiv3\\pmod4$ and all three fail, then $\\alpha=\\beta=-\\gamma$. The two remaining sign patterns can be handled by the universal identity of Proposition~\\ref{prop:universal}.","There are $(p+1)/4$ elements $t$ for which both $t$ and $1-t$ are nonsquares. Here is a direct count. Extend the quadratic character $\\chi_p$ by zero. Counting $(x-y)(x+y)=d$ for $d\\ne0$ gives\n\\[\n\\sum_x\\chi_p(x^2-d)=-1.\n\\]\nCompleting the square then gives $\\sum_t\\chi_p(t(1-t))=-\\chi_p(-1)=1$. After omitting $0,1$, both single character sums are $-1$. Expanding the indicator $(1-\\chi_p(t))(1-\\chi_p(1-t))/4$ over the remaining points gives $(p+1)/4>0$.","both $t$ and $1-t$ are nonsquares. Here is a direct count. Extend the quadratic character $\\chi_p$ by zero. Counting $(x-y)(x+y)=d$ for $d\\ne0$ gives\n\\[\n\\sum_x\\chi_p(x^2-d)=-1.\n\\]\nCompleting the square then gives $\\sum_t\\chi_p(t(1-t))=-\\chi_p(-1)=1$. After omitting $0,1$, both single character sums are $-1$. Expanding the indicator $(1-\\chi_p(t))(1-\\chi_p(1-t))/4$ over the remaining points gives $(p+1)/4>0$.","For such a $t$, the three bases in Proposition~\\ref{prop:universal} have characters $(1,1,-1)$. Multiplication by the weighted powers of a nonsquare $\\lambda$ changes all three signs and gives $(-1,-1,1)$. In either needed sign pattern, the ratios of these bases to $a,b,c$ are squares. Choosing their square roots gives a solution of \\eqref{eq:strata} with all bases units. The lifting argument completes the proof at these primes.","For sufficiency, fix the selected even root to be one when its coefficient is even, and two when the coefficient is odd. Fix one odd root to be one. After moving the terms and dividing by the remaining coefficient power, the required remaining root power is a unit congruent to one modulo eight. Such a unit is a square in $\\Z_2$, and every odd power map on $\\Z_2^\\times$ is bijective. It consequently has the required sixth, tenth, or fourteenth root.","The square assertion has a direct lifting proof. For $n\\ge3$, changing an odd trial square root by $2^{n-1}$ changes its square by $2^n$ modulo $2^{n+1}$, allowing the next digit to be selected. Odd power maps have odd derivative and lift uniquely. Thus the dyadic criterion also covers every higher precision.","\\subsection{Two families that survive every local refinement}\nThe coefficients $(a,b,c)=(3,5,2)$ pass the even $C$ row because $3+5\\equiv0\\pmod8$. At three, $bc=10$ is a square residue. At five, $ac=6$ is a square residue. Hence\n\\[\n27u^6+3125v^{10}=128w^{14}\n\\]\nhas a primitive solution at every prime. Likewise $(2,3,11)$ passes the even $A$ row since $3\\equiv11\\pmod8$. At three, $ac=22\\equiv1$, and at eleven, $-ab=-6\\equiv5=4^2$. Thus\n\\[\n8u^6+243v^{10}=11^7w^{14}\n\\]\nalso has a primitive solution at every prime. Both have positive real solutions. Neither is asserted to have an integer solution.","o families that survive every local refinement}\nThe coefficients $(a,b,c)=(3,5,2)$ pass the even $C$ row because $3+5\\equiv0\\pmod8$. At three, $bc=10$ is a square residue. At five, $ac=6$ is a square residue. Hence\n\\[\n27u^6+3125v^{10}=128w^{14}\n\\]\nhas a primitive solution at every prime. Likewise $(2,3,11)$ passes the even $A$ row since $3\\equiv11\\pmod8$. At three, $ac=22\\equiv1$, and at eleven, $-ab=-6\\equiv5=4^2$. Thus\n\\[\n8u^6+243v^{10}=11^7w^{14}\n\\]\nalso has a primitive solution at every prime. Both have positive real solutions. Neither is asserted to have an integer solution.","ry local refinement}\nThe coefficients $(a,b,c)=(3,5,2)$ pass the even $C$ row because $3+5\\equiv0\\pmod8$. At three, $bc=10$ is a square residue. At five, $ac=6$ is a square residue. Hence\n\\[\n27u^6+3125v^{10}=128w^{14}\n\\]\nhas a primitive solution at every prime. Likewise $(2,3,11)$ passes the even $A$ row since $3\\equiv11\\pmod8$. At three, $ac=22\\equiv1$, and at eleven, $-ab=-6\\equiv5=4^2$. Thus\n\\[\n8u^6+243v^{10}=11^7w^{14}\n\\]\nalso has a primitive solution at every prime. Both have positive real solutions. Neither is asserted to have an integer solution.","For either coefficient triple and any modulus $M$, there are positive integer roots giving pairwise coprime bases and satisfying the equation modulo $M$. Enlarge $M$ to contain the coefficient primes, choose primitive local residues, and combine them by the Chinese remainder theorem. Choose $u$ in its positive progression. Choose $v$ in its progression with $v\\equiv1$ at each prime dividing $u$ outside the enlarged modulus. Choose $w$ with the analogous conditions at primes dividing $uv$. The auxiliary moduli are coprime to the original one. Inside the modulus, local primitivity prevents shared base primes. Outside it, the avoidance conditions do so. This proves the finite congruence assertion, while exact equality remains unproved.","y modulus $M$, there are positive integer roots giving pairwise coprime bases and satisfying the equation modulo $M$. Enlarge $M$ to contain the coefficient primes, choose primitive local residues, and combine them by the Chinese remainder theorem. Choose $u$ in its positive progression. Choose $v$ in its progression with $v\\equiv1$ at each prime dividing $u$ outside the enlarged modulus. Choose $w$ with the analogous conditions at primes dividing $uv$. The auxiliary moduli are coprime to the original one. Inside the modulus, local primitivity prevents shared base primes. Outside it, the avoidance conditions do so. This proves the finite congruence assertion, while exact equality remains unproved.","g pairwise coprime bases and satisfying the equation modulo $M$. Enlarge $M$ to contain the coefficient primes, choose primitive local residues, and combine them by the Chinese remainder theorem. Choose $u$ in its positive progression. Choose $v$ in its progression with $v\\equiv1$ at each prime dividing $u$ outside the enlarged modulus. Choose $w$ with the analogous conditions at primes dividing $uv$. The auxiliary moduli are coprime to the original one. Inside the modulus, local primitivity prevents shared base primes. Outside it, the avoidance conditions do so. This proves the finite congruence assertion, while exact equality remains unproved.","\\section{A curve that retains the exponent seven}\\label{sec:frey}\nThe signature route uses the integral curve\n\\begin{equation}\\label{eq:integralcurve}\ny^2+(X^3+B)y=2BX^3+3AX+B^2.\n\\end{equation}\nCompleting the square with $Z=2y+X^3+B$ gives\n\\begin{equation}\\label{eq:sextic}\nZ^2=f_{A,B}(X)=X^6+10BX^3+12AX+5B^2.\n\\end{equation}\nThis is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as $(a,b,c,p)=(B,-C,A,7)$ \\cite{pacetti-villagra2025}. Explicitly, put $t=-B^5/A^3$, $d=-B^2/A$, and substitute $x_{\\mathrm{PV}}=d/X$, $y_{\\mathrm{PV}}=tZ/X^3$ into their equation $y^2=5x^6-12x^5+10tx^3+t^2$. Multiplication by $X^6/t^2$ gives \\eqref{eq:sextic}. The map is rational for $AB\\ne0$. It does not on its own verify a twist, integral frame, or stable","\\section{A curve that retains the exponent seven}\\label{sec:frey}\nThe signature route uses the integral curve\n\\begin{equation}\\label{eq:integralcurve}\ny^2+(X^3+B)y=2BX^3+3AX+B^2.\n\\end{equation}\nCompleting the square with $Z=2y+X^3+B$ gives\n\\begin{equation}\\label{eq:sextic}\nZ^2=f_{A,B}(X)=X^6+10BX^3+12AX+5B^2.\n\\end{equation}\nThis is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as $(a,b,c,p)=(B,-C,A,7)$ \\cite{pacetti-villagra2025}. Explicitly, put $t=-B^5/A^3$, $d=-B^2/A$, and substitute $x_{\\mathrm{PV}}=d/X$, $y_{\\mathrm{PV}}=tZ/X^3$ into their equation $y^2=5x^6-12x^5+10tx^3+t^2$. Multiplication by $X^6/t^2$ gives \\eqref{eq:sextic}. The map is rational for $AB\\ne0$. It does not on its own verify a twist, integral frame, or stable model. Their general theorems whose hypotheses exclu","\\section{A curve that retains the exponent seven}\\label{sec:frey}\nThe signature route uses the integral curve\n\\begin{equation}\\label{eq:integralcurve}\ny^2+(X^3+B)y=2BX^3+3AX+B^2.\n\\end{equation}\nCompleting the square with $Z=2y+X^3+B$ gives\n\\begin{equation}\\label{eq:sextic}\nZ^2=f_{A,B}(X)=X^6+10BX^3+12AX+5B^2.\n\\end{equation}\nThis is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as $(a,b,c,p)=(B,-C,A,7)$ \\cite{pacetti-villagra2025}. Explicitly, put $t=-B^5/A^3$, $d=-B^2/A$, and substitute $x_{\\mathrm{PV}}=d/X$, $y_{\\mathrm{PV}}=tZ/X^3$ into their equation $y^2=5x^6-12x^5+10tx^3+t^2$. Multiplication by $X^6/t^2$ gives \\eqref{eq:sextic}. The map is rational for $AB\\ne0$. It does not on its own verify a twist, integral frame, or stable model. Their general theorems whose hypotheses exclude seven cannot be applied at seven merely because the family is the same. The","on}\\label{eq:integralcurve}\ny^2+(X^3+B)y=2BX^3+3AX+B^2.\n\\end{equation}\nCompleting the square with $Z=2y+X^3+B$ gives\n\\begin{equation}\\label{eq:sextic}\nZ^2=f_{A,B}(X)=X^6+10BX^3+12AX+5B^2.\n\\end{equation}\nThis is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as $(a,b,c,p)=(B,-C,A,7)$ \\cite{pacetti-villagra2025}. Explicitly, put $t=-B^5/A^3$, $d=-B^2/A$, and substitute $x_{\\mathrm{PV}}=d/X$, $y_{\\mathrm{PV}}=tZ/X^3$ into their equation $y^2=5x^6-12x^5+10tx^3+t^2$. Multiplication by $X^6/t^2$ gives \\eqref{eq:sextic}. The map is rational for $AB\\ne0$. It does not on its own verify a twist, integral frame, or stable model. Their general theorems whose hypotheses exclude seven cannot be applied at seven merely because the family is the same. The versioned theorem hypotheses are recorded in the literature report.","n}\nCompleting the square with $Z=2y+X^3+B$ gives\n\\begin{equation}\\label{eq:sextic}\nZ^2=f_{A,B}(X)=X^6+10BX^3+12AX+5B^2.\n\\end{equation}\nThis is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as $(a,b,c,p)=(B,-C,A,7)$ \\cite{pacetti-villagra2025}. Explicitly, put $t=-B^5/A^3$, $d=-B^2/A$, and substitute $x_{\\mathrm{PV}}=d/X$, $y_{\\mathrm{PV}}=tZ/X^3$ into their equation $y^2=5x^6-12x^5+10tx^3+t^2$. Multiplication by $X^6/t^2$ gives \\eqref{eq:sextic}. The map is rational for $AB\\ne0$. It does not on its own verify a twist, integral frame, or stable model. Their general theorems whose hypotheses exclude seven cannot be applied at seven merely because the family is the same. The versioned theorem hypotheses are recorded in the literature report.","the square with $Z=2y+X^3+B$ gives\n\\begin{equation}\\label{eq:sextic}\nZ^2=f_{A,B}(X)=X^6+10BX^3+12AX+5B^2.\n\\end{equation}\nThis is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as $(a,b,c,p)=(B,-C,A,7)$ \\cite{pacetti-villagra2025}. Explicitly, put $t=-B^5/A^3$, $d=-B^2/A$, and substitute $x_{\\mathrm{PV}}=d/X$, $y_{\\mathrm{PV}}=tZ/X^3$ into their equation $y^2=5x^6-12x^5+10tx^3+t^2$. Multiplication by $X^6/t^2$ gives \\eqref{eq:sextic}. The map is rational for $AB\\ne0$. It does not on its own verify a twist, integral frame, or stable model. Their general theorems whose hypotheses exclude seven cannot be applied at seven merely because the family is the same. The versioned theorem hypotheses are recorded in the literature report.","$ gives\n\\begin{equation}\\label{eq:sextic}\nZ^2=f_{A,B}(X)=X^6+10BX^3+12AX+5B^2.\n\\end{equation}\nThis is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as $(a,b,c,p)=(B,-C,A,7)$ \\cite{pacetti-villagra2025}. Explicitly, put $t=-B^5/A^3$, $d=-B^2/A$, and substitute $x_{\\mathrm{PV}}=d/X$, $y_{\\mathrm{PV}}=tZ/X^3$ into their equation $y^2=5x^6-12x^5+10tx^3+t^2$. Multiplication by $X^6/t^2$ gives \\eqref{eq:sextic}. The map is rational for $AB\\ne0$. It does not on its own verify a twist, integral frame, or stable model. Their general theorems whose hypotheses exclude seven cannot be applied at seven merely because the family is the same. The versioned theorem hypotheses are recorded in the literature report.","}\\label{eq:sextic}\nZ^2=f_{A,B}(X)=X^6+10BX^3+12AX+5B^2.\n\\end{equation}\nThis is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as $(a,b,c,p)=(B,-C,A,7)$ \\cite{pacetti-villagra2025}. Explicitly, put $t=-B^5/A^3$, $d=-B^2/A$, and substitute $x_{\\mathrm{PV}}=d/X$, $y_{\\mathrm{PV}}=tZ/X^3$ into their equation $y^2=5x^6-12x^5+10tx^3+t^2$. Multiplication by $X^6/t^2$ gives \\eqref{eq:sextic}. The map is rational for $AB\\ne0$. It does not on its own verify a twist, integral frame, or stable model. Their general theorems whose hypotheses exclude seven cannot be applied at seven merely because the family is the same. The versioned theorem hypotheses are recorded in the literature report.","12AX+5B^2.\n\\end{equation}\nThis is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as $(a,b,c,p)=(B,-C,A,7)$ \\cite{pacetti-villagra2025}. Explicitly, put $t=-B^5/A^3$, $d=-B^2/A$, and substitute $x_{\\mathrm{PV}}=d/X$, $y_{\\mathrm{PV}}=tZ/X^3$ into their equation $y^2=5x^6-12x^5+10tx^3+t^2$. Multiplication by $X^6/t^2$ gives \\eqref{eq:sextic}. The map is rational for $AB\\ne0$. It does not on its own verify a twist, integral frame, or stable model. Their general theorems whose hypotheses exclude seven cannot be applied at seven merely because the family is the same. The versioned theorem hypotheses are recorded in the literature report.","family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as $(a,b,c,p)=(B,-C,A,7)$ \\cite{pacetti-villagra2025}. Explicitly, put $t=-B^5/A^3$, $d=-B^2/A$, and substitute $x_{\\mathrm{PV}}=d/X$, $y_{\\mathrm{PV}}=tZ/X^3$ into their equation $y^2=5x^6-12x^5+10tx^3+t^2$. Multiplication by $X^6/t^2$ gives \\eqref{eq:sextic}. The map is rational for $AB\\ne0$. It does not on its own verify a twist, integral frame, or stable model. Their general theorems whose hypotheses exclude seven cannot be applied at seven merely because the family is the same. The versioned theorem hypotheses are recorded in the literature report.","he family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as $(a,b,c,p)=(B,-C,A,7)$ \\cite{pacetti-villagra2025}. Explicitly, put $t=-B^5/A^3$, $d=-B^2/A$, and substitute $x_{\\mathrm{PV}}=d/X$, $y_{\\mathrm{PV}}=tZ/X^3$ into their equation $y^2=5x^6-12x^5+10tx^3+t^2$. Multiplication by $X^6/t^2$ gives \\eqref{eq:sextic}. The map is rational for $AB\\ne0$. It does not on its own verify a twist, integral frame, or stable model. Their general theorems whose hypotheses exclude seven cannot be applied at seven merely because the family is the same. The versioned theorem hypotheses are recorded in the literature report.","\\begin{proposition}[Polynomial discriminant]\\label{prop:disc}\nThe sextic in \\eqref{eq:sextic} satisfies\n\\[\n\\operatorname{disc}(f_{A,B})=2^{12}3^65^5(A^3+B^5)^2.\n\\]\nFor a hypothetical $(3,5,7)$ solution this becomes $2^{12}3^65^5C^{14}$.\n\\end{proposition}\n\\begin{proof}\nGive $X,A,B$ weights $1,5,3$. Then\n\\[\nf_{\\lambda^5A,\\lambda^3B}(\\lambda X)=\\lambda^6f_{A,B}(X).\n\\]\nThe six roots scale by $\\lambda$, so their fifteen squared differences give discriminant weight thirty. The only nonnegative integer solutions of $5i+3j=30$ are $(i,j)=(6,0),(3,5),(0,10)$. The discriminant therefore has the form $uA^6+vA^3B^5+wB^{10}$.","rname{disc}(f_{A,B})=2^{12}3^65^5(A^3+B^5)^2.\n\\]\nFor a hypothetical $(3,5,7)$ solution this becomes $2^{12}3^65^5C^{14}$.\n\\end{proposition}\n\\begin{proof}\nGive $X,A,B$ weights $1,5,3$. Then\n\\[\nf_{\\lambda^5A,\\lambda^3B}(\\lambda X)=\\lambda^6f_{A,B}(X).\n\\]\nThe six roots scale by $\\lambda$, so their fifteen squared differences give discriminant weight thirty. The only nonnegative integer solutions of $5i+3j=30$ are $(i,j)=(6,0),(3,5),(0,10)$. The discriminant therefore has the form $uA^6+vA^3B^5+wB^{10}$.","A,B})=2^{12}3^65^5(A^3+B^5)^2.\n\\]\nFor a hypothetical $(3,5,7)$ solution this becomes $2^{12}3^65^5C^{14}$.\n\\end{proposition}\n\\begin{proof}\nGive $X,A,B$ weights $1,5,3$. Then\n\\[\nf_{\\lambda^5A,\\lambda^3B}(\\lambda X)=\\lambda^6f_{A,B}(X).\n\\]\nThe six roots scale by $\\lambda$, so their fifteen squared differences give discriminant weight thirty. The only nonnegative integer solutions of $5i+3j=30$ are $(i,j)=(6,0),(3,5),(0,10)$. The discriminant therefore has the form $uA^6+vA^3B^5+wB^{10}$.","ion this becomes $2^{12}3^65^5C^{14}$.\n\\end{proposition}\n\\begin{proof}\nGive $X,A,B$ weights $1,5,3$. Then\n\\[\nf_{\\lambda^5A,\\lambda^3B}(\\lambda X)=\\lambda^6f_{A,B}(X).\n\\]\nThe six roots scale by $\\lambda$, so their fifteen squared differences give discriminant weight thirty. The only nonnegative integer solutions of $5i+3j=30$ are $(i,j)=(6,0),(3,5),(0,10)$. The discriminant therefore has the form $uA^6+vA^3B^5+wB^{10}$.","Exact Sylvester determinants at $(A,B)=(1,0),(0,1),(1,1)$ give respectively $K,K,4K$, where $K=9\\,331\\,200\\,000=2^{12}3^65^5$. These values force $u=w=K$ and $v=2K$. This proves the identity in the entire polynomial ring. The finite calculation determines three coefficients in a proved three dimensional monomial space, rather than extrapolating a formula from a sample.\n\\end{proof}\nThis is the discriminant of a displayed polynomial. It has not been identified with a minimal discriminant, a field discriminant, or the conductor of the Jacobian. What it does show directly is that, outside $2,3,5$, every possible bad reduction prime divides $C$. At a prime at least seven not dividing $C$, the sextic has distinct roots and supplies a good reduction mod","Exact Sylvester determinants at $(A,B)=(1,0),(0,1),(1,1)$ give respectively $K,K,4K$, where $K=9\\,331\\,200\\,000=2^{12}3^65^5$. These values force $u=w=K$ and $v=2K$. This proves the identity in the entire polynomial ring. The finite calculation determines three coefficients in a proved three dimensional monomial space, rather than extrapolating a formula from a sample.\n\\end{proof}\nThis is the discriminant of a displayed polynomial. It has not been identified with a minimal discriminant, a field discriminant, or the conductor of the Jacobian. What it does show directly is that, outside $2,3,5$, every possible bad reduction prime divides $C$. At a prime at least seven not dividing $C$, the sextic has distinct roots and supplies a good reduction model.","the entire polynomial ring. The finite calculation determines three coefficients in a proved three dimensional monomial space, rather than extrapolating a formula from a sample.\n\\end{proof}\nThis is the discriminant of a displayed polynomial. It has not been identified with a minimal discriminant, a field discriminant, or the conductor of the Jacobian. What it does show directly is that, outside $2,3,5$, every possible bad reduction prime divides $C$. At a prime at least seven not dividing $C$, the sextic has distinct roots and supplies a good reduction model.","determines three coefficients in a proved three dimensional monomial space, rather than extrapolating a formula from a sample.\n\\end{proof}\nThis is the discriminant of a displayed polynomial. It has not been identified with a minimal discriminant, a field discriminant, or the conductor of the Jacobian. What it does show directly is that, outside $2,3,5$, every possible bad reduction prime divides $C$. At a prime at least seven not dividing $C$, the sextic has distinct roots and supplies a good reduction model.","ree dimensional monomial space, rather than extrapolating a formula from a sample.\n\\end{proof}\nThis is the discriminant of a displayed polynomial. It has not been identified with a minimal discriminant, a field discriminant, or the conductor of the Jacobian. What it does show directly is that, outside $2,3,5$, every possible bad reduction prime divides $C$. At a prime at least seven not dividing $C$, the sextic has distinct roots and supplies a good reduction model.","At a variable prime dividing $C$, primitivity makes $A,B$ units. In the residue field put $u=-B^2/A$. Then $u^3=B$, $u^5=-A$, and\n\\begin{equation}\\label{eq:nodalfactor}\nf_{A,B}(uX)=u^6(X^2+X-1)^2(X^2-2X+5).\n\\end{equation}\nThe discriminants of the two quadratics are $5$ and $-16$, and their resultant is $45$. Away from $2,3,5$, the special fiber has exactly two ordinary nodes, rational normalization, and smooth infinity. This factorization is the local geometric entry point for the period and monodromy calculations.","\\subsection{A characteristic seven cancellation}\nIn characteristic seven, the Cartier coefficient calculation comes from $f^3$. Its relevant coefficients are\n\\begin{align*}\n[X^6]f^3&=1575B^4,& [X^5]f^3&=4320A^2B,","\\subsection{A characteristic seven cancellation}\nIn characteristic seven, the Cartier coefficient calculation comes from $f^3$. Its relevant coefficients are\n\\begin{align*}\n[X^6]f^3&=1575B^4,& [X^5]f^3&=4320A^2B,\\\\\n[X^{13}]f^3&=36A,& [X^{12}]f^3&=315B^2.\n\\end{align*}\nThe diagonal lanes vanish modulo seven and the off diagonal lanes become\n\\begin{equation}\\label{eq:cartier}\nM=\\begin{pmatrix}0&A^2B","[X^{13}]f^3&=36A,& [X^{12}]f^3&=315B^2.\n\\end{align*}\nThe diagonal lanes vanish modulo seven and the off diagonal lanes become\n\\begin{equation}\\label{eq:cartier}\nM=\\begin{pmatrix}0&A^2B\\\\A&0\\end{pmatrix},\\qquad M^2=A^3B I_2.\n\\end{equation}\nThese are polynomial and matrix identities. Their geometric interpretation requires the correct Cartier and Hasse--Witt conventions. On a good reduction fiber at seven they imply ordinarity when $AB$ is nonzero modulo seven and seven rank zero when $AB$ is zero. There is no rank one stratum in this calculation. The retained sources and the formal report distinguish these algebraic statements from the associated finite flat and Frobenius applications.","A&0\\end{pmatrix},\\qquad M^2=A^3B I_2.\n\\end{equation}\nThese are polynomial and matrix identities. Their geometric interpretation requires the correct Cartier and Hasse--Witt conventions. On a good reduction fiber at seven they imply ordinarity when $AB$ is nonzero modulo seven and seven rank zero when $AB$ is zero. There is no rank one stratum in this calculation. The retained sources and the formal report distinguish these algebraic statements from the associated finite flat and Frobenius applications.","\\subsection{What the direct irreducibility argument proves}\nThe canonical recovered result is absolute irreducibility of the associated residual rank two RM representation when $14\\nmid C$, subject to its declared realization, conductor, local weight, and finite flat inputs. Its two branches use different information.","When $C$ is odd, the integral model has good reduction at two. Over the norm four prime of $K=\\Q(\\sqrt5)$, the full Frobenius polynomial is either $(T^2+T+4)^2$ or $(T^2+4)^2$. A hypothetical degree one constituent, after the conductor and global unit argument, must have eighth power one. Neither quadratic has a root with eighth power one modulo seven. For $T^2+4$, a root has square three, hence eighth power four. For $T^2+T+4$, reduction of powers gives fourth power five, again eighth power four. The contradiction uses the full polynomial and the same global constituent.","tegral model has good reduction at two. Over the norm four prime of $K=\\Q(\\sqrt5)$, the full Frobenius polynomial is either $(T^2+T+4)^2$ or $(T^2+4)^2$. A hypothetical degree one constituent, after the conductor and global unit argument, must have eighth power one. Neither quadratic has a root with eighth power one modulo seven. For $T^2+4$, a root has square three, hence eighth power four. For $T^2+T+4$, reduction of powers gives fourth power five, again eighth power four. The contradiction uses the full polynomial and the same global constituent.","The global unit step uses $\\phi=(1+\\sqrt5)/2$, whose residue has order sixteen in $\\F_{49}^\\times$. The local weight possibilities at seven are $0,1,7,8$, up to a common reciprocity sign. Applying reciprocity to the eighth power of each constituent gives $\\overline\\phi^{8e}=1$ and rules out the mixed weights $1,7$. The determinant norm relation $\\delta^8=\\chi^2$ then leaves one constituent unramified at seven, without assuming an unproved rank two determinant normalization in this argument. Its eighth power is everywhere unramified, and class number one makes it trivial. This is why the local quadratic contradiction applies to that constituent.","When $C$ is even and seven does not divide $C$, the two adic torus calculation replaces good reduction at two. The global character has Frobenius value $-1$ at two. Reciprocity on $\\phi^4$ and $2/7$ transports that value to seven. The finite flat comparison at good seven, together with \\eqref{eq:cartier}, gives the value $A^3B$. Thus $A^3B=-1$ modulo seven. Since both bases are nonzero, $B=-A^3$ and $A^3+B^5=0$ modulo seven, contradicting $7\\nmid C$.","$C$ is even and seven does not divide $C$, the two adic torus calculation replaces good reduction at two. The global character has Frobenius value $-1$ at two. Reciprocity on $\\phi^4$ and $2/7$ transports that value to seven. The finite flat comparison at good seven, together with \\eqref{eq:cartier}, gives the value $A^3B$. Thus $A^3B=-1$ modulo seven. Since both bases are nonzero, $B=-A^3$ and $A^3+B^5=0$ modulo seven, contradicting $7\\nmid C$.","C$, the two adic torus calculation replaces good reduction at two. The global character has Frobenius value $-1$ at two. Reciprocity on $\\phi^4$ and $2/7$ transports that value to seven. The finite flat comparison at good seven, together with \\eqref{eq:cartier}, gives the value $A^3B$. Thus $A^3B=-1$ modulo seven. Since both bases are nonzero, $B=-A^3$ and $A^3+B^5=0$ modulo seven, contradicting $7\\nmid C$.","The conclusion removes an obstruction to a modular method. It does not eliminate the modular candidates. Its uncovered parity and divisibility branch is precisely $14\\mid C$. The later reducible research concerns a further stipulated branch with $3990\\mid C$, retaining the same curve and its specified twist.","Suppose a Hecke eigenvalue $a_{\\mathfrak q}$ is replaced by its minimal polynomial. A resultant against a possible curve trace polynomial asks whether some conjugate can match some allowable trace. Repeating the test at a different prime can choose a different conjugate. If the original representation requires one residue embedding for every prime, this relaxed sequence of tests may preserve candidates that the labelled comparison would exclude. An exclusion obtained from the relaxation remains valid. Survival becomes less informative.","The Block 47 audit identified this mechanism in the pinned Pacetti--Villagra Torcomian code. It also identified a parameter dictionary issue at eleven: the unlabelled multisets agreed, but individual labels matched after $t\\mapsto t^{-1}$. A refinement attached to the wrong labels could incorrectly combine local constraints. The new reader therefore distinguishes direct point counts on \\eqref{eq:sextic} from imported hypergeometric parameter tables.","Two exact local restrictions were established under the stated modular and multiplicative prime comparisons. A rank two trace zero comparison at a prime above eleven forces $11\\mid A$. At a prime above nineteen it forces $19\\mid B$. In either case the full four dimensional reduction to check is\n\\[\n(T^2+q)^2\\pmod7,\n\\]\nwhich is stronger than zero four dimensional trace. At bad $q$, the multiplicative comparison gives $\\pm(q+1)$, nonzero modulo seven for these two primes. At good $q$, the direct curve table selects the stated base zero stratum. These are divisibility restrictions, not contradictions.","rictions were established under the stated modular and multiplicative prime comparisons. A rank two trace zero comparison at a prime above eleven forces $11\\mid A$. At a prime above nineteen it forces $19\\mid B$. In either case the full four dimensional reduction to check is\n\\[\n(T^2+q)^2\\pmod7,\n\\]\nwhich is stronger than zero four dimensional trace. At bad $q$, the multiplicative comparison gives $\\pm(q+1)$, nonzero modulo seven for these two primes. At good $q$, the direct curve table selects the stated base zero stratum. These are divisibility restrictions, not contradictions.","cative prime comparisons. A rank two trace zero comparison at a prime above eleven forces $11\\mid A$. At a prime above nineteen it forces $19\\mid B$. In either case the full four dimensional reduction to check is\n\\[\n(T^2+q)^2\\pmod7,\n\\]\nwhich is stronger than zero four dimensional trace. At bad $q$, the multiplicative comparison gives $\\pm(q+1)$, nonzero modulo seven for these two primes. At good $q$, the direct curve table selects the stated base zero stratum. These are divisibility restrictions, not contradictions.","\\subsection{From point counts to a common polynomial}\nFor a smooth genus two curve over $\\F_q$, let $N_1,N_2$ count its points over $\\F_q,\\F_{q^2}$, including infinity. Write\n\\[\ns_1=q+1-N_1,\\qquad\ns_2=\\frac{s_1^2-(q^2+1-N_2)}2.\n\\]\nIts Frobenius polynomial is\n\\begin{equation}\\label{eq:quartic}\nT^4-s_1T^3+s_2T^2-qs_1T+q^2.\n\\end{equation}\nIn the RM setting, the rank two traces are the roots of\n\\[\nX^2-s_1X+(s_2-2q).\n\\]\nThe two roots belong to one common quartic. Choosing one root from one fiber and another root from an unrelated fiber can enlarge a necessary locus, but does not give a compatible simultaneous reduction of the original curve. The later Frobenius sieves explicitly retain this common polynomial condition.","\\subsection{The remaining modular branches}\nBlock 48 and the accumulated Block 49 notes retain conditional local type and divisibility reductions, including the branch $3990\\mid C$. They do not certify every Hilbert modular candidate as eliminated. The imported authors' transcripts have their own index conventions and some aggregate exception lists do not identify a complete seven specific set. A form index in such a transcript is not an intrinsic arithmetic identifier.","\\section{The natural lattice and the depth of an opposite extension}\\label{sec:depth}\nThe reducible branch uses two different kinds of radical. The base field is\n\\[\nK=\\Q(\\sqrt5),\\qquad\\phi=(1+\\sqrt5)/2.