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Psi","display":"\\Psi","kind":"use","occurrences":[100,104,107,108,115,119,121,150,151]},{"id":"symbol-06351a0b74281088","tex":"\\Psi^2","aliases":"Ψ^2 Psi","display":"\\Psi^2","kind":"use","occurrences":[108,119,151]},{"id":"symbol-0f5d7c21634eb0e9","tex":"\\Psi^3","aliases":"Ψ^3 Psi","display":"\\Psi^3","kind":"use","occurrences":[121]},{"id":"symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","occurrences":[10,55,57,64,66,67,157,159]},{"id":"symbol-f2f53155f7e59bd3","tex":"\\omega^2","aliases":"ω^2 omega","display":"\\omega^2","kind":"use","occurrences":[10,55,64,66,67,159]},{"id":"symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","occurrences":[126]},{"id":"symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","occurrences":[147]},{"id":"symbol-f8468cc287609c57","tex":"\\prod_{u=1-n}^0","aliases":"∏_u=1-n^0 prod","display":"\\prod_{u=1-n}^0","kind":"use","occurrences":[147]},{"id":"symbol-f18de406d9a1804e","tex":"\\prod_{v=0}^{n-1}","aliases":"∏_v=0^n-1 prod","display":"\\prod_{v=0}^{n-1}","kind":"use","occurrences":[147]},{"id":"symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","occurrences":[123]},{"id":"symbol-0db207badc6ca948","tex":"\\sum_{\\substack{n\\ge1\\\\5\\nmid n}}","aliases":"∑_substackn≥1 5nmid n sum substack ge nmid","display":null,"kind":"use","occurrences":[123]},{"id":"symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","occurrences":[100]},{"id":"symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","occurrences":[76,112]},{"id":"symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","occurrences":[128]},{"id":"symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","occurrences":[123]}],"contexts":["For the number field \\(K=\\mathbb Q(\\beta)\\), \\(\\beta^3=7\\), the previously constructed unramified cubic is generated by a root of \\(f(X)=X^3-\\beta X^2+1\\). Its cyclic Galois action extends to a globally defined order three Möbius transformation on \\(\\mathbb P^1_K\\). The quotient coordinate \\(d(X)\\) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at \\(d=0\\) is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deformed norm and cube formulas. A rational inverse of the earlier quadratic arithmetic coordinate then turns the elliptic gamma function into a canonical meromorphic candidate for the cubic genera","For the number field \\(K=\\mathbb Q(\\beta)\\), \\(\\beta^3=7\\), the previously constructed unramified cubic is generated by a root of \\(f(X)=X^3-\\beta X^2+1\\). Its cyclic Galois action extends to a globally defined order three Möbius transformation on \\(\\mathbb P^1_K\\). The quotient coordinate \\(d(X)\\) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at \\(d=0\\) is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deformed norm and cube formulas. A rational inverse of the earlier quadratic arithmetic coordinate then turns the elliptic gamma function into a canonical meromorphic candidate for the cubic generator and supplie","For the number field \\(K=\\mathbb Q(\\beta)\\), \\(\\beta^3=7\\), the previously constructed unramified cubic is generated by a root of \\(f(X)=X^3-\\beta X^2+1\\). Its cyclic Galois action extends to a globally defined order three Möbius transformation on \\(\\mathbb P^1_K\\). The quotient coordinate \\(d(X)\\) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at \\(d=0\\) is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deformed norm and cube formulas. A rational inverse of the earlier quadratic arithmetic coordinate then turns the elliptic gamma function into a canonical meromorphic