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zeta","display":"\\zeta","kind":"use","occurrences":[115,116,117,121,123]},{"id":"symbol-c7d778a23b2effd4","tex":"\\zeta^{a\\cdot x}","aliases":"ζ^a· x zeta cdot","display":"\\zeta^{a\\cdot x}","kind":"use","occurrences":[123]},{"id":"symbol-79a0be0094ccc021","tex":"\\zeta_7","aliases":"ζ_7 zeta","display":"\\zeta_7","kind":"use","occurrences":[115]},{"id":"symbol-893414d3dc2a0f92","tex":"\\zeta_7^{3(4k+5\\ell+j)}","aliases":"ζ_7^3(4k+5ℓ+j) zeta ell","display":"\\zeta_7^{3(4k+5\\ell+j)}","kind":"use","occurrences":[117]},{"id":"symbol-76738c7c39098112","tex":"\\zeta_7^{3k}","aliases":"ζ_7^3k zeta","display":"\\zeta_7^{3k}","kind":"use","occurrences":[116]},{"id":"symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","occurrences":[36,38,39,40,44,109]},{"id":"symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","occurrences":[8]},{"id":"symbol-c32bc1bd750febf9","tex":"\\theta_0","aliases":"θ_0 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ldots","display":"\\ldots","kind":"use","occurrences":[39]},{"id":"symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","occurrences":[38,39,45,49,51,52,54,55,56,57,63,72,73,109,116,117]},{"id":"symbol-8aaa13d7254b3cae","tex":"\\ell'","aliases":"ℓ' ell","display":"\\ell'","kind":"use","occurrences":[54,55]},{"id":"symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","occurrences":[84,85]},{"id":"symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"use","occurrences":[39,72,77,123,124,125]},{"id":"symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","occurrences":[81,87,91,111]},{"id":"symbol-7acbb14efa7a9854","tex":"\\prod_{(u,v,j)\\ne(0,0,0)}","aliases":"∏_(u,v,j)≠(0,0,0) prod ne","display":"\\prod_{(u,v,j)\\ne(0,0,0)}","kind":"use","occurrences":[111]},{"id":"symbol-0bafcdd9e08caf20","tex":"\\prod_{j=0}^{n-1}","aliases":"∏_j=0^n-1 prod","display":"\\prod_{j=0}^{n-1}","kind":"use","occurrences":[91]},{"id":"symbol-f5a40c774f37b582","tex":"\\prod_{m=1}^{\\infty}","aliases":"∏_m=1^∞ prod infty","display":"\\prod_{m=1}^{\\infty}","kind":"use","occurrences":[87]},{"id":"symbol-e70803265aa2607c","tex":"\\prod_{r=0}^{n-1}","aliases":"∏_r=0^n-1 prod","display":"\\prod_{r=0}^{n-1}","kind":"use","occurrences":[81]},{"id":"symbol-2ab6ac29f7239138","tex":"\\prod_{r=n}^{-1}","aliases":"∏_r=n^-1 prod","display":"\\prod_{r=n}^{-1}","kind":"use","occurrences":[81]},{"id":"symbol-d085977da2e32d60","tex":"\\prod_{u=1-n}^{0}","aliases":"∏_u=1-n^0 prod","display":"\\prod_{u=1-n}^{0}","kind":"use","occurrences":[91]},{"id":"symbol-f18de406d9a1804e","tex":"\\prod_{v=0}^{n-1}","aliases":"∏_v=0^n-1 prod","display":"\\prod_{v=0}^{n-1}","kind":"use","occurrences":[91]},{"id":"symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","occurrences":[89,97,98,123,125]},{"id":"symbol-965e6a0927763af5","tex":"\\sum_{b\\in V}","aliases":"∑_b∈ V sum in","display":"\\sum_{b\\in