\n\\]\nThe coefficient ring and field are\n\\[\n\\Oloc=\\Z_7[w]/(w^2-5),\\qquad k=\\F_7[w]/(w^2-5)=\\F_{49}.\n\\]\nThe base unit $\\phi$ defines Kummer classes. The coefficient $w$ acts on a rank two representation. Although both involve a square root of five, they have different roles and are never identified by an implicit residue substitution.","The inherited characteristic zero representation is the specified RM factor of the twisted Jacobian on the retained $3990\\mid C$ branch, with determinant the cyclotomic character $\\chi$. Its natural residual orientation is\n\\[\n0\\longrightarrow k(\\chi)\\longrightarrow\\overline V\\longrightarrow k\\longrightarrow0.\n\\]\nThe opposite endpoint has a nonzero class $\\beta\\in H^1(K,k(\\chi^{-1}))$. A stable lattice change can alter residual inertia. Therefore properties of the natural Jacobian lattice cannot automatically be transferred to an arbitrary lattice supplied by Ribet's lemma \\cite{ophir-weiss2024}.","\\subsection{Why a new eligible prime is necessary}\nThe opposite orientation forces an additional prime divisor of $C$. Here is the conditional argument, retaining the class group and local geometric inputs. Suppose a characteristic zero local toric line has reduction $\\chi$, while the opposite residual extension has subline one and quotient $\\chi$. Intersect the actual toric line with its stable lattice. The intersection is saturated, so it reduces to a nonzero local $\\chi$ line. If the local character $\\chi$ is nontrivial, that line is distinct from the global subline one. They split the residual extension locally. This proof survives stable lattice changes because it starts with the characteristic zero line.","on forces an additional prime divisor of $C$. Here is the conditional argument, retaining the class group and local geometric inputs. Suppose a characteristic zero local toric line has reduction $\\chi$, while the opposite residual extension has subline one and quotient $\\chi$. Intersect the actual toric line with its stable lattice. The intersection is saturated, so it reduces to a nonzero local $\\chi$ line. If the local character $\\chi$ is nontrivial, that line is distinct from the global subline one. They split the residual extension locally. This proof survives stable lattice changes because it starts with the characteristic zero line.","r of $C$. Here is the conditional argument, retaining the class group and local geometric inputs. Suppose a characteristic zero local toric line has reduction $\\chi$, while the opposite residual extension has subline one and quotient $\\chi$. Intersect the actual toric line with its stable lattice. The intersection is saturated, so it reduces to a nonzero local $\\chi$ line. If the local character $\\chi$ is nontrivial, that line is distinct from the global subline one. They split the residual extension locally. This proof survives stable lattice changes because it starts with the characteristic zero line.","At seven, the specified twist makes the toric characters $\\chi$ and one, and cyclotomic inertia is nontrivial. At a variable bad place with $Q\\not\\equiv1\\pmod7$, the same argument applies. At the places over two, three and five, a separate local cohomology calculation excludes opposite extensions. For a non seven local field of residue cardinality $Q$, an inertia cocycle with coefficients $k(\\chi^{-1})$ can be nonzero only when $Q^2\\equiv1\\pmod7$: the tame Frobenius relation multiplies its inertia value by $Q$, while the coefficient action multiplies it by $Q^{-1}$. The unramified quotient is zero when $Q\\not\\equiv1$. Thus $Q\\not\\equiv\\pm1$ forces local $H^1$ to vanish. Here the relevant norms are four, nine and five, all outside those two residue clas","At seven, the specified twist makes the toric characters $\\chi$ and one, and cyclotomic inertia is nontrivial. At a variable bad place with $Q\\not\\equiv1\\pmod7$, the same argument applies. At the places over two, three and five, a separate local cohomology calculation excludes opposite extensions. For a non seven local field of residue cardinality $Q$, an inertia cocycle with coefficients $k(\\chi^{-1})$ can be nonzero only when $Q^2\\equiv1\\pmod7$: the tame Frobenius relation multiplies its inertia value by $Q$, while the coefficient action multiplies it by $Q^{-1}$. The unramified quotient is zero when $Q\\not\\equiv1$. Thus $Q\\not\\equiv\\pm1$ forces local $H^1$ to vanish. Here the relevant norms are four, nine and five, all outside those two residue classes.","es the toric characters $\\chi$ and one, and cyclotomic inertia is nontrivial. At a variable bad place with $Q\\not\\equiv1\\pmod7$, the same argument applies. At the places over two, three and five, a separate local cohomology calculation excludes opposite extensions. For a non seven local field of residue cardinality $Q$, an inertia cocycle with coefficients $k(\\chi^{-1})$ can be nonzero only when $Q^2\\equiv1\\pmod7$: the tame Frobenius relation multiplies its inertia value by $Q$, while the coefficient action multiplies it by $Q^{-1}$. The unramified quotient is zero when $Q\\not\\equiv1$. Thus $Q\\not\\equiv\\pm1$ forces local $H^1$ to vanish. Here the relevant norms are four, nine and five, all outside those two residue classes.","a variable bad place with $Q\\not\\equiv1\\pmod7$, the same argument applies. At the places over two, three and five, a separate local cohomology calculation excludes opposite extensions. For a non seven local field of residue cardinality $Q$, an inertia cocycle with coefficients $k(\\chi^{-1})$ can be nonzero only when $Q^2\\equiv1\\pmod7$: the tame Frobenius relation multiplies its inertia value by $Q$, while the coefficient action multiplies it by $Q^{-1}$. The unramified quotient is zero when $Q\\not\\equiv1$. Thus $Q\\not\\equiv\\pm1$ forces local $H^1$ to vanish. Here the relevant norms are four, nine and five, all outside those two residue classes.","e same argument applies. At the places over two, three and five, a separate local cohomology calculation excludes opposite extensions. For a non seven local field of residue cardinality $Q$, an inertia cocycle with coefficients $k(\\chi^{-1})$ can be nonzero only when $Q^2\\equiv1\\pmod7$: the tame Frobenius relation multiplies its inertia value by $Q$, while the coefficient action multiplies it by $Q^{-1}$. The unramified quotient is zero when $Q\\not\\equiv1$. Thus $Q\\not\\equiv\\pm1$ forces local $H^1$ to vanish. Here the relevant norms are four, nine and five, all outside those two residue classes.",", three and five, a separate local cohomology calculation excludes opposite extensions. For a non seven local field of residue cardinality $Q$, an inertia cocycle with coefficients $k(\\chi^{-1})$ can be nonzero only when $Q^2\\equiv1\\pmod7$: the tame Frobenius relation multiplies its inertia value by $Q$, while the coefficient action multiplies it by $Q^{-1}$. The unramified quotient is zero when $Q\\not\\equiv1$. Thus $Q\\not\\equiv\\pm1$ forces local $H^1$ to vanish. Here the relevant norms are four, nine and five, all outside those two residue classes.","eparate local cohomology calculation excludes opposite extensions. For a non seven local field of residue cardinality $Q$, an inertia cocycle with coefficients $k(\\chi^{-1})$ can be nonzero only when $Q^2\\equiv1\\pmod7$: the tame Frobenius relation multiplies its inertia value by $Q$, while the coefficient action multiplies it by $Q^{-1}$. The unramified quotient is zero when $Q\\not\\equiv1$. Thus $Q\\not\\equiv\\pm1$ forces local $H^1$ to vanish. Here the relevant norms are four, nine and five, all outside those two residue classes.","ation excludes opposite extensions. For a non seven local field of residue cardinality $Q$, an inertia cocycle with coefficients $k(\\chi^{-1})$ can be nonzero only when $Q^2\\equiv1\\pmod7$: the tame Frobenius relation multiplies its inertia value by $Q$, while the coefficient action multiplies it by $Q^{-1}$. The unramified quotient is zero when $Q\\not\\equiv1$. Thus $Q\\not\\equiv\\pm1$ forces local $H^1$ to vanish. Here the relevant norms are four, nine and five, all outside those two residue classes.","If no variable $C$ prime had $Q\\equiv1\\pmod7$, beta would restrict to an everywhere unramified additive class over $L=K(\\zeta_7)$. Restriction is injective because $[L:K]=6$ is prime to seven. The inherited Masley class number result gives $h(\\Q(\\zeta_{35}))=1$ \\cite{masley1976}. The field $L$ is a subfield of index two. Extension and norm on ideal classes compose to multiplication by two, so the seven primary class group of $L$ is zero. An everywhere unramified nonzero additive seven class is therefore impossible by class field theory.","Q\\equiv1\\pmod7$, beta would restrict to an everywhere unramified additive class over $L=K(\\zeta_7)$. Restriction is injective because $[L:K]=6$ is prime to seven. The inherited Masley class number result gives $h(\\Q(\\zeta_{35}))=1$ \\cite{masley1976}. The field $L$ is a subfield of index two. Extension and norm on ideal classes compose to multiplication by two, so the seven primary class group of $L$ is zero. An everywhere unramified nonzero additive seven class is therefore impossible by class field theory.","At least one variable prime above a divisor of $C$ must consequently have norm one modulo seven. A rational prime splits in $K$ when it is one or four modulo five, giving residue degree one. At an inert prime the degree is two. Combining $q^f\\equiv1\\pmod7$ with these splitting possibilities gives exactly\n\\[\nq\\pmod{35}\\in\\{1,8,13,22,27,29\\}.\n\\]\nNone of $2,3,5,7,19$, the primes already in $3990$, supplies this new variable prime. The conclusion asks for an eligible additional prime, not divisibility by a particular fixed prime. Once such primes are allowed, their ray class columns can support beta. Ordinary class number one does not make that ramified character space vanish.","\\subsection{An actual finite segment of stable lattices}\nThe inherited split toric calculation at three supplies Frobenius eigenvalues nine and one. Their difference eight is a seven adic unit. The projectors\n\\[\nE_9=\\frac{\\rho(g_0)-I}{8},\\qquad E_1=\\frac{9I-\\rho(g_0)}8\n\\]\nare complementary integral idempotents. Choose generators of their images. In this basis, $\\rho(g_0)=\\operatorname{diag}(9,1)$ and every lower entry is divisible by seven. Natural upper monodromy at three has a unit entry. Absolute irreducibility in characteristic zero forces some lower entry to be nonzero.","Define\n\\begin{equation}\\label{eq:actualdepth}\nn=\\min_{g:\\rho(g)_{21}\\ne0}v_7(\\rho(g)_{21}).\n\\end{equation}\nThis positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every $\\Oloc$ lattice stable under $\\rho$ is preserved by the two projectors, so up to homothety it has the form\n\\[\n\\Lambda_j=\\Oloc e_1+\\Oloc 7^j e_2.\n\\]\nThe unit upper monodromy forces $j\\ge0$. The attained lower entry minimum forces $j\\le n$. Conversely every $j$ in that interval is stable. Thus the stable lattices form a finite segment with the natural and opposite endpoints at $0,n$.","n}\\label{eq:actualdepth}\nn=\\min_{g:\\rho(g)_{21}\\ne0}v_7(\\rho(g)_{21}).\n\\end{equation}\nThis positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every $\\Oloc$ lattice stable under $\\rho$ is preserved by the two projectors, so up to homothety it has the form\n\\[\n\\Lambda_j=\\Oloc e_1+\\Oloc 7^j e_2.\n\\]\nThe unit upper monodromy forces $j\\ge0$. The attained lower entry minimum forces $j\\le n$. Conversely every $j$ in that interval is stable. Thus the stable lattices form a finite segment with the natural and opposite endpoints at $0,n$.","e0}v_7(\\rho(g)_{21}).\n\\end{equation}\nThis positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every $\\Oloc$ lattice stable under $\\rho$ is preserved by the two projectors, so up to homothety it has the form\n\\[\n\\Lambda_j=\\Oloc e_1+\\Oloc 7^j e_2.\n\\]\nThe unit upper monodromy forces $j\\ge0$. The attained lower entry minimum forces $j\\le n$. Conversely every $j$ in that interval is stable. Thus the stable lattices form a finite segment with the natural and opposite endpoints at $0,n$.","equation}\nThis positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every $\\Oloc$ lattice stable under $\\rho$ is preserved by the two projectors, so up to homothety it has the form\n\\[\n\\Lambda_j=\\Oloc e_1+\\Oloc 7^j e_2.\n\\]\nThe unit upper monodromy forces $j\\ge0$. The attained lower entry minimum forces $j\\le n$. Conversely every $j$ in that interval is stable. Thus the stable lattices form a finite segment with the natural and opposite endpoints at $0,n$.","a computation. Every $\\Oloc$ lattice stable under $\\rho$ is preserved by the two projectors, so up to homothety it has the form\n\\[\n\\Lambda_j=\\Oloc e_1+\\Oloc 7^j e_2.\n\\]\nThe unit upper monodromy forces $j\\ge0$. The attained lower entry minimum forces $j\\le n$. Conversely every $j$ in that interval is stable. Thus the stable lattices form a finite segment with the natural and opposite endpoints at $0,n$.","At the opposite endpoint, the attained minimum gives a nonzero lower residual entry. A split complement would have to be the first coordinate line, because $g_0$ has distinct residual eigenvalues. That line is not stable. Hence the opposite residual extension is nonsplit. This constructs the relevant class on a compatible endpoint rather than importing natural inertia through an unjustified lattice change.","\\subsection{A local depth theorem}\nHere is the matrix argument controlling where this opposite class can ramify. Let $\\Oloc$ be a discrete valuation ring with uniformizer $\\pi$, let $F\\in\\mathrm{GL}_2(\\Oloc)$, $Q\\in\\Oloc^\\times$ with $Q\\ne1$, and let $n\\ge1$. Suppose $N\\in M_2(\\Oloc)$ is a nonzero nilpotent matrix of the form\n\\[\nN=\\mu\\begin{pmatrix}-xy&x^2","\\subsection{A local depth theorem}\nHere is the matrix argument controlling where this opposite class can ramify. Let $\\Oloc$ be a discrete valuation ring with uniformizer $\\pi$, let $F\\in\\mathrm{GL}_2(\\Oloc)$, $Q\\in\\Oloc^\\times$ with $Q\\ne1$, and let $n\\ge1$. Suppose $N\\in M_2(\\Oloc)$ is a nonzero nilpotent matrix of the form\n\\[\nN=\\mu\\begin{pmatrix}-xy&x^2\\\\-y^2&xy\\end{pmatrix},\n\\]\nwhere $\\mu\\in\\Oloc\\setminus\\{0\\}$, $(x,y)\\in\\Oloc^2$ is primitive, $t=v(\\mu)\\ge0$, and\n\\[\nFN=QN,\\qquad NF=N,\\qquad F_{21}\\in\\pi^n\\Oloc,\\qquad v(N_{21})=n.\n\\]\nSet $h=v(Q-1)$. Then\n\\begin{equation}\\label{eq:depthlocal}\nn=t+2s,\\qquad s=v(y),\\qquad0\\le s\\le h.\n\\end{equation}\n\\begin{proof}\nThe lower nilpotent entry is $-\\mu y^2$, so its valuation gives $n=t+2s$. The common image and kernel line of $N$ is generated by $(x,y)^T$. The two identities for $F$ show that $F-I$ has image in this line and kills no extra quotient direction. Write\n\\[\nF=I+\\begin{pmatrix}x","-y^2&xy\\end{pmatrix},\n\\]\nwhere $\\mu\\in\\Oloc\\setminus\\{0\\}$, $(x,y)\\in\\Oloc^2$ is primitive, $t=v(\\mu)\\ge0$, and\n\\[\nFN=QN,\\qquad NF=N,\\qquad F_{21}\\in\\pi^n\\Oloc,\\qquad v(N_{21})=n.\n\\]\nSet $h=v(Q-1)$. Then\n\\begin{equation}\\label{eq:depthlocal}\nn=t+2s,\\qquad s=v(y),\\qquad0\\le s\\le h.\n\\end{equation}\n\\begin{proof}\nThe lower nilpotent entry is $-\\mu y^2$, so its valuation gives $n=t+2s$. The common image and kernel line of $N$ is generated by $(x,y)^T$. The two identities for $F$ show that $F-I$ has image in this line and kills no extra quotient direction. Write\n\\[\nF=I+\\begin{pmatrix}x","\\]\nwhere $\\mu\\in\\Oloc\\setminus\\{0\\}$, $(x,y)\\in\\Oloc^2$ is primitive, $t=v(\\mu)\\ge0$, and\n\\[\nFN=QN,\\qquad NF=N,\\qquad F_{21}\\in\\pi^n\\Oloc,\\qquad v(N_{21})=n.\n\\]\nSet $h=v(Q-1)$. Then\n\\begin{equation}\\label{eq:depthlocal}\nn=t+2s,\\qquad s=v(y),\\qquad0\\le s\\le h.\n\\end{equation}\n\\begin{proof}\nThe lower nilpotent entry is $-\\mu y^2$, so its valuation gives $n=t+2s$. The common image and kernel line of $N$ is generated by $(x,y)^T$. The two identities for $F$ show that $F-I$ has image in this line and kills no extra quotient direction. Write\n\\[\nF=I+\\begin{pmatrix}x","c\\setminus\\{0\\}$, $(x,y)\\in\\Oloc^2$ is primitive, $t=v(\\mu)\\ge0$, and\n\\[\nFN=QN,\\qquad NF=N,\\qquad F_{21}\\in\\pi^n\\Oloc,\\qquad v(N_{21})=n.\n\\]\nSet $h=v(Q-1)$. Then\n\\begin{equation}\\label{eq:depthlocal}\nn=t+2s,\\qquad s=v(y),\\qquad0\\le s\\le h.\n\\end{equation}\n\\begin{proof}\nThe lower nilpotent entry is $-\\mu y^2$, so its valuation gives $n=t+2s$. The common image and kernel line of $N$ is generated by $(x,y)^T$. The two identities for $F$ show that $F-I$ has image in this line and kills no extra quotient direction. Write\n\\[\nF=I+\\begin{pmatrix}x","$ is primitive, $t=v(\\mu)\\ge0$, and\n\\[\nFN=QN,\\qquad NF=N,\\qquad F_{21}\\in\\pi^n\\Oloc,\\qquad v(N_{21})=n.\n\\]\nSet $h=v(Q-1)$. Then\n\\begin{equation}\\label{eq:depthlocal}\nn=t+2s,\\qquad s=v(y),\\qquad0\\le s\\le h.\n\\end{equation}\n\\begin{proof}\nThe lower nilpotent entry is $-\\mu y^2$, so its valuation gives $n=t+2s$. The common image and kernel line of $N$ is generated by $(x,y)^T$. The two identities for $F$ show that $F-I$ has image in this line and kills no extra quotient direction. Write\n\\[\nF=I+\\begin{pmatrix}x",", $t=v(\\mu)\\ge0$, and\n\\[\nFN=QN,\\qquad NF=N,\\qquad F_{21}\\in\\pi^n\\Oloc,\\qquad v(N_{21})=n.\n\\]\nSet $h=v(Q-1)$. Then\n\\begin{equation}\\label{eq:depthlocal}\nn=t+2s,\\qquad s=v(y),\\qquad0\\le s\\le h.\n\\end{equation}\n\\begin{proof}\nThe lower nilpotent entry is $-\\mu y^2$, so its valuation gives $n=t+2s$. The common image and kernel line of $N$ is generated by $(x,y)^T$. The two identities for $F$ show that $F-I$ has image in this line and kills no extra quotient direction. Write\n\\[\nF=I+\\begin{pmatrix}x","d v(N_{21})=n.\n\\]\nSet $h=v(Q-1)$. Then\n\\begin{equation}\\label{eq:depthlocal}\nn=t+2s,\\qquad s=v(y),\\qquad0\\le s\\le h.\n\\end{equation}\n\\begin{proof}\nThe lower nilpotent entry is $-\\mu y^2$, so its valuation gives $n=t+2s$. The common image and kernel line of $N$ is generated by $(x,y)^T$. The two identities for $F$ show that $F-I$ has image in this line and kills no extra quotient direction. Write\n\\[\nF=I+\\begin{pmatrix}x\\\\y\\end{pmatrix}(a,b),\\qquad ax+by=Q-1.\n\\]\nThe lower entry is $ya$, so $a\\in\\pi^{n-s}\\Oloc=\\pi^{t+s}\\Oloc$. Both $ax$ and $by$ are divisible by $\\pi^s$. Their sum $Q-1$ is therefore divisible by $\\pi^s$, giving $s\\le h$.\n\\end{proof}\nThe square on $y$ explains the factor two. A bound $n\\le t+h$ would omit the geometry of the rank one nilpotent and is false. The examples\n\\[\nN=7^t\\begin{pmatrix}-7^s&1","y\\end{pmatrix}(a,b),\\qquad ax+by=Q-1.\n\\]\nThe lower entry is $ya$, so $a\\in\\pi^{n-s}\\Oloc=\\pi^{t+s}\\Oloc$. Both $ax$ and $by$ are divisible by $\\pi^s$. Their sum $Q-1$ is therefore divisible by $\\pi^s$, giving $s\\le h$.\n\\end{proof}\nThe square on $y$ explains the factor two. A bound $n\\le t+h$ would omit the geometry of the rank one nilpotent and is false. The examples\n\\[\nN=7^t\\begin{pmatrix}-7^s&1","y\\end{pmatrix}(a,b),\\qquad ax+by=Q-1.\n\\]\nThe lower entry is $ya$, so $a\\in\\pi^{n-s}\\Oloc=\\pi^{t+s}\\Oloc$. Both $ax$ and $by$ are divisible by $\\pi^s$. Their sum $Q-1$ is therefore divisible by $\\pi^s$, giving $s\\le h$.\n\\end{proof}\nThe square on $y$ explains the factor two. A bound $n\\le t+h$ would omit the geometry of the rank one nilpotent and is false. The examples\n\\[\nN=7^t\\begin{pmatrix}-7^s&1\\\\-7^{2s}&7^s\\end{pmatrix},\\qquad\nF=\\begin{pmatrix}1&(Q-1)/7^s\\\\0&Q\\end{pmatrix}\n\\]\nrealize every allowed $s$ when $v_7(Q-1)=h$. They are integral local matrix controls. They are not automatically global Galois representations or Jacobians.","0&Q\\end{pmatrix}\n\\]\nrealize every allowed $s$ when $v_7(Q-1)=h$. They are integral local matrix controls. They are not automatically global Galois representations or Jacobians.","For the natural curve, the variable node thickness is $7v_q(C)$. The natural monodromy content is consequently\n\\[\nt=1+v_7(v_q(C)).\n\\]\nThis inherited geometric bridge uses the ordinary node smoothing parameter and the identification of the graph pairing with the period valuation \\cite{baker-rabinoff2013}. Applied to an actually ramified beta prime, \\eqref{eq:depthlocal} becomes\n\\begin{equation}\\label{eq:arithdepth}\nn=1+v_7(v_q(C))+2s,\\qquad0\\le s\\le v_7(q^f-1).\n\\end{equation}\nHere $f$ is the residue degree in $K$. If $7\\nmid v_q(C)$ at every eligible prime, the actual depth is odd. If in addition every eligible residue norm is nontrivial modulo forty nine, the only possible depths are one and three. These are necessary conditions within the inherited ar","For the natural curve, the variable node thickness is $7v_q(C)$. The natural monodromy content is consequently\n\\[\nt=1+v_7(v_q(C)).\n\\]\nThis inherited geometric bridge uses the ordinary node smoothing parameter and the identification of the graph pairing with the period valuation \\cite{baker-rabinoff2013}. Applied to an actually ramified beta prime, \\eqref{eq:depthlocal} becomes\n\\begin{equation}\\label{eq:arithdepth}\nn=1+v_7(v_q(C))+2s,\\qquad0\\le s\\le v_7(q^f-1).\n\\end{equation}\nHere $f$ is the residue degree in $K$. If $7\\nmid v_q(C)$ at every eligible prime, the actual depth is odd. If in addition every eligible residue norm is nontrivial modulo forty nine, the only possible depths are one and three. These are necessary conditions within the inherited arithmetic contract, not a uniform depth bound for arbitrary $C$.","content is consequently\n\\[\nt=1+v_7(v_q(C)).\n\\]\nThis inherited geometric bridge uses the ordinary node smoothing parameter and the identification of the graph pairing with the period valuation \\cite{baker-rabinoff2013}. Applied to an actually ramified beta prime, \\eqref{eq:depthlocal} becomes\n\\begin{equation}\\label{eq:arithdepth}\nn=1+v_7(v_q(C))+2s,\\qquad0\\le s\\le v_7(q^f-1).\n\\end{equation}\nHere $f$ is the residue degree in $K$. If $7\\nmid v_q(C)$ at every eligible prime, the actual depth is odd. If in addition every eligible residue norm is nontrivial modulo forty nine, the only possible depths are one and three. These are necessary conditions within the inherited arithmetic contract, not a uniform depth bound for arbitrary $C$.","=1+v_7(v_q(C)).\n\\]\nThis inherited geometric bridge uses the ordinary node smoothing parameter and the identification of the graph pairing with the period valuation \\cite{baker-rabinoff2013}. Applied to an actually ramified beta prime, \\eqref{eq:depthlocal} becomes\n\\begin{equation}\\label{eq:arithdepth}\nn=1+v_7(v_q(C))+2s,\\qquad0\\le s\\le v_7(q^f-1).\n\\end{equation}\nHere $f$ is the residue degree in $K$. If $7\\nmid v_q(C)$ at every eligible prime, the actual depth is odd. If in addition every eligible residue norm is nontrivial modulo forty nine, the only possible depths are one and three. These are necessary conditions within the inherited arithmetic contract, not a uniform depth bound for arbitrary $C$.","_q(C)).\n\\]\nThis inherited geometric bridge uses the ordinary node smoothing parameter and the identification of the graph pairing with the period valuation \\cite{baker-rabinoff2013}. Applied to an actually ramified beta prime, \\eqref{eq:depthlocal} becomes\n\\begin{equation}\\label{eq:arithdepth}\nn=1+v_7(v_q(C))+2s,\\qquad0\\le s\\le v_7(q^f-1).\n\\end{equation}\nHere $f$ is the residue degree in $K$. If $7\\nmid v_q(C)$ at every eligible prime, the actual depth is odd. If in addition every eligible residue norm is nontrivial modulo forty nine, the only possible depths are one and three. These are necessary conditions within the inherited arithmetic contract, not a uniform depth bound for arbitrary $C$.","v_7(v_q(C))+2s,\\qquad0\\le s\\le v_7(q^f-1).\n\\end{equation}\nHere $f$ is the residue degree in $K$. If $7\\nmid v_q(C)$ at every eligible prime, the actual depth is odd. If in addition every eligible residue norm is nontrivial modulo forty nine, the only possible depths are one and three. These are necessary conditions within the inherited arithmetic contract, not a uniform depth bound for arbitrary $C$.","\\subsection{The local cup endpoint}\nAt an eligible beta prime, $t\\ge1$ and $h\\ge1$, so $\\chi$ is trivial modulo seven. Natural inertia is trivial modulo seven, while opposite inertia is nonzero. Below the upper local endpoint $n=t+2h$, the same matrix calculation forces natural Frobenius $\\alpha(F)$ to vanish. At the endpoint it is nonzero.","If $\\alpha\\cup\\beta$ were a coboundary in the endpoint case, the two adjacent entries would extend to a representation in the three by three upper unitriangular group over $k$. That group has exponent seven. The tame relation $FTF^{-1}=T^Q$ with $Q\\equiv1\\pmod7$ would force the images to commute. Their central commutator is $\\alpha(F)\\beta(T)-\\alpha(T)\\beta(F)=\\alpha(F)\\beta(T)\\ne0$. This contradiction proves local nonvanishing at the endpoint. The zero below the endpoint and nonzero at it are statements about the","If $\\alpha\\cup\\beta$ were a coboundary in the endpoint case, the two adjacent entries would extend to a representation in the three by three upper unitriangular group over $k$. That group has exponent seven. The tame relation $FTF^{-1}=T^Q$ with $Q\\equiv1\\pmod7$ would force the images to commute. Their central commutator is $\\alpha(F)\\beta(T)-\\alpha(T)\\beta(F)=\\alpha(F)\\beta(T)\\ne0$. This contradiction proves local nonvanishing at the endpoint. The zero below the endpoint and nonzero at it are statements about the same labelled local representation.","\\section{Global coefficients, diagonal characters, and Frobenius precision}\\label{sec:coefficients}\nThe natural class has a fixed global expansion\n\\begin{equation}\\label{eq:globalalpha}\n\\alpha=a_\\phi[\\phi]+a_3[3]+a_5[\\sqrt5],\\qquad a_3a_5\\ne0.\n\\end{equation}\nThis lies in a Kummer group tensored with $k$. It does not mean that an element of $\\F_{49}$ is an exponent of a base field unit. The three coefficients are normalized together, up to one common scalar. Using a different scalar at each completion would discard the global information that makes the reciprocity tests effective.","The actual matrices have the form\n\\[\n\\rho(g)=\\begin{pmatrix}a(g)&b(g)\\\\7^nc(g)&d(g)\\end{pmatrix}.\n\\]\nModulo $7^n$, the diagonal entries are characters. Determinant $\\chi$ lets us write them as $\\chi\\eta_n$ and $\\eta_n^{-1}$. The natural monodromy restrictions and fixed local traces constrain the same $\\eta_n$ at every place. A locally chosen ray character with the right residue does not prove it occurs as this actual diagonal character.","7^nc(g)&d(g)\\end{pmatrix}.\n\\]\nModulo $7^n$, the diagonal entries are characters. Determinant $\\chi$ lets us write them as $\\chi\\eta_n$ and $\\eta_n^{-1}$. The natural monodromy restrictions and fixed local traces constrain the same $\\eta_n$ at every place. A locally chosen ray character with the right residue does not prove it occurs as this actual diagonal character.","The sequence of Blocks 52 through 58 tightened this coupling. Scalar Frobenius at good thirteen first forced $a_\\phi=0$ on that subcase. Later, torsion exponent information at eighty three forced $a_\\phi\\ne0$ in the retained actual branch. Hence thirteen must be bad there. The conclusion is not a contradiction from two incompatible coefficients. The first coefficient statement assumed good thirteen, so the combined result is $13\\mid C$.","ugh 58 tightened this coupling. Scalar Frobenius at good thirteen first forced $a_\\phi=0$ on that subcase. Later, torsion exponent information at eighty three forced $a_\\phi\\ne0$ in the retained actual branch. Hence thirteen must be bad there. The conclusion is not a contradiction from two incompatible coefficients. The first coefficient statement assumed good thirteen, so the combined result is $13\\mid C$.","At good auxiliary primes, the actual rank two trace obeys a precision condition of the form\n\\[\n\\tau\\equiv Q+1\\pmod{7^r\\Oloc}.\n\\]\nThe exponent $r$ is supplied by the depth and local diagonal estimates. It is not inferred merely from a rank four congruence. This stronger comparison yields a particularly useful universal lemma.","\\begin{lemma}[Discriminant collapse]\\label{lem:collapse}\nSuppose a good Frobenius polynomial has integral coefficients and factorization\n\\[\nP(T)=(T^2-\\tau T+Q)(T^2-\\sigma(\\tau)T+Q)\n =T^4-ST^3+C_2T^2-QST+Q^2,\n\\]\nwhere $\\Oloc/\\Z_7$ is unramified quadratic, $\\sigma$ is its nontrivial automorphism, all complex roots have absolute value $\\sqrt Q$, and $\\tau\\equiv Q+1\\pmod{7^r\\Oloc}$. Then the integer\n\\[\nD=S^2-4(C_2-2Q)\n\\]\nsatisfies $7^{2r}\\mid D$ and $0\\le D\\le16Q$. If $7^{2r}>16Q$, then\n\\[\nP(T)=(T^2-aT+Q)^2,\n\\quad a\\in\\Z,\\quad |a|\\le2\\sqrt Q,\n\\quad a\\equiv Q+1\\pmod{7^r}.