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rangle","display":"\\rangle_p","kind":"use","occurrences":[146,158,161,164,940,953,1129]},{"id":"symbol-b13d0f69099cf79e","tex":"\\rangle_p^{p-1}","aliases":"⟩_p^p-1 rangle","display":"\\rangle_p^{p-1}","kind":"use","occurrences":[160]},{"id":"symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","occurrences":[262,520,569,605,719,732,802,808,816,838,839,852,943,1080,1089,1133,1152,1178,1242,1268,1286,1289,1325,1334]},{"id":"symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","occurrences":[541,729,852,1178,1275]}],"contexts":["\\appendix\n\\PartChapter{Worked Examples, Exact Calculations, and Applications}{One Triple Through the Entire Calculus: \\texorpdfstring{$3+125=128$}{3+125=128}}","\\section{Arithmetic data}\nConsider\n\\[\n a=3,\n \\qquad\n b=125=5^3,\n \\qquad\n c=128=2^7.\n\\]\nThe radical is\n\\[\n R=2\\cdot3\\cdot5=30.\n\\]\nThe quality is\n\\[\n q=\\frac{\\log128}{\\log30}.\n\\]\nThe output contact modulus is\n\\[\n M=\\frac{128}{2}=64.\n\\]\nThis example is especially useful because each sector consists of one prime, the radical divisor is attained sharply, the output has a long dyadic filtration, and every later object can be written explicitly.","\\section{Prime derivative}\nThe additive row is\n\\[\n \\lambda=(1,75,-448).\n\\]\nThe phase row is\n\\[\n \\lambda^{\\mathrm{ph}}=(1,25,-64).\n\\]\nThe coarse shortest transverse packet\n\\[\n x=(-2,6,1)\n\\]\nsatisfies\n\\[\n -2+450=448.\n\\]\nThe Wronskian is\n\\[\n W_x=1600=\\frac{abc}{R}.\n\\]\nThe coarse height is six.","The phase packet\n\\[\n y=(-8,8,3)\n\\]\nsatisfies\n\\[\n D_y(3)=-8,\n \\qquad\n D_y(125)=200,\n \\qquad\n D_y(128)=192,\n\\]\nand\n\\[\n -8+200=192.\n\\]\nIts Wronskian is also $1600$. The phase height is eight. The phase inequality is sharp:\n\\[\n 128\n =\n 30\\left(\\frac13+\\frac15\\right)8.\n\\]\nThis illustrates that the exponent-free phase channel can sharpen the bound even when its packet norm is larger.","\\section{Obstruction complex}\nThe joint exponent-free phase--Wronskian matrix has Smith form\n\\[\n \\operatorname{diag}(1,1600).\n\\]\nThe additive lattice maps onto $1600\\Z$ through the Wronskian. The primitive obstruction section exists algebraically. Its height remains the metric problem.","The stable real height is computed from\n\\[\n \\sigma_a=\\frac13,\n \\qquad\n \\sigma_b=\\frac15,\n \\qquad\n \\sigma_c=\\frac12.\n\\]\nThe smallest sector pair is\n\\[\n \\sigma_a+\\sigma_b=\\frac8{15}.\n\\]\nThus\n\\[\n \\overline H\n =\n \\frac{128}{30(8/15)}\n =8.\n\\]\nThe phase packet has height eight, so the integral optimum reaches the real lower barrier exactly.","\\section{Moving search}\nThe primitive obstruction phase is already optimal. The certified stopping bound contains no later coset capable of improving height eight because\n\\[\n k\\overline H\\ge8\n\\]\nfor every $k\\ge1$. This example is therefore a clean calibration of the no-escape identity.","\\section{Signal and Frey layers}\nThe prime-shell energy of $c$ is\n\\[\n \\log128=7\\log2,\n\\]\nwhile the activated output bandwidth is\n\\[\n \\log2.\n\\]\nThe output persistence energy is\n\\[\n 6\\log2=\\log64.\n\\]","The marked Frey curve is\n\\[\n y^2=x(x-3)(x+125).\n\\]\nIts discriminant is\n\\[\n 16\\cdot3^2\\cdot125^2\\cdot128^2.\n\\]\nAt the odd primes, the marked depths are two at $3$ and six at $5$. The dyadic depth requires the weighted branch.","The rational automorphic channel alone does not determine the marked dyadic depth of the displayed Frey representative. Any passage to an isogenous or twisted representative must therefore carry the differential scaling and the marked local $2$-adic data explicitly.","\\section{Filtered contact and Witt packet}\nThe output contact complex is\n\\[\n [128\\Z\\hookrightarrow2\\Z]\n \\simeq\n [\\Z\\xrightarrow{64}\\Z].\n\\]\nIts cohomology is\n\\[\n \\Jcal_c\\cong\\Z/64\\Z.\n\\]\nThe associated graded is\n\\[\n (\\F_2)^6.\n\\]\nThe six binary layers are joined by five nontrivial carry extensions. A cyclic target of order $64$ can carry the filtered source exactly, while no cyclic target can contain the split rank-six source injectively.","The local Witt coordinate is\n\\[\n W_6(\\F_2)\n \\cong\n \\Z/64\\Z.\n\\]\nThe weighted multiplicative dyadic coordinate is\n\\[\n (1+4\\Z_2)/(1+2^8\\Z_2),\n\\]\nwhich has the same order.","\\section{Kummer and Fermat packets}\nThe global Kummer map is\n\\[\n \\kappa_{128}:\\Z/64\\Z\\to H^1(\\Q,\\mu_{64}).\n\\]\nSince\n\\[\n \\gcd(64,7)=1,\n\\]\nit is injective.","For the balanced ratio\n\\[\n \\beta=-\\frac{125}{3},\n\\]\nthe signed exponent vector on the input primes is\n\\[\n \\nu=(-1,3).\n\\]\nIts Kummer order modulo $64$ is full. Choosing $64$th roots gives a point on\n\\[\n X^{64}+Y^{64}=1.\n\\]\nThe coordinate height is\n\\[\n \\frac{\\log128}{64}.\n\\]\nWith the tensor-power metric induced from $\\mathcal O(1)$, the full plane-model canonical scale is\n\\[\n \\frac{61}{64}\\log128.\n\\]\nFor an arbitrary compatible Weil-height normalization, this formula holds up to an additive $O(1)$ term. It records the line-bundle cost of retaining the complete Fermat curve.","mes is\n\\[\n \\nu=(-1,3).\n\\]\nIts Kummer order modulo $64$ is full. Choosing $64$th roots gives a point on\n\\[\n X^{64}+Y^{64}=1.\n\\]\nThe coordinate height is\n\\[\n \\frac{\\log128}{64}.\n\\]\nWith the tensor-power metric induced from $\\mathcal O(1)$, the full plane-model canonical scale is\n\\[\n \\frac{61}{64}\\log128.\n\\]\nFor an arbitrary compatible Weil-height normalization, this formula holds up to an additive $O(1)$ term. It records the line-bundle cost of retaining the complete Fermat curve.","\\section{Polylogarithmic path}\nSet\n\\[\n z=\\frac3{128},\n \\qquad\n 1-z=\\frac{125}{128},\n \\qquad\n \\beta=1-\\frac1z=-\\frac{125}{3}.\n\\]\nThe depth-three functional relation closes on these three arguments. At the output prime two,\n\\[\n v_2(\\beta-1)\n =\n v_2\\left(-\\frac{128}{3}\\right)\n =7.\n\\]\nThe unscaled principal logarithm has $2$-adic valuation seven. After division by four, the weighted principal coordinate occupies five filtration levels, while the separate sign packet supplies the first dyadic layer.","\\section{Coherence lattice}\nAfter removing the sign layer,\n\\[\n (\\epsilon_2,K,W,C)=(2,1,1,32).\n\\]\nThe residual row is\n\\[\n g=(29,31)\\pmod{32}.\n\\]\nThe lattice is\n\\[\n \\Lambda_{\\mathrm{coh}}\n =\n \\{(x_3,x_5):29x_3+31x_5\\equiv0\\pmod{32}\\}.\n\\]\nThe input vector lies in the kernel because\n\\[\n 29(-1)+31(3)=64.\n\\]\nThe full repeated output depth has now been transformed into one arithmetic cyclic character of order $32$, after the bounded sign layer has been separated.","n{Coherence lattice}\nAfter removing the sign layer,\n\\[\n (\\epsilon_2,K,W,C)=(2,1,1,32).\n\\]\nThe residual row is\n\\[\n g=(29,31)\\pmod{32}.\n\\]\nThe lattice is\n\\[\n \\Lambda_{\\mathrm{coh}}\n =\n \\{(x_3,x_5):29x_3+31x_5\\equiv0\\pmod{32}\\}.\n\\]\nThe input vector lies in the kernel because\n\\[\n 29(-1)+31(3)=64.\n\\]\nThe full repeated output depth has now been transformed into one arithmetic cyclic character of order $32$, after the bounded sign layer has been separated.","\\section{\\texorpdfstring{$1+2400=2401$}{1+2400=2401} and dyadic Landen descent}\nThe triple is\n\\[\n 1+2400=2401=7^4.\n\\]\nThe radical is\n\\[\n R=2\\cdot3\\cdot5\\cdot7=210.\n\\]\nThe output contact modulus is\n\\[\n M=7^3=343.\n\\]\nThe global Kummer kernel is one because\n\\[\n \\gcd(343,4)=1.\n\\]","The triple belongs to the Landen family with\n\\[\n u=49,\n \\qquad\n v=1.\n\\]\nThe dyadic child is\n\\[\n 49+576=625.\n\\]\nBoth triples have radical $210$. The child has lower output height. The corresponding Frey curves occupy the same two-isogeny packet, while their marked integral representatives have different heights and local depths.","The output persistence of the child is\n\\[\n \\frac{625}{5}=125.\n\\]\nThe modular degree of one associated minimal curve is a power of two in the stored example. This disproves the naive claim that the output persistence must divide one modular degree. The local marked depth and global eigenline separation are distinct channels.","\\section{\\texorpdfstring{$289+6272=6561$}{289+6272=6561}}\nHere\n\\[\n 289=17^2,\n \\qquad\n 6272=2^7\\cdot7^2,\n \\qquad\n 6561=3^8.\n\\]\nThe radical is\n\\[\n R=2\\cdot3\\cdot7\\cdot17=714.\n\\]\nThe output contact modulus is\n\\[\n M=3^7=2187.\n\\]\nThe Kummer kernel is\n\\[\n \\gcd(2187,8)=1.\n\\]\nThus the output contact packet globalizes injectively.","\\section{\\texorpdfstring{$1+4374=4375$}{1+4374=4375}}\nThe output is\n\\[\n 4375=5^4\\cdot7.\n\\]\nThe output radical is $35$, and\n\\[\n M=5^3=125\n\\]\nThe output contact packet has order $125$. This one-prime output is a clean setting for separating Kummer preload, local logarithmic defect, and residual coherence.","\\section{A large high-quality triple}\nConsider the stored high-quality example\n\\[\n 24833\n +\n 5020969537415167\n =\n 5020969537440000.\n\\]\nThe output factorization contains\n\\[\n 2^8 3^{22}5^4.\n\\]\nThe output contact modulus is\n\\[\n M=2^7 3^{21}5^3\n =167365651248000.\n\\]\nThis exceeds many fixed-width intermediate scales. It is the canonical regression case for arbitrary-precision arithmetic.","The split associated graded has rank\n\\[\n 7+21+3=31.\n\\]\nThe filtered source is one cyclic group of order $M$. The global Kummer kernel is only two in the finite data set. Thus almost the entire enormous contact packet survives globalization. The unresolved arithmetic is the economical global synthesis of the remaining coherence data.","\\section{A generalized Wieferich example}\nFor\n\\[\n 1+57121=57122,\n\\]\nwith\n\\[\n 57121=239^2,\n \\qquad\n 57122=2\\cdot13^4,\n\\]\nthe output contact modulus is\n\\[\n M=13^3.\n\\]\nThe local logarithm of the input prime $239$ at $13$ has exceptional depth. In the four-factor decomposition,\n\\[\n (\\epsilon_2,K,W,C)=(1,1,13^3,1).\n\\]\nThe entire repeated output depth is explained by one generalized Wieferich edge. This is a rare mechanism. Most finite examples have $W=1$ and large collective coherence factor $C$.","1+57121=57122,\n\\]\nwith\n\\[\n 57121=239^2,\n \\qquad\n 57122=2\\cdot13^4,\n\\]\nthe output contact modulus is\n\\[\n M=13^3.\n\\]\nThe local logarithm of the input prime $239$ at $13$ has exceptional depth. In the four-factor decomposition,\n\\[\n (\\epsilon_2,K,W,C)=(1,1,13^3,1).\n\\]\nThe entire repeated output depth is explained by one generalized Wieferich edge. This is a rare mechanism. Most finite examples have $W=1$ and large collective coherence factor $C$.","quad\n 57122=2\\cdot13^4,\n\\]\nthe output contact modulus is\n\\[\n M=13^3.\n\\]\nThe local logarithm of the input prime $239$ at $13$ has exceptional depth. In the four-factor decomposition,\n\\[\n (\\epsilon_2,K,W,C)=(1,1,13^3,1).\n\\]\nThe entire repeated output depth is explained by one generalized Wieferich edge. This is a rare mechanism. Most finite examples have $W=1$ and large collective coherence factor $C$.","\\begin{enumerate}[leftmargin=1.5em]\n\\item The universal prime-packet differential obeys Leibniz.\n\\item The additive condition is one integer row.\n\\item The arithmetic Wronskian is one additional row.\n\\item The divisor $abc/R$ divides every additive Wronskian.\n\\item Sector compression reduces each prime block to one aggregate coordinate plus lift gauges.\n\\item The joint obstruction matrix has torsion invariant $abc/R$.\n\\item The additive lattice fits into\n\\[\n 0\\to L_0\\to L\\to\\Z\\to0.\n\\]\n\\item The sharp aggregate solutions form an affine integer line.\n\\item Each local lift reduces to centered CRT residues and one quotient-sum equation.\n\\item The first feasible interval intersection gives the exact coset height.\n\\item The continuous lower bound gives finite stopping in the obstruction multiple.\n\\item The filtered contact quotient is $R_c\\Z/c\\Z$.\n\\item The associated graded is the direct sum of residue-field layers.","te}[leftmargin=1.5em]\n\\item The universal prime-packet differential obeys Leibniz.\n\\item The additive condition is one integer row.\n\\item The arithmetic Wronskian is one additional row.\n\\item The divisor $abc/R$ divides every additive Wronskian.\n\\item Sector compression reduces each prime block to one aggregate coordinate plus lift gauges.\n\\item The joint obstruction matrix has torsion invariant $abc/R$.\n\\item The additive lattice fits into\n\\[\n 0\\to L_0\\to L\\to\\Z\\to0.\n\\]\n\\item The sharp aggregate solutions form an affine integer line.\n\\item Each local lift reduces to centered CRT residues and one quotient-sum equation.\n\\item The first feasible interval intersection gives the exact coset height.\n\\item The continuous lower bound gives finite stopping in the obstruction multiple.\n\\item The filtered contact quotient is $R_c\\Z/c\\Z$.\n\\item The associated graded is the direct sum of residue-field layers.\n\\item The maximal image between two cyclic packets has order equal to their gcd.\n\\item The global Kummer kernel is the gcd of the modulus with the valuation vector.\n\\item The Fermat charact…","-packet differential obeys Leibniz.\n\\item The additive condition is one integer row.\n\\item The arithmetic Wronskian is one additional row.\n\\item The divisor $abc/R$ divides every additive Wronskian.\n\\item Sector compression reduces each prime block to one aggregate coordinate plus lift gauges.\n\\item The joint obstruction matrix has torsion invariant $abc/R$.\n\\item The additive lattice fits into\n\\[\n 0\\to L_0\\to L\\to\\Z\\to0.\n\\]\n\\item The sharp aggregate solutions form an affine integer line.\n\\item Each local lift reduces to centered CRT residues and one quotient-sum equation.\n\\item The first feasible interval intersection gives the exact coset height.\n\\item The continuous lower bound gives finite stopping in the obstruction multiple.\n\\item The filtered contact quotient is $R_c\\Z/c\\Z$.\n\\item The associated graded is the direct sum of residue-field layers.\n\\item The maximal image between two cyclic packets has order equal to their gcd.\n\\item The global Kummer kernel is the gcd of the modulus with the valuation vector.\n\\item The Fermat character projectors satisfy exact orthogonality and c…","attice fits into\n\\[\n 0\\to L_0\\to L\\to\\Z\\to0.\n\\]\n\\item The sharp aggregate solutions form an affine integer line.\n\\item Each local lift reduces to centered CRT residues and one quotient-sum equation.\n\\item The first feasible interval intersection gives the exact coset height.\n\\item The continuous lower bound gives finite stopping in the obstruction multiple.\n\\item The filtered contact quotient is $R_c\\Z/c\\Z$.\n\\item The associated graded is the direct sum of residue-field layers.\n\\item The maximal image between two cyclic packets has order equal to their gcd.\n\\item The global Kummer kernel is the gcd of the modulus with the valuation vector.\n\\item The Fermat character projectors satisfy exact orthogonality and completeness.\n\\item The coherence lattice has index equal to the coherence quotient.\n\\end{enumerate}","The Atkin--Lehner Walsh bank is a finite product of commuting involutions. Its identities are polynomial consequences of $W_j^2=I$.","\\section{Local logarithmic layer}\nFor odd $p$, the identity\n\\[\n v_p(\\log_p(1+u))=v_p(u)\n\\]\nholds when $v_p(u)>0$. This yields the exact output notch depth. The coefficient formula\n\\[\n v_p(q^{p-1}-1)-1\n\\]\ncomes from the principal-unit logarithm of the input prime.","At $2$, the logarithm begins on $1+4\\Z_2$. The separate sign bit and the shifted principal filtration must be retained. This is the source of the weighted dyadic normalization.","\\section{Conditional implications}\nEvery conditional $abc$ theorem in this paper has the same logical shape.","\\begin{enumerate}[leftmargin=1.5em]\n\\item Construct a finite source of order comparable to $c/R_c$.\n\\item Map it into a global target with kernel at most $R^\\eps$.\n\\item Bound the target size by $R_aR_bR^\\eps$.\n\\item Multiply by $R_c$ and absorb logarithmic factors.\n\\end{enumerate}","\\section{Factorization and data normalization}\nFor each triple, the canonical record contains\n\\[\n (a,b,c),\n \\quad\n \\{v_p(a)\\},\n \\quad\n \\{v_p(b)\\},\n \\quad\n \\{v_p(c)\\},\n \\quad\n R,\n \\quad\n q(a,b,c).\n\\]\nDerived objects must be computed from this record. No displayed factorization is accepted as input without verification.","\\begin{enumerate}[leftmargin=1.5em]\n\\item checking $a+b=c$\n\\item checking $\\gcd(a,b)=1$\n\\item ordering $a\\le b$ when a search convention requires it\n\\item computing radicals from exact prime sets\n\\item separating dyadic and odd-adic branches\n\\end{enumerate}","\\begin{enumerate}[leftmargin=1.5em]\n\\item Compute $r_u$, $P_u$, $R$, $Q$, and the exact stable height $\\overline H$.\n\\item Compute the primitive obstruction height $H(1)$ by binary search on the three interval intersections.\n\\item Set\n\\[\n K_{\\max}\n =\n \\left\\lfloor\\frac{H(1)-1}{\\overline H}\\right\\rfloor.\n\\]\n\\item For $2\\le k\\le K_{\\max}$, skip a coset when $k\\overline H$ already reaches the current best height.\n\\item Build centered CRT phase residues for the three sectors.\n\\item Compute the coarse affine line.\n\\item Binary-search the smallest $H$ whose three admissible $n$-intervals meet.\n\\item Construct one witness, verify additivity and obstruction multiple, and update the stopping bound.\n\\item Store the preceding empty radius as the no-smalle","\\begin{enumerate}[leftmargin=1.5em]\n\\item Compute $r_u$, $P_u$, $R$, $Q$, and the exact stable height $\\overline H$.\n\\item Compute the primitive obstruction height $H(1)$ by binary search on the three interval intersections.\n\\item Set\n\\[\n K_{\\max}\n =\n \\left\\lfloor\\frac{H(1)-1}{\\overline H}\\right\\rfloor.\n\\]\n\\item For $2\\le k\\le K_{\\max}$, skip a coset when $k\\overline H$ already reaches the current best height.\n\\item Build centered CRT phase residues for the three sectors.\n\\item Compute the coarse affine line.\n\\item Binary-search the smallest $H$ whose three admissible $n$-intervals meet.\n\\item Construct one witness, verify additivity and obstruction multiple, and update the stopping bound.\n\\item Store the preceding empty radius as the no-smaller-box c","\\begin{enumerate}[leftmargin=1.5em]\n\\item Compute $r_u$, $P_u$, $R$, $Q$, and the exact stable height $\\overline H$.\n\\item Compute the primitive obstruction height $H(1)$ by binary search on the three interval intersections.\n\\item Set\n\\[\n K_{\\max}\n =\n \\left\\lfloor\\frac{H(1)-1}{\\overline H}\\right\\rfloor.\n\\]\n\\item For $2\\le k\\le K_{\\max}$, skip a coset when $k\\overline H$ already reaches the current best height.\n\\item Build centered CRT phase residues for the three sectors.\n\\item Compute the coarse affine line.\n\\item Binary-search the smallest $H$ whose three admissible $n$-intervals meet.\n\\item Construct one witness, verify additivity and obstruction multiple, and update the stopping bound.\n\\item Store the preceding empty radius as the no-smaller-box certif","\\begin{enumerate}[leftmargin=1.5em]\n\\item Compute $r_u$, $P_u$, $R$, $Q$, and the exact stable height $\\overline H$.\n\\item Compute the primitive obstruction height $H(1)$ by binary search on the three interval intersections.\n\\item Set\n\\[\n K_{\\max}\n =\n \\left\\lfloor\\frac{H(1)-1}{\\overline H}\\right\\rfloor.\n\\]\n\\item For $2\\le k\\le K_{\\max}$, skip a coset when $k\\overline H$ already reaches the current best height.\n\\item Build centered CRT phase residues for the three sectors.\n\\item Compute the coarse affine line.\n\\item Binary-search the smallest $H$ whose three admissible $n$-intervals meet.\n\\item Construct one witness, verify additivity and obstruction multiple, and update the stopping bound.\n\\item Store the preceding empty radius as the no-smaller-box certificate","\\begin{enumerate}[leftmargin=1.5em]\n\\item Compute $r_u$, $P_u$, $R$, $Q$, and the exact stable height $\\overline H$.\n\\item Compute the primitive obstruction height $H(1)$ by binary search on the three interval intersections.\n\\item Set\n\\[\n K_{\\max}\n =\n \\left\\lfloor\\frac{H(1)-1}{\\overline H}\\right\\rfloor.\n\\]\n\\item For $2\\le k\\le K_{\\max}$, skip a coset when $k\\overline H$ already reaches the current best height.\n\\item Build centered CRT phase residues for the three sectors.\n\\item Compute the coarse affine line.\n\\item Binary-search the smallest $H$ whose three admissible $n$-intervals meet.\n\\item Construct one witness, verify additivity and obstruction multiple, and update the stopping bound.\n\\item Store the preceding empty radius as the no-smaller-box certificate.\n\\end{enumerate}","ruction height $H(1)$ by binary search on the three interval intersections.\n\\item Set\n\\[\n K_{\\max}\n =\n \\left\\lfloor\\frac{H(1)-1}{\\overline H}\\right\\rfloor.\n\\]\n\\item For $2\\le k\\le K_{\\max}$, skip a coset when $k\\overline H$ already reaches the current best height.\n\\item Build centered CRT phase residues for the three sectors.\n\\item Compute the coarse affine line.\n\\item Binary-search the smallest $H$ whose three admissible $n$-intervals meet.\n\\item Construct one witness, verify additivity and obstruction multiple, and update the stopping bound.\n\\item Store the preceding empty radius as the no-smaller-box certificate.\n\\end{enumerate}","nary search on the three interval intersections.\n\\item Set\n\\[\n K_{\\max}\n =\n \\left\\lfloor\\frac{H(1)-1}{\\overline H}\\right\\rfloor.\n\\]\n\\item For $2\\le k\\le K_{\\max}$, skip a coset when $k\\overline H$ already reaches the current best height.\n\\item Build centered CRT phase residues for the three sectors.\n\\item Compute the coarse affine line.\n\\item Binary-search the smallest $H$ whose three admissible $n$-intervals meet.\n\\item Construct one witness, verify additivity and obstruction multiple, and update the stopping bound.\n\\item Store the preceding empty radius as the no-smaller-box certificate.\n\\end{enumerate}","\\begin{enumerate}[leftmargin=1.5em]\n\\item Form the signed input exponent vector $\\nu$.\n\\item Remove the common Kummer preload $K$.\n\\item For every output prime, construct the finite logarithmic row.\n\\item Divide each row by its common coefficient depth to obtain the local defect $W_p$.\n\\item Remove the dyadic sign layer separately.\n\\item Combine primitive residual rows by CRT.\n\\item Verify that the normalized exponent vector lies in the kernel.\n\\item Compute the exact quotient order $C$ and the factorization\n\\[\n M=\\epsilon_2KWC.\n\\]\n\\end{enumerate}","tem Remove the common Kummer preload $K$.\n\\item For every output prime, construct the finite logarithmic row.\n\\item Divide each row by its common coefficient depth to obtain the local defect $W_p$.\n\\item Remove the dyadic sign layer separately.\n\\item Combine primitive residual rows by CRT.\n\\item Verify that the normalized exponent vector lies in the kernel.\n\\item Compute the exact quotient order $C$ and the factorization\n\\[\n M=\\epsilon_2KWC.\n\\]\n\\end{enumerate}","r preload $K$.\n\\item For every output prime, construct the finite logarithmic row.\n\\item Divide each row by its common coefficient depth to obtain the local defect $W_p$.\n\\item Remove the dyadic sign layer separately.\n\\item Combine primitive residual rows by CRT.\n\\item Verify that the normalized exponent vector lies in the kernel.\n\\item Compute the exact quotient order $C$ and the factorization\n\\[\n M=\\epsilon_2KWC.\n\\]\n\\end{enumerate}","\\section{Simultaneous Wieferich networks}\nDefine a directed prime network by the edge depth\n\\[\n w_p(q)=v_p(q^{p-1}-1)-1\n\\]\nfor odd $p$, with the weighted dyadic analogue. An additive relation selects a finite bipartite subnetwork from input primes to output primes.","\\section{Exceptional \\texorpdfstring{$abc$}{abc} classification}\nA structural classifier should attach to every high-quality triple the tuple\n\\[\n (\\epsilon_2,K,W,C)\n\\]\nand additional minor data from the cross-prime logarithm matrix. A high-quality exceptional set may then be partitioned into:","Even without a uniform $abc$ proof, a theorem showing that one or more mechanisms occur in a density-zero family would be meaningful progress.","\\section{Generalized Fermat}\nFor\n\\[\n x^p+y^q=z^r,\n\\]\nattach filtered contact packets to the three terms. The exponents multiply the local valuation depths. Balanced Kummer ratios and cross-prime logarithmic filters can eliminate many local patterns before a Frey or hypergeometric motive is attached.","\\section{Beal-type equations}\nA primitive counterexample to\n\\[\n A^x+B^y=C^z,\n \\qquad x,y,z>2,\n\\]\nwould contain three deep contact packets on disjoint prime supports. This is an especially strong setting for prime-contact filtering.","\\section{Pillai and Hall}\nFor\n\\[\n x^m-y^n=k,\n\\]\nthe residual support is fixed by $k$. The two perfect-power packets must cancel to high precision while producing a small fixed output. This is a natural two-sector coherence problem.","Hall's equation\n\\[\n x^3-y^2=k\n\\]\nis particularly attractive because the dyadic and triadic branches are built into the exponents. The polylogarithmic weight-three resonance may be relevant, but a rigorous connection has not yet been established.","\\section{Bernoulli--Witt and Erd\\H{o}s--Moser}\nFaulhaber's formula rewrites\n\\[\n 1^k+\\cdots+(m-1)^k=m^k\n\\]\nas a Bernoulli-polynomial identity. The denominator primes of Bernoulli numbers, the finite-difference divisibility, and the separate $2$-adic and odd-adic lift structures suggest a different contact object:\n\\[\n \\boxed{\n \\text{Bernoulli contact crystal}\n +\n \\text{denominator prime packet}\n +\n \\text{finite-difference coherence lattice}.\n }\n\\]\nThis branch should be developed independently before it is merged with the logarithmic prime-contact theory.","\\section{Combined logarithmic--coherence conductor}\nFor the actual rows constructed from the input primes and output-prime logarithms, prove\n\\[\n WC\\le C_\\eps R_aR_bR^\\eps.\n\\]\nThe theorem must control both entrywise logarithmic degeneration and collective coherence and must exploit the addition law. On the subfamily $W=1$, it reduces to a pure bound for $C$. The corresponding statement is false for arbitrary primitive congruence rows.","\\section{Sequential normalization}\nLet\n\\[\n p^e\\parallel c.\n\\]\nThe local output contact packet has order\n\\[\n p^{e-1}.\n\\]\nThe final factorization separates this depth into distinct mechanisms. The separation must be sequential because the same prime-power depth can otherwise be counted more than once.","Start with the signed input exponent vector\n\\[\n \\nu=(\\nu_q)_{q\\mid ab},\n \\qquad\n \\nu_q=v_q(b)-v_q(a).\n\\]\nLet $K_p$ be the $p$-primary part of the common Kummer factor. Divide the exponent vector and the local modulus by this common power. The normalized vector is primitive relative to the remaining modulus.","For odd $p$, form the logarithmic row\n\\[\n \\ell_{p,e}(q)\n =\n \\frac1p\\log_p\\langle q\\rangle_p\n \\pmod{p^{e-1}}.\n\\]\nLet\n\\[\n w_p\n =\n \\min_{q\\mid ab}\n v_p(\\ell_{p,e}(q))\n\\]\nafter Kummer normalization. Divide the row and the modulus by $p^{w_p}$. The remaining row is primitive modulo the residual power $p^{\\chi_p}$.","The local residual character is\n\\[\n \\chi_p(x)\n =\n \\exp\\left(\n \\frac{2\\pi i}{p^{\\chi_p}}g_p\\cdot x\n \\right).\n\\]\nIts kernel contains the normalized exponent vector. The local factors are therefore\n\\[\n K_p,\n \\qquad\n W_p=p^{w_p},\n \\qquad\n C_p=p^{\\chi_p}.\n\\]\nAt two, one sign layer $\\epsilon_{2,p}$ is removed before the weighted logarithmic row is formed.","Multiplying over output primes gives\n\\[\n \\boxed{\n M=\\epsilon_2KWC.\n }\n\\]\nThe equality follows because the local exponents partition the original contact depth:\n\\[\n e-1\n =\n v_p(\\epsilon_2)\n +\n v_p(K)\n +\n v_p(W)\n +\n v_p(C).\n\\]","\\section{Why the Kummer layer comes first}\nSuppose every input exponent is divisible by $p^k$. Then the balanced rational packet is already a $p^k$th power in the valuation lattice. The corresponding output contact depth should not also be attributed to a local logarithm defect.","\\section{Odd-prime coefficient depth}\nLet $q\\neq p$ be prime. Write\n\\[\n q=\\omega(q)\\langle q\\rangle_p,\n\\]\nwhere $\\omega(q)$ is the Teichm\\\"uller factor. Since\n\\[\n \\langle q\\rangle_p^{p-1}=q^{p-1},\n\\]\nwe have\n\\[\n \\log_p\\langle q\\rangle_p\n =\n \\frac1{p-1}\\log_p(q^{p-1}).\n\\]\nIf\n\\[\n s=v_p(q^{p-1}-1)>0,\n\\]\nthen\n\\[\n v_p(\\log_p(q^{p-1}))=s.\n\\]\nTherefore\n\\[\n v_p\\left(\\frac1p\\log_p\\langle q\\rangle_p\\right)\n =s-1.\n\\]\nTruncating at the available contact depth gives\n\\[\n v_p(\\ell_{p,e}(q))\n =\n \\min\\{e-1,s-1\\}.\n\\]",". Write\n\\[\n q=\\omega(q)\\langle q\\rangle_p,\n\\]\nwhere $\\omega(q)$ is the Teichm\\\"uller factor. Since\n\\[\n \\langle q\\rangle_p^{p-1}=q^{p-1},\n\\]\nwe have\n\\[\n \\log_p\\langle q\\rangle_p\n =\n \\frac1{p-1}\\log_p(q^{p-1}).\n\\]\nIf\n\\[\n s=v_p(q^{p-1}-1)>0,\n\\]\nthen\n\\[\n v_p(\\log_p(q^{p-1}))=s.\n\\]\nTherefore\n\\[\n v_p\\left(\\frac1p\\log_p\\langle q\\rangle_p\\right)\n =s-1.\n\\]\nTruncating at the available contact depth gives\n\\[\n v_p(\\ell_{p,e}(q))\n =\n \\min\\{e-1,s-1\\}.\n\\]","\\section{Dyadic coefficient depth}\nFor odd $q$, separate the sign by writing\n\\[\n q=(-1)^{(q-1)/2}\\langle q\\rangle_2,\n \\qquad\n \\langle q\\rangle_2\\in1+4\\Z_2.\n\\]\nThen\n\\[\n \\langle q\\rangle_2^2=q^2.\n\\]\nThe weighted coefficient is\n\\[\n \\ell_{2,e}(q)\n =\n \\frac14\\log_2\\langle q\\rangle_2.\n\\]\nIf\n\\[\n s=v_2(q^2-1),\n\\]\nthen\n\\[\n v_2(\\ell_{2,e}(q))=s-3,\n\\]\ntruncated at $e-2$. The missing first layer is the separate sign bit. This is the final form of the weighted dyadic branch.","\\section{CRT synthesis of the global row}\nThe local residual moduli $C_p$ are pairwise coprime. Let\n\\[\n C=\\prod_{p\\mid c}C_p.\n\\]\nFor each input coordinate $q$, the Chinese remainder theorem gives a unique residue\n\\[\n g_q\\pmod C\n\\]\nwhose reduction modulo $C_p$ is the local coefficient $g_{p,q}$.","The resulting row\n\\[\n g=(g_q)_{q\\mid ab}\n\\]\nis primitive modulo $C$ because every local row is primitive. Its character\n\\[\n \\chi_{\\mathrm{coh}}(x)\n =\n e^{2\\pi i g\\cdot x/C}\n\\]\nhas exact order $C$. The coherence lattice is its kernel.","\\section{A determinant viewpoint}\nWhen several independent exponent packets are available, replace the one-row character by an exterior-power regulator. Let $L$ be a normalized cross-prime logarithm matrix. A nonzero $r\\times r$ minor\n\\[\n \\det L_{I,J}\n\\]\nmeasures the failure of $r$ input packets to cancel simultaneously across $r$ output channels.","\\section{One-row Smith normal form}\nFor an integer row\n\\[\n a=(a_1,\\ldots,a_s),\n\\]\nlet\n\\[\n d=\\gcd(a_1,\\ldots,a_s).\n\\]\nThere exists a unimodular matrix $U\\in\\mathrm{GL}_s(\\Z)$ such that\n\\[\n aU=(d,0,\\ldots,0).\n\\]\nTherefore the image of the row is $d\\Z$, the kernel has rank $s-1$, and the quotient by the kernel is infinite cyclic scaled by $d$.","\\section{Two-row Smith data}\nFor a two-row matrix $M$, the first Smith factor is the gcd of all entries. The product of the two factors is the gcd of all nonzero $2\\times2$ minors. This allows the obstruction torsion to be computed without constructing the full unimodular reduction.","For the exponent-free phase obstruction matrix, every nonzero cross-sector minor is\n\\[\n \\pm\\frac{abc}{pq}\n =\n \\pm\\frac{abc}{R}\\frac{R}{pq}.