candidate for the cubic generator and supplies a single scalar analytic obstruction \\(d_\\Gamma(z)\\). The three existing gamma charts agree nu","For the number field \\(K=\\mathbb Q(\\beta)\\), \\(\\beta^3=7\\), the previously constructed unramified cubic is generated by a root of \\(f(X)=X^3-\\beta X^2+1\\). Its cyclic Galois action extends to a globally defined order three Möbius transformation on \\(\\mathbb P^1_K\\). The quotient coordinate \\(d(X)\\) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at \\(d=0\\) is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deformed norm and cube formulas. A rational inverse of the earlier quadratic arithmetic coordinate then turns the elliptic gamma function into a canonical meromorphic candidate for the cubic generator and supplies a single scalar analytic obstruction \\(d_\\Gamma(z)\\). The three existing gamma charts agree numerically with the expected Möbius orbit at \\(z=1/3\\), but not at \\(z=1/4\\). Proving the required theta or gerb","For the number field \\(K=\\mathbb Q(\\beta)\\), \\(\\beta^3=7\\), the previously constructed unramified cubic is generated by a root of \\(f(X)=X^3-\\beta X^2+1\\). Its cyclic Galois action extends to a globally defined order three Möbius transformation on \\(\\mathbb P^1_K\\). The quotient coordinate \\(d(X)\\) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at \\(d=0\\) is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deformed norm and cube formulas. A rational inverse of the earlier quadratic arithmetic coordinate then turns the elliptic gamma function into a canonical meromorphic candidate for the cubic generator and supplies a single scalar analytic obstruction \\(d_\\Gamma(z)\\). The three existing gamma charts agree numerically with the expected Möbius orbit at \\(z=1/3\\), but not at \\(z=1/4\\). Proving the required theta or gerbe normalization that forces \\(d_\\G","Q(\\beta)\\), \\(\\beta^3=7\\), the previously constructed unramified cubic is generated by a root of \\(f(X)=X^3-\\beta X^2+1\\). Its cyclic Galois action extends to a globally defined order three Möbius transformation on \\(\\mathbb P^1_K\\). The quotient coordinate \\(d(X)\\) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at \\(d=0\\) is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deformed norm and cube formulas. A rational inverse of the earlier quadratic arithmetic coordinate then turns the elliptic gamma function into a canonical meromorphic candidate for the cubic generator and supplies a single scalar analytic obstruction \\(d_\\Gamma(z)\\). The three existing gamma charts agree numerically with the expected Möbius orbit at \\(z=1/3\\), but not at \\(z=1/4\\). Proving the required theta or gerbe normalization that forces \\(d_\\Gamma(1/3)=0\\) remains open.","ect squares. Its special fiber at \\(d=0\\) is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deformed norm and cube formulas. A rational inverse of the earlier quadratic arithmetic coordinate then turns the elliptic gamma function into a canonical meromorphic candidate for the cubic generator and supplies a single scalar analytic obstruction \\(d_\\Gamma(z)\\). The three existing gamma charts agree numerically with the expected Möbius orbit at \\(z=1/3\\), but not at \\(z=1/4\\). Proving the required theta or gerbe normalization that forces \\(d_\\Gamma(1/3)=0\\) remains open.","ive Fourier resolvents have exact, explicitly deformed norm and cube formulas. A rational inverse of the earlier quadratic arithmetic coordinate then turns the elliptic gamma function into a canonical meromorphic candidate for the