\n\\]\n\\end{lemma}\n\\begin{proof}\nExpansion gives $S=\\tau+\\sigma(\\tau)$, $C_2-2Q=\\tau\\sigma(\\tau)$, and $D=(\\tau-\\sigma(\\tau))^2$. The congruence is preserved by $\\sigma$, so this difference is divisible by $7^r$. Unramifiedness makes its square give ordinary integer divisibility by $7^{2r}$.","begin{lemma}[Discriminant collapse]\\label{lem:collapse}\nSuppose a good Frobenius polynomial has integral coefficients and factorization\n\\[\nP(T)=(T^2-\\tau T+Q)(T^2-\\sigma(\\tau)T+Q)\n =T^4-ST^3+C_2T^2-QST+Q^2,\n\\]\nwhere $\\Oloc/\\Z_7$ is unramified quadratic, $\\sigma$ is its nontrivial automorphism, all complex roots have absolute value $\\sqrt Q$, and $\\tau\\equiv Q+1\\pmod{7^r\\Oloc}$. Then the integer\n\\[\nD=S^2-4(C_2-2Q)\n\\]\nsatisfies $7^{2r}\\mid D$ and $0\\le D\\le16Q$. If $7^{2r}>16Q$, then\n\\[\nP(T)=(T^2-aT+Q)^2,\n\\quad a\\in\\Z,\\quad |a|\\le2\\sqrt Q,\n\\quad a\\equiv Q+1\\pmod{7^r}.\n\\]\n\\end{lemma}\n\\begin{proof}\nExpansion gives $S=\\tau+\\sigma(\\tau)$, $C_2-2Q=\\tau\\sigma(\\tau)$, and $D=(\\tau-\\sigma(\\tau))^2$. The congruence is preserved by $\\sigma$, so this difference is divisible by $7^r$. Unramifiedness makes its square give ordinary integer divisibility by $7^{2r}$.","pse]\\label{lem:collapse}\nSuppose a good Frobenius polynomial has integral coefficients and factorization\n\\[\nP(T)=(T^2-\\tau T+Q)(T^2-\\sigma(\\tau)T+Q)\n =T^4-ST^3+C_2T^2-QST+Q^2,\n\\]\nwhere $\\Oloc/\\Z_7$ is unramified quadratic, $\\sigma$ is its nontrivial automorphism, all complex roots have absolute value $\\sqrt Q$, and $\\tau\\equiv Q+1\\pmod{7^r\\Oloc}$. Then the integer\n\\[\nD=S^2-4(C_2-2Q)\n\\]\nsatisfies $7^{2r}\\mid D$ and $0\\le D\\le16Q$. If $7^{2r}>16Q$, then\n\\[\nP(T)=(T^2-aT+Q)^2,\n\\quad a\\in\\Z,\\quad |a|\\le2\\sqrt Q,\n\\quad a\\equiv Q+1\\pmod{7^r}.\n\\]\n\\end{lemma}\n\\begin{proof}\nExpansion gives $S=\\tau+\\sigma(\\tau)$, $C_2-2Q=\\tau\\sigma(\\tau)$, and $D=(\\tau-\\sigma(\\tau))^2$. The congruence is preserved by $\\sigma$, so this difference is divisible by $7^r$. Unramifiedness makes its square give ordinary integer divisibility by $7^{2r}$.","apse}\nSuppose a good Frobenius polynomial has integral coefficients and factorization\n\\[\nP(T)=(T^2-\\tau T+Q)(T^2-\\sigma(\\tau)T+Q)\n =T^4-ST^3+C_2T^2-QST+Q^2,\n\\]\nwhere $\\Oloc/\\Z_7$ is unramified quadratic, $\\sigma$ is its nontrivial automorphism, all complex roots have absolute value $\\sqrt Q$, and $\\tau\\equiv Q+1\\pmod{7^r\\Oloc}$. Then the integer\n\\[\nD=S^2-4(C_2-2Q)\n\\]\nsatisfies $7^{2r}\\mid D$ and $0\\le D\\le16Q$. If $7^{2r}>16Q$, then\n\\[\nP(T)=(T^2-aT+Q)^2,\n\\quad a\\in\\Z,\\quad |a|\\le2\\sqrt Q,\n\\quad a\\equiv Q+1\\pmod{7^r}.\n\\]\n\\end{lemma}\n\\begin{proof}\nExpansion gives $S=\\tau+\\sigma(\\tau)$, $C_2-2Q=\\tau\\sigma(\\tau)$, and $D=(\\tau-\\sigma(\\tau))^2$. The congruence is preserved by $\\sigma$, so this difference is divisible by $7^r$. Unramifiedness makes its square give ordinary integer divisibility by $7^{2r}$.","d Frobenius polynomial has integral coefficients and factorization\n\\[\nP(T)=(T^2-\\tau T+Q)(T^2-\\sigma(\\tau)T+Q)\n =T^4-ST^3+C_2T^2-QST+Q^2,\n\\]\nwhere $\\Oloc/\\Z_7$ is unramified quadratic, $\\sigma$ is its nontrivial automorphism, all complex roots have absolute value $\\sqrt Q$, and $\\tau\\equiv Q+1\\pmod{7^r\\Oloc}$. Then the integer\n\\[\nD=S^2-4(C_2-2Q)\n\\]\nsatisfies $7^{2r}\\mid D$ and $0\\le D\\le16Q$. If $7^{2r}>16Q$, then\n\\[\nP(T)=(T^2-aT+Q)^2,\n\\quad a\\in\\Z,\\quad |a|\\le2\\sqrt Q,\n\\quad a\\equiv Q+1\\pmod{7^r}.\n\\]\n\\end{lemma}\n\\begin{proof}\nExpansion gives $S=\\tau+\\sigma(\\tau)$, $C_2-2Q=\\tau\\sigma(\\tau)$, and $D=(\\tau-\\sigma(\\tau))^2$. The congruence is preserved by $\\sigma$, so this difference is divisible by $7^r$. Unramifiedness makes its square give ordinary integer divisibility by $7^{2r}$.","al has integral coefficients and factorization\n\\[\nP(T)=(T^2-\\tau T+Q)(T^2-\\sigma(\\tau)T+Q)\n =T^4-ST^3+C_2T^2-QST+Q^2,\n\\]\nwhere $\\Oloc/\\Z_7$ is unramified quadratic, $\\sigma$ is its nontrivial automorphism, all complex roots have absolute value $\\sqrt Q$, and $\\tau\\equiv Q+1\\pmod{7^r\\Oloc}$. Then the integer\n\\[\nD=S^2-4(C_2-2Q)\n\\]\nsatisfies $7^{2r}\\mid D$ and $0\\le D\\le16Q$. If $7^{2r}>16Q$, then\n\\[\nP(T)=(T^2-aT+Q)^2,\n\\quad a\\in\\Z,\\quad |a|\\le2\\sqrt Q,\n\\quad a\\equiv Q+1\\pmod{7^r}.\n\\]\n\\end{lemma}\n\\begin{proof}\nExpansion gives $S=\\tau+\\sigma(\\tau)$, $C_2-2Q=\\tau\\sigma(\\tau)$, and $D=(\\tau-\\sigma(\\tau))^2$. The congruence is preserved by $\\sigma$, so this difference is divisible by $7^r$. Unramifiedness makes its square give ordinary integer divisibility by $7^{2r}$.","loc/\\Z_7$ is unramified quadratic, $\\sigma$ is its nontrivial automorphism, all complex roots have absolute value $\\sqrt Q$, and $\\tau\\equiv Q+1\\pmod{7^r\\Oloc}$. Then the integer\n\\[\nD=S^2-4(C_2-2Q)\n\\]\nsatisfies $7^{2r}\\mid D$ and $0\\le D\\le16Q$. If $7^{2r}>16Q$, then\n\\[\nP(T)=(T^2-aT+Q)^2,\n\\quad a\\in\\Z,\\quad |a|\\le2\\sqrt Q,\n\\quad a\\equiv Q+1\\pmod{7^r}.\n\\]\n\\end{lemma}\n\\begin{proof}\nExpansion gives $S=\\tau+\\sigma(\\tau)$, $C_2-2Q=\\tau\\sigma(\\tau)$, and $D=(\\tau-\\sigma(\\tau))^2$. The congruence is preserved by $\\sigma$, so this difference is divisible by $7^r$. Unramifiedness makes its square give ordinary integer divisibility by $7^{2r}$.","quadratic, $\\sigma$ is its nontrivial automorphism, all complex roots have absolute value $\\sqrt Q$, and $\\tau\\equiv Q+1\\pmod{7^r\\Oloc}$. Then the integer\n\\[\nD=S^2-4(C_2-2Q)\n\\]\nsatisfies $7^{2r}\\mid D$ and $0\\le D\\le16Q$. If $7^{2r}>16Q$, then\n\\[\nP(T)=(T^2-aT+Q)^2,\n\\quad a\\in\\Z,\\quad |a|\\le2\\sqrt Q,\n\\quad a\\equiv Q+1\\pmod{7^r}.\n\\]\n\\end{lemma}\n\\begin{proof}\nExpansion gives $S=\\tau+\\sigma(\\tau)$, $C_2-2Q=\\tau\\sigma(\\tau)$, and $D=(\\tau-\\sigma(\\tau))^2$. The congruence is preserved by $\\sigma$, so this difference is divisible by $7^r$. Unramifiedness makes its square give ordinary integer divisibility by $7^{2r}$.","trivial automorphism, all complex roots have absolute value $\\sqrt Q$, and $\\tau\\equiv Q+1\\pmod{7^r\\Oloc}$. Then the integer\n\\[\nD=S^2-4(C_2-2Q)\n\\]\nsatisfies $7^{2r}\\mid D$ and $0\\le D\\le16Q$. If $7^{2r}>16Q$, then\n\\[\nP(T)=(T^2-aT+Q)^2,\n\\quad a\\in\\Z,\\quad |a|\\le2\\sqrt Q,\n\\quad a\\equiv Q+1\\pmod{7^r}.\n\\]\n\\end{lemma}\n\\begin{proof}\nExpansion gives $S=\\tau+\\sigma(\\tau)$, $C_2-2Q=\\tau\\sigma(\\tau)$, and $D=(\\tau-\\sigma(\\tau))^2$. The congruence is preserved by $\\sigma$, so this difference is divisible by $7^r$. Unramifiedness makes its square give ordinary integer divisibility by $7^{2r}$.","e $\\sqrt Q$, and $\\tau\\equiv Q+1\\pmod{7^r\\Oloc}$. Then the integer\n\\[\nD=S^2-4(C_2-2Q)\n\\]\nsatisfies $7^{2r}\\mid D$ and $0\\le D\\le16Q$. If $7^{2r}>16Q$, then\n\\[\nP(T)=(T^2-aT+Q)^2,\n\\quad a\\in\\Z,\\quad |a|\\le2\\sqrt Q,\n\\quad a\\equiv Q+1\\pmod{7^r}.\n\\]\n\\end{lemma}\n\\begin{proof}\nExpansion gives $S=\\tau+\\sigma(\\tau)$, $C_2-2Q=\\tau\\sigma(\\tau)$, and $D=(\\tau-\\sigma(\\tau))^2$. The congruence is preserved by $\\sigma$, so this difference is divisible by $7^r$. Unramifiedness makes its square give ordinary integer divisibility by $7^{2r}$.","c}$. Then the integer\n\\[\nD=S^2-4(C_2-2Q)\n\\]\nsatisfies $7^{2r}\\mid D$ and $0\\le D\\le16Q$. If $7^{2r}>16Q$, then\n\\[\nP(T)=(T^2-aT+Q)^2,\n\\quad a\\in\\Z,\\quad |a|\\le2\\sqrt Q,\n\\quad a\\equiv Q+1\\pmod{7^r}.\n\\]\n\\end{lemma}\n\\begin{proof}\nExpansion gives $S=\\tau+\\sigma(\\tau)$, $C_2-2Q=\\tau\\sigma(\\tau)$, and $D=(\\tau-\\sigma(\\tau))^2$. The congruence is preserved by $\\sigma$, so this difference is divisible by $7^r$. Unramifiedness makes its square give ordinary integer divisibility by $7^{2r}$.","e D\\le16Q$. If $7^{2r}>16Q$, then\n\\[\nP(T)=(T^2-aT+Q)^2,\n\\quad a\\in\\Z,\\quad |a|\\le2\\sqrt Q,\n\\quad a\\equiv Q+1\\pmod{7^r}.\n\\]\n\\end{lemma}\n\\begin{proof}\nExpansion gives $S=\\tau+\\sigma(\\tau)$, $C_2-2Q=\\tau\\sigma(\\tau)$, and $D=(\\tau-\\sigma(\\tau))^2$. The congruence is preserved by $\\sigma$, so this difference is divisible by $7^r$. Unramifiedness makes its square give ordinary integer divisibility by $7^{2r}$.","The two traces are algebraic roots of $X^2-SX+(C_2-2Q)$. Under their complex embeddings, the corresponding quadratic factors have roots of product $Q$ and magnitude $\\sqrt Q$. Their sums are real and lie in $[-2\\sqrt Q,2\\sqrt Q]$. Their difference therefore has magnitude at most $4\\sqrt Q$, proving the bound. If the divisibility exceeds the bound, $D=0$. The common trace is rational and integral, hence an integer $a$, with the stated congruence and magnitude.\n\\end{proof}\nFor $Q=2549$ and $r=3$, one has $7^6=117649>40784=16Q$. The allowed integral trace would lie in $[-100,100]$ and be congruent to $149$ modulo $343$. Neither nearest representative, $149$ or $-194$, lies in the interval. Therefore the inherited precision theorem forces $2549\\mid","The two traces are algebraic roots of $X^2-SX+(C_2-2Q)$. Under their complex embeddings, the corresponding quadratic factors have roots of product $Q$ and magnitude $\\sqrt Q$. Their sums are real and lie in $[-2\\sqrt Q,2\\sqrt Q]$. Their difference therefore has magnitude at most $4\\sqrt Q$, proving the bound. If the divisibility exceeds the bound, $D=0$. The common trace is rational and integral, hence an integer $a$, with the stated congruence and magnitude.\n\\end{proof}\nFor $Q=2549$ and $r=3$, one has $7^6=117649>40784=16Q$. The allowed integral trace would lie in $[-100,100]$ and be congruent to $149$ modulo $343$. Neither nearest representative, $149$ or $-194$, lies in the interval. Therefore the inherited precision theorem forces $2549\\mid C$ whenever it applies at depth at least three.","algebraic roots of $X^2-SX+(C_2-2Q)$. Under their complex embeddings, the corresponding quadratic factors have roots of product $Q$ and magnitude $\\sqrt Q$. Their sums are real and lie in $[-2\\sqrt Q,2\\sqrt Q]$. Their difference therefore has magnitude at most $4\\sqrt Q$, proving the bound. If the divisibility exceeds the bound, $D=0$. The common trace is rational and integral, hence an integer $a$, with the stated congruence and magnitude.\n\\end{proof}\nFor $Q=2549$ and $r=3$, one has $7^6=117649>40784=16Q$. The allowed integral trace would lie in $[-100,100]$ and be congruent to $149$ modulo $343$. Neither nearest representative, $149$ or $-194$, lies in the interval. Therefore the inherited precision theorem forces $2549\\mid C$ whenever it applies at depth at least three.","ddings, the corresponding quadratic factors have roots of product $Q$ and magnitude $\\sqrt Q$. Their sums are real and lie in $[-2\\sqrt Q,2\\sqrt Q]$. Their difference therefore has magnitude at most $4\\sqrt Q$, proving the bound. If the divisibility exceeds the bound, $D=0$. The common trace is rational and integral, hence an integer $a$, with the stated congruence and magnitude.\n\\end{proof}\nFor $Q=2549$ and $r=3$, one has $7^6=117649>40784=16Q$. The allowed integral trace would lie in $[-100,100]$ and be congruent to $149$ modulo $343$. Neither nearest representative, $149$ or $-194$, lies in the interval. Therefore the inherited precision theorem forces $2549\\mid C$ whenever it applies at depth at least three.","orresponding quadratic factors have roots of product $Q$ and magnitude $\\sqrt Q$. Their sums are real and lie in $[-2\\sqrt Q,2\\sqrt Q]$. Their difference therefore has magnitude at most $4\\sqrt Q$, proving the bound. If the divisibility exceeds the bound, $D=0$. The common trace is rational and integral, hence an integer $a$, with the stated congruence and magnitude.\n\\end{proof}\nFor $Q=2549$ and $r=3$, one has $7^6=117649>40784=16Q$. The allowed integral trace would lie in $[-100,100]$ and be congruent to $149$ modulo $343$. Neither nearest representative, $149$ or $-194$, lies in the interval. Therefore the inherited precision theorem forces $2549\\mid C$ whenever it applies at depth at least three.","adratic factors have roots of product $Q$ and magnitude $\\sqrt Q$. Their sums are real and lie in $[-2\\sqrt Q,2\\sqrt Q]$. Their difference therefore has magnitude at most $4\\sqrt Q$, proving the bound. If the divisibility exceeds the bound, $D=0$. The common trace is rational and integral, hence an integer $a$, with the stated congruence and magnitude.\n\\end{proof}\nFor $Q=2549$ and $r=3$, one has $7^6=117649>40784=16Q$. The allowed integral trace would lie in $[-100,100]$ and be congruent to $149$ modulo $343$. Neither nearest representative, $149$ or $-194$, lies in the interval. Therefore the inherited precision theorem forces $2549\\mid C$ whenever it applies at depth at least three.","$. Their sums are real and lie in $[-2\\sqrt Q,2\\sqrt Q]$. Their difference therefore has magnitude at most $4\\sqrt Q$, proving the bound. If the divisibility exceeds the bound, $D=0$. The common trace is rational and integral, hence an integer $a$, with the stated congruence and magnitude.\n\\end{proof}\nFor $Q=2549$ and $r=3$, one has $7^6=117649>40784=16Q$. The allowed integral trace would lie in $[-100,100]$ and be congruent to $149$ modulo $343$. Neither nearest representative, $149$ or $-194$, lies in the interval. Therefore the inherited precision theorem forces $2549\\mid C$ whenever it applies at depth at least three.","$[-2\\sqrt Q,2\\sqrt Q]$. Their difference therefore has magnitude at most $4\\sqrt Q$, proving the bound. If the divisibility exceeds the bound, $D=0$. The common trace is rational and integral, hence an integer $a$, with the stated congruence and magnitude.\n\\end{proof}\nFor $Q=2549$ and $r=3$, one has $7^6=117649>40784=16Q$. The allowed integral trace would lie in $[-100,100]$ and be congruent to $149$ modulo $343$. Neither nearest representative, $149$ or $-194$, lies in the interval. Therefore the inherited precision theorem forces $2549\\mid C$ whenever it applies at depth at least three.","2\\sqrt Q]$. Their difference therefore has magnitude at most $4\\sqrt Q$, proving the bound. If the divisibility exceeds the bound, $D=0$. The common trace is rational and integral, hence an integer $a$, with the stated congruence and magnitude.\n\\end{proof}\nFor $Q=2549$ and $r=3$, one has $7^6=117649>40784=16Q$. The allowed integral trace would lie in $[-100,100]$ and be congruent to $149$ modulo $343$. Neither nearest representative, $149$ or $-194$, lies in the interval. Therefore the inherited precision theorem forces $2549\\mid C$ whenever it applies at depth at least three.","has magnitude at most $4\\sqrt Q$, proving the bound. If the divisibility exceeds the bound, $D=0$. The common trace is rational and integral, hence an integer $a$, with the stated congruence and magnitude.\n\\end{proof}\nFor $Q=2549$ and $r=3$, one has $7^6=117649>40784=16Q$. The allowed integral trace would lie in $[-100,100]$ and be congruent to $149$ modulo $343$. Neither nearest representative, $149$ or $-194$, lies in the interval. Therefore the inherited precision theorem forces $2549\\mid C$ whenever it applies at depth at least three.","tude at most $4\\sqrt Q$, proving the bound. If the divisibility exceeds the bound, $D=0$. The common trace is rational and integral, hence an integer $a$, with the stated congruence and magnitude.\n\\end{proof}\nFor $Q=2549$ and $r=3$, one has $7^6=117649>40784=16Q$. The allowed integral trace would lie in $[-100,100]$ and be congruent to $149$ modulo $343$. Neither nearest representative, $149$ or $-194$, lies in the interval. Therefore the inherited precision theorem forces $2549\\mid C$ whenever it applies at depth at least three.","nd, $D=0$. The common trace is rational and integral, hence an integer $a$, with the stated congruence and magnitude.\n\\end{proof}\nFor $Q=2549$ and $r=3$, one has $7^6=117649>40784=16Q$. The allowed integral trace would lie in $[-100,100]$ and be congruent to $149$ modulo $343$. Neither nearest representative, $149$ or $-194$, lies in the interval. Therefore the inherited precision theorem forces $2549\\mid C$ whenever it applies at depth at least three.","The rank four congruence alone is weaker. The retained negative control\n\\[\nP(T)=T^4+45T^3+2603T^2+114705T+6497401\n\\]\nis a generic reciprocal Weil polynomial at $2549$ satisfying $(T-1)^2(T-Q)^2$ modulo $343$. Its trace discriminant is $5\\cdot49^2$, not divisible by $7^6$. It does not satisfy the actual rank two precision condition. It is not claimed to be a Frey reduction. Its role is to falsify the shortcut that would replace the stronger hypothesis by an unlabelled quartic congruence.","Other fixed primes forced into $C$ at stated depths include the good reduction tests at $41,59,197,491$. Combining these bad primes with reciprocity rows eventually trivialized the diagonal character and reduced depth on declared finite high support classes. The finite support hypotheses remain essential: a new conductor outside the declared universe changes the global ray character presentation.","At a variable place away from $2,3,5,7$, write $C=q^cC_0$ with $C_0$ a unit. In the marked two node basis, the actual leading period coordinates are\n\\begin{align*}\n\\Pi_{aa}&=-\\frac{\\eps\\phi^2}{675\\sqrt5}(1+\\mathfrak m),","At a variable place away from $2,3,5,7$, write $C=q^cC_0$ with $C_0$ a unit. In the marked two node basis, the actual leading period coordinates are\n\\begin{align*}\n\\Pi_{aa}&=-\\frac{\\eps\\phi^2}{675\\sqrt5}(1+\\mathfrak m),\\\\\n\\Pi_{bb}&=\\frac{\\eps\\phi^{-2}}{675\\sqrt5}(1+\\mathfrak m),\\\\\n\\Pi_{ab}&=\\Pi_{ba}=-\\phi^{-4}(1+\\mathfrak m),\n\\end{align*}\nwhere $\\eps=C^7/A^3$ and the notation means that the quotient by the displayed leading term is a principal unit. The analytic uniformization and marked smoothing comparison are inherited written inputs, with precise sources in the literature report \\cite{mumford1972,ichikawa1993}. The principal unit qualification is essential at seven, where those units may carry higher information. Away from seven they are divisible by every power of seven.","\\Pi_{ab}&=\\Pi_{ba}=-\\phi^{-4}(1+\\mathfrak m),\n\\end{align*}\nwhere $\\eps=C^7/A^3$ and the notation means that the quotient by the displayed leading term is a principal unit. The analytic uniformization and marked smoothing comparison are inherited written inputs, with precise sources in the literature report \\cite{mumford1972,ichikawa1993}. The principal unit qualification is essential at seven, where those units may carry higher information. Away from seven they are divisible by every power of seven.","Put $M=7^j$ and write $[u]$ additively in $F^\\times/F^{\\times M}$ for the local field $F=K_v$. Let\n\\[\nX=[\\phi],\\qquad Z=7[C]-3[A]-3[3]-5[\\sqrt5].\n\\]\nThe entire marked period matrix becomes\n\\begin{equation}\\label{eq:periodmatrix}\n([\\Pi_{ij}])=\n\\begin{pmatrix}Z+2X&-4X","Put $M=7^j$ and write $[u]$ additively in $F^\\times/F^{\\times M}$ for the local field $F=K_v$. Let\n\\[\nX=[\\phi],\\qquad Z=7[C]-3[A]-3[3]-5[\\sqrt5].\n\\]\nThe entire marked period matrix becomes\n\\begin{equation}\\label{eq:periodmatrix}\n([\\Pi_{ij}])=\n\\begin{pmatrix}Z+2X&-4X\\\\-4X&Z-2X\\end{pmatrix}\n=ZI+2XR,\\qquad R=\\begin{pmatrix}1&-2\\\\-2&-1\\end{pmatrix},\\quad R^2=5I.\n\\end{equation}\nSince four is invertible modulo $M$, every marked period coordinate vanishes if and only if $X=Z=0$. The matrix $R$ belongs to this marked local period calculation. It is not silently identified with a global endomorphism or with an integral semilinear descent operator.","-2&-1\\end{pmatrix},\\quad R^2=5I.\n\\end{equation}\nSince four is invertible modulo $M$, every marked period coordinate vanishes if and only if $X=Z=0$. The matrix $R$ belongs to this marked local period calculation. It is not silently identified with a global endomorphism or with an integral semilinear descent operator.","\\begin{proposition}[Full torsion and period powers]\\label{prop:torsionperiods}\nFor the stipulated split uniformization $J^{\\mathrm{an}}=\\mathbb G_m^2/\\Lambda$ with rational period lattice, full $J[M]$ is $F$ rational if and only if $\\mu_M\\subset F$ and every coordinate of a basis of $\\Lambda$ is an $M$th power in $F$.\n\\end{proposition}\n\\begin{proof}\nThe uniformization gives\n\\[\n0\\longrightarrow\\mu_M^2\\longrightarrow J[M]\n\\longrightarrow\\Lambda/M\\Lambda\\longrightarrow0.\n\\]\nIf all torsion points are rational, so are $\\mu_M^2$. For $\\lambda\\in\\Lambda$, choose an $M$th root $u$ in the torus. Rationality of its torsion class gives $g(u)/u\\in\\Lambda$ for every $g\\in G_F$. Since $\\lambda$ is rational, this quotient also lies in $\\mu_M^2$. The lattice is torsion free, so the intersection is trivial. Thus $u$ is rational. Every period coordinate is an $M$","\\begin{proposition}[Full torsion and period powers]\\label{prop:torsionperiods}\nFor the stipulated split uniformization $J^{\\mathrm{an}}=\\mathbb G_m^2/\\Lambda$ with rational period lattice, full $J[M]$ is $F$ rational if and only if $\\mu_M\\subset F$ and every coordinate of a basis of $\\Lambda$ is an $M$th power in $F$.\n\\end{proposition}\n\\begin{proof}\nThe uniformization gives\n\\[\n0\\longrightarrow\\mu_M^2\\longrightarrow J[M]\n\\longrightarrow\\Lambda/M\\Lambda\\longrightarrow0.\n\\]\nIf all torsion points are rational, so are $\\mu_M^2$. For $\\lambda\\in\\Lambda$, choose an $M$th root $u$ in the torus. Rationality of its torsion class gives $g(u)/u\\in\\Lambda$ for every $g\\in G_F$. Since $\\lambda$ is rational, this quotient also lies in $\\mu_M^2$. The lattice is torsion free, so the intersection is trivial. Thus $u$ is rational. Every period coordinate is an $M$th power.","J^{\\mathrm{an}}=\\mathbb G_m^2/\\Lambda$ with rational period lattice, full $J[M]$ is $F$ rational if and only if $\\mu_M\\subset F$ and every coordinate of a basis of $\\Lambda$ is an $M$th power in $F$.\n\\end{proposition}\n\\begin{proof}\nThe uniformization gives\n\\[\n0\\longrightarrow\\mu_M^2\\longrightarrow J[M]\n\\longrightarrow\\Lambda/M\\Lambda\\longrightarrow0.\n\\]\nIf all torsion points are rational, so are $\\mu_M^2$. For $\\lambda\\in\\Lambda$, choose an $M$th root $u$ in the torus. Rationality of its torsion class gives $g(u)/u\\in\\Lambda$ for every $g\\in G_F$. Since $\\lambda$ is rational, this quotient also lies in $\\mu_M^2$. The lattice is torsion free, so the intersection is trivial. Thus $u$ is rational. Every period coordinate is an $M$th power.","=\\mathbb G_m^2/\\Lambda$ with rational period lattice, full $J[M]$ is $F$ rational if and only if $\\mu_M\\subset F$ and every coordinate of a basis of $\\Lambda$ is an $M$th power in $F$.\n\\end{proposition}\n\\begin{proof}\nThe uniformization gives\n\\[\n0\\longrightarrow\\mu_M^2\\longrightarrow J[M]\n\\longrightarrow\\Lambda/M\\Lambda\\longrightarrow0.\n\\]\nIf all torsion points are rational, so are $\\mu_M^2$. For $\\lambda\\in\\Lambda$, choose an $M$th root $u$ in the torus. Rationality of its torsion class gives $g(u)/u\\in\\Lambda$ for every $g\\in G_F$. Since $\\lambda$ is rational, this quotient also lies in $\\mu_M^2$. The lattice is torsion free, so the intersection is trivial. Thus $u$ is rational. Every period coordinate is an $M$th power.","tional period lattice, full $J[M]$ is $F$ rational if and only if $\\mu_M\\subset F$ and every coordinate of a basis of $\\Lambda$ is an $M$th power in $F$.\n\\end{proposition}\n\\begin{proof}\nThe uniformization gives\n\\[\n0\\longrightarrow\\mu_M^2\\longrightarrow J[M]\n\\longrightarrow\\Lambda/M\\Lambda\\longrightarrow0.\n\\]\nIf all torsion points are rational, so are $\\mu_M^2$. For $\\lambda\\in\\Lambda$, choose an $M$th root $u$ in the torus. Rationality of its torsion class gives $g(u)/u\\in\\Lambda$ for every $g\\in G_F$. Since $\\lambda$ is rational, this quotient also lies in $\\mu_M^2$. The lattice is torsion free, so the intersection is trivial. Thus $u$ is rational. Every period coordinate is an $M$th power.","od lattice, full $J[M]$ is $F$ rational if and only if $\\mu_M\\subset F$ and every coordinate of a basis of $\\Lambda$ is an $M$th power in $F$.\n\\end{proposition}\n\\begin{proof}\nThe uniformization gives\n\\[\n0\\longrightarrow\\mu_M^2\\longrightarrow J[M]\n\\longrightarrow\\Lambda/M\\Lambda\\longrightarrow0.\n\\]\nIf all torsion points are rational, so are $\\mu_M^2$. For $\\lambda\\in\\Lambda$, choose an $M$th root $u$ in the torus. Rationality of its torsion class gives $g(u)/u\\in\\Lambda$ for every $g\\in G_F$. Since $\\lambda$ is rational, this quotient also lies in $\\mu_M^2$. The lattice is torsion free, so the intersection is trivial. Thus $u$ is rational. Every period coordinate is an $M$th power.","mu_M\\subset F$ and every coordinate of a basis of $\\Lambda$ is an $M$th power in $F$.\n\\end{proposition}\n\\begin{proof}\nThe uniformization gives\n\\[\n0\\longrightarrow\\mu_M^2\\longrightarrow J[M]\n\\longrightarrow\\Lambda/M\\Lambda\\longrightarrow0.\n\\]\nIf all torsion points are rational, so are $\\mu_M^2$. For $\\lambda\\in\\Lambda$, choose an $M$th root $u$ in the torus. Rationality of its torsion class gives $g(u)/u\\in\\Lambda$ for every $g\\in G_F$. Since $\\lambda$ is rational, this quotient also lies in $\\mu_M^2$. The lattice is torsion free, so the intersection is trivial. Thus $u$ is rational. Every period coordinate is an $M$th power.","dinate of a basis of $\\Lambda$ is an $M$th power in $F$.\n\\end{proposition}\n\\begin{proof}\nThe uniformization gives\n\\[\n0\\longrightarrow\\mu_M^2\\longrightarrow J[M]\n\\longrightarrow\\Lambda/M\\Lambda\\longrightarrow0.\n\\]\nIf all torsion points are rational, so are $\\mu_M^2$. For $\\lambda\\in\\Lambda$, choose an $M$th root $u$ in the torus. Rationality of its torsion class gives $g(u)/u\\in\\Lambda$ for every $g\\in G_F$. Since $\\lambda$ is rational, this quotient also lies in $\\mu_M^2$. The lattice is torsion free, so the intersection is trivial. Thus $u$ is rational. Every period coordinate is an $M$th power.","of $\\Lambda$ is an $M$th power in $F$.\n\\end{proposition}\n\\begin{proof}\nThe uniformization gives\n\\[\n0\\longrightarrow\\mu_M^2\\longrightarrow J[M]\n\\longrightarrow\\Lambda/M\\Lambda\\longrightarrow0.\n\\]\nIf all torsion points are rational, so are $\\mu_M^2$. For $\\lambda\\in\\Lambda$, choose an $M$th root $u$ in the torus. Rationality of its torsion class gives $g(u)/u\\in\\Lambda$ for every $g\\in G_F$. Since $\\lambda$ is rational, this quotient also lies in $\\mu_M^2$. The lattice is torsion free, so the intersection is trivial. Thus $u$ is rational. Every period coordinate is an $M$th power.","ition}\n\\begin{proof}\nThe uniformization gives\n\\[\n0\\longrightarrow\\mu_M^2\\longrightarrow J[M]\n\\longrightarrow\\Lambda/M\\Lambda\\longrightarrow0.\n\\]\nIf all torsion points are rational, so are $\\mu_M^2$. For $\\lambda\\in\\Lambda$, choose an $M$th root $u$ in the torus. Rationality of its torsion class gives $g(u)/u\\in\\Lambda$ for every $g\\in G_F$. Since $\\lambda$ is rational, this quotient also lies in $\\mu_M^2$. The lattice is torsion free, so the intersection is trivial. Thus $u$ is rational. Every period coordinate is an $M$th power.","rightarrow J[M]\n\\longrightarrow\\Lambda/M\\Lambda\\longrightarrow0.\n\\]\nIf all torsion points are rational, so are $\\mu_M^2$. For $\\lambda\\in\\Lambda$, choose an $M$th root $u$ in the torus. Rationality of its torsion class gives $g(u)/u\\in\\Lambda$ for every $g\\in G_F$. Since $\\lambda$ is rational, this quotient also lies in $\\mu_M^2$. The lattice is torsion free, so the intersection is trivial. Thus $u$ is rational. Every period coordinate is an $M$th power.","\\longrightarrow0.\n\\]\nIf all torsion points are rational, so are $\\mu_M^2$. For $\\lambda\\in\\Lambda$, choose an $M$th root $u$ in the torus. Rationality of its torsion class gives $g(u)/u\\in\\Lambda$ for every $g\\in G_F$. Since $\\lambda$ is rational, this quotient also lies in $\\mu_M^2$. The lattice is torsion free, so the intersection is trivial. Thus $u$ is rational. Every period coordinate is an $M$th power.","Conversely, take rational $M$th roots of each period basis vector. Their classes, together with $\\mu_M^2$, generate all $M$ torsion by the exact sequence. Every generator is rational.\n\\end{proof}\nAt residue characteristic different from seven, a unit is an $M$th power precisely when its residue is, by simple root lifting. A general element also needs valuation divisible by $M$. If $Q$ is the residue cardinality, $\\mu_M\\subset F$ precisely when $M\\mid Q-1$. Applying Proposition~\\ref{prop:torsionperiods} to \\eqref{eq:periodmatrix} gives the complete local test\n\\begin{equation}\\label{eq:full-local-torsion}\n\\begin{gathered}\n7^j\\mid Q-1,\\qquad7^j\\mid7c,","rational $M$th roots of each period basis vector. Their classes, together with $\\mu_M^2$, generate all $M$ torsion by the exact sequence. Every generator is rational.\n\\end{proof}\nAt residue characteristic different from seven, a unit is an $M$th power precisely when its residue is, by simple root lifting. A general element also needs valuation divisible by $M$. If $Q$ is the residue cardinality, $\\mu_M\\subset F$ precisely when $M\\mid Q-1$. Applying Proposition~\\ref{prop:torsionperiods} to \\eqref{eq:periodmatrix} gives the complete local test\n\\begin{equation}\\label{eq:full-local-torsion}\n\\begin{gathered}\n7^j\\mid Q-1,\\qquad7^j\\mid7c,","iod basis vector. Their classes, together with $\\mu_M^2$, generate all $M$ torsion by the exact sequence. Every generator is rational.\n\\end{proof}\nAt residue characteristic different from seven, a unit is an $M$th power precisely when its residue is, by simple root lifting. A general element also needs valuation divisible by $M$. If $Q$ is the residue cardinality, $\\mu_M\\subset F$ precisely when $M\\mid Q-1$. Applying Proposition~\\ref{prop:torsionperiods} to \\eqref{eq:periodmatrix} gives the complete local test\n\\begin{equation}\\label{eq:full-local-torsion}\n\\begin{gathered}\n7^j\\mid Q-1,\\qquad7^j\\mid7c,","l.