V}","kind":"use","occurrences":[125]},{"id":"symbol-99f2cf78baa9ddf5","tex":"\\sum_{\\substack{m\\ge1\\\\5\\nmid m}}","aliases":"∑_substackm≥1 5nmid m sum substack ge nmid","display":null,"kind":"use","occurrences":[89]},{"id":"symbol-886fc2f13956afc2","tex":"\\sum_{\\substack{q\\ge1\\\\5\\nmid nq}}","aliases":"∑_substackq≥1 5nmid nq sum substack ge nmid","display":null,"kind":"use","occurrences":[98]},{"id":"symbol-518929e214b8ff43","tex":"\\sum_{u=1-n}^{0}","aliases":"∑_u=1-n^0 sum","display":"\\sum_{u=1-n}^{0}","kind":"use","occurrences":[97]},{"id":"symbol-33fd4d4704aa1e77","tex":"\\sum_{v=0}^{n-1}","aliases":"∑_v=0^n-1 sum","display":"\\sum_{v=0}^{n-1}","kind":"use","occurrences":[97]},{"id":"symbol-9923075fc1158c85","tex":"\\sum_{x\\in V}","aliases":"∑_x∈ V sum in","display":"\\sum_{x\\in V}","kind":"use","occurrences":[123]},{"id":"symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","occurrences":[87]},{"id":"symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","occurrences":[9,18,24,111]},{"id":"symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","occurrences":[74,89,90,98]},{"id":"symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","occurrences":[31,32]},{"id":"symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","occurrences":[31,32]},{"id":"symbol-79956f9daec39523","tex":"\\rangle_H","aliases":"⟩_H rangle","display":"\\rangle_H","kind":"use","occurrences":[31,32]}],"contexts":["This research package continues the pure cubic experiment \\(K=\\mathbb Q(\\beta)\\), \\(\\beta^3=7\\).","\\[\n\\omega^2+\\omega+1=0,\\quad \\beta^3=7,\\quad\n\\varepsilon=2-\\beta,\\quad \\rho=\\varepsilon^{-1}=\\beta^2+2\\beta+4.\n\\]","Choose the complex embedding \\(\\beta=7^{1/3}\\omega^2\\), so \\(|\\rho|<1\\). Put","\\[\n\\tau=\\frac{\\beta^2+2\\beta}{15},\\qquad\n\\sigma=-\\frac{\\beta+2}{15},\\qquad t=-\\tau,\n\\quad z_0=\\frac13.\n\\]","\\[\nF(z)=\\frac{\\Gamma_{\\mathrm{ell}}(z;\\tau,\\sigma)^5}\n{\\Gamma_{\\mathrm{ell}}(5z;5\\tau,5\\sigma)},\\qquad U_\\Gamma=F(z_0).\n\\]","\\[\nF(z+1)=F(z),\\qquad\nF(z+\\tau)=q_\\sigma(z)F(z),\\qquad\nF(z+\\sigma)=q_t(z)^{-1}F(z),\n\\]","\\[\nq_a(z)=\\frac{\\theta_0(z;a)^5}{\\theta_0(5z;5a)}.\n\\]","The previously established zero-anchor normalization is \\(F(0)=\\varepsilon^2\\ne0\\). The divisor support is contained in\n\\(\\Lambda_5=\\mathbb Z\\tau+\\mathbb Z\\sigma+\\tfrac15\\mathbb Z\\). These analytic facts are inherited inputs, not freshly formalized in Lean here.","The arithmetic lattice is \\(\\Lambda=\\mathbb Z\\tau+\\mathbb Z\\sigma+\\mathbb Z\\). Multiplication by \\(\\varepsilon\\) on its ordered basis is","\\[\nM=\\begin{pmatrix}0&1&0\\\\-4&2&15\\\\-1&0&4\\end{pmatrix},\\qquad\nR=M^{-1}=\\begin{pmatrix}8&-4&15\\\\1&0&0\\\\2&-1&4\\end{pmatrix}.\n\\]","We use \\(R\\) for the integral matrix and \\(\\rho\\) for the corresponding scalar multiplication.","Let \\(N=M-2I\\pmod7\\). Then \\(N^3=0\\) and \\(N^2\\ne0\\). Define","\\[\nu=(1,2,4)^T,\\quad v=(0,1,2)^T,\\quad w=(0,0,1)^T.