\n\\]\nAll such minors are divisible by $abc/R$. For each prime $\\ell\\mid R$, choose one column with $p=\\ell$ and one column from a different nonempty sector. The reduced minor $R/(pq)$ is then not divisible by $\\ell$. Hence the gcd of the reduced minors is one and the second Smith factor is exactly $abc/R$.","n matrix, every nonzero cross-sector minor is\n\\[\n \\pm\\frac{abc}{pq}\n =\n \\pm\\frac{abc}{R}\\frac{R}{pq}.\n\\]\nAll such minors are divisible by $abc/R$. For each prime $\\ell\\mid R$, choose one column with $p=\\ell$ and one column from a different nonempty sector. The reduced minor $R/(pq)$ is then not divisible by $\\ell$. Hence the gcd of the reduced minors is one and the second Smith factor is exactly $abc/R$.","\\section{Local CRT lift in closed form}\nLet\n\\[\n r=\\prod_{i=1}^{m}p_i,\n \\qquad\n w_i=r/p_i.\n\\]\nThe equation\n\\[\n \\sum_iw_ix_i=t\n\\]\nhas a solution because\n\\[\n \\gcd(w_1,\\ldots,w_m)=1.\n\\]\nModulo $p_i$,\n\\[\n x_i\\equiv t w_i^{-1}\\pmod{p_i}.\n\\]\nLet $\\xi_i$ be the centered residue. Then\n\\[\n \\sum_iw_i\\xi_i-t\n\\]\nis divisible by $r$. Set\n\\[\n q=\\frac{t-\\sum_iw_i\\xi_i}{r}.\n\\]\nWriting $x_i=\\xi_i+p_i z_i$ gives\n\\[\n \\sum_i z_i=q.\n\\]","For a box $|x_i|\\le H$, the permitted $z_i$ form intervals. The sum of independent integer intervals is every integer between the sum of lower endpoints and the sum of upper endpoints. This proves the exact feasibility test.","\\section{Binary search certificate}\nThe lower and upper endpoint sums are monotone in $H$. Therefore the least feasible height is found by binary search. The result includes both:","\\begin{enumerate}[leftmargin=1.5em]\n\\item a witness at height $H$\n\\item an interval-separation certificate at height $H-1$\n\\end{enumerate}","\\section{Polyphase circulant determinant}\nFor the valuation word\n\\[\n V_e(T)=1+T+\\cdots+T^{e-1}\n\\]\nin $A_N=\\Z[T]/(T^N-1)$, multiplication is circulant. Its determinant is\n\\[\n \\prod_{\\zeta^N=1}V_e(\\zeta).\n\\]\nLet $I_N$ be the augmentation ideal of $A_N$. When $\\gcd(e,N)=1$, the nontrivial roots are permuted by $\\zeta\\mapsto\\zeta^e$, so multiplication by $V_e$ has determinant one on the integral lattice $I_N$ and is therefore an automorphism of it. On $A_N/I_N\\cong\\Z$, the map is multiplication by $e$. The exact sequence for $I_N\\subset A_N$ then gives cokernel $\\Z/e\\Z$ and Smith form $\\operatorname{diag}(1,\\ldots,1,e)$.","tion{Polyphase circulant determinant}\nFor the valuation word\n\\[\n V_e(T)=1+T+\\cdots+T^{e-1}\n\\]\nin $A_N=\\Z[T]/(T^N-1)$, multiplication is circulant. Its determinant is\n\\[\n \\prod_{\\zeta^N=1}V_e(\\zeta).\n\\]\nLet $I_N$ be the augmentation ideal of $A_N$. When $\\gcd(e,N)=1$, the nontrivial roots are permuted by $\\zeta\\mapsto\\zeta^e$, so multiplication by $V_e$ has determinant one on the integral lattice $I_N$ and is therefore an automorphism of it. On $A_N/I_N\\cong\\Z$, the map is multiplication by $e$. The exact sequence for $I_N\\subset A_N$ then gives cokernel $\\Z/e\\Z$ and Smith form $\\operatorname{diag}(1,\\ldots,1,e)$.","uation word\n\\[\n V_e(T)=1+T+\\cdots+T^{e-1}\n\\]\nin $A_N=\\Z[T]/(T^N-1)$, multiplication is circulant. Its determinant is\n\\[\n \\prod_{\\zeta^N=1}V_e(\\zeta).\n\\]\nLet $I_N$ be the augmentation ideal of $A_N$. When $\\gcd(e,N)=1$, the nontrivial roots are permuted by $\\zeta\\mapsto\\zeta^e$, so multiplication by $V_e$ has determinant one on the integral lattice $I_N$ and is therefore an automorphism of it. On $A_N/I_N\\cong\\Z$, the map is multiplication by $e$. The exact sequence for $I_N\\subset A_N$ then gives cokernel $\\Z/e\\Z$ and Smith form $\\operatorname{diag}(1,\\ldots,1,e)$.","$A_N=\\Z[T]/(T^N-1)$, multiplication is circulant. Its determinant is\n\\[\n \\prod_{\\zeta^N=1}V_e(\\zeta).\n\\]\nLet $I_N$ be the augmentation ideal of $A_N$. When $\\gcd(e,N)=1$, the nontrivial roots are permuted by $\\zeta\\mapsto\\zeta^e$, so multiplication by $V_e$ has determinant one on the integral lattice $I_N$ and is therefore an automorphism of it. On $A_N/I_N\\cong\\Z$, the map is multiplication by $e$. The exact sequence for $I_N\\subset A_N$ then gives cokernel $\\Z/e\\Z$ and Smith form $\\operatorname{diag}(1,\\ldots,1,e)$.","lication is circulant. Its determinant is\n\\[\n \\prod_{\\zeta^N=1}V_e(\\zeta).\n\\]\nLet $I_N$ be the augmentation ideal of $A_N$. When $\\gcd(e,N)=1$, the nontrivial roots are permuted by $\\zeta\\mapsto\\zeta^e$, so multiplication by $V_e$ has determinant one on the integral lattice $I_N$ and is therefore an automorphism of it. On $A_N/I_N\\cong\\Z$, the map is multiplication by $e$. The exact sequence for $I_N\\subset A_N$ then gives cokernel $\\Z/e\\Z$ and Smith form $\\operatorname{diag}(1,\\ldots,1,e)$.","t is\n\\[\n \\prod_{\\zeta^N=1}V_e(\\zeta).\n\\]\nLet $I_N$ be the augmentation ideal of $A_N$. When $\\gcd(e,N)=1$, the nontrivial roots are permuted by $\\zeta\\mapsto\\zeta^e$, so multiplication by $V_e$ has determinant one on the integral lattice $I_N$ and is therefore an automorphism of it. On $A_N/I_N\\cong\\Z$, the map is multiplication by $e$. The exact sequence for $I_N\\subset A_N$ then gives cokernel $\\Z/e\\Z$ and Smith form $\\operatorname{diag}(1,\\ldots,1,e)$.","1}V_e(\\zeta).\n\\]\nLet $I_N$ be the augmentation ideal of $A_N$. When $\\gcd(e,N)=1$, the nontrivial roots are permuted by $\\zeta\\mapsto\\zeta^e$, so multiplication by $V_e$ has determinant one on the integral lattice $I_N$ and is therefore an automorphism of it. On $A_N/I_N\\cong\\Z$, the map is multiplication by $e$. The exact sequence for $I_N\\subset A_N$ then gives cokernel $\\Z/e\\Z$ and Smith form $\\operatorname{diag}(1,\\ldots,1,e)$.","\\section{Synthesis-normalized refinement}\nSuppose a coarse integer $t$ is split into $e$ shell coordinates. The minimum child sup norm is roughly $|t|/e$. The shell-sum synthesis map has norm $e$. The product of analysis norm and synthesis norm remains at least one. Any claimed gain must therefore be stated in a synthesis-normalized norm.","\\section{Exact ratio semantics}\nThree ratios recur in the computation:\n\\[\n \\frac{\\Phi}{\\overline H},\n \\qquad\n \\frac{\\Phi}{k_*\\overline H},\n \\qquad\n \\frac{H(1)}{\\Phi}.\n\\]\nThey are all at least one by definition. A value greater than one is a discrete overhead, not an error. Exact rational strings should accompany decimal displays when the integers are large.","\\midrule\n$a,b,c$ & primitive additive triple","$R$ & global radical","$q$ & standard quality","$M=c/R_c$ & output contact modulus","$\\Theta,\\Phi$ & coarse and phase packet heights when computed","$H(1),k_*,H(k_*)$ & exact moving-search data","$\\epsilon_2,K,W,C$ & sequential final factorization","$\\nu_0$ & normalized input exponent vector","\\section{What a high-quality row does not prove}\nA large $C$, a short relation vector, or a striking phase resonance in one row does not imply a uniform law. The row can suggest a mechanism. A theorem requires a family definition and a proof that the mechanism persists or is rare.","\\section{Recommended finite experiments}\nThe most informative next experiments are not simply larger bounds on $c$. They should stratify by mechanism.","\\begin{enumerate}[leftmargin=1.5em]\n\\item Fix the number of input primes and vary output depth.\n\\item Fix the output prime and search for large local coefficient defects.\n\\item Separate $W>1$ rows from $W=1$ coherence rows.\n\\item Measure normalized logarithmic minors rather than only one coherence quotient.\n\\item Search families generated by recurrences or perfect powers.\n\\item Compare exact lattice invariants before and after gluing two relations.\n\\end{enumerate}","\\begin{enumerate}[leftmargin=1.5em]\n\\item Prime-packet Leibniz rule.\n\\item One-row additive lattice.\n\\item Arithmetic Wronskian and radical divisibility.\n\\item Three-sector Smith compression.\n\\item Polyphase Smith form with one aggregate invariant.\n\\item Joint obstruction torsion $abc/R$.\n\\item Exact lattice extension $0\\to L_0\\to L\\to\\Z\\to0$.\n\\item Exact real stable height.\n\\item Formal smoothness and all-order integral jet extension.\n\\item Exact moving-interval search and finite stopping.\n\\item Prime-shell energy identities.\n\\item Divisor-polyphase perfect reconstruction.\n\\item Atkin--Lehner scaled Walsh bank.\n\\item Integral projector denominator theorem.\n\\item Filtered contact quotient and carry correction.\n\\item Exact cyclic gcd transfer.\n\\item Witt realization of contact depth.\n\\item Exact Kummer kernel and logarithmic bound.\n\\item Fermat height compression and character bank.\n\\item Full-plane Fermat line-bundle cost.\n\\item Polylogarithmic distribution and three-point","\\begin{enumerate}[leftmargin=1.5em]\n\\item Prime-packet Leibniz rule.\n\\item One-row additive lattice.\n\\item Arithmetic Wronskian and radical divisibility.\n\\item Three-sector Smith compression.\n\\item Polyphase Smith form with one aggregate invariant.\n\\item Joint obstruction torsion $abc/R$.\n\\item Exact lattice extension $0\\to L_0\\to L\\to\\Z\\to0$.\n\\item Exact real stable height.\n\\item Formal smoothness and all-order integral jet extension.\n\\item Exact moving-interval search and finite stopping.\n\\item Prime-shell energy identities.\n\\item Divisor-polyphase perfect reconstruction.\n\\item Atkin--Lehner scaled Walsh bank.\n\\item Integral projector denominator theorem.\n\\item Filtered contact quotient and carry correction.\n\\item Exact cyclic gcd transfer.\n\\item Witt realization of contact depth.\n\\item Exact Kummer kernel and logarithmic bound.\n\\item Fermat height compression and character bank.\n\\item Full-plane Fermat line-bundle cost.\n\\item Polylogarithmic distribution and three-point closure.\n\\item Exact finite-place logarithmic notch.\n\\i","\\begin{enumerate}[leftmargin=1.5em]\n\\item small transverse packet implies $abc$\n\\item filtered cusp-to-Hecke transfer with two size bounds implies $abc$\n\\item sparse Kummer--Jacobi quotient with two size bounds implies $abc$\n\\item coherence conductor bound implies $abc$\n\\item suitable adelic logarithmic minor bound implies $abc$\n\\end{enumerate}","\\section{Arithmetic endpoint}\nThe arithmetic endpoint is the coherence character\n\\[\n \\chi_{\\mathrm{coh}}:\n \\Z^{S_{ab}}\\to\\Z/C\\Z\n\\]\nwhose row is built from simultaneous logarithms of the actual input primes and whose kernel contains the normalized exponent vector forced by $a+b=c$.","\\section{Why exact synthesis prevents free coordinate splitting}\nLet $X$ and $Y$ be normed spaces, and suppose\n\\[\n J:X\\longrightarrow Y,\n \\qquad\n S:Y\\longrightarrow X\n\\]\nsatisfy\n\\[\n SJ=I_X.\n\\]\nThen\n\\[\n \\norm{x}\n =\n \\norm{SJx}\n \\le\n \\norm{S}\\norm{Jx}.\n\\]\nTherefore\n\\[\n \\boxed{\n \\norm{Jx}\n \\ge\n \\frac{\\norm{x}}{\\norm{S}}.\n }\n\\]\nAn analysis map may make coordinates smaller only by increasing the synthesis norm. This elementary inequality is the common source of several no-free-refinement results.","For a scalar split\n\\[\n t=y_1+\\cdots+y_e,\n\\]\nthe child sup norm can fall to\n\\[\n \\left\\lceil\\frac{|t|}{e}\\right\\rceil.\n\\]\nThe synthesis map has norm $e$, so\n\\[\n e\\left\\lceil\\frac{|t|}{e}\\right\\rceil\\ge|t|.\n\\]\nThis explains why raw shell splitting cannot lower the intrinsic packet height.","\\section{Homogeneous higher obstructions}\nLet $\\mathcal O$ be homogeneous of degree $m$:\n\\[\n \\mathcal O(tx)=t^m\\mathcal O(x).\n\\]\nThen\n\\[\n \\frac{\\norm{x}}\n {|\\mathcal O(x)|^{1/m}}\n\\]\nis scale invariant. Replacing one large coordinate by several coordinates of size roughly the $m$th root does not improve a bound whose obstruction estimate raises those coordinates back to degree $m$.","\\section{Derivation of the stable height}\nThe real relaxation can be derived directly from aggregate sector variables. Let $A$, $B$, and $C$ denote the aggregate derivative values on the three sectors. Additivity gives\n\\[\n A+B=C.\n\\]\nThe normalized Wronskian is\n\\[\n -bA+aB=kQ.\n\\]\nFor one obstruction unit $k=1$, the real minimization seeks to minimize $H$ subject to\n\\[\n |A|\\le a\\sigma_aH,\n \\qquad\n |B|\\le b\\sigma_bH,\n \\qquad\n |C|\\le c\\sigma_cH,\n \\qquad\n A+B=C.\n\\]\nThe Wronskian has the three equivalent representations\n\\[\n -bA+aB,\n \\qquad\n aC-cA,\n \\qquad\n cB-bC.\n\\]\nThe triangle inequality therefore gives\n\\[\n |W|\\le H\\min\\left\\{\n ab(\\sigma_a+\\sigma_b),\n ac(\\sigma_a+\\sigma_c),\n bc(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThis bound is attained. For example, if the first displayed quantity is minimal, comparison with the other two","\\section{Derivation of the stable height}\nThe real relaxation can be derived directly from aggregate sector variables. Let $A$, $B$, and $C$ denote the aggregate derivative values on the three sectors. Additivity gives\n\\[\n A+B=C.\n\\]\nThe normalized Wronskian is\n\\[\n -bA+aB=kQ.\n\\]\nFor one obstruction unit $k=1$, the real minimization seeks to minimize $H$ subject to\n\\[\n |A|\\le a\\sigma_aH,\n \\qquad\n |B|\\le b\\sigma_bH,\n \\qquad\n |C|\\le c\\sigma_cH,\n \\qquad\n A+B=C.\n\\]\nThe Wronskian has the three equivalent representations\n\\[\n -bA+aB,\n \\qquad\n aC-cA,\n \\qquad\n cB-bC.\n\\]\nThe triangle inequality therefore gives\n\\[\n |W|\\le H\\min\\left\\{\n ab(\\sigma_a+\\sigma_b),\n ac(\\sigma_a+\\sigma_c),\n bc(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThis bound is attained. For example, if the first displayed quantity is minimal, comparison with the other two quan","\\section{Derivation of the stable height}\nThe real relaxation can be derived directly from aggregate sector variables. Let $A$, $B$, and $C$ denote the aggregate derivative values on the three sectors. Additivity gives\n\\[\n A+B=C.\n\\]\nThe normalized Wronskian is\n\\[\n -bA+aB=kQ.\n\\]\nFor one obstruction unit $k=1$, the real minimization seeks to minimize $H$ subject to\n\\[\n |A|\\le a\\sigma_aH,\n \\qquad\n |B|\\le b\\sigma_bH,\n \\qquad\n |C|\\le c\\sigma_cH,\n \\qquad\n A+B=C.\n\\]\nThe Wronskian has the three equivalent representations\n\\[\n -bA+aB,\n \\qquad\n aC-cA,\n \\qquad\n cB-bC.\n\\]\nThe triangle inequality therefore gives\n\\[\n |W|\\le H\\min\\left\\{\n ab(\\sigma_a+\\sigma_b),\n ac(\\sigma_a+\\sigma_c),\n bc(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThis bound is attained. For example, if the first displayed quantity is minimal, comparison with the other two quantities im","\\section{Derivation of the stable height}\nThe real relaxation can be derived directly from aggregate sector variables. Let $A$, $B$, and $C$ denote the aggregate derivative values on the three sectors. Additivity gives\n\\[\n A+B=C.\n\\]\nThe normalized Wronskian is\n\\[\n -bA+aB=kQ.\n\\]\nFor one obstruction unit $k=1$, the real minimization seeks to minimize $H$ subject to\n\\[\n |A|\\le a\\sigma_aH,\n \\qquad\n |B|\\le b\\sigma_bH,\n \\qquad\n |C|\\le c\\sigma_cH,\n \\qquad\n A+B=C.\n\\]\nThe Wronskian has the three equivalent representations\n\\[\n -bA+aB,\n \\qquad\n aC-cA,\n \\qquad\n cB-bC.\n\\]\nThe triangle inequality therefore gives\n\\[\n |W|\\le H\\min\\left\\{\n ab(\\sigma_a+\\sigma_b),\n ac(\\sigma_a+\\sigma_c),\n bc(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThis bound is attained. For example, if the first displayed quantity is minimal, comparison with the other two quantities implies\n\\[\n a\\sigma_a\\le b\\sigma_b+c\\sigma_c,\n \\qquad\n b\\sigma_b\\le a\\sigma_a+c\\sigma_c.\n\\]\nH","\\section{Derivation of the stable height}\nThe real relaxation can be derived directly from aggregate sector variables. Let $A$, $B$, and $C$ denote the aggregate derivative values on the three sectors. Additivity gives\n\\[\n A+B=C.\n\\]\nThe normalized Wronskian is\n\\[\n -bA+aB=kQ.\n\\]\nFor one obstruction unit $k=1$, the real minimization seeks to minimize $H$ subject to\n\\[\n |A|\\le a\\sigma_aH,\n \\qquad\n |B|\\le b\\sigma_bH,\n \\qquad\n |C|\\le c\\sigma_cH,\n \\qquad\n A+B=C.\n\\]\nThe Wronskian has the three equivalent representations\n\\[\n -bA+aB,\n \\qquad\n aC-cA,\n \\qquad\n cB-bC.\n\\]\nThe triangle inequality therefore gives\n\\[\n |W|\\le H\\min\\left\\{\n ab(\\sigma_a+\\sigma_b),\n ac(\\sigma_a+\\sigma_c),\n bc(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThis bound is attained. For example, if the first displayed quantity is minimal, comparison with the other two quantities implies\n\\[\n a\\sigma_a\\le b\\sigma_b+c\\sigma_c,\n \\qquad\n b\\sigma_b\\le a\\sigma_a+c\\sigma_c.\n\\]\nHence the vertex\n\\[\n A=-a\\sigma_aH,\n \\qquad\n B=","\\section{Derivation of the stable height}\nThe real relaxation can be derived directly from aggregate sector variables. Let $A$, $B$, and $C$ denote the aggregate derivative values on the three sectors. Additivity gives\n\\[\n A+B=C.\n\\]\nThe normalized Wronskian is\n\\[\n -bA+aB=kQ.\n\\]\nFor one obstruction unit $k=1$, the real minimization seeks to minimize $H$ subject to\n\\[\n |A|\\le a\\sigma_aH,\n \\qquad\n |B|\\le b\\sigma_bH,\n \\qquad\n |C|\\le c\\sigma_cH,\n \\qquad\n A+B=C.\n\\]\nThe Wronskian has the three equivalent representations\n\\[\n -bA+aB,\n \\qquad\n aC-cA,\n \\qquad\n cB-bC.\n\\]\nThe triangle inequality therefore gives\n\\[\n |W|\\le H\\min\\left\\{\n ab(\\sigma_a+\\sigma_b),\n ac(\\sigma_a+\\sigma_c),\n bc(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThis bound is attained. For example, if the first displayed quantity is minimal, comparison with the other two quantities implies\n\\[\n a\\sigma_a\\le b\\sigma_b+c\\sigma_c,\n \\qquad\n b\\sigma_b\\le a\\sigma_a+c\\sigma_c.\n\\]\nHence the vertex\n\\[\n A=-a\\sigma_aH,\n \\qquad\n B=b\\sigma_bH\n\\]\nis compatible with","\\section{Derivation of the stable height}\nThe real relaxation can be derived directly from aggregate sector variables. Let $A$, $B$, and $C$ denote the aggregate derivative values on the three sectors. Additivity gives\n\\[\n A+B=C.\n\\]\nThe normalized Wronskian is\n\\[\n -bA+aB=kQ.\n\\]\nFor one obstruction unit $k=1$, the real minimization seeks to minimize $H$ subject to\n\\[\n |A|\\le a\\sigma_aH,\n \\qquad\n |B|\\le b\\sigma_bH,\n \\qquad\n |C|\\le c\\sigma_cH,\n \\qquad\n A+B=C.\n\\]\nThe Wronskian has the three equivalent representations\n\\[\n -bA+aB,\n \\qquad\n aC-cA,\n \\qquad\n cB-bC.\n\\]\nThe triangle inequality therefore gives\n\\[\n |W|\\le H\\min\\left\\{\n ab(\\sigma_a+\\sigma_b),\n ac(\\sigma_a+\\sigma_c),\n bc(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThis bound is attained. For example, if the first displayed quantity is minimal, comparison with the other two quantities implies\n\\[\n a\\sigma_a\\le b\\sigma_b+c\\sigma_c,\n \\qquad\n b\\sigma_b\\le a\\sigma_a+c\\sigma_c.\n\\]\nHence the vertex\n\\[\n A=-a\\sigma_aH,\n \\qquad\n B=b\\sigma_bH\n\\]\nis compatible with the third constraint $|A+B|\\le c\\sigma_cH$ a","\\section{Derivation of the stable height}\nThe real relaxation can be derived directly from aggregate sector variables. Let $A$, $B$, and $C$ denote the aggregate derivative values on the three sectors. Additivity gives\n\\[\n A+B=C.\n\\]\nThe normalized Wronskian is\n\\[\n -bA+aB=kQ.\n\\]\nFor one obstruction unit $k=1$, the real minimization seeks to minimize $H$ subject to\n\\[\n |A|\\le a\\sigma_aH,\n \\qquad\n |B|\\le b\\sigma_bH,\n \\qquad\n |C|\\le c\\sigma_cH,\n \\qquad\n A+B=C.\n\\]\nThe Wronskian has the three equivalent representations\n\\[\n -bA+aB,\n \\qquad\n aC-cA,\n \\qquad\n cB-bC.\n\\]\nThe triangle inequality therefore gives\n\\[\n |W|\\le H\\min\\left\\{\n ab(\\sigma_a+\\sigma_b),\n ac(\\sigma_a+\\sigma_c),\n bc(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThis bound is attained. For example, if the first displayed quantity is minimal, comparison with the other two quantities implies\n\\[\n a\\sigma_a\\le b\\sigma_b+c\\sigma_c,\n \\qquad\n b\\sigma_b\\le a\\sigma_a+c\\sigma_c.\n\\]\nHence the vertex\n\\[\n A=-a\\sigma_aH,\n \\qquad\n B=b\\sigma_bH\n\\]\nis compatible with the third constraint $|A+B|\\le c\\sigma_cH$ and attains the first bound. The other two cases…","t $A$, $B$, and $C$ denote the aggregate derivative values on the three sectors. Additivity gives\n\\[\n A+B=C.\n\\]\nThe normalized Wronskian is\n\\[\n -bA+aB=kQ.\n\\]\nFor one obstruction unit $k=1$, the real minimization seeks to minimize $H$ subject to\n\\[\n |A|\\le a\\sigma_aH,\n \\qquad\n |B|\\le b\\sigma_bH,\n \\qquad\n |C|\\le c\\sigma_cH,\n \\qquad\n A+B=C.\n\\]\nThe Wronskian has the three equivalent representations\n\\[\n -bA+aB,\n \\qquad\n aC-cA,\n \\qquad\n cB-bC.\n\\]\nThe triangle inequality therefore gives\n\\[\n |W|\\le H\\min\\left\\{\n ab(\\sigma_a+\\sigma_b),\n ac(\\sigma_a+\\sigma_c),\n bc(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThis bound is attained. For example, if the first displayed quantity is minimal, comparison with the other two quantities implies\n\\[\n a\\sigma_a\\le b\\sigma_b+c\\sigma_c,\n \\qquad\n b\\sigma_b\\le a\\sigma_a+c\\sigma_c.\n\\]\nHence the vertex\n\\[\n A=-a\\sigma_aH,\n \\qquad\n B=b\\sigma_bH\n\\]\nis compatible with the third constraint $|A+B|\\le c\\sigma_cH$ and attains the first bound. The other two cases follow by cyclic permutation.","vity gives\n\\[\n A+B=C.\n\\]\nThe normalized Wronskian is\n\\[\n -bA+aB=kQ.\n\\]\nFor one obstruction unit $k=1$, the real minimization seeks to minimize $H$ subject to\n\\[\n |A|\\le a\\sigma_aH,\n \\qquad\n |B|\\le b\\sigma_bH,\n \\qquad\n |C|\\le c\\sigma_cH,\n \\qquad\n A+B=C.\n\\]\nThe Wronskian has the three equivalent representations\n\\[\n -bA+aB,\n \\qquad\n aC-cA,\n \\qquad\n cB-bC.\n\\]\nThe triangle inequality therefore gives\n\\[\n |W|\\le H\\min\\left\\{\n ab(\\sigma_a+\\sigma_b),\n ac(\\sigma_a+\\sigma_c),\n bc(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThis bound is attained. For example, if the first displayed quantity is minimal, comparison with the other two quantities implies\n\\[\n a\\sigma_a\\le b\\sigma_b+c\\sigma_c,\n \\qquad\n b\\sigma_b\\le a\\sigma_a+c\\sigma_c.\n\\]\nHence the vertex\n\\[\n A=-a\\sigma_aH,\n \\qquad\n B=b\\sigma_bH\n\\]\nis compatible with the third constraint $|A+B|\\le c\\sigma_cH$ and attains the first bound. The other two cases follow by cyclic permutation.","qquad\n A+B=C.\n\\]\nThe Wronskian has the three equivalent representations\n\\[\n -bA+aB,\n \\qquad\n aC-cA,\n \\qquad\n cB-bC.\n\\]\nThe triangle inequality therefore gives\n\\[\n |W|\\le H\\min\\left\\{\n ab(\\sigma_a+\\sigma_b),\n ac(\\sigma_a+\\sigma_c),\n bc(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThis bound is attained. For example, if the first displayed quantity is minimal, comparison with the other two quantities implies\n\\[\n a\\sigma_a\\le b\\sigma_b+c\\sigma_c,\n \\qquad\n b\\sigma_b\\le a\\sigma_a+c\\sigma_c.\n\\]\nHence the vertex\n\\[\n A=-a\\sigma_aH,\n \\qquad\n B=b\\sigma_bH\n\\]\nis compatible with the third constraint $|A+B|\\le c\\sigma_cH$ and attains the first bound. The other two cases follow by cyclic permutation.","\\qquad\n cB-bC.\n\\]\nThe triangle inequality therefore gives\n\\[\n |W|\\le H\\min\\left\\{\n ab(\\sigma_a+\\sigma_b),\n ac(\\sigma_a+\\sigma_c),\n bc(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThis bound is attained. For example, if the first displayed quantity is minimal, comparison with the other two quantities implies\n\\[\n a\\sigma_a\\le b\\sigma_b+c\\sigma_c,\n \\qquad\n b\\sigma_b\\le a\\sigma_a+c\\sigma_c.\n\\]\nHence the vertex\n\\[\n A=-a\\sigma_aH,\n \\qquad\n B=b\\sigma_bH\n\\]\nis compatible with the third constraint $|A+B|\\le c\\sigma_cH$ and attains the first bound. The other two cases follow by cyclic permutation.","ab(\\sigma_a+\\sigma_b),\n ac(\\sigma_a+\\sigma_c),\n bc(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThis bound is attained. For example, if the first displayed quantity is minimal, comparison with the other two quantities implies\n\\[\n a\\sigma_a\\le b\\sigma_b+c\\sigma_c,\n \\qquad\n b\\sigma_b\\le a\\sigma_a+c\\sigma_c.\n\\]\nHence the vertex\n\\[\n A=-a\\sigma_aH,\n \\qquad\n B=b\\sigma_bH\n\\]\nis compatible with the third constraint $|A+B|\\le c\\sigma_cH$ and attains the first bound. The other two cases follow by cyclic permutation.","Since $Q=abc/R$, the exact minimum is\n\\[\n \\overline H\n =\n \\frac{c}{R\\Delta(a,b,c)},\n\\]\nwhere\n\\[\n \\Delta\n =\n \\min\\left\\{\n \\sigma_a+\\sigma_b,\n \\frac cb(\\sigma_a+\\sigma_c),\n \\frac ca(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThe real problem is therefore completely solved. The unresolved content lies in the integral departure from this stable width.","\\section{Formal jet recursion}\nLet\n\\[\n F(\\mathbf T)=0\n\\]\nbe the shell equation and let\n\\[\n \\mathbf T(t)=x^{(1)}t+x^{(2)}t^2+\\cdots.\n\\]\nSuppose the gradient row $\\Lambda$ is primitive. Choose an integral vector $v$ satisfying\n\\[\n \\Lambda\\cdot v=1.\n\\]\nAt order $m$, substitution gives\n\\[\n \\Lambda\\cdot x^{(m)}\n +\n G_m(x^{(1)},\\ldots,x^{(m-1)})\n =0.\n\\]\nThe integral correction\n\\[\n \\boxed{\n x^{(m)}\n =\n -G_m v\n }\n\\]\nsolves the equation at that order. Induction produces an integral formal curve.","\\section{Exact Kummer kernel for a rational packet}\nLet\n\\[\n x=\\pm\\prod_{q\\in S}q^{\\nu_q}\n\\]\nand consider its class modulo $M$th powers. For positive $x$, the order is\n\\[\n \\frac{M}{\\gcd(M,\\{\\nu_q\\})}.\n\\]\nIf $x<0$ and $M$ is even, the sign contributes a separate element of order two. The exact order is therefore the least common multiple of the valuation order and the sign order.","The kernel is bounded by the gcd of $M$ with the valuation vector. Since each nonzero valuation is at most $\\log_2|x|$, the kernel is logarithmic in the height of the packet. This is the precise reason Kummer globalization is almost lossless.","\\section{The mean-preserving Hecke cascade}\nAt a Tate curve, an $\\ell$-isogeny packet contains one branch multiplying depth by $\\ell$ and $\\ell$ branches multiplying depth by $1/\\ell$. Under uniform averaging,\n\\[\n \\frac{\\ell+\\ell(1/\\ell)}{\\ell+1}=1.\n\\]\nAfter $n$ independent refinements, if the deep branch is chosen $k$ times, the normalized depth is\n\\[\n \\ell^{2k-n}\n\\]\nwith multiplicity\n\\[\n \\binom nk\\ell^{n-k}.\n\\]\nTherefore\n\\[\n \\sum_{k=0}^n\n \\frac{\\binom nk\\ell^{n-k}}{(\\ell+1)^n}\n \\ell^{2k-n}\n =1.\n\\]\nThe first moment is an exact martingale. The mean logarithmic depth is\n\\[\n -n\\frac{\\ell-1}{\\ell+1}\\log\\ell.\n\\]\nMost branches become shallow, while rare branches retain the mean. Linear averaging cannot produce a uniform contact contraction.","Hecke cascade}\nAt a Tate curve, an $\\ell$-isogeny packet contains one branch multiplying depth by $\\ell$ and $\\ell$ branches multiplying depth by $1/\\ell$. Under uniform averaging,\n\\[\n \\frac{\\ell+\\ell(1/\\ell)}{\\ell+1}=1.\n\\]\nAfter $n$ independent refinements, if the deep branch is chosen $k$ times, the normalized depth is\n\\[\n \\ell^{2k-n}\n\\]\nwith multiplicity\n\\[\n \\binom nk\\ell^{n-k}.\n\\]\nTherefore\n\\[\n \\sum_{k=0}^n\n \\frac{\\binom nk\\ell^{n-k}}{(\\ell+1)^n}\n \\ell^{2k-n}\n =1.\n\\]\nThe first moment is an exact martingale. The mean logarithmic depth is\n\\[\n -n\\frac{\\ell-1}{\\ell+1}\\log\\ell.\n\\]\nMost branches become shallow, while rare branches retain the mean. Linear averaging cannot produce a uniform contact contraction.","\\ell$. Under uniform averaging,\n\\[\n \\frac{\\ell+\\ell(1/\\ell)}{\\ell+1}=1.\n\\]\nAfter $n$ independent refinements, if the deep branch is chosen $k$ times, the normalized depth is\n\\[\n \\ell^{2k-n}\n\\]\nwith multiplicity\n\\[\n \\binom nk\\ell^{n-k}.\n\\]\nTherefore\n\\[\n \\sum_{k=0}^n\n \\frac{\\binom nk\\ell^{n-k}}{(\\ell+1)^n}\n \\ell^{2k-n}\n =1.\n\\]\nThe first moment is an exact martingale. The mean logarithmic depth is\n\\[\n -n\\frac{\\ell-1}{\\ell+1}\\log\\ell.\n\\]\nMost branches become shallow, while rare branches retain the mean. Linear averaging cannot produce a uniform contact contraction.","\\section{Arbitrary-index congruence lattices}\nLet $v\\in\\Z^s$ be primitive. Complete $v$ to a unimodular basis\n\\[\n v,v_2,\\ldots,v_s.\n\\]\nIn the dual basis, choose the row that vanishes on $v$ and takes value one on $v_2$. Reduce it modulo an arbitrary integer $C$. The resulting row is primitive modulo $C$ and annihilates $v$.","Thus\n\\[\n \\Lambda_C\n =\n \\{x:g_C\\cdot x\\equiv0\\pmod C\\}\n\\]\nhas index $C$ and contains the same fixed vector $v$. This proves that no bound on $C$ can follow from the length or support of $v$ alone.","\\section{Direct positive divisibility}\nSuppose $A,B>0$ and $c\\mid A+B$. Then\n\\[\n A+B\\ge c.\n\\]\nThe larger of the two is at least half their sum, so\n\\[\n \\max(A,B)\\ge c/2.\n\\]\nThis elementary inequality rules out a wide class of direct constructions in which one attempts to build a small positive $S$-unit relation whose sum is divisible by the output.","Consider\n\\[\n A+B=C\n\\]\nand\n\\[\n C+D=E.\n\\]\nThe term $C$ is an output in the first relation and an input in the second. Its contact packet should not be duplicated independently. A functorial theory should identify the two copies and record how depth moves across the diagram.","\\section{Contact pushout}\nLet $\\Jcal_C^{\\mathrm{out}}$ be the contact packet of $C$ in the first equation and $\\Jcal_C^{\\mathrm{in}}$ its packet in the second. Both are canonically\n\\[\n \\rad(C)\\Z/C\\Z.\n\\]\nThe first gluing operation identifies them through the identity map. The total contact object is the pushout of the two relation diagrams along this common packet.","\\section{Kummer fiber product}\nBalanced Kummer ratios from the two equations define classes\n\\[\n \\beta_1=-B/A\n\\]\nand\n\\[\n \\beta_2=-D/C.\n\\]\nThe shared term $C$ creates a multiplicative compatibility. The natural global object is a fiber product of the two Kummer torsors over the Kummer class of $C$.","\\section{Logarithmic matrix composition}\nLet $L_1$ and $L_2$ be the cross-prime logarithmic matrices of the two relations. The shared term determines a common set of prime channels. The combined system has block form\n\\[\n \\begin{pmatrix}\n L_1&0","\\section{Logarithmic matrix composition}\nLet $L_1$ and $L_2$ be the cross-prime logarithmic matrices of the two relations. The shared term determines a common set of prime channels. The combined system has block form\n\\[\n \\begin{pmatrix}\n L_1&0\\\\\n *&L_2\n \\end{pmatrix}\n\\]\nwith a coupling block induced by the shared valuations of $C$.","*&L_2\n \\end{pmatrix}\n\\]\nwith a coupling block induced by the shared valuations of $C$.","\\section{Coherence-character composition}\nSuppose the two normalized coherence characters have orders $C_1$ and $C_2$. Their product character has order dividing\n\\[\n \\lcm(C_1,C_2).\n\\]\nIf the shared term forces a compatibility relation, the actual order can be strictly smaller. Conversely, the combined system can expose a higher-rank determinant even when each individual quotient is cyclic.","A useful first target is:\n\\[\n \\boxed{\n C_{\\mathrm{glued}}\n \\le\n \\frac{C_1C_2}{\\rad(C)}\n \\times\n \\text{subpolynomial correction}.\n }\n\\]\nThe displayed formula is schematic. It expresses the hope that the shared term cancels one full support layer. No such theorem is proved.","\\begin{enumerate}[leftmargin=1.5em]\n\\item recurrence sequences with addition formulas\n\\item chains of perfect-power gaps\n\\item arithmetic dynamics and orbit collisions\n\\item decompositions of generalized Fermat equations\n\\item additive combinatorial identities among $S$-units\n\\end{enumerate}","\\section{Simultaneous Wieferich theorems}\nThe edge weight\n\\[\n w_p(q)=v_p(q^{p-1}-1)-1\n\\]\nmeasures local degeneracy. A first family of theorems could bound the number of pairs $(p,q)$ with $w_p(q)\\ge k$ in a declared box. A more relevant target would control simultaneous rows:\n\\[\n \\#\\left\\{\n (P,Q):\n \\operatorname{rank}(\\mathcal L_{P,Q}\\bmod p^k)\n \\text{ is deficient}\n \\right\\}.