\n\\end{proof}\nAt residue characteristic different from seven, a unit is an $M$th power precisely when its residue is, by simple root lifting. A general element also needs valuation divisible by $M$. If $Q$ is the residue cardinality, $\\mu_M\\subset F$ precisely when $M\\mid Q-1$. Applying Proposition~\\ref{prop:torsionperiods} to \\eqref{eq:periodmatrix} gives the complete local test\n\\begin{equation}\\label{eq:full-local-torsion}\n\\begin{gathered}\n7^j\\mid Q-1,\\qquad7^j\\mid7c,\\\\\n\\phi\\in(\\kappa_v^\\times)^{7^j},\\qquad\n\\frac{C_0^7}{3^3A^3(\\sqrt5)^5}\\in(\\kappa_v^\\times)^{7^j}.\n\\end{gathered}\n\\end{equation}\nThe valuation condition on $c$ is not lost when replacing $C$ by its unit part $C_0$.","\\phi\\in(\\kappa_v^\\times)^{7^j},\\qquad\n\\frac{C_0^7}{3^3A^3(\\sqrt5)^5}\\in(\\kappa_v^\\times)^{7^j}.\n\\end{gathered}\n\\end{equation}\nThe valuation condition on $c$ is not lost when replacing $C$ by its unit part $C_0$.","At level seven, the last unit condition reduces to $75A$ being a seventh power. At level forty nine, exact multiplicative identities reduce it to\n\\[\n3\\cdot5^9A/C_0^{35}\\in(\\kappa_v^\\times)^{49}.\n\\]\nThus full forty nine torsion is rational if and only if $49\\mid Q-1$, $7\\mid c$, $\\phi$ is a forty ninth power residue, and the displayed rational unit is as well. These are two way local criteria for the stipulated local Jacobian.","They do not produce one global beta class or determine its actual depth. The archive retains positive local Frey families at $197,883,1373,1667$ satisfying the full criterion. Their survival is compatible with the later exclusion of a global beta supported only at those primes.","That exclusion illustrates why the common global coefficients matter. Correct inverse cyclotomic reciprocity on this four prime support leaves only inertia multiples of $(1,0,0,3)$. A nonzero class must therefore ramify at $197$ and $1667$. The natural rows in $(\\phi,3,\\sqrt5)$ coordinates are $(0,6,6)$ and $(0,3,4)$. They force both\n\\[\n6a_3+6a_5=0,\\qquad3a_3+4a_5=0.\n\\]\nThe determinant is six modulo seven, so $a_3=a_5=0$, contradicting \\eqref{eq:globalalpha}. No value of $a_\\phi$ can repair the contradiction because both unit symbols vanish. This is a global common coefficient exclusion. It does not retract any local positive family or prohibit an individual prime from a","That exclusion illustrates why the common global coefficients matter. Correct inverse cyclotomic reciprocity on this four prime support leaves only inertia multiples of $(1,0,0,3)$. A nonzero class must therefore ramify at $197$ and $1667$. The natural rows in $(\\phi,3,\\sqrt5)$ coordinates are $(0,6,6)$ and $(0,3,4)$. They force both\n\\[\n6a_3+6a_5=0,\\qquad3a_3+4a_5=0.\n\\]\nThe determinant is six modulo seven, so $a_3=a_5=0$, contradicting \\eqref{eq:globalalpha}. No value of $a_\\phi$ can repair the contradiction because both unit symbols vanish. This is a global common coefficient exclusion. It does not retract any local positive family or prohibit an individual prime from appearing in a larger conductor.","illustrates why the common global coefficients matter. Correct inverse cyclotomic reciprocity on this four prime support leaves only inertia multiples of $(1,0,0,3)$. A nonzero class must therefore ramify at $197$ and $1667$. The natural rows in $(\\phi,3,\\sqrt5)$ coordinates are $(0,6,6)$ and $(0,3,4)$. They force both\n\\[\n6a_3+6a_5=0,\\qquad3a_3+4a_5=0.\n\\]\nThe determinant is six modulo seven, so $a_3=a_5=0$, contradicting \\eqref{eq:globalalpha}. No value of $a_\\phi$ can repair the contradiction because both unit symbols vanish. This is a global common coefficient exclusion. It does not retract any local positive family or prohibit an individual prime from appearing in a larger conductor.","inverse cyclotomic reciprocity on this four prime support leaves only inertia multiples of $(1,0,0,3)$. A nonzero class must therefore ramify at $197$ and $1667$. The natural rows in $(\\phi,3,\\sqrt5)$ coordinates are $(0,6,6)$ and $(0,3,4)$. They force both\n\\[\n6a_3+6a_5=0,\\qquad3a_3+4a_5=0.\n\\]\nThe determinant is six modulo seven, so $a_3=a_5=0$, contradicting \\eqref{eq:globalalpha}. No value of $a_\\phi$ can repair the contradiction because both unit symbols vanish. This is a global common coefficient exclusion. It does not retract any local positive family or prohibit an individual prime from appearing in a larger conductor.","\\section{Twisted reciprocity and actual trace products}\\label{sec:reciprocity}\nThe opposite class has coefficient action $\\chi^{-1}$. Its ray relations therefore use the inverse cyclotomic eigenspace, not the ordinary character rows that will appear later for differences of primitives.","In the descended all inert regime, let $M=\\Q(\\zeta_7)$ and let $\\gamma\\in H^1(\\Q,\\F_7(\\chi^{-1}))$ restrict to the relevant beta class. Over $M$, the restriction is additive and satisfies\n\\[\n\\gamma(\\delta_a g\\delta_a^{-1})=a^{-1}\\gamma(g).\n\\]\nAt a conductor prime $q\\equiv1\\pmod7$, there are six degree one primes in $M$. At $q\\equiv-1\\pmod7$, there are three degree two primes. If $z_q$ is a chosen seventh root of unity and $\\ell_q$ is the corresponding seventh power logarithm, the local functional at the embedding $\\zeta_7\\mapsto z_q^b$ is $b s_q\\ell_q$.","regime, let $M=\\Q(\\zeta_7)$ and let $\\gamma\\in H^1(\\Q,\\F_7(\\chi^{-1}))$ restrict to the relevant beta class. Over $M$, the restriction is additive and satisfies\n\\[\n\\gamma(\\delta_a g\\delta_a^{-1})=a^{-1}\\gamma(g).\n\\]\nAt a conductor prime $q\\equiv1\\pmod7$, there are six degree one primes in $M$. At $q\\equiv-1\\pmod7$, there are three degree two primes. If $z_q$ is a chosen seventh root of unity and $\\ell_q$ is the corresponding seventh power logarithm, the local functional at the embedding $\\zeta_7\\mapsto z_q^b$ is $b s_q\\ell_q$.","ant beta class. Over $M$, the restriction is additive and satisfies\n\\[\n\\gamma(\\delta_a g\\delta_a^{-1})=a^{-1}\\gamma(g).\n\\]\nAt a conductor prime $q\\equiv1\\pmod7$, there are six degree one primes in $M$. At $q\\equiv-1\\pmod7$, there are three degree two primes. If $z_q$ is a chosen seventh root of unity and $\\ell_q$ is the corresponding seventh power logarithm, the local functional at the embedding $\\zeta_7\\mapsto z_q^b$ is $b s_q\\ell_q$.","the restriction is additive and satisfies\n\\[\n\\gamma(\\delta_a g\\delta_a^{-1})=a^{-1}\\gamma(g).\n\\]\nAt a conductor prime $q\\equiv1\\pmod7$, there are six degree one primes in $M$. At $q\\equiv-1\\pmod7$, there are three degree two primes. If $z_q$ is a chosen seventh root of unity and $\\ell_q$ is the corresponding seventh power logarithm, the local functional at the embedding $\\zeta_7\\mapsto z_q^b$ is $b s_q\\ell_q$.","For degree two, the indices $b$ and $-b$ represent the same prime. Frobenius sends a residue $x$ to $x^q$, and $\\ell_q(x^q)=-\\ell_q(x)$. Therefore\n\\[\n(-b)\\ell_q(x^q)=b\\ell_q(x).\n\\]\nThe functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator $\\pi_l$ outside the conductor,\n\\begin{equation}\\label{eq:orbitreciprocity}\nR_{l,q}=-\\sum_{b\\in\\mathrm{reps}_q}b\\ell_q(\\pi_l(z_q^b))\n=-f_q^{-1}\\sum_{b=1}^6b\\ell_q(\\pi_l(z_q^b)),\n\\end{equation}\nwhere $f_q=1$ or two. Using all six terms without dividing by two in the second case would double count each prime. Constant weights or inverse $b$ weights give different modules.","_q(x^q)=-\\ell_q(x)$. Therefore\n\\[\n(-b)\\ell_q(x^q)=b\\ell_q(x).\n\\]\nThe functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator $\\pi_l$ outside the conductor,\n\\begin{equation}\\label{eq:orbitreciprocity}\nR_{l,q}=-\\sum_{b\\in\\mathrm{reps}_q}b\\ell_q(\\pi_l(z_q^b))\n=-f_q^{-1}\\sum_{b=1}^6b\\ell_q(\\pi_l(z_q^b)),\n\\end{equation}\nwhere $f_q=1$ or two. Using all six terms without dividing by two in the second case would double count each prime. Constant weights or inverse $b$ weights give different modules.","dings. Principal reciprocity gives, for a test prime generator $\\pi_l$ outside the conductor,\n\\begin{equation}\\label{eq:orbitreciprocity}\nR_{l,q}=-\\sum_{b\\in\\mathrm{reps}_q}b\\ell_q(\\pi_l(z_q^b))\n=-f_q^{-1}\\sum_{b=1}^6b\\ell_q(\\pi_l(z_q^b)),\n\\end{equation}\nwhere $f_q=1$ or two. Using all six terms without dividing by two in the second case would double count each prime. Constant weights or inverse $b$ weights give different modules.","\\subsection{The same representation trace product}\nAt depth one and an unramified place of residue norm $Q\\equiv1\\pmod7$, write the actual Frobenius in the fixed natural frame as\n\\[\nF=\\begin{pmatrix}a&b","\\subsection{The same representation trace product}\nAt depth one and an unramified place of residue norm $Q\\equiv1\\pmod7$, write the actual Frobenius in the fixed natural frame as\n\\[\nF=\\begin{pmatrix}a&b\\\\7c&d\\end{pmatrix},\\qquad\\det F=Q.\n\\]\nPut $d=1+7D$ and $\\tr(F)=Q+1+7T$. Then $a=Q+7(T-D)$, and the determinant equation gives\n\\begin{equation}\\label{eq:trace-product-exact}\nbc=T+(Q-1)D+7TD-7D^2.\n\\end{equation}\nModulo seven this becomes\n\\begin{equation}\\label{eq:trace-product}\n\\frac{\\tr(F)-Q-1}{7}=\\alpha(F)\\beta(F)\\quad\\text{in }k.\n\\end{equation}\nThis is the coefficient field rank two trace of the actual representation. It is not half the rational four dimensional trace. The hypothesis $Q\\equiv1\\pmod7$ makes the extra diagonal term disappear. At other residue norms, \\eqref{eq:trace-product-exact} must be retained.","7c&d\\end{pmatrix},\\qquad\\det F=Q.\n\\]\nPut $d=1+7D$ and $\\tr(F)=Q+1+7T$. Then $a=Q+7(T-D)$, and the determinant equation gives\n\\begin{equation}\\label{eq:trace-product-exact}\nbc=T+(Q-1)D+7TD-7D^2.\n\\end{equation}\nModulo seven this becomes\n\\begin{equation}\\label{eq:trace-product}\n\\frac{\\tr(F)-Q-1}{7}=\\alpha(F)\\beta(F)\\quad\\text{in }k.\n\\end{equation}\nThis is the coefficient field rank two trace of the actual representation. It is not half the rational four dimensional trace. The hypothesis $Q\\equiv1\\pmod7$ makes the extra diagonal term disappear. At other residue norms, \\eqref{eq:trace-product-exact} must be retained.","and the determinant equation gives\n\\begin{equation}\\label{eq:trace-product-exact}\nbc=T+(Q-1)D+7TD-7D^2.\n\\end{equation}\nModulo seven this becomes\n\\begin{equation}\\label{eq:trace-product}\n\\frac{\\tr(F)-Q-1}{7}=\\alpha(F)\\beta(F)\\quad\\text{in }k.\n\\end{equation}\nThis is the coefficient field rank two trace of the actual representation. It is not half the rational four dimensional trace. The hypothesis $Q\\equiv1\\pmod7$ makes the extra diagonal term disappear. At other residue norms, \\eqref{eq:trace-product-exact} must be retained.","At good places, the right side must equal one of the corrections extracted from the common quartic \\eqref{eq:quartic}. At bad toric places, exact trace $Q+1$ makes the product zero. The zero product condition does not always imply $\\beta(F)=0$, because $\\alpha(F)$ may already vanish. This is the error corrected in the Block 70 draft. It was consequential: imposing the stronger zero beta condition would falsely remove valid necessary system data.","If the actual polynomial is\n\\[\nP(X)=X^4-SX^3+BX^2-QSX+Q^2,\n\\]\nthe correction $u=(\\tau-Q-1)/7$ has\n\\[\n\\operatorname{Tr}(u)=\\frac{S-2(Q+1)}7,\\qquad\n\\operatorname{Nm}(u)=\\frac{(B-2Q)-(Q+1)S+(Q+1)^2}{49}\n\\]\nmodulo seven, under the stated divisibility assumptions. The two implementations extract these values using different methods, including an integer trace discriminant and an enumeration of $k$ by trace and norm. Neither assigns a Frobenius value at a beta conductor prime by pretending beta is unramified there. Such conductor values require the separate self unit calculation of Section~\\ref{sec:self-conductor}.","\\subsection{Coefficient rigidity without radical identification}\nThe recovered coefficient theorem, initially in the descended depth one regime and later in the all depth semilinear formulation, normalizes the natural class as\n\\begin{equation}\\label{eq:rigidalpha}\n\\alpha=vw[\\phi]+r[3]+[\\sqrt5],\\qquad v,r\\in\\F_7^\\times.\n\\end{equation}\nIts mechanism retains the actual joint finite quotient. The three Kummer classes are independent after cyclotomic restriction. Distinct cyclotomic eigenspaces separate the natural and opposite images. Chebotarev then realizes joint vectors at good split primes, where the same quartic at the paired $K$ places forces products to be coefficient conjugate. The resulting conditions are $u^7=-u$ and $r^7=r$. In the descended subcase beta has an $\\F_7$ representative.","uation}\\label{eq:rigidalpha}\n\\alpha=vw[\\phi]+r[3]+[\\sqrt5],\\qquad v,r\\in\\F_7^\\times.\n\\end{equation}\nIts mechanism retains the actual joint finite quotient. The three Kummer classes are independent after cyclotomic restriction. Distinct cyclotomic eigenspaces separate the natural and opposite images. Chebotarev then realizes joint vectors at good split primes, where the same quartic at the paired $K$ places forces products to be coefficient conjugate. The resulting conditions are $u^7=-u$ and $r^7=r$. In the descended subcase beta has an $\\F_7$ representative.","\\end{equation}\nIts mechanism retains the actual joint finite quotient. The three Kummer classes are independent after cyclotomic restriction. Distinct cyclotomic eigenspaces separate the natural and opposite images. Chebotarev then realizes joint vectors at good split primes, where the same quartic at the paired $K$ places forces products to be coefficient conjugate. The resulting conditions are $u^7=-u$ and $r^7=r$. In the descended subcase beta has an $\\F_7$ representative.","}\nIts mechanism retains the actual joint finite quotient. The three Kummer classes are independent after cyclotomic restriction. Distinct cyclotomic eigenspaces separate the natural and opposite images. Chebotarev then realizes joint vectors at good split primes, where the same quartic at the paired $K$ places forces products to be coefficient conjugate. The resulting conditions are $u^7=-u$ and $r^7=r$. In the descended subcase beta has an $\\F_7$ representative.","e quotient. The three Kummer classes are independent after cyclotomic restriction. Distinct cyclotomic eigenspaces separate the natural and opposite images. Chebotarev then realizes joint vectors at good split primes, where the same quartic at the paired $K$ places forces products to be coefficient conjugate. The resulting conditions are $u^7=-u$ and $r^7=r$. In the descended subcase beta has an $\\F_7$ representative.","This is a cohomological coefficient statement. It does not identify the base radical with $w$ and does not descend an arbitrary Tate lattice. In the more general semilinear formulation, beta may decompose as a descended plus component and a minus component twisted by the quadratic character of $K/\\Q$. The bounded all inert computation uses the stipulated plus regime. The minus component remains a separate arithmetic obligation.","The early twenty five prime depth one system, using inert rational conductors congruent to one modulo seven and bounded by 2000, retained 112 normalized tuples. Further same polynomial tests reduced these to $38,12,6,2$, then a test at $1051$ removed that restricted conductor class. Degree two conductor primes $307,797,1483$, congruent to minus one modulo seven, could not be discarded. Their natural Kummer rows vanish, so they supply genuine extra columns in the necessary system.","The enlarged calculation retained 94 tuples after the tests at $701,911,1009$. Further tests at $1289,1429,1471,1499,1709$ reduced that list to 18. Finally the tests at $2129,2269,2311$ left eight. These are historical exact finite necessary system counts. The fresh reproduction scope and input pins are reported separately in Section~\\ref{sec:evidence}. A count quoted from an immutable report is not silently called a new run.","Every final tuple has natural ratio $r=6$, nonzero unit coefficient, and exactly nine ramifying rational primes. Six primes are mandatory:\n\\[\n307,337,547,883,1483,1933.\n\\]\nThe other three belong to $197,673,743,797$ in the four support patterns below. Each row has two coefficient conjugate tuples, according to the sign of $u$.\n\\begin{center}\\small\n\\begin{tabular}{clp{82mm}}\\toprule\nPattern&$u$&Ordered beta support and inertia vector","midrule\nA&$\\pm3w$&$307,337,547,673,743,797,883,1483,1933$\\newline $(4,4,2,4,3,6,3,1,3)$","[5pt]\nB&$\\pm3w$&$197,307,337,547,673,797,883,1483,1933$\\newline $(5,1,4,2,6,4,1,4,2)$","[5pt]\nC&$\\pm w$&$197,307,337,547,673,743,883,1483,1933$\\newline $(1,5,4,1,4,1,6,5,4)$","[5pt]\nD&$\\pm w$&$197,307,337,547,743,797,883,1483,1933$\\newline $(4,5,5,6,2,4,2,1,6)$","account for the eight coefficient conjugate tuples in the frozen bounded regime. Additional beta ramification changes the column system.}\nThe table is a necessary cover under the inherited geometry, actual depth one, all inert regime, and the requirement that every rational prime ramifying in beta is at most 2000. It is not a finite cover of all primitive integer triples. Additional divisors of $C$ unramified in beta remain allowed. New beta ramification is outside the cover and must add its columns everywhere, including in the Frobenius evaluations.","\\subsection{Why a homogeneous finite sieve can saturate}\nBlock 70 proved a limitation for the specified zero or good test family. Chebotarev supplies arbitrarily many eligible inert conductor primes where the natural Kummer rows all vanish. On $N$ such primes, the inverse cyclotomic ray character space has dimension $N$ over $\\F_7$ under the declared class and unit inputs. Each prescribed test zero value adds at most one linear equation. If $N$ exceeds the number of equations, a nonzero kernel remains. Every trace product at those tests is zero, so the bad alternative accepts the abstract data.","saturate}\nBlock 70 proved a limitation for the specified zero or good test family. Chebotarev supplies arbitrarily many eligible inert conductor primes where the natural Kummer rows all vanish. On $N$ such primes, the inverse cyclotomic ray character space has dimension $N$ over $\\F_7$ under the declared class and unit inputs. Each prescribed test zero value adds at most one linear equation. If $N$ exceeds the number of equations, a nonzero kernel remains. Every trace product at those tests is zero, so the bad alternative accepts the abstract data.","\\section{The fixed seven period target}\\label{sec:fixedseven}\nAt seven, principal units are not automatically seventh powers. The residue class of $8=1+7$ is a simple warning. The away from seven period simplification cannot be copied without an error bound. The inherited integral marked family supplies an error in $1+7^{7c}\\Oloc_F$, where $c=v_7(C)\\ge1$ and $F=\\Q_7(\\sqrt5)$. That specific error is sufficiently deep for the stated period window.","Let $X=[\\phi]$ and $H=[3]$ in the mod seven local Kummer group. Exact logarithmic first digits give\n\\[\nL(\\phi)=5\\overline{\\sqrt5},\\qquad L(3)=6,\\qquad L(\\sqrt5)=3,\n\\quad L(u)=\\frac{u^{48}-1}{7}\\pmod7.\n\\]\nThe two unit coordinates $X,H$ are independent, and $[\\sqrt5]=4H$. For a rational seven adic unit $A$, put $q_7(A)=(A^6-1)/7$ modulo seven. The local marked period comparison has\n\\[\nP=ZI+2XR,\\qquad Z=(3q_7(A)+5)H,\n\\]\nwith $R$ from \\eqref{eq:periodmatrix}.","in the mod seven local Kummer group. Exact logarithmic first digits give\n\\[\nL(\\phi)=5\\overline{\\sqrt5},\\qquad L(3)=6,\\qquad L(\\sqrt5)=3,\n\\quad L(u)=\\frac{u^{48}-1}{7}\\pmod7.\n\\]\nThe two unit coordinates $X,H$ are independent, and $[\\sqrt5]=4H$. For a rational seven adic unit $A$, put $q_7(A)=(A^6-1)/7$ modulo seven. The local marked period comparison has\n\\[\nP=ZI+2XR,\\qquad Z=(3q_7(A)+5)H,\n\\]\nwith $R$ from \\eqref{eq:periodmatrix}.","The actual coefficient action must intertwine the toric submodule and quotient period matrices. Their valuation matrix is exactly $7cI$, so the two action matrices agree after cancellation over $\\Z_7$. Their reduction commutes with $R$ and has characteristic polynomial $T^2-5$. The retained centralizer argument then gives one orientation $\\epsilon\\in\\{1,-1\\}$ and local scalar class\n\\begin{equation}\\label{eq:kappa7}\n\\kappa_7=2\\epsilon wX+(3q_7(A)+5)H.\n\\end{equation}\nMerely knowing $R^2=5I$ would not identify this as the actual coefficient action. The period intertwiner is the extra geometric input.","he actual coefficient action must intertwine the toric submodule and quotient period matrices. Their valuation matrix is exactly $7cI$, so the two action matrices agree after cancellation over $\\Z_7$. Their reduction commutes with $R$ and has characteristic polynomial $T^2-5$. The retained centralizer argument then gives one orientation $\\epsilon\\in\\{1,-1\\}$ and local scalar class\n\\begin{equation}\\label{eq:kappa7}\n\\kappa_7=2\\epsilon wX+(3q_7(A)+5)H.\n\\end{equation}\nMerely knowing $R^2=5I$ would not identify this as the actual coefficient action. The period intertwiner is the extra geometric input.","matrices. Their valuation matrix is exactly $7cI$, so the two action matrices agree after cancellation over $\\Z_7$. Their reduction commutes with $R$ and has characteristic polynomial $T^2-5$. The retained centralizer argument then gives one orientation $\\epsilon\\in\\{1,-1\\}$ and local scalar class\n\\begin{equation}\\label{eq:kappa7}\n\\kappa_7=2\\epsilon wX+(3q_7(A)+5)H.\n\\end{equation}\nMerely knowing $R^2=5I$ would not identify this as the actual coefficient action. The period intertwiner is the extra geometric input.","The globally normalized natural class \\eqref{eq:rigidalpha} restricts to $vwX+(r+4)H$. Comparing it with \\eqref{eq:kappa7} allows one common nonzero graded scalar $g=v/(2\\epsilon)$. The remaining coordinate gives\n\\[\n2r+1=\\epsilon v(3q_7(A)+5),\n\\]\nand hence\n\\begin{equation}\\label{eq:fermatquotient}\n(2r+1)^2=v^2(3q_7(A)+5)^2\\quad\\text{in }\\F_7.\n\\end{equation}\nThe scalar is not independently adjustable in the two unit coordinates.","For the final bounded patterns $r=6$. Patterns A and B, with $u=\\pm3w$, require $q_7(A)\\in\\{0,6\\}$ and $A^6\\equiv1$ or $43$ modulo $49$. Patterns C and D, with $u=\\pm w$, require $q_7(A)\\in\\{1,5\\}$ and $A^6\\equiv8$ or $36$. Each tuple still permits twelve rational unit residues $A$ modulo $49$. None is eliminated.","\\subsection{Higher precision retains a valuation coordinate}\nFor $1\\le j\\le7c-1$, the deep period errors vanish in $F^\\times/F^{\\times7^j}$. This follows because seventh powering takes $1+7^s\\Oloc_F$ bijectively onto $1+7^{s+1}\\Oloc_F$ for $s\\ge1$. The guaranteed window does not assert that actual errors become nontrivial at the next precision.","Use the three coordinate basis\n\\[\nX_j=[\\phi],\\qquad H_j=[3],\\qquad T_j=[7].\n\\]\nThe third coordinate is valuation. The actual local scalar class in the window is\n\\begin{equation}\\label{eq:higherkappa}\n\\kappa_j=2\\epsilon wX_j+\\Lambda_jH_j+7cT_j,\n\\end{equation}\nwhere\n\\[\n\\Lambda_j=7\\ell_j(C_0)-3\\ell_j(A)-3-5d_j,\n\\quad \\ell_j(a)=\\frac{\\log(a^6)}{\\log(3^6)}\\pmod{7^j},\n\\quad d_j=\\frac{\\ell_j(5)}2.\n\\]\nAt level $49$, with $Q_A=(A^6-1)/7$ modulo $49$ and $q_0=(C_0^6-1)/7$ modulo seven,\n\\begin{equation}\\label{eq:level49}\n\\Lambda_2=47+38Q_A+14Q_A^2-7q_0\\pmod{49}.\n\\end{equation}\nBoth $C_0$ and the coordinate $7c[7]$ disappeared modulo seven. Neither may be omitted modulo forty nine.","three coordinate basis\n\\[\nX_j=[\\phi],\\qquad H_j=[3],\\qquad T_j=[7].\n\\]\nThe third coordinate is valuation. The actual local scalar class in the window is\n\\begin{equation}\\label{eq:higherkappa}\n\\kappa_j=2\\epsilon wX_j+\\Lambda_jH_j+7cT_j,\n\\end{equation}\nwhere\n\\[\n\\Lambda_j=7\\ell_j(C_0)-3\\ell_j(A)-3-5d_j,\n\\quad \\ell_j(a)=\\frac{\\log(a^6)}{\\log(3^6)}\\pmod{7^j},\n\\quad d_j=\\frac{\\ell_j(5)}2.\n\\]\nAt level $49$, with $Q_A=(A^6-1)/7$ modulo $49$ and $q_0=(C_0^6-1)/7$ modulo seven,\n\\begin{equation}\\label{eq:level49}\n\\Lambda_2=47+38Q_A+14Q_A^2-7q_0\\pmod{49}.\n\\end{equation}\nBoth $C_0$ and the coordinate $7c[7]$ disappeared modulo seven. Neither may be omitted modulo forty nine.","dinate basis\n\\[\nX_j=[\\phi],\\qquad H_j=[3],\\qquad T_j=[7].\n\\]\nThe third coordinate is valuation. The actual local scalar class in the window is\n\\begin{equation}\\label{eq:higherkappa}\n\\kappa_j=2\\epsilon wX_j+\\Lambda_jH_j+7cT_j,\n\\end{equation}\nwhere\n\\[\n\\Lambda_j=7\\ell_j(C_0)-3\\ell_j(A)-3-5d_j,\n\\quad \\ell_j(a)=\\frac{\\log(a^6)}{\\log(3^6)}\\pmod{7^j},\n\\quad d_j=\\frac{\\ell_j(5)}2.\n\\]\nAt level $49$, with $Q_A=(A^6-1)/7$ modulo $49$ and $q_0=(C_0^6-1)/7$ modulo seven,\n\\begin{equation}\\label{eq:level49}\n\\Lambda_2=47+38Q_A+14Q_A^2-7q_0\\pmod{49}.\n\\end{equation}\nBoth $C_0$ and the coordinate $7c[7]$ disappeared modulo seven. Neither may be omitted modulo forty nine.","hi],\\qquad H_j=[3],\\qquad T_j=[7].\n\\]\nThe third coordinate is valuation. The actual local scalar class in the window is\n\\begin{equation}\\label{eq:higherkappa}\n\\kappa_j=2\\epsilon wX_j+\\Lambda_jH_j+7cT_j,\n\\end{equation}\nwhere\n\\[\n\\Lambda_j=7\\ell_j(C_0)-3\\ell_j(A)-3-5d_j,\n\\quad \\ell_j(a)=\\frac{\\log(a^6)}{\\log(3^6)}\\pmod{7^j},\n\\quad d_j=\\frac{\\ell_j(5)}2.\n\\]\nAt level $49$, with $Q_A=(A^6-1)/7$ modulo $49$ and $q_0=(C_0^6-1)/7$ modulo seven,\n\\begin{equation}\\label{eq:level49}\n\\Lambda_2=47+38Q_A+14Q_A^2-7q_0\\pmod{49}.\n\\end{equation}\nBoth $C_0$ and the coordinate $7c[7]$ disappeared modulo seven. Neither may be omitted modulo forty nine.","d H_j=[3],\\qquad T_j=[7].\n\\]\nThe third coordinate is valuation. The actual local scalar class in the window is\n\\begin{equation}\\label{eq:higherkappa}\n\\kappa_j=2\\epsilon wX_j+\\Lambda_jH_j+7cT_j,\n\\end{equation}\nwhere\n\\[\n\\Lambda_j=7\\ell_j(C_0)-3\\ell_j(A)-3-5d_j,\n\\quad \\ell_j(a)=\\frac{\\log(a^6)}{\\log(3^6)}\\pmod{7^j},\n\\quad d_j=\\frac{\\ell_j(5)}2.\n\\]\nAt level $49$, with $Q_A=(A^6-1)/7$ modulo $49$ and $q_0=(C_0^6-1)/7$ modulo seven,\n\\begin{equation}\\label{eq:level49}\n\\Lambda_2=47+38Q_A+14Q_A^2-7q_0\\pmod{49}.\n\\end{equation}\nBoth $C_0$ and the coordinate $7c[7]$ disappeared modulo seven. Neither may be omitted modulo forty nine.","te is valuation. The actual local scalar class in the window is\n\\begin{equation}\\label{eq:higherkappa}\n\\kappa_j=2\\epsilon wX_j+\\Lambda_jH_j+7cT_j,\n\\end{equation}\nwhere\n\\[\n\\Lambda_j=7\\ell_j(C_0)-3\\ell_j(A)-3-5d_j,\n\\quad \\ell_j(a)=\\frac{\\log(a^6)}{\\log(3^6)}\\pmod{7^j},\n\\quad d_j=\\frac{\\ell_j(5)}2.\n\\]\nAt level $49$, with $Q_A=(A^6-1)/7$ modulo $49$ and $q_0=(C_0^6-1)/7$ modulo seven,\n\\begin{equation}\\label{eq:level49}\n\\Lambda_2=47+38Q_A+14Q_A^2-7q_0\\pmod{49}.\n\\end{equation}\nBoth $C_0$ and the coordinate $7c[7]$ disappeared modulo seven. Neither may be omitted modulo forty nine.","abel{eq:higherkappa}\n\\kappa_j=2\\epsilon wX_j+\\Lambda_jH_j+7cT_j,\n\\end{equation}\nwhere\n\\[\n\\Lambda_j=7\\ell_j(C_0)-3\\ell_j(A)-3-5d_j,\n\\quad \\ell_j(a)=\\frac{\\log(a^6)}{\\log(3^6)}\\pmod{7^j},\n\\quad d_j=\\frac{\\ell_j(5)}2.\n\\]\nAt level $49$, with $Q_A=(A^6-1)/7$ modulo $49$ and $q_0=(C_0^6-1)/7$ modulo seven,\n\\begin{equation}\\label{eq:level49}\n\\Lambda_2=47+38Q_A+14Q_A^2-7q_0\\pmod{49}.\n\\end{equation}\nBoth $C_0$ and the coordinate $7c[7]$ disappeared modulo seven. Neither may be omitted modulo forty nine.","pa_j=2\\epsilon wX_j+\\Lambda_jH_j+7cT_j,\n\\end{equation}\nwhere\n\\[\n\\Lambda_j=7\\ell_j(C_0)-3\\ell_j(A)-3-5d_j,\n\\quad \\ell_j(a)=\\frac{\\log(a^6)}{\\log(3^6)}\\pmod{7^j},\n\\quad d_j=\\frac{\\ell_j(5)}2.\n\\]\nAt level $49$, with $Q_A=(A^6-1)/7$ modulo $49$ and $q_0=(C_0^6-1)/7$ modulo seven,\n\\begin{equation}\\label{eq:level49}\n\\Lambda_2=47+38Q_A+14Q_A^2-7q_0\\pmod{49}.\n\\end{equation}\nBoth $C_0$ and the coordinate $7c[7]$ disappeared modulo seven. Neither may be omitted modulo forty nine.","At higher actual lattice depth, put $m=\\lfloor n/2\\rfloor+1$. The inherited diagonal theorem supplies an actual global upper extension $\\alpha_j$ for $1\\le j\\le m$. In the common range $1\\le j\\le\\min(m,7c-1)$, its local coordinates satisfy $U_j=2\\epsilon wg_j$, $V_j=\\Lambda_jg_j$, $W_j=7cg_j$ for one unit $g_j$. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision $1\\le j\\le m$, even beyond the proved unit period window. Consequently\n\\[\nv_7^{\\mathrm{trunc}}(W_j)=\\min(j,1+v_7(c)).\n\\]\nAt depth one, only the residual global triangular precision is supplied. Replacing the actual mod forty nine representation by an arbitrary triangular Kummer lift would erase its lower $7\\beta$ entry. The local higher class","At higher actual lattice depth, put $m=\\lfloor n/2\\rfloor+1$. The inherited diagonal theorem supplies an actual global upper extension $\\alpha_j$ for $1\\le j\\le m$. In the common range $1\\le j\\le\\min(m,7c-1)$, its local coordinates satisfy $U_j=2\\epsilon wg_j$, $V_j=\\Lambda_jg_j$, $W_j=7cg_j$ for one unit $g_j$. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision $1\\le j\\le m$, even beyond the proved unit period window. Consequently\n\\[\nv_7^{\\mathrm{trunc}}(W_j)=\\min(j,1+v_7(c)).\n\\]\nAt depth one, only the residual global triangular precision is supplied. Replacing the actual mod forty nine representation by an arbitrary triangular Kummer lift would erase its lower $7\\beta$ entry. The local higher class exists, but its global comparison must retain the nontriangular deformation.","lattice depth, put $m=\\lfloor n/2\\rfloor+1$. The inherited diagonal theorem supplies an actual global upper extension $\\alpha_j$ for $1\\le j\\le m$. In the common range $1\\le j\\le\\min(m,7c-1)$, its local coordinates satisfy $U_j=2\\epsilon wg_j$, $V_j=\\Lambda_jg_j$, $W_j=7cg_j$ for one unit $g_j$. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision $1\\le j\\le m$, even beyond the proved unit period window. Consequently\n\\[\nv_7^{\\mathrm{trunc}}(W_j)=\\min(j,1+v_7(c)).\n\\]\nAt depth one, only the residual global triangular precision is supplied. Replacing the actual mod forty nine representation by an arbitrary triangular Kummer lift would erase its lower $7\\beta$ entry. The local higher class exists, but its global comparison must retain the nontriangular deformation.","orem supplies an actual global upper extension $\\alpha_j$ for $1\\le j\\le m$. In the common range $1\\le j\\le\\min(m,7c-1)$, its local coordinates satisfy $U_j=2\\epsilon wg_j$, $V_j=\\Lambda_jg_j$, $W_j=7cg_j$ for one unit $g_j$. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision $1\\le j\\le m$, even beyond the proved unit period window. Consequently\n\\[\nv_7^{\\mathrm{trunc}}(W_j)=\\min(j,1+v_7(c)).\n\\]\nAt depth one, only the residual global triangular precision is supplied. Replacing the actual mod forty nine representation by an arbitrary triangular Kummer lift would erase its lower $7\\beta$ entry. The local higher class exists, but its global comparison must retain the nontriangular deformation.","t valuation matrix preserves the valuation identity at every legitimate triangular precision $1\\le j\\le m$, even beyond the proved unit period window. Consequently\n\\[\nv_7^{\\mathrm{trunc}}(W_j)=\\min(j,1+v_7(c)).\n\\]\nAt depth one, only the residual global triangular precision is supplied. Replacing the actual mod forty nine representation by an arbitrary triangular Kummer lift would erase its lower $7\\beta$ entry. The local higher class exists, but its global comparison must retain the nontriangular deformation.","The archive contains exact seven adic fifth root positive controls for all admitted local residues. Their $B$ values solve $B^5=C^7-A^3$ in $\\Q_7$. They are not asserted rational or ordinary integers. The higher local period criterion therefore provides real conditions while still leaving every bounded tuple locally possible.","\\section{The coupled adjoint equations}\\label{sec:deformation}\nChoose one reference Kummer representation over $\\Oloc/49$,\n\\begin{equation}\\label{eq:rho0}\n\\rho_0(g)=\\begin{pmatrix}\\chi(g)&b_0(g)","\\section{The coupled adjoint equations}\\label{sec:deformation}\nChoose one reference Kummer representation over $\\Oloc/49$,\n\\begin{equation}\\label{eq:rho0}\n\\rho_0(g)=\\begin{pmatrix}\\chi(g)&b_0(g)\\\\0&1\\end{pmatrix},\n\\end{equation}\nwhose reduction has the same residual $\\alpha$ as the actual natural representation. The reference is built by lifting the coefficients of the same global classes $[\\phi],[3],[\\sqrt5]$. It has no seven valuation coordinate at the fixed place. It is an auxiliary representation, not the actual one.","0&1\\end{pmatrix},\n\\end{equation}\nwhose reduction has the same residual $\\alpha$ as the actual natural representation. The reference is built by lifting the coefficients of the same global classes $[\\phi],[3],[\\sqrt5]$. It has no seven valuation coordinate at the fixed place. It is an auxiliary representation, not the actual one.","Every determinant $\\chi$ framed lift of this residual representation has a unique expression\n\\begin{equation}\\label{eq:adjoint}\n\\rho(g)=(I+7X(g))\\rho_0(g),\\qquad\nX(g)=\\begin{pmatrix}p(g)&q(g)","Every determinant $\\chi$ framed lift of this residual representation has a unique expression\n\\begin{equation}\\label{eq:adjoint}\n\\rho(g)=(I+7X(g))\\rho_0(g),\\qquad\nX(g)=\\begin{pmatrix}p(g)&q(g)\\\\\\beta(g)&-p(g)\\end{pmatrix}\\in\\mathfrak{sl}_2(k).\n\\end{equation}\nThe determinant condition makes $X$ traceless. Multiplication shows that $\\rho$ is a representation if and only if\n\\[\nX(gh)=X(g)+\\Ad(\\overline\\rho_0(g))X(h).\n\\]\nFor $\\overline\\rho_0(g)=\\left(\\begin{smallmatrix}x&a","beta(g)&-p(g)\\end{pmatrix}\\in\\mathfrak{sl}_2(k).\n\\end{equation}\nThe determinant condition makes $X$ traceless. Multiplication shows that $\\rho$ is a representation if and only if\n\\[\nX(gh)=X(g)+\\Ad(\\overline\\rho_0(g))X(h).\n\\]\nFor $\\overline\\rho_0(g)=\\left(\\begin{smallmatrix}x&a","beta(g)&-p(g)\\end{pmatrix}\\in\\mathfrak{sl}_2(k).\n\\end{equation}\nThe determinant condition makes $X$ traceless. Multiplication shows that $\\rho$ is a representation if and only if\n\\[\nX(gh)=X(g)+\\Ad(\\overline\\rho_0(g))X(h).\n\\]\nFor $\\overline\\rho_0(g)=\\left(\\begin{smallmatrix}x&a\\\\0&1\\end{smallmatrix}\\right)$, direct conjugation sends $(p,q,b)$ to\n\\[\n(p+ax^{-1}b,\\;xq-2ap-a^2x^{-1}b,\\;x^{-1}b).\n\\]\nThus the three coupled equations are\n\\begin{align}\n\\beta(gh)&=\\beta(g)+\\chi(g)^{-1}\\beta(h),\\label{eq:beta-cocycle}","0&1\\end{smallmatrix}\\right)$, direct conjugation sends $(p,q,b)$ to\n\\[\n(p+ax^{-1}b,\\;xq-2ap-a^2x^{-1}b,\\;x^{-1}b).\n\\]\nThus the three coupled equations are\n\\begin{align}\n\\beta(gh)&=\\beta(g)+\\chi(g)^{-1}\\beta(h),\\label{eq:beta-cocycle}","0&1\\end{smallmatrix}\\right)$, direct conjugation sends $(p,q,b)$ to\n\\[\n(p+ax^{-1}b,\\;xq-2ap-a^2x^{-1}b,\\;x^{-1}b).\n\\]\nThus the three coupled equations are\n\\begin{align}\n\\beta(gh)&=\\beta(g)+\\chi(g)^{-1}\\beta(h),\\label{eq:beta-cocycle}\\\\\np(gh)&=p(g)+p(h)+\\alpha(g)\\chi(g)^{-1}\\beta(h),\\label{eq:p-cocycle}\\\\\nq(gh)&=q(g)+\\chi(g)q(h)-2\\alpha(g)p(h)\n-\\alpha(g)^2\\chi(g)^{-1}\\beta(h).\\label{eq:q-cocycle}\n\\end{align}\nAll coefficients in these equations lie in $k$. The reference itself is a representation over $\\Oloc/49$, so no characteristic seven carry term has been silently discarded in deriving them.","q(gh)&=q(g)+\\chi(g)q(h)-2\\alpha(g)p(h)\n-\\alpha(g)^2\\chi(g)^{-1}\\beta(h).\\label{eq:q-cocycle}\n\\end{align}\nAll coefficients in these equations lie in $k$. The reference itself is a representation over $\\Oloc/49$, so no characteristic seven carry term has been silently discarded in deriving them.","Throughout, $\\alpha$ and beta are normalized continuous cocycles with their displayed coefficient actions. Their cup has trivial target action and is explicitly $(\\alpha\\cup\\beta)(g,h)=\\alpha(g)\\chi(g)^{-1}\\beta(h)$. Define the ordinary differential by\n\\[\ndp(g,h)=p(g)+p(h)-p(gh)\n\\]\nand the twisted differential by $d_\\chi q(g,h)=q(g)+\\chi(g)q(h)-q(gh)$. Then\n\\begin{align}\ndp&=-\\alpha\\cup\\beta,\\label{eq:p-equation}","Throughout, $\\alpha$ and beta are normalized continuous cocycles with their displayed coefficient actions. Their cup has trivial target action and is explicitly $(\\alpha\\cup\\beta)(g,h)=\\alpha(g)\\chi(g)^{-1}\\beta(h)$. Define the ordinary differential by\n\\[\ndp(g,h)=p(g)+p(h)-p(gh)\n\\]\nand the twisted differential by $d_\\chi q(g,h)=q(g)+\\chi(g)q(h)-q(gh)$. Then\n\\begin{align}\ndp&=-\\alpha\\cup\\beta,\\label{eq:p-equation}\\\\\nd_\\chi q(g,h)&=2\\alpha(g)p(h)+\\alpha(g)^2\\chi(g)^{-1}\\beta(h).\\label{eq:q-equation}\n\\end{align}\nThe right side of \\eqref{eq:q-equation} is a two cocycle once the preceding equations hold. For a general fixed $p$, its vanishing in cohomology is the next bare existence problem. Solving the $p$ equation alone does not formally solve the $q$ equation. The project does not establish that this bare upper class is nonzero in the actual branches. Prescribed upper geometry and integral lifting retain additional conditions.","d_\\chi q(g,h)&=2\\alpha(g)p(h)+\\alpha(g)^2\\chi(g)^{-1}\\beta(h).\\label{eq:q-equation}\n\\end{align}\nThe right side of \\eqref{eq:q-equation} is a two cocycle once the preceding equations hold. For a general fixed $p$, its vanishing in cohomology is the next bare existence problem. Solving the $p$ equation alone does not formally solve the $q$ equation. The project does not establish that this bare upper class is nonzero in the actual branches. Prescribed upper geometry and integral lifting retain additional conditions.","The actual trace is\n\\begin{equation}\\label{eq:adjointtrace}\n\\tr\\rho(g)=\\chi(g)+1\n+7\\bigl((\\chi(g)-1)p(g)+\\alpha(g)\\beta(g)\\bigr)\\pmod{49}.\n\\end{equation}\nThe upper cochain $q$ is invisible to trace at this precision. On $\\chi(g)=1$, the diagonal cochain $p$ is also invisible and only the product $\\alpha\\beta$ remains. This explains why a homogeneous Frobenius sieve can leave essential deformation coordinates undetermined.","Strict conjugation by $I+7Y$ changes $X$ by $Y-\\Ad(\\overline\\rho_0)Y$. Independent local changes can simplify representatives but do not create a common global integral frame. The affine target in the last part of this reader is formulated after one frame and one reference have been fixed.","\\subsection{A delayed tame obstruction}\nSuppose at a beta support prime the actual inertia and Frobenius have\n\\[\nI+7M\\pmod{49},\\qquad I+7A\\pmod{49},\\qquad Q=1+7h\\pmod{49}.\n\\]\nThe tame relation is automatically insensitive to some first digit changes modulo forty nine. One precision later it forces\n\\begin{equation}\\label{eq:tame-next}\n[A,M]=hM\\quad\\text{over }k.\n\\end{equation}\nA trace invisible correction modulo forty nine need not pass \\eqref{eq:tame-next}. The archive retains a seven valuation correction that fails this next relation at $883$ when complete inertia is held fixed. This excludes that strengthening inside the stipulated deformation fiber. It does not","\\subsection{A delayed tame obstruction}\nSuppose at a beta support prime the actual inertia and Frobenius have\n\\[\nI+7M\\pmod{49},\\qquad I+7A\\pmod{49},\\qquad Q=1+7h\\pmod{49}.\n\\]\nThe tame relation is automatically insensitive to some first digit changes modulo forty nine. One precision later it forces\n\\begin{equation}\\label{eq:tame-next}\n[A,M]=hM\\quad\\text{over }k.\n\\end{equation}\nA trace invisible correction modulo forty nine need not pass \\eqref{eq:tame-next}. The archive retains a seven valuation correction that fails this next relation at $883$ when complete inertia is held fixed. This excludes that strengthening inside the stipulated deformation fiber. It does not prove that the actual valuation coordinate is zero in every candidate, because equality of a finite trace list is weaker than equality of the entire framed inertia and trace data used by the","9},\\qquad Q=1+7h\\pmod{49}.\n\\]\nThe tame relation is automatically insensitive to some first digit changes modulo forty nine. One precision later it forces\n\\begin{equation}\\label{eq:tame-next}\n[A,M]=hM\\quad\\text{over }k.\n\\end{equation}\nA trace invisible correction modulo forty nine need not pass \\eqref{eq:tame-next}. The archive retains a seven valuation correction that fails this next relation at $883$ when complete inertia is held fixed. This excludes that strengthening inside the stipulated deformation fiber. It does not prove that the actual valuation coordinate is zero in every candidate, because equality of a finite trace list is weaker than equality of the entire framed inertia and trace data used by the fiber theorem.","\\section{A mixed trace that detects the valuation}\\label{sec:mixedtrace}\nThe valuation coordinate can be read from a joint word in the same representation. For $T(g)=\\tr_{\\Oloc}\\rho(g)$, define\n\\[\nD(\\tau,g)=T(\\tau g)-T(\\tau)-T(g)+2.\n\\]\nThe elementary identity\n\\begin{equation}\\label{eq:mixedidentity}\nD(\\tau,g)=\\tr\\bigl((\\rho(\\tau)-I)(\\rho(g)-I)\\bigr)\n\\end{equation}\nshows why the product label matters. Two lists of individual traces do not determine the trace of the product.","The local Kummer basis $[\\phi],[3],[7]$ supplies a unit annihilating probe $\\tau_j$ after adjoining the required $7^j$ roots of unity. It kills the unit coordinates and evaluates the uniformizer coordinate to one. The inherited construction uses the cyclotomic action to show restriction remains injective. At higher cyclotomic levels this is not a blanket prime to seven degree argument, because the extension also has a seven primary part.","To define the valuation functional used here, strictly align the actual local toric cyclotomic line at $K_7$ with the reference line. The aligned local representative of $X$ has $p=\\beta=0$ and upper entry $q$ a $k(\\chi)$ Kummer cocycle. Define $\\nu_7(X)$ to be its $[7]$ valuation coordinate. Remaining aligned strict gauges change $q$ by a coboundary and a scalar multiple of residual $\\alpha$, both of valuation zero. Thus the coordinate is well defined under these strict equivalences. The reference $b_0$ has valuation zero and the actual difference is $b_7-b_0=7q$, giving the fixed value below. This local alignment never erases beta from the global deformation.","_7$ with the reference line. The aligned local representative of $X$ has $p=\\beta=0$ and upper entry $q$ a $k(\\chi)$ Kummer cocycle. Define $\\nu_7(X)$ to be its $[7]$ valuation coordinate. Remaining aligned strict gauges change $q$ by a coboundary and a scalar multiple of residual $\\alpha$, both of valuation zero. Thus the coordinate is well defined under these strict equivalences. The reference $b_0$ has valuation zero and the actual difference is $b_7-b_0=7q$, giving the fixed value below. This local alignment never erases beta from the global deformation.","ntative of $X$ has $p=\\beta=0$ and upper entry $q$ a $k(\\chi)$ Kummer cocycle. Define $\\nu_7(X)$ to be its $[7]$ valuation coordinate. Remaining aligned strict gauges change $q$ by a coboundary and a scalar multiple of residual $\\alpha$, both of valuation zero. Thus the coordinate is well defined under these strict equivalences. The reference $b_0$ has valuation zero and the actual difference is $b_7-b_0=7q$, giving the fixed value below. This local alignment never erases beta from the global deformation.","The actual period valuation matrix is $7cI$, so this probe has\n\\[\n\\rho(\\tau_j)=I+7cPE_{12}P^{-1}\\pmod{7^j},\n\\]\nwhere the same local toric line has the required alignment with the natural line. At actual depth one and a detecting $g$ in the residual kernel, \\eqref{eq:mixedidentity} gives\n\\begin{equation}\\label{eq:mixedvalue}\nD(\\tau,g)=49\\nu_7(X)\\beta(g)\\pmod{343}.\n\\end{equation}\nThe actual fixed seven period comparison requires\n\\[\n\\nu_7(X)=c\\frac{v}{2\\epsilon}\\quad\\text{in }\\F_7.\n\\]\nIf $\\beta(g)\\ne0$, the mixed trace determines this value.","x is $7cI$, so this probe has\n\\[\n\\rho(\\tau_j)=I+7cPE_{12}P^{-1}\\pmod{7^j},\n\\]\nwhere the same local toric line has the required alignment with the natural line. At actual depth one and a detecting $g$ in the residual kernel, \\eqref{eq:mixedidentity} gives\n\\begin{equation}\\label{eq:mixedvalue}\nD(\\tau,g)=49\\nu_7(X)\\beta(g)\\pmod{343}.\n\\end{equation}\nThe actual fixed seven period comparison requires\n\\[\n\\nu_7(X)=c\\frac{v}{2\\epsilon}\\quad\\text{in }\\F_7.\n\\]\nIf $\\beta(g)\\ne0$, the mixed trace determines this value.","^{-1}\\pmod{7^j},\n\\]\nwhere the same local toric line has the required alignment with the natural line. At actual depth one and a detecting $g$ in the residual kernel, \\eqref{eq:mixedidentity} gives\n\\begin{equation}\\label{eq:mixedvalue}\nD(\\tau,g)=49\\nu_7(X)\\beta(g)\\pmod{343}.\n\\end{equation}\nThe actual fixed seven period comparison requires\n\\[\n\\nu_7(X)=c\\frac{v}{2\\epsilon}\\quad\\text{in }\\F_7.\n\\]\nIf $\\beta(g)\\ne0$, the mixed trace determines this value.","The all depth formula uses $t=1+v_7(c)$, precision $j\\ge n+t+1$, the probe $\\tau_j$, and a detecting $g\\in\\ker\\overline\\rho$ with $\\beta(g)\\ne0$. It is\n\\[\nv_7(D(\\tau_j,g))=n+t,\n\\qquad\n\\frac{D(\\tau_j,g)}{7^{n+t}}\n=\\frac{c}{7^{t-1}}\\frac{v}{2\\epsilon}\\beta(g)\\pmod7.\n\\]\nThe coefficient on the right is nonzero in the detecting case. This is an exact valuation statement under the probe and actual geometric hypotheses. It does not bound $n$ or $c$.","c)$, precision $j\\ge n+t+1$, the probe $\\tau_j$, and a detecting $g\\in\\ker\\overline\\rho$ with $\\beta(g)\\ne0$. It is\n\\[\nv_7(D(\\tau_j,g))=n+t,\n\\qquad\n\\frac{D(\\tau_j,g)}{7^{n+t}}\n=\\frac{c}{7^{t-1}}\\frac{v}{2\\epsilon}\\beta(g)\\pmod7.\n\\]\nThe coefficient on the right is nonzero in the detecting case. This is an exact valuation statement under the probe and actual geometric hypotheses. It does not bound $n$ or $c$.","ecision $j\\ge n+t+1$, the probe $\\tau_j$, and a detecting $g\\in\\ker\\overline\\rho$ with $\\beta(g)\\ne0$. It is\n\\[\nv_7(D(\\tau_j,g))=n+t,\n\\qquad\n\\frac{D(\\tau_j,g)}{7^{n+t}}\n=\\frac{c}{7^{t-1}}\\frac{v}{2\\epsilon}\\beta(g)\\pmod7.\n\\]\nThe coefficient on the right is nonzero in the detecting case. This is an exact valuation statement under the probe and actual geometric hypotheses. It does not bound $n$ or $c$.","\\begin{example}\nLet $U_a=\\left(\\begin{smallmatrix}1&7a\\\\0&1\\end{smallmatrix}\\right)$ and $V=\\left(\\begin{smallmatrix}1&0","0&1\\end{smallmatrix}\\right)$ and $V=\\left(\\begin{smallmatrix}1&0\\\\7&1\\end{smallmatrix}\\right)$. Both have trace two and determinant one, for every $a$. Their product has trace $2+49a$. Thus $a=1,2$ give product traces $51,100$ modulo $343$, while every individual trace and every trace modulo forty nine loses this distinction.\n\\end{example}\nThese matrices are exact controls for the information loss. They are not claimed to be the actual labelled probe and detecting element. The current eight tuples do not supply $T(\\tau)$, $T(g)$, and $T(\\tau g)$ from one numerical global realization. The required mixed value is a target, not an observed contradiction.","7&1\\end{smallmatrix}\\right)$. Both have trace two and determinant one, for every $a$. Their product has trace $2+49a$. Thus $a=1,2$ give product traces $51,100$ modulo $343$, while every individual trace and every trace modulo forty nine loses this distinction.\n\\end{example}\nThese matrices are exact controls for the information loss. They are not claimed to be the actual labelled probe and detecting element. The current eight tuples do not supply $T(\\tau)$, $T(g)$, and $T(\\tau g)$ from one numerical global realization. The required mixed value is a target, not an observed contradiction.","d determinant one, for every $a$. Their product has trace $2+49a$. Thus $a=1,2$ give product traces $51,100$ modulo $343$, while every individual trace and every trace modulo forty nine loses this distinction.\n\\end{example}\nThese matrices are exact controls for the information loss. They are not claimed to be the actual labelled probe and detecting element. The current eight tuples do not supply $T(\\tau)$, $T(g)$, and $T(\\tau g)$ from one numerical global realization. The required mixed value is a target, not an observed contradiction.","nt one, for every $a$. Their product has trace $2+49a$. Thus $a=1,2$ give product traces $51,100$ modulo $343$, while every individual trace and every trace modulo forty nine loses this distinction.\n\\end{example}\nThese matrices are exact controls for the information loss. They are not claimed to be the actual labelled probe and detecting element. The current eight tuples do not supply $T(\\tau)$, $T(g)$, and $T(\\tau g)$ from one numerical global realization. The required mixed value is a target, not an observed contradiction.","every $a$. Their product has trace $2+49a$. Thus $a=1,2$ give product traces $51,100$ modulo $343$, while every individual trace and every trace modulo forty nine loses this distinction.\n\\end{example}\nThese matrices are exact controls for the information loss. They are not claimed to be the actual labelled probe and detecting element. The current eight tuples do not supply $T(\\tau)$, $T(g)$, and $T(\\tau g)$ from one numerical global realization. The required mixed value is a target, not an observed contradiction.","For instance, $c=1,v=3,\\epsilon=-1,\\beta(g)=1$ requires $D\\equiv98\\pmod{343}$. Writing that number does not evaluate the trace of an unknown candidate. A future global calculation must identify the words and the representation together.","\\section{Integral semilinearity and the unlabelled image}\\label{sec:image}\nThe integral semilinear theorem supplies more than equality of residual cohomology classes. Under the retained geometric inputs, an actual map between the conjugate curves induces an integral operator $U=C\\sigma$ on the Tate lattice, with\n\\[\n\\rho(jgj^{-1})=C\\sigma(\\rho(g))C^{-1},\\qquad\nC\\sigma(C)=-\\rho(j^2),\\qquad C_{21}\\in7^n\\Oloc.\n\\]\nIt preserves the natural endpoint, the opposite endpoint, and the stable lattice segment. The minus sign in the square comes from the actual conjugate curve map. It cannot be replaced by a plus sign simply to obtain a preferred descent datum.","The proof separates the ordinary coefficient linear part of the actual geometric operator from its semilinear part. An ordinary intertwiner would make the normalized natural classes $(u,r,1)$ and $(-u,r,1)$ proportional. The final coordinate fixes the scalar to one, forcing $u=0$, contrary to the retained nonvanishing theorem. The ordinary part must vanish. The actual integral origin and endpoint uniqueness then retain a semilinear comparison on the lattices, not just over their fraction field.","With a compatible local choice of $j$, one has $\\rho(j^2)=I$. The coefficient unit $(1+w)/2$ has norm minus one, so dividing $U$ by it gives a semilinear involution. An adapted basis of its plus lattice produces a global endpoint preserving integral frame with\n\\[\n\\rho(jgj^{-1})=\\sigma(\\rho(g)).\n\\]\nThis is an existential normal form. The unknown candidate frame has not been numerically constructed. Its plus lattice need not be $G_K$ stable, so the statement is not ordinary rank two $\\Q_7$ descent.","of $j$, one has $\\rho(j^2)=I$. The coefficient unit $(1+w)/2$ has norm minus one, so dividing $U$ by it gives a semilinear involution. An adapted basis of its plus lattice produces a global endpoint preserving integral frame with\n\\[\n\\rho(jgj^{-1})=\\sigma(\\rho(g)).\n\\]\nThis is an existential normal form. The unknown candidate frame has not been numerically constructed. Its plus lattice need not be $G_K$ stable, so the statement is not ordinary rank two $\\Q_7$ descent.","w)/2$ has norm minus one, so dividing $U$ by it gives a semilinear involution. An adapted basis of its plus lattice produces a global endpoint preserving integral frame with\n\\[\n\\rho(jgj^{-1})=\\sigma(\\rho(g)).\n\\]\nThis is an existential normal form. The unknown candidate frame has not been numerically constructed. Its plus lattice need not be $G_K$ stable, so the statement is not ordinary rank two $\\Q_7$ descent.","\\subsection{The depth one image theorem}\nLet $K_\\infty$ be the cyclotomic tower. Write $B$ for the additive image of beta on its Galois group and $d=\\dim_{\\F_7}B$. The inherited group theorem at actual depth one identifies\n\\begin{equation}\\label{eq:imagegroup}\nH=\\rho(G_{K_\\infty})=\n\\left\\{M\\in\\mathrm{SL}_2(\\Oloc):\n\\begin{array}{l}\nM\\bmod7=\\left(\\begin{smallmatrix}1&x","\\subsection{The depth one image theorem}\nLet $K_\\infty$ be the cyclotomic tower. Write $B$ for the additive image of beta on its Galois group and $d=\\dim_{\\F_7}B$. The inherited group theorem at actual depth one identifies\n\\begin{equation}\\label{eq:imagegroup}\nH=\\rho(G_{K_\\infty})=\n\\left\\{M\\in\\mathrm{SL}_2(\\Oloc):\n\\begin{array}{l}\nM\\bmod7=\\left(\\begin{smallmatrix}1&x\\\\0&1\\end{smallmatrix}\\right),\\ x\\in k,\\\\\n(M_{21}/7)\\bmod7\\in B\n\\end{array}\\right\\}.\n\\end{equation}\nIn particular it contains the full kernel of reduction modulo forty nine. Its finite quotient orders are\n\\[\n|H\\bmod7^j|=7^{6j-6+d}\\quad(j\\ge2),\\qquad|H\\bmod7|=49.\n\\]\nFor the bounded patterns $d=1$, the orders at forty nine and $343$ are $7^7$ and $7^{13}$. The latter is $96\\,889\\,010\\,407$. This is a structural count, not enumeration of that many matrices or arithmetic representations.","(M_{21}/7)\\bmod7\\in B\n\\end{array}\\right\\}.\n\\end{equation}\nIn particular it contains the full kernel of reduction modulo forty nine. Its finite quotient orders are\n\\[\n|H\\bmod7^j|=7^{6j-6+d}\\quad(j\\ge2),\\qquad|H\\bmod7|=49.\n\\]\nFor the bounded patterns $d=1$, the orders at forty nine and $343$ are $7^7$ and $7^{13}$. The latter is $96\\,889\\,010\\,407$. This is a structural count, not enumeration of that many matrices or arithmetic representations.","The abelianization is $k\\oplus B$. The independent global Kummer class $[7]$ cannot factor through that abelianization with its required cyclotomic weight, giving the stated global noncontainment of the uniformizer Kummer extension in the torsion field over $K_\\infty$. This field statement does not imply local noncontainment or zero local period valuation.","The image group contains upper and lower elements $T_a,G_b$ with\n\\[\n\\tr(T_aG_b)-\\tr(T_a)-\\tr(G_b)+2=49ab\\pmod{343}.\n\\]\nAll leading residue values occur as the elements vary. This flexibility does not allow a fixed local uniformizer probe to be replaced by any convenient upper element. Preimages in the Galois group may have different local membership and different arithmetic labels. Knowledge of the unlabelled image is therefore weaker than the quotient map and compatible local embeddings required by the mixed trace problem.","The theorem also retains its depth hypothesis and full residual upper image. A single $\\F_7$ upper direction does not force the same congruence subgroup. At depth greater than one, the lower ideal may be too small to contain a desired element at the previous precision. The archive supplies explicit controls against both invalid strengthenings.","\\section{Cup vanishing and the secondary obstruction}\\label{sec:secondary}\nThe first equation $dp=-\\alpha\\cup\\beta$ has an existence obstruction in $H^2(K,k)$. Once it vanishes, a further obstruction remains for the upper equation. The distinction is algebraic and can be proved without assuming an arithmetic realization.","Put $s(g)=\\alpha(g)\\beta(g)$. Expansion of the two cocycle laws gives\n\\begin{equation}\\label{eq:ds}\nds=-(\\alpha\\cup\\beta+\\beta\\cup\\alpha).\n\\end{equation}\nFor a chosen primitive $p$, set $D_{12}=-p$ and $D_{23}=p-s$. They form a defining system with differentials $\\alpha\\cup\\beta$ and $\\beta\\cup\\alpha$. The associated twisted Massey type cocycle is\n\\begin{equation}\\label{eq:mu}\n\\mu_p=\\alpha\\cup(p-s)-p\\cup\\alpha.\n\\end{equation}\nExplicitly it equals\n\\[\n\\mu_p(g,h)=\\alpha(g)(p(h)-\\alpha(h)\\beta(h))\n-p(g)\\chi(g)\\alpha(h).\n\\]\nLet $R_p$ denote the right side of \\eqref{eq:q-equation}. Direct expansion of $f(g)=\\alpha(g)p(g)$ gives\n\\begin{equation}\\label{eq:secondaryidentity}\nR_p+d_\\chi f=\\mu_p.\n\\end{equation}\nThus $[R_p]=[\\mu_","Put $s(g)=\\alpha(g)\\beta(g)$. Expansion of the two cocycle laws gives\n\\begin{equation}\\label{eq:ds}\nds=-(\\alpha\\cup\\beta+\\beta\\cup\\alpha).\n\\end{equation}\nFor a chosen primitive $p$, set $D_{12}=-p$ and $D_{23}=p-s$. They form a defining system with differentials $\\alpha\\cup\\beta$ and $\\beta\\cup\\alpha$. The associated twisted Massey type cocycle is\n\\begin{equation}\\label{eq:mu}\n\\mu_p=\\alpha\\cup(p-s)-p\\cup\\alpha.\n\\end{equation}\nExplicitly it equals\n\\[\n\\mu_p(g,h)=\\alpha(g)(p(h)-\\alpha(h)\\beta(h))\n-p(g)\\chi(g)\\alpha(h).\n\\]\nLet $R_p$ denote the right side of \\eqref{eq:q-equation}. Direct expansion of $f(g)=\\alpha(g)p(g)$ gives\n\\begin{equation}\\label{eq:secondaryidentity}\nR_p+d_\\chi f=\\mu_p.\n\\end{equation}\nThus $[R_p]=[\\mu_p]$ in $H^2(K,k(\\chi))$. For this fixed $p$, the upper equation is soluble if and only if this class is zero.","cycle laws gives\n\\begin{equation}\\label{eq:ds}\nds=-(\\alpha\\cup\\beta+\\beta\\cup\\alpha).\n\\end{equation}\nFor a chosen primitive $p$, set $D_{12}=-p$ and $D_{23}=p-s$. They form a defining system with differentials $\\alpha\\cup\\beta$ and $\\beta\\cup\\alpha$. The associated twisted Massey type cocycle is\n\\begin{equation}\\label{eq:mu}\n\\mu_p=\\alpha\\cup(p-s)-p\\cup\\alpha.\n\\end{equation}\nExplicitly it equals\n\\[\n\\mu_p(g,h)=\\alpha(g)(p(h)-\\alpha(h)\\beta(h))\n-p(g)\\chi(g)\\alpha(h).\n\\]\nLet $R_p$ denote the right side of \\eqref{eq:q-equation}. Direct expansion of $f(g)=\\alpha(g)p(g)$ gives\n\\begin{equation}\\label{eq:secondaryidentity}\nR_p+d_\\chi f=\\mu_p.\n\\end{equation}\nThus $[R_p]=[\\mu_p]$ in $H^2(K,k(\\chi))$. For this fixed $p$, the upper equation is soluble if and only if this class is zero.","\\alpha).\n\\end{equation}\nFor a chosen primitive $p$, set $D_{12}=-p$ and $D_{23}=p-s$. They form a defining system with differentials $\\alpha\\cup\\beta$ and $\\beta\\cup\\alpha$. The associated twisted Massey type cocycle is\n\\begin{equation}\\label{eq:mu}\n\\mu_p=\\alpha\\cup(p-s)-p\\cup\\alpha.\n\\end{equation}\nExplicitly it equals\n\\[\n\\mu_p(g,h)=\\alpha(g)(p(h)-\\alpha(h)\\beta(h))\n-p(g)\\chi(g)\\alpha(h).\n\\]\nLet $R_p$ denote the right side of \\eqref{eq:q-equation}. Direct expansion of $f(g)=\\alpha(g)p(g)$ gives\n\\begin{equation}\\label{eq:secondaryidentity}\nR_p+d_\\chi f=\\mu_p.\n\\end{equation}\nThus $[R_p]=[\\mu_p]$ in $H^2(K,k(\\chi))$. For this fixed $p$, the upper equation is soluble if and only if this class is zero.","_{23}=p-s$. They form a defining system with differentials $\\alpha\\cup\\beta$ and $\\beta\\cup\\alpha$. The associated twisted Massey type cocycle is\n\\begin{equation}\\label{eq:mu}\n\\mu_p=\\alpha\\cup(p-s)-p\\cup\\alpha.\n\\end{equation}\nExplicitly it equals\n\\[\n\\mu_p(g,h)=\\alpha(g)(p(h)-\\alpha(h)\\beta(h))\n-p(g)\\chi(g)\\alpha(h).