cubic generator and supplies a single scalar analytic obstruction \\(d_\\Gamma(z)\\). The three existing gamma charts agree numerically with the expected Möbius orbit at \\(z=1/3\\), but not at \\(z=1/4\\). Proving the required theta or gerbe normalization that forces \\(d_\\Gamma(1/3)=0\\) remains open.","have exact, explicitly deformed norm and cube formulas. A rational inverse of the earlier quadratic arithmetic coordinate then turns the elliptic gamma function into a canonical meromorphic candidate for the cubic generator and supplies a single scalar analytic obstruction \\(d_\\Gamma(z)\\). The three existing gamma charts agree numerically with the expected Möbius orbit at \\(z=1/3\\), but not at \\(z=1/4\\). Proving the required theta or gerbe normalization that forces \\(d_\\Gamma(1/3)=0\\) remains open.","se of the earlier quadratic arithmetic coordinate then turns the elliptic gamma function into a canonical meromorphic candidate for the cubic generator and supplies a single scalar analytic obstruction \\(d_\\Gamma(z)\\). The three existing gamma charts agree numerically with the expected Möbius orbit at \\(z=1/3\\), but not at \\(z=1/4\\). Proving the required theta or gerbe normalization that forces \\(d_\\Gamma(1/3)=0\\) remains open.","\\[\n\\omega^2+\\omega+1=0,\\qquad \\beta=7^{1/3}\\omega^2,\\qquad J=2+3\\omega,\n\\qquad \\overline J=-1-3\\omega.\n\\]","\\[\nK=\\mathbb Q(\\beta),\\qquad f(X)=X^3-\\beta X^2+1.\n\\]","The existing class field construction supplies an irreducible, everywhere unramified cyclic cubic model generated by a root \\(v\\) of \\(f\\). Its field and ramification properties are inherited from the earlier archive and are not re-proved here. The polynomial discriminant is \\(1\\).","\\[\n\\tau=\\frac{\\beta^2+2\\beta}{15},\\qquad\ns=-\\frac{\\beta+2}{15},\\qquad t=-\\tau,\n\\]","\\[\nF(z)=\\frac{\\Gamma_{\\rm ell}(z;\\tau,s)^5}{\\Gamma_{\\rm ell}(5z;5\\tau,5s)},\n\\qquad U_\\Gamma=F(1/3),\\qquad Y_\\Gamma=U_\\Gamma+U_\\Gamma^{-1}.\n\\]","The previously constructed algebraic candidate satisfies \\(Y_{\\rm alg}=Av^2+Bv+C\\) where","\\[\nA=4\\beta^2+\\beta-14,\\quad B=11\\beta-23,\\quad C=-\\beta^2+3\\beta-3.\n\\]","\\[\nR(Y)=Y^3+(6\\beta^2-14\\beta+2)Y^2\n +(4\\beta^2-6\\beta-1)Y+94\\beta^2-124\\beta-107.\n\\]","Thus the original analytic membership problem is \\(R(Y_\\Gamma)=0\\) together with the established branch selection. The existing modulus theorem does not establish this membership.","\\[\nG=\\begin{pmatrix}5&-3\\beta\\\\ \\beta^2&-4\\end{pmatrix}\n\\]","has determinant one and trace one. In particular \\(G^2-G+I=0\\) and \\(G^3=-I\\). Its action on \\(\\mathbb P^1_K\\) is","\\[\n\\Phi(X)=\\frac{5X-3\\beta}{\\beta^2X-4},\\qquad\n\\Phi^2(X)=\\frac{4X-3\\beta}{\\beta^2X-5},\\qquad \\Phi^3(X)=X.\n\\]","For \\(f(v)=0\\), these are exactly the three cyclic conjugates. On these roots the same action has polynomial representatives","\\[\n\\Phi(v)=\\beta^2v^2-3v-\\beta,\\qquad\n\\Phi^2(v)=-\\beta^2v^2+2v+2\\beta.\n\\]","*Proof.* Direct calculation gives \\(\\det G= -20+3\\beta^3=1\\) and \\(\\operatorname{tr}G=1\\). Cayley–Hamilton proves projective order three. Multiplying each fractional formula by its denominator and reducing modulo \\(v^3-\\beta v^2+1\\) proves the polynomial expressions. The three roots are distinct because the discriminant is one. The nontrivial maps cyclically permute them. \\(\\square\\)","\\[\n\\boxed{d(X)=\\frac{f(X)}{(\\beta^2X-4)(\\beta^2X-5)}.}\n\\]","**Theorem 2.** The coordinate \\(d\\) is invariant under \\(\\Phi\\):","\\[\n\\boxed{d(\\Phi(X))=d(X).