\n\\]\nThe coherence lattice suggests studying exterior powers rather than entrywise defects alone.","\\section{Exceptional \\texorpdfstring{$abc$}{abc} by mechanism}\nFor a quality threshold $1+\\delta$, define subclasses:","\\begin{align*}\n \\mathcal E_K(\\delta)&=\\{\\text{large Kummer preload}\\},\\\\\n \\mathcal E_W(\\delta)&=\\{\\text{large local logarithmic defect}\\},\\\\\n \\mathcal E_C(\\delta)&=\\{\\text{large collective coherence}\\},\\\\\n \\mathcal E_A(\\delta)&=\\{\\text{large automorphic congruence defect}\\}.\n\\end{align*}","\\section{Generalized Fermat signatures}\nFor a signature $(p,q,r)$, local contact depths are multiplied by the corresponding exponents. The first computational filter should test whether one output-prime logarithmic row has a unit minor after exponent normalization. Such a minor would prevent the required deep cancellation.","\\section{Beal under a coherence hypothesis}\nSuppose\n\\[\n A^x+B^y=C^z\n\\]\nis primitive. The input exponent vector contains blocks scaled by $x$ and $y$. At an output prime $p\\mid C$, the logarithmic row must annihilate that vector to depth $zv_p(C)-1$.","A nondegenerate $2\\times2$ minor involving one prime from $A$ and one from $B$ would bound the possible cancellation depth. This suggests a theorem of the form:","\\section{Hall and the dyadic-triadic interface}\nFor\n\\[\n x^3-y^2=k,\n\\]\nthe prime-contact systems of the square and cube interact through exactly the two exceptional primes of the depth-three Smith form. The dyadic and triadic corrections are not incidental. They are the primary channels in which the weight-three orbit lattice departs from a split $(\\Z/M\\Z)^3$ form.","\\begin{enumerate}[leftmargin=1.5em]\n\\item the $2$-adic contact filtration of $y^2$\n\\item the $3$-adic contact filtration of $x^3$\n\\item the fixed support of $k$\n\\item the depth-three polylogarithmic packet\n\\item the elliptic curve $Y^2=X^3-k$\n\\end{enumerate}","\\section{Pillai with fixed residual support}\nFor fixed $k$, the equation\n\\[\n x^m-y^n=k\n\\]\nhas a fixed residual radical. As $x$ and $y$ grow, the two perfect-power contact packets must cancel against a bounded output support. This makes the problem closer to a regulator inequality than unrestricted $abc$.","\\section{Bernoulli--Witt direction}\nThe Bernoulli branch begins with the identity\n\\[\n \\sum_{j=1}^{m-1}j^k\n =\n \\frac{B_{k+1}(m)-B_{k+1}}{k+1}.\n\\]\nThe equation with $m^k$ imposes strong congruences on Bernoulli values. A filtered prime-contact treatment should separate:","\\begin{enumerate}[leftmargin=1.5em]\n\\item denominator primes from the von Staudt--Clausen theorem\n\\item repeated congruence depth in the numerator\n\\item finite-difference packets across $m$\n\\item a separate dyadic branch\n\\item Witt carries in the Bernoulli polynomial evaluation\n\\end{enumerate}","\\begin{enumerate}[leftmargin=1.5em]\n\\item Arithmetic derivatives and their relation to $abc$ have an existing literature.\n\\item Prony-type annihilation and finite-rate signal recovery motivate some signal language but are not part of the number-theory proofs.\n\\item The Frey curve, modularity, and local Tate uniformization are established arithmetic tools.\n\\item Atkin--Lehner involutions and Hecke congruence modules are standard automorphic objects.\n\\item Wiles defects compare cotangent and congruence data in suitable Hecke algebras.\n\\item Kummer theory gives the global contact class.\n\\item Fermat character decompositions and Jacobi sums are classical.\n\\item Polylogarithmic distribution and three-point functional identities are established analytic and motivic relations.\n\\item Witt","\\begin{enumerate}[leftmargin=1.5em]\n\\item the strong $abc$ conjecture\n\\item a uniform coherence conductor bound\n\\item a radical-scale adelic logarithmic determinant theorem\n\\item a filtered cusp-to-Hecke map with the required kernel bound\n\\item a sparse motivic quotient with radical-scale integral cost\n\\item a functorial gluing theorem\n\\item any full major-conjecture application beyond the exact reductions stated\n\\end{enumerate}","\\begin{enumerate}[leftmargin=1.5em]\n\\item The first-order search was solved exactly and shown to be circular as a route to a new $abc$ gain.\n\\item Pure shell splitting was shown to be synthesis-normalized.\n\\item The split persistence packet was shown to have the wrong type for a cyclic target.\n\\item Restoring carry extensions corrected the source to a filtered cyclic object.\n\\item The full plane Fermat bank carries a line-bundle factor of order $M$, so it does not yield a free intrinsic height saving.\n\\item A small complex period was shown not to control the integral lattice.\n\\item Abstract lattice geometry was shown unable to bound the coherence index.\n\\item Positive divisibility constructions were shown to have height at least $c/2$.\n\\item Linear Hecke refinement was shown to preserve mean contact depth.\n\\end{enumerate}","t-order search was solved exactly and shown to be circular as a route to a new $abc$ gain.\n\\item Pure shell splitting was shown to be synthesis-normalized.\n\\item The split persistence packet was shown to have the wrong type for a cyclic target.\n\\item Restoring carry extensions corrected the source to a filtered cyclic object.\n\\item The full plane Fermat bank carries a line-bundle factor of order $M$, so it does not yield a free intrinsic height saving.\n\\item A small complex period was shown not to control the integral lattice.\n\\item Abstract lattice geometry was shown unable to bound the coherence index.\n\\item Positive divisibility constructions were shown to have height at least $c/2$.\n\\item Linear Hecke refinement was shown to preserve mean contact depth.\n\\end{enumerate}","e source to a filtered cyclic object.\n\\item The full plane Fermat bank carries a line-bundle factor of order $M$, so it does not yield a free intrinsic height saving.\n\\item A small complex period was shown not to control the integral lattice.\n\\item Abstract lattice geometry was shown unable to bound the coherence index.\n\\item Positive divisibility constructions were shown to have height at least $c/2$.\n\\item Linear Hecke refinement was shown to preserve mean contact depth.\n\\end{enumerate}","\\bibitem{MochizukiIUT}\nS. Mochizuki,\n``Inter-universal Teichm\\\"uller theory I--IV,''\n\\emph{Publications of the Research Institute for Mathematical Sciences} 57 (2021), 3--723.\nThe present paper does not use these papers as an established proof of $abc$.","\\bibitem{ScholzeStix}\nP. Scholze and J. Stix,\n``Why $abc$ is still a conjecture,''\nmanuscript, 2018.","\\bibitem{BCDT}\nC. Breuil, B. Conrad, F. Diamond, and R. Taylor,\n``On the modularity of elliptic curves over $\\mathbb Q$: wild 3-adic exercises,''\n\\emph{Journal of the American Mathematical Society} 14 (2001), 843--939.\n\\href{https://doi.org/10.1090/S0894-0347-01-00370-8}{doi:10.1090/S0894-0347-01-00370-8}.","\\bibitem{PastenShimura}\nH. Pasten,\n``Shimura curves and the $abc$ conjecture,''\narXiv:1705.09251, 2017.","\\bibitem{AKN}\nB. Antieau, A. Krause, and T. Nikolaus,\n``On the K-theory of $\\mathbb Z/p^n$,''\narXiv:2405.04329, 2024.","\\section{The prime-coordinate module}\nLet\n\\[\n a+b=c,\n \\qquad\n \\gcd(a,b)=1.\n\\]\nThe three integers are pairwise coprime. Write\n\\[\n S=S_a\\sqcup S_b\\sqcup S_c\n =\\{p:p\\mid abc\\}.\n\\]\nThe first object of the calculus is the free prime-packet module\n\\[\n \\Omega_S=\\bigoplus_{p\\in S}\\Z\\xi_p.\n\\]\nFor a positive integer $n$, define the universal multiplicative differential\n\\[\n \\boxed{\n \\mathrm dn\n =\n n\\sum_{p\\mid n}\\frac{v_p(n)}p\\,\\xi_p.\n }\n\\]\nEvery coefficient is an integer because $p\\mid n$. The map obeys the exact Leibniz law\n\\[\n \\mathrm d(mn)=m\\,\\mathrm dn+n\\,\\mathrm dm.\n\\]\nThe verification is local. For each prime $p$,\n\\[\n v_p(mn)=v_p(m)+v_p(n),\n\\]\nso the $p$-coordinate of the right side is\n\\[\n mn\\frac{v_p(m)+v_p(n)}p.\n\\]","\\]\nThe three integers are pairwise coprime. Write\n\\[\n S=S_a\\sqcup S_b\\sqcup S_c\n =\\{p:p\\mid abc\\}.\n\\]\nThe first object of the calculus is the free prime-packet module\n\\[\n \\Omega_S=\\bigoplus_{p\\in S}\\Z\\xi_p.\n\\]\nFor a positive integer $n$, define the universal multiplicative differential\n\\[\n \\boxed{\n \\mathrm dn\n =\n n\\sum_{p\\mid n}\\frac{v_p(n)}p\\,\\xi_p.\n }\n\\]\nEvery coefficient is an integer because $p\\mid n$. The map obeys the exact Leibniz law\n\\[\n \\mathrm d(mn)=m\\,\\mathrm dn+n\\,\\mathrm dm.\n\\]\nThe verification is local. For each prime $p$,\n\\[\n v_p(mn)=v_p(m)+v_p(n),\n\\]\nso the $p$-coordinate of the right side is\n\\[\n mn\\frac{v_p(m)+v_p(n)}p.\n\\]","e\n\\[\n S=S_a\\sqcup S_b\\sqcup S_c\n =\\{p:p\\mid abc\\}.\n\\]\nThe first object of the calculus is the free prime-packet module\n\\[\n \\Omega_S=\\bigoplus_{p\\in S}\\Z\\xi_p.\n\\]\nFor a positive integer $n$, define the universal multiplicative differential\n\\[\n \\boxed{\n \\mathrm dn\n =\n n\\sum_{p\\mid n}\\frac{v_p(n)}p\\,\\xi_p.\n }\n\\]\nEvery coefficient is an integer because $p\\mid n$. The map obeys the exact Leibniz law\n\\[\n \\mathrm d(mn)=m\\,\\mathrm dn+n\\,\\mathrm dm.\n\\]\nThe verification is local. For each prime $p$,\n\\[\n v_p(mn)=v_p(m)+v_p(n),\n\\]\nso the $p$-coordinate of the right side is\n\\[\n mn\\frac{v_p(m)+v_p(n)}p.\n\\]","he free prime-packet module\n\\[\n \\Omega_S=\\bigoplus_{p\\in S}\\Z\\xi_p.\n\\]\nFor a positive integer $n$, define the universal multiplicative differential\n\\[\n \\boxed{\n \\mathrm dn\n =\n n\\sum_{p\\mid n}\\frac{v_p(n)}p\\,\\xi_p.\n }\n\\]\nEvery coefficient is an integer because $p\\mid n$. The map obeys the exact Leibniz law\n\\[\n \\mathrm d(mn)=m\\,\\mathrm dn+n\\,\\mathrm dm.\n\\]\nThe verification is local. For each prime $p$,\n\\[\n v_p(mn)=v_p(m)+v_p(n),\n\\]\nso the $p$-coordinate of the right side is\n\\[\n mn\\frac{v_p(m)+v_p(n)}p.\n\\]","e prime-packet module\n\\[\n \\Omega_S=\\bigoplus_{p\\in S}\\Z\\xi_p.\n\\]\nFor a positive integer $n$, define the universal multiplicative differential\n\\[\n \\boxed{\n \\mathrm dn\n =\n n\\sum_{p\\mid n}\\frac{v_p(n)}p\\,\\xi_p.\n }\n\\]\nEvery coefficient is an integer because $p\\mid n$. The map obeys the exact Leibniz law\n\\[\n \\mathrm d(mn)=m\\,\\mathrm dn+n\\,\\mathrm dm.\n\\]\nThe verification is local. For each prime $p$,\n\\[\n v_p(mn)=v_p(m)+v_p(n),\n\\]\nso the $p$-coordinate of the right side is\n\\[\n mn\\frac{v_p(m)+v_p(n)}p.\n\\]","bigoplus_{p\\in S}\\Z\\xi_p.\n\\]\nFor a positive integer $n$, define the universal multiplicative differential\n\\[\n \\boxed{\n \\mathrm dn\n =\n n\\sum_{p\\mid n}\\frac{v_p(n)}p\\,\\xi_p.\n }\n\\]\nEvery coefficient is an integer because $p\\mid n$. The map obeys the exact Leibniz law\n\\[\n \\mathrm d(mn)=m\\,\\mathrm dn+n\\,\\mathrm dm.\n\\]\nThe verification is local. For each prime $p$,\n\\[\n v_p(mn)=v_p(m)+v_p(n),\n\\]\nso the $p$-coordinate of the right side is\n\\[\n mn\\frac{v_p(m)+v_p(n)}p.\n\\]","sitive integer $n$, define the universal multiplicative differential\n\\[\n \\boxed{\n \\mathrm dn\n =\n n\\sum_{p\\mid n}\\frac{v_p(n)}p\\,\\xi_p.\n }\n\\]\nEvery coefficient is an integer because $p\\mid n$. The map obeys the exact Leibniz law\n\\[\n \\mathrm d(mn)=m\\,\\mathrm dn+n\\,\\mathrm dm.\n\\]\nThe verification is local. For each prime $p$,\n\\[\n v_p(mn)=v_p(m)+v_p(n),\n\\]\nso the $p$-coordinate of the right side is\n\\[\n mn\\frac{v_p(m)+v_p(n)}p.\n\\]","A scalar packet is an integer vector\n\\[\n x=(x_p)_{p\\in S}\\in\\Z^S.\n\\]\nPairing with $x$ gives the arithmetic derivative\n\\[\n \\boxed{\n \\mathrm d_x(n)\n =\n n\\sum_{p\\mid n}\\frac{v_p(n)}p x_p.\n }\n\\]\nThe Leibniz rule survives scalarization.","Arithmetic derivatives and their connection with $abc$ have an established literature. The present construction is organized here in prime-packet coordinates and later refined into filtered shell objects~\\cite{PastenDerivatives,VojtaABC}.","\\section{The additive row}\nThe condition\n\\[\n \\mathrm d_x(a)+\\mathrm d_x(b)=\\mathrm d_x(c)\n\\]\nis one integer equation. Define\n\\[\n \\lambda_p=\n \\begin{cases}\n \\dfrac{a v_p(a)}p,&p\\mid a,","\\section{The additive row}\nThe condition\n\\[\n \\mathrm d_x(a)+\\mathrm d_x(b)=\\mathrm d_x(c)\n\\]\nis one integer equation. Define\n\\[\n \\lambda_p=\n \\begin{cases}\n \\dfrac{a v_p(a)}p,&p\\mid a,\\\\[6pt]\n \\dfrac{b v_p(b)}p,&p\\mid b,\\\\[6pt]\n -\\dfrac{c v_p(c)}p,&p\\mid c.\n \\end{cases}\n\\]\nThen\n\\[\n \\boxed{\\lambda\\cdot x=0.}\n\\]\nThe additive derivative lattice is\n\\[\n \\boxed{\n \\mathcal T_{a,b,c}\n =\n \\ker(\\lambda)\\cap\\Z^S.\n }\n\\]\nSince the row is nonzero, its rank is $|S|-1$. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by $a$ and $b$.","[6pt]\n -\\dfrac{c v_p(c)}p,&p\\mid c.\n \\end{cases}\n\\]\nThen\n\\[\n \\boxed{\\lambda\\cdot x=0.}\n\\]\nThe additive derivative lattice is\n\\[\n \\boxed{\n \\mathcal T_{a,b,c}\n =\n \\ker(\\lambda)\\cap\\Z^S.\n }\n\\]\nSince the row is nonzero, its rank is $|S|-1$. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by $a$ and $b$.","]\n -\\dfrac{c v_p(c)}p,&p\\mid c.\n \\end{cases}\n\\]\nThen\n\\[\n \\boxed{\\lambda\\cdot x=0.}\n\\]\nThe additive derivative lattice is\n\\[\n \\boxed{\n \\mathcal T_{a,b,c}\n =\n \\ker(\\lambda)\\cap\\Z^S.\n }\n\\]\nSince the row is nonzero, its rank is $|S|-1$. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by $a$ and $b$.","ac{c v_p(c)}p,&p\\mid c.\n \\end{cases}\n\\]\nThen\n\\[\n \\boxed{\\lambda\\cdot x=0.}\n\\]\nThe additive derivative lattice is\n\\[\n \\boxed{\n \\mathcal T_{a,b,c}\n =\n \\ker(\\lambda)\\cap\\Z^S.\n }\n\\]\nSince the row is nonzero, its rank is $|S|-1$. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by $a$ and $b$.","\\section{The arithmetic Wronskian}\nDefine\n\\[\n \\boxed{\n W_x(a,b)\n =a\\,\\mathrm d_x(b)-b\\,\\mathrm d_x(a).\n }\n\\]\nWriting\n\\[\n \\mu_p=\n \\begin{cases}\n -\\dfrac{v_p(a)}p,&p\\mid a,","\\section{The arithmetic Wronskian}\nDefine\n\\[\n \\boxed{\n W_x(a,b)\n =a\\,\\mathrm d_x(b)-b\\,\\mathrm d_x(a).\n }\n\\]\nWriting\n\\[\n \\mu_p=\n \\begin{cases}\n -\\dfrac{v_p(a)}p,&p\\mid a,\\\\[6pt]\n \\dfrac{v_p(b)}p,&p\\mid b,\\\\[6pt]\n 0,&p\\mid c,\n \\end{cases}\n\\]\nwe obtain\n\\[\n W_x(a,b)=ab\\,\\mu\\cdot x.\n\\]\nAn additive packet is \\emph{transverse} when $W_x(a,b)\\neq0$. It is degenerate when the two input terms remain derivative-proportional.","[6pt]\n 0,&p\\mid c,\n \\end{cases}\n\\]\nwe obtain\n\\[\n W_x(a,b)=ab\\,\\mu\\cdot x.\n\\]\nAn additive packet is \\emph{transverse} when $W_x(a,b)\\neq0$. It is degenerate when the two input terms remain derivative-proportional.","Over $\\R$, the additive projection is\n\\[\n P_{\\mathrm{add}}\n =\n I-\n \\frac{\\lambda\\lambda^{\\mathsf T}}{\\norm{\\lambda}_2^2}.\n\\]\nThe obstruction direction inside the additive hyperplane is\n\\[\n \\mu_\\perp=P_{\\mathrm{add}}\\mu.\n\\]\nWhen $\\mu_\\perp\\neq0$, the rank-one obstruction projector is\n\\[\n P_{\\mathrm{obs}}\n =\n \\frac{\\mu_\\perp\\mu_\\perp^{\\mathsf T}}\n {\\norm{\\mu_\\perp}_2^2}.\n\\]\nThis is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation direction, one obstruction direction, and the remaining null packets.","\\section{Radical divisibility}\nSet\n\\[\n R=\\rad(abc).\n\\]\nThe central first-order theorem is an exact divisibility law.","\\begin{theorem}[Wronskian divisibility]\nFor every additive packet $x\\in\\mathcal T_{a,b,c}$,\n\\[\n \\boxed{\n \\frac{abc}{R}\\mid W_x(a,b).\n }\n\\]\n\\end{theorem}","\\begin{proof}\nLet $p^e\\parallel a$. Every term of $\\mathrm d_x(a)$ is divisible by $p^{e-1}$. Since $p\\nmid b$, the term $b\\,\\mathrm d_x(a)$ is divisible by $p^{e-1}$. Also $p^e\\mid a$, so $a\\,\\mathrm d_x(b)$ is divisible by $p^e$, hence by $p^{e-1}$. Therefore\n\\[\n p^{e-1}\\mid W_x(a,b).\n\\]\nThe same argument applies to primes dividing $b$.","Now let $p^e\\parallel c$. Additivity gives\n\\[\n \\mathrm d_x(c)=\\mathrm d_x(a)+\\mathrm d_x(b).\n\\]\nUsing $c=a+b$,\n\\[\n \\begin{aligned}\n W_x(a,b)\n &=a\\,\\mathrm d_x(b)-b\\,\\mathrm d_x(a)","Now let $p^e\\parallel c$. Additivity gives\n\\[\n \\mathrm d_x(c)=\\mathrm d_x(a)+\\mathrm d_x(b).\n\\]\nUsing $c=a+b$,\n\\[\n \\begin{aligned}\n W_x(a,b)\n &=a\\,\\mathrm d_x(b)-b\\,\\mathrm d_x(a)\\\\\n &=a\\bigl(\\mathrm d_x(c)-\\mathrm d_x(a)\\bigr)-b\\,\\mathrm d_x(a)\\\\\n &=a\\,\\mathrm d_x(c)-c\\,\\mathrm d_x(a).\n \\end{aligned}\n\\]\nThe first term is divisible by $p^{e-1}$ because $\\mathrm d_x(c)$ is. The second is divisible by $p^e$. Hence $p^{e-1}\\mid W_x(a,b)$.","&=a\\,\\mathrm d_x(c)-c\\,\\mathrm d_x(a).\n \\end{aligned}\n\\]\nThe first term is divisible by $p^{e-1}$ because $\\mathrm d_x(c)$ is. The second is divisible by $p^e$. Hence $p^{e-1}\\mid W_x(a,b)$.","The prime supports of $a$, $b$, and $c$ are disjoint. Multiplying all local divisibilities gives\n\\[\n \\prod_{p^e\\parallel abc}p^{e-1}\n =\n \\frac{abc}{R}\n \\mid W_x(a,b).\n\\]\n\\end{proof}","For a transverse packet,\n\\[\n \\boxed{\n |W_x(a,b)|\\ge\\frac{abc}{R}.\n }\n\\]\nThis is the integer analogue of the repeated-root divisor appearing in the polynomial Mason--Stothers theorem.","\\section{The master height inequality}\nFor every positive $n$,\n\\[\n \\begin{aligned}\n |\\mathrm d_x(n)|\n &\\le\n n\\norm{x}_\\infty\n \\sum_{p\\mid n}\\frac{v_p(n)}p.\n \\end{aligned}\n\\]\nSince $p\\ge2$,\n\\[\n \\frac1p\\le\\frac1{2\\log2}\\log p.\n\\]\nTherefore\n\\[\n \\sum_{p\\mid n}\\frac{v_p(n)}p\n \\le\n \\frac{\\log n}{2\\log2},\n\\]\nand\n\\[\n |\\mathrm d_x(n)|\n \\le\n \\frac{n\\log n}{2\\log2}\\norm{x}_\\infty.\n\\]\nConsequently,\n\\[\n \\begin{aligned}\n |W_x(a,b)|\n &\\le\n a|\\mathrm d_x(b)|+b|\\mathrm d_x(a)|","\\section{The master height inequality}\nFor every positive $n$,\n\\[\n \\begin{aligned}\n |\\mathrm d_x(n)|\n &\\le\n n\\norm{x}_\\infty\n \\sum_{p\\mid n}\\frac{v_p(n)}p.\n \\end{aligned}\n\\]\nSince $p\\ge2$,\n\\[\n \\frac1p\\le\\frac1{2\\log2}\\log p.\n\\]\nTherefore\n\\[\n \\sum_{p\\mid n}\\frac{v_p(n)}p\n \\le\n \\frac{\\log n}{2\\log2},\n\\]\nand\n\\[\n |\\mathrm d_x(n)|\n \\le\n \\frac{n\\log n}{2\\log2}\\norm{x}_\\infty.\n\\]\nConsequently,\n\\[\n \\begin{aligned}\n |W_x(a,b)|\n &\\le\n a|\\mathrm d_x(b)|+b|\\mathrm d_x(a)|\\\\\n &\\le\n \\frac{ab}{\\log2}\\norm{x}_\\infty\\log c.\n \\end{aligned}\n\\]\nCombining this with radical divisibility gives\n\\[\n \\boxed{\n c\n \\le\n \\frac{R}{\\log2}\\norm{x}_\\infty\\log c.\n }\n\\]","&\\le\n \\frac{ab}{\\log2}\\norm{x}_\\infty\\log c.\n \\end{aligned}\n\\]\nCombining this with radical divisibility gives\n\\[\n \\boxed{\n c\n \\le\n \\frac{R}{\\log2}\\norm{x}_\\infty\\log c.\n }\n\\]","Define the shortest transverse packet\n\\[\n \\boxed{\n \\Theta(a,b,c)\n =\n \\min\\left\\{\n \\norm{x}_\\infty:\n x\\in\\mathcal T_{a,b,c},\n W_x(a,b)\\neq0\n \\right\\}.\n }\n\\]\nThen\n\\[\n c\n \\le\n \\frac{R}{\\log2}\\Theta(a,b,c)\\log c.\n\\]\nA subpolynomial estimate\n\\[\n \\Theta(a,b,c)=c^{o(1)}\n\\]\nwould imply the strong $abc$ inequality. Later chapters explain why this target is too close to the conjecture to be reached by generic lattice geometry alone.","\\section{Three-sector compression}\nWithin the $a$-sector, define the integer row\n\\[\n \\lambda^{(a)}\n =\n \\left(\\frac{a v_p(a)}p\\right)_{p\\mid a}\n\\]\nand let\n\\[\n g_a=\\gcd\\left\\{\\frac{a v_p(a)}p:p\\mid a\\right\\}.\n\\]\nDefine $g_b$ and $g_c$ similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes\n\\[\n \\boxed{\n g_at_a+g_bt_b-g_ct_c=0.\n }\n\\]\nThe Wronskian becomes\n\\[\n \\boxed{\n a g_bt_b-b g_at_a\\neq0.\n }\n\\]\nAll remaining sector coordinates are lift gauges. They matter only because they determine the cost of realizing an aggregate channel by an integral prime packet.","ion{Three-sector compression}\nWithin the $a$-sector, define the integer row\n\\[\n \\lambda^{(a)}\n =\n \\left(\\frac{a v_p(a)}p\\right)_{p\\mid a}\n\\]\nand let\n\\[\n g_a=\\gcd\\left\\{\\frac{a v_p(a)}p:p\\mid a\\right\\}.\n\\]\nDefine $g_b$ and $g_c$ similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes\n\\[\n \\boxed{\n g_at_a+g_bt_b-g_ct_c=0.\n }\n\\]\nThe Wronskian becomes\n\\[\n \\boxed{\n a g_bt_b-b g_at_a\\neq0.\n }\n\\]\nAll remaining sector coordinates are lift gauges. They matter only because they determine the cost of realizing an aggregate channel by an integral prime packet.","nteger row\n\\[\n \\lambda^{(a)}\n =\n \\left(\\frac{a v_p(a)}p\\right)_{p\\mid a}\n\\]\nand let\n\\[\n g_a=\\gcd\\left\\{\\frac{a v_p(a)}p:p\\mid a\\right\\}.\n\\]\nDefine $g_b$ and $g_c$ similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes\n\\[\n \\boxed{\n g_at_a+g_bt_b-g_ct_c=0.\n }\n\\]\nThe Wronskian becomes\n\\[\n \\boxed{\n a g_bt_b-b g_at_a\\neq0.\n }\n\\]\nAll remaining sector coordinates are lift gauges. They matter only because they determine the cost of realizing an aggregate channel by an integral prime packet.","For each term define\n\\[\n \\rho_a(t)\n =\n \\min\\left\\{\n \\norm{x^{(a)}}_\\infty:\n \\lambda^{(a)}\\cdot x^{(a)}=g_at\n \\right\\},\n\\]\nwith analogous functions for $b$ and $c$. Then\n\\[\n \\Theta(a,b,c)\n =\n \\min\n \\max\\{\\rho_a(t_a),\\rho_b(t_b),\\rho_c(t_c)\\}\n\\]\nover aggregate solutions satisfying additivity and transversality.","\\section{Worked example: \\texorpdfstring{$3+125=128$}{3+125=128}}\nThe prime packets are $3$, $5$, and $2$. The additive row is\n\\[\n \\lambda=(1,75,-448).\n\\]\nA shortest transverse packet is\n\\[\n x=(-2,6,1).\n\\]\nIt gives\n\\[\n \\mathrm d_x(3)=-2,\n \\qquad\n \\mathrm d_x(125)=450,\n \\qquad\n \\mathrm d_x(128)=448.\n\\]\nThus additivity holds. The Wronskian is\n\\[\n W_x(3,125)\n =3(450)-125(-2)\n =1600.\n\\]\nThe radical is\n\\[\n R=2\\cdot3\\cdot5=30,\n\\]\nand\n\\[\n \\frac{3\\cdot125\\cdot128}{30}=1600.\n\\]\nHence the radical lower bound is attained exactly.","ction{Worked example: \\texorpdfstring{$3+125=128$}{3+125=128}}\nThe prime packets are $3$, $5$, and $2$. The additive row is\n\\[\n \\lambda=(1,75,-448).\n\\]\nA shortest transverse packet is\n\\[\n x=(-2,6,1).\n\\]\nIt gives\n\\[\n \\mathrm d_x(3)=-2,\n \\qquad\n \\mathrm d_x(125)=450,\n \\qquad\n \\mathrm d_x(128)=448.\n\\]\nThus additivity holds. The Wronskian is\n\\[\n W_x(3,125)\n =3(450)-125(-2)\n =1600.\n\\]\nThe radical is\n\\[\n R=2\\cdot3\\cdot5=30,\n\\]\nand\n\\[\n \\frac{3\\cdot125\\cdot128}{30}=1600.\n\\]\nHence the radical lower bound is attained exactly.","string{$3+125=128$}{3+125=128}}\nThe prime packets are $3$, $5$, and $2$. The additive row is\n\\[\n \\lambda=(1,75,-448).\n\\]\nA shortest transverse packet is\n\\[\n x=(-2,6,1).\n\\]\nIt gives\n\\[\n \\mathrm d_x(3)=-2,\n \\qquad\n \\mathrm d_x(125)=450,\n \\qquad\n \\mathrm d_x(128)=448.\n\\]\nThus additivity holds. The Wronskian is\n\\[\n W_x(3,125)\n =3(450)-125(-2)\n =1600.\n\\]\nThe radical is\n\\[\n R=2\\cdot3\\cdot5=30,\n\\]\nand\n\\[\n \\frac{3\\cdot125\\cdot128}{30}=1600.\n\\]\nHence the radical lower bound is attained exactly.","\\section{Replacing one exponent by a valuation word}\nThe coefficient $v_p(n)$ compresses a run of valuation shells into one scalar. For a prime power $p^e$, define the valuation word\n\\[\n \\boxed{\n V_e(T)=1+T+\\cdots+T^{e-1}.\n }\n\\]\nIt obeys the concatenation identity\n\\[\n V_{e+f}(T)=V_e(T)+T^eV_f(T).\n\\]\nThe refined partial derivative\n\\[\n \\partial_{p,T}(n)\n =\n \\frac npV_{v_p(n)}(T)\n\\]\ntherefore satisfies the twisted product rule\n\\[\n \\boxed{\n \\partial_{p,T}(mn)\n =\n n\\partial_{p,T}(m)\n +\n mT^{v_p(m)}\\partial_{p,T}(n).\n }\n\\]\nAt $T=1$, the ordinary arithmetic partial derivative is recovered.","t by a valuation word}\nThe coefficient $v_p(n)$ compresses a run of valuation shells into one scalar. For a prime power $p^e$, define the valuation word\n\\[\n \\boxed{\n V_e(T)=1+T+\\cdots+T^{e-1}.\n }\n\\]\nIt obeys the concatenation identity\n\\[\n V_{e+f}(T)=V_e(T)+T^eV_f(T).\n\\]\nThe refined partial derivative\n\\[\n \\partial_{p,T}(n)\n =\n \\frac npV_{v_p(n)}(T)\n\\]\ntherefore satisfies the twisted product rule\n\\[\n \\boxed{\n \\partial_{p,T}(mn)\n =\n n\\partial_{p,T}(m)\n +\n mT^{v_p(m)}\\partial_{p,T}(n).\n }\n\\]\nAt $T=1$, the ordinary arithmetic partial derivative is recovered.","lar. For a prime power $p^e$, define the valuation word\n\\[\n \\boxed{\n V_e(T)=1+T+\\cdots+T^{e-1}.\n }\n\\]\nIt obeys the concatenation identity\n\\[\n V_{e+f}(T)=V_e(T)+T^eV_f(T).\n\\]\nThe refined partial derivative\n\\[\n \\partial_{p,T}(n)\n =\n \\frac npV_{v_p(n)}(T)\n\\]\ntherefore satisfies the twisted product rule\n\\[\n \\boxed{\n \\partial_{p,T}(mn)\n =\n n\\partial_{p,T}(m)\n +\n mT^{v_p(m)}\\partial_{p,T}(n).\n }\n\\]\nAt $T=1$, the ordinary arithmetic partial derivative is recovered.","\\section{Finite polyphase rings}\nFix an integer $N\\ge2$ and work in\n\\[\n A_N=\\Z[T]/(T^N-1).\n\\]\nThe residue of $V_e(T)$ records the number of valuation shells in each congruence class modulo $N$. Multiplication by $V_e$ is represented by a circulant matrix $C_{e,N}$.","\\begin{theorem}[Polyphase Smith normal form]\nIf $\\gcd(e,N)=1$, then\n\\[\n \\boxed{\n \\SNF(C_{e,N})\n =\n \\operatorname{diag}(1,\\ldots,1,e).\n }\n\\]\n\\end{theorem}","\\begin{proof}\nLet\n\\[\n I_N=\\ker\\bigl(A_N\\xrightarrow{\\,T\\mapsto1\\,}\\Z\\bigr)\n\\]\nbe the augmentation ideal. It is a free abelian group of rank $N-1$, and multiplication by $V_e$ preserves it. After extension to $\\C$, the eigenvalues on $I_N\\otimes\\C$ are\n\\[\n V_e(\\zeta)\n =\n \\frac{1-\\zeta^e}{1-\\zeta},\n \\qquad\n \\zeta^N=1,\n \\quad\n \\zeta\\neq1.\n\\]\nBecause $\\gcd(e,N)=1$, the map $\\zeta\\mapsto\\zeta^e$ permutes the nontrivial $N$th roots. Therefore\n\\[\n \\det\\bigl(V_e\\mid I_N\\bigr)\n =\n \\prod_{\\substack{\\zeta^N=1","[\n I_N=\\ker\\bigl(A_N\\xrightarrow{\\,T\\mapsto1\\,}\\Z\\bigr)\n\\]\nbe the augmentation ideal. It is a free abelian group of rank $N-1$, and multiplication by $V_e$ preserves it. After extension to $\\C$, the eigenvalues on $I_N\\otimes\\C$ are\n\\[\n V_e(\\zeta)\n =\n \\frac{1-\\zeta^e}{1-\\zeta},\n \\qquad\n \\zeta^N=1,\n \\quad\n \\zeta\\neq1.\n\\]\nBecause $\\gcd(e,N)=1$, the map $\\zeta\\mapsto\\zeta^e$ permutes the nontrivial $N$th roots. Therefore\n\\[\n \\det\\bigl(V_e\\mid I_N\\bigr)\n =\n \\prod_{\\substack{\\zeta^N=1","ghtarrow{\\,T\\mapsto1\\,}\\Z\\bigr)\n\\]\nbe the augmentation ideal. It is a free abelian group of rank $N-1$, and multiplication by $V_e$ preserves it. After extension to $\\C$, the eigenvalues on $I_N\\otimes\\C$ are\n\\[\n V_e(\\zeta)\n =\n \\frac{1-\\zeta^e}{1-\\zeta},\n \\qquad\n \\zeta^N=1,\n \\quad\n \\zeta\\neq1.\n\\]\nBecause $\\gcd(e,N)=1$, the map $\\zeta\\mapsto\\zeta^e$ permutes the nontrivial $N$th roots. Therefore\n\\[\n \\det\\bigl(V_e\\mid I_N\\bigr)\n =\n \\prod_{\\substack{\\zeta^N=1\\\\\\zeta\\neq1}}\n \\frac{1-\\zeta^e}{1-\\zeta}\n =1.\n\\]\nThus multiplication by $V_e$ is an automorphism of the integral lattice $I_N$. On the quotient\n\\[\n A_N/I_N\\cong\\Z,\n\\]\nit acts as multiplication by $V_e(1)=e$. Applying the snake lemma to\n\\[\n 0\\longrightarrow I_N\\longrightarrow A_N\\longrightarrow\\Z\\longrightarrow0\n\\]\nshows that the full map is injective and has cokernel $\\Z/e\\Z$. Hence its Smith normal form is\n\\[\n \\operatorname{diag}(1,\\ldots,1,e).\n\\]\n\\end{proof}","zeta\\neq1}}\n \\frac{1-\\zeta^e}{1-\\zeta}\n =1.\n\\]\nThus multiplication by $V_e$ is an automorphism of the integral lattice $I_N$. On the quotient\n\\[\n A_N/I_N\\cong\\Z,\n\\]\nit acts as multiplication by $V_e(1)=e$. Applying the snake lemma to\n\\[\n 0\\longrightarrow I_N\\longrightarrow A_N\\longrightarrow\\Z\\longrightarrow0\n\\]\nshows that the full map is injective and has cokernel $\\Z/e\\Z$. Hence its Smith normal form is\n\\[\n \\operatorname{diag}(1,\\ldots,1,e).\n\\]\n\\end{proof}","\\frac{1-\\zeta^e}{1-\\zeta}\n =1.\n\\]\nThus multiplication by $V_e$ is an automorphism of the integral lattice $I_N$. On the quotient\n\\[\n A_N/I_N\\cong\\Z,\n\\]\nit acts as multiplication by $V_e(1)=e$. Applying the snake lemma to\n\\[\n 0\\longrightarrow I_N\\longrightarrow A_N\\longrightarrow\\Z\\longrightarrow0\n\\]\nshows that the full map is injective and has cokernel $\\Z/e\\Z$. Hence its Smith normal form is\n\\[\n \\operatorname{diag}(1,\\ldots,1,e).\n\\]\n\\end{proof}","The refined block has $N-1$ integrally invertible phase channels and one aggregate multiplicity channel. Every local denominator lies in the aggregate channel.","\\section{Dyadic and odd-adic primary lifts}\nFactor\n\\[\n e=2^\\alpha\\prod_{\\ell\\text{ odd}}\\ell^{\\alpha_\\ell}.\n\\]\nThe cokernel of the valuation block is\n\\[\n \\Z/e\\Z\n \\cong\n \\Z/2^\\alpha\\Z\n \\times\n \\prod_{\\ell\\text{ odd}}\n \\Z/\\ell^{\\alpha_\\ell}\\Z.\n\\]\nThe primary factors specify the local lifting problems for the aggregate multiplicity channel. Its $2$-primary component must be treated $2$-adically, while the odd components admit the corresponding odd-adic lifts. The phase channels require no local denominator correction.","This Smith decomposition identifies where separate $2$-primary bookkeeping first enters. The additional one-level shift in multiplicative principal-unit coordinates at the prime $2$ comes from the local $2$-adic logarithm and is established later. It is not a consequence of the Smith theorem alone.","\\section{Coarse and phase derivatives}\nChoose a polyphase order $N$ coprime to every active valuation exponent. After the Smith decomposition, each refined prime packet has one coarse coordinate and integrally invertible phase coordinates in the common augmentation lattice. The additive equation separates accordingly.","The coarse equation recovers the original exponent-weighted row\n\\[\n \\sum_{p\\mid a}\\frac{av_p(a)}p t_p\n +\n \\sum_{p\\mid b}\\frac{bv_p(b)}p t_p\n -\n \\sum_{p\\mid c}\\frac{cv_p(c)}p t_p\n =0.\n\\]\nThe phase equation is exponent free:\n\\[\n \\boxed{\n \\sum_{p\\mid a}\\frac ap y_p\n +\n \\sum_{p\\mid b}\\frac bp y_p\n -\n \\sum_{p\\mid c}\\frac cp y_p\n =0.\n }\n\\]\nDefine the phase derivative\n\\[\n D_y(n)=n\\sum_{p\\mid n}\\frac{y_p}{p}.\n\\]\nThe same radical divisibility argument gives\n\\[\n \\frac{abc}{R}\\mid aD_y(b)-bD_y(a).\n\\]","hted row\n\\[\n \\sum_{p\\mid a}\\frac{av_p(a)}p t_p\n +\n \\sum_{p\\mid b}\\frac{bv_p(b)}p t_p\n -\n \\sum_{p\\mid c}\\frac{cv_p(c)}p t_p\n =0.\n\\]\nThe phase equation is exponent free:\n\\[\n \\boxed{\n \\sum_{p\\mid a}\\frac ap y_p\n +\n \\sum_{p\\mid b}\\frac bp y_p\n -\n \\sum_{p\\mid c}\\frac cp y_p\n =0.\n }\n\\]\nDefine the phase derivative\n\\[\n D_y(n)=n\\sum_{p\\mid n}\\frac{y_p}{p}.\n\\]\nThe same radical divisibility argument gives\n\\[\n \\frac{abc}{R}\\mid aD_y(b)-bD_y(a).\n\\]","Let\n\\[\n \\sigma(a,b)\n =\n \\sum_{p\\mid a}\\frac1p\n +\n \\sum_{p\\mid b}\\frac1p.\n\\]\nIf $\\Phi(a,b,c)$ is the shortest transverse phase packet, then\n\\[\n \\boxed{\n c\n \\le\n R\\,\\sigma(a,b)\\,\\Phi(a,b,c).\n }\n\\]\nThis can be sharper than the exponent-weighted inequality on individual triples, but finite evidence still shows positive power growth rather than a subpolynomial law.","\\section{First-order no-free-refinement principle}\nThe shell-refined factor\n\\[\n \\prod_{r=0}^{e-1}\n \\left(1+\\frac{T_r}{p}\\right)\n\\]\nhas first-order expansion\n\\[\n 1+\\frac1p\\sum_rT_r+O(T^2).\n\\]\nOnly the shell sum appears. Every internal phase in the kernel of shell synthesis is invisible to both first-order additivity and the first-order Wronskian.","\\begin{theorem}[First-order no-free-polyphase principle]\nLet\n\\[\n S:(T_{p,r})_{p,r}\\longmapsto\n \\left(\\sum_rT_{p,r}\\right)_p\n\\]\nbe shell synthesis. The first-order additive map and first-order Wronskian factor through $S$. Hence\n\\[\n \\Ker S\n\\]\nis contained in the kernels of both first-order maps.