\n\\]\nLet $R_p$ denote the right side of \\eqref{eq:q-equation}. Direct expansion of $f(g)=\\alpha(g)p(g)$ gives\n\\begin{equation}\\label{eq:secondaryidentity}\nR_p+d_\\chi f=\\mu_p.\n\\end{equation}\nThus $[R_p]=[\\mu_p]$ in $H^2(K,k(\\chi))$. For this fixed $p$, the upper equation is soluble if and only if this class is zero.","th differentials $\\alpha\\cup\\beta$ and $\\beta\\cup\\alpha$. The associated twisted Massey type cocycle is\n\\begin{equation}\\label{eq:mu}\n\\mu_p=\\alpha\\cup(p-s)-p\\cup\\alpha.\n\\end{equation}\nExplicitly it equals\n\\[\n\\mu_p(g,h)=\\alpha(g)(p(h)-\\alpha(h)\\beta(h))\n-p(g)\\chi(g)\\alpha(h).\n\\]\nLet $R_p$ denote the right side of \\eqref{eq:q-equation}. Direct expansion of $f(g)=\\alpha(g)p(g)$ gives\n\\begin{equation}\\label{eq:secondaryidentity}\nR_p+d_\\chi f=\\mu_p.\n\\end{equation}\nThus $[R_p]=[\\mu_p]$ in $H^2(K,k(\\chi))$. For this fixed $p$, the upper equation is soluble if and only if this class is zero.","ed twisted Massey type cocycle is\n\\begin{equation}\\label{eq:mu}\n\\mu_p=\\alpha\\cup(p-s)-p\\cup\\alpha.\n\\end{equation}\nExplicitly it equals\n\\[\n\\mu_p(g,h)=\\alpha(g)(p(h)-\\alpha(h)\\beta(h))\n-p(g)\\chi(g)\\alpha(h).\n\\]\nLet $R_p$ denote the right side of \\eqref{eq:q-equation}. Direct expansion of $f(g)=\\alpha(g)p(g)$ gives\n\\begin{equation}\\label{eq:secondaryidentity}\nR_p+d_\\chi f=\\mu_p.\n\\end{equation}\nThus $[R_p]=[\\mu_p]$ in $H^2(K,k(\\chi))$. For this fixed $p$, the upper equation is soluble if and only if this class is zero.","ype cocycle is\n\\begin{equation}\\label{eq:mu}\n\\mu_p=\\alpha\\cup(p-s)-p\\cup\\alpha.\n\\end{equation}\nExplicitly it equals\n\\[\n\\mu_p(g,h)=\\alpha(g)(p(h)-\\alpha(h)\\beta(h))\n-p(g)\\chi(g)\\alpha(h).\n\\]\nLet $R_p$ denote the right side of \\eqref{eq:q-equation}. Direct expansion of $f(g)=\\alpha(g)p(g)$ gives\n\\begin{equation}\\label{eq:secondaryidentity}\nR_p+d_\\chi f=\\mu_p.\n\\end{equation}\nThus $[R_p]=[\\mu_p]$ in $H^2(K,k(\\chi))$. For this fixed $p$, the upper equation is soluble if and only if this class is zero.","abel{eq:mu}\n\\mu_p=\\alpha\\cup(p-s)-p\\cup\\alpha.\n\\end{equation}\nExplicitly it equals\n\\[\n\\mu_p(g,h)=\\alpha(g)(p(h)-\\alpha(h)\\beta(h))\n-p(g)\\chi(g)\\alpha(h).\n\\]\nLet $R_p$ denote the right side of \\eqref{eq:q-equation}. Direct expansion of $f(g)=\\alpha(g)p(g)$ gives\n\\begin{equation}\\label{eq:secondaryidentity}\nR_p+d_\\chi f=\\mu_p.\n\\end{equation}\nThus $[R_p]=[\\mu_p]$ in $H^2(K,k(\\chi))$. For this fixed $p$, the upper equation is soluble if and only if this class is zero.","Every other first primitive is $p+z$ with $z\\in Z^1(K,k)$ an ordinary character. Formula \\eqref{eq:mu} gives\n\\[\n\\mu_{p+z}-\\mu_p=\\alpha\\cup z-z\\cup\\alpha.\n\\]\nGraded commutativity in odd characteristic makes its cohomology class $2[\\alpha]\\cup[z]$. Therefore the choice independent object at this stage is the affine coset\n\\begin{equation}\\label{eq:masseycoset}\n[\\mu_p]+2[\\alpha]\\cup H^1(K,k).\n\\end{equation}\nThere exists some first primitive for which the upper equation is soluble precisely when zero belongs to this coset. There is not one canonical class independent of the primitive without taking this quotient by indeterminacy. With restricted ramification or prescribed Stage P local conditions, only the corresponding allowed ch","Every other first primitive is $p+z$ with $z\\in Z^1(K,k)$ an ordinary character. Formula \\eqref{eq:mu} gives\n\\[\n\\mu_{p+z}-\\mu_p=\\alpha\\cup z-z\\cup\\alpha.\n\\]\nGraded commutativity in odd characteristic makes its cohomology class $2[\\alpha]\\cup[z]$. Therefore the choice independent object at this stage is the affine coset\n\\begin{equation}\\label{eq:masseycoset}\n[\\mu_p]+2[\\alpha]\\cup H^1(K,k).\n\\end{equation}\nThere exists some first primitive for which the upper equation is soluble precisely when zero belongs to this coset. There is not one canonical class independent of the primitive without taking this quotient by indeterminacy. With restricted ramification or prescribed Stage P local conditions, only the corresponding allowed character subspace may","Every other first primitive is $p+z$ with $z\\in Z^1(K,k)$ an ordinary character. Formula \\eqref{eq:mu} gives\n\\[\n\\mu_{p+z}-\\mu_p=\\alpha\\cup z-z\\cup\\alpha.\n\\]\nGraded commutativity in odd characteristic makes its cohomology class $2[\\alpha]\\cup[z]$. Therefore the choice independent object at this stage is the affine coset\n\\begin{equation}\\label{eq:masseycoset}\n[\\mu_p]+2[\\alpha]\\cup H^1(K,k).\n\\end{equation}\nThere exists some first primitive for which the upper equation is soluble precisely when zero belongs to this coset. There is not one canonical class independent of the primitive without taking this quotient by indeterminacy. With restricted ramification or prescribed Stage P local conditions, only the corresponding allowed character subspace may vary $p$. After the final Stage P reduction this is $\\ker A$, so the larger absolute indeterminacy","Every other first primitive is $p+z$ with $z\\in Z^1(K,k)$ an ordinary character. Formula \\eqref{eq:mu} gives\n\\[\n\\mu_{p+z}-\\mu_p=\\alpha\\cup z-z\\cup\\alpha.\n\\]\nGraded commutativity in odd characteristic makes its cohomology class $2[\\alpha]\\cup[z]$. Therefore the choice independent object at this stage is the affine coset\n\\begin{equation}\\label{eq:masseycoset}\n[\\mu_p]+2[\\alpha]\\cup H^1(K,k).\n\\end{equation}\nThere exists some first primitive for which the upper equation is soluble precisely when zero belongs to this coset. There is not one canonical class independent of the primitive without taking this quotient by indeterminacy. With restricted ramification or prescribed Stage P local conditions, only the corresponding allowed character subspace may vary $p$. After the final Stage P reduction this is $\\ker A$, so the larger absolute indeterminacy cannot discharge those affine conditions.","mu_p]+2[\\alpha]\\cup H^1(K,k).\n\\end{equation}\nThere exists some first primitive for which the upper equation is soluble precisely when zero belongs to this coset. There is not one canonical class independent of the primitive without taking this quotient by indeterminacy. With restricted ramification or prescribed Stage P local conditions, only the corresponding allowed character subspace may vary $p$. After the final Stage P reduction this is $\\ker A$, so the larger absolute indeterminacy cannot discharge those affine conditions.","ere exists some first primitive for which the upper equation is soluble precisely when zero belongs to this coset. There is not one canonical class independent of the primitive without taking this quotient by indeterminacy. With restricted ramification or prescribed Stage P local conditions, only the corresponding allowed character subspace may vary $p$. After the final Stage P reduction this is $\\ker A$, so the larger absolute indeterminacy cannot discharge those affine conditions.","\\subsection{An exact control separating the two stages}\nTake the finite group $U_3(\\F_7)$ with coordinates $(a,b,c)$ and multiplication\n\\[\n(a,b,c)(A,B,C)=(a+A,b+B,c+C+aB).\n\\]\nUse trivial cyclotomic action, $\\alpha=a$, $\\beta=b$, and $p=c$. Then $dp=-\\alpha\\cup\\beta$, so the first obstruction vanishes. The elements $x=(1,0,0)$ and $z=(0,0,1)$ commute, but\n\\[\nR_p(x,z)=2,\\qquad R_p(z,x)=0.\n\\]\nFor every ordinary cochain $q$, commuting arguments give $dq(x,z)=dq(z,x)$. Hence no $q$ solves the second equation. This is a finite group control, not an absolute Galois group or a Frey representation. It proves that cup zero alone does not formally imply upper solvability.","control separating the two stages}\nTake the finite group $U_3(\\F_7)$ with coordinates $(a,b,c)$ and multiplication\n\\[\n(a,b,c)(A,B,C)=(a+A,b+B,c+C+aB).\n\\]\nUse trivial cyclotomic action, $\\alpha=a$, $\\beta=b$, and $p=c$. Then $dp=-\\alpha\\cup\\beta$, so the first obstruction vanishes. The elements $x=(1,0,0)$ and $z=(0,0,1)$ commute, but\n\\[\nR_p(x,z)=2,\\qquad R_p(z,x)=0.\n\\]\nFor every ordinary cochain $q$, commuting arguments give $dq(x,z)=dq(z,x)$. Hence no $q$ solves the second equation. This is a finite group control, not an absolute Galois group or a Frey representation. It proves that cup zero alone does not formally imply upper solvability.","ges}\nTake the finite group $U_3(\\F_7)$ with coordinates $(a,b,c)$ and multiplication\n\\[\n(a,b,c)(A,B,C)=(a+A,b+B,c+C+aB).\n\\]\nUse trivial cyclotomic action, $\\alpha=a$, $\\beta=b$, and $p=c$. Then $dp=-\\alpha\\cup\\beta$, so the first obstruction vanishes. The elements $x=(1,0,0)$ and $z=(0,0,1)$ commute, but\n\\[\nR_p(x,z)=2,\\qquad R_p(z,x)=0.\n\\]\nFor every ordinary cochain $q$, commuting arguments give $dq(x,z)=dq(z,x)$. Hence no $q$ solves the second equation. This is a finite group control, not an absolute Galois group or a Frey representation. It proves that cup zero alone does not formally imply upper solvability.","U_3(\\F_7)$ with coordinates $(a,b,c)$ and multiplication\n\\[\n(a,b,c)(A,B,C)=(a+A,b+B,c+C+aB).\n\\]\nUse trivial cyclotomic action, $\\alpha=a$, $\\beta=b$, and $p=c$. Then $dp=-\\alpha\\cup\\beta$, so the first obstruction vanishes. The elements $x=(1,0,0)$ and $z=(0,0,1)$ commute, but\n\\[\nR_p(x,z)=2,\\qquad R_p(z,x)=0.\n\\]\nFor every ordinary cochain $q$, commuting arguments give $dq(x,z)=dq(z,x)$. Hence no $q$ solves the second equation. This is a finite group control, not an absolute Galois group or a Frey representation. It proves that cup zero alone does not formally imply upper solvability.","\\subsection{The fixed primitive local to global theorem}\nFor $K=\\Q(\\sqrt5)$, localization on $H^2(K,k(\\chi))$ is injective. To see the coefficient translation, restrict to $L=K(\\zeta_7)$, whose degree over $K$ is six. Restriction followed by corestriction multiplies by six, which is invertible on seven primary cohomology. Over $L$, the module is two trivial $\\F_7$ copies and $\\mu_7$ is present. Kummer theory identifies each copy of $H^2(L,\\F_7)$ with $\\operatorname{Br}(L)[7]$. Brauer--Hasse--Noether gives local to global injectivity for the Brauer group. A class zero at every completion over $K$ is therefore zero over $L$ and then over $K$ \\cite{milneCFT2020}. The exact leaves are II.1.22 and II.1.30 for Kummer and restriction with corestriction, and VIII.4.2 for","\\subsection{The fixed primitive local to global theorem}\nFor $K=\\Q(\\sqrt5)$, localization on $H^2(K,k(\\chi))$ is injective. To see the coefficient translation, restrict to $L=K(\\zeta_7)$, whose degree over $K$ is six. Restriction followed by corestriction multiplies by six, which is invertible on seven primary cohomology. Over $L$, the module is two trivial $\\F_7$ copies and $\\mu_7$ is present. Kummer theory identifies each copy of $H^2(L,\\F_7)$ with $\\operatorname{Br}(L)[7]$. Brauer--Hasse--Noether gives local to global injectivity for the Brauer group. A class zero at every completion over $K$ is therefore zero over $L$ and then over $K$ \\cite{milneCFT2020}. The exact leaves are II.1.22 and II.1.30 for Kummer and restriction with corestriction, and VIII.4.2 for the Brauer sequence.","to global theorem}\nFor $K=\\Q(\\sqrt5)$, localization on $H^2(K,k(\\chi))$ is injective. To see the coefficient translation, restrict to $L=K(\\zeta_7)$, whose degree over $K$ is six. Restriction followed by corestriction multiplies by six, which is invertible on seven primary cohomology. Over $L$, the module is two trivial $\\F_7$ copies and $\\mu_7$ is present. Kummer theory identifies each copy of $H^2(L,\\F_7)$ with $\\operatorname{Br}(L)[7]$. Brauer--Hasse--Noether gives local to global injectivity for the Brauer group. A class zero at every completion over $K$ is therefore zero over $L$ and then over $K$ \\cite{milneCFT2020}. The exact leaves are II.1.22 and II.1.30 for Kummer and restriction with corestriction, and VIII.4.2 for the Brauer sequence.","For $K=\\Q(\\sqrt5)$, localization on $H^2(K,k(\\chi))$ is injective. To see the coefficient translation, restrict to $L=K(\\zeta_7)$, whose degree over $K$ is six. Restriction followed by corestriction multiplies by six, which is invertible on seven primary cohomology. Over $L$, the module is two trivial $\\F_7$ copies and $\\mu_7$ is present. Kummer theory identifies each copy of $H^2(L,\\F_7)$ with $\\operatorname{Br}(L)[7]$. Brauer--Hasse--Noether gives local to global injectivity for the Brauer group. A class zero at every completion over $K$ is therefore zero over $L$ and then over $K$ \\cite{milneCFT2020}. The exact leaves are II.1.22 and II.1.30 for Kummer and restriction with corestriction, and VIII.4.2 for the Brauer sequence.","over $K$ is six. Restriction followed by corestriction multiplies by six, which is invertible on seven primary cohomology. Over $L$, the module is two trivial $\\F_7$ copies and $\\mu_7$ is present. Kummer theory identifies each copy of $H^2(L,\\F_7)$ with $\\operatorname{Br}(L)[7]$. Brauer--Hasse--Noether gives local to global injectivity for the Brauer group. A class zero at every completion over $K$ is therefore zero over $L$ and then over $K$ \\cite{milneCFT2020}. The exact leaves are II.1.22 and II.1.30 for Kummer and restriction with corestriction, and VIII.4.2 for the Brauer sequence.","on followed by corestriction multiplies by six, which is invertible on seven primary cohomology. Over $L$, the module is two trivial $\\F_7$ copies and $\\mu_7$ is present. Kummer theory identifies each copy of $H^2(L,\\F_7)$ with $\\operatorname{Br}(L)[7]$. Brauer--Hasse--Noether gives local to global injectivity for the Brauer group. A class zero at every completion over $K$ is therefore zero over $L$ and then over $K$ \\cite{milneCFT2020}. The exact leaves are II.1.22 and II.1.30 for Kummer and restriction with corestriction, and VIII.4.2 for the Brauer sequence.","estriction multiplies by six, which is invertible on seven primary cohomology. Over $L$, the module is two trivial $\\F_7$ copies and $\\mu_7$ is present. Kummer theory identifies each copy of $H^2(L,\\F_7)$ with $\\operatorname{Br}(L)[7]$. Brauer--Hasse--Noether gives local to global injectivity for the Brauer group. A class zero at every completion over $K$ is therefore zero over $L$ and then over $K$ \\cite{milneCFT2020}. The exact leaves are II.1.22 and II.1.30 for Kummer and restriction with corestriction, and VIII.4.2 for the Brauer sequence.","At a fixed place where beta is the coboundary of $\\lambda_v$ and $p=p_v^{\\mathrm{fix}}$, the explicit local upper primitive\n\\[\nq_v^{\\mathrm{fix}}(g)=-\\lambda_v\\chi(g)^{-1}\\alpha(g)^2\n\\]\nsatisfies $d_\\chi q_v^{\\mathrm{fix}}=R_p$ by direct expansion. At support, $\\alpha=0$ also kills the local upper right side. These facts prevent treating a formal secondary obstruction as an established nonzero obstruction in the actual local branches. Full prescribed upper values and geometric compatibility remain separate.","The local to global theorem uses the same global $p$ at every place. Independently chosen local primitives can have vanishing secondary classes without supplying one such global primitive. Even once $q$ exists globally, its local values must match the actual toric, endpoint, semilinear, and fixed seven conditions. Those conditions form another affine localization problem.","General Massey vanishing theorems do not erase these qualifications. The all prime theorem of Min\\'{a}\\v{c} and T\\^{a}n concerns ordinary products over absolute Galois groups with trivial $\\F_p$ coefficients \\cite{minac-tan2016}. The JEMS theorem cited in some older discussion has a mod two statement \\cite{minac-tan2017}. The 2026 strong vanishing theorem likewise concerns its stated ordinary coefficient setting and may introduce additional ramification \\cite{maire-et-al2026}. None is a theorem that the one twisted defining system prescribed here, on a fixed ramification set and with actual affine local targets, has a compatible integral lift.","For a finite module $A$, let $A'=\\operatorname{Hom}(A,\\mu_7)$ and let $G'$ be the finite group of the field cut out by that dual. The cited sufficient criterion is $H^1(G',A')=0$. For $A=k$ with trivial action, $A'\\simeq k(\\chi)$ and $G'=C_6$. Averaging kills its positive cohomology because six is invertible modulo seven. For $A=k(\\chi)$, the source action cancels the cyclotomic target action, so the dual is trivial and $G'=1$. Both satisfy the criterion.","let $A'=\\operatorname{Hom}(A,\\mu_7)$ and let $G'$ be the finite group of the field cut out by that dual. The cited sufficient criterion is $H^1(G',A')=0$. For $A=k$ with trivial action, $A'\\simeq k(\\chi)$ and $G'=C_6$. Averaging kills its positive cohomology because six is invertible modulo seven. For $A=k(\\chi)$, the source action cancels the cyclotomic target action, so the dual is trivial and $G'=1$. Both satisfy the criterion.","Consequently the unrestricted maps\n\\[\nH^1(K,k)\\longrightarrow\\prod_{v\\in S}H^1(K_v,k),\\qquad\nH^1(K,k(\\chi))\\longrightarrow\\prod_{v\\in S}H^1(K_v,k(\\chi))\n\\]\nare surjective for every finite set $S$ under the cited localization theorem. The precise source formulation and its fixed additional finite set refinement are recorded in the literature audit. This statement does not keep the character unramified at every other place.","If $p_0$ is one global bare primitive and $p_v$ are desired finite local primitives, their differences $p_v-p_0|_v$ are ordinary characters. For trivial coefficients there are no nonzero coboundaries. Surjectivity supplies one global character matching every prescribed finite difference exactly. Then $p=p_0+z$ meets those local equations, but may ramify at new primes.","For the upper equation, once one global $q_0$ exists, prescribed local differences are $k(\\chi)$ cocycles. Their cohomology classes can likewise be matched at finitely many places. Local coboundaries correspond to strict upper conjugation. Thus conditions invariant under such conjugation cannot alone create an unrestricted finite localization obstruction when each is locally attainable.","\\subsection{The local toric hyperplane}\nAt a beta support prime, use a pro seven tame generator $T$ and arithmetic Frobenius $F$. Put\n\\[\nc=\\beta(T)\\ne0,\\qquad b=\\beta(F),\\qquad h=(Q-1)/7\\pmod7.\n\\]\nThe local first digit normal form is\n\\[\nM=c\\begin{pmatrix}\\theta&-\\theta^2","\\subsection{The local toric hyperplane}\nAt a beta support prime, use a pro seven tame generator $T$ and arithmetic Frobenius $F$. Put\n\\[\nc=\\beta(T)\\ne0,\\qquad b=\\beta(F),\\qquad h=(Q-1)/7\\pmod7.\n\\]\nThe local first digit normal form is\n\\[\nM=c\\begin{pmatrix}\\theta&-\\theta^2\\\\1&-\\theta\\end{pmatrix},\\qquad\nA=\\begin{pmatrix}\\theta b&\\theta h-\\theta^2b\\\\b&h-\\theta b\\end{pmatrix}.\n\\]\nRelative to the chosen reference,\n\\[\np(T)=\\theta c,\\qquad p(F)=\\theta b-h.\n\\]\nEliminating $\\theta$ gives the invariant affine equation\n\\begin{equation}\\label{eq:toricp}\ncp(F)-bp(T)=-hc.\n\\end{equation}\nConversely any pair satisfying this equation determines $\\theta=p(T)/c$ and reconstructs the displayed first order packet. The packet converse does not construct a global curve or establish every higher integral relation.","b&h-\\theta b\\end{pmatrix}.\n\\]\nRelative to the chosen reference,\n\\[\np(T)=\\theta c,\\qquad p(F)=\\theta b-h.\n\\]\nEliminating $\\theta$ gives the invariant affine equation\n\\begin{equation}\\label{eq:toricp}\ncp(F)-bp(T)=-hc.\n\\end{equation}\nConversely any pair satisfying this equation determines $\\theta=p(T)/c$ and reconstructs the displayed first order packet. The packet converse does not construct a global curve or establish every higher integral relation.","Changing $F$ to $T^mF$ adds $mc$ to $b$ and $mp(T)$ to $p(F)$, so the left side of \\eqref{eq:toricp} is unchanged. A consistent Frobenius lift change cannot remove the affine right side.","\\item[H1.] The actual primitive Frey family, its specified twist and integral $\\Oloc$ action, determinant $\\chi$, and natural depth one endpoint conventions are fixed.\n\\item[H2.] At every variable bad place the actual inertia has $I+N$ with $N^2=0$ and $N$ in $7M_2(\\Oloc)$. Its primitive line factorization and same frame Frobenius obey the toric equations. Good inertia is trivial.\n\\item[H3.] Outside the four fixed rational primes $2,3,5,7$, every possible bad place divides $C$. The reference is unramified there.\n\\item[H4.] At each fixed place over $2,3,5,7$, the actual rank two trace is $\\chi+1$, and a local element has $\\chi-1$ a unit modulo seven.\n\\item[H5.] The four frozen beta supports, its stated all inert descent, and local natural class zero at every beta support pla","\\item[H1.] The actual primitive Frey family, its specified twist and integral $\\Oloc$ action, determinant $\\chi$, and natural depth one endpoint conventions are fixed.\n\\item[H2.] At every variable bad place the actual inertia has $I+N$ with $N^2=0$ and $N$ in $7M_2(\\Oloc)$. Its primitive line factorization and same frame Frobenius obey the toric equations. Good inertia is trivial.\n\\item[H3.] Outside the four fixed rational primes $2,3,5,7$, every possible bad place divides $C$. The reference is unramified there.\n\\item[H4.] At each fixed place over $2,3,5,7$, the actual rank two trace is $\\chi+1$, and a local element has $\\chi-1$ a unit modulo seven.\n\\item[H5.] The four frozen beta supports, its stated all inert descent, and local natural class zero at every beta support place are retained.\n\\item[H6.]","\\item[H2.] At every variable bad place the actual inertia has $I+N$ with $N^2=0$ and $N$ in $7M_2(\\Oloc)$. Its primitive line factorization and same frame Frobenius obey the toric equations. Good inertia is trivial.\n\\item[H3.] Outside the four fixed rational primes $2,3,5,7$, every possible bad place divides $C$. The reference is unramified there.\n\\item[H4.] At each fixed place over $2,3,5,7$, the actual rank two trace is $\\chi+1$, and a local element has $\\chi-1$ a unit modulo seven.\n\\item[H5.] The four frozen beta supports, its stated all inert descent, and local natural class zero at every beta support place are retained.\n\\item[H6.] The complete level seven period criterion and natural node thickness are used for the local Frey positive control only.\n\\e","\\item[H2.] At every variable bad place the actual inertia has $I+N$ with $N^2=0$ and $N$ in $7M_2(\\Oloc)$. Its primitive line factorization and same frame Frobenius obey the toric equations. Good inertia is trivial.\n\\item[H3.] Outside the four fixed rational primes $2,3,5,7$, every possible bad place divides $C$. The reference is unramified there.\n\\item[H4.] At each fixed place over $2,3,5,7$, the actual rank two trace is $\\chi+1$, and a local element has $\\chi-1$ a unit modulo seven.\n\\item[H5.] The four frozen beta supports, its stated all inert descent, and local natural class zero at every beta support place are retained.\n\\item[H6.] The complete level seven period criterion and natural node thickness are used for the local Frey positive control only.\n\\end{descriptio","\\item[H2.] At every variable bad place the actual inertia has $I+N$ with $N^2=0$ and $N$ in $7M_2(\\Oloc)$. Its primitive line factorization and same frame Frobenius obey the toric equations. Good inertia is trivial.\n\\item[H3.] Outside the four fixed rational primes $2,3,5,7$, every possible bad place divides $C$. The reference is unramified there.\n\\item[H4.] At each fixed place over $2,3,5,7$, the actual rank two trace is $\\chi+1$, and a local element has $\\chi-1$ a unit modulo seven.\n\\item[H5.] The four frozen beta supports, its stated all inert descent, and local natural class zero at every beta support place are retained.\n\\item[H6.] The complete level seven period criterion and natural node thickness are used for the local Frey positive control only.\n\\end{description}\nThe e","\\item[H2.] At every variable bad place the actual inertia has $I+N$ with $N^2=0$ and $N$ in $7M_2(\\Oloc)$. Its primitive line factorization and same frame Frobenius obey the toric equations. Good inertia is trivial.\n\\item[H3.] Outside the four fixed rational primes $2,3,5,7$, every possible bad place divides $C$. The reference is unramified there.\n\\item[H4.] At each fixed place over $2,3,5,7$, the actual rank two trace is $\\chi+1$, and a local element has $\\chi-1$ a unit modulo seven.\n\\item[H5.] The four frozen beta supports, its stated all inert descent, and local natural class zero at every beta support place are retained.\n\\item[H6.] The complete level seven period criterion and natural node thickness are used for the local Frey positive control only.\n\\end{description}\nThe exact original con","\\item[H3.] Outside the four fixed rational primes $2,3,5,7$, every possible bad place divides $C$. The reference is unramified there.\n\\item[H4.] At each fixed place over $2,3,5,7$, the actual rank two trace is $\\chi+1$, and a local element has $\\chi-1$ a unit modulo seven.\n\\item[H5.] The four frozen beta supports, its stated all inert descent, and local natural class zero at every beta support place are retained.\n\\item[H6.] The complete level seven period criterion and natural node thickness are used for the local Frey positive control only.\n\\end{description}\nThe exact original contracts remain in the source archive. Preservation of their bytes is not independent mathematical acceptance of the geometric chain.","\\item[H4.] At each fixed place over $2,3,5,7$, the actual rank two trace is $\\chi+1$, and a local element has $\\chi-1$ a unit modulo seven.\n\\item[H5.] The four frozen beta supports, its stated all inert descent, and local natural class zero at every beta support place are retained.\n\\item[H6.] The complete level seven period criterion and natural node thickness are used for the local Frey positive control only.\n\\end{description}\nThe exact original contracts remain in the source archive. Preservation of their bytes is not independent mathematical acceptance of the geometric chain.","\\begin{proposition}[No auxiliary diagonal inertia]\\label{prop:pinertia}\nUnder H1--H3, the first cochain $p$ is unramified outside the fixed primes and the complete beta ramification support. It factors as a cochain through the corresponding restricted Galois group.\n\\end{proposition}\n\\begin{proof}\nAt a variable bad place, the reference inertia is identity. Reduce $N/7$ modulo seven. Its traceless matrix is\n\\[\n\\begin{pmatrix}p(T)&q(T)","osition}[No auxiliary diagonal inertia]\\label{prop:pinertia}\nUnder H1--H3, the first cochain $p$ is unramified outside the fixed primes and the complete beta ramification support. It factors as a cochain through the corresponding restricted Galois group.\n\\end{proposition}\n\\begin{proof}\nAt a variable bad place, the reference inertia is identity. Reduce $N/7$ modulo seven. Its traceless matrix is\n\\[\n\\begin{pmatrix}p(T)&q(T)\\\\\\beta(T)&-p(T)\\end{pmatrix},\n\\]\nand nilpotence gives $p(T)^2+q(T)\\beta(T)=0$. If beta is unramified, its value is zero on every inertia element. Thus $p(T)^2=0$ in the field $k$, implying $p(T)=0$. This square zero inference is in a field, not in the ring $\\Oloc/49$, which has nilpotents. At good places both representations are unramified, giving the same conclusion.","beta(T)&-p(T)\\end{pmatrix},\n\\]\nand nilpotence gives $p(T)^2+q(T)\\beta(T)=0$. If beta is unramified, its value is zero on every inertia element. Thus $p(T)^2=0$ in the field $k$, implying $p(T)=0$. This square zero inference is in a field, not in the ring $\\Oloc/49$, which has nilpotents. At good places both representations are unramified, giving the same conclusion.","Let $I$ be an omitted inertia subgroup. The residual classes and $p$ all vanish on $I$. The multiplication law \\eqref{eq:p-cocycle} yields $p(gi)=p(g)=p(ig)$ for $i\\in I$. It also gives vanishing on conjugates and products of omitted inertia elements. Continuity extends this to their closed normal closure. The cochain descends to the restricted quotient.\n\\end{proof}\nThis result permits a finite Stage P computation even if $C$ has many extra divisors unramified in beta. The set of all those divisors need not be bounded. The theorem controls only the diagonal first cochain.","ia subgroup. The residual classes and $p$ all vanish on $I$. The multiplication law \\eqref{eq:p-cocycle} yields $p(gi)=p(g)=p(ig)$ for $i\\in I$. It also gives vanishing on conjugates and products of omitted inertia elements. Continuity extends this to their closed normal closure. The cochain descends to the restricted quotient.\n\\end{proof}\nThis result permits a finite Stage P computation even if $C$ has many extra divisors unramified in beta. The set of all those divisors need not be bounded. The theorem controls only the diagonal first cochain.","If beta is unramified and $q(T)$ is nonzero, the first digit inertia is purely upper. It can survive in the actual toric local category. The archive retains an exact local Frey positive control at $5167$, with $A=3238$, $C=5167$ and a local fifth root $B$ satisfying the equation. Its leading residue is $3658$, and simple root lifting supplies the root to every precision. This $B$ is a local number, not an ordinary integer solution.","\\subsection{Upper values at beta support are fixed after \\texorpdfstring{$p$}{p}}\nAt beta support, the natural class is identically zero locally, and the coefficient action is trivial modulo seven. Relative to the same reference, put $B_0(F)=b_0(F)/7$ modulo seven. The local packet yields\n\\begin{align}\ncp(F)-bp(T)&=-hc,\\label{eq:p-support}","\\subsection{Upper values at beta support are fixed after \\texorpdfstring{$p$}{p}}\nAt beta support, the natural class is identically zero locally, and the coefficient action is trivial modulo seven. Relative to the same reference, put $B_0(F)=b_0(F)/7$ modulo seven. The local packet yields\n\\begin{align}\ncp(F)-bp(T)&=-hc,\\label{eq:p-support}\\\\\ncq(T)&=-p(T)^2,\\label{eq:q-inertia}\\\\\nc(q(F)+B_0(F))&=-p(T)p(F).