}\n\\]","For generic \\(X\\), the orbit \\(X,\\Phi(X),\\Phi^2(X)\\) has elementary symmetric functions","\\[\n\\begin{aligned}\ne_1&=\\beta(1+7d),\\\\\ne_2&=9\\beta^2d,\\\\\ne_3&=-1+20d.\n\\end{aligned}\n\\]","\\[\n\\boxed{P_d(Z)=Z^3-\\beta(1+7d)Z^2+9\\beta^2dZ+1-20d.}\n\\]","\\[\n\\boxed{\\operatorname{Disc}_Z(P_d)=(49d^2-13d+1)^2.}\n\\]","The rational function \\(d(X)\\) has degree three, hence \\([K(X):K(d)]=3\\). The three distinct iterates of \\(\\Phi\\) give the full automorphism group, so the generic function field extension \\(K(X)/K(d)\\) is cyclic Galois of degree three. Its fiber at \\(d=0\\) is precisely \\(f(Z)=0\\), the existing arithmetic cubic.","*Proof.* Work over \\(K(X)\\). Write \\(D=(\\beta^2X-4)(\\beta^2X-5)\\). Expanding the symmetric functions gives the three exact numerator identities","\\[\n\\begin{aligned}\nD(e_1-\\beta)&=7\\beta f(X),\\\\\nDe_2&=9\\beta^2 f(X),\\\\\nD(e_3+1)&=20 f(X).\n\\end{aligned}\n\\]","They imply the asserted polynomial. Since its root set is the orbit, applying \\(\\Phi\\) leaves \\(d\\) unchanged. The discriminant follows by the standard cubic formula using \\(\\beta^3=7\\). The degree of the quotient function proves the field statement. \\(\\square\\)","\\[\nX_0=X,\\quad X_1=\\Phi(X),\\quad X_2=\\Phi^2(X),\n\\]","\\[\nL=X_0+\\omega X_1+\\omega^2X_2,\\qquad\nL'=X_0+\\omega^2X_1+\\omega X_2.\n\\]","Define \\(q(d)=1-13d+49d^2\\).","**Theorem 3.** As identities of rational functions over \\(K(\\omega)\\),","\\[\n\\boxed{LL'=\\beta^2q(d),}\n\\]","\\[\n\\boxed{L^3=(J^2+49d)q(d),\\qquad\n(L')^3=(\\overline J^{\\,2}+49d)q(d).}\n\\]","*Proof.* The norm identity follows from \\(LL'=e_1^2-3e_2\\). The discriminant gives the square of the Vandermonde. Its orientation is fixed by the root ordering \\(X,\\Phi(X),\\Phi^2(X)\\), which at \\(d=0\\) has Vandermonde \\(+1\\). Hence","\\[\nL^3-(L')^3=3(\\omega-\\omega^2)q(d).\n\\]","\\[\nL^3+(L')^3=27e_3-e_1^3+3e_1LL'=(98d-13)q(d).\n\\]","Adding and subtracting yields the displayed cubes because\n\\(J^2=(-13+3(\\omega-\\omega^2))/2\\) and\n\\(\\overline J^{\\,2}=(-13-3(\\omega-\\omega^2))/2\\). \\(\\square\\)","At \\(d=0\\) the equations become exactly","\\[\nLL'=\\beta^2,\\qquad L^3=J^2,\\qquad (L')^3=\\overline J^{\\,2}.\n\\]","But importantly the same symmetry and deformed Fourier laws exist for every generic \\(d\\). Therefore the cyclic symmetry alone **cannot** select the arithmetic fiber \\(d=0\\).","\\[\n\\begin{aligned}\nD_Y(Y)&=A(Y-C)+B^2+\\beta AB,\\\\\nN_Y(Y)&=(Y-C)(B+\\beta A)-A^2,\\\\\nc&=3(652\\beta^2-1230\\beta-33).\n\\end{aligned}\n\\]","**Theorem 4.** The following identity holds in \\(K[Y]\\):","\\[\n\\boxed{\nN_Y(Y)^3-\\beta N_Y(Y)^2D_Y(Y)+D_Y(Y)^3=cR(Y).\n}\n\\]","\\[\n\\boxed{N_Y'(Y)D_Y(Y)-N_Y(Y)D_Y'(Y)=c\\ne0.}\n\\]","Hence \\(h(Y)=N_Y(Y)/D_Y(Y)\\) is a Möbius transformation. On the algebraic cubic it inverts \\(Y=Av^2+Bv+C\\).","*Proof.* The elimination equation \\(Av^2+Bv+C=Y\\) gives \\(Av^2=(Y-C)-Bv\\). Substitution into \\(v^3-\\beta v^2+1\\) yields the exact polynomial identity","\\[\nA^2f(v)=D_Y(Y)v-N_Y(Y)\n\\]","whenever \\(Y=Av^2+Bv+C\\). The cubic polynomial identity and determinant follow by coefficient comparison over \\(K\\) using \\(\\beta^3=7\\), as independently checked by the attached verifier. Nonzero \\(c\\) follows from the irreducibility of \\(\\beta^3-7\\), since \\(c\\) is represented by a nonzero polynomial of degree at most two. \\(\\square\\)","\\[\n\\boxed{\nv_\\Gamma(z)=\\frac{N_Y(F(z)+F(z)^{-1})}{D_Y(F(z)+F(z)^{-1})}.\n}\n\\]","This is a rational construction in \\(F(z)\\), with no numeric root fitting or cubic-branch choice. There is no assertion that it lies in the desired number field for arbitrary \\(z\\).","\\[\n\\boxed{d_\\Gamma(z)=d(v_\\Gamma(z)).