\n\\end{theorem}","\\section{Second-order phase energy}\nAt second order,\n\\[\n \\prod_{r=0}^{e-1}\n \\left(1+\\frac{T_r}{p}\\right)\n =\n 1+\n \\frac1p\\sum_rT_r+\n \\frac1{p^2}\\sum_{r<s}T_rT_s+\n O(T^3).\n\\]\nLet $\\widehat T_j$ be the discrete Fourier transform of the shell coordinates. Parseval gives\n\\[\n \\sum_{r<s}T_rT_s\n =\n \\frac12\\left[\n \\widehat T_0^2\n -\n \\frac1e\\sum_j\n \\widehat T_j\\widehat T_{-j}\n \\right].\n\\]\nFor real coordinates,\n\\[\n \\sum_{r<s}T_rT_s\n =\n \\frac12\\left[\n \\left(1-\\frac1e\\right)|\\widehat T_0|^2\n -\n \\frac1e\\sum_{j\\neq0}|\\widehat T_j|^2\n \\right].\n\\]\nThus nonzero polyphase energy first becomes visible at second order. Later formal-smoothness results show that this curvature is not itself a lifting obstruction.","on{Second-order phase energy}\nAt second order,\n\\[\n \\prod_{r=0}^{e-1}\n \\left(1+\\frac{T_r}{p}\\right)\n =\n 1+\n \\frac1p\\sum_rT_r+\n \\frac1{p^2}\\sum_{r<s}T_rT_s+\n O(T^3).\n\\]\nLet $\\widehat T_j$ be the discrete Fourier transform of the shell coordinates. Parseval gives\n\\[\n \\sum_{r<s}T_rT_s\n =\n \\frac12\\left[\n \\widehat T_0^2\n -\n \\frac1e\\sum_j\n \\widehat T_j\\widehat T_{-j}\n \\right].\n\\]\nFor real coordinates,\n\\[\n \\sum_{r<s}T_rT_s\n =\n \\frac12\\left[\n \\left(1-\\frac1e\\right)|\\widehat T_0|^2\n -\n \\frac1e\\sum_{j\\neq0}|\\widehat T_j|^2\n \\right].\n\\]\nThus nonzero polyphase energy first becomes visible at second order. Later formal-smoothness results show that this curvature is not itself a lifting obstruction.","\\section{Exact local lift gauges}\nLet\n\\[\n r_u=\\rad(u),\n \\qquad\n P_u=\\frac{u}{r_u}.\n\\]\nA shell packet $y=(y_{p,r})$ produces\n\\[\n D_y(u)\n =\n u\\sum_{p\\mid u}\\frac1p\n \\sum_{r=0}^{v_p(u)-1}y_{p,r}\n =P_ut_u,\n\\]\nwhere\n\\[\n t_u=\n \\sum_{p\\mid u}\n \\frac{r_u}{p}\n \\sum_r y_{p,r}.\n\\]\nBecause\n\\[\n \\gcd_{p\\mid u}\\frac{r_u}{p}=1,\n\\]\nevery integer $t_u$ is attainable when $u>1$.","The shell lift gauge is\n\\[\n \\rho_u^{\\mathrm{sh}}(t)\n =\n \\min\\left\\{\n \\norm{y}_\\infty:\n \\sum_{p\\mid u}\n \\frac{r_u}{p}\n \\sum_r y_{p,r}=t\n \\right\\}.\n\\]\nAt the aggregate level,\n\\[\n P_at_a+P_bt_b=P_ct_c,\n\\]\nand\n\\[\n W_y(a,b)\n =\n P_aP_b(r_at_b-r_bt_a).\n\\]\nThe radical divisor is attained sharply when\n\\[\n r_at_b-r_bt_a=P_c.\n\\]\nWhen $a,b>1$, all sharp aggregate solutions form a one-parameter family\n\\[\n \\begin{aligned}\n t_a(n)&=t_a^{(0)}+nr_a,","The shell lift gauge is\n\\[\n \\rho_u^{\\mathrm{sh}}(t)\n =\n \\min\\left\\{\n \\norm{y}_\\infty:\n \\sum_{p\\mid u}\n \\frac{r_u}{p}\n \\sum_r y_{p,r}=t\n \\right\\}.\n\\]\nAt the aggregate level,\n\\[\n P_at_a+P_bt_b=P_ct_c,\n\\]\nand\n\\[\n W_y(a,b)\n =\n P_aP_b(r_at_b-r_bt_a).\n\\]\nThe radical divisor is attained sharply when\n\\[\n r_at_b-r_bt_a=P_c.\n\\]\nWhen $a,b>1$, all sharp aggregate solutions form a one-parameter family\n\\[\n \\begin{aligned}\n t_a(n)&=t_a^{(0)}+nr_a,\\\\\n t_b(n)&=t_b^{(0)}+nr_b,\\\\\n t_c(n)&=t_c^{(0)}+nr_c.\n \\end{aligned}\n\\]\nThe unit-sector cases have no moving parameter. If $a=1$, then\n\\[\n (t_a,t_b,t_c)=(0,P_c,P_b),\n\\]\nand if $b=1$, then\n\\[\n (t_a,t_b,t_c)=(-P_c,0,-P_a).\n\\]\nFor obstruction multiple $k$, these fixed triples are multiplied by $k$. Thus the high-dimensional sharp search reduces either to one integer parameter plus three local lift gauges, or to one fixed aggregate triple in a unit-sector case.","t_c(n)&=t_c^{(0)}+nr_c.\n \\end{aligned}\n\\]\nThe unit-sector cases have no moving parameter. If $a=1$, then\n\\[\n (t_a,t_b,t_c)=(0,P_c,P_b),\n\\]\nand if $b=1$, then\n\\[\n (t_a,t_b,t_c)=(-P_c,0,-P_a).\n\\]\nFor obstruction multiple $k$, these fixed triples are multiplied by $k$. Thus the high-dimensional sharp search reduces either to one integer parameter plus three local lift gauges, or to one fixed aggregate triple in a unit-sector case.","\\section{A joint exponent-free two-row complex}\nThe Smith obstruction is attached to the exponent-free phase, or shell-synthesis, rows rather than to the exponent-weighted derivative row. For every prime $p\\mid abc$, define\n\\[\n \\Lambda_p=\n \\begin{cases}\n a/p,&p\\mid a,","\\section{A joint exponent-free two-row complex}\nThe Smith obstruction is attached to the exponent-free phase, or shell-synthesis, rows rather than to the exponent-weighted derivative row. For every prime $p\\mid abc$, define\n\\[\n \\Lambda_p=\n \\begin{cases}\n a/p,&p\\mid a,\\\\\n b/p,&p\\mid b,\\\\\n -c/p,&p\\mid c,\n \\end{cases}\n \\qquad\n \\mathsf w_p=\n \\begin{cases}\n -ab/p,&p\\mid a,\\\\\n ab/p,&p\\mid b,\\\\\n 0,&p\\mid c.\n \\end{cases}\n\\]\nConsider\n\\[\n \\mathsf M_{a,b,c}\n =\n \\begin{pmatrix}\n \\Lambda","0,&p\\mid c.\n \\end{cases}\n\\]\nConsider\n\\[\n \\mathsf M_{a,b,c}\n =\n \\begin{pmatrix}\n \\Lambda\\\\\n \\mathsf w\n \\end{pmatrix}:\n \\Z^S\\longrightarrow\\Z^2.\n\\]\nThe first row is the exponent-free additive phase row. The second is its integer Wronskian row. Set\n\\[\n Q=\\frac{abc}{R}.\n\\]","\\mathsf w\n \\end{pmatrix}:\n \\Z^S\\longrightarrow\\Z^2.\n\\]\nThe first row is the exponent-free additive phase row. The second is its integer Wronskian row. Set\n\\[\n Q=\\frac{abc}{R}.\n\\]","\\begin{theorem}[Smith obstruction theorem]\\label{thm:phase-smith}\nLet $a+b=c$ be primitive, with at least two of $a,b,c$ greater than one. Then\n\\[\n \\boxed{\n \\SNF(\\mathsf M_{a,b,c})\n =\n \\operatorname{diag}(1,Q).\n }\n\\]\nConsequently,\n\\[\n \\boxed{\n \\operatorname{coker}(\\mathsf M_{a,b,c})_{\\mathrm{tors}}\n \\cong\n \\Z/Q\\Z.\n }\n\\]\nThe exceptional triple $(1,1,2)$ has only one nonempty prime sector, so the matrix has rank one and is excluded from the two-row statement.\n\\end{theorem}","\\begin{proof}\nThe gcd of the entries of $\\Lambda$ is one. Indeed, every prime dividing one sector is absent from every other nonempty sector, and an entry from another sector is not divisible by it. Hence the first Smith factor is one.","Rows restricted to one sector are proportional, so same-sector $2\\times2$ minors vanish. If $p$ and $q$ lie in two different nonempty sectors, the corresponding minor is\n\\[\n \\pm\\frac{abc}{pq}\n =\n \\pm Q\\frac{R}{pq}.\n\\]\nEvery nonzero minor is therefore divisible by $Q$. Conversely, for each prime $\\ell\\mid R$, choose $p=\\ell$ and choose $q$ in a different nonempty sector. Since $R$ is squarefree, $R/(pq)$ is not divisible by $\\ell$. Thus the gcd of the reduced minors $R/(pq)$ is one. The gcd of all nonzero minors is exactly $Q$. The determinantal-divisor description of Smith normal form now gives the result.\n\\end{proof}","r are proportional, so same-sector $2\\times2$ minors vanish. If $p$ and $q$ lie in two different nonempty sectors, the corresponding minor is\n\\[\n \\pm\\frac{abc}{pq}\n =\n \\pm Q\\frac{R}{pq}.\n\\]\nEvery nonzero minor is therefore divisible by $Q$. Conversely, for each prime $\\ell\\mid R$, choose $p=\\ell$ and choose $q$ in a different nonempty sector. Since $R$ is squarefree, $R/(pq)$ is not divisible by $\\ell$. Thus the gcd of the reduced minors $R/(pq)$ is one. The gcd of all nonzero minors is exactly $Q$. The determinantal-divisor description of Smith normal form now gives the result.\n\\end{proof}","2$ minors vanish. If $p$ and $q$ lie in two different nonempty sectors, the corresponding minor is\n\\[\n \\pm\\frac{abc}{pq}\n =\n \\pm Q\\frac{R}{pq}.\n\\]\nEvery nonzero minor is therefore divisible by $Q$. Conversely, for each prime $\\ell\\mid R$, choose $p=\\ell$ and choose $q$ in a different nonempty sector. Since $R$ is squarefree, $R/(pq)$ is not divisible by $\\ell$. Thus the gcd of the reduced minors $R/(pq)$ is one. The gcd of all nonzero minors is exactly $Q$. The determinantal-divisor description of Smith normal form now gives the result.\n\\end{proof}","mpty sectors, the corresponding minor is\n\\[\n \\pm\\frac{abc}{pq}\n =\n \\pm Q\\frac{R}{pq}.\n\\]\nEvery nonzero minor is therefore divisible by $Q$. Conversely, for each prime $\\ell\\mid R$, choose $p=\\ell$ and choose $q$ in a different nonempty sector. Since $R$ is squarefree, $R/(pq)$ is not divisible by $\\ell$. Thus the gcd of the reduced minors $R/(pq)$ is one. The gcd of all nonzero minors is exactly $Q$. The determinantal-divisor description of Smith normal form now gives the result.\n\\end{proof}","\\section{The exact lattice extension}\nDefine\n\\[\n L=\\ker(\\Lambda)\\cap\\Z^S\n\\]\nand\n\\[\n L_0=L\\cap\\ker(\\mathsf w).\n\\]\nBecause $\\Lambda$ is primitive, choose $t\\in\\Z^S$ with $\\Lambda(t)=1$. Then\n\\[\n \\Z^S=L\\oplus\\Z t.\n\\]\nThe image of $\\mathsf M_{a,b,c}$ is generated by $(1,\\mathsf w(t))$ together with $(0,\\mathsf w(L))$. Its index in $\\Z^2$ is therefore the index of $\\mathsf w(L)$ in $\\Z$. Theorem~\\ref{thm:phase-smith} shows that this index is $Q$, so\n\\[\n \\mathsf w(L)=Q\\Z.\n\\]\nHence\n\\[\n \\omega(x)=\\frac{\\mathsf w\\cdot x}{Q}\n\\]\nis an integer-valued surjection and gives the exact sequence\n\\[\n \\boxed{\n 0\\longrightarrow L_0\n \\longrightarrow L\n \\xrightarrow{\\omega}\n \\Z\n \\longrightarrow0.\n }\n\\]\nThe extension splits algebraically. The difficult quantity is the norm of a section,","\\section{The exact lattice extension}\nDefine\n\\[\n L=\\ker(\\Lambda)\\cap\\Z^S\n\\]\nand\n\\[\n L_0=L\\cap\\ker(\\mathsf w).\n\\]\nBecause $\\Lambda$ is primitive, choose $t\\in\\Z^S$ with $\\Lambda(t)=1$. Then\n\\[\n \\Z^S=L\\oplus\\Z t.\n\\]\nThe image of $\\mathsf M_{a,b,c}$ is generated by $(1,\\mathsf w(t))$ together with $(0,\\mathsf w(L))$. Its index in $\\Z^2$ is therefore the index of $\\mathsf w(L)$ in $\\Z$. Theorem~\\ref{thm:phase-smith} shows that this index is $Q$, so\n\\[\n \\mathsf w(L)=Q\\Z.\n\\]\nHence\n\\[\n \\omega(x)=\\frac{\\mathsf w\\cdot x}{Q}\n\\]\nis an integer-valued surjection and gives the exact sequence\n\\[\n \\boxed{\n 0\\longrightarrow L_0\n \\longrightarrow L\n \\xrightarrow{\\omega}\n \\Z\n \\longrightarrow0.\n }\n\\]\nThe extension splits algebraically. The difficult quantity is the norm of a section, not its existence.","ne\n\\[\n L=\\ker(\\Lambda)\\cap\\Z^S\n\\]\nand\n\\[\n L_0=L\\cap\\ker(\\mathsf w).\n\\]\nBecause $\\Lambda$ is primitive, choose $t\\in\\Z^S$ with $\\Lambda(t)=1$. Then\n\\[\n \\Z^S=L\\oplus\\Z t.\n\\]\nThe image of $\\mathsf M_{a,b,c}$ is generated by $(1,\\mathsf w(t))$ together with $(0,\\mathsf w(L))$. Its index in $\\Z^2$ is therefore the index of $\\mathsf w(L)$ in $\\Z$. Theorem~\\ref{thm:phase-smith} shows that this index is $Q$, so\n\\[\n \\mathsf w(L)=Q\\Z.\n\\]\nHence\n\\[\n \\omega(x)=\\frac{\\mathsf w\\cdot x}{Q}\n\\]\nis an integer-valued surjection and gives the exact sequence\n\\[\n \\boxed{\n 0\\longrightarrow L_0\n \\longrightarrow L\n \\xrightarrow{\\omega}\n \\Z\n \\longrightarrow0.\n }\n\\]\nThe extension splits algebraically. The difficult quantity is the norm of a section, not its existence.","\\ker(\\Lambda)\\cap\\Z^S\n\\]\nand\n\\[\n L_0=L\\cap\\ker(\\mathsf w).\n\\]\nBecause $\\Lambda$ is primitive, choose $t\\in\\Z^S$ with $\\Lambda(t)=1$. Then\n\\[\n \\Z^S=L\\oplus\\Z t.\n\\]\nThe image of $\\mathsf M_{a,b,c}$ is generated by $(1,\\mathsf w(t))$ together with $(0,\\mathsf w(L))$. Its index in $\\Z^2$ is therefore the index of $\\mathsf w(L)$ in $\\Z$. Theorem~\\ref{thm:phase-smith} shows that this index is $Q$, so\n\\[\n \\mathsf w(L)=Q\\Z.\n\\]\nHence\n\\[\n \\omega(x)=\\frac{\\mathsf w\\cdot x}{Q}\n\\]\nis an integer-valued surjection and gives the exact sequence\n\\[\n \\boxed{\n 0\\longrightarrow L_0\n \\longrightarrow L\n \\xrightarrow{\\omega}\n \\Z\n \\longrightarrow0.\n }\n\\]\nThe extension splits algebraically. The difficult quantity is the norm of a section, not its existence.","L_0=L\\cap\\ker(\\mathsf w).\n\\]\nBecause $\\Lambda$ is primitive, choose $t\\in\\Z^S$ with $\\Lambda(t)=1$. Then\n\\[\n \\Z^S=L\\oplus\\Z t.\n\\]\nThe image of $\\mathsf M_{a,b,c}$ is generated by $(1,\\mathsf w(t))$ together with $(0,\\mathsf w(L))$. Its index in $\\Z^2$ is therefore the index of $\\mathsf w(L)$ in $\\Z$. Theorem~\\ref{thm:phase-smith} shows that this index is $Q$, so\n\\[\n \\mathsf w(L)=Q\\Z.\n\\]\nHence\n\\[\n \\omega(x)=\\frac{\\mathsf w\\cdot x}{Q}\n\\]\nis an integer-valued surjection and gives the exact sequence\n\\[\n \\boxed{\n 0\\longrightarrow L_0\n \\longrightarrow L\n \\xrightarrow{\\omega}\n \\Z\n \\longrightarrow0.\n }\n\\]\nThe extension splits algebraically. The difficult quantity is the norm of a section, not its existence.","n\n\\[\n \\Z^S=L\\oplus\\Z t.\n\\]\nThe image of $\\mathsf M_{a,b,c}$ is generated by $(1,\\mathsf w(t))$ together with $(0,\\mathsf w(L))$. Its index in $\\Z^2$ is therefore the index of $\\mathsf w(L)$ in $\\Z$. Theorem~\\ref{thm:phase-smith} shows that this index is $Q$, so\n\\[\n \\mathsf w(L)=Q\\Z.\n\\]\nHence\n\\[\n \\omega(x)=\\frac{\\mathsf w\\cdot x}{Q}\n\\]\nis an integer-valued surjection and gives the exact sequence\n\\[\n \\boxed{\n 0\\longrightarrow L_0\n \\longrightarrow L\n \\xrightarrow{\\omega}\n \\Z\n \\longrightarrow0.\n }\n\\]\nThe extension splits algebraically. The difficult quantity is the norm of a section, not its existence.","For $k\\in\\Z$, define\n\\[\n H(k)\n =\n \\min\\left\\{\n \\norm{x}_\\infty:\n x\\in L,\n \\omega(x)=k\n \\right\\}.\n\\]\nThree heights must be distinguished:\n\\[\n H(1),\n \\qquad\n \\Phi=\\min_{k\\neq0}H(k),\n \\qquad\n \\overline H=\n \\inf_{k\\ge1}\\frac{H(k)}k.\n\\]\nThe shortest nonzero detector may live in a nonprimitive obstruction coset.","\\section{The stable real height}\nRelax the integral packet to a real vector in the additive hyperplane. The exact continuous width can be computed from the three sector weights. Let\n\\[\n \\sigma_u=\\sum_{p\\mid u}\\frac1p\n\\]\nand\n\\[\n \\Delta(a,b,c)\n =\n \\min\\left\\{\n \\sigma_a+\\sigma_b,\n \\frac cb(\\sigma_a+\\sigma_c),\n \\frac ca(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThen\n\\[\n \\boxed{\n \\overline H\n =\n \\frac{c}{R\\Delta(a,b,c)}.\n }\n\\]\nThe discrete master inequality becomes\n\\[\n c\\le R\\Delta\\Phi.\n\\]\nSince\n\\[\n \\Phi=\\overline H\\,k_*\\Gamma\n\\]\nfor the optimal obstruction multiple $k_*$ and an integrality factor $\\Gamma\\ge1$,\n\\[\n \\boxed{\n \\Delta\\Phi\n =\n \\frac cR k_*\\Gamma.\n }\n\\]\nThis is the no-escape identity. Optimizing the same first-order invariant cannot lower the continuous floor $c/R$.","to a real vector in the additive hyperplane. The exact continuous width can be computed from the three sector weights. Let\n\\[\n \\sigma_u=\\sum_{p\\mid u}\\frac1p\n\\]\nand\n\\[\n \\Delta(a,b,c)\n =\n \\min\\left\\{\n \\sigma_a+\\sigma_b,\n \\frac cb(\\sigma_a+\\sigma_c),\n \\frac ca(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThen\n\\[\n \\boxed{\n \\overline H\n =\n \\frac{c}{R\\Delta(a,b,c)}.\n }\n\\]\nThe discrete master inequality becomes\n\\[\n c\\le R\\Delta\\Phi.\n\\]\nSince\n\\[\n \\Phi=\\overline H\\,k_*\\Gamma\n\\]\nfor the optimal obstruction multiple $k_*$ and an integrality factor $\\Gamma\\ge1$,\n\\[\n \\boxed{\n \\Delta\\Phi\n =\n \\frac cR k_*\\Gamma.\n }\n\\]\nThis is the no-escape identity. Optimizing the same first-order invariant cannot lower the continuous floor $c/R$.","ve hyperplane. The exact continuous width can be computed from the three sector weights. Let\n\\[\n \\sigma_u=\\sum_{p\\mid u}\\frac1p\n\\]\nand\n\\[\n \\Delta(a,b,c)\n =\n \\min\\left\\{\n \\sigma_a+\\sigma_b,\n \\frac cb(\\sigma_a+\\sigma_c),\n \\frac ca(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThen\n\\[\n \\boxed{\n \\overline H\n =\n \\frac{c}{R\\Delta(a,b,c)}.\n }\n\\]\nThe discrete master inequality becomes\n\\[\n c\\le R\\Delta\\Phi.\n\\]\nSince\n\\[\n \\Phi=\\overline H\\,k_*\\Gamma\n\\]\nfor the optimal obstruction multiple $k_*$ and an integrality factor $\\Gamma\\ge1$,\n\\[\n \\boxed{\n \\Delta\\Phi\n =\n \\frac cR k_*\\Gamma.\n }\n\\]\nThis is the no-escape identity. Optimizing the same first-order invariant cannot lower the continuous floor $c/R$.","e sector weights. Let\n\\[\n \\sigma_u=\\sum_{p\\mid u}\\frac1p\n\\]\nand\n\\[\n \\Delta(a,b,c)\n =\n \\min\\left\\{\n \\sigma_a+\\sigma_b,\n \\frac cb(\\sigma_a+\\sigma_c),\n \\frac ca(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThen\n\\[\n \\boxed{\n \\overline H\n =\n \\frac{c}{R\\Delta(a,b,c)}.\n }\n\\]\nThe discrete master inequality becomes\n\\[\n c\\le R\\Delta\\Phi.\n\\]\nSince\n\\[\n \\Phi=\\overline H\\,k_*\\Gamma\n\\]\nfor the optimal obstruction multiple $k_*$ and an integrality factor $\\Gamma\\ge1$,\n\\[\n \\boxed{\n \\Delta\\Phi\n =\n \\frac cR k_*\\Gamma.\n }\n\\]\nThis is the no-escape identity. Optimizing the same first-order invariant cannot lower the continuous floor $c/R$.","_u=\\sum_{p\\mid u}\\frac1p\n\\]\nand\n\\[\n \\Delta(a,b,c)\n =\n \\min\\left\\{\n \\sigma_a+\\sigma_b,\n \\frac cb(\\sigma_a+\\sigma_c),\n \\frac ca(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThen\n\\[\n \\boxed{\n \\overline H\n =\n \\frac{c}{R\\Delta(a,b,c)}.\n }\n\\]\nThe discrete master inequality becomes\n\\[\n c\\le R\\Delta\\Phi.\n\\]\nSince\n\\[\n \\Phi=\\overline H\\,k_*\\Gamma\n\\]\nfor the optimal obstruction multiple $k_*$ and an integrality factor $\\Gamma\\ge1$,\n\\[\n \\boxed{\n \\Delta\\Phi\n =\n \\frac cR k_*\\Gamma.\n }\n\\]\nThis is the no-escape identity. Optimizing the same first-order invariant cannot lower the continuous floor $c/R$.","u}\\frac1p\n\\]\nand\n\\[\n \\Delta(a,b,c)\n =\n \\min\\left\\{\n \\sigma_a+\\sigma_b,\n \\frac cb(\\sigma_a+\\sigma_c),\n \\frac ca(\\sigma_b+\\sigma_c)\n \\right\\}.\n\\]\nThen\n\\[\n \\boxed{\n \\overline H\n =\n \\frac{c}{R\\Delta(a,b,c)}.\n }\n\\]\nThe discrete master inequality becomes\n\\[\n c\\le R\\Delta\\Phi.\n\\]\nSince\n\\[\n \\Phi=\\overline H\\,k_*\\Gamma\n\\]\nfor the optimal obstruction multiple $k_*$ and an integrality factor $\\Gamma\\ge1$,\n\\[\n \\boxed{\n \\Delta\\Phi\n =\n \\frac cR k_*\\Gamma.\n }\n\\]\nThis is the no-escape identity. Optimizing the same first-order invariant cannot lower the continuous floor $c/R$.","\\overline H\n =\n \\frac{c}{R\\Delta(a,b,c)}.\n }\n\\]\nThe discrete master inequality becomes\n\\[\n c\\le R\\Delta\\Phi.\n\\]\nSince\n\\[\n \\Phi=\\overline H\\,k_*\\Gamma\n\\]\nfor the optimal obstruction multiple $k_*$ and an integrality factor $\\Gamma\\ge1$,\n\\[\n \\boxed{\n \\Delta\\Phi\n =\n \\frac cR k_*\\Gamma.\n }\n\\]\nThis is the no-escape identity. Optimizing the same first-order invariant cannot lower the continuous floor $c/R$.","\\section{The full shell deformation}\nIntroduce one variable for every valuation shell and define\n\\[\n \\mathscr J_n(\\mathbf T)\n =\n \\prod_{p^e\\parallel n}\n \\prod_{r=0}^{e-1}(p+T_{p,r}).\n\\]\nAt $\\mathbf T=0$,\n\\[\n \\mathscr J_n(0)=n.\n\\]\nFor $a+b=c$, set\n\\[\n F_{a,b,c}(\\mathbf T)\n =\n \\mathscr J_a(\\mathbf T)\n +\n \\mathscr J_b(\\mathbf T)\n -\n \\mathscr J_c(\\mathbf T).\n\\]\nThe shell gradient has entries\n\\[\n \\frac ap,\n \\qquad\n \\frac bp,\n \\qquad\n -\\frac cp\n\\]\nrepeated according to valuation depth. The gcd of all entries is one.","\\begin{theorem}[Formal affine-space theorem]\nLet $d$ be the number of shell variables. After a unimodular integral change of shell coordinates, the $(\\mathbf T)$-adic completion of\n\\[\n \\Z[\\mathbf T]/(F_{a,b,c})\n\\]\nat the zero section is isomorphic to a formal power-series ring $\\Z[[U_2,\\ldots,U_d]]$.\n\\end{theorem}","\\begin{proof}\nA primitive gradient row can be completed to a unimodular matrix. In the new coordinates, one variable has linear coefficient one, which is a unit in $\\Z$. The formal implicit-function recursion therefore solves uniquely for that variable at every degree with integral coefficients, giving the stated isomorphism of completed rings.\n\\end{proof}","\\begin{corollary}[Integral jet extension]\nEvery integral first-order tangent vector to $F_{a,b,c}=0$ extends to an integral formal curve through all orders.\n\\end{corollary}","\\section{Gauge collapse}\nWithin one sector, the product\n\\[\n \\prod_{p,r}(p+T_{p,r})\n\\]\ndepends only on the aggregate product of its shell variables. Internal phase directions that preserve this product form a gauge torus. The full shell equation factors through three aggregate sector products\n\\[\n A,\n \\qquad\n B,\n \\qquad\n C,\n\\]\nwith one linear relation\n\\[\n aA+bB=cC.\n\\]\nThus all formal complexity collapses to a two-dimensional aggregate plane plus gauge directions. This explains why local refinements repeatedly fail to create a new global obstruction.","\\section{Obstruction cosets}\nChoose a section $x_0\\in L$ with $\\omega(x_0)=1$. Then\n\\[\n L\n =\n \\bigsqcup_{k\\in\\Z}\n (kx_0+L_0).\n\\]\nFor fixed $k$, the problem is a closest-vector problem to the moving target $-kx_0$ in the lattice $L_0$. The three-sector compression makes this search much more explicit.","\\section{Aggregate affine line}\nWrite\n\\[\n u=P_ur_u,\n \\qquad\n r_u=\\rad(u),\n \\qquad\n P_u=u/r_u.\n\\]\nThe sharp aggregate conditions are\n\\[\n P_at_a+P_bt_b=P_ct_c\n\\]\nand\n\\[\n r_at_b-r_bt_a=kP_c.\n\\]\nSince $\\gcd(r_a,r_b)=1$, choose $s,t$ with\n\\[\n sr_a+tr_b=1.\n\\]\nOne solution is\n\\[\n t_a^{(0)}=-tkP_c,\n \\qquad\n t_b^{(0)}=skP_c.\n\\]\nEvery aggregate solution is\n\\[\n \\boxed{\n \\begin{aligned}\n t_a(k,n)&=t_a^{(0)}+nr_a,","\\section{Aggregate affine line}\nWrite\n\\[\n u=P_ur_u,\n \\qquad\n r_u=\\rad(u),\n \\qquad\n P_u=u/r_u.\n\\]\nThe sharp aggregate conditions are\n\\[\n P_at_a+P_bt_b=P_ct_c\n\\]\nand\n\\[\n r_at_b-r_bt_a=kP_c.\n\\]\nSince $\\gcd(r_a,r_b)=1$, choose $s,t$ with\n\\[\n sr_a+tr_b=1.\n\\]\nOne solution is\n\\[\n t_a^{(0)}=-tkP_c,\n \\qquad\n t_b^{(0)}=skP_c.\n\\]\nEvery aggregate solution is\n\\[\n \\boxed{\n \\begin{aligned}\n t_a(k,n)&=t_a^{(0)}+nr_a,\\\\\n t_b(k,n)&=t_b^{(0)}+nr_b,\\\\\n t_c(k,n)&=t_c^{(0)}+nr_c,\n \\end{aligned}\n \\qquad n\\in\\Z.\n }\n\\]\nWhen $a,b>1$, the global search in one obstruction coset is therefore one-dimensional. If $a=1$, the empty $a$-sector forces $t_a=0$ and fixes the coarse parameter, giving\n\\[\n (t_a,t_b,t_c)=(0,kP_c,kP_b).\n\\]\nIf $b=1$, the corresponding fixed triple is\n\\[\n (t_a,t_b,t_c)=(-kP_c,0,-kP_a).\n\\]\nThus a unit-sector coset has one aggregate candidate rather than a moving affine line.","t_c(k,n)&=t_c^{(0)}+nr_c,\n \\end{aligned}\n \\qquad n\\in\\Z.\n }\n\\]\nWhen $a,b>1$, the global search in one obstruction coset is therefore one-dimensional. If $a=1$, the empty $a$-sector forces $t_a=0$ and fixes the coarse parameter, giving\n\\[\n (t_a,t_b,t_c)=(0,kP_c,kP_b).\n\\]\nIf $b=1$, the corresponding fixed triple is\n\\[\n (t_a,t_b,t_c)=(-kP_c,0,-kP_a).\n\\]\nThus a unit-sector coset has one aggregate candidate rather than a moving affine line.","\\section{Local CRT normal form}\nFix one sector with primes\n\\[\n p_1,\\ldots,p_m\n\\]\nand radical\n\\[\n r=\\prod_i p_i.\n\\]\nFor a nonempty sector, the local equation is\n\\[\n \\sum_i\\frac r{p_i}x_i=t.\n\\]\nAn empty unit sector has no prime coordinates and is feasible exactly when $t=0$.\nModulo $p_i$, the coordinate $x_i$ is forced:\n\\[\n x_i\n \\equiv\n t\\left(\\frac r{p_i}\\right)^{-1}\n \\pmod{p_i}.\n\\]\nChoose the centered residue $\\xi_i$ and write\n\\[\n x_i=\\xi_i+p_i z_i.\n\\]\nSubstitution gives one quotient equation\n\\[\n \\sum_i z_i=q(t).\n\\]\nFor a box radius $H$, each $z_i$ lies in the interval\n\\[\n \\ell_i(H)\n =\n \\left\\lceil\\frac{-H-\\xi_i}{p_i}\\right\\rceil\n \\le z_i\\le\n \\left\\lfloor\\frac{H-\\xi_i}{p_i}\\right\\rfloor\n =u_i(H).\n\\]\nThe local lift exists exactly when\n\\[\n \\boxed{\n \\sum_i\\ell_i(H)\n \\le","\\section{Local CRT normal form}\nFix one sector with primes\n\\[\n p_1,\\ldots,p_m\n\\]\nand radical\n\\[\n r=\\prod_i p_i.\n\\]\nFor a nonempty sector, the local equation is\n\\[\n \\sum_i\\frac r{p_i}x_i=t.\n\\]\nAn empty unit sector has no prime coordinates and is feasible exactly when $t=0$.\nModulo $p_i$, the coordinate $x_i$ is forced:\n\\[\n x_i\n \\equiv\n t\\left(\\frac r{p_i}\\right)^{-1}\n \\pmod{p_i}.\n\\]\nChoose the centered residue $\\xi_i$ and write\n\\[\n x_i=\\xi_i+p_i z_i.\n\\]\nSubstitution gives one quotient equation\n\\[\n \\sum_i z_i=q(t).\n\\]\nFor a box radius $H$, each $z_i$ lies in the interval\n\\[\n \\ell_i(H)\n =\n \\left\\lceil\\frac{-H-\\xi_i}{p_i}\\right\\rceil\n \\le z_i\\le\n \\left\\lfloor\\frac{H-\\xi_i}{p_i}\\right\\rfloor\n =u_i(H).\n\\]\nThe local lift exists exactly when\n\\[\n \\boxed{\n \\sum_i\\ell_i(H)\n \\le q(t)\\le\n \\sum_i u_i(H).\n }\n\\]\nThis","\\section{Local CRT normal form}\nFix one sector with primes\n\\[\n p_1,\\ldots,p_m\n\\]\nand radical\n\\[\n r=\\prod_i p_i.\n\\]\nFor a nonempty sector, the local equation is\n\\[\n \\sum_i\\frac r{p_i}x_i=t.\n\\]\nAn empty unit sector has no prime coordinates and is feasible exactly when $t=0$.\nModulo $p_i$, the coordinate $x_i$ is forced:\n\\[\n x_i\n \\equiv\n t\\left(\\frac r{p_i}\\right)^{-1}\n \\pmod{p_i}.\n\\]\nChoose the centered residue $\\xi_i$ and write\n\\[\n x_i=\\xi_i+p_i z_i.\n\\]\nSubstitution gives one quotient equation\n\\[\n \\sum_i z_i=q(t).\n\\]\nFor a box radius $H$, each $z_i$ lies in the interval\n\\[\n \\ell_i(H)\n =\n \\left\\lceil\\frac{-H-\\xi_i}{p_i}\\right\\rceil\n \\le z_i\\le\n \\left\\lfloor\\frac{H-\\xi_i}{p_i}\\right\\rfloor\n =u_i(H).\n\\]\nThe local lift exists exactly when\n\\[\n \\boxed{\n \\sum_i\\ell_i(H)\n \\le q(t)\\le\n \\sum_i u_i(H).\n }\n\\]\nThis is an exact interval criterion.","CRT normal form}\nFix one sector with primes\n\\[\n p_1,\\ldots,p_m\n\\]\nand radical\n\\[\n r=\\prod_i p_i.\n\\]\nFor a nonempty sector, the local equation is\n\\[\n \\sum_i\\frac r{p_i}x_i=t.\n\\]\nAn empty unit sector has no prime coordinates and is feasible exactly when $t=0$.\nModulo $p_i$, the coordinate $x_i$ is forced:\n\\[\n x_i\n \\equiv\n t\\left(\\frac r{p_i}\\right)^{-1}\n \\pmod{p_i}.\n\\]\nChoose the centered residue $\\xi_i$ and write\n\\[\n x_i=\\xi_i+p_i z_i.\n\\]\nSubstitution gives one quotient equation\n\\[\n \\sum_i z_i=q(t).\n\\]\nFor a box radius $H$, each $z_i$ lies in the interval\n\\[\n \\ell_i(H)\n =\n \\left\\lceil\\frac{-H-\\xi_i}{p_i}\\right\\rceil\n \\le z_i\\le\n \\left\\lfloor\\frac{H-\\xi_i}{p_i}\\right\\rfloor\n =u_i(H).\n\\]\nThe local lift exists exactly when\n\\[\n \\boxed{\n \\sum_i\\ell_i(H)\n \\le q(t)\\le\n \\sum_i u_i(H).\n }\n\\]\nThis is an exact interval criterion.","ix one sector with primes\n\\[\n p_1,\\ldots,p_m\n\\]\nand radical\n\\[\n r=\\prod_i p_i.\n\\]\nFor a nonempty sector, the local equation is\n\\[\n \\sum_i\\frac r{p_i}x_i=t.\n\\]\nAn empty unit sector has no prime coordinates and is feasible exactly when $t=0$.\nModulo $p_i$, the coordinate $x_i$ is forced:\n\\[\n x_i\n \\equiv\n t\\left(\\frac r{p_i}\\right)^{-1}\n \\pmod{p_i}.\n\\]\nChoose the centered residue $\\xi_i$ and write\n\\[\n x_i=\\xi_i+p_i z_i.\n\\]\nSubstitution gives one quotient equation\n\\[\n \\sum_i z_i=q(t).\n\\]\nFor a box radius $H$, each $z_i$ lies in the interval\n\\[\n \\ell_i(H)\n =\n \\left\\lceil\\frac{-H-\\xi_i}{p_i}\\right\\rceil\n \\le z_i\\le\n \\left\\lfloor\\frac{H-\\xi_i}{p_i}\\right\\rfloor\n =u_i(H).\n\\]\nThe local lift exists exactly when\n\\[\n \\boxed{\n \\sum_i\\ell_i(H)\n \\le q(t)\\le\n \\sum_i u_i(H).\n }\n\\]\nThis is an exact interval criterion.","prod_i p_i.\n\\]\nFor a nonempty sector, the local equation is\n\\[\n \\sum_i\\frac r{p_i}x_i=t.\n\\]\nAn empty unit sector has no prime coordinates and is feasible exactly when $t=0$.\nModulo $p_i$, the coordinate $x_i$ is forced:\n\\[\n x_i\n \\equiv\n t\\left(\\frac r{p_i}\\right)^{-1}\n \\pmod{p_i}.\n\\]\nChoose the centered residue $\\xi_i$ and write\n\\[\n x_i=\\xi_i+p_i z_i.\n\\]\nSubstitution gives one quotient equation\n\\[\n \\sum_i z_i=q(t).\n\\]\nFor a box radius $H$, each $z_i$ lies in the interval\n\\[\n \\ell_i(H)\n =\n \\left\\lceil\\frac{-H-\\xi_i}{p_i}\\right\\rceil\n \\le z_i\\le\n \\left\\lfloor\\frac{H-\\xi_i}{p_i}\\right\\rfloor\n =u_i(H).\n\\]\nThe local lift exists exactly when\n\\[\n \\boxed{\n \\sum_i\\ell_i(H)\n \\le q(t)\\le\n \\sum_i u_i(H).\n }\n\\]\nThis is an exact interval criterion.","e local equation is\n\\[\n \\sum_i\\frac r{p_i}x_i=t.\n\\]\nAn empty unit sector has no prime coordinates and is feasible exactly when $t=0$.\nModulo $p_i$, the coordinate $x_i$ is forced:\n\\[\n x_i\n \\equiv\n t\\left(\\frac r{p_i}\\right)^{-1}\n \\pmod{p_i}.\n\\]\nChoose the centered residue $\\xi_i$ and write\n\\[\n x_i=\\xi_i+p_i z_i.\n\\]\nSubstitution gives one quotient equation\n\\[\n \\sum_i z_i=q(t).\n\\]\nFor a box radius $H$, each $z_i$ lies in the interval\n\\[\n \\ell_i(H)\n =\n \\left\\lceil\\frac{-H-\\xi_i}{p_i}\\right\\rceil\n \\le z_i\\le\n \\left\\lfloor\\frac{H-\\xi_i}{p_i}\\right\\rfloor\n =u_i(H).\n\\]\nThe local lift exists exactly when\n\\[\n \\boxed{\n \\sum_i\\ell_i(H)\n \\le q(t)\\le\n \\sum_i u_i(H).\n }\n\\]\nThis is an exact interval criterion.","uation is\n\\[\n \\sum_i\\frac r{p_i}x_i=t.\n\\]\nAn empty unit sector has no prime coordinates and is feasible exactly when $t=0$.\nModulo $p_i$, the coordinate $x_i$ is forced:\n\\[\n x_i\n \\equiv\n t\\left(\\frac r{p_i}\\right)^{-1}\n \\pmod{p_i}.\n\\]\nChoose the centered residue $\\xi_i$ and write\n\\[\n x_i=\\xi_i+p_i z_i.\n\\]\nSubstitution gives one quotient equation\n\\[\n \\sum_i z_i=q(t).\n\\]\nFor a box radius $H$, each $z_i$ lies in the interval\n\\[\n \\ell_i(H)\n =\n \\left\\lceil\\frac{-H-\\xi_i}{p_i}\\right\\rceil\n \\le z_i\\le\n \\left\\lfloor\\frac{H-\\xi_i}{p_i}\\right\\rfloor\n =u_i(H).\n\\]\nThe local lift exists exactly when\n\\[\n \\boxed{\n \\sum_i\\ell_i(H)\n \\le q(t)\\le\n \\sum_i u_i(H).\n }\n\\]\nThis is an exact interval criterion.","p_i}x_i=t.\n\\]\nAn empty unit sector has no prime coordinates and is feasible exactly when $t=0$.\nModulo $p_i$, the coordinate $x_i$ is forced:\n\\[\n x_i\n \\equiv\n t\\left(\\frac r{p_i}\\right)^{-1}\n \\pmod{p_i}.\n\\]\nChoose the centered residue $\\xi_i$ and write\n\\[\n x_i=\\xi_i+p_i z_i.\n\\]\nSubstitution gives one quotient equation\n\\[\n \\sum_i z_i=q(t).\n\\]\nFor a box radius $H$, each $z_i$ lies in the interval\n\\[\n \\ell_i(H)\n =\n \\left\\lceil\\frac{-H-\\xi_i}{p_i}\\right\\rceil\n \\le z_i\\le\n \\left\\lfloor\\frac{H-\\xi_i}{p_i}\\right\\rfloor\n =u_i(H).\n\\]\nThe local lift exists exactly when\n\\[\n \\boxed{\n \\sum_i\\ell_i(H)\n \\le q(t)\\le\n \\sum_i u_i(H).\n }\n\\]\nThis is an exact interval criterion.","rac r{p_i}\\right)^{-1}\n \\pmod{p_i}.\n\\]\nChoose the centered residue $\\xi_i$ and write\n\\[\n x_i=\\xi_i+p_i z_i.\n\\]\nSubstitution gives one quotient equation\n\\[\n \\sum_i z_i=q(t).\n\\]\nFor a box radius $H$, each $z_i$ lies in the interval\n\\[\n \\ell_i(H)\n =\n \\left\\lceil\\frac{-H-\\xi_i}{p_i}\\right\\rceil\n \\le z_i\\le\n \\left\\lfloor\\frac{H-\\xi_i}{p_i}\\right\\rfloor\n =u_i(H).\n\\]\nThe local lift exists exactly when\n\\[\n \\boxed{\n \\sum_i\\ell_i(H)\n \\le q(t)\\le\n \\sum_i u_i(H).\n }\n\\]\nThis is an exact interval criterion.","\\section{Moving-interval theorem}\nFor fixed $k$ and $H$, the three sectors produce three integer intervals for the coarse parameter $n$. A packet of height at most $H$ exists if and only if those intervals share an integer.","\\begin{theorem}[Moving-interval theorem]\nFor fixed obstruction multiple $k$ and radius $H$,\n\\[\n H(k)\\le H\n\\]\nif and only if\n\\[\n I_a(H,k)\n \\cap\n I_b(H,k)\n \\cap\n I_c(H,k)\n \\cap\\Z\n \\neq\\varnothing.\n\\]\n\\end{theorem}","The first feasible $H$ gives the exact coset height. The preceding radius is an explicit no-smaller-box certificate.","\\section{Finite global stopping}\nThe continuous lower bound gives\n\\[\n H(k)\\ge k\\overline H.\n\\]\nIf the best exact height found so far is $B$, then no coset with\n\\[\n k\\overline H\\ge B\n\\]\ncan improve it. Therefore only finitely many obstruction multiples must be searched:\n\\[\n \\boxed{\n 1\\le k\\le\n \\left\\lfloor\\frac{B-1}{\\overline H}\\right\\rfloor.\n }\n\\]\nEvery unsearched coset is excluded by a proved lower bound.","\\section{Phase and coarse separation}\nThe local target has the form\n\\[\n t_u(k,n)=\\alpha_uk+r_un.\n\\]\nFor a local prime $p\\mid r_u$,\n\\[\n t_u(k,n)\\equiv\\alpha_uk\\pmod p.\n\\]\nThus the obstruction multiple $k$ determines the finite phase packet. The coarse parameter $n$ changes only the quotient load. The exact search is a moving polyphase mechanism rather than a generic mixed-integer program.","\\section{Arithmetic correction and arbitrary precision}\nSome large high-quality triples exceed signed $64$-bit range, and intermediate quantities such as\n\\[\n ab,\n \\qquad\n \\frac{abc}{R},\n \\qquad\n w\\cdot x\n\\]\ncan overflow even when the input terms individually fit. Every public arithmetic entry point must therefore promote integer-like inputs to arbitrary-precision integers before multiplication. Floating-point values are display-only. All logical comparisons and stopping bounds are exact rational or integer computations.","Several reported ratios are mathematically at least one:\n\\[\n \\frac{\\Phi}{\\overline H}\\ge1,\n \\qquad\n \\frac{\\Phi}{k_*\\overline H}\\ge1,\n \\qquad\n \\frac{H(1)}{\\Phi}\\ge1.\n\\]\nValues above one are not probability errors.","\\section{What the exact search proved}\nThe moving search found nonprimitive optima on many finite examples. It also found increasingly deep phase resonances. This showed that a fixed cutoff in $k$ is unjustified. More importantly, the no-escape identity proved that even a complete exact search of the first-order invariant cannot imply $abc$ independently. The search became a diagnostic and classification engine rather than the final proof mechanism.","\\section{Prime support as bandwidth and valuation as persistence}\nFor\n\\[\n n=\\prod_p p^{e_p},\n\\]\ndefine the prime-shell signal\n\\[\n \\Psi_n(p,r)\n =\n \\sqrt{\\log p}\\,\\mathbf 1_{0\\le r<e_p}.\n\\]\nIts total energy is\n\\[\n \\norm{\\Psi_n}_2^2\n =\n \\sum_p e_p\\log p\n =\n \\boxed{\\log n}.\n\\]\nLet $P_{\\mathrm{act}}$ retain only the first shell in each active prime channel. Then\n\\[\n \\norm{P_{\\mathrm{act}}\\Psi_n}_2^2\n =\n \\sum_{p\\mid n}\\log p\n =\n \\boxed{\\log\\rad(n)}.\n\\]\nThus the radical is activated bandwidth, while the difference\n\\[\n \\log n-\\log\\rad(n)\n\\]\nis repeated persistence depth.","ell signal\n\\[\n \\Psi_n(p,r)\n =\n \\sqrt{\\log p}\\,\\mathbf 1_{0\\le r<e_p}.\n\\]\nIts total energy is\n\\[\n \\norm{\\Psi_n}_2^2\n =\n \\sum_p e_p\\log p\n =\n \\boxed{\\log n}.\n\\]\nLet $P_{\\mathrm{act}}$ retain only the first shell in each active prime channel. Then\n\\[\n \\norm{P_{\\mathrm{act}}\\Psi_n}_2^2\n =\n \\sum_{p\\mid n}\\log p\n =\n \\boxed{\\log\\rad(n)}.\n\\]\nThus the radical is activated bandwidth, while the difference\n\\[\n \\log n-\\log\\rad(n)\n\\]\nis repeated persistence depth.","For a primitive triple, the direct-sum signal\n\\[\n \\Psi_{a,b,c}\n =\n \\Psi_a\\oplus\\Psi_b\\oplus\\Psi_c\n\\]\nhas disjoint prime support in the three sectors. The standard quality becomes the exact energy ratio\n\\[\n \\boxed{\n q(a,b,c)\n =\n \\frac{\\log c}{\\log\\rad(abc)}\n =\n \\frac{\\norm{P_c\\Psi_{a,b,c}}_2^2}\n {\\norm{P_{\\mathrm{act}}\\Psi_{a,b,c}}_2^2}.\n }\n\\]\nHigh quality means large output persistence relative to activated global bandwidth.","\\begin{proposition}[Unitary no-gain principle]\nLet $U$ be unitary on the shell coordinate of every active prime channel. Then\n\\[\n \\norm{U\\Psi_n}_2=\\norm{\\Psi_n}_2.\n\\]\nNo inequality that depends only on these two energies becomes stronger after applying $U$.\n\\end{proposition}","\\section{The three-spike \\texorpdfstring{$p$-adic}{p-adic} coherence signal}\nRepresent the additive triple by\n\\[\n \\nu_{a,b,c}=\\delta_0+\\delta_a+\\delta_c.\n\\]\nThe pairwise differences are $a$, $b$, and $c$. At a prime dividing exactly one of these integers, exactly one pair of spikes is $p$-adically close. The depth of that edge is the corresponding valuation.","The triple therefore determines a prime-indexed coherence triangle. The radical counts the active triangles. The valuations count how long a pair remains coupled through the $p$-adic refinement tree. This picture anticipates the later cross-prime logarithmic network.","\\section{The Frey curve}\nAttach the marked Frey curve\n\\[\n \\boxed{\n E_{a,b,c}:y^2=x(x-a)(x+b).\n }\n\\]\nThe roots $0$, $a$, and $-b$ retain the three differences $a$, $b$, and $c$. The basic invariants are\n\\[\n \\Delta_E=16a^2b^2c^2\n\\]\nand\n\\[\n c_4(E)=16(a^2+ab+b^2).\n\\]\nAt every odd prime $p\\mid abc$, the curve has multiplicative reduction. The conductor records that the channel is active, while the minimal discriminant records the full depth.","In signal language, the Frey curve is a nonlinear encoder that converts:\n\\[\n \\begin{aligned}\n \\text{prime support}&\\longrightarrow\\text{conductor},\\\\\n \\text{valuation depth}&\\longrightarrow\\text{minimal discriminant depth},\\\\\n \\text{additive relation}&\\longrightarrow\\text{elliptic geometry}.\n \\end{aligned}\n\\]\nThis is precisely the kind of nonlinear transform that a unitary shell system could not provide.","\\section{The automorphic eigenchannel}\nModularity associates a weight-two newform $f_E$ to the rational isogeny class of $E$. Its Hecke coefficients\n\\[\n a_\\ell(E)\n\\]\nform a global frequency signal. The conductor is the automorphic bandwidth. The newform, however, remembers the rational isogeny class rather than the marked integral representative.","\\section{Dyadic Landen descent}\nLet $u>v>0$ be coprime odd integers. Consider\n\\[\n T_0=(v^2,u^2-v^2,u^2)\n\\]\nand\n\\[\n T_1=\n \\left(\n uv,\n \\left(\\frac{u-v}{2}\\right)^2,\n \\left(\\frac{u+v}{2}\\right)^2\n \\right).\n\\]\nBoth are primitive triples. Their radicals are equal:\n\\[\n \\rad(T_0)=\\rad(T_1)=\\rad\\bigl(uv(u-v)(u+v)\\bigr).\n\\]\nThe second output is strictly smaller:\n\\[\n \\left(\\frac{u+v}{2}\\right)^2<u^2.\n\\]\nHence the second quality is smaller.","The Legendre parameter of the parent is\n\\[\n \\lambda=t^2,\n \\qquad\n t=v/u.\n\\]\nThe degree-two Landen transformation gives\n\\[\n \\boxed{\n \\lambda\n \\longmapsto\n \\frac{4\\sqrt\\lambda}{(1+\\sqrt\\lambda)^2}.\n }\n\\]\nThis is exactly the child ratio $a_1/c_1$. The transformation preserves the automorphic channel up to the corresponding two-isogeny while lowering the integral height. It proves that the integral representative inside an isogeny class is relevant to $abc$.","^2,\n \\qquad\n t=v/u.\n\\]\nThe degree-two Landen transformation gives\n\\[\n \\boxed{\n \\lambda\n \\longmapsto\n \\frac{4\\sqrt\\lambda}{(1+\\sqrt\\lambda)^2}.\n }\n\\]\nThis is exactly the child ratio $a_1/c_1$. The transformation preserves the automorphic channel up to the corresponding two-isogeny while lowering the integral height. It proves that the integral representative inside an isogeny class is relevant to $abc$.","\\section{Exact automorphic analysis--synthesis identity}\nLet\n\\[\n \\phi_E:X_0(N_E)\\longrightarrow E\n\\]\nbe an optimal modular parametrization, and let\n\\[\n \\phi_E^*\\omega_E\n =\n 2\\pi i\\,c_E f_E(z)\\,dz.\n\\]\nDefine the differential area\n\\[\n A(E)\n =\n \\frac{i}{2}\n \\int_{E(\\C)}\n \\omega_E\\wedge\\overline{\\omega_E}.\n\\]\nPulling the area form back gives\n\\[\n \\boxed{\n \\deg(\\phi_E)A(E)\n =\n 4\\pi^2 c_E^2\n \\norm{f_E}_{\\mathrm{Pet}}^2.\n }\n\\]\nFor the Frey model, scaling $x=cX$, $y=c^{3/2}Y$ gives an area factor proportional to $1/c$. Thus the height $c$ can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Petersson energy.","ity}\nLet\n\\[\n \\phi_E:X_0(N_E)\\longrightarrow E\n\\]\nbe an optimal modular parametrization, and let\n\\[\n \\phi_E^*\\omega_E\n =\n 2\\pi i\\,c_E f_E(z)\\,dz.\n\\]\nDefine the differential area\n\\[\n A(E)\n =\n \\frac{i}{2}\n \\int_{E(\\C)}\n \\omega_E\\wedge\\overline{\\omega_E}.\n\\]\nPulling the area form back gives\n\\[\n \\boxed{\n \\deg(\\phi_E)A(E)\n =\n 4\\pi^2 c_E^2\n \\norm{f_E}_{\\mathrm{Pet}}^2.\n }\n\\]\nFor the Frey model, scaling $x=cX$, $y=c^{3/2}Y$ gives an area factor proportional to $1/c$. Thus the height $c$ can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Petersson energy.","\\[\n \\phi_E:X_0(N_E)\\longrightarrow E\n\\]\nbe an optimal modular parametrization, and let\n\\[\n \\phi_E^*\\omega_E\n =\n 2\\pi i\\,c_E f_E(z)\\,dz.\n\\]\nDefine the differential area\n\\[\n A(E)\n =\n \\frac{i}{2}\n \\int_{E(\\C)}\n \\omega_E\\wedge\\overline{\\omega_E}.\n\\]\nPulling the area form back gives\n\\[\n \\boxed{\n \\deg(\\phi_E)A(E)\n =\n 4\\pi^2 c_E^2\n \\norm{f_E}_{\\mathrm{Pet}}^2.\n }\n\\]\nFor the Frey model, scaling $x=cX$, $y=c^{3/2}Y$ gives an area factor proportional to $1/c$. Thus the height $c$ can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Petersson energy.","imal modular parametrization, and let\n\\[\n \\phi_E^*\\omega_E\n =\n 2\\pi i\\,c_E f_E(z)\\,dz.\n\\]\nDefine the differential area\n\\[\n A(E)\n =\n \\frac{i}{2}\n \\int_{E(\\C)}\n \\omega_E\\wedge\\overline{\\omega_E}.\n\\]\nPulling the area form back gives\n\\[\n \\boxed{\n \\deg(\\phi_E)A(E)\n =\n 4\\pi^2 c_E^2\n \\norm{f_E}_{\\mathrm{Pet}}^2.\n }\n\\]\nFor the Frey model, scaling $x=cX$, $y=c^{3/2}Y$ gives an area factor proportional to $1/c$. Thus the height $c$ can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Petersson energy.","ation, and let\n\\[\n \\phi_E^*\\omega_E\n =\n 2\\pi i\\,c_E f_E(z)\\,dz.\n\\]\nDefine the differential area\n\\[\n A(E)\n =\n \\frac{i}{2}\n \\int_{E(\\C)}\n \\omega_E\\wedge\\overline{\\omega_E}.\n\\]\nPulling the area form back gives\n\\[\n \\boxed{\n \\deg(\\phi_E)A(E)\n =\n 4\\pi^2 c_E^2\n \\norm{f_E}_{\\mathrm{Pet}}^2.\n }\n\\]\nFor the Frey model, scaling $x=cX$, $y=c^{3/2}Y$ gives an area factor proportional to $1/c$. Thus the height $c$ can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Petersson energy.","\\section{Exact-order divisor packets}\nLet\n\\[\n n=\\prod_p p^{e_p}.\n\\]\nFor every $d\\ge1$, define\n\\[\n \\boxed{\n B_d(n)\n =\n \\prod_{\\substack{p\\mid n","\\section{Exact-order divisor packets}\nLet\n\\[\n n=\\prod_p p^{e_p}.\n\\]\nFor every $d\\ge1$, define\n\\[\n \\boxed{\n B_d(n)\n =\n \\prod_{\\substack{p\\mid n\\\\d\\mid e_p}}p.\n }\n\\]\nSince\n\\[\n \\sum_{d\\mid e}\\varphi(d)=e,\n\\]\nwe have the perfect reconstruction identity\n\\[\n \\boxed{\n n\n =\n \\prod_{d\\ge1}\n B_d(n)^{\\varphi(d)}.\n }\n\\]\nThe first packet is\n\\[\n B_1(n)=\\rad(n).\n\\]\nThe powerful part is\n\\[\n \\boxed{\n \\frac{n}{\\rad(n)}\n =\n \\prod_{d\\ge2}\n B_d(n)^{\\varphi(d)}.\n }\n\\]\nThis is a multiplicative polyphase decomposition indexed by exact character orders.","d\\mid e_p}}p.\n }\n\\]\nSince\n\\[\n \\sum_{d\\mid e}\\varphi(d)=e,\n\\]\nwe have the perfect reconstruction identity\n\\[\n \\boxed{\n n\n =\n \\prod_{d\\ge1}\n B_d(n)^{\\varphi(d)}.\n }\n\\]\nThe first packet is\n\\[\n B_1(n)=\\rad(n).\n\\]\nThe powerful part is\n\\[\n \\boxed{\n \\frac{n}{\\rad(n)}\n =\n \\prod_{d\\ge2}\n B_d(n)^{\\varphi(d)}.\n }\n\\]\nThis is a multiplicative polyphase decomposition indexed by exact character orders.","\\section{Quality as input-output persistence imbalance}\nFor a primitive triple,\n\\[\n q-1\n =\n \\frac{\n \\log(c/R_c)-\\log(R_aR_b)\n }{\n \\log R\n }.\n\\]\nUsing the divisor packets,\n\\[\n \\boxed{\n q-1\n =\n \\frac{\n \\sum_{d\\ge2}\\varphi(d)\\log B_d(c)\n -\n \\log(R_aR_b)\n }{\\log R}.\n }\n\\]\nA high-quality triple has more repeated-depth persistence in the output than the input radicals supply as activated bandwidth.","Strong $abc$ is equivalent to\n\\[\n \\prod_{d\\ge2}\n B_d(c)^{\\varphi(d)}\n \\le\n C_\\eps R_aR_bR^\\eps.\n\\]\nThe decomposition makes the target transparent, but not easier by itself. A global channel must explain why the exact-order packets cannot all be large simultaneously.","\\section{Local Tate-depth interpretation}\nAt an odd prime $p^e\\parallel c$, the Frey discriminant has local depth $2e$. The divisors of $e$ organize the levels at which prime-power coefficient systems can become insensitive to the local extension. The exact-order divisor packet therefore has a local representation-theoretic interpretation, although local disappearance does not automatically produce a global low-level eigenform.","\\section{Atkin--Lehner Walsh packets}\nLet\n\\[\n W_1,\\ldots,W_r\n\\]\nbe commuting Atkin--Lehner involutions. For a sign vector\n\\[\n \\varepsilon\\in\\{\\pm1\\}^r,\n\\]\ndefine\n\\[\n A_\\varepsilon\n =\n \\prod_{j=1}^{r}(I+\\varepsilon_jW_j).\n\\]\nThen\n\\[\n \\boxed{\n A_\\varepsilon^2=2^rA_\\varepsilon,\n }\n\\]\n\\[\n A_\\varepsilon A_{\\varepsilon'}=0\n \\quad(\\varepsilon\\neq\\varepsilon'),\n\\]\nand\n\\[\n \\boxed{\n \\sum_\\varepsilon A_\\varepsilon=2^rI.\n }\n\\]\nThe unnormalized analysis packets are integral. The common synthesis denominator is $2^r$, which is subpolynomial in a squarefree conductor. Local sign separation is therefore not the large persistence carrier.","For a sign vector\n\\[\n \\varepsilon\\in\\{\\pm1\\}^r,\n\\]\ndefine\n\\[\n A_\\varepsilon\n =\n \\prod_{j=1}^{r}(I+\\varepsilon_jW_j).\n\\]\nThen\n\\[\n \\boxed{\n A_\\varepsilon^2=2^rA_\\varepsilon,\n }\n\\]\n\\[\n A_\\varepsilon A_{\\varepsilon'}=0\n \\quad(\\varepsilon\\neq\\varepsilon'),\n\\]\nand\n\\[\n \\boxed{\n \\sum_\\varepsilon A_\\varepsilon=2^rI.\n }\n\\]\nThe unnormalized analysis packets are integral. The common synthesis denominator is $2^r$, which is subpolynomial in a squarefree conductor. Local sign separation is therefore not the large persistence carrier.","\\section{The same-sign separation problem}\nAfter fixing the local signs, several eigenforms can remain congruent inside the same packet. Let $\\Lambda$ be an integral Hecke lattice and let $e_f$ be the rational projector onto the Frey eigenline. Define\n\\[\n \\eta_f\n =\n \\min\\{m\\ge1:m e_f(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nThis is the integral eigenchannel separation denominator. It is an arithmetic analogue of leakage in a filter bank. The difficulty is not rational spectral isolation. It is isolation inside the integral lattice.","\\section{Projector denominator as a congruence exponent}\nLet $V=U\\oplus W$ be a rational decomposition and let $\\Lambda\\subset V$ be an integral lattice. Let $e_U$ be the rational projector onto $U$. Define\n\\[\n \\Lambda_U=\\Lambda\\cap U,\n \\qquad\n \\Lambda_W=\\Lambda\\cap W,\n\\]\nand\n\\[\n C_U=\\Lambda/(\\Lambda_U+\\Lambda_W).\n\\]","\\begin{theorem}[Integral projector theorem]\nThe least positive integer $\\eta_U$ satisfying\n\\[\n \\eta_U e_U(\\Lambda)\\subseteq\\Lambda\n\\]\nis the exponent of the finite quotient $C_U$.\n\\end{theorem}","\\begin{proof}\nFor $x=u+w\\in\\Lambda$, the condition $m e_U(x)=mu\\in\\Lambda_U$ is equivalent to\n\\[\n mx=mu+mw\\in\\Lambda_U+\\Lambda_W.\n\\]\nThus $m$ clears every projector denominator exactly when it annihilates every class in $C_U$.\n\\end{proof}","For a primitive rank-one vector $v\\in\\Z^n$, the orthogonal projector denominator is\n\\[\n \\langle v,v\\rangle,\n\\]\nand the congruence quotient is cyclic of that order.","\\section{Why one cyclic quotient is not enough for split persistence}\nThe split persistence module\n\\[\n \\mathcal P_c\n \\cong\n \\bigoplus_{p^e\\parallel c}\n (\\Z/p\\Z)^{e-1}\n\\]\nhas large $p$-rank. The $p$-torsion of a cyclic target has rank at most one. Therefore every map from $\\mathcal P_c$ to a cyclic group has a forced kernel of order at least\n\\[\n \\prod_{p^e\\parallel c}p^{\\max(e-2,0)}.\n\\]\nThis is a true obstruction to the split source. It motivates the Hecke cotangent module\n\\[\n \\mathfrak p/\\mathfrak p^2,\n\\]\nwhich can retain multiple independent congruence directions before cyclic compression.","persistence module\n\\[\n \\mathcal P_c\n \\cong\n \\bigoplus_{p^e\\parallel c}\n (\\Z/p\\Z)^{e-1}\n\\]\nhas large $p$-rank. The $p$-torsion of a cyclic target has rank at most one. Therefore every map from $\\mathcal P_c$ to a cyclic group has a forced kernel of order at least\n\\[\n \\prod_{p^e\\parallel c}p^{\\max(e-2,0)}.\n\\]\nThis is a true obstruction to the split source. It motivates the Hecke cotangent module\n\\[\n \\mathfrak p/\\mathfrak p^2,\n\\]\nwhich can retain multiple independent congruence directions before cyclic compression.","\\section{A toy Hecke order}\nConsider\n\\[\n \\mathbb T(\\mathbf m)\n =\n \\{(x_0,x_1,\\ldots,x_r)\\in\\Z^{r+1}:x_i\\equiv x_0\\pmod{m_i}\\}.\n\\]\nFor the target coordinate $x_0$, the cotangent module is\n\\[\n \\boxed{\n \\mathfrak p/\\mathfrak p^2\n \\cong\n \\bigoplus_{i=1}^{r}\\Z/m_i\\Z.\n }\n\\]\nThe cyclic congruence quotient is\n\\[\n \\boxed{\n \\Z/\\lcm(m_1,\\ldots,m_r)\\Z.\n }\n\\]\nThe quotient of cotangent order by congruence order is\n\\[\n \\frac{\\prod_i m_i}{\\lcm(m_i)}.\n\\]\nThis is a toy Wiles-defect factor. It measures the multiplicity discarded by cyclic compression.","on{A toy Hecke order}\nConsider\n\\[\n \\mathbb T(\\mathbf m)\n =\n \\{(x_0,x_1,\\ldots,x_r)\\in\\Z^{r+1}:x_i\\equiv x_0\\pmod{m_i}\\}.\n\\]\nFor the target coordinate $x_0$, the cotangent module is\n\\[\n \\boxed{\n \\mathfrak p/\\mathfrak p^2\n \\cong\n \\bigoplus_{i=1}^{r}\\Z/m_i\\Z.\n }\n\\]\nThe cyclic congruence quotient is\n\\[\n \\boxed{\n \\Z/\\lcm(m_1,\\ldots,m_r)\\Z.\n }\n\\]\nThe quotient of cotangent order by congruence order is\n\\[\n \\frac{\\prod_i m_i}{\\lcm(m_i)}.\n\\]\nThis is a toy Wiles-defect factor. It measures the multiplicity discarded by cyclic compression.","\\section{Complete-intersection control case}\nLet\n\\[\n \\mathbb T=\\Z[X]/(F)\n\\]\nand evaluate at a simple root $a_0$. Then the congruence ideal and cotangent module both have order\n\\[\n |F'(a_0)|.\n\\]\nThe defect is one. Extra multiplicity is therefore tied to non-complete-intersection structure rather than being a normalization artifact.","\\section{Two prime indices}\nThe persistence is indexed by conductor primes $q\\mid abc$. Congruence depth is indexed by coefficient primes $\\ell$. The natural local signal is a matrix\n\\[\n D_f(\\ell,q).\n\\]\nAt a split Tate prime $q\\neq\\ell$, scalarity modulo $\\ell^n$ forces\n\\[\n n\\le\\min\\{v_\\ell(q-1),v_\\ell(v_q(q_E))\\}.\n\\]\nThese off-diagonal channels carry tame congruence information. They do not naturally reproduce the full base-$q$ factor $q^{e-1}$. This mismatch motivates the later filtered contact correction and the diagonal coefficient-prime channel.","e persistence is indexed by conductor primes $q\\mid abc$. Congruence depth is indexed by coefficient primes $\\ell$. The natural local signal is a matrix\n\\[\n D_f(\\ell,q).\n\\]\nAt a split Tate prime $q\\neq\\ell$, scalarity modulo $\\ell^n$ forces\n\\[\n n\\le\\min\\{v_\\ell(q-1),v_\\ell(v_q(q_E))\\}.\n\\]\nThese off-diagonal channels carry tame congruence information. They do not naturally reproduce the full base-$q$ factor $q^{e-1}$. This mismatch motivates the later filtered contact correction and the diagonal coefficient-prime channel.","ce is indexed by conductor primes $q\\mid abc$. Congruence depth is indexed by coefficient primes $\\ell$. The natural local signal is a matrix\n\\[\n D_f(\\ell,q).\n\\]\nAt a split Tate prime $q\\neq\\ell$, scalarity modulo $\\ell^n$ forces\n\\[\n n\\le\\min\\{v_\\ell(q-1),v_\\ell(v_q(q_E))\\}.\n\\]\nThese off-diagonal channels carry tame congruence information. They do not naturally reproduce the full base-$q$ factor $q^{e-1}$. This mismatch motivates the later filtered contact correction and the diagonal coefficient-prime channel.","\\section{The three-cusp section}\nView a primitive triple as the integral section\n\\[\n \\sigma_{a,b,c}:\n \\Spec\\Z\\longrightarrow\\mathbb P^1_\\Z,\n \\qquad\n \\sigma=[a:c].\n\\]\nThe pullbacks of the boundary divisors $0$, $1$, and $\\infty$ are\n\\[\n \\operatorname{div}(a),\n \\qquad\n \\operatorname{div}(b),\n \\qquad\n \\operatorname{div}(c).\n\\]\nFor the output cusp, let\n\\[\n C=\\operatorname{div}(c),\n \\qquad\n C_{\\mathrm{red}}=\\operatorname{div}(R_c).\n\\]\nDefine the two-term contact complex\n\\[\n \\boxed{\n \\mathbf J_c\n =\n [\\mathcal O(-C)\\longrightarrow\\mathcal O(-C_{\\mathrm{red}})].\n }\n\\]\nOn global sections,\n\\[\n \\mathbf J_c\n =\n [c\\Z\\hookrightarrow R_c\\Z]\n \\simeq\n [\\Z\\xrightarrow{c/R_c}\\Z].\n\\]","section\n\\[\n \\sigma_{a,b,c}:\n \\Spec\\Z\\longrightarrow\\mathbb P^1_\\Z,\n \\qquad\n \\sigma=[a:c].\n\\]\nThe pullbacks of the boundary divisors $0$, $1$, and $\\infty$ are\n\\[\n \\operatorname{div}(a),\n \\qquad\n \\operatorname{div}(b),\n \\qquad\n \\operatorname{div}(c).\n\\]\nFor the output cusp, let\n\\[\n C=\\operatorname{div}(c),\n \\qquad\n C_{\\mathrm{red}}=\\operatorname{div}(R_c).\n\\]\nDefine the two-term contact complex\n\\[\n \\boxed{\n \\mathbf J_c\n =\n [\\mathcal O(-C)\\longrightarrow\\mathcal O(-C_{\\mathrm{red}})].\n }\n\\]\nOn global sections,\n\\[\n \\mathbf J_c\n =\n [c\\Z\\hookrightarrow R_c\\Z]\n \\simeq\n [\\Z\\xrightarrow{c/R_c}\\Z].\n\\]","undary divisors $0$, $1$, and $\\infty$ are\n\\[\n \\operatorname{div}(a),\n \\qquad\n \\operatorname{div}(b),\n \\qquad\n \\operatorname{div}(c).\n\\]\nFor the output cusp, let\n\\[\n C=\\operatorname{div}(c),\n \\qquad\n C_{\\mathrm{red}}=\\operatorname{div}(R_c).\n\\]\nDefine the two-term contact complex\n\\[\n \\boxed{\n \\mathbf J_c\n =\n [\\mathcal O(-C)\\longrightarrow\\mathcal O(-C_{\\mathrm{red}})].\n }\n\\]\nOn global sections,\n\\[\n \\mathbf J_c\n =\n [c\\Z\\hookrightarrow R_c\\Z]\n \\simeq\n [\\Z\\xrightarrow{c/R_c}\\Z].\n\\]","\\begin{theorem}[Filtered contact quotient]\n\\[\n \\boxed{\n H^0(\\mathbf J_c)\n =\n R_c\\Z/c\\Z\n \\cong\n \\Z/(c/R_c)\\Z.\n }\n\\]\n\\end{theorem}","\\section{Associated graded and carry extensions}\nAt $p^e\\parallel c$,\n\\[\n \\Jcal_{p,e}\n =\n p\\Z_p/p^e\\Z_p\n \\cong\n \\Z/p^{e-1}\\Z.\n\\]\nIts filtration\n\\[\n p^j\\Z_p/p^e\\Z_p\n\\]\nhas successive quotients $\\F_p$. Thus\n\\[\n \\gr\\Jcal_{p,e}\\cong(\\F_p)^{e-1}.\n\\]\nThe exact sequences\n\\[\n 0\\longrightarrow\n \\Z/p^{j-1}\\Z\n \\xrightarrow{\\times p}\n \\Z/p^j\\Z\n \\longrightarrow\n \\F_p\n \\longrightarrow0\n\\]\nare nonsplit. These carry extensions are precisely what the split persistence packet forgot.","\\begin{corollary}[Correction of the cyclic-rank barrier]\nThe $p$-rank obstruction applies to the associated graded source. It does not obstruct a filtered map from $\\Z/p^{e-1}\\Z$ to a cyclic target of matching depth.\n\\end{corollary}","\\section{Exact cyclic transducer}\nLet\n\\[\n C(m)=[\\Z\\xrightarrow{m}\\Z]\n\\]\nand\n\\[\n C(\\eta)=[\\Z\\xrightarrow{\\eta}\\Z].\n\\]\nSet $g=\\gcd(m,\\eta)$. Multiplication by $m/g$ in degree $-1$ and by $\\eta/g$ in degree $0$ defines a chain map. On cohomology,\n\\[\n \\boxed{\n |\\ker|=m/g,\n \\qquad\n |\\im|=g,\n \\qquad\n |\\coker|=\\eta/g.\n }\n\\]\nThe maximal transferable packet is the common prime-power depth.","For the output source $m=c/R_c$, the corrected algebraic defect against a cyclic target of order $\\eta_f$ is\n\\[\n \\boxed{\n \\delta_{\\mathrm{alg}}(c,f)\n =\n \\frac{c/R_c}{\\gcd(c/R_c,\\eta_f)}.\n }\n\\]\nPrime by prime,\n\\[\n \\delta_{\\mathrm{alg}}(c,f)\n =\n \\prod_{p^e\\parallel c}\n p^{\\max(e-1-v_p(\\eta_f),0)}.\n\\]\nThe problem is missing depth, not missing rank.","\\section{Witt realization}\nThe local packet has a natural additive identification\n\\[\n \\boxed{\n \\Jcal_{p,e}\n \\cong\n W_{e-1}(\\F_p).\n }\n\\]\nThe Verschiebung filtration matches multiplication by successive powers of $p$. This is the typed local object underlying the carry chain.","At odd $p$, the $p$-adic logarithm identifies\n\\[\n (1+p\\Z_p)/(1+p^e\\Z_p)\n \\cong\n p\\Z_p/p^e\\Z_p.\n\\]\nAt $2$, the additive contact packet remains uniform, while the multiplicative coordinate begins on $1+4\\Z_2$ and therefore carries the familiar weighted dyadic shift.","\\section{The filtered local-global target}\nThe intended transducer is a filtered chain\n\\[\n \\mathbf J_c\n \\longrightarrow\n \\bigoplus_{p\\mid c}\\mathbf L_p^{\\mathrm{st}}\n \\longrightarrow\n \\mathbf L_{R^{\\mathrm{glob}}}\n \\xrightarrow{R=\\mathbb T}\n \\mathbf L_{\\mathbb T_f}\n \\longrightarrow\n \\mathbf C_f.\n\\]\nIt must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archimedean size.","\\begin{theorem}[Conditional filtered-transfer criterion]\nSuppose a canonical map\n\\[\n H^0(\\Theta_f):\\Z/(c/R_c)\\Z\\longrightarrow H^0(\\mathbf C_f)\n\\]\nsatisfies, for every $\\eps>0$,\n\\[\n |\\ker H^0(\\Theta_f)|\\le C_\\eps R^\\eps\n\\]\nand\n\\[\n |H^0(\\mathbf C_f)|\\le C'_\\eps R_aR_bR^\\eps.\n\\]\nThen the strong $abc$ conjecture follows.\n\\end{theorem}","\\begin{proof}\nThe image has order\n\\[\n \\frac{c/R_c}{|\\ker|}\n\\]\nand is bounded by the target order. Therefore\n\\[\n \\frac{c/R_c}{C_\\eps R^\\eps}\n \\le\n C'_\\eps R_aR_bR^\\eps.\n\\]\nMultiplying by $R_c$ and replacing $\\eps$ by $\\eps/2$ gives the strong bound.\n\\end{proof}","\\section{The contact modulus}\nLet\n\\[\n c=\\prod_{i=1}^{r}p_i^{e_i},\n \\qquad\n R_c=\\prod_i p_i,\n \\qquad\n M=\\frac{c}{R_c}\n =\\prod_i p_i^{e_i-1}.\n\\]\nThe filtered output contact packet is the cyclic group\n\\[\n \\Z/M\\Z.\n\\]\nThe first global question is whether this packet can be realized by one arithmetic class without losing most of its order.","\\section{The global Kummer map}\nKummer theory gives\n\\[\n H^1(\\Q,\\mu_M)\n \\cong\n \\Q^\\times/(\\Q^\\times)^M.\n\\]\nDefine\n\\[\n \\boxed{\n \\kappa_c:\n \\Z/M\\Z\n \\longrightarrow\n H^1(\\Q,\\mu_M),\n \\qquad\n k\\longmapsto[c^k].\n }\n\\]\nThe valuation vector of $c$ is\n\\[\n (e_1,\\ldots,e_r)\n \\in\n (\\Z/M\\Z)^r.\n\\]","\\begin{theorem}[Exact Kummer kernel]\nLet\n\\[\n g_c=\\gcd(M,e_1,\\ldots,e_r).\n\\]\nThen\n\\[\n \\boxed{\n |\\ker\\kappa_c|=g_c,\n \\qquad\n |\\im\\kappa_c|=M/g_c.\n }\n\\]\n\\end{theorem}","\\begin{proof}\nThe class $[c]^k$ is trivial exactly when every prime valuation $ke_i$ is divisible by $M$. The least positive such $k$ is\n\\[\n \\operatorname{lcm}_i\n \\frac{M}{\\gcd(M,e_i)}\n =\n \\frac{M}{\\gcd(M,e_1,\\ldots,e_r)}.\n\\]\nThis is the order of the image. The kernel size follows.\n\\end{proof}","Because\n\\[\n g_c\\le\\min_i e_i\\le\\log_2 c,\n\\]\nwe obtain the universal near-injectivity estimate\n\\[\n \\boxed{\n |\\ker\\kappa_c|\\le\\log_2c.\n }\n\\]\nThis closes the global-occupancy problem for the contact packet up to logarithmic loss.","\\section{Balanced Kummer class}\nThe class of $c$ does not remember the additive relation. A better coordinate is\n\\[\n \\beta=-\\frac ba=1-\\frac ca.\n\\]\nLet\n\\[\n \\nu_q=v_q(b)-v_q(a)\n\\]\nfor input primes $q\\mid ab$. The order of $[\\beta]$ in\n\\[\n \\Q^\\times/(\\Q^\\times)^M\n\\]\nis determined by the signed valuation vector, with one possible dyadic sign correction when $M$ is even. If\n\\[\n g_{\\mathrm{val}}\n =\n \\gcd(M,\\{\\nu_q\\}),\n \\qquad\n m_0=M/g_{\\mathrm{val}},\n\\]\nthen the order is $m_0$ unless $M$ is even and $m_0$ is odd, in which case it is $2m_0$.","dditive relation. A better coordinate is\n\\[\n \\beta=-\\frac ba=1-\\frac ca.\n\\]\nLet\n\\[\n \\nu_q=v_q(b)-v_q(a)\n\\]\nfor input primes $q\\mid ab$. The order of $[\\beta]$ in\n\\[\n \\Q^\\times/(\\Q^\\times)^M\n\\]\nis determined by the signed valuation vector, with one possible dyadic sign correction when $M$ is even. If\n\\[\n g_{\\mathrm{val}}\n =\n \\gcd(M,\\{\\nu_q\\}),\n \\qquad\n m_0=M/g_{\\mathrm{val}},\n\\]\nthen the order is $m_0$ unless $M$ is even and $m_0$ is odd, in which case it is $2m_0$.","ion. A better coordinate is\n\\[\n \\beta=-\\frac ba=1-\\frac ca.\n\\]\nLet\n\\[\n \\nu_q=v_q(b)-v_q(a)\n\\]\nfor input primes $q\\mid ab$. The order of $[\\beta]$ in\n\\[\n \\Q^\\times/(\\Q^\\times)^M\n\\]\nis determined by the signed valuation vector, with one possible dyadic sign correction when $M$ is even. If\n\\[\n g_{\\mathrm{val}}\n =\n \\gcd(M,\\{\\nu_q\\}),\n \\qquad\n m_0=M/g_{\\mathrm{val}},\n\\]\nthen the order is $m_0$ unless $M$ is even and $m_0$ is odd, in which case it is $2m_0$.","ordinate is\n\\[\n \\beta=-\\frac ba=1-\\frac ca.\n\\]\nLet\n\\[\n \\nu_q=v_q(b)-v_q(a)\n\\]\nfor input primes $q\\mid ab$. The order of $[\\beta]$ in\n\\[\n \\Q^\\times/(\\Q^\\times)^M\n\\]\nis determined by the signed valuation vector, with one possible dyadic sign correction when $M$ is even. If\n\\[\n g_{\\mathrm{val}}\n =\n \\gcd(M,\\{\\nu_q\\}),\n \\qquad\n m_0=M/g_{\\mathrm{val}},\n\\]\nthen the order is $m_0$ unless $M$ is even and $m_0$ is odd, in which case it is $2m_0$.","\\frac ca.\n\\]\nLet\n\\[\n \\nu_q=v_q(b)-v_q(a)\n\\]\nfor input primes $q\\mid ab$. The order of $[\\beta]$ in\n\\[\n \\Q^\\times/(\\Q^\\times)^M\n\\]\nis determined by the signed valuation vector, with one possible dyadic sign correction when $M$ is even. If\n\\[\n g_{\\mathrm{val}}\n =\n \\gcd(M,\\{\\nu_q\\}),\n \\qquad\n m_0=M/g_{\\mathrm{val}},\n\\]\nthen the order is $m_0$ unless $M$ is even and $m_0$ is odd, in which case it is $2m_0$.","\\section{The Kummer--Fermat lift}\nChoose algebraic numbers $u$ and $v$ satisfying\n\\[\n u^M=\\frac ac,\n \\qquad\n v^M=\\frac bc.\n\\]\nThen\n\\[\n \\boxed{u^M+v^M=1.}\n\\]\nThus\n\\[\n P_M=[u:v:1]\n\\]\nlies on the Fermat curve\n\\[\n F_M:X^M+Y^M=Z^M.\n\\]\nThe construction is canonical up to the action of $\\mu_M^2$.","\\begin{theorem}[Exact height compression]\n\\[\n \\boxed{\n h_{\\mathcal O(1)}(P_M)=\\frac{\\log c}{M}.\n }\n\\]\n\\end{theorem}","\\begin{proof}\nThe coordinatewise $M$th-power map sends $P_M$ to $[a:b:c]$. Absolute logarithmic projective height multiplies by $M$ under this map. Since the primitive positive triple has height $\\log c$, the result follows.\n\\end{proof}","\\section{Fermat character polyphase}\nThe group $\\mu_M^2$ acts by\n\\[\n (\\xi,\\eta)\\cdot(X,Y)=(\\xi X,\\eta Y).\n\\]\nLet $\\sigma$ and $\\tau$ denote the two coordinate rotations. For\n\\[\n (r,s)\\in(\\Z/M\\Z)^2,\n\\]\ndefine\n\\[\n P_{r,s}\n =\n \\frac1{M^2}\n \\sum_{i,j=0}^{M-1}\n \\zeta_M^{-ri-sj}\n \\sigma^i\\tau^j.\n\\]\nCharacter orthogonality gives\n\\[\n P_{r,s}P_{r',s'}\n =\n \\delta_{r,r'}\\delta_{s,s'}P_{r,s}\n\\]\nand\n\\[\n \\boxed{\n \\sum_{r,s}P_{r,s}=I.\n }\n\\]\nThis is an exact two-dimensional character filter bank.","\\section{Line-bundle cost of the full Fermat model}\nFor $M\\ge4$, adjunction gives\n\\[\n K_{F_M}\\cong\\mathcal O_{F_M}(M-3).\n\\]\nFor compatible Weil-height normalizations,\n\\[\n h_{K_{F_M}}(P_M)\n =\n (M-3)h_{\\mathcal O(1)}(P_M)+O(1)\n =\n \\frac{M-3}{M}\\log c+O(1).\n\\]\nIf the metric on $K_{F_M}$ is defined by the induced tensor-power metric on $\\mathcal O(1)^{\\otimes(M-3)}$, the displayed equality holds without the $O(1)$ term. Thus the full plane-curve polarization reintroduces a factor of order $M$ into the height accounting.","{Line-bundle cost of the full Fermat model}\nFor $M\\ge4$, adjunction gives\n\\[\n K_{F_M}\\cong\\mathcal O_{F_M}(M-3).\n\\]\nFor compatible Weil-height normalizations,\n\\[\n h_{K_{F_M}}(P_M)\n =\n (M-3)h_{\\mathcal O(1)}(P_M)+O(1)\n =\n \\frac{M-3}{M}\\log c+O(1).\n\\]\nIf the metric on $K_{F_M}$ is defined by the induced tensor-power metric on $\\mathcal O(1)^{\\otimes(M-3)}$, the displayed equality holds without the $O(1)$ term. Thus the full plane-curve polarization reintroduces a factor of order $M$ into the height accounting.","M}\\cong\\mathcal O_{F_M}(M-3).\n\\]\nFor compatible Weil-height normalizations,\n\\[\n h_{K_{F_M}}(P_M)\n =\n (M-3)h_{\\mathcal O(1)}(P_M)+O(1)\n =\n \\frac{M-3}{M}\\log c+O(1).\n\\]\nIf the metric on $K_{F_M}$ is defined by the induced tensor-power metric on $\\mathcal O(1)^{\\otimes(M-3)}$, the displayed equality holds without the $O(1)$ term. Thus the full plane-curve polarization reintroduces a factor of order $M$ into the height accounting.","\\begin{proposition}[Line-bundle cost of the full plane model]\nThe Kummer lift reduces the $\\mathcal O(1)$ coordinate height by $M$, while the canonical line bundle of the complete Fermat curve has degree factor $M-3$. This rules out a height saving obtained merely by retaining the full plane-curve carrier with its induced polarization. It does not rule out a lower-complexity quotient equipped with a different arithmetic metric.\n\\end{proposition}","\\section{Root-distribution compression}\nLet $u^M=z$ and let $\\zeta_M$ be a primitive $M$th root of unity. For $n\\ge1$,\n\\[\n \\boxed{\n \\sum_{j=0}^{M-1}\n \\operatorname{Li}_n(\\zeta_M^ju)\n =\n M^{1-n}\\operatorname{Li}_n(u^M).\n }\n\\]\nInside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.","The inverse synthesis gain at weight $n$ is $M^{n-1}$. Through weights one, two, and three, the total gain is\n\\[\n \\boxed{1\\cdot M\\cdot M^2=M^3.}\n\\]\nThe analytic branch count becomes fixed, but the integral transition matrix retains a large determinant.","\\section{Three-point functional closure}\nSet\n\\[\n z=\\frac ac,\n \\qquad\n 1-z=\\frac bc,\n \\qquad\n \\beta=1-\\frac1z=-\\frac ba.\n\\]\nThe Rogers dilogarithm\n\\[\n L_2(z)\n =\n \\operatorname{Li}_2(z)\n +\n \\frac12\\log z\\log(1-z)\n\\]\nsatisfies\n\\[\n \\boxed{\n L_2(z)+L_2(1-z)=\\frac{\\pi^2}{6}.\n }\n\\]\nAt weight three,\n\\[\n \\boxed{\n \\begin{aligned}\n &\\operatorname{Li}_3(z)\n +\\operatorname{Li}_3(1-z)\n +\\operatorname{Li}_3(\\beta)","\\section{Three-point functional closure}\nSet\n\\[\n z=\\frac ac,\n \\qquad\n 1-z=\\frac bc,\n \\qquad\n \\beta=1-\\frac1z=-\\frac ba.\n\\]\nThe Rogers dilogarithm\n\\[\n L_2(z)\n =\n \\operatorname{Li}_2(z)\n +\n \\frac12\\log z\\log(1-z)\n\\]\nsatisfies\n\\[\n \\boxed{\n L_2(z)+L_2(1-z)=\\frac{\\pi^2}{6}.\n }\n\\]\nAt weight three,\n\\[\n \\boxed{\n \\begin{aligned}\n &\\operatorname{Li}_3(z)\n +\\operatorname{Li}_3(1-z)\n +\\operatorname{Li}_3(\\beta)\\\\\n &\\qquad=\n \\zeta(3)\n +\\frac16\\log^3z\n +\\frac{\\pi^2}{6}\\log z\n -\\frac12\\log^2z\\log(1-z).\n \\end{aligned}\n }\n\\]\nThe balanced Kummer coordinate is therefore exactly the third argument in the depth-three functional closure.","A fixed path packet is\n\\[\n \\Pi_{\\le3}(z)\n =\n \\left(\n \\log z,\n \\log(1-z),\n L_2(z),\n \\operatorname{Li}_3(z),\n \\operatorname{Li}_3(1-z)\n \\right).\n\\]\nIts coordinate count is independent of $M$.","\\section{Archimedean envelope}\nFor $0<z<1$,\n\\[\n |\\log z|,\n |\\log(1-z)|\n \\le\\log c.\n\\]\nThe positive dilogarithm and trilogarithm values on $(0,1)$ are bounded by $\\zeta(2)$ and $\\zeta(3)$. The three-point identity gives\n\\[\n \\left|\\operatorname{Li}_3\\left(-\\frac ba\\right)\\right|\n \\le\n 3\\zeta(3)\n +\n \\frac{\\pi^2}{6}\\log c\n +\n \\frac23(\\log c)^3.\n\\]\nThus the complex packet has only cubic logarithmic growth. The surrounding motivic and regulator language is informed by the polylogarithmic complexes of Goncharov, the regulator framework discussed by Zagier, and Borel's rank theorem~\\cite{Goncharov,ZagierPolylog,Borel}.","\\section{Fixed-rank integral uncertainty}\nLet $A\\in M_d(\\Z)$ be nonsingular and suppose $D\\mid\\det A$. If every entry satisfies $|A_{ij}|\\le B$, then Hadamard's inequality gives\n\\[\n D\\le|\\det A|\\le d^{d/2}B^d.\n\\]\nHence\n\\[\n \\boxed{\n B\\ge\\frac{D^{1/d}}{\\sqrt d}.\n }\n\\]\nA small complex period packet does not imply a small integral realization.","\\section{Exact finite-place notch}\nAt an output prime $p^e\\parallel c$,\n\\[\n \\beta=1-\\frac ca.\n\\]\nSince $p\\nmid a$,\n\\[\n v_p(\\beta-1)=e.\n\\]\nFor odd $p$,\n\\[\n \\boxed{\n v_p(\\log_p\\beta)=e.\n }\n\\]\nThe corresponding weighted dyadic identity holds on the principal branch when $e\\ge2$.","Writing\n\\[\n \\beta=-\\prod_{q\\mid ab}q^{\\nu_q},\n\\]\nwe obtain the cross-prime logarithmic filter relation\n\\[\n \\boxed{\n \\log_p\\beta\n =\n \\sum_{q\\mid ab}\n \\nu_q\\log_p\\langle q\\rangle_p.\n }\n\\]\nThe same depth-three path therefore has a small complex envelope and exact deep finite notches.","\\section{Conditional motivic transfer}\nLet $K_\\beta$ be the cyclic group generated by the balanced Kummer class. Suppose there exists an integral finite object $\\mathcal H_3(a,b,c)$ and a canonical map\n\\[\n \\Theta_3:K_\\beta\\longrightarrow\\mathcal H_3(a,b,c)\n\\]\nsatisfying\n\\[\n |\\ker\\Theta_3|\\le C_\\eps R^\\eps\n\\]\nand\n\\[\n |\\mathcal H_3(a,b,c)|\\le C'_\\eps R_aR_bR^\\eps.\n\\]\nThen the strong $abc$ conjecture follows after absorbing the logarithmic Kummer kernel.","\\section{The signed input exponent signal}\nLet\n\\[\n S_{ab}=\\{q:q\\mid ab\\}.\n\\]\nDefine\n\\[\n \\nu_q=v_q(b)-v_q(a).\n\\]\nThen\n\\[\n -\\frac ba=-\\prod_{q\\mid ab}q^{\\nu_q}.\n\\]\nFor every output prime $p\\mid c$, the same input exponent packet produces the deep principal unit\n\\[\n 1-\\frac ca.\n\\]\nThe output persistence is therefore a simultaneous cross-prime cancellation phenomenon.","\\section{Odd logarithmic rows}\nFor odd $p^e\\parallel c$, define the finite-depth row coefficient\n\\[\n \\ell_{p,e}(q)\n =\n \\frac1p\\log_p\\langle q\\rangle_p\n \\pmod{p^{e-1}}.\n\\]\nThe balanced relation gives\n\\[\n \\boxed{\n \\sum_{q\\mid ab}\n \\nu_q\\ell_{p,e}(q)\n \\equiv0\n \\pmod{p^{e-1}}.\n }\n\\]\nDefine the truncated valuation on $\\Z/p^{e-1}\\Z$ by\n\\[\n \\bar v_p(0)=e-1,\n \\qquad\n \\bar v_p(x)=\\min\\{e-1,v_p(\\widetilde x)\\}\n\\]\nfor any integer lift $\\widetilde x$ of a nonzero residue. Then\n\\[\n \\boxed{\n \\bar v_p(\\ell_{p,e}(q))\n =\n \\min\\{e-1,v_p(q^{p-1}-1)-1\\}.\n }\n\\]\nExceptional entrywise depth is a generalized Wieferich event.","}\nFor odd $p^e\\parallel c$, define the finite-depth row coefficient\n\\[\n \\ell_{p,e}(q)\n =\n \\frac1p\\log_p\\langle q\\rangle_p\n \\pmod{p^{e-1}}.\n\\]\nThe balanced relation gives\n\\[\n \\boxed{\n \\sum_{q\\mid ab}\n \\nu_q\\ell_{p,e}(q)\n \\equiv0\n \\pmod{p^{e-1}}.\n }\n\\]\nDefine the truncated valuation on $\\Z/p^{e-1}\\Z$ by\n\\[\n \\bar v_p(0)=e-1,\n \\qquad\n \\bar v_p(x)=\\min\\{e-1,v_p(\\widetilde x)\\}\n\\]\nfor any integer lift $\\widetilde x$ of a nonzero residue. Then\n\\[\n \\boxed{\n \\bar v_p(\\ell_{p,e}(q))\n =\n \\min\\{e-1,v_p(q^{p-1}-1)-1\\}.\n }\n\\]\nExceptional entrywise depth is a generalized Wieferich event.","te-depth row coefficient\n\\[\n \\ell_{p,e}(q)\n =\n \\frac1p\\log_p\\langle q\\rangle_p\n \\pmod{p^{e-1}}.\n\\]\nThe balanced relation gives\n\\[\n \\boxed{\n \\sum_{q\\mid ab}\n \\nu_q\\ell_{p,e}(q)\n \\equiv0\n \\pmod{p^{e-1}}.\n }\n\\]\nDefine the truncated valuation on $\\Z/p^{e-1}\\Z$ by\n\\[\n \\bar v_p(0)=e-1,\n \\qquad\n \\bar v_p(x)=\\min\\{e-1,v_p(\\widetilde x)\\}\n\\]\nfor any integer lift $\\widetilde x$ of a nonzero residue. Then\n\\[\n \\boxed{\n \\bar v_p(\\ell_{p,e}(q))\n =\n \\min\\{e-1,v_p(q^{p-1}-1)-1\\}.\n }\n\\]\nExceptional entrywise depth is a generalized Wieferich event.","\\section{Weighted dyadic row}\nFor an odd input prime $q$, define\n\\[\n \\ell_{2,e}(q)\n =\n \\frac14\n \\log_2\\left((-1)^{(q-1)/2}q\\right)\n \\pmod{2^{e-2}}.\n\\]\nWith the analogous truncated valuation on $\\Z/2^{e-2}\\Z$,\n\\[\n \\bar v_2(\\ell_{2,e}(q))\n =\n \\min\\{e-2,v_2(q^2-1)-3\\}.\n\\]\nA separate sign packet supplies the missing first dyadic layer. This is the weighted dyadic branch in its final logarithmic form.","\\section{Sequential factorization of the powerful part}\nThe factorization is defined prime by prime. For an odd repeated output prime $p^e\\parallel c$, set\n\\[\n d_p=e-1.\n\\]\nFor a repeated dyadic prime $2^e\\parallel c$, remove one sign layer and set\n\\[\n s_2=1,\n \\qquad\n d_2=e-2.\n\\]\nIf $2\\nmid M$, take $s_2=0$. Put $\\epsilon_2=2^{s_2}$. Let $K_{\\mathrm{Kum}}$ be the kernel size of the balanced Kummer class and put\n\\[\n r_p=\\min\\{d_p,v_p(K_{\\mathrm{Kum}})\\}.\n\\]\nThe balanced order formula implies\n\\[\n K_{\\mathrm{Kum}}\\mid M/\\epsilon_2,\n\\]\nso $K_{\\mathrm{Kum}}=\\prod_{p\\mid c}p^{r_p}$. Write $K=K_{\\mathrm{Kum}}$. After dividing the input exponent vector by $K$, let $a_{p,q}$ denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define\n\\[\n w_p\n =\n \\min\\left\n \\{\n d_p-r_p,\n \\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi","\\section{Sequential factorization of the powerful part}\nThe factorization is defined prime by prime. For an odd repeated output prime $p^e\\parallel c$, set\n\\[\n d_p=e-1.\n\\]\nFor a repeated dyadic prime $2^e\\parallel c$, remove one sign layer and set\n\\[\n s_2=1,\n \\qquad\n d_2=e-2.\n\\]\nIf $2\\nmid M$, take $s_2=0$. Put $\\epsilon_2=2^{s_2}$. Let $K_{\\mathrm{Kum}}$ be the kernel size of the balanced Kummer class and put\n\\[\n r_p=\\min\\{d_p,v_p(K_{\\mathrm{Kum}})\\}.\n\\]\nThe balanced order formula implies\n\\[\n K_{\\mathrm{Kum}}\\mid M/\\epsilon_2,\n\\]\nso $K_{\\mathrm{Kum}}=\\prod_{p\\mid c}p^{r_p}$. Write $K=K_{\\mathrm{Kum}}$. After dividing the input exponent vector by $K$, let $a_{p,q}$ denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define\n\\[\n w_p\n =\n \\min\\left\n \\{\n d_p-r_p,\n \\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi_p=d_p-r_p-w_p.\n\\]\nT","\\section{Sequential factorization of the powerful part}\nThe factorization is defined prime by prime. For an odd repeated output prime $p^e\\parallel c$, set\n\\[\n d_p=e-1.\n\\]\nFor a repeated dyadic prime $2^e\\parallel c$, remove one sign layer and set\n\\[\n s_2=1,\n \\qquad\n d_2=e-2.\n\\]\nIf $2\\nmid M$, take $s_2=0$. Put $\\epsilon_2=2^{s_2}$. Let $K_{\\mathrm{Kum}}$ be the kernel size of the balanced Kummer class and put\n\\[\n r_p=\\min\\{d_p,v_p(K_{\\mathrm{Kum}})\\}.\n\\]\nThe balanced order formula implies\n\\[\n K_{\\mathrm{Kum}}\\mid M/\\epsilon_2,\n\\]\nso $K_{\\mathrm{Kum}}=\\prod_{p\\mid c}p^{r_p}$. Write $K=K_{\\mathrm{Kum}}$. After dividing the input exponent vector by $K$, let $a_{p,q}$ denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define\n\\[\n w_p\n =\n \\min\\left\n \\{\n d_p-r_p,\n \\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi_p=d_p-r_p-w_p.\n\\]\nThe normalized global factors are\n\\[\n K=\\prod_{","\\section{Sequential factorization of the powerful part}\nThe factorization is defined prime by prime. For an odd repeated output prime $p^e\\parallel c$, set\n\\[\n d_p=e-1.\n\\]\nFor a repeated dyadic prime $2^e\\parallel c$, remove one sign layer and set\n\\[\n s_2=1,\n \\qquad\n d_2=e-2.\n\\]\nIf $2\\nmid M$, take $s_2=0$. Put $\\epsilon_2=2^{s_2}$. Let $K_{\\mathrm{Kum}}$ be the kernel size of the balanced Kummer class and put\n\\[\n r_p=\\min\\{d_p,v_p(K_{\\mathrm{Kum}})\\}.\n\\]\nThe balanced order formula implies\n\\[\n K_{\\mathrm{Kum}}\\mid M/\\epsilon_2,\n\\]\nso $K_{\\mathrm{Kum}}=\\prod_{p\\mid c}p^{r_p}$. Write $K=K_{\\mathrm{Kum}}$. After dividing the input exponent vector by $K$, let $a_{p,q}$ denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define\n\\[\n w_p\n =\n \\min\\left\n \\{\n d_p-r_p,\n \\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi_p=d_p-r_p-w_p.\n\\]\nThe normalized global factors are\n\\[\n K=\\prod_{p\\mid c}p^{r_p},\n \\qquad\n W=\\prod_{p\\mid c}p^{w_p},\n \\qquad\n C","\\section{Sequential factorization of the powerful part}\nThe factorization is defined prime by prime. For an odd repeated output prime $p^e\\parallel c$, set\n\\[\n d_p=e-1.\n\\]\nFor a repeated dyadic prime $2^e\\parallel c$, remove one sign layer and set\n\\[\n s_2=1,\n \\qquad\n d_2=e-2.\n\\]\nIf $2\\nmid M$, take $s_2=0$. Put $\\epsilon_2=2^{s_2}$. Let $K_{\\mathrm{Kum}}$ be the kernel size of the balanced Kummer class and put\n\\[\n r_p=\\min\\{d_p,v_p(K_{\\mathrm{Kum}})\\}.\n\\]\nThe balanced order formula implies\n\\[\n K_{\\mathrm{Kum}}\\mid M/\\epsilon_2,\n\\]\nso $K_{\\mathrm{Kum}}=\\prod_{p\\mid c}p^{r_p}$. Write $K=K_{\\mathrm{Kum}}$. After dividing the input exponent vector by $K$, let $a_{p,q}$ denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define\n\\[\n w_p\n =\n \\min\\left\n \\{\n d_p-r_p,\n \\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi_p=d_p-r_p-w_p.\n\\]\nThe normalized global factors are\n\\[\n K=\\prod_{p\\mid c}p^{r_p},\n \\qquad\n W=\\prod_{p\\mid c}p^{w_p},\n \\qquad\n C=\\prod_{p\\mid c","\\section{Sequential factorization of the powerful part}\nThe factorization is defined prime by prime. For an odd repeated output prime $p^e\\parallel c$, set\n\\[\n d_p=e-1.\n\\]\nFor a repeated dyadic prime $2^e\\parallel c$, remove one sign layer and set\n\\[\n s_2=1,\n \\qquad\n d_2=e-2.\n\\]\nIf $2\\nmid M$, take $s_2=0$. Put $\\epsilon_2=2^{s_2}$. Let $K_{\\mathrm{Kum}}$ be the kernel size of the balanced Kummer class and put\n\\[\n r_p=\\min\\{d_p,v_p(K_{\\mathrm{Kum}})\\}.\n\\]\nThe balanced order formula implies\n\\[\n K_{\\mathrm{Kum}}\\mid M/\\epsilon_2,\n\\]\nso $K_{\\mathrm{Kum}}=\\prod_{p\\mid c}p^{r_p}$. Write $K=K_{\\mathrm{Kum}}$. After dividing the input exponent vector by $K$, let $a_{p,q}$ denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define\n\\[\n w_p\n =\n \\min\\left\n \\{\n d_p-r_p,\n \\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi_p=d_p-r_p-w_p.\n\\]\nThe normalized global factors are\n\\[\n K=\\prod_{p\\mid c}p^{r_p},\n \\qquad\n W=\\prod_{p\\mid c}p^{w_p},\n \\qquad\n C=\\prod_{p\\mid c}p^{\\chi_p}.\n\\","\\section{Sequential factorization of the powerful part}\nThe factorization is defined prime by prime. For an odd repeated output prime $p^e\\parallel c$, set\n\\[\n d_p=e-1.\n\\]\nFor a repeated dyadic prime $2^e\\parallel c$, remove one sign layer and set\n\\[\n s_2=1,\n \\qquad\n d_2=e-2.\n\\]\nIf $2\\nmid M$, take $s_2=0$. Put $\\epsilon_2=2^{s_2}$. Let $K_{\\mathrm{Kum}}$ be the kernel size of the balanced Kummer class and put\n\\[\n r_p=\\min\\{d_p,v_p(K_{\\mathrm{Kum}})\\}.\n\\]\nThe balanced order formula implies\n\\[\n K_{\\mathrm{Kum}}\\mid M/\\epsilon_2,\n\\]\nso $K_{\\mathrm{Kum}}=\\prod_{p\\mid c}p^{r_p}$. Write $K=K_{\\mathrm{Kum}}$. After dividing the input exponent vector by $K$, let $a_{p,q}$ denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define\n\\[\n w_p\n =\n \\min\\left\n \\{\n d_p-r_p,\n \\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi_p=d_p-r_p-w_p.\n\\]\nThe normalized global factors are\n\\[\n K=\\prod_{p\\mid c}p^{r_p},\n \\qquad\n W=\\prod_{p\\mid c}p^{w_p},\n \\qquad\n C=\\prod_{p\\mid c}p^{\\chi_p}.\n\\]\nThe balanced Kummer orde","\\section{Sequential factorization of the powerful part}\nThe factorization is defined prime by prime. For an odd repeated output prime $p^e\\parallel c$, set\n\\[\n d_p=e-1.\n\\]\nFor a repeated dyadic prime $2^e\\parallel c$, remove one sign layer and set\n\\[\n s_2=1,\n \\qquad\n d_2=e-2.\n\\]\nIf $2\\nmid M$, take $s_2=0$. Put $\\epsilon_2=2^{s_2}$. Let $K_{\\mathrm{Kum}}$ be the kernel size of the balanced Kummer class and put\n\\[\n r_p=\\min\\{d_p,v_p(K_{\\mathrm{Kum}})\\}.\n\\]\nThe balanced order formula implies\n\\[\n K_{\\mathrm{Kum}}\\mid M/\\epsilon_2,\n\\]\nso $K_{\\mathrm{Kum}}=\\prod_{p\\mid c}p^{r_p}$. Write $K=K_{\\mathrm{Kum}}$. After dividing the input exponent vector by $K$, let $a_{p,q}$ denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define\n\\[\n w_p\n =\n \\min\\left\n \\{\n d_p-r_p,\n \\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi_p=d_p-r_p-w_p.\n\\]\nThe normalized global factors are\n\\[\n K=\\prod_{p\\mid c}p^{r_p},\n \\qquad\n W=\\prod_{p\\mid c}p^{w_p},\n \\qquad\n C=\\prod_{p\\mid c}p^{\\chi_p}.\n\\]\nThe balanced Kummer order formula guarantees tha","ial factorization of the powerful part}\nThe factorization is defined prime by prime. For an odd repeated output prime $p^e\\parallel c$, set\n\\[\n d_p=e-1.\n\\]\nFor a repeated dyadic prime $2^e\\parallel c$, remove one sign layer and set\n\\[\n s_2=1,\n \\qquad\n d_2=e-2.\n\\]\nIf $2\\nmid M$, take $s_2=0$. Put $\\epsilon_2=2^{s_2}$. Let $K_{\\mathrm{Kum}}$ be the kernel size of the balanced Kummer class and put\n\\[\n r_p=\\min\\{d_p,v_p(K_{\\mathrm{Kum}})\\}.\n\\]\nThe balanced order formula implies\n\\[\n K_{\\mathrm{Kum}}\\mid M/\\epsilon_2,\n\\]\nso $K_{\\mathrm{Kum}}=\\prod_{p\\mid c}p^{r_p}$. Write $K=K_{\\mathrm{Kum}}$. After dividing the input exponent vector by $K$, let $a_{p,q}$ denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define\n\\[\n w_p\n =\n \\min\\left\n \\{\n d_p-r_p,\n \\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi_p=d_p-r_p-w_p.\n\\]\nThe normalized global factors are\n\\[\n K=\\prod_{p\\mid c}p^{r_p},\n \\qquad\n W=\\prod_{p\\mid c}p^{w_p},\n \\qquad\n C=\\prod_{p\\mid c}p^{\\chi_p}.\n\\]\nThe balanced Kummer order formula guarantees that the dyadic sign layer is removed before the principal $2$-…","me. For an odd repeated output prime $p^e\\parallel c$, set\n\\[\n d_p=e-1.\n\\]\nFor a repeated dyadic prime $2^e\\parallel c$, remove one sign layer and set\n\\[\n s_2=1,\n \\qquad\n d_2=e-2.\n\\]\nIf $2\\nmid M$, take $s_2=0$. Put $\\epsilon_2=2^{s_2}$. Let $K_{\\mathrm{Kum}}$ be the kernel size of the balanced Kummer class and put\n\\[\n r_p=\\min\\{d_p,v_p(K_{\\mathrm{Kum}})\\}.\n\\]\nThe balanced order formula implies\n\\[\n K_{\\mathrm{Kum}}\\mid M/\\epsilon_2,\n\\]\nso $K_{\\mathrm{Kum}}=\\prod_{p\\mid c}p^{r_p}$. Write $K=K_{\\mathrm{Kum}}$. After dividing the input exponent vector by $K$, let $a_{p,q}$ denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define\n\\[\n w_p\n =\n \\min\\left\n \\{\n d_p-r_p,\n \\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi_p=d_p-r_p-w_p.\n\\]\nThe normalized global factors are\n\\[\n K=\\prod_{p\\mid c}p^{r_p},\n \\qquad\n W=\\prod_{p\\mid c}p^{w_p},\n \\qquad\n C=\\prod_{p\\mid c}p^{\\chi_p}.\n\\]\nThe balanced Kummer order formula guarantees that the dyadic sign layer is removed before the principal $2$-adic Kummer depth, so these exponents partition the contact depth. Hence\n\\[\n \\box…","rallel c$, set\n\\[\n d_p=e-1.\n\\]\nFor a repeated dyadic prime $2^e\\parallel c$, remove one sign layer and set\n\\[\n s_2=1,\n \\qquad\n d_2=e-2.\n\\]\nIf $2\\nmid M$, take $s_2=0$. Put $\\epsilon_2=2^{s_2}$. Let $K_{\\mathrm{Kum}}$ be the kernel size of the balanced Kummer class and put\n\\[\n r_p=\\min\\{d_p,v_p(K_{\\mathrm{Kum}})\\}.\n\\]\nThe balanced order formula implies\n\\[\n K_{\\mathrm{Kum}}\\mid M/\\epsilon_2,\n\\]\nso $K_{\\mathrm{Kum}}=\\prod_{p\\mid c}p^{r_p}$. Write $K=K_{\\mathrm{Kum}}$. After dividing the input exponent vector by $K$, let $a_{p,q}$ denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define\n\\[\n w_p\n =\n \\min\\left\n \\{\n d_p-r_p,\n \\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi_p=d_p-r_p-w_p.\n\\]\nThe normalized global factors are\n\\[\n K=\\prod_{p\\mid c}p^{r_p},\n \\qquad\n W=\\prod_{p\\mid c}p^{w_p},\n \\qquad\n C=\\prod_{p\\mid c}p^{\\chi_p}.\n\\]\nThe balanced Kummer order formula guarantees that the dyadic sign layer is removed before the principal $2$-adic Kummer depth, so these exponents partition the contact depth. Hence\n\\[\n \\boxed{\n M=\\epsilon_2KWC.\n }\n\\]\nThe order of rem…","dic prime $2^e\\parallel c$, remove one sign layer and set\n\\[\n s_2=1,\n \\qquad\n d_2=e-2.\n\\]\nIf $2\\nmid M$, take $s_2=0$. Put $\\epsilon_2=2^{s_2}$. Let $K_{\\mathrm{Kum}}$ be the kernel size of the balanced Kummer class and put\n\\[\n r_p=\\min\\{d_p,v_p(K_{\\mathrm{Kum}})\\}.\n\\]\nThe balanced order formula implies\n\\[\n K_{\\mathrm{Kum}}\\mid M/\\epsilon_2,\n\\]\nso $K_{\\mathrm{Kum}}=\\prod_{p\\mid c}p^{r_p}$. Write $K=K_{\\mathrm{Kum}}$. After dividing the input exponent vector by $K$, let $a_{p,q}$ denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define\n\\[\n w_p\n =\n \\min\\left\n \\{\n d_p-r_p,\n \\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi_p=d_p-r_p-w_p.\n\\]\nThe normalized global factors are\n\\[\n K=\\prod_{p\\mid c}p^{r_p},\n \\qquad\n W=\\prod_{p\\mid c}p^{w_p},\n \\qquad\n C=\\prod_{p\\mid c}p^{\\chi_p}.\n\\]\nThe balanced Kummer order formula guarantees that the dyadic sign layer is removed before the principal $2$-adic Kummer depth, so these exponents partition the contact depth. Hence\n\\[\n \\boxed{\n M=\\epsilon_2KWC.\n }\n\\]\nThe order of removal matters. The local logarithmic defect is mea…","1,\n \\qquad\n d_2=e-2.\n\\]\nIf $2\\nmid M$, take $s_2=0$. Put $\\epsilon_2=2^{s_2}$. Let $K_{\\mathrm{Kum}}$ be the kernel size of the balanced Kummer class and put\n\\[\n r_p=\\min\\{d_p,v_p(K_{\\mathrm{Kum}})\\}.\n\\]\nThe balanced order formula implies\n\\[\n K_{\\mathrm{Kum}}\\mid M/\\epsilon_2,\n\\]\nso $K_{\\mathrm{Kum}}=\\prod_{p\\mid c}p^{r_p}$. Write $K=K_{\\mathrm{Kum}}$. After dividing the input exponent vector by $K$, let $a_{p,q}$ denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define\n\\[\n w_p\n =\n \\min\\left\n \\{\n d_p-r_p,\n \\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi_p=d_p-r_p-w_p.\n\\]\nThe normalized global factors are\n\\[\n K=\\prod_{p\\mid c}p^{r_p},\n \\qquad\n W=\\prod_{p\\mid c}p^{w_p},\n \\qquad\n C=\\prod_{p\\mid c}p^{\\chi_p}.\n\\]\nThe balanced Kummer order formula guarantees that the dyadic sign layer is removed before the principal $2$-adic Kummer depth, so these exponents partition the contact depth. Hence\n\\[\n \\boxed{\n M=\\epsilon_2KWC.\n }\n\\]\nThe order of removal matters. The local logarithmic defect is measured only after the Kummer preload has been divided out. The resu…","d\n d_2=e-2.\n\\]\nIf $2\\nmid M$, take $s_2=0$. Put $\\epsilon_2=2^{s_2}$. Let $K_{\\mathrm{Kum}}$ be the kernel size of the balanced Kummer class and put\n\\[\n r_p=\\min\\{d_p,v_p(K_{\\mathrm{Kum}})\\}.\n\\]\nThe balanced order formula implies\n\\[\n K_{\\mathrm{Kum}}\\mid M/\\epsilon_2,\n\\]\nso $K_{\\mathrm{Kum}}=\\prod_{p\\mid c}p^{r_p}$. Write $K=K_{\\mathrm{Kum}}$. After dividing the input exponent vector by $K$, let $a_{p,q}$ denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define\n\\[\n w_p\n =\n \\min\\left\n \\{\n d_p-r_p,\n \\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi_p=d_p-r_p-w_p.\n\\]\nThe normalized global factors are\n\\[\n K=\\prod_{p\\mid c}p^{r_p},\n \\qquad\n W=\\prod_{p\\mid c}p^{w_p},\n \\qquad\n C=\\prod_{p\\mid c}p^{\\chi_p}.\n\\]\nThe balanced Kummer order formula guarantees that the dyadic sign layer is removed before the principal $2$-adic Kummer depth, so these exponents partition the contact depth. Hence\n\\[\n \\boxed{\n M=\\epsilon_2KWC.\n }\n\\]\nThe order of removal matters. The local logarithmic defect is measured only after the Kummer preload has been divided out. The resulting row…","be the kernel size of the balanced Kummer class and put\n\\[\n r_p=\\min\\{d_p,v_p(K_{\\mathrm{Kum}})\\}.\n\\]\nThe balanced order formula implies\n\\[\n K_{\\mathrm{Kum}}\\mid M/\\epsilon_2,\n\\]\nso $K_{\\mathrm{Kum}}=\\prod_{p\\mid c}p^{r_p}$. Write $K=K_{\\mathrm{Kum}}$. After dividing the input exponent vector by $K$, let $a_{p,q}$ denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define\n\\[\n w_p\n =\n \\min\\left\n \\{\n d_p-r_p,\n \\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi_p=d_p-r_p-w_p.\n\\]\nThe normalized global factors are\n\\[\n K=\\prod_{p\\mid c}p^{r_p},\n \\qquad\n W=\\prod_{p\\mid c}p^{w_p},\n \\qquad\n C=\\prod_{p\\mid c}p^{\\chi_p}.\n\\]\nThe balanced Kummer order formula guarantees that the dyadic sign layer is removed before the principal $2$-adic Kummer depth, so these exponents partition the contact depth. Hence\n\\[\n \\boxed{\n M=\\epsilon_2KWC.\n }\n\\]\nThe order of removal matters. The local logarithmic defect is measured only after the Kummer preload has been divided out. The resulting row is then divided by its common logarithmic depth, leaving a primitive row modulo $p^{\\chi_p}$…","athrm{Kum}}\\mid M/\\epsilon_2,\n\\]\nso $K_{\\mathrm{Kum}}=\\prod_{p\\mid c}p^{r_p}$. Write $K=K_{\\mathrm{Kum}}$. After dividing the input exponent vector by $K$, let $a_{p,q}$ denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define\n\\[\n w_p\n =\n \\min\\left\n \\{\n d_p-r_p,\n \\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi_p=d_p-r_p-w_p.\n\\]\nThe normalized global factors are\n\\[\n K=\\prod_{p\\mid c}p^{r_p},\n \\qquad\n W=\\prod_{p\\mid c}p^{w_p},\n \\qquad\n C=\\prod_{p\\mid c}p^{\\chi_p}.\n\\]\nThe balanced Kummer order formula guarantees that the dyadic sign layer is removed before the principal $2$-adic Kummer depth, so these exponents partition the contact depth. Hence\n\\[\n \\boxed{\n M=\\epsilon_2KWC.\n }\n\\]\nThe order of removal matters. The local logarithmic defect is measured only after the Kummer preload has been divided out. The resulting row is then divided by its common logarithmic depth, leaving a primitive row modulo $p^{\\chi_p}$.","adic logarithmic row coefficients. Define\n\\[\n w_p\n =\n \\min\\left\n \\{\n d_p-r_p,\n \\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi_p=d_p-r_p-w_p.\n\\]\nThe normalized global factors are\n\\[\n K=\\prod_{p\\mid c}p^{r_p},\n \\qquad\n W=\\prod_{p\\mid c}p^{w_p},\n \\qquad\n C=\\prod_{p\\mid c}p^{\\chi_p}.\n\\]\nThe balanced Kummer order formula guarantees that the dyadic sign layer is removed before the principal $2$-adic Kummer depth, so these exponents partition the contact depth. Hence\n\\[\n \\boxed{\n M=\\epsilon_2KWC.\n }\n\\]\nThe order of removal matters. The local logarithmic defect is measured only after the Kummer preload has been divided out. The resulting row is then divided by its common logarithmic depth, leaving a primitive row modulo $p^{\\chi_p}$.","\\min_{q\\mid ab}\\bar v_p(a_{p,q})\n \\right\\},\n \\qquad\n \\chi_p=d_p-r_p-w_p.\n\\]\nThe normalized global factors are\n\\[\n K=\\prod_{p\\mid c}p^{r_p},\n \\qquad\n W=\\prod_{p\\mid c}p^{w_p},\n \\qquad\n C=\\prod_{p\\mid c}p^{\\chi_p}.\n\\]\nThe balanced Kummer order formula guarantees that the dyadic sign layer is removed before the principal $2$-adic Kummer depth, so these exponents partition the contact depth. Hence\n\\[\n \\boxed{\n M=\\epsilon_2KWC.\n }\n\\]\nThe order of removal matters. The local logarithmic defect is measured only after the Kummer preload has been divided out. The resulting row is then divided by its common logarithmic depth, leaving a primitive row modulo $p^{\\chi_p}$.","ntees that the dyadic sign layer is removed before the principal $2$-adic Kummer depth, so these exponents partition the contact depth. Hence\n\\[\n \\boxed{\n M=\\epsilon_2KWC.\n }\n\\]\nThe order of removal matters. The local logarithmic defect is measured only after the Kummer preload has been divided out. The resulting row is then divided by its common logarithmic depth, leaving a primitive row modulo $p^{\\chi_p}$.","\\section{The coherence character}\nAfter normalization, every output prime produces a primitive row\n\\[\n g_p\n \\in\n (\\Z/p^{\\chi_p}\\Z)^{S_{ab}}\n\\]\nsatisfying\n\\[\n g_p\\cdot\\nu_0\n \\equiv0\n \\pmod{p^{\\chi_p}},\n\\]\nwhere $\\nu_0$ is the Kummer-normalized input exponent vector. The Chinese remainder theorem combines the local rows into one primitive row\n\\[\n g\\in(\\Z/C\\Z)^{S_{ab}}.\n\\]\nDefine the coherence character\n\\[\n \\boxed{\n \\chi_{\\mathrm{coh}}(x)\n =\n \\exp\\left(\\frac{2\\pi i}{C}g\\cdot x\\right).\n }\n\\]\nIts kernel is the adelic coherence lattice\n\\[\n \\boxed{\n \\Lambda_{\\mathrm{coh}}\n =\n \\{x\\in\\Z^{S_{ab}}:g\\cdot x\\equiv0\\pmod C\\}.\n }\n\\]\nSince $g$ is primitive modulo $C$,\n\\[\n [\\Z^{S_{ab}}:\\Lambda_{\\mathrm{coh}}]=C\n\\]\nand\n\\[\n \\Z^{S_{ab}}/\\Lambda_{\\mathrm{coh}}\\cong\\Z/C\\Z.\n\\]\nThe normalized exponent vector lies in the lattice.","on{The coherence character}\nAfter normalization, every output prime produces a primitive row\n\\[\n g_p\n \\in\n (\\Z/p^{\\chi_p}\\Z)^{S_{ab}}\n\\]\nsatisfying\n\\[\n g_p\\cdot\\nu_0\n \\equiv0\n \\pmod{p^{\\chi_p}},\n\\]\nwhere $\\nu_0$ is the Kummer-normalized input exponent vector. The Chinese remainder theorem combines the local rows into one primitive row\n\\[\n g\\in(\\Z/C\\Z)^{S_{ab}}.\n\\]\nDefine the coherence character\n\\[\n \\boxed{\n \\chi_{\\mathrm{coh}}(x)\n =\n \\exp\\left(\\frac{2\\pi i}{C}g\\cdot x\\right).\n }\n\\]\nIts kernel is the adelic coherence lattice\n\\[\n \\boxed{\n \\Lambda_{\\mathrm{coh}}\n =\n \\{x\\in\\Z^{S_{ab}}:g\\cdot x\\equiv0\\pmod C\\}.\n }\n\\]\nSince $g$ is primitive modulo $C$,\n\\[\n [\\Z^{S_{ab}}:\\Lambda_{\\mathrm{coh}}]=C\n\\]\nand\n\\[\n \\Z^{S_{ab}}/\\Lambda_{\\mathrm{coh}}\\cong\\Z/C\\Z.\n\\]\nThe normalized exponent vector lies in the lattice.","}\n\\]\nsatisfying\n\\[\n g_p\\cdot\\nu_0\n \\equiv0\n \\pmod{p^{\\chi_p}},\n\\]\nwhere $\\nu_0$ is the Kummer-normalized input exponent vector. The Chinese remainder theorem combines the local rows into one primitive row\n\\[\n g\\in(\\Z/C\\Z)^{S_{ab}}.\n\\]\nDefine the coherence character\n\\[\n \\boxed{\n \\chi_{\\mathrm{coh}}(x)\n =\n \\exp\\left(\\frac{2\\pi i}{C}g\\cdot x\\right).\n }\n\\]\nIts kernel is the adelic coherence lattice\n\\[\n \\boxed{\n \\Lambda_{\\mathrm{coh}}\n =\n \\{x\\in\\Z^{S_{ab}}:g\\cdot x\\equiv0\\pmod C\\}.\n }\n\\]\nSince $g$ is primitive modulo $C$,\n\\[\n [\\Z^{S_{ab}}:\\Lambda_{\\mathrm{coh}}]=C\n\\]\nand\n\\[\n \\Z^{S_{ab}}/\\Lambda_{\\mathrm{coh}}\\cong\\Z/C\\Z.\n\\]\nThe normalized exponent vector lies in the lattice.","ormalized input exponent vector. The Chinese remainder theorem combines the local rows into one primitive row\n\\[\n g\\in(\\Z/C\\Z)^{S_{ab}}.\n\\]\nDefine the coherence character\n\\[\n \\boxed{\n \\chi_{\\mathrm{coh}}(x)\n =\n \\exp\\left(\\frac{2\\pi i}{C}g\\cdot x\\right).\n }\n\\]\nIts kernel is the adelic coherence lattice\n\\[\n \\boxed{\n \\Lambda_{\\mathrm{coh}}\n =\n \\{x\\in\\Z^{S_{ab}}:g\\cdot x\\equiv0\\pmod C\\}.\n }\n\\]\nSince $g$ is primitive modulo $C$,\n\\[\n [\\Z^{S_{ab}}:\\Lambda_{\\mathrm{coh}}]=C\n\\]\nand\n\\[\n \\Z^{S_{ab}}/\\Lambda_{\\mathrm{coh}}\\cong\\Z/C\\Z.\n\\]\nThe normalized exponent vector lies in the lattice.","vector. The Chinese remainder theorem combines the local rows into one primitive row\n\\[\n g\\in(\\Z/C\\Z)^{S_{ab}}.\n\\]\nDefine the coherence character\n\\[\n \\boxed{\n \\chi_{\\mathrm{coh}}(x)\n =\n \\exp\\left(\\frac{2\\pi i}{C}g\\cdot x\\right).\n }\n\\]\nIts kernel is the adelic coherence lattice\n\\[\n \\boxed{\n \\Lambda_{\\mathrm{coh}}\n =\n \\{x\\in\\Z^{S_{ab}}:g\\cdot x\\equiv0\\pmod C\\}.\n }\n\\]\nSince $g$ is primitive modulo $C$,\n\\[\n [\\Z^{S_{ab}}:\\Lambda_{\\mathrm{coh}}]=C\n\\]\nand\n\\[\n \\Z^{S_{ab}}/\\Lambda_{\\mathrm{coh}}\\cong\\Z/C\\Z.\n\\]\nThe normalized exponent vector lies in the lattice.","r. The Chinese remainder theorem combines the local rows into one primitive row\n\\[\n g\\in(\\Z/C\\Z)^{S_{ab}}.\n\\]\nDefine the coherence character\n\\[\n \\boxed{\n \\chi_{\\mathrm{coh}}(x)\n =\n \\exp\\left(\\frac{2\\pi i}{C}g\\cdot x\\right).\n }\n\\]\nIts kernel is the adelic coherence lattice\n\\[\n \\boxed{\n \\Lambda_{\\mathrm{coh}}\n =\n \\{x\\in\\Z^{S_{ab}}:g\\cdot x\\equiv0\\pmod C\\}.\n }\n\\]\nSince $g$ is primitive modulo $C$,\n\\[\n [\\Z^{S_{ab}}:\\Lambda_{\\mathrm{coh}}]=C\n\\]\nand\n\\[\n \\Z^{S_{ab}}/\\Lambda_{\\mathrm{coh}}\\cong\\Z/C\\Z.\n\\]\nThe normalized exponent vector lies in the lattice.","l rows into one primitive row\n\\[\n g\\in(\\Z/C\\Z)^{S_{ab}}.\n\\]\nDefine the coherence character\n\\[\n \\boxed{\n \\chi_{\\mathrm{coh}}(x)\n =\n \\exp\\left(\\frac{2\\pi i}{C}g\\cdot x\\right).\n }\n\\]\nIts kernel is the adelic coherence lattice\n\\[\n \\boxed{\n \\Lambda_{\\mathrm{coh}}\n =\n \\{x\\in\\Z^{S_{ab}}:g\\cdot x\\equiv0\\pmod C\\}.\n }\n\\]\nSince $g$ is primitive modulo $C$,\n\\[\n [\\Z^{S_{ab}}:\\Lambda_{\\mathrm{coh}}]=C\n\\]\nand\n\\[\n \\Z^{S_{ab}}/\\Lambda_{\\mathrm{coh}}\\cong\\Z/C\\Z.\n\\]\nThe normalized exponent vector lies in the lattice.","\\section{Worked lattice for \\texorpdfstring{$3+125=128$}{3+125=128}}\nHere\n\\[\n M=64\n\\]\nand\n\\[\n (\\epsilon_2,K,W,C)=(2,1,1,32).\n\\]\nThe input vector is\n\\[\n \\nu_0=(-1,3)\n\\]\non the prime coordinates $(3,5)$. The residual row is\n\\[\n (29,31)\\pmod{32}.\n\\]\nThus\n\\[\n \\boxed{\n \\Lambda_{\\mathrm{coh}}\n =\n \\{(x_3,x_5)\\in\\Z^2:29x_3+31x_5\\equiv0\\pmod{32}\\}.\n }\n\\]\nIndeed,\n\\[\n 29(-1)+31(3)=64\\equiv0\\pmod{32}.\n\\]\nThe five residual dyadic contact layers are exactly the index of this lattice.","\\begin{enumerate}[leftmargin=1.5em]\n\\item An entrywise defect can make every coefficient in one row divisible by a large power of $p$. This is the factor $W$.\n\\item Ordinary coefficients can cancel coherently against the specific exponent vector. This is the factor $C$.\n\\end{enumerate}","Finite data shows that $W$ is usually one, while $C$ is often large. The typical difficult triple is therefore not explained by one exceptional Wieferich edge. It is explained by coordinated cancellation among several ordinary edges.","\\section{Why abstract lattice geometry is insufficient}\nLet $v\\in\\Z^s$ be primitive, with $s\\ge2$. For every $C\\ge2$, there exists a primitive row $g\\pmod C$ satisfying\n\\[\n g\\cdot v\\equiv0\\pmod C.\n\\]\nThus a fixed short vector can lie in primitive congruence lattices of arbitrarily large index.","The proof completes $v$ to a unimodular basis and chooses a primitive row annihilating its first basis coordinate modulo $C$.","The missing theorem must use the arithmetic origin of $g$: its entries are simultaneous $p$-adic logarithms of the actual input primes, and the output primes are exactly the divisors of $a+b$.","\\section{Why positive divisibility is circular}\nIf positive integers $A$ and $B$ satisfy $c\\mid A+B$, then\n\\[\n A+B\\ge c,\n\\]\nso\n\\[\n \\max(A,B)\\ge c/2.\n\\]\nA direct search for a small positive $S$-unit relation divisible by $c$ cannot produce an $abc$ saving. Signed cancellation, a determinant, a regulator, or a functional equation is necessary.","\\section{Exact reformulation of strong \\texorpdfstring{$abc$}{abc}}\nSince\n\\[\n c=R_c\\epsilon_2KWC\n\\]\nand $\\epsilon_2\\le2$ while $K\\le\\log_2c$, the all-$\\eps$ strong $abc$ statement is equivalent, after the usual absorption of bounded and logarithmic factors into an arbitrarily small power, to the target\n\\[\n \\boxed{\n WC\n \\le\n C_\\eps R_aR_bR^\\eps.\n }\n\\]\nIn the generic finite examples $W=1$, leaving the coherence conductor bound\n\\[\n C\\le C_\\eps R_aR_bR^\\eps.\n\\]","strong \\texorpdfstring{$abc$}{abc}}\nSince\n\\[\n c=R_c\\epsilon_2KWC\n\\]\nand $\\epsilon_2\\le2$ while $K\\le\\log_2c$, the all-$\\eps$ strong $abc$ statement is equivalent, after the usual absorption of bounded and logarithmic factors into an arbitrarily small power, to the target\n\\[\n \\boxed{\n WC\n \\le\n C_\\eps R_aR_bR^\\eps.\n }\n\\]\nIn the generic finite examples $W=1$, leaving the coherence conductor bound\n\\[\n C\\le C_\\eps R_aR_bR^\\eps.\n\\]","\\section{Adelic coherence regulator object}\nPackage the surviving data as\n\\[\n \\boxed{\n \\mathfrak R_{a,b,c}\n =\n \\left(\n U_{ab},\n \\beta^{1/K},\n \\{\\chi_p\\}_{p\\mid c},\n \\chi_{\\mathrm{coh}},\n \\Lambda_{\\mathrm{coh}},\n \\norm{\\cdot}_\\infty\n \\right).\n }\n\\]\nHere $U_{ab}$ is the input $S$-unit group, the local characters are the normalized logarithmic rows, and the archimedean norm records input height.","The open theorem is an adelic slope or regulator inequality bounding the finite conductor $\\log C$ by the input radical and an arbitrarily small global leakage term. General simultaneous linear-form theory contains the right kinds of local and archimedean quantities, but known general estimates are much too coarse for the $R^\\eps$ scale.","\\section{Sparse additive relations}\nLet\n\\[\n \\epsilon_1n_1+\\cdots+\\epsilon_rn_r=0,\n \\qquad\n \\epsilon_i\\in\\{\\pm1\\},\n\\]\nbe primitive. For every term define\n\\[\n R_i=\\rad(n_i),\n \\qquad\n M_i=n_i/R_i.\n\\]\nThe prime-contact coherence calculus assigns:","\\subsection{Sign reversal}\nMultiplying the relation by $-1$ reverses the signed exponent packet but leaves the kernel lattice invariant.","\\section{Prime-contact data structure}\nThe reusable object attached to a sparse relation is\n\\[\n \\boxed{\n \\mathfrak A(\\mathbf n,\\boldsymbol\\epsilon)\n =\n \\left(\n \\mathbb W_{\\mathbf n},\n \\kappa_{\\mathbf n},\n \\mathcal L_{\\mathbf n},\n \\Lambda_{\\mathrm{coh}},\n \\mathcal R_\\infty,\n \\mathcal M_{\\mathrm{glob}}\n \\right).\n }\n\\]\nThe first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being built into the definition.","\\section{Why the calculus extends beyond \\texorpdfstring{$abc$}{abc}}\nThe essential input is not the particular equation $a+b=c$. It is a sparse additive relation with repeated prime depth. This occurs naturally in generalized Fermat equations, perfect-power gaps, recurrence identities, and Bernoulli congruence problems. The later application chapters develop these directions at a programmatic level.","\\[\n\\boxed{\n\\begin{aligned}\n&\\text{primitive sparse relation}\\\\\n&\\quad\\Downarrow\\\\\n&\\text{prime-packet differential and Wronskian}\\\\\n&\\quad\\Downarrow\\\\\n&\\text{valuation-word and shell refinement}\\\\\n&\\quad\\Downarrow\\\\\n&\\text{joint obstruction complex and exact search}\\\\\n&\\quad\\Downarrow\\\\\n&\\text{prime-shell signal and Frey transform}\\\\\n&\\quad\\Downarrow\\\\\n&\\text{divisor-polyphase persistence and Hecke filters}\\\\\n&\\quad\\Downarrow\\\\\n&\\text{filtered cusp-contact and Witt packet}\\\\\n&\\quad\\Downarrow\\\\\n&\\text{global Kummer and Fermat character packet}\\\\\n&\\quad\\Downarrow\\\\\n&\\text{polylogarithmic path and cross-prime logarithms}\\\\\n&\\quad\\Downarrow\\\\\n&\\text{adelic coherence character and kernel lattice}.\n\\end{aligned}}\n\\]","\\midrule\n\\endhead\nPrime derivative & $\\mathcal T_{a,b,c}$ and $W_x$ & prime support, valuation weights, additive tangent, repeated-root divisor & uniform small integral transversality","Polyphase & $V_e(T)$ and phase packets & valuation-shell residues, dyadic and odd-adic primary lifts & global small-height synchronization","Obstruction complex & $\\mathsf M_{a,b,c}$, $L$, $L_0$ & exact torsion $abc/R$, primitive obstruction class & metric size of a section","Moving search & $H(k)$ and interval certificates & exact optimum in every obstruction coset & no gain below the real first-order floor","Prime-shell signal & $\\Psi_n$ & support energy and persistence energy & nonlinear additive arithmetic","Frey packet & $(E,f_E,N_E,\\Delta_E)$ & conductor, discriminant depth, automorphic eigenchannel & marked integral representative within isogeny or twist class","Divisor packets & $B_d(n)$ & exact-order persistence spectrum & a global inequality controlling all output packets","Hecke filter & $A_\\varepsilon$, $e_f$, $C_f$ & local signs and integral eigenline separation & output contact depth by itself","Filtered contact & $\\mathbf J_c$ & cyclic prime-power depth with carries & geometry-induced global transfer","Witt packet & $W_{e-1}(\\F_p)$ & typed Frobenius and Verschiebung carry structure & conductor-scale global synthesis","Kummer packet & $\\kappa_c$ or $[\\beta]$ & almost all contact order in one global Galois class & additive relation unless paired with another object","Fermat packet & $(P_M,\\mu_M^2)$ & additive relation and exact character bank & geometric degree grows with $M$","Polylog path & $\\Pi_{\\le3}(z)$ & fixed complex packet and three-point closure & integral lattice denominator or covolume","Coherence object & $(\\chi_{\\mathrm{coh}},\\Lambda_{\\mathrm{coh}})$ & exact residual simultaneous logarithmic conductor & radical-scale arithmetic bound","\\begin{enumerate}[leftmargin=1.5em]\n\\item Pairwise coprimality of a primitive triple separates prime sectors.\n\\item The prime-packet differential produces one additive row.\n\\item The arithmetic Wronskian produces one transverse row.\n\\item Local valuation divisibility gives $abc/R\\mid W$.\n\\item Smith reduction gives the exact obstruction torsion and the lattice extension.\n\\item Sector compression and CRT reduce exact search to one moving integer parameter.\n\\item The continuous relaxation gives the exact stable height and finite stopping.\n\\item The no-escape identity closes the first-order search as a proof route.\n\\item Filtered contact restores the carry data lost by split shell packets.\n\\item Kummer theory globalizes the cyclic contact packet with logarithmic loss.\n\\item The additive relation yields a Fermat point and exact character bank.\n\\item Polylogarithmic distribution compresses the complex branch bank but leaves integral gain.\n\\item Simultaneous local logarithms pr","\\midrule\n\\endhead\n$abc/R$ divides every additive Wronskian & Exact & local valuation proof","No independent $abc$ gain from first-order search & Structural limitation & no-escape identity","Filtered contact source is cyclic & Exact & ideal quotient $R_c\\Z/c\\Z$","No near-injective map from split persistence to one cyclic target & Structural limitation & $p$-rank obstruction","\\begin{conjecture}[Combined logarithmic--coherence bound]\nFor every $\\eps>0$,\n\\[\n WC\\le C_\\eps R_aR_bR^\\eps.\n\\]\n\\end{conjecture}","\\begin{conjecture}[Integral adelic regulator]\nThere exists a canonical global object carrying the balanced contact packet whose finite kernel and integral archimedean synthesis cost are bounded by $R^\\eps$ and $R_aR_bR^\\eps$, respectively.\n\\end{conjecture}","Either the combined bound or the stated integral-regulator theorem, together with the exact factorization and logarithmic Kummer bound, yields strong $abc$.","8\\textwidth}\n\\large\nThis paper develops a single calculus for sparse additive relations with repeated prime-power depth. The construction begins with prime-coordinate differentials and exact obstruction lattices, passes through valuation words, filtered contact modules, Kummer globalization, and polylogarithmic path packets, and culminates in an adelic coherence character built from simultaneous $p$-adic logarithms of the actual input primes. Exact theorems, finite calculations, conditional implications, and structural limitations are distinguished throughout.\n\\end{minipage}\n\\vfill\n{\\Large\\bfseries Komorebi P.\\par}\n\\vspace{0.30in}\n{\\large July 2026\\par}\n\\end{titlepage}","\\chapter*{Abstract}\n\\addcontentsline{toc}{chapter}{Abstract}\nLet\n\\[\n a+b=c,\n \\qquad\n \\gcd(a,b)=1,\n \\qquad\n R=\\rad(abc).\n\\]\nThis paper develops a prime-contact coherence calculus for the interaction between sparse addition, squarefree prime support, and repeated valuation depth. The first layer is a prime-packet differential whose additive row and arithmetic Wronskian yield the exact divisibility\n\\[\n \\frac{abc}{R}\\mid W.\n\\]\nValuation words and polyphase refinement separate aggregate multiplicity from dyadic and odd-adic phase data. Smith reduction identifies the exact obstruction torsion, while a moving-interval search computes the first-order optimum and proves that the corresponding master inequality cannot fall below its real relaxation.","chapter*{Abstract}\n\\addcontentsline{toc}{chapter}{Abstract}\nLet\n\\[\n a+b=c,\n \\qquad\n \\gcd(a,b)=1,\n \\qquad\n R=\\rad(abc).\n\\]\nThis paper develops a prime-contact coherence calculus for the interaction between sparse addition, squarefree prime support, and repeated valuation depth. The first layer is a prime-packet differential whose additive row and arithmetic Wronskian yield the exact divisibility\n\\[\n \\frac{abc}{R}\\mid W.\n\\]\nValuation words and polyphase refinement separate aggregate multiplicity from dyadic and odd-adic phase data. Smith reduction identifies the exact obstruction torsion, while a moving-interval search computes the first-order optimum and proves that the corresponding master inequality cannot fall below its real relaxation.","For each output prime, simultaneous $p$-adic logarithms of the input primes define a finite filter row. After removing the dyadic sign layer, common Kummer powers, and entrywise logarithmic defects, the residual rows combine into a cyclic coherence character\n\\[\n \\chi_{\\mathrm{coh}}:\\Z^{S_{ab}}\\longrightarrow\\Z/C\\Z\n\\]\nwhose kernel contains the normalized signed exponent vector. The output powerful part factors exactly as\n\\[\n \\frac{c}{\\rad(c)}=\\epsilon_2KWC.\n\\]\nThe remaining uniform problem is an arithmetic bound on the local logarithmic defect $W$ and coherence conductor $C$. The paper proves the exact reductions and finite identities used to reach this object. It does not prove the $abc$ conjecture or the other major conjectures","For each output prime, simultaneous $p$-adic logarithms of the input primes define a finite filter row. After removing the dyadic sign layer, common Kummer powers, and entrywise logarithmic defects, the residual rows combine into a cyclic coherence character\n\\[\n \\chi_{\\mathrm{coh}}:\\Z^{S_{ab}}\\longrightarrow\\Z/C\\Z\n\\]\nwhose kernel contains the normalized signed exponent vector. The output powerful part factors exactly as\n\\[\n \\frac{c}{\\rad(c)}=\\epsilon_2KWC.\n\\]\nThe remaining uniform problem is an arithmetic bound on the local logarithmic defect $W$ and coherence conductor $C$. The paper proves the exact reductions and finite identities used to reach this object. It does not prove the $abc$ conjecture or the other major conjectures discussed in the applications.","ultaneous $p$-adic logarithms of the input primes define a finite filter row. After removing the dyadic sign layer, common Kummer powers, and entrywise logarithmic defects, the residual rows combine into a cyclic coherence character\n\\[\n \\chi_{\\mathrm{coh}}:\\Z^{S_{ab}}\\longrightarrow\\Z/C\\Z\n\\]\nwhose kernel contains the normalized signed exponent vector. The output powerful part factors exactly as\n\\[\n \\frac{c}{\\rad(c)}=\\epsilon_2KWC.\n\\]\nThe remaining uniform problem is an arithmetic bound on the local logarithmic defect $W$ and coherence conductor $C$. The paper proves the exact reductions and finite identities used to reach this object. It does not prove the $abc$ conjecture or the other major conjectures discussed in the applications.","ummer powers, and entrywise logarithmic defects, the residual rows combine into a cyclic coherence character\n\\[\n \\chi_{\\mathrm{coh}}:\\Z^{S_{ab}}\\longrightarrow\\Z/C\\Z\n\\]\nwhose kernel contains the normalized signed exponent vector. The output powerful part factors exactly as\n\\[\n \\frac{c}{\\rad(c)}=\\epsilon_2KWC.\n\\]\nThe remaining uniform problem is an arithmetic bound on the local logarithmic defect $W$ and coherence conductor $C$. The paper proves the exact reductions and finite identities used to reach this object. It does not prove the $abc$ conjecture or the other major conjectures discussed in the applications.","logarithmic defects, the residual rows combine into a cyclic coherence character\n\\[\n \\chi_{\\mathrm{coh}}:\\Z^{S_{ab}}\\longrightarrow\\Z/C\\Z\n\\]\nwhose kernel contains the normalized signed exponent vector. The output powerful part factors exactly as\n\\[\n \\frac{c}{\\rad(c)}=\\epsilon_2KWC.\n\\]\nThe remaining uniform problem is an arithmetic bound on the local logarithmic defect $W$ and coherence conductor $C$. The paper proves the exact reductions and finite identities used to reach this object. It does not prove the $abc$ conjecture or the other major conjectures discussed in the applications.","}}\\longrightarrow\\Z/C\\Z\n\\]\nwhose kernel contains the normalized signed exponent vector. The output powerful part factors exactly as\n\\[\n \\frac{c}{\\rad(c)}=\\epsilon_2KWC.\n\\]\nThe remaining uniform problem is an arithmetic bound on the local logarithmic defect $W$ and coherence conductor $C$. The paper proves the exact reductions and finite identities used to reach this object. It does not prove the $abc$ conjecture or the other major conjectures discussed in the applications.","\\textbf{Conditional} & A theorem proves that a stated estimate would imply $abc$ or another target, while the estimate itself remains open.","\\textbf{Structural path.} Read the opening chapter, object comparison, coherence factorization, and central open theorems.\n\\item \\textbf{Proof path.} Read the detailed parts in order, beginning with prime-packet divisibility and ending with the coherence lattice.\n\\item \\textbf{Applications path.} Read the general functor, computational methods, and the sections on Wieferich networks, exceptional $abc$, generalized Fermat, Hall--Pillai, and Bernoulli--Witt problems.\n\\end{enumerate}","Coherence lattice & Simultaneous logarithmic character & Exact factorization $M=\\epsilon_2KWC$ and cyclic kernel lattice & Radical-scale bound for $WC$ remains open","\\section{Reusable core}\nThe construction extends to primitive sparse additive relations beyond three terms:\n\\[\n\\boxed{\n\\begin{aligned}\n\\text{prime support}\n&\\longrightarrow\n\\text{filtered contact}\n\\longrightarrow\n\\text{Kummer globalization}\\\\\n&\\longrightarrow\n\\text{cross-prime logarithmic coherence}\n\\longrightarrow\n\\text{equation-specific global geometry}.\n\\end{aligned}\n}\n\\]\nThe general functorial form is developed after the three-term case.","\\section{Primitive sparse additive relations}\nLet\n\\[\n\\epsilon_1n_1+\\cdots+\\epsilon_rn_r=0,\n\\qquad\n\\epsilon_i\\in\\{\\pm1\\},\n\\]\nbe primitive in the sense that the integers have no common prime divisor. The $abc$ relation is the case\n\\[\na+b-c=0.\n\\]\nFor each term define\n\\[\nR_i=\\rad(n_i),\n\\qquad\nM_i=\\frac{n_i}{R_i}.\n\\]\nThe radical $R_i$ records activated prime support. The powerful part $M_i$ records repeated valuation depth.","\\section{Filtered contact packets}\nFor a prime power\n\\[\np^e\\parallel n_i,\n\\]\ndefine the local filtered contact packet\n\\[\n\\Jcal_{i,p}=p\\Z_p/p^e\\Z_p.\n\\]\nIts order is\n\\[\n\\abs{\\Jcal_{i,p}}=p^{e-1}.\n\\]\nIts associated graded is\n\\[\n\\gr\\Jcal_{i,p}\\cong(\\F_p)^{e-1}.\n\\]\nThe distinction is essential. The filtered packet is cyclic. The associated graded has $p$-rank $e-1$. The non-split extensions between successive grades are the arithmetic carries.","Globally, the contact packet of one term is\n\\[\n\\Jcal_{n_i}\n=\nR_i\\Z/n_i\\Z\n\\cong\n\\Z/M_i\\Z.\n\\]","\\begin{theorem}[Exact contact quotient]\nFor every positive integer $n$,\n\\[\n\\boxed{\n\\rad(n)\\Z/n\\Z\n\\cong\n\\Z/(n/\\rad(n))\\Z.\n}\n\\]\nIts associated graded is\n\\[\n\\boxed{\n\\gr\\left(\\rad(n)\\Z/n\\Z\\right)\n\\cong\n\\bigoplus_{p^e\\parallel n}(\\F_p)^{e-1}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nMultiplication by $\\rad(n)$ identifies $\\Z/(n/\\rad(n))\\Z$ with $\\rad(n)\\Z/n\\Z$. The primary filtration is the product of the filtrations $p^j\\Z_p/p^e\\Z_p$. Each consecutive quotient is one-dimensional over $\\F_p$.\n\\end{proof}","\\section{Witt coordinates}\nThe local packet has a natural $p$-typical coordinate description\n\\[\n\\Jcal_{i,p}\n\\cong\nW_{e-1}(\\F_p)\n\\]\nas an additive filtered object. Frobenius and Verschiebung make the prime type explicit. The dyadic branch uses the same additive depth but a shifted multiplicative logarithm coordinate on principal units.","\\section{Kummer globalization}\nLet\n\\[\nx=\\prod_{q\\in S}q^{\\nu_q}\\in\\Q^\\times\n\\]\nand fix a modulus $M$. The Kummer class of $x$ lies in\n\\[\nH^1(\\Q,\\mu_M)\n\\cong\n\\Q^\\times/(\\Q^\\times)^M.\n\\]\nIts order is controlled exactly by the valuation vector.","\\begin{theorem}[Kummer order from the valuation vector]\nAssume $x>0$. Then\n\\[\n\\boxed{\n\\ord([x])\n=\n\\frac{M}{\\gcd(M,\\{\\nu_q\\}_{q\\in S})}.\n}\n\\]\nFor a negative rational number and even $M$, one additional dyadic sign layer may occur.\n\\end{theorem}","\\begin{proof}\nThe class $[x]^k$ is trivial exactly when every $k\\nu_q$ is divisible by $M$. The least positive such $k$ is the least common multiple of $M/\\gcd(M,\\nu_q)$, which is $M/\\gcd(M,\\{\\nu_q\\})$.\n\\end{proof}","\\section{Cross-prime logarithmic filters}\nLet $Q$ be a set of input primes and $P$ a set of output primes. At odd $p\\in P$, define\n\\[\nL_{p,q}\n=\n\\frac1p\\log_p\\langle q\\rangle_p\n\\]\nwith the appropriate finite-depth truncation. At $p=2$, use the weighted principal-unit coordinate together with a separate sign bit.","The matrix\n\\[\n\\Lcal_{P,Q}=(L_{p,q})\n\\]\nis a cross-prime signal operator. A sparse additive relation supplies a short signed valuation vector $\\nu$ satisfying deep congruences in every output row.","\\section{The coherence lattice}\nAfter removing common Kummer powers and local logarithmic image defects, the surviving rows combine by the Chinese remainder theorem into a primitive cyclic character\n\\[\n\\chi_{\\mathrm{coh}}:\\Z^Q\\longrightarrow\\Z/C\\Z.\n\\]\nThe coherence lattice is\n\\[\n\\boxed{\n\\Lambda_{\\mathrm{coh}}\n=\n\\Ker\\chi_{\\mathrm{coh}}.\n}\n\\]\nWhen the row is primitive,\n\\[\n[\\Z^Q:\\Lambda_{\\mathrm{coh}}]=C.\n\\]\nThe normalized exponent vector of the relation lies in this kernel.","\\section{The prime-contact coherence data structure}\nThe reusable data attached to a primitive sparse relation is\n\\[\n\\boxed{\n\\mathfrak A(\\mathbf n,\\boldsymbol\\epsilon)\n=\n\\left(\n\\mathbf R,\n\\mathbf M,\n\\Witt_{\\mathbf n},\n\\kappa_{\\mathbf n},\n\\Lcal_{\\mathbf n},\n\\Lambda_{\\mathrm{coh}},\n\\chi_{\\mathrm{coh}},\n\\mathcal G_{\\mathbf n}\n\\right).\n}\n\\]\nThe calculus consists of exact transformations among these layers, plus the norm and determinant questions required to compare them globally.","\\section{Prime-packet derivative and radical divisibility}\nFor a primitive triple\n\\[\na+b=c,\n\\qquad\nR=\\rad(abc),\n\\]\nthe prime-packet construction attaches one coordinate to every prime dividing $abc$. An arithmetic derivative produces an additivity row, while an arithmetic Wronskian produces a nondegeneracy row. The key local divisibility theorem is\n\\[\n\\boxed{\n\\frac{abc}{R}\n\\mid\nW_x(a,b)\n}\n\\]\nfor every additive packet $x$. A nonzero Wronskian therefore satisfies\n\\[\n\\abs{W_x(a,b)}\\ge\\frac{abc}{R}.\n\\]\nCombining this with the derivative-size bound yields a valid master inequality involving the shortest transverse packet. The theorem is exact. The missing estimate on the packet size was not proved.","derivative and radical divisibility}\nFor a primitive triple\n\\[\na+b=c,\n\\qquad\nR=\\rad(abc),\n\\]\nthe prime-packet construction attaches one coordinate to every prime dividing $abc$. An arithmetic derivative produces an additivity row, while an arithmetic Wronskian produces a nondegeneracy row. The key local divisibility theorem is\n\\[\n\\boxed{\n\\frac{abc}{R}\n\\mid\nW_x(a,b)\n}\n\\]\nfor every additive packet $x$. A nonzero Wronskian therefore satisfies\n\\[\n\\abs{W_x(a,b)}\\ge\\frac{abc}{R}.\n\\]\nCombining this with the derivative-size bound yields a valid master inequality involving the shortest transverse packet. The theorem is exact. The missing estimate on the packet size was not proved.","mitive triple\n\\[\na+b=c,\n\\qquad\nR=\\rad(abc),\n\\]\nthe prime-packet construction attaches one coordinate to every prime dividing $abc$. An arithmetic derivative produces an additivity row, while an arithmetic Wronskian produces a nondegeneracy row. The key local divisibility theorem is\n\\[\n\\boxed{\n\\frac{abc}{R}\n\\mid\nW_x(a,b)\n}\n\\]\nfor every additive packet $x$. A nonzero Wronskian therefore satisfies\n\\[\n\\abs{W_x(a,b)}\\ge\\frac{abc}{R}.\n\\]\nCombining this with the derivative-size bound yields a valid master inequality involving the shortest transverse packet. The theorem is exact. The missing estimate on the packet size was not proved.","\\section{Valuation-word Smith form}\nFor the valuation word\n\\[\nV_e(T)=1+T+\\cdots+T^{e-1}\n\\]\nacting on\n\\[\n\\Z[T]/(T^N-1),\n\\]\nthe phase/coarse block has Smith form\n\\[\n\\boxed{\n\\SNF(C_{e,N})\n=\n\\operatorname{diag}(1,\\ldots,1,e)\n}\n\\]\nwhen $\\gcd(e,N)=1$. Thus all nonconstant phase channels are integrally invertible and the entire multiplicity is concentrated in one aggregate channel. The proof uses the augmentation ideal: multiplication by $V_e$ has determinant one on that integral rank-$N-1$ lattice and acts as multiplication by $e$ on the quotient.","For the valuation word\n\\[\nV_e(T)=1+T+\\cdots+T^{e-1}\n\\]\nacting on\n\\[\n\\Z[T]/(T^N-1),\n\\]\nthe phase/coarse block has Smith form\n\\[\n\\boxed{\n\\SNF(C_{e,N})\n=\n\\operatorname{diag}(1,\\ldots,1,e)\n}\n\\]\nwhen $\\gcd(e,N)=1$. Thus all nonconstant phase channels are integrally invertible and the entire multiplicity is concentrated in one aggregate channel. The proof uses the augmentation ideal: multiplication by $V_e$ has determinant one on that integral rank-$N-1$ lattice and acts as multiplication by $e$ on the quotient.","-1}\n\\]\nacting on\n\\[\n\\Z[T]/(T^N-1),\n\\]\nthe phase/coarse block has Smith form\n\\[\n\\boxed{\n\\SNF(C_{e,N})\n=\n\\operatorname{diag}(1,\\ldots,1,e)\n}\n\\]\nwhen $\\gcd(e,N)=1$. Thus all nonconstant phase channels are integrally invertible and the entire multiplicity is concentrated in one aggregate channel. The proof uses the augmentation ideal: multiplication by $V_e$ has determinant one on that integral rank-$N-1$ lattice and acts as multiplication by $e$ on the quotient.","ase/coarse block has Smith form\n\\[\n\\boxed{\n\\SNF(C_{e,N})\n=\n\\operatorname{diag}(1,\\ldots,1,e)\n}\n\\]\nwhen $\\gcd(e,N)=1$. Thus all nonconstant phase channels are integrally invertible and the entire multiplicity is concentrated in one aggregate channel. The proof uses the augmentation ideal: multiplication by $V_e$ has determinant one on that integral rank-$N-1$ lattice and acts as multiplication by $e$ on the quotient.","\\section{Exact obstruction complex}\nThe exponent-free phase, or shell-synthesis, additivity and Wronskian rows define an integer matrix whose Smith form is\n\\[\n\\boxed{\n\\operatorname{diag}\\left(1,\\frac{abc}{R}\\right).\n}\n\\]\nThis two-row statement applies whenever at least two prime sectors are nonempty. The degenerate triple $(1,1,2)$ has rank one. On the additive phase lattice $L$, the normalized Wronskian gives an exact sequence\n\\[\n0\\longrightarrow L_0\n\\longrightarrow L\n\\longrightarrow\\Z\n\\longrightarrow0.\n\\]\nThe sequence splits as an exact sequence of abelian groups, and all local lift equations are solvable. The full shell deformation is formally smooth over $\\Z$. Every compatible first-order tangent extends to all formal orders.","}\nThe exponent-free phase, or shell-synthesis, additivity and Wronskian rows define an integer matrix whose Smith form is\n\\[\n\\boxed{\n\\operatorname{diag}\\left(1,\\frac{abc}{R}\\right).\n}\n\\]\nThis two-row statement applies whenever at least two prime sectors are nonempty. The degenerate triple $(1,1,2)$ has rank one. On the additive phase lattice $L$, the normalized Wronskian gives an exact sequence\n\\[\n0\\longrightarrow L_0\n\\longrightarrow L\n\\longrightarrow\\Z\n\\longrightarrow0.\n\\]\nThe sequence splits as an exact sequence of abelian groups, and all local lift equations are solvable. The full shell deformation is formally smooth over $\\Z$. Every compatible first-order tangent extends to all formal orders.","t two prime sectors are nonempty. The degenerate triple $(1,1,2)$ has rank one. On the additive phase lattice $L$, the normalized Wronskian gives an exact sequence\n\\[\n0\\longrightarrow L_0\n\\longrightarrow L\n\\longrightarrow\\Z\n\\longrightarrow0.\n\\]\nThe sequence splits as an exact sequence of abelian groups, and all local lift equations are solvable. The full shell deformation is formally smooth over $\\Z$. Every compatible first-order tangent extends to all formal orders.","\\section{Exact first-order moving search}\nThe moving-coset search solves\n\\[\nH(k)\n=\n\\min\\{\\norm{x}_\\infty:x\\in L,\\ \\omega(x)=k\\}\n\\]\nexactly for every supplied factored triple. Sector compression and the Chinese remainder theorem reduce each fixed-$k$ problem to the overlap of three integer intervals. A real lower bound supplies a finite stopping rule in $k$.","The decisive identity is\n\\[\n\\boxed{\n\\overline H\n=\n\\frac{c}{R\\Delta}\n}\n\\]\nfor the exact real relaxation. The master inequality\n\\[\nc\\le R\\Delta\\Phi\n\\]\ntherefore becomes\n\\[\n\\boxed{\nc\n\\le\nc\\frac{\\Phi}{\\overline H},\n\\qquad\n\\frac{\\Phi}{\\overline H}\\ge1.\n}\n\\]\nThe first-order search is complete and cannot create a new $abc$ gain.","\\section{Prime-shell energy identities}\nFor\n\\[\nn=\\prod_pp^{e_p},\n\\]\ndefine the shell signal\n\\[\n\\Psi_n(p,r)=\\sqrt{\\log p}\\,\\mathbf 1_{0\\le r<e_p}.\n\\]\nThen\n\\[\n\\boxed{\n\\norm{\\Psi_n}_2^2=\\log n.\n}\n\\]\nIf $P_{\\mathrm{act}}$ retains only the first shell at every active prime, then\n\\[\n\\boxed{\n\\norm{P_{\\mathrm{act}}\\Psi_n}_2^2\n=\n\\log\\rad(n).\n}\n\\]\nFor the direct-sum signal of an $abc$ triple, the standard quality is the output-sector energy divided by activated bandwidth.","\\section{Divisor-polyphase perfect reconstruction}\nFor\n\\[\nn=\\prod_pp^{e_p},\n\\]\ndefine\n\\[\nB_d(n)=\\prod_{p\\mid n,\\ d\\mid e_p}p.\n\\]\nThen\n\\[\n\\boxed{\nn=\\prod_{d\\ge1}B_d(n)^{\\varphi(d)}\n}\n\\]\nand\n\\[\n\\boxed{\n\\frac{n}{\\rad(n)}\n=\n\\prod_{d\\ge2}B_d(n)^{\\varphi(d)}.\n}\n\\]\nThis is an exact multiplicative polyphase decomposition of prime-power persistence.","\\section{Filtered contact correction}\nThe split module\n\\[\n\\bigoplus_{p^e\\parallel c}(\\F_p)^{e-1}\n\\]\nis the associated graded of the cyclic contact module\n\\[\n\\boxed{\nR_c\\Z/c\\Z\n\\cong\n\\Z/(c/R_c)\\Z.\n}\n\\]\nThe direct split-to-cyclic rank obstruction is therefore a theorem about the graded approximation, not the filtered source.","For cyclic packets of orders $m$ and $\\eta$, the largest possible image has order\n\\[\n\\gcd(m,\\eta).\n\\]\nThe exact source loss is\n\\[\n\\boxed{\n\\frac{m}{\\gcd(m,\\eta)}.\n}\n\\]\nThus the correct cyclic obstruction is prime-power depth shortfall.","\\section{Witt--Kummer globalization}\nThe local contact packet is naturally a truncated Witt packet\n\\[\nW_{e-1}(\\F_p).\n\\]\nGlobally, the Kummer map\n\\[\n\\kappa_c:\\Z/M\\Z\\longrightarrow H^1(\\Q,\\mu_M),\n\\qquad\nk\\longmapsto[c^k]\n\\]\nhas exact kernel\n\\[\n\\boxed{\n\\gcd(M,e_1,\\ldots,e_r)\n}\n\\]\nwhen\n\\[\nc=\\prod_ip_i^{e_i}.\n\\]\nThe kernel is at most $\\log_2c$. Therefore most of the contact packet is already a global Galois class.","\\section{Fermat lift and full-plane line-bundle cost}\nChoose\n\\[\nu^M=\\frac ac,\n\\qquad\nv^M=\\frac bc.\n\\]\nThen\n\\[\nu^M+v^M=1.\n\\]\nThe point $[u:v:1]$ on the Fermat curve has exact projective height\n\\[\n\\boxed{\n\\frac{\\log c}{M}.\n}\n\\]\nHowever\n\\[\nK_{F_M}\\cong\\mathcal O_{F_M}(M-3).\n\\]\nFor compatible Weil-height normalizations,\n\\[\nh_{K_{F_M}}(P_M)\n=\n\\frac{M-3}{M}\\log c+O(1).\n\\]\nThus the full plane-curve polarization reintroduces a factor of order $M$. This observation concerns the complete Fermat model with its induced line bundle and does not exclude a lower-complexity quotient with a different arithmetic metric.","-bundle cost}\nChoose\n\\[\nu^M=\\frac ac,\n\\qquad\nv^M=\\frac bc.\n\\]\nThen\n\\[\nu^M+v^M=1.\n\\]\nThe point $[u:v:1]$ on the Fermat curve has exact projective height\n\\[\n\\boxed{\n\\frac{\\log c}{M}.\n}\n\\]\nHowever\n\\[\nK_{F_M}\\cong\\mathcal O_{F_M}(M-3).\n\\]\nFor compatible Weil-height normalizations,\n\\[\nh_{K_{F_M}}(P_M)\n=\n\\frac{M-3}{M}\\log c+O(1).\n\\]\nThus the full plane-curve polarization reintroduces a factor of order $M$. This observation concerns the complete Fermat model with its induced line bundle and does not exclude a lower-complexity quotient with a different arithmetic metric.","\\section{Polylogarithmic path compression}\nThe root-of-unity distribution identity\n\\[\n\\boxed{\n\\sum_{j=0}^{M-1}\\operatorname{Li}_n(\\zeta_M^ju)\n=\nM^{1-n}\\operatorname{Li}_n(u^M)\n}\n\\]\ncompresses an $M$-branch orbit to one weight-$n$ coordinate. Through weights one, two, and three, the synthesis gains multiply to\n\\[\nM^3.\n\\]\nThe complex depth-three path packet has at most cubic logarithmic growth. A fixed-rank integral presentation carrying order $M$ must nevertheless have a large coefficient, denominator, degree, or covolume. Small complex periods are not enough.","root-of-unity distribution identity\n\\[\n\\boxed{\n\\sum_{j=0}^{M-1}\\operatorname{Li}_n(\\zeta_M^ju)\n=\nM^{1-n}\\operatorname{Li}_n(u^M)\n}\n\\]\ncompresses an $M$-branch orbit to one weight-$n$ coordinate. Through weights one, two, and three, the synthesis gains multiply to\n\\[\nM^3.\n\\]\nThe complex depth-three path packet has at most cubic logarithmic growth. A fixed-rank integral presentation carrying order $M$ must nevertheless have a large coefficient, denominator, degree, or covolume. Small complex periods are not enough.","\\section{Adelic coherence factorization}\nFor the balanced input ratio, the output contact modulus factors exactly as\n\\[\n\\boxed{\nM=\\epsilon_2KWC.\n}\n\\]\nHere $\\epsilon_2$ is the weighted dyadic sign layer, $K$ is the common Kummer power, $W$ is the residual local logarithmic image defect, and $C$ is the genuine cross-prime coherence quotient.","The coherence quotient is the index of an explicit lattice\n\\[\n\\Lambda_{\\mathrm{coh}}\n\\subset\n\\Z^{S_{ab}}.\n\\]\nThe normalized input exponent vector lies in this lattice. The corresponding dual character has order $C$.","\\section{No-go theorem: abstract lattice geometry}\nLet $v\\in\\Z^s$ be primitive with $s\\ge2$. For every prescribed modulus $C$, one can choose a primitive row modulo $C$ whose kernel contains $v$. Therefore a fixed short primitive relation can lie in a primitive congruence lattice of arbitrarily large index.","\\[\n\\boxed{\n\\text{Short vector plus large lattice index is possible abstractly at every scale.}\n}\n\\]\nThe arithmetic origin of the logarithmic row is essential.","\\section{No-go theorem: direct positive contact}\nIf positive integers $A$ and $B$ satisfy\n\\[\nc\\mid A+B,\n\\]\nthen\n\\[\nA+B\\ge c\n\\]\nand\n\\[\n\\boxed{\n\\max(A,B)\\ge\\frac c2.\n}\n\\]\nA direct positive $S$-unit relation divisible by $c$ is already at the scale one is trying to bound.","\\section{No-go theorem: linear Hecke averaging}\nIn the local Tate model, the $\\ell+1$ isogeny branches multiply contact depth by\n\\[\n\\ell,\n\\quad\n\\underbrace{\\ell^{-1},\\ldots,\\ell^{-1}}_{\\ell\\text{ branches}}.\n\\]\nUniform averaging preserves the linear mean exactly. Typical logarithmic depth falls, but rare deep branches retain the mean. Linear Hecke averaging is redistribution rather than contraction.","\\section{Information retained by signal transforms}\nThe signal representation asks which transforms retain the squarefree support, repeated valuation depth, additive marking, and integral normalization needed by the $abc$ inequality. The conclusions are sharp.","\\section{Prime-shell signals}\nFor a positive integer\n\\[\nn=\\prod_pp^{e_p},\n\\]\nthe shell signal\n\\[\n\\Psi_n(p,r)=\\sqrt{\\log p}\\,\\mathbf1_{0\\le r<e_p}\n\\]\nis a rectangular pulse of length $e_p$ in each prime channel.","Its depth Fourier transform is a Dirichlet packet:\n\\[\n\\widehat\\Psi_{n,p}(\\theta)\n=\n\\sqrt{\\log p}\\,\n\\exp\\left(-\\frac{i(e_p-1)\\theta}{2}\\right)\n\\frac{\\sin(e_p\\theta/2)}{\\sin(\\theta/2)}.\n\\]\nThe zero phase records the exponent. Nonzero phases record the endpoint and residue geometry of the valuation run.","This explains why polyphase refinement was natural. It also proves why a unitary depth transform is not enough. Parseval preserves the total shell energy\n\\[\n\\log n\n\\]\nand the activated first-shell energy\n\\[\n\\log\\rad(n).\n\\]\nThe transform changes coordinates, not the underlying imbalance.","\\section{The Frey curve as a nonlinear signal transform}\nAttach the marked Frey curve\n\\[\nE_{a,b,c}:y^2=x(x-a)(x+b).\n\\]\nIts discriminant is\n\\[\n\\Delta_E=16a^2b^2c^2.\n\\]\nAt odd primes dividing $abc$, the conductor records the activated channel while the minimal discriminant records the local depth. Thus the Frey transform converts the prime-shell split into a standard automorphic split:\n\\[\n\\begin{aligned}\n\\text{conductor}\n&\\longleftrightarrow\n\\text{activated bandwidth},","\\section{The Frey curve as a nonlinear signal transform}\nAttach the marked Frey curve\n\\[\nE_{a,b,c}:y^2=x(x-a)(x+b).\n\\]\nIts discriminant is\n\\[\n\\Delta_E=16a^2b^2c^2.\n\\]\nAt odd primes dividing $abc$, the conductor records the activated channel while the minimal discriminant records the local depth. Thus the Frey transform converts the prime-shell split into a standard automorphic split:\n\\[\n\\begin{aligned}\n\\text{conductor}\n&\\longleftrightarrow\n\\text{activated bandwidth},\\\\\n\\text{minimal discriminant}\n&\\longleftrightarrow\n\\text{persistence depth}.\n\\end{aligned}\n\\]","A useful exact family appears when $u>v>0$ are coprime odd integers. Define\n\\[\nT_0\n=\n\\left(v^2,u^2-v^2,u^2\\right)\n\\]\nand\n\\[\nT_1\n=\n\\left(\nuv,\n\\left(\\frac{u-v}{2}\\right)^2,\n\\left(\\frac{u+v}{2}\\right)^2\n\\right).\n\\]\nBoth are primitive additive triples. They have the same radical\n\\[\n\\rad\\bigl(uv(u-v)(u+v)\\bigr),\n\\]\nwhile the second output is smaller:\n\\[\n\\left(\\frac{u+v}{2}\\right)^2<u^2.\n\\]","Their Legendre parameters are linked by the standard dyadic Landen transformation. If\n\\[\n\\lambda_0=\\left(\\frac vu\\right)^2,\n\\]\nthen the complementary child parameter is\n\\[\n\\lambda_1\n=\n\\frac{4uv}{(u+v)^2}\n=\n\\frac{4\\sqrt{\\lambda_0}}{(1+\\sqrt{\\lambda_0})^2}.\n\\]\nThis is the exact two-isogeny refinement law behind the family.","The family shows that one rational automorphic channel can contain several integral representatives with the same radical and different additive heights. Ordinary eigenvalues alone do not encode the $abc$ quantity.","\\section{The integral automorphic packet}\nThe resulting integral automorphic object is\n\\[\n\\boxed{\n\\Acal^{\\sharp}(E,\\iota)\n=\n\\left(\n f_E,\n \\Lambda_E,\n \\omega_E,\n \\{q_p\\}_{p\\mid N_E},\n \\iota,\n \\phi_E\n\\right).\n}\n\\]\nThe components are:\n\\begin{itemize}[leftmargin=1.5em]\n\\item the rational Hecke eigen-signal $f_E$\n\\item an integral Hecke, modular-symbol, or homology lattice $\\Lambda_E$\n\\item the minimal N\\'eron differential $\\omega_E$\n\\item local Tate-depth packets $q_p$\n\\item the marking $\\iota$ of the three rational two-torsion differences\n\\item the modular synthesis map $\\phi_E$\n\\end{itemize}","utomorphic packet}\nThe resulting integral automorphic object is\n\\[\n\\boxed{\n\\Acal^{\\sharp}(E,\\iota)\n=\n\\left(\n f_E,\n \\Lambda_E,\n \\omega_E,\n \\{q_p\\}_{p\\mid N_E},\n \\iota,\n \\phi_E\n\\right).\n}\n\\]\nThe components are:\n\\begin{itemize}[leftmargin=1.5em]\n\\item the rational Hecke eigen-signal $f_E$\n\\item an integral Hecke, modular-symbol, or homology lattice $\\Lambda_E$\n\\item the minimal N\\'eron differential $\\omega_E$\n\\item local Tate-depth packets $q_p$\n\\item the marking $\\iota$ of the three rational two-torsion differences\n\\item the modular synthesis map $\\phi_E$\n\\end{itemize}","automorphic object is\n\\[\n\\boxed{\n\\Acal^{\\sharp}(E,\\iota)\n=\n\\left(\n f_E,\n \\Lambda_E,\n \\omega_E,\n \\{q_p\\}_{p\\mid N_E},\n \\iota,\n \\phi_E\n\\right).\n}\n\\]\nThe components are:\n\\begin{itemize}[leftmargin=1.5em]\n\\item the rational Hecke eigen-signal $f_E$\n\\item an integral Hecke, modular-symbol, or homology lattice $\\Lambda_E$\n\\item the minimal N\\'eron differential $\\omega_E$\n\\item local Tate-depth packets $q_p$\n\\item the marking $\\iota$ of the three rational two-torsion differences\n\\item the modular synthesis map $\\phi_E$\n\\end{itemize}","\\boxed{\n\\Acal^{\\sharp}(E,\\iota)\n=\n\\left(\n f_E,\n \\Lambda_E,\n \\omega_E,\n \\{q_p\\}_{p\\mid N_E},\n \\iota,\n \\phi_E\n\\right).\n}\n\\]\nThe components are:\n\\begin{itemize}[leftmargin=1.5em]\n\\item the rational Hecke eigen-signal $f_E$\n\\item an integral Hecke, modular-symbol, or homology lattice $\\Lambda_E$\n\\item the minimal N\\'eron differential $\\omega_E$\n\\item local Tate-depth packets $q_p$\n\\item the marking $\\iota$ of the three rational two-torsion differences\n\\item the modular synthesis map $\\phi_E$\n\\end{itemize}","N_E},\n \\iota,\n \\phi_E\n\\right).\n}\n\\]\nThe components are:\n\\begin{itemize}[leftmargin=1.5em]\n\\item the rational Hecke eigen-signal $f_E$\n\\item an integral Hecke, modular-symbol, or homology lattice $\\Lambda_E$\n\\item the minimal N\\'eron differential $\\omega_E$\n\\item local Tate-depth packets $q_p$\n\\item the marking $\\iota$ of the three rational two-torsion differences\n\\item the modular synthesis map $\\phi_E$\n\\end{itemize}","\\section{Exact automorphic analysis--synthesis identity}\nLet\n\\[\n\\phi_E:X_0(N_E)\\longrightarrow E\n\\]\nbe an optimal modular parametrization, and write\n\\[\n\\phi_E^*\\omega_E\n=\n2\\pi i\\,c_Ef_E(z)\\,dz.\n\\]\nIf\n\\[\nA(E)\n=\n\\frac{i}{2}\n\\int_{E(\\C)}\\omega_E\\wedge\\overline{\\omega_E},\n\\]\nthen change of variables under the finite map gives\n\\[\n\\boxed{\n\\deg(\\phi_E)A(E)\n=\n4\\pi^2c_E^2\\norm{f_E}_{\\mathrm{Pet}}^2.\n}\n\\]\nThis is an exact analysis--synthesis energy identity.","For a marked Frey representative, the scaling from the normalized Legendre shape to the integral curve introduces the additive height. Schematically,\n\\[\n\\boxed{\n\\text{height}\n=\n\\text{shape factor}\n\\times\n\\frac{\\text{synthesis multiplicity}}\n{\\text{integral normalization}\\times\\text{spectral energy}}.\n}\n\\]\nThe formula explains why modular degree, Manin constant, N\\'eron scaling, and Petersson energy all enter the integral synthesis identity.","\\section{Definition of the calculus}\nA prime-contact coherence system attached to a primitive sparse additive relation consists of the data\n\\[\n\\boxed{\n\\mathfrak A(\\mathbf n,\\boldsymbol\\epsilon)\n=\n\\left(\n\\mathbf R,\n\\mathbf M,\n\\Witt_{\\mathbf n},\n\\kappa_{\\mathbf n},\n\\Lcal_{\\mathbf n},\n\\Lambda_{\\mathrm{coh}},\n\\chi_{\\mathrm{coh}},\n\\mathcal G_{\\mathbf n}\n\\right).\n}\n\\]\nHere:\n\\begin{itemize}[leftmargin=1.5em]\n\\item $\\mathbf R$ records squarefree prime support\n\\item $\\mathbf M$ records powerful parts\n\\item $\\Witt_{\\mathbf n}$ is the product of filtered Witt contact crystals\n\\item $\\kappa_{\\mathbf n}$ is the system of global Kummer classes\n\\item $\\Lcal_{\\mathbf n}$ is the cross-prime logarithmic filter matrix\n\\item $\\Lambda_{\\mathrm{coh}}$ is the simultaneous congruence kernel\n\\item $\\chi_{\\mathrm{coh}}$ is the dual cyclic coherence character\n\\item $\\mathcal G_{\\mathbf n}$ is an optional equation-specific geometric or automorphic realization\n\\end{itemize}","inition of the calculus}\nA prime-contact coherence system attached to a primitive sparse additive relation consists of the data\n\\[\n\\boxed{\n\\mathfrak A(\\mathbf n,\\boldsymbol\\epsilon)\n=\n\\left(\n\\mathbf R,\n\\mathbf M,\n\\Witt_{\\mathbf n},\n\\kappa_{\\mathbf n},\n\\Lcal_{\\mathbf n},\n\\Lambda_{\\mathrm{coh}},\n\\chi_{\\mathrm{coh}},\n\\mathcal G_{\\mathbf n}\n\\right).\n}\n\\]\nHere:\n\\begin{itemize}[leftmargin=1.5em]\n\\item $\\mathbf R$ records squarefree prime support\n\\item $\\mathbf M$ records powerful parts\n\\item $\\Witt_{\\mathbf n}$ is the product of filtered Witt contact crystals\n\\item $\\kappa_{\\mathbf n}$ is the system of global Kummer classes\n\\item $\\Lcal_{\\mathbf n}$ is the cross-prime logarithmic filter matrix\n\\item $\\Lambda_{\\mathrm{coh}}$ is the simultaneous congruence kernel\n\\item $\\chi_{\\mathrm{coh}}$ is the dual cyclic coherence character\n\\item $\\mathcal G_{\\mathbf n}$ is an optional equation-specific geometric or automorphic realization\n\\end{itemize}","system attached to a primitive sparse additive relation consists of the data\n\\[\n\\boxed{\n\\mathfrak A(\\mathbf n,\\boldsymbol\\epsilon)\n=\n\\left(\n\\mathbf R,\n\\mathbf M,\n\\Witt_{\\mathbf n},\n\\kappa_{\\mathbf n},\n\\Lcal_{\\mathbf n},\n\\Lambda_{\\mathrm{coh}},\n\\chi_{\\mathrm{coh}},\n\\mathcal G_{\\mathbf n}\n\\right).\n}\n\\]\nHere:\n\\begin{itemize}[leftmargin=1.5em]\n\\item $\\mathbf R$ records squarefree prime support\n\\item $\\mathbf M$ records powerful parts\n\\item $\\Witt_{\\mathbf n}$ is the product of filtered Witt contact crystals\n\\item $\\kappa_{\\mathbf n}$ is the system of global Kummer classes\n\\item $\\Lcal_{\\mathbf n}$ is the cross-prime logarithmic filter matrix\n\\item $\\Lambda_{\\mathrm{coh}}$ is the simultaneous congruence kernel\n\\item $\\chi_{\\mathrm{coh}}$ is the dual cyclic coherence character\n\\item $\\mathcal G_{\\mathbf n}$ is an optional equation-specific geometric or automorphic realization\n\\end{itemize}","itive relation consists of the data\n\\[\n\\boxed{\n\\mathfrak A(\\mathbf n,\\boldsymbol\\epsilon)\n=\n\\left(\n\\mathbf R,\n\\mathbf M,\n\\Witt_{\\mathbf n},\n\\kappa_{\\mathbf n},\n\\Lcal_{\\mathbf n},\n\\Lambda_{\\mathrm{coh}},\n\\chi_{\\mathrm{coh}},\n\\mathcal G_{\\mathbf n}\n\\right).\n}\n\\]\nHere:\n\\begin{itemize}[leftmargin=1.5em]\n\\item $\\mathbf R$ records squarefree prime support\n\\item $\\mathbf M$ records powerful parts\n\\item $\\Witt_{\\mathbf n}$ is the product of filtered Witt contact crystals\n\\item $\\kappa_{\\mathbf n}$ is the system of global Kummer classes\n\\item $\\Lcal_{\\mathbf n}$ is the cross-prime logarithmic filter matrix\n\\item $\\Lambda_{\\mathrm{coh}}$ is the simultaneous congruence kernel\n\\item $\\chi_{\\mathrm{coh}}$ is the dual cyclic coherence character\n\\item $\\mathcal G_{\\mathbf n}$ is an optional equation-specific geometric or automorphic realization\n\\end{itemize}","ymbol\\epsilon)\n=\n\\left(\n\\mathbf R,\n\\mathbf M,\n\\Witt_{\\mathbf n},\n\\kappa_{\\mathbf n},\n\\Lcal_{\\mathbf n},\n\\Lambda_{\\mathrm{coh}},\n\\chi_{\\mathrm{coh}},\n\\mathcal G_{\\mathbf n}\n\\right).\n}\n\\]\nHere:\n\\begin{itemize}[leftmargin=1.5em]\n\\item $\\mathbf R$ records squarefree prime support\n\\item $\\mathbf M$ records powerful parts\n\\item $\\Witt_{\\mathbf n}$ is the product of filtered Witt contact crystals\n\\item $\\kappa_{\\mathbf n}$ is the system of global Kummer classes\n\\item $\\Lcal_{\\mathbf n}$ is the cross-prime logarithmic filter matrix\n\\item $\\Lambda_{\\mathrm{coh}}$ is the simultaneous congruence kernel\n\\item $\\chi_{\\mathrm{coh}}$ is the dual cyclic coherence character\n\\item $\\mathcal G_{\\mathbf n}$ is an optional equation-specific geometric or automorphic realization\n\\end{itemize}","kappa_{\\mathbf n},\n\\Lcal_{\\mathbf n},\n\\Lambda_{\\mathrm{coh}},\n\\chi_{\\mathrm{coh}},\n\\mathcal G_{\\mathbf n}\n\\right).\n}\n\\]\nHere:\n\\begin{itemize}[leftmargin=1.5em]\n\\item $\\mathbf R$ records squarefree prime support\n\\item $\\mathbf M$ records powerful parts\n\\item $\\Witt_{\\mathbf n}$ is the product of filtered Witt contact crystals\n\\item $\\kappa_{\\mathbf n}$ is the system of global Kummer classes\n\\item $\\Lcal_{\\mathbf n}$ is the cross-prime logarithmic filter matrix\n\\item $\\Lambda_{\\mathrm{coh}}$ is the simultaneous congruence kernel\n\\item $\\chi_{\\mathrm{coh}}$ is the dual cyclic coherence character\n\\item $\\mathcal G_{\\mathbf n}$ is an optional equation-specific geometric or automorphic realization\n\\end{itemize}","thrm{coh}},\n\\mathcal G_{\\mathbf n}\n\\right).\n}\n\\]\nHere:\n\\begin{itemize}[leftmargin=1.5em]\n\\item $\\mathbf R$ records squarefree prime support\n\\item $\\mathbf M$ records powerful parts\n\\item $\\Witt_{\\mathbf n}$ is the product of filtered Witt contact crystals\n\\item $\\kappa_{\\mathbf n}$ is the system of global Kummer classes\n\\item $\\Lcal_{\\mathbf n}$ is the cross-prime logarithmic filter matrix\n\\item $\\Lambda_{\\mathrm{coh}}$ is the simultaneous congruence kernel\n\\item $\\chi_{\\mathrm{coh}}$ is the dual cyclic coherence character\n\\item $\\mathcal G_{\\mathbf n}$ is an optional equation-specific geometric or automorphic realization\n\\end{itemize}","}[leftmargin=1.5em]\n\\item $\\mathbf R$ records squarefree prime support\n\\item $\\mathbf M$ records powerful parts\n\\item $\\Witt_{\\mathbf n}$ is the product of filtered Witt contact crystals\n\\item $\\kappa_{\\mathbf n}$ is the system of global Kummer classes\n\\item $\\Lcal_{\\mathbf n}$ is the cross-prime logarithmic filter matrix\n\\item $\\Lambda_{\\mathrm{coh}}$ is the simultaneous congruence kernel\n\\item $\\chi_{\\mathrm{coh}}$ is the dual cyclic coherence character\n\\item $\\mathcal G_{\\mathbf n}$ is an optional equation-specific geometric or automorphic realization\n\\end{itemize}","ort\n\\item $\\mathbf M$ records powerful parts\n\\item $\\Witt_{\\mathbf n}$ is the product of filtered Witt contact crystals\n\\item $\\kappa_{\\mathbf n}$ is the system of global Kummer classes\n\\item $\\Lcal_{\\mathbf n}$ is the cross-prime logarithmic filter matrix\n\\item $\\Lambda_{\\mathrm{coh}}$ is the simultaneous congruence kernel\n\\item $\\chi_{\\mathrm{coh}}$ is the dual cyclic coherence character\n\\item $\\mathcal G_{\\mathbf n}$ is an optional equation-specific geometric or automorphic realization\n\\end{itemize}","\\subsection{Sign reversal}\nMultiplying the full relation by $-1$ changes the signed exponent vector but not the contact depths, Kummer orders, or coherence indices.","\\subsection{Primary decomposition}\nEvery filtered contact object decomposes exactly into its $p$-primary components. The dyadic branch may require a shifted multiplicative coordinate, but the additive filtered length remains $e-1$.","\\section{The master factorization for an output term}\nFor the three-term $abc$ relation, the final exact factorization is\n\\[\n\\boxed{\nM=\\epsilon_2KWC.\n}\n\\]\nThe factors distinguish four mechanisms.","\\midrule\n\\endhead\n$\\epsilon_2$ & Dyadic sign layer & The separate weighted two-adic sign contribution","$K$ & Kummer preload & Depth removed by a common power in the signed input valuation vector","$W$ & Local logarithmic defect & Depth already present because every active logarithmic coefficient is exceptionally divisible","$C$ & Coherence quotient & Depth created by genuine simultaneous cancellation among the normalized cross-prime logarithmic channels","In the finite data set, the Kummer and Wieferich layers were usually trivial. Most difficult triples placed their repeated depth in $C$.","\\section{The simultaneous coherence problem}\nLet $Q$ be the input prime set and $P$ the repeated output prime set. After normalization, each output row is primitive modulo its remaining prime-power depth. The rows assemble into one cyclic map\n\\[\n\\chi_{\\mathrm{coh}}:\\Z^Q\\longrightarrow\\Z/C\\Z.\n\\]\nThe exponent packet of the additive relation lies in the kernel.","The missing theorem is not a generic short-vector theorem. It is an arithmetic conductor estimate for a row whose entries are reductions of $p$-adic logarithms of the actual input primes and whose kernel vector is forced by the additive equation.","A model target is\n\\[\n\\boxed{\nWC\n\\le\nC_\\eps\nR_{\\mathrm{in}}R^\\eps,\n}\n\\]\nwhere $R_{\\mathrm{in}}$ is the activated input bandwidth. For $abc$, $R_{\\mathrm{in}}=R_aR_b$.","\\section{A general contact functor}\nThe construction extends to a sparse relation\n\\[\n\\sum_{j=1}^{r}\\epsilon_jn_j=0.\n\\]\nChoose one or more output terms whose repeated depth is to be explained. The assignment\n\\[\n(\\mathbf n,\\boldsymbol\\epsilon)\n\\longmapsto\n\\mathfrak A(\\mathbf n,\\boldsymbol\\epsilon)\n\\]\nis canonical under permutations, sign reversal, primitive reduction, and primary decomposition. The optional geometric lift depends on the equation, but the support, contact, Kummer, and logarithmic layers do not.","Exact arithmetic requires the rule:\n\\[\n\\boxed{\n\\text{Every public arithmetic input is promoted to an arbitrary-precision integer before multiplication.}\n}\n\\]\nDisplay values may be converted to floating point only after the exact rational or integer quantity has been stored.","Ratios such as\n\\[\n\\frac{\\Phi}{\\overline H},\n\\qquad\n\\frac{\\Phi}{k\\overline H},\n\\qquad\n\\frac{H(1)}{\\Phi}\n\\]\nare not probabilities. They are constrained to be at least one by their definitions. Values above one are meaningful integrality overheads or improvement factors.","wo outputs:\n\\begin{enumerate}[leftmargin=1.5em]\n\\item a witness realizing the claimed optimum\n\\item a certificate that no smaller witness exists\n\\end{enumerate}\nExamples include:\n\\begin{itemize}[leftmargin=1.5em]\n\\item exact mixed-integer and CRT interval solutions for transverse packets\n\\item Smith normal forms and determinantal divisors\n\\item finite stopping bounds for the obstruction multiple $k$\n\\item exact Kummer kernel formulas checked against brute force on small cases\n\\item exact reconstruction of contact orders and filtered defects\n\\item symbolic verification of polynomial and Witt carry identities\n\\end{itemize}","\\section{The 361-triple finite data set}\nThe calculations use a finite set of 361 distinct primitive triples assembled from:\n\\begin{itemize}[leftmargin=1.5em]\n\\item all selected $abc$ hits through $10^4$\n\\item a broader quality-filtered scan through $10^5$\n\\item a high-quality reference set\n\\end{itemize}\nThe data set was used to verify exact identities, not to infer an asymptotic distribution.","\\section{The cross-prime resonance network}\nThe most direct generalization is a directed network on the prime numbers. For distinct primes $q$ and $p$, attach the depth\n\\[\n\\boxed{\nw_p(q)=v_p(q^{p-1}-1)-1\n}\n\\]\nfor odd $p$, together with the weighted dyadic analogue. The edge\n\\[\nq\\longrightarrow p\n\\]\nmeasures how deeply the input prime $q$ lies in the principal-unit filtration at the output prime $p$.","A sparse additive relation selects a finite bipartite subnetwork\n\\[\n\\{q:q\\mid\\text{input terms}\\}\n\\longrightarrow\n\\{p:p\\mid\\text{repeated output terms}\\}.\n\\]\nIf $p^e$ divides an output term exactly, the signed input-exponent signal must cancel through the $p$-row to the required depth after the Kummer and local-image layers have been removed.","\\section{Simultaneous Wieferich and regulator theory}\nThis is the closest immediate field because the cross-prime logarithmic matrix is already the main object. For finite sets of primes $P$ and $Q$, define\n\\[\n\\Lcal_{P,Q}\n=\n\\left(\n\\ell_p(q)\n\\right)_{p\\in P,q\\in Q}.\n\\]\nNatural theorem targets include:\n\\begin{enumerate}[leftmargin=1.5em]\n\\item a rank criterion for the matrix modulo prescribed prime powers\n\\item an exact decomposition of depth into entry defects, minor defects, and relation-specific cancellation\n\\item a density-zero or power-saving theorem for simultaneous deep degeneration\n\\item a regulator lower bound that is uniform over sparse prime supports\n\\item an effective classification of low-rank cross-prime networks\n\\end{enumerate}","\\section{Exceptional abc classification}\nThe full uniform inequality may be approached through the exceptional set. A structural target is:\n\\[\nq(a,b,c)>1+\\delta\n\\]\nimplies at least one of the following:\n\\begin{enumerate}[leftmargin=1.5em]\n\\item a large generalized Wieferich defect $W$\n\\item a highly divisible logarithmic minor\n\\item an unusually short relation in the normalized coherence lattice\n\\item an exceptional automorphic congruence or regulator degeneration\n\\end{enumerate}","\\section{Generalized Fermat and Beal}\nConsider\n\\[\nx^p+y^q=z^r.\n\\]\nEvery term carries repeated depth by construction. The prime-contact calculus attaches:\n\\begin{itemize}[leftmargin=1.5em]\n\\item filtered contact crystals for the three powered terms\n\\item balanced Kummer ratios\n\\item cross-prime logarithmic filter matrices\n\\item one or more coherence lattices\n\\item an equation-specific Frey curve, higher Frey variety, or hypergeometric motive when available\n\\end{itemize}","For the Beal equation\n\\[\nA^x+B^y=C^z,\n\\qquad\nx,y,z>2,\n\\]\na primitive counterexample would require three deep contact systems with disjoint support. A concrete intermediate theorem is:","\\section{Pillai and fixed perfect-power gaps}\nFor\n\\[\nx^m-y^n=k,\n\\qquad\nk\\ne0\\text{ fixed},\n\\]\ntwo deep contact packets collide through a residual channel whose prime support is fixed. This is more rigid than unrestricted $abc$ because the output bandwidth is prescribed by $\\rad(k)$.","The coherence matrix has input columns from the prime factors of $x$ and $y$, while the output rows are controlled by the prime factors of the perfect-power terms and of $k$. A first effective theorem should impose a nondegenerate logarithmic-minor condition and derive explicit bounds for $x$, $y$, $m$, or $n$ in terms of $k$.","The calculus suggests a separation into:\n\\begin{itemize}[leftmargin=1.5em]\n\\item local preloaded depth\n\\item individual Wieferich depth\n\\item collective cancellation depth\n\\item fixed residual bandwidth from $k$\n\\end{itemize}\nThis decomposition is available before invoking a general linear-form theorem.","\\section{Hall's square-cube gap}\nHall's problem concerns\n\\[\nx^3-y^2=k.\n\\]\nThe exponents two and three make the dyadic and triadic branches intrinsic. The phase/coarse Smith calculations and the later depth-three polylogarithmic resonance suggest a natural two-three contact object.","A serious Hall branch should begin by defining:\n\\[\n\\begin{aligned}\n&\\text{the square contact crystal of }y^2,\\\\\n&\\text{the cube contact crystal of }x^3,\\\\\n&\\text{their joint dyadic-triadic coherence lattice},\\\\\n&\\text{the elliptic curve and its marked integral packet}.\n\\end{aligned}\n\\]\nThe weight-three resonance of the path construction is suggestive but not yet a theorem about Hall's conjecture. The first goal is a precise correspondence between the cube-side contact determinant and the elliptic regulator.","\\section{A Bernoulli--Witt branch for Erd\\H{o}s--Moser}\nThe equation\n\\[\n1^k+2^k+\\cdots+(m-1)^k=m^k\n\\]\nis controlled by Faulhaber and Bernoulli polynomials. The broader seed calculus already contains Euler and Bernoulli families, residue packets, finite-difference divisibility, and dyadic/odd-adic lifting.","A dedicated object should combine:\n\\[\n\\boxed{\n\\text{Bernoulli denominator support}\n+\n\\text{Witt depth}\n+\n\\text{finite-difference coherence}.\n}\n\\]\nThis branch should remain separate from the logarithmic prime network until an exact transfer between the Bernoulli and contact packets is proved.","and regulator theory};\n\\node[target,fill=PaleTeal,right=37mm of core] (abc) {Exceptional $abc$","\\section{The abc endpoint}\nFor the output contact modulus\n\\[\nM=\\frac{c}{R_c},\n\\]\nthe final exact factorization is\n\\[\n\\boxed{\nM=\\epsilon_2KWC.\n}\n\\]\nThe dyadic sign factor $\\epsilon_2$ is bounded. The Kummer factor $K$ is at most logarithmic. The local logarithmic defect $W$ and the coherence quotient $C$ contain the remaining arithmetic depth.","Strong $abc$ would follow from\n\\[\n\\boxed{\nWC\n\\le\nC_\\eps R_aR_bR^\\eps.\n}\n\\]\nThe equation is exact as a target because\n\\[\nc=R_c\\epsilon_2KWC.\n\\]\nNo estimate of this strength is proved.","\\section{Open theorem A: combined logarithmic--coherence conductor}\nLet\n\\[\n\\chi_{\\mathrm{coh}}:\\Z^{S_{ab}}\\longrightarrow\\Z/C\\Z\n\\]\nbe the normalized coherence character constructed from the actual input primes and output-prime logarithmic filters.","\\begin{conjecture}[Combined logarithmic--coherence bound]\nFor every $\\eps>0$,\n\\[\n\\boxed{\nWC\n\\le\nC_\\eps R_aR_bR^\\eps.\n}\n\\]\n\\end{conjecture}","A proof must control both entrywise logarithmic degeneration and collective coherence, using the arithmetic origin of the rows and the relation $a+b=c$. Abstract lattice geometry cannot imply it. On the subfamily $W=1$, this reduces to the pure coherence-conductor estimate\n\\[\n C\\le C_\\eps R_aR_bR^\\eps.\n\\]\nA counterexample to the combined bound would require a family in which the local defect, the collective coherence, or their product exceeds every radical-scale estimate.","\\begin{conjecture}[Adelic minor nondegeneracy]\nFor every $\\eps>0$, a suitable primitive minor or exterior-power regulator of the normalized matrix satisfies an adelic lower bound whose denominator is at most\n\\[\nC_\\eps R_aR_bR^\\eps.\n\\]\n\\end{conjecture}","This would force the coherence quotient to remain at radical scale after entrywise normalization. To imply the combined endpoint, it must be paired with a bound for the removed local factor $W$. The correct minor size may depend on the input support and on the rank of the signed exponent relations.","\\section{Open theorem C: filtered local-global transfer}\nThe filtered contact construction proposes a map\n\\[\n\\mathbf J_c\n\\longrightarrow\n\\mathbf H_f\n\\]\nfrom the cusp-contact complex into a Hecke, Selmer, or deformation complex.","The required theorem has two parts:\n\\[\n\\abs{\\Ker H^0}\\le R^\\eps\n\\]\nand\n\\[\n\\abs{H^0(\\mathbf H_f)}\\le R_aR_bR^\\eps.\n\\]\nThe first is depth occupancy. The second is integral automorphic size. Existing modularity and $R=\\mathbb T$ theorems do not by themselves construct this filtered map or give these bounds.","\\section{Open theorem D: sparse motivic or Jacobi quotient}\nThe Kummer--Fermat and polylogarithmic constructions produce a full character bank and then a fixed-depth path quotient. The full bank has a line-bundle and synthesis cost of order $M$. The fixed complex packet is small but has an uncontrolled integral lattice.","\\section{Open theorem E: functorial gluing}\nA general prime-contact calculus should assign objects not only to individual relations but also to diagrams of relations. If\n\\[\nA+B=C\n\\]\nand\n\\[\nC+D=E,\n\\]\none should understand how the contact, Kummer, and coherence objects glue across the shared term $C$.","The full Fermat character bank & Its full-plane line bundle reintroduces a factor of order $M$.","Direct positive contact relations & Divisibility by $c$ forces height at least $c/2$.","\\midrule\n\\endhead\n$R=\\rad(abc)$ & Activated prime support of a primitive triple","$R_u=\\rad(u)$ & Radical of one term","$M=c/R_c$ & Output powerful part or contact modulus","$\\Theta$ & Coarse shortest transverse derivative packet","$\\Phi$ & Phase-refined shortest transverse packet. The same symbol denotes the shortest nonzero detector when the context is clear","$L$ & Additive packet lattice","$L_0$ & Wronskian-zero sublattice","$H(k)$ & Exact shortest height in obstruction coset $k$","$\\overline H$ & Stable real relaxation of the obstruction height","$B_d(n)$ & Exact-order divisor-polyphase bandwidth packet","$\\Jcal_c$ & Filtered cusp-contact module or complex","$\\Witt_c$ & Product of local truncated Witt contact crystals","$\\kappa_c$ & Global Kummer map carrying the contact packet","$\\Lcal_{P,Q}$ & Cross-prime logarithmic filter matrix","$\\epsilon_2$ & Weighted dyadic sign layer","$K$ & Common Kummer preload factor","$W$ & Local logarithmic or generalized Wieferich defect","$C$ & Cross-channel coherence quotient","$\\Lambda_{\\mathrm{coh}}$ & Kernel lattice of the coherence character","$\\chi_{\\mathrm{coh}}$ & Dual cyclic coherence character","\\section{Research priorities}\nThe most direct next problems are:\n\\begin{enumerate}[leftmargin=1.5em]\n\\item establish a simultaneous coherence bound for a nontrivial restricted family\n\\item classify high-quality triples by the exact factors $\\epsilon_2$, $K$, $W$, and $C$\n\\item develop cross-prime minor estimates for generalized Wieferich networks\n\\item apply contact filtering to selected generalized Fermat signatures\n\\item construct the Bernoulli--Witt contact object for Erd\\H{o}s--Moser-type equations\n\\end{enumerate}","\\[\n\\boxed{\n\\begin{aligned}\n\\text{first-order formulation: }&\\text{find a short point in a large lattice},\\\\\n\\text{arithmetic formulation: }&\\text{bound the conductor of a specific arithmetic character}\\\\\n&\\text{whose kernel contains the additive exponent packet}.\n\\end{aligned}\n}\n\\]"]}