\\label{eq:q-frob}\n\\end{align}\nThe reference term in the last equation is retained. It is not reset to zero by an independent local basis choice. Once $p$ and beta are fixed, all upper values on the full beta supporting local group are fixed at this precision. Two upper corrections with the same data therefore have zero local difference there.","c(q(F)+B_0(F))&=-p(T)p(F).\\label{eq:q-frob}\n\\end{align}\nThe reference term in the last equation is retained. It is not reset to zero by an independent local basis choice. Once $p$ and beta are fixed, all upper values on the full beta supporting local group are fixed at this precision. Two upper corrections with the same data therefore have zero local difference there.","At beta unramified bad places, upper inertia may still occur when the local beta class splits. The exact matrices $N=7uE_{12}$ and $F=\\left(\\begin{smallmatrix}Q&R","At beta unramified bad places, upper inertia may still occur when the local beta class splits. The exact matrices $N=7uE_{12}$ and $F=\\left(\\begin{smallmatrix}Q&R\\\\0&1\\end{smallmatrix}\\right)$ provide all precision toric controls. Hence the attempted theorem confining \\emph{every} adjoint ramification to beta support is false. The finite restricted set is justified for Stage P. Stage Q retains its own auxiliary arithmetic problem.","\\section{A bare primitive on the justified ramification set}\\label{sec:bareprimitive}\nFix one of the nine prime beta supports $S$ in Section~\\ref{sec:patterns}. Let $\\Sigma$ consist of $S$, the places above $2,3,5,7$, and the two real places. The residual classes factor through $\\Gsig$. At every beta support place, $\\chi$ is trivial modulo seven and the natural local class is zero. Hence its normalized cocycle is identically zero there. At the fixed places, beta is a local coboundary under the inherited contract.","There are three questions to keep distinct:\n\\begin{enumerate}\n\\item Does the bare restricted equation $dp_0=-\\alpha\\cup\\beta$ have a solution?\n\\item Does one such global solution satisfy all prescribed first trace and toric conditions?\n\\item Does a matching first solution extend to the upper and higher integral geometric stages?\n\\end{enumerate}\nThe first question can now be answered using finite set duality. Its answer does not assume an actual higher representation in order to construct the first primitive.","\\begin{lemma}[The dual Kummer kernel]\\label{lem:sha1}\nFor every finite $\\Sigma$ containing the prime above seven and all real places,\n\\[\n\\Sha^1_\\Sigma(K,\\mu_7)\n=\\ker\\left(H^1(\\Gsig,\\mu_7)\n\\longrightarrow\\prod_{v\\in\\Sigma}H^1(K_v,\\mu_7)\\right)=0.\n\\]\n\\end{lemma}\n\\begin{proof}\nThe discriminant of $K$ is five. Minkowski's ideal bound is $\\sqrt5/2<2$, so every ideal class has a representative of norm one. Thus $\\Cl(K)=1$ \\cite{milneANT2020}.","ma}[The dual Kummer kernel]\\label{lem:sha1}\nFor every finite $\\Sigma$ containing the prime above seven and all real places,\n\\[\n\\Sha^1_\\Sigma(K,\\mu_7)\n=\\ker\\left(H^1(\\Gsig,\\mu_7)\n\\longrightarrow\\prod_{v\\in\\Sigma}H^1(K_v,\\mu_7)\\right)=0.\n\\]\n\\end{lemma}\n\\begin{proof}\nThe discriminant of $K$ is five. Minkowski's ideal bound is $\\sqrt5/2<2$, so every ideal class has a representative of norm one. Thus $\\Cl(K)=1$ \\cite{milneANT2020}.","Its units are $\\{\\pm\\phi^e:e\\in\\Z\\}$. For an elementary verification, scale a unit by a power of $\\phi$ and by a sign so that its first real embedding lies in $[1,\\phi)$. If its value $a$ were strictly between the endpoints, a norm minus one unit would have integral trace $a-a^{-1}$ strictly between zero and one. A norm plus one unit would have trace $a+a^{-1}$ strictly between two and $\\sqrt5<3$. Both are impossible. The normalized unit is therefore one, proving the assertion.","A Kummer class in the restricted group has valuations divisible by seven outside $\\Sigma$. Class number one lets us remove those valuations by a global seventh power and leave a $\\Sigma$ unit. If its local class is zero at every place in $\\Sigma$, the remaining prime valuations are also divisible by seven. Remove these as well. Since minus one is its own seventh power, the class is represented by $\\phi^e$.","Kummer class in the restricted group has valuations divisible by seven outside $\\Sigma$. Class number one lets us remove those valuations by a global seventh power and leave a $\\Sigma$ unit. If its local class is zero at every place in $\\Sigma$, the remaining prime valuations are also divisible by seven. Remove these as well. Since minus one is its own seventh power, the class is represented by $\\phi^e$.","At $F=\\Q_7(\\sqrt5)$, exact arithmetic modulo $49$ gives\n\\begin{equation}\\label{eq:phi-nonseventh}\n\\frac{\\phi^{48}-1}{7}=5\\overline{\\sqrt5}\\ne0\\quad\\text{in }\\F_{49}.\n\\end{equation}\nIf a unit is a seventh power, its forty eighth power is one modulo forty nine. Indeed, any unit has forty eighth power in $1+7\\Oloc_F$, and the seventh power of this group is contained in $1+49\\Oloc_F$. Formula \\eqref{eq:phi-nonseventh} shows that $\\phi$ is not a seventh power in $F$.","thmetic modulo $49$ gives\n\\begin{equation}\\label{eq:phi-nonseventh}\n\\frac{\\phi^{48}-1}{7}=5\\overline{\\sqrt5}\\ne0\\quad\\text{in }\\F_{49}.\n\\end{equation}\nIf a unit is a seventh power, its forty eighth power is one modulo forty nine. Indeed, any unit has forty eighth power in $1+7\\Oloc_F$, and the seventh power of this group is contained in $1+49\\Oloc_F$. Formula \\eqref{eq:phi-nonseventh} shows that $\\phi$ is not a seventh power in $F$.","equation}\\label{eq:phi-nonseventh}\n\\frac{\\phi^{48}-1}{7}=5\\overline{\\sqrt5}\\ne0\\quad\\text{in }\\F_{49}.\n\\end{equation}\nIf a unit is a seventh power, its forty eighth power is one modulo forty nine. Indeed, any unit has forty eighth power in $1+7\\Oloc_F$, and the seventh power of this group is contained in $1+49\\Oloc_F$. Formula \\eqref{eq:phi-nonseventh} shows that $\\phi$ is not a seventh power in $F$.","If $e$ is nonzero modulo seven and $\\phi^e$ is a seventh power, a Bezout identity between $e$ and seven makes $\\phi$ a seventh power too. Therefore local triviality forces $e\\equiv0\\pmod7$, and the global Kummer class is zero.\n\\end{proof}\nThe lemma is uniform in finite $\\Sigma$. It does not use a prime sample to test arbitrary global units. The single seven adic symbol is sufficient only after the class number and unit reductions have removed every other possible coordinate.","\\begin{theorem}[Conditional restricted bare primitive]\\label{thm:bareprimitive}\nAssume the stated residual and local properties for $\\alpha\\in Z^1(\\Gsig,k(\\chi))$ and $\\beta\\in Z^1(\\Gsig,k(\\chi^{-1}))$. Then there is a normalized continuous cochain $p_0:\\Gsig\\to k$ satisfying\n\\[\ndp_0=-\\alpha\\cup\\beta.\n\\]\nNo auxiliary ramification outside $\\Sigma$ is needed.\n\\end{theorem}\n\\begin{proof}\nThe Cartier dual of trivial $\\F_7$ is $\\mu_7$. The nondegenerate $\\F_7$ trace pairing on $k$ identifies the dual of trivial $k$ with $k(\\chi)$. Lemma~\\ref{lem:sha1} in two $\\F_7$ coordinates therefore gives $\\Sha^1_\\Sigma(K,k(\\chi))=0$.","onditional restricted bare primitive]\\label{thm:bareprimitive}\nAssume the stated residual and local properties for $\\alpha\\in Z^1(\\Gsig,k(\\chi))$ and $\\beta\\in Z^1(\\Gsig,k(\\chi^{-1}))$. Then there is a normalized continuous cochain $p_0:\\Gsig\\to k$ satisfying\n\\[\ndp_0=-\\alpha\\cup\\beta.\n\\]\nNo auxiliary ramification outside $\\Sigma$ is needed.\n\\end{theorem}\n\\begin{proof}\nThe Cartier dual of trivial $\\F_7$ is $\\mu_7$. The nondegenerate $\\F_7$ trace pairing on $k$ identifies the dual of trivial $k$ with $k(\\chi)$. Lemma~\\ref{lem:sha1} in two $\\F_7$ coordinates therefore gives $\\Sha^1_\\Sigma(K,k(\\chi))=0$.","restricted bare primitive]\\label{thm:bareprimitive}\nAssume the stated residual and local properties for $\\alpha\\in Z^1(\\Gsig,k(\\chi))$ and $\\beta\\in Z^1(\\Gsig,k(\\chi^{-1}))$. Then there is a normalized continuous cochain $p_0:\\Gsig\\to k$ satisfying\n\\[\ndp_0=-\\alpha\\cup\\beta.\n\\]\nNo auxiliary ramification outside $\\Sigma$ is needed.\n\\end{theorem}\n\\begin{proof}\nThe Cartier dual of trivial $\\F_7$ is $\\mu_7$. The nondegenerate $\\F_7$ trace pairing on $k$ identifies the dual of trivial $k$ with $k(\\chi)$. Lemma~\\ref{lem:sha1} in two $\\F_7$ coordinates therefore gives $\\Sha^1_\\Sigma(K,k(\\chi))=0$.","\\label{thm:bareprimitive}\nAssume the stated residual and local properties for $\\alpha\\in Z^1(\\Gsig,k(\\chi))$ and $\\beta\\in Z^1(\\Gsig,k(\\chi^{-1}))$. Then there is a normalized continuous cochain $p_0:\\Gsig\\to k$ satisfying\n\\[\ndp_0=-\\alpha\\cup\\beta.\n\\]\nNo auxiliary ramification outside $\\Sigma$ is needed.\n\\end{theorem}\n\\begin{proof}\nThe Cartier dual of trivial $\\F_7$ is $\\mu_7$. The nondegenerate $\\F_7$ trace pairing on $k$ identifies the dual of trivial $k$ with $k(\\chi)$. Lemma~\\ref{lem:sha1} in two $\\F_7$ coordinates therefore gives $\\Sha^1_\\Sigma(K,k(\\chi))=0$.","}\nAssume the stated residual and local properties for $\\alpha\\in Z^1(\\Gsig,k(\\chi))$ and $\\beta\\in Z^1(\\Gsig,k(\\chi^{-1}))$. Then there is a normalized continuous cochain $p_0:\\Gsig\\to k$ satisfying\n\\[\ndp_0=-\\alpha\\cup\\beta.\n\\]\nNo auxiliary ramification outside $\\Sigma$ is needed.\n\\end{theorem}\n\\begin{proof}\nThe Cartier dual of trivial $\\F_7$ is $\\mu_7$. The nondegenerate $\\F_7$ trace pairing on $k$ identifies the dual of trivial $k$ with $k(\\chi)$. Lemma~\\ref{lem:sha1} in two $\\F_7$ coordinates therefore gives $\\Sha^1_\\Sigma(K,k(\\chi))=0$.","cal properties for $\\alpha\\in Z^1(\\Gsig,k(\\chi))$ and $\\beta\\in Z^1(\\Gsig,k(\\chi^{-1}))$. Then there is a normalized continuous cochain $p_0:\\Gsig\\to k$ satisfying\n\\[\ndp_0=-\\alpha\\cup\\beta.\n\\]\nNo auxiliary ramification outside $\\Sigma$ is needed.\n\\end{theorem}\n\\begin{proof}\nThe Cartier dual of trivial $\\F_7$ is $\\mu_7$. The nondegenerate $\\F_7$ trace pairing on $k$ identifies the dual of trivial $k$ with $k(\\chi)$. Lemma~\\ref{lem:sha1} in two $\\F_7$ coordinates therefore gives $\\Sha^1_\\Sigma(K,k(\\chi))=0$.","rties for $\\alpha\\in Z^1(\\Gsig,k(\\chi))$ and $\\beta\\in Z^1(\\Gsig,k(\\chi^{-1}))$. Then there is a normalized continuous cochain $p_0:\\Gsig\\to k$ satisfying\n\\[\ndp_0=-\\alpha\\cup\\beta.\n\\]\nNo auxiliary ramification outside $\\Sigma$ is needed.\n\\end{theorem}\n\\begin{proof}\nThe Cartier dual of trivial $\\F_7$ is $\\mu_7$. The nondegenerate $\\F_7$ trace pairing on $k$ identifies the dual of trivial $k$ with $k(\\chi)$. Lemma~\\ref{lem:sha1} in two $\\F_7$ coordinates therefore gives $\\Sha^1_\\Sigma(K,k(\\chi))=0$.","$ and $\\beta\\in Z^1(\\Gsig,k(\\chi^{-1}))$. Then there is a normalized continuous cochain $p_0:\\Gsig\\to k$ satisfying\n\\[\ndp_0=-\\alpha\\cup\\beta.\n\\]\nNo auxiliary ramification outside $\\Sigma$ is needed.\n\\end{theorem}\n\\begin{proof}\nThe Cartier dual of trivial $\\F_7$ is $\\mu_7$. The nondegenerate $\\F_7$ trace pairing on $k$ identifies the dual of trivial $k$ with $k(\\chi)$. Lemma~\\ref{lem:sha1} in two $\\F_7$ coordinates therefore gives $\\Sha^1_\\Sigma(K,k(\\chi))=0$.","1}))$. Then there is a normalized continuous cochain $p_0:\\Gsig\\to k$ satisfying\n\\[\ndp_0=-\\alpha\\cup\\beta.\n\\]\nNo auxiliary ramification outside $\\Sigma$ is needed.\n\\end{theorem}\n\\begin{proof}\nThe Cartier dual of trivial $\\F_7$ is $\\mu_7$. The nondegenerate $\\F_7$ trace pairing on $k$ identifies the dual of trivial $k$ with $k(\\chi)$. Lemma~\\ref{lem:sha1} in two $\\F_7$ coordinates therefore gives $\\Sha^1_\\Sigma(K,k(\\chi))=0$.","Finite set Poitou--Tate duality pairs this group perfectly with $\\Sha^2_\\Sigma(K,k)$. Every prime dividing the module order belongs to $\\Sigma$, as required by the cited theorem. Odd primary real Tate terms vanish. Thus localization\n\\begin{equation}\\label{eq:restrictedH2}\nH^2(\\Gsig,k)\\longrightarrow\\bigoplus_{v\\in\\Sigma}H^2(K_v,k)\n\\end{equation}\nis injective \\cite{milneADT2006}.","At beta support, $\\alpha$ is identically zero locally, so its cup with beta is zero. At a fixed place, beta is a coboundary, so the local cup class is zero. The real terms vanish. Injectivity of \\eqref{eq:restrictedH2} gives $[\\alpha\\cup\\beta]=0$ globally in the restricted group. A continuous one cochain primitive exists by the definition of cohomological vanishing. Evaluating the primitive equation at the identity gives $p_0(1)=0$.\n\\end{proof}\nIt would be invalid to replace \\eqref{eq:restrictedH2} by an assertion about unrestricted absolute cohomology and infer the same ramification control. The dual kernel has been killed inside the restricted problem itself.","is identically zero locally, so its cup with beta is zero. At a fixed place, beta is a coboundary, so the local cup class is zero. The real terms vanish. Injectivity of \\eqref{eq:restrictedH2} gives $[\\alpha\\cup\\beta]=0$ globally in the restricted group. A continuous one cochain primitive exists by the definition of cohomological vanishing. Evaluating the primitive equation at the identity gives $p_0(1)=0$.\n\\end{proof}\nIt would be invalid to replace \\eqref{eq:restrictedH2} by an assertion about unrestricted absolute cohomology and infer the same ramification control. The dual kernel has been killed inside the restricted problem itself.","For the frozen supports, each supporting local field has residue norm one modulo seven and contains $\\mu_7$. Local duality gives one $k$ dimensional $H^2$ term there. The four fixed fields do not contain $\\mu_7$ and have zero $H^2(K_v,k)$. Global dual invariants also vanish. The final portion of the Poitou--Tate sequence then identifies the restricted $H^2$ with $k^9$. The dimension nine specializes the frozen support, whereas Lemma~\\ref{lem:sha1} held for arbitrary finite $\\Sigma$.","o seven and contains $\\mu_7$. Local duality gives one $k$ dimensional $H^2$ term there. The four fixed fields do not contain $\\mu_7$ and have zero $H^2(K_v,k)$. Global dual invariants also vanish. The final portion of the Poitou--Tate sequence then identifies the restricted $H^2$ with $k^9$. The dimension nine specializes the frozen support, whereas Lemma~\\ref{lem:sha1} held for arbitrary finite $\\Sigma$.","The theorem proves existence of an origin in the bare primitive torsor. It does not compute values of $p_0$ on labelled Frobenius or inertia, a defining field for $p_0$, or its affine compatibility class. Those numerical values are exactly what the final evaluator still lacks.","\\subsection{The unique local fixed trace primitive}\nAt a fixed place $v$ above $2,3,5,7$, write\n\\[\n\\beta(g)=(\\chi(g)^{-1}-1)\\lambda_v.\n\\]\nThe local separating element makes $\\lambda_v$ unique. Define\n\\begin{equation}\\label{eq:pfix}\np_v^{\\mathrm{fix}}(g)=\\lambda_v\\chi(g)^{-1}\\alpha(g).\n\\end{equation}\nDirect cocycle expansion yields\n\\begin{align}\ndp_v^{\\mathrm{fix}}(g,h)&=-\\alpha(g)\\chi(g)^{-1}\\beta(h),\\label{eq:pfixdiff}","\\subsection{The unique local fixed trace primitive}\nAt a fixed place $v$ above $2,3,5,7$, write\n\\[\n\\beta(g)=(\\chi(g)^{-1}-1)\\lambda_v.\n\\]\nThe local separating element makes $\\lambda_v$ unique. Define\n\\begin{equation}\\label{eq:pfix}\np_v^{\\mathrm{fix}}(g)=\\lambda_v\\chi(g)^{-1}\\alpha(g).\n\\end{equation}\nDirect cocycle expansion yields\n\\begin{align}\ndp_v^{\\mathrm{fix}}(g,h)&=-\\alpha(g)\\chi(g)^{-1}\\beta(h),\\label{eq:pfixdiff}\\\\\n(\\chi(g)-1)p_v^{\\mathrm{fix}}(g)+\\alpha(g)\\beta(g)&=0.\\label{eq:pfixtrace}\n\\end{align}\nThe second equation is precisely the first trace condition in \\eqref{eq:adjointtrace}.","If another local primitive satisfies the same trace condition, their difference $z$ is an ordinary character and $(\\chi(g)-1)z(g)=0$ for every $g$. If $\\chi(g)\\ne1$, this already gives $z(g)=0$. For $\\chi(g)=1$, choose $h$ with $\\chi(h)\\ne1$. Both $z(h)$ and $z(hg)$ vanish, so additivity gives $z(g)=0$. Thus \\eqref{eq:pfix} is the unique local cochain satisfying both equations in this fixed frame. It is not uniqueness of a global compatible primitive.","For a bare global $p_0$, the difference $p_v^{\\mathrm{fix}}-p_0|_v$ is an ordinary local character. Only that difference has the class field coordinates used below. Neither inhomogeneous cochain can be treated individually as an ordinary character.","\\section{The complete ordinary localization map}\\label{sec:localization}\nLet $z$ be an ordinary $k$ valued character of $\\Gsig$. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order:\n\\begin{equation}\\label{eq:localcoords}\nF_2,F_3,F_5,\\quad7:\\phi,3,7,\\quad\nI_{q_1},F_{q_1},\\ldots,I_{q_9},F_{q_9}.\n\\end{equation}\nWe use the local Artin convention sending a selected uniformizer to arithmetic Frobenius. At five the uniformizer is $\\sqrt5$. At support primes it is the rational prime $q$. Every sign in the reciprocity rows and every value of $h=(Nv-1)/7$ below uses this convention \\cite{milneCFT2020}.","$\\Gsig$. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order:\n\\begin{equation}\\label{eq:localcoords}\nF_2,F_3,F_5,\\quad7:\\phi,3,7,\\quad\nI_{q_1},F_{q_1},\\ldots,I_{q_9},F_{q_9}.\n\\end{equation}\nWe use the local Artin convention sending a selected uniformizer to arithmetic Frobenius. At five the uniformizer is $\\sqrt5$. At support primes it is the rational prime $q$. Every sign in the reciprocity rows and every value of $h=(Nv-1)/7$ below uses this convention \\cite{milneCFT2020}.","beta and not either primitive itself. Local class field theory gives the following coordinate order:\n\\begin{equation}\\label{eq:localcoords}\nF_2,F_3,F_5,\\quad7:\\phi,3,7,\\quad\nI_{q_1},F_{q_1},\\ldots,I_{q_9},F_{q_9}.\n\\end{equation}\nWe use the local Artin convention sending a selected uniformizer to arithmetic Frobenius. At five the uniformizer is $\\sqrt5$. At support primes it is the rational prime $q$. Every sign in the reciprocity rows and every value of $h=(Nv-1)/7$ below uses this convention \\cite{milneCFT2020}.","ory gives the following coordinate order:\n\\begin{equation}\\label{eq:localcoords}\nF_2,F_3,F_5,\\quad7:\\phi,3,7,\\quad\nI_{q_1},F_{q_1},\\ldots,I_{q_9},F_{q_9}.\n\\end{equation}\nWe use the local Artin convention sending a selected uniformizer to arithmetic Frobenius. At five the uniformizer is $\\sqrt5$. At support primes it is the rational prime $q$. Every sign in the reciprocity rows and every value of $h=(Nv-1)/7$ below uses this convention \\cite{milneCFT2020}.","At $2,3,5$, the residue norms are $4,9,5$, whose unit orders have no seven factor. Only the unramified coordinate remains. At seven, the two independent pro seven unit coordinates are $\\phi,3$, whose normalized logarithmic first digits are $5\\sqrt5,6$, and the third coordinate is the uniformizer seven. Each of the nine inert support primes has one seventh power residue coordinate and one unramified coordinate. Real odd primary characters vanish. The total local dimension is\n\\[\n3+3+2\\cdot9=24\\quad\\text{over }k.\n\\]\nThe chosen $g_q=\\phi+a_q$ generating the residue seventh power quotient supplies $I_q$. It need not generate the whole multiplicative group. Only its nonzero seventh power symbol and specifi","At $2,3,5$, the residue norms are $4,9,5$, whose unit orders have no seven factor. Only the unramified coordinate remains. At seven, the two independent pro seven unit coordinates are $\\phi,3$, whose normalized logarithmic first digits are $5\\sqrt5,6$, and the third coordinate is the uniformizer seven. Each of the nine inert support primes has one seventh power residue coordinate and one unramified coordinate. Real odd primary characters vanish. The total local dimension is\n\\[\n3+3+2\\cdot9=24\\quad\\text{over }k.\n\\]\nThe chosen $g_q=\\phi+a_q$ generating the residue seventh power quotient supplies $I_q$. It need not generate the whole multiplicative group. Only its nonzero seventh power symbol and specified normalization are required.",". Only the unramified coordinate remains. At seven, the two independent pro seven unit coordinates are $\\phi,3$, whose normalized logarithmic first digits are $5\\sqrt5,6$, and the third coordinate is the uniformizer seven. Each of the nine inert support primes has one seventh power residue coordinate and one unramified coordinate. Real odd primary characters vanish. The total local dimension is\n\\[\n3+3+2\\cdot9=24\\quad\\text{over }k.\n\\]\nThe chosen $g_q=\\phi+a_q$ generating the residue seventh power quotient supplies $I_q$. It need not generate the whole multiplicative group. Only its nonzero seventh power symbol and specified normalization are required.",", the two independent pro seven unit coordinates are $\\phi,3$, whose normalized logarithmic first digits are $5\\sqrt5,6$, and the third coordinate is the uniformizer seven. Each of the nine inert support primes has one seventh power residue coordinate and one unramified coordinate. Real odd primary characters vanish. The total local dimension is\n\\[\n3+3+2\\cdot9=24\\quad\\text{over }k.\n\\]\nThe chosen $g_q=\\phi+a_q$ generating the residue seventh power quotient supplies $I_q$. It need not generate the whole multiplicative group. Only its nonzero seventh power symbol and specified normalization are required.","ormalized logarithmic first digits are $5\\sqrt5,6$, and the third coordinate is the uniformizer seven. Each of the nine inert support primes has one seventh power residue coordinate and one unramified coordinate. Real odd primary characters vanish. The total local dimension is\n\\[\n3+3+2\\cdot9=24\\quad\\text{over }k.\n\\]\nThe chosen $g_q=\\phi+a_q$ generating the residue seventh power quotient supplies $I_q$. It need not generate the whole multiplicative group. Only its nonzero seventh power symbol and specified normalization are required.","\\subsection{Fourteen principal relations are exhaustive}\nThe $\\Sigma$ unit group modulo seventh powers has basis\n\\begin{equation}\\label{eq:sigmabasis}\n\\phi,2,3,\\sqrt5,7,q_1,\\ldots,q_9.\n\\end{equation}\nPrime valuations prove independence of the thirteen prime coordinates, and \\eqref{eq:phi-nonseventh} proves the unit coordinate is nontrivial. Class number one removes extra ideal class variables. Principal reciprocity pairs these fourteen units with the twenty four local characters and gives a matrix $L$ of size $14$ by $24$.","as basis\n\\begin{equation}\\label{eq:sigmabasis}\n\\phi,2,3,\\sqrt5,7,q_1,\\ldots,q_9.\n\\end{equation}\nPrime valuations prove independence of the thirteen prime coordinates, and \\eqref{eq:phi-nonseventh} proves the unit coordinate is nontrivial. Class number one removes extra ideal class variables. Principal reciprocity pairs these fourteen units with the twenty four local characters and gives a matrix $L$ of size $14$ by $24$.","gin{equation}\\label{eq:sigmabasis}\n\\phi,2,3,\\sqrt5,7,q_1,\\ldots,q_9.\n\\end{equation}\nPrime valuations prove independence of the thirteen prime coordinates, and \\eqref{eq:phi-nonseventh} proves the unit coordinate is nontrivial. Class number one removes extra ideal class variables. Principal reciprocity pairs these fourteen units with the twenty four local characters and gives a matrix $L$ of size $14$ by $24$.","tion}\\label{eq:sigmabasis}\n\\phi,2,3,\\sqrt5,7,q_1,\\ldots,q_9.\n\\end{equation}\nPrime valuations prove independence of the thirteen prime coordinates, and \\eqref{eq:phi-nonseventh} proves the unit coordinate is nontrivial. Class number one removes extra ideal class variables. Principal reciprocity pairs these fourteen units with the twenty four local characters and gives a matrix $L$ of size $14$ by $24$.","There are two complementary reasons why this matrix has rank fourteen. Lemma~\\ref{lem:sha1} proves that global Kummer localization is injective. Under the local perfect pairings, $L$ is the transpose of the global Kummer localization matrix, so the fourteen global Kummer coordinates give fourteen independent functionals. The finite arithmetic then reconstructs their declared coordinate entries and verifies this rank with explicit certificates. This order avoids treating a computational rank as the only reason a global reciprocity presentation is complete.","The finite set Poitou--Tate sequence identifies the entire ordinary localization image with $\\ker L$. Equivalently, the ray class presentation says that a compatible family of local characters extends exactly when it annihilates the principal $\\Sigma$ units \\cite{milneADT2006,milneCFT2020}. This is a restricted image theorem. It is not unrestricted finite localization surjectivity.","Ordinary global localization is injective here. A character with zero local classes at every allowed ramification place is everywhere unramified. Odd real terms vanish, and the ordinary class group is trivial. Thus it is zero. The ordinary global space is therefore ten dimensional, since $24-14=10$.","\\subsection{An explicit image parametrization}\nPut $a=z_7(3)$ and $x_i=z(I_{q_i})$. Let $\\ell_i$ be the seventh power logarithm normalized by $\\ell_i(g_{q_i})=1$. The supporting symbols of $\\phi$ vanish. For rational seven adic units define\n\\[\n\\ell_7(t)=\\frac{\\log(t^6)}{\\log(3^6)}\\pmod7\n=6\\frac{t^6-1}{7}\\pmod7.\n\\]\nPrincipal reciprocity determines the remaining coordinates:\n\\begin{align}\nz_7(\\phi)&=0,\\label{eq:Jphi}","\\subsection{An explicit image parametrization}\nPut $a=z_7(3)$ and $x_i=z(I_{q_i})$. Let $\\ell_i$ be the seventh power logarithm normalized by $\\ell_i(g_{q_i})=1$. The supporting symbols of $\\phi$ vanish. For rational seven adic units define\n\\[\n\\ell_7(t)=\\frac{\\log(t^6)}{\\log(3^6)}\\pmod7\n=6\\frac{t^6-1}{7}\\pmod7.\n\\]\nPrincipal reciprocity determines the remaining coordinates:\n\\begin{align}\nz_7(\\phi)&=0,\\label{eq:Jphi}\\\\\nz(F_2)&=-5a-\\sum_i\\ell_i(2)x_i,\\label{eq:J2}\\\\\nz(F_3)&=-a-\\sum_i\\ell_i(3)x_i,\\label{eq:J3}\\\\\nz(F_5)&=-4a-\\sum_i\\ell_i(\\sqrt5)x_i,\\label{eq:J5}\\\\\nz_7(7)&=-\\sum_i\\ell_i(7)x_i,\\label{eq:J7}\\\\\nz(F_q)&=-\\ell_7(q)a-\\sum_{i:q_i\\ne q}\\ell_i(q)x_i.\\label{eq:Jq}\n\\end{align}\nAt its own conductor place, the rational prime $q$ is the chosen uniformizer with unit part one. Its self residue logarithm in the ordinary character presentation is zero, not the undefined logarithm of zero. Every other support prime contributes simultaneously.","z(F_q)&=-\\ell_7(q)a-\\sum_{i:q_i\\ne q}\\ell_i(q)x_i.\\label{eq:Jq}\n\\end{align}\nAt its own conductor place, the rational prime $q$ is the chosen uniformizer with unit part one. Its self residue logarithm in the ordinary character presentation is zero, not the undefined logarithm of zero. Every other support prime contributes simultaneously.","These equations define a $24$ by $10$ matrix $J$. The ten free parameters occur in their declared local coordinates, so $J$ is injective. Direct substitution gives $LJ=0$. Since both $\\im J$ and $\\ker L$ have dimension ten,\n\\begin{equation}\\label{eq:Jcomplete}\n\\im J=\\ker L.\n\\end{equation}\nEvery parameter vector gives a global ordinary character. This is not merely an overapproximation by necessary equations.","The matrices have entries in $\\F_7$ and are extended to $k$. Their use does not assume coefficient descent. A smaller common pointwise semilinear problem can be imposed afterward, but it requires the same $j$, frame, and reference for the compared cochains.","\\section{Frobenius at a conductor prime}\\label{sec:self-conductor}\nThe toric row $cz(F)-bz(T)$ needs $b=\\beta(F)$ even where beta ramifies. An unramified test prime formula is insufficient. The final calculation includes the self conductor unit and converts the old twisted inertia normalization to the ordinary Artin generator of Section~\\ref{sec:localization}.","Let $M=\\Q(\\zeta_7)$. For a rational conductor $q\\equiv1\\pmod7$, choose a generator $\\pi_q$ of norm $q$ for the selected degree one prime. The source gives exact coefficient vectors in $1,\\zeta_7,\\ldots,\\zeta_7^5$ for $q=197,337,547,673,743,883,1933$. For example,\n\\[\n\\pi_{197}=\\zeta_7+2\\zeta_7^2+2\\zeta_7^4,\n\\]\nand\n\\[\n\\pi_{337}=-1+\\zeta_7-\\zeta_7^2+3\\zeta_7^3-\\zeta_7^4+\\zeta_7^5.\n\\]\nThe norm equality identifies one prime ideal. It does not leave an unspecified product of several primes.","At the chosen $q$ place, write $\\pi_q=q u_q$ with $u_q$ a unit. The same norm identity gives\n\\begin{equation}\\label{eq:selfunit}\n\\ell_q(u_q)=-\\sum_{b=2}^6\\ell_q(\\pi_q(z_q^b)).\n\\end{equation}\nPrincipal reciprocity on $\\pi_q$ therefore supplies the own conductor column\n\\begin{equation}\\label{eq:selfcolumn}\nD_{q,q}=\\sum_{b=2}^6(1-b)\\ell_q(\\pi_q(z_q^b)).\n\\end{equation}\nFor $r\\ne q$, the column is\n\\begin{equation}\\label{eq:othercolumn}\nD_{q,r}=-\\sum_{b\\in\\mathrm{reps}_r}b\\ell_r(\\pi_q(z_r^b)).\n\\end{equation}\nThe degree two representative convention is the one proved in \\eqref{eq:orbitreciprocity}. These formulas use the same arithmetic Frobenius principal idele convention as the ordinary localization rows.","n $q$ place, write $\\pi_q=q u_q$ with $u_q$ a unit. The same norm identity gives\n\\begin{equation}\\label{eq:selfunit}\n\\ell_q(u_q)=-\\sum_{b=2}^6\\ell_q(\\pi_q(z_q^b)).\n\\end{equation}\nPrincipal reciprocity on $\\pi_q$ therefore supplies the own conductor column\n\\begin{equation}\\label{eq:selfcolumn}\nD_{q,q}=\\sum_{b=2}^6(1-b)\\ell_q(\\pi_q(z_q^b)).\n\\end{equation}\nFor $r\\ne q$, the column is\n\\begin{equation}\\label{eq:othercolumn}\nD_{q,r}=-\\sum_{b\\in\\mathrm{reps}_r}b\\ell_r(\\pi_q(z_r^b)).\n\\end{equation}\nThe degree two representative convention is the one proved in \\eqref{eq:orbitreciprocity}. These formulas use the same arithmetic Frobenius principal idele convention as the ordinary localization rows.","The direct independent check lifts the chosen root of $\\Phi_7$ modulo $q^2$ by Hensel, then computes $\\pi_q(\\widetilde z_q)/q$ modulo $q$. It verifies the self unit rather than assigning its missing value from the norm relation. The norm calculation and Hensel calculation share the input generator but use different arithmetic routes.","Because each support prime is inert in $K/\\Q$, restriction at a rational $q\\equiv1\\pmod7$ multiplies the compatible rational Frobenius value by two:\n\\begin{equation}\\label{eq:betaF}\nb_q=2\\sum_rD_{q,r}s_r.\n\\end{equation}\nFor $q\\equiv-1\\pmod7$, the compatible squared rational Frobenius has beta value zero, from the cocycle factor $1+\\chi(q)^{-1}=0$. This does not say beta is unramified there.","The ordinary inertia generator differs from the frozen twisted residue convention. Its beta value is\n\\[\nc_q=s_q\\ell_q(\\operatorname{Nm}_{\\F_{q^2}/\\F_q}g_q)\n\\quad(q\\equiv1\\pmod7),\n\\]\nand $c_q=s_q\\ell_q(g_q)$ for $q\\equiv-1\\pmod7$. The conversion factors are\n\\begin{center}\n\\begin{tabular}{rrrrr}\\toprule\n$197:5$&$307:2$&$337:5$&$547:4$&$673:1$","$743:1$&$797:4$&$883:3$&$1483:1$&$1933:1$","bottomrule\n\\end{tabular}\n\\end{center}\nSetting $c_q=s_q$ without this conversion mixes two local generators and changes the problem.","Finally, the expression $c_qz(F_q)-b_qz(I_q)$ is invariant under a compatible Frobenius lift change. Both $b_q$ and $z(F_q)$ acquire their respective inertia multiples, and the extra terms cancel. This invariance does not permit omitting the self conductor unit or changing only one normalization.","On the local space $k^{24}$, let $E$ have fifteen rows. Its first six rows select all fixed local character coordinates in \\eqref{eq:localcoords}. Its remaining nine rows are\n\\[\nz\\longmapsto c_qz(F_q)-b_qz(I_q).\n\\]\nDefine\n\\begin{equation}\\label{eq:A}\nA=EJ:k^{10}\\longrightarrow k^{15}.\n\\end{equation}\nThe self conductor values and inertia conversions in Section~\\ref{sec:self-conductor} are part of this operator. Dropping them would reproduce a different matrix.","\\subsection{The homogeneous results}\nThe complete exact matrices, row reductions, null bases, and rank minors give\n\\begin{center}\n\\begin{tabular}{crrrl}\\toprule\nPattern&$\\operatorname{rank}A$&$\\dim_k\\ker A$&$\\dim_k\\coker A$&Remaining difference support","midrule\nA&7&3&8&$307,797,1483$","B&7&3&8&$307,797,1483$","C&8&2&7&$307,1483$","D&7&3&8&$307,797,1483$","bottomrule\n\\end{tabular}\n\\end{center}\n\\figplaceholder{19_affine_dimensions}{The kernel measures ambiguity after an affine solution exists. The cokernel measures possible first stage incompatibility. Neither dimension evaluates the actual target.}\nIn the global parameter order $(a,x_{q_1},\\ldots,x_{q_9})$, the kernel is exactly the span of the unit vectors at the listed minus congruence primes. After all toric rows are imposed, the plus congruence inertia differences vanish. This is a theorem about differences. It does not imply that the absolute inertia of a bare primitive vanishes at those plus primes.","Pattern C improves the previous homogeneous dimension from four to two. Its two former plus directions, evaluated on the added toric rows at $197,337$, give\n\\[\n\\begin{pmatrix}3&1","Pattern C improves the previous homogeneous dimension from four to two. Its two former plus directions, evaluated on the added toric rows at $197,337$, give\n\\[\n\\begin{pmatrix}3&1\\\\5&2\\end{pmatrix},\\qquad\\det=1\\quad\\text{in }\\F_7.\n\\]\nThus both directions are removed. Nonzero minors prove rank lower bounds, while full null bases prove the matching upper bounds. Exact arithmetic, not numerical singular value thresholds, determines the ranks.","\\subsection{The arithmetic target}\nChoose a bare global primitive $p_0$ from Theorem~\\ref{thm:bareprimitive}. This is one continuous global cochain. Its values have not been computed. Define $t(p_0)\\in k^{15}$ as follows. Its first six coordinates are the ordinary local character coordinates of $p_v^{\\mathrm{fix}}-p_0|_v$. At each support prime, put\n\\begin{equation}\\label{eq:target}\nt_q(p_0)=-h_qc_q-c_qp_0(F_q)+b_qp_0(T_q),\n\\qquad h_q=(q^2-1)/7\\pmod7.\n\\end{equation}\nAt support, $\\alpha$ is zero and $\\chi$ trivial, so the restriction of $p_0$ is an ordinary character. At fixed places only its difference from the prescribed primitive is ordinary. The target therefore has the declared coordinates without mistyping either inhomogeneous cochain as an ordinary chara","\\subsection{The arithmetic target}\nChoose a bare global primitive $p_0$ from Theorem~\\ref{thm:bareprimitive}. This is one continuous global cochain. Its values have not been computed. Define $t(p_0)\\in k^{15}$ as follows. Its first six coordinates are the ordinary local character coordinates of $p_v^{\\mathrm{fix}}-p_0|_v$. At each support prime, put\n\\begin{equation}\\label{eq:target}\nt_q(p_0)=-h_qc_q-c_qp_0(F_q)+b_qp_0(T_q),\n\\qquad h_q=(q^2-1)/7\\pmod7.\n\\end{equation}\nAt support, $\\alpha$ is zero and $\\chi$ trivial, so the restriction of $p_0$ is an ordinary character. At fixed places only its difference from the prescribed primitive is ordinary. The target therefore has the declared coordinates without mistyping either inhomogeneous cochain as an ordinary character.","~\\ref{thm:bareprimitive}. This is one continuous global cochain. Its values have not been computed. Define $t(p_0)\\in k^{15}$ as follows. Its first six coordinates are the ordinary local character coordinates of $p_v^{\\mathrm{fix}}-p_0|_v$. At each support prime, put\n\\begin{equation}\\label{eq:target}\nt_q(p_0)=-h_qc_q-c_qp_0(F_q)+b_qp_0(T_q),\n\\qquad h_q=(q^2-1)/7\\pmod7.\n\\end{equation}\nAt support, $\\alpha$ is zero and $\\chi$ trivial, so the restriction of $p_0$ is an ordinary character. At fixed places only its difference from the prescribed primitive is ordinary. The target therefore has the declared coordinates without mistyping either inhomogeneous cochain as an ordinary character.","ve}. This is one continuous global cochain. Its values have not been computed. Define $t(p_0)\\in k^{15}$ as follows. Its first six coordinates are the ordinary local character coordinates of $p_v^{\\mathrm{fix}}-p_0|_v$. At each support prime, put\n\\begin{equation}\\label{eq:target}\nt_q(p_0)=-h_qc_q-c_qp_0(F_q)+b_qp_0(T_q),\n\\qquad h_q=(q^2-1)/7\\pmod7.\n\\end{equation}\nAt support, $\\alpha$ is zero and $\\chi$ trivial, so the restriction of $p_0$ is an ordinary character. At fixed places only its difference from the prescribed primitive is ordinary. The target therefore has the declared coordinates without mistyping either inhomogeneous cochain as an ordinary character.","hain. Its values have not been computed. Define $t(p_0)\\in k^{15}$ as follows. Its first six coordinates are the ordinary local character coordinates of $p_v^{\\mathrm{fix}}-p_0|_v$. At each support prime, put\n\\begin{equation}\\label{eq:target}\nt_q(p_0)=-h_qc_q-c_qp_0(F_q)+b_qp_0(T_q),\n\\qquad h_q=(q^2-1)/7\\pmod7.\n\\end{equation}\nAt support, $\\alpha$ is zero and $\\chi$ trivial, so the restriction of $p_0$ is an ordinary character. At fixed places only its difference from the prescribed primitive is ordinary. The target therefore has the declared coordinates without mistyping either inhomogeneous cochain as an ordinary character.","\\begin{theorem}[Exact Stage P criterion]\\label{thm:affine}\nFix the same residual cocycles, full support, natural frame, and reference, together with the accepted inputs used above. Let $N$ be any full left null matrix of $A$, with independent rows and\n\\[\nNA=0,\\qquad\\operatorname{rank}N=15-\\operatorname{rank}A.\n\\]\nThere is a global restricted primitive meeting all fixed first trace and support toric conditions if and only if\n\\begin{equation}\\label{eq:obstruction}\nNt(p_0)=0.\n\\end{equation}\nThe class $[t(p_0)]\\in\\coker A$ is independent of the chosen bare primitive.\n\\end{theorem}\n\\begin{proof}\nTwo bare primitives have the same differential, so their difference is an ordinary global character. Conversely adding any such character preserves the differential. By Section~\\ref{sec:localization}, every character has a unique parameter vector $z\\in k^{10}$ with local vector $Jz$.","el{thm:affine}\nFix the same residual cocycles, full support, natural frame, and reference, together with the accepted inputs used above. Let $N$ be any full left null matrix of $A$, with independent rows and\n\\[\nNA=0,\\qquad\\operatorname{rank}N=15-\\operatorname{rank}A.\n\\]\nThere is a global restricted primitive meeting all fixed first trace and support toric conditions if and only if\n\\begin{equation}\\label{eq:obstruction}\nNt(p_0)=0.\n\\end{equation}\nThe class $[t(p_0)]\\in\\coker A$ is independent of the chosen bare primitive.\n\\end{theorem}\n\\begin{proof}\nTwo bare primitives have the same differential, so their difference is an ordinary global character. Conversely adding any such character preserves the differential. By Section~\\ref{sec:localization}, every character has a unique parameter vector $z\\in k^{10}$ with local vector $Jz$.","natural frame, and reference, together with the accepted inputs used above. Let $N$ be any full left null matrix of $A$, with independent rows and\n\\[\nNA=0,\\qquad\\operatorname{rank}N=15-\\operatorname{rank}A.\n\\]\nThere is a global restricted primitive meeting all fixed first trace and support toric conditions if and only if\n\\begin{equation}\\label{eq:obstruction}\nNt(p_0)=0.\n\\end{equation}\nThe class $[t(p_0)]\\in\\coker A$ is independent of the chosen bare primitive.\n\\end{theorem}\n\\begin{proof}\nTwo bare primitives have the same differential, so their difference is an ordinary global character. Conversely adding any such character preserves the differential. By Section~\\ref{sec:localization}, every character has a unique parameter vector $z\\in k^{10}$ with local vector $Jz$.","abel{eq:obstruction}\nNt(p_0)=0.\n\\end{equation}\nThe class $[t(p_0)]\\in\\coker A$ is independent of the chosen bare primitive.\n\\end{theorem}\n\\begin{proof}\nTwo bare primitives have the same differential, so their difference is an ordinary global character. Conversely adding any such character preserves the differential. By Section~\\ref{sec:localization}, every character has a unique parameter vector $z\\in k^{10}$ with local vector $Jz$.","\\end{equation}\nThe class $[t(p_0)]\\in\\coker A$ is independent of the chosen bare primitive.\n\\end{theorem}\n\\begin{proof}\nTwo bare primitives have the same differential, so their difference is an ordinary global character. Conversely adding any such character preserves the differential. By Section~\\ref{sec:localization}, every character has a unique parameter vector $z\\in k^{10}$ with local vector $Jz$.","A candidate $p=p_0+z_{\\mathrm{char}}$ meets the fixed local condition precisely when the selected local character coordinates equal those of $p_v^{\\mathrm{fix}}-p_0|_v$. At support, substitution in \\eqref{eq:toricp} gives\n\\[\nc_qz_{\\mathrm{char}}(F_q)-b_qz_{\\mathrm{char}}(T_q)=t_q(p_0).\n\\]\nTogether, these fifteen equations are exactly $Az=t(p_0)$. Thus Stage P is soluble if and only if $t(p_0)\\in\\im A$.","Since $NA=0$, the image is contained in $\\ker N$. Rank nullity gives\n\\[\n\\dim\\ker N=15-\\operatorname{rank}N=\\operatorname{rank}A=\\dim\\im A.\n\\]\nHence $\\ker N=\\im A$, proving the equivalence with \\eqref{eq:obstruction}. Merely having some left null rows would prove necessity, not sufficiency. Completeness of $N$ is part of the theorem.","Now replace the origin by $p'_0=p_0+z_{0,\\mathrm{char}}$, where its localization is $Jz_0$. At the fixed places, the target differences decrease by the selected coordinates of this character. At support, \\eqref{eq:target} decreases by $c_qz_{0,\\mathrm{char}}(F_q)-b_qz_{0,\\mathrm{char}}(T_q)$. Therefore\n\\begin{equation}\\label{eq:originshift}\nt(p'_0)=t(p_0)-Az_0.\n\\end{equation}\nThe quotient class and $Nt$ are unchanged. The coordinate basis for $N$ itself is optional, but its vanishing criterion and the quotient class are canonical.\n\\end{proof}\nThe result concerns the stated first stage. It does not identify its solution set with actual representations, solve the upper equations with their prescribed geometry, or prove higher integral liftability.","replace the origin by $p'_0=p_0+z_{0,\\mathrm{char}}$, where its localization is $Jz_0$. At the fixed places, the target differences decrease by the selected coordinates of this character. At support, \\eqref{eq:target} decreases by $c_qz_{0,\\mathrm{char}}(F_q)-b_qz_{0,\\mathrm{char}}(T_q)$. Therefore\n\\begin{equation}\\label{eq:originshift}\nt(p'_0)=t(p_0)-Az_0.\n\\end{equation}\nThe quotient class and $Nt$ are unchanged. The coordinate basis for $N$ itself is optional, but its vanishing criterion and the quotient class are canonical.\n\\end{proof}\nThe result concerns the stated first stage. It does not identify its solution set with actual representations, solve the upper equations with their prescribed geometry, or prove higher integral liftability.","{char}}$, where its localization is $Jz_0$. At the fixed places, the target differences decrease by the selected coordinates of this character. At support, \\eqref{eq:target} decreases by $c_qz_{0,\\mathrm{char}}(F_q)-b_qz_{0,\\mathrm{char}}(T_q)$. Therefore\n\\begin{equation}\\label{eq:originshift}\nt(p'_0)=t(p_0)-Az_0.\n\\end{equation}\nThe quotient class and $Nt$ are unchanged. The coordinate basis for $N$ itself is optional, but its vanishing criterion and the quotient class are canonical.\n\\end{proof}\nThe result concerns the stated first stage. It does not identify its solution set with actual representations, solve the upper equations with their prescribed geometry, or prove higher integral liftability.","\\begin{corollary}[Affine fibers]\nIf Stage P has one solution with parameter $z_*$, its complete solution set for these linear conditions is $z_*+\\ker A$. It has $49^3,49^3,49^2,49^3$ choices in patterns A through D, respectively. If the target is not in the image, the set is empty.\n\\end{corollary}\n\\begin{proof}\n$Az=t$ is equivalent to $A(z-z_*)=0$. The exact kernel dimensions give the counts over $k$.\n\\end{proof}\nAn additional common pointwise semilinear $j$, frame, and reference condition makes the relevant minus conductor coefficients lie in $w\\F_7$. Under that extra hypothesis, and only after compatible nonemptiness is known, the corresponding first stage ambiguity bounds are $343,343,49,343$. They are neither representation counts nor integer solution counts. The ba","\\begin{corollary}[Affine fibers]\nIf Stage P has one solution with parameter $z_*$, its complete solution set for these linear conditions is $z_*+\\ker A$. It has $49^3,49^3,49^2,49^3$ choices in patterns A through D, respectively. If the target is not in the image, the set is empty.\n\\end{corollary}\n\\begin{proof}\n$Az=t$ is equivalent to $A(z-z_*)=0$. The exact kernel dimensions give the counts over $k$.\n\\end{proof}\nAn additional common pointwise semilinear $j$, frame, and reference condition makes the relevant minus conductor coefficients lie in $w\\F_7$. Under that extra hypothesis, and only after compatible nonemptiness is known, the corresponding first stage ambiguity bounds are $343,343,49,343$. They are neither representation counts nor integer solution counts. The bare theorem does not prove the semilinear affine fiber nonempty.","begin{corollary}[Affine fibers]\nIf Stage P has one solution with parameter $z_*$, its complete solution set for these linear conditions is $z_*+\\ker A$. It has $49^3,49^3,49^2,49^3$ choices in patterns A through D, respectively. If the target is not in the image, the set is empty.\n\\end{corollary}\n\\begin{proof}\n$Az=t$ is equivalent to $A(z-z_*)=0$. The exact kernel dimensions give the counts over $k$.\n\\end{proof}\nAn additional common pointwise semilinear $j$, frame, and reference condition makes the relevant minus conductor coefficients lie in $w\\F_7$. Under that extra hypothesis, and only after compatible nonemptiness is known, the corresponding first stage ambiguity bounds are $343,343,49,343$. They are neither representation counts nor integer solution counts. The bare theorem does not prove the semilinear affine fiber nonempty.","with parameter $z_*$, its complete solution set for these linear conditions is $z_*+\\ker A$. It has $49^3,49^3,49^2,49^3$ choices in patterns A through D, respectively. If the target is not in the image, the set is empty.\n\\end{corollary}\n\\begin{proof}\n$Az=t$ is equivalent to $A(z-z_*)=0$. The exact kernel dimensions give the counts over $k$.\n\\end{proof}\nAn additional common pointwise semilinear $j$, frame, and reference condition makes the relevant minus conductor coefficients lie in $w\\F_7$. Under that extra hypothesis, and only after compatible nonemptiness is known, the corresponding first stage ambiguity bounds are $343,343,49,343$. They are neither representation counts nor integer solution counts. The bare theorem does not prove the semilinear affine fiber nonempty.","$. It has $49^3,49^3,49^2,49^3$ choices in patterns A through D, respectively. If the target is not in the image, the set is empty.\n\\end{corollary}\n\\begin{proof}\n$Az=t$ is equivalent to $A(z-z_*)=0$. The exact kernel dimensions give the counts over $k$.\n\\end{proof}\nAn additional common pointwise semilinear $j$, frame, and reference condition makes the relevant minus conductor coefficients lie in $w\\F_7$. Under that extra hypothesis, and only after compatible nonemptiness is known, the corresponding first stage ambiguity bounds are $343,343,49,343$. They are neither representation counts nor integer solution counts. The bare theorem does not prove the semilinear affine fiber nonempty.","orollary}\n\\begin{proof}\n$Az=t$ is equivalent to $A(z-z_*)=0$. The exact kernel dimensions give the counts over $k$.\n\\end{proof}\nAn additional common pointwise semilinear $j$, frame, and reference condition makes the relevant minus conductor coefficients lie in $w\\F_7$. Under that extra hypothesis, and only after compatible nonemptiness is known, the corresponding first stage ambiguity bounds are $343,343,49,343$. They are neither representation counts nor integer solution counts. The bare theorem does not prove the semilinear affine fiber nonempty.","\\subsection{A small model of the mistake to avoid}\nConsider $A:k\\to k^2$ given by $A(z)=(z,0)$. Its kernel is zero, its rank is one, and it has a full left null functional $N(u,v)=v$. For target $(a,b)$, a solution exists precisely when $b=0$. The map has a one dimensional image and a zero kernel, but this gives no information about the target's second coordinate. Substituting $(0,0)$ because it is convenient manufactures solvability of a different problem.","The final Beal operator has the same logic with fifteen target coordinates. Synthetic image targets are useful tests of an evaluator. They cannot be substituted for the arithmetic values of $p_0$.","\\subsection{What the evaluator actually returns}\nThe supplied program takes a pattern, a field basis, fifteen encoded values, a kind tag, and provenance. It returns the obstruction coordinates, a character correction when one exists, and a homogeneous kernel basis. Field values use the encoding $a+7b$ for $a+bw$, with $0\\le a,b<7$. The program records arithmetic target authentication as false. It verifies linear arithmetic on supplied data. It neither constructs a defining extension for $p_0$ nor authenticates its values.","d basis, fifteen encoded values, a kind tag, and provenance. It returns the obstruction coordinates, a character correction when one exists, and a homogeneous kernel basis. Field values use the encoding $a+7b$ for $a+bw$, with $0\\le a,b<7$. The program records arithmetic target authentication as false. It verifies linear arithmetic on supplied data. It neither constructs a defining extension for $p_0$ nor authenticates its values.","No actual $t(p_0)$ or $Nt(p_0)$ has been supplied for any of the eight patterns. Some obstruction coordinates may vanish for further arithmetic reasons, but none has been evaluated here. The next arithmetic obligation is therefore concrete: construct a defining extension or a duality evaluation of this same canonical class. A nonzero result excludes the first stage branch under its inputs. A zero result supplies a first stage family for the prescribed upper and integral compatibility problem.\n\\figplaceholder{24_","No actual $t(p_0)$ or $Nt(p_0)$ has been supplied for any of the eight patterns. Some obstruction coordinates may vanish for further arithmetic reasons, but none has been evaluated here. The next arithmetic obligation is therefore concrete: construct a defining extension or a duality evaluation of this same canonical class. A nonzero result excludes the first stage branch under its inputs. A zero result supplies a first stage family for the prescribed upper and integral compatibility problem.\n\\figplaceholder{24_research_boun","The audit also found a legacy Block 71 display metadata error. One primary record said the unit domain modulo $49$ contained 2304 elements, but its loop checks $49^2-7^2=2352$ units. The independent record already reports 2352. The mod seventh power quotient size forty nine and its forty eight element fibers are unchanged. The original record is preserved, while the corrected domain count is stated here.","The final target remains mathematical rather than editorial. A defining arithmetic origin or equivalent duality calculation must evaluate $[t(p_0)]\\in\\coker A$. Successful linear algebra cannot replace this missing input. After a compatible first primitive is obtained, the prescribed upper geometry and higher integral realization remain to be solved. Broader conductors, higher depths, the minus component, other modular branches, and other signatures require their own exhaustive coverage.","A current source check also found Chocian's preprint of 22 September 2026, which reports a computer assisted proof for the primitive equation $x^3+y^5=z^7$ \\cite{chocian2026}. Its existence and reported scope are recorded here. This release does not independently audit that claimed proof or import it as a theorem closing the recovered project route. Earlier project descriptions of the route as incomplete should therefore be read as descriptions of this project's evidence, not as an exhaustive assertion about all current literature.","ty, 2005, 325--340. DOI: \\href{https://doi.org/10.4171/009-1/21}{10.4171/009-1/21}.\n\\bibitem{bilu2005} Y. F. Bilu, Catalan without logarithmic forms (after Bugeaud, Hanrot and Mih\\u{a}ilescu), \\emph{J. Th\\'{e}or. Nombres Bordeaux} \\textbf{17} (2005), 69--85. DOI: \\href{https://doi.org/10.5802/jtnb.478}{10.5802/jtnb.478}.\n\\bibitem{darmon-granville1995} H. Darmon and A. Granville, On the equations $z^m=F(x,y)$ and $Ax^p+By^q=Cz^r$, \\emph{Bull. London Math. Soc.} \\textbf{27} (1995), 513--543. DOI: \\href{https://doi.org/10.1112/blms/27.6.513}{10.1112/blms/27.6.513}.\n\\bibitem{dahmen-siksek2024} S. R. Dahmen and S. Siksek, \\emph{On the generalized Fermat equation $x^5+y^3=z^7$}, author working paper, July 2024. \\url{https://few.vu.nl/~sdn249/GFE357.pdf}.\n\\bibitem{pacetti-villagra2025} A. Pacetti and L. Villagra Torcomian, \\emph{On the generalized Fermat equation of signature $(5,p,3)$}, arXiv:2512.17845","0. DOI: \\href{https://doi.org/10.4171/009-1/21}{10.4171/009-1/21}.\n\\bibitem{bilu2005} Y. F. Bilu, Catalan without logarithmic forms (after Bugeaud, Hanrot and Mih\\u{a}ilescu), \\emph{J. Th\\'{e}or. Nombres Bordeaux} \\textbf{17} (2005), 69--85. DOI: \\href{https://doi.org/10.5802/jtnb.478}{10.5802/jtnb.478}.\n\\bibitem{darmon-granville1995} H. Darmon and A. Granville, On the equations $z^m=F(x,y)$ and $Ax^p+By^q=Cz^r$, \\emph{Bull. London Math. Soc.} \\textbf{27} (1995), 513--543. DOI: \\href{https://doi.org/10.1112/blms/27.6.513}{10.1112/blms/27.6.513}.\n\\bibitem{dahmen-siksek2024} S. R. Dahmen and S. Siksek, \\emph{On the generalized Fermat equation $x^5+y^3=z^7$}, author working paper, July 2024. \\url{https://few.vu.nl/~sdn249/GFE357.pdf}.\n\\bibitem{pacetti-villagra2025} A. Pacetti and L. Villagra Torcomian, \\emph{On the generalized Fermat equation of signature $(5,p,3)$}, arXiv:2512.17845v1, 19 December 2025.","ef{https://doi.org/10.5802/jtnb.478}{10.5802/jtnb.478}.\n\\bibitem{darmon-granville1995} H. Darmon and A. Granville, On the equations $z^m=F(x,y)$ and $Ax^p+By^q=Cz^r$, \\emph{Bull. London Math. Soc.} \\textbf{27} (1995), 513--543. DOI: \\href{https://doi.org/10.1112/blms/27.6.513}{10.1112/blms/27.6.513}.\n\\bibitem{dahmen-siksek2024} S. R. Dahmen and S. Siksek, \\emph{On the generalized Fermat equation $x^5+y^3=z^7$}, author working paper, July 2024. \\url{https://few.vu.nl/~sdn249/GFE357.pdf}.\n\\bibitem{pacetti-villagra2025} A. Pacetti and L. Villagra Torcomian, \\emph{On the generalized Fermat equation of signature $(5,p,3)$}, arXiv:2512.17845v1, 19 December 2025. \\url{https://arxiv.org/abs/2512.17845v1}.\n\\bibitem{chocian2026} P. Chocian, \\emph{The primitive generalized Fermat equation $x^3+y^5=z^7$: A computer-assisted proof}, arXiv:2609.26996v1, 22 September 2026. \\url{https://arxiv.org/abs/2609.26996v1}",", 513--543. DOI: \\href{https://doi.org/10.1112/blms/27.6.513}{10.1112/blms/27.6.513}.\n\\bibitem{dahmen-siksek2024} S. R. Dahmen and S. Siksek, \\emph{On the generalized Fermat equation $x^5+y^3=z^7$}, author working paper, July 2024. \\url{https://few.vu.nl/~sdn249/GFE357.pdf}.\n\\bibitem{pacetti-villagra2025} A. Pacetti and L. Villagra Torcomian, \\emph{On the generalized Fermat equation of signature $(5,p,3)$}, arXiv:2512.17845v1, 19 December 2025. \\url{https://arxiv.org/abs/2512.17845v1}.\n\\bibitem{chocian2026} P. Chocian, \\emph{The primitive generalized Fermat equation $x^3+y^5=z^7$: A computer-assisted proof}, arXiv:2609.26996v1, 22 September 2026. \\url{https://arxiv.org/abs/2609.26996v1}. Recent preprint, not used here as an independently audited theorem.\n\\bibitem{milneADT2006} J. S. Milne, \\emph{Arithmetic Duality Theorems}, second edition, BookSurge, 2006. \\url{https://www.jmilne.org/math/Book","equation $x^5+y^3=z^7$}, author working paper, July 2024. \\url{https://few.vu.nl/~sdn249/GFE357.pdf}.\n\\bibitem{pacetti-villagra2025} A. Pacetti and L. Villagra Torcomian, \\emph{On the generalized Fermat equation of signature $(5,p,3)$}, arXiv:2512.17845v1, 19 December 2025. \\url{https://arxiv.org/abs/2512.17845v1}.\n\\bibitem{chocian2026} P. Chocian, \\emph{The primitive generalized Fermat equation $x^3+y^5=z^7$: A computer-assisted proof}, arXiv:2609.26996v1, 22 September 2026. \\url{https://arxiv.org/abs/2609.26996v1}. Recent preprint, not used here as an independently audited theorem.\n\\bibitem{milneADT2006} J. S. Milne, \\emph{Arithmetic Duality Theorems}, second edition, BookSurge, 2006. \\url{https://www.jmilne.org/math/Books/ADTnot.pdf}.\n\\bibitem{milneCFT2020} J. S. Milne, \\emph{Class Field Theory}, course notes, version 4.03, 6 August 2020. \\url{https://www.jmilne.org/math/CourseNotes/CFT.pdf}.\n\\b","curves over complete local rings, \\emph{Compositio Math.} \\textbf{24} (1972), 129--174. \\url{https://www.numdam.org/item/CM_1972__24_2_129_0/}.\n\\bibitem{ichikawa1993} T. Ichikawa, The universal periods of curves and the Schottky problem, \\emph{Compositio Math.} \\textbf{85} (1993), 1--8. \\url{https://www.numdam.org/item/CM_1993__85_1_1_0/}.\n\\bibitem{gregory-liedtke2019} O. Gregory and C. Liedtke, $p$-Adic Tate conjectures and abeloid varieties, \\emph{Documenta Math.} \\textbf{24} (2019), 1879--1934. DOI: \\href{https://doi.org/10.4171/DM/718}{10.4171/DM/718}.\n\\bibitem{baker-rabinoff2013} M. Baker and J. Rabinoff, \\emph{The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves}, arXiv:1308.3864v1, 18 August 2013. \\url{https://arxiv.org/abs/1308.3864v1}.\n\\bibitem{dolgachev-lehavi2008} I. Dolgachev and D. Lehavi, \\emph{On iso","morphic functions from tropical to algebraic curves}, arXiv:1308.3864v1, 18 August 2013. \\url{https://arxiv.org/abs/1308.3864v1}.\n\\bibitem{dolgachev-lehavi2008} I. Dolgachev and D. Lehavi, \\emph{On isogenous principally polarized abelian surfaces}, arXiv:0710.1298v3, 8 March 2008. \\url{https://arxiv.org/abs/0710.1298v3}.\n\\bibitem{ophir-weiss2024} A. Ophir and A. Weiss, \\emph{On Ribet's lemma for $\\mathrm{GL}_2$ modulo prime powers}, arXiv:2111.01559v3, 1 February 2024. \\url{https://arxiv.org/abs/2111.01559v3}.\n\\bibitem{galbraith2012} S. D. Galbraith, \\emph{Mathematics of Public Key Cryptography}, Cambridge University Press, 2012. Chapter 10 checked in the author's extended and corrected version 2.0. \\url{https://www.math.auckland.ac.nz/~sgal018/crypto-book/ch10.pdf}.\n\\bibitem{minac-tan2016} J. Min\\'{a}\\v{c} and N. D. T\\^{a}n, Triple Massey products vanish over all fields, \\emph{J. London Math. Soc.}"]}