\n\\]","\\[\nNu=0,\\qquad Nv=u,\\qquad Nw=v.\n\\]","The matrix with columns \\(u,v,w\\) has determinant one. Consequently the three vectors form an integral basis.","\\[\nR=(2I+N)^{-1}=4I+5N+N^2.\n\\]","\\[\nR^3-I=2N+5N^2\\pmod7,\\qquad\nR^7=4I\\pmod7,\\qquad R^{21}=I\\pmod7.\n\\]","Because \\(R^3-I\\ne0\\), the order of \\(R\\) in \\(\\mathrm{GL}_3(\\mathbb F_7)\\) is exactly 21. Its one-dimensional eigenline has scalar action \\(4\\), of order three. Thus the order-three torsion packet is the semisimple slice of a nonsemisimple order-21 arithmetic operator.","\\[\nH=\\begin{pmatrix}4&-4&15\\\\-4&1&0\\\\15&0&0\\end{pmatrix},\\qquad\nM^T H=HM,\n\\]","and, in the ordered basis \\(u,v,w\\), its reduction is","\\[\n\\begin{pmatrix}0&0&1\\\\0&1&0\\\\1&0&0\\end{pmatrix}\\pmod7.\n\\]","The essential duality is \\(\\langle u,w\\rangle_H=1\\) while \\(\\langle u,u\\rangle_H=\\langle u,v\\rangle_H=0\\) modulo seven. A stable linear subspace containing \\(w\\) also contains \\(v=Nw\\) and \\(u=N^2w\\). Thus both the full unit action and this nontrivial socle-to-top pairing require the complete three-dimensional module.","\\[\n\\delta=\\frac{\\tau+2\\sigma+4}{7},\\qquad\n\\eta=\\frac{\\sigma+2}{7},\\qquad\n\\xi=\\frac17.\n\\]","The seven-point packet has arguments \\(z_0+k\\delta\\). The first refinement consists of \\(z_0+k\\delta+\\ell\\eta\\), with 49 labels. The full ambient packet is","\\[\n\\boxed{z_{k,\\ell,j}=z_0+k\\delta+\\ell\\eta+j\\xi,\\qquad\nk,\\ell,j\\in\\{0,\\ldots,6\\}.}\n\\]","Since \\(\\delta,\\eta,\\xi\\) are the images of \\(u/7,v/7,w/7\\), and \\(\\det[u\\ v\\ w]=1\\), the 343 labels represent all of \\(\\tfrac17\\Lambda/\\Lambda\\). Here this is torsion of the rank-three lattice quotient, not a claim that the nondiscrete rank-three quotient is an ordinary elliptic curve.","\\[\n\\begin{aligned}\n\\rho z_0&=z_0+5\\tau+1,\\\\\n\\rho\\delta&=4\\delta+8\\tau-\\sigma,\\\\\n\\rho\\eta&=5\\delta+4\\eta+3\\tau-2\\sigma-3,\\\\\n\\rho\\xi&=\\delta+5\\eta+4\\xi+2\\tau-\\sigma-2.\n\\end{aligned}\n\\]","\\[\n\\boxed{T(k,\\ell,j)=(4k+5\\ell+j,\\ 4\\ell+5j,\\ 4j)\\pmod7.}\n\\]","\\[\n\\boxed{343=1+2\\cdot3+16\\cdot21.}\n\\]","Restricting to \\(j=0\\) gives \\(49=1+2\\cdot3+2\\cdot21\\). Restricting again to \\(\\ell=j=0\\) gives \\(7=1+2\\cdot3\\). All orbit decompositions were enumerated exactly. The quotient action, not a heuristic spectral analogy, produces these numbers.","For \\((k,\\ell,j)\\), set","\\[\nA_0=4k+5\\ell+j,\\qquad B_0=4\\ell+5j,\\qquad C_0=4j,\n\\]","\\[\nq_1=\\lfloor A_0/7\\rfloor,\\quad\nq_2=\\lfloor B_0/7\\rfloor,\\quad\nq_3=\\lfloor C_0/7\\rfloor.\n\\]","Write \\((k',\\ell',j')=(A_0,B_0,C_0)\\bmod7\\). Then","\\[\n\\boxed{\n\\rho z_{k,\\ell,j}=\n z_{k',\\ell',j'}+a\\tau+b\\sigma+c\n}\n\\]","\\[\n\\begin{aligned}\na&=5+8k+3\\ell+2j+q_1,\\\\\nb&=-k-2\\ell-j+2q_1+q_2,\\\\\nc&=1-3\\ell-2j+4q_1+2q_2+q_3.\n\\end{aligned}\n\\]","No \\(z_{k,\\ell,j}\\) lies in \\(\\Lambda_5\\). To see this, compare the rationally independent coefficients of \\(\\tau,\\sigma,1\\). The coefficient of \\(\\tau\\) forces \\(k=0\\), that of \\(\\sigma\\) then forces \\(\\ell=0\\), while \\(1/3+j/7\\) is never in \\(\\tfrac15\\mathbb Z\\). The unit \\(\\rho\\) preserves \\(\\Lambda_5\\) since","\\[\n\\rho\\tau=8\\tau+\\sigma+2,\\qquad\n\\rho\\sigma=-4\\tau-1,\\qquad\n\\rho/5=3\\tau+4/5,\n\\]","and the inverse \\(\\varepsilon\\) also preserves \\(\\Lambda_5\\). Thus every contracting orbit avoids the divisor of \\(F\\).","Let \\(x=(k,\\ell,j)\\in V=(\\mathbb Z/7)^3\\) and \\(z_x=z_{k,\\ell,j}\\). Define","\\[\nf_x^{(m)}=F(\\rho^m z_x),\\qquad m\\ge0.\n\\]","Let \\(d_x=(a_x,b_x,c_x)^T\\) be the integer carry vector from Section 3. At stage \\(m\\), the carry is \\(d_{x,m}=R^{m-1}d_x\\in\\mathbb Z^3\\). Set \\(W_{x,m}=\\rho^{m-1}z_{T x}\\). By the arithmetic transport formula,","\\[\n\\rho^m z_x=W_{x,m}+a_{x,m}\\tau+b_{x,m}\\sigma+c_{x,m}.\n\\]","For an integer \\(n\\), define the oriented finite product","\\[\n\\mathcal P(g;n,w,h)=\n\\begin{cases}\n\\prod_{r=0}^{n-1}g(w+rh),&n>0,\\\\\n1,&n=0,\\\\\n\\prod_{r=n}^{-1}g(w+rh)^{-1},&n<0.\n\\end{cases}\n\\]","\\[\n\\boxed{\nC_x^{(m)}=\n\\mathcal P(q_t^{-1};b_{x,m},W_{x,m},\\sigma)\\,\n\\mathcal P(q_\\sigma;a_{x,m},W_{x,m}+b_{x,m}\\sigma,\\tau).\n}\n\\]","\\[\n\\boxed{f_x^{(m)}=C_x^{(m)} f_{T x}^{(m-1)}.}\n\\]","As \\(\\rho^m z_x\\to0\\), all \\(f_x^{(m)}\\to F(0)=\\varepsilon^2\\). Every multiplier is nonzero at the chosen arguments, and\n\\(C_x^{(m)}=1+O(|\\rho|^{m-1})\\).","\\[\n\\boxed{\nF(z_x)=\\varepsilon^2\n\\prod_{m=1}^{\\infty}\n\\left(C_{T^{-m}x}^{(m)}\\right)^{-1}.\n}\n\\]","This exact expression is branch-free: it uses only ordinary meromorphic theta quotients and their products. For the seven-point eigenline \\(T^{-m}k=2^m k\\bmod7\\), recovering the previous seven-packet identity. The full 343-point form also transports all unipotent directions.","\\[\nL_{t,\\sigma}(z)=-5\\sum_{\\substack{m\\ge1\\\\5\\nmid m}}\n\\frac{P^m e^{2\\pi imz}-Q^m e^{-2\\pi imz}}\n{m(1-P^m)(1-Q^m)},\\qquad\nP=e^{2\\pi it},\\ Q=e^{2\\pi i\\sigma}.\n\\]","For \\(n\\ge1\\), define the complete rank-three distribution","\\[\n\\mathcal N_n F(z)=\n\\prod_{u=1-n}^{0}\\prod_{v=0}^{n-1}\\prod_{j=0}^{n-1}\nF\\left(z+\\frac{ut+v\\sigma+j}{n}\\right).\n\\]","\\[\n\\boxed{\n\\mathcal N_nF(z)=\n\\begin{cases}F(nz),&5\\nmid n,\\\\1,&5\\mid n.\\end{cases}\n}\n\\]","**Proof.** Work first where all logarithmic series converge normally, which includes real \\(z\\) in the fixed upper half-plane conventions. Summing over \\(j\\) gives zero unless the Fourier index is divisible by \\(n\\). For Fourier index \\(nq\\), the two other finite geometric series yield","\\[\n\\sum_{u=1-n}^{0}P^{qu}\n=P^{-(n-1)q}\\frac{1-P^{nq}}{1-P^q},\\qquad\n\\sum_{v=0}^{n-1}Q^{qv}\n=\\frac{1-Q^{nq}}{1-Q^q}.\n\\]","\\[\n-5\\sum_{\\substack{q\\ge1\\\\5\\nmid nq}}\n\\frac{P^q e^{2\\pi i nq z}-Q^q e^{-2\\pi i nq z}}\n{q(1-P^q)(1-Q^q)}.\n\\]","When \\(5\\nmid n\\), the condition \\(5\\nmid nq\\) is exactly \\(5\\nmid q\\), giving \\(L_{t,\\sigma}(nz)\\). When \\(5\\mid n\\), no terms remain and the logarithm is zero. Exponentiating proves the equality in the convergence region. Meromorphic continuation gives the stated identities. \\(\\square\\)","At \\(n=7\\) and \\(z_0=1/3\\), period-one invariance gives","\\[\n\\boxed{\\mathcal N_7F(1/3)=F(7/3)=F(1/3)=U_\\Gamma.}\n\\]","The chosen shifts \\((ut+v\\sigma+j)/7\\) represent the same full 343-element torsion module as \\(k\\delta+\\ell\\eta+j\\xi\\), but their selected representatives differ by integer periods, which introduces explicit theta factors. In the canonical distribution representatives the fixed, zero-shift factor is \\(U_\\Gamma\\). Removing it gives the exact 342-factor identity","\\[\n\\boxed{\n\\prod_{(u,v,j)\\ne(0,0,0)}\nF\\left(\\frac13+\\frac{ut+v\\sigma+j}{7}\\right)=1.\n}\n\\]","**Interpretation.** The untwisted complete norm cannot determine the arithmetic phase of \\(U_\\Gamma\\), because that factor cancels. Extra finite torsion alone does not remove the need for an oriented or character-sensitive comparison.","In the integral Jordan basis \\(u,v,w\\), the perfect codifferent pairing modulo seven has matrix","\\[\nG_H=\\begin{pmatrix}0&0&1\\\\0&1&0\\\\1&0&0\\end{pmatrix}.\n\\]","Fix the oriented seventh-root phase \\(\\zeta_7=e^{2\\pi i/7}\\). The additive character against the top dual direction is","\\[\n\\boxed{\\psi_w(k,\\ell,j)=\\zeta_7^{3k}.}\n\\]","\\[\n\\boxed{\\psi_w(T(k,\\ell,j))=\\zeta_7^{3(4k+5\\ell+j)}.}\n\\]","A successful next theorem would compute a character-sensitive, correctly based analytic transport and prove that its arithmetic evaluation agrees with the oriented Gauss/Jacobi coordinate \\(J=2+3\\omega\\). The fixed desired scalar identity remains","\\[\n\\operatorname{Im}(C_{\\mathrm{pol}}U_\\Gamma^3)=0\n\\]","Let \\(V=(\\mathbb Z/7)^3\\), \\(\\zeta=\\exp(2\\pi i/7)\\), and let \\(\\mathcal F\\) be the ordinary finite Fourier transform","\\[\n\\widehat f(a)=\\sum_{x\\in V}\\zeta^{a\\cdot x}f(x),\\qquad a\\in V.\n\\]","\\[\nf_m(x)=C_m(x)f_{m-1}(Tx),\\qquad x\\in V.\n\\]","\\[\n\\boxed{\n\\widehat f_m(a)=\\frac1{343}\\sum_{b\\in V}\n\\widehat C_m(a-b)\\,\\widehat f_{m-1}(T^{-T}b).\n}\n\\]","In the basis \\(u,v,w\\), the dual transport matrix is","\\[\nT^{-T}=\\begin{pmatrix}2&0&0\\\\1&2&0\\\\0&1&2\\end{pmatrix}\\pmod7.\n\\]","The covector \\(a_0=(1,0,0)^T\\), corresponding to the oriented complementary pairing with \\(w\\), has an orbit of length **21** under this matrix. Its first three iterates are","\\[\n(2,1,0)^T,\\qquad (4,4,1)^T,\\qquad(1,5,6)^T.\n\\]","An independent numerical distribution test covers \\(n=2,3,5,7\\) with 38-digit arithmetic and finite truncation. The absolute logarithmic residual in the n=7 test was approximately \\(10^{-37}\\). These are regression tests and not directed interval certificates. The multiplication theorem itself is the exact Fourier-series proof above."]}