}\n\\]","\\[\n\\boxed{\nd_\\Gamma(z)=\n\\frac{c\\,R(Y(z))}{D_Y(Y(z))\\,[\\beta^2N_Y(Y(z))-4D_Y(Y(z))]\\,[\\beta^2N_Y(Y(z))-5D_Y(Y(z))]},\n}\n\\]","where \\(Y(z)=F(z)+F(z)^{-1}\\). Every denominator is interpreted meromorphically. At a regular point with nonzero denominator, \\(d_\\Gamma(z)=0\\) if and only if \\(R(Y(z))=0\\).","Since \\(Q(U)=U^3R(U+U^{-1})\\), this is exactly the original reciprocal-sextic membership target, now realized as an ambient cubic invariant.","The transported order three action on \\(Y\\) is the exact Möbius map","\\[\n\\Psi=h^{-1}\\circ\\Phi\\circ h,\n\\]","where \\(h(Y)=N_Y(Y)/D_Y(Y)\\). If \\(M_h\\) is its \\(2\\times2\\) coefficient matrix, then the matrix for \\(\\Psi\\) is \\(M_h^{-1}GM_h\\). Its determinant is one and trace is one, so it also has order three.","Let \\(Y_0=Y\\), \\(Y_1=\\Psi(Y)\\), \\(Y_2=\\Psi^2(Y)\\). Their trace is a rational function of the invariant \\(d=d(h(Y))\\). Exact elimination gives","\\[\n\\boxed{\nY_0+Y_1+Y_2\n=-6\\beta^2+14\\beta-2\n+\\frac{\\kappa d}{1+\\theta d},\n}\n\\]","\\[\n\\kappa=-\\frac{1300\\beta^2-3486\\beta+2723}{41},\n\\qquad\n\\theta=-\\frac{14(2\\beta^2+7\\beta+45)}{41}.\n\\]","Both formulas were checked by polynomial coefficient comparison. In particular \\(\\kappa\\ne0\\). For a finite orbit and nonzero denominator, the single trace normalization","\\[\n\\boxed{Y_0+Y_1+Y_2=-6\\beta^2+14\\beta-2}\n\\]","forces \\(d=0\\), **provided the three values really belong to the same \\(\\Psi\\) orbit**.","\\[\n(t_0,s_0)=(t,s),\\quad\n(t_{10},s_{10})=((t-1)/10,(s+3)/10),\\quad\n(t_{23},s_{23})=((t-5)/23,(s-2)/23).\n\\]","Define \\(U_n=F_{t_n,s_n}(1/3)\\) and \\(Y_n=U_n+U_n^{-1}\\). The exact sufficient analytic obligations now are","\\[\n\\boxed{Y_{23}=\\Psi(Y_0),\\qquad Y_{10}=\\Psi^2(Y_0),}\n\\]","\\[\n\\boxed{Y_0+Y_{10}+Y_{23}=-6\\beta^2+14\\beta-2.}\n\\]","The orbit laws are **conjectural at the analytic gamma special values**, even though \\(\\Psi^3=1\\) is an exact algebraic theorem. If all three special-value identities hold, the established branch separation identifies \\(U_\\Gamma=U_{\\mathrm{alg}}\\). The nontrivial analytical content has not been proved by the algebraic construction.","\\[\n\\log F_{t,s}(z)=-5\\sum_{\\substack{n\\ge1\\\\5\\nmid n}}\n\\frac{e^{2\\pi in(t+z)}-e^{2\\pi in(s-z)}}\n{n(1-e^{2\\pi int})(1-e^{2\\pi ins})}.\n\\]","For real \\(z\\), writing \\(r=\\max(|e^{2\\pi it}|,|e^{2\\pi is}|)\\) and \\(q=\\min(\\ldots)\\), the tail after \\(N\\) terms is bounded by","\\[\n|\\text{tail}|\\le\\frac{10r^{N+1}}{(N+1)(1-r)^2(1-q)}.\n\\]","The fresh 64-digit run used \\(N=4000,19000,34000\\) for charts \\(1,10,23\\). At \\(z=1/3\\), the observed \\(|d_\\Gamma|\\) for the three charts was approximately \\(3\\times10^{-64}\\), \\(1.6\\times10^{-55}\\), and \\(2.3\\times10^{-43}\\). The last two are smaller than the conservative bounds on the corresponding gamma logarithm tails, and must not be interpreted as certified agreement to 43 or 55 places. The gamma chart orbit discrepancies were around \\(8\\times10^{-44}\\) and \\(4\\times10^{-56}\\) numerically, again without an interval certificate. The three-chart \\(Y\\)-trace discrepancy was about \\(1.4\\times10^{-42}\\).","00\\) for charts \\(1,10,23\\). At \\(z=1/3\\), the observed \\(|d_\\Gamma|\\) for the three charts was approximately \\(3\\times10^{-64}\\), \\(1.6\\times10^{-55}\\), and \\(2.3\\times10^{-43}\\). The last two are smaller than the conservative bounds on the corresponding gamma logarithm tails, and must not be interpreted as certified agreement to 43 or 55 places. The gamma chart orbit discrepancies were around \\(8\\times10^{-44}\\) and \\(4\\times10^{-56}\\) numerically, again without an interval certificate. The three-chart \\(Y\\)-trace discrepancy was about \\(1.4\\times10^{-42}\\).","3\\). At \\(z=1/3\\), the observed \\(|d_\\Gamma|\\) for the three charts was approximately \\(3\\times10^{-64}\\), \\(1.6\\times10^{-55}\\), and \\(2.3\\times10^{-43}\\). The last two are smaller than the conservative bounds on the corresponding gamma logarithm tails, and must not be interpreted as certified agreement to 43 or 55 places. The gamma chart orbit discrepancies were around \\(8\\times10^{-44}\\) and \\(4\\times10^{-56}\\) numerically, again without an interval certificate. The three-chart \\(Y\\)-trace discrepancy was about \\(1.4\\times10^{-42}\\).","3\\times10^{-64}\\), \\(1.6\\times10^{-55}\\), and \\(2.3\\times10^{-43}\\). The last two are smaller than the conservative bounds on the corresponding gamma logarithm tails, and must not be interpreted as certified agreement to 43 or 55 places. The gamma chart orbit discrepancies were around \\(8\\times10^{-44}\\) and \\(4\\times10^{-56}\\) numerically, again without an interval certificate. The three-chart \\(Y\\)-trace discrepancy was about \\(1.4\\times10^{-42}\\).","-55}\\), and \\(2.3\\times10^{-43}\\). The last two are smaller than the conservative bounds on the corresponding gamma logarithm tails, and must not be interpreted as certified agreement to 43 or 55 places. The gamma chart orbit discrepancies were around \\(8\\times10^{-44}\\) and \\(4\\times10^{-56}\\) numerically, again without an interval certificate. The three-chart \\(Y\\)-trace discrepancy was about \\(1.4\\times10^{-42}\\).","At \\(z=1/4\\), the corresponding invariant magnitudes were approximately","\\[\n0.1280579,\\quad 0.1428319,\\quad 0.1430370.\n\\]","The chart-orbit discrepancies were approximately \\(0.3429346\\) and \\(0.2098520\\). Thus the proposed exact equalities are **not generic functional identities** at arbitrary \\(z\\). They require the distinguished conductor-three special point.","The first high-precision numeric pass incorrectly formed \\(1/3\\) at the default 15-digit precision, producing spurious \\(10^{-15}\\)-scale residuals. This error was corrected by creating the rational evaluation point only *after* setting the precision. The accepted verification log comes from the corrected run. All exact algebraic checks are independent of this issue.","\\[\nF_{(t-a)/n,(s+b)/n}(z)\n=\\prod_{u=1-n}^0\\prod_{v=0}^{n-1}\nF_{t,s}\\left(z+u\\frac{t-a}{n}+v\\frac{s+b}{n}\\right).\n\\]","The untwisted principal seven norm alone does not determine \\(F(1/3)\\). At the special point, its factor \\(U_\\Gamma\\) cancels from both sides, leaving an exact identity among six shifted gamma values and theta products. This earlier negative control rules out treating the norm as a completed scalar comparison.","1. **Analytic orbit step:** Derive the identities \\(Y_{23}=\\Psi(Y_0)\\) and \\(Y_{10}=\\Psi^2(Y_0)\\) by a properly normalized transport, if true.\n2. **Scalar normalization step:** Independently prove the trace of the coherent analytic orbit is \\(-6\\beta^2+14\\beta-2\\). Under the orbit step, this forces \\(d_\\Gamma(1/3)=0\\).\n3. **Branch step:** Apply the already retained analytic enclosure or another rigorous branch selector to identify the intended root of the reciprocal sextic.","This identifies an obstruction rather than suppressing it. Every generic fiber \\(d\\) carries the same abstract \\(C_3\\) symmetry. Symmetry, perfect-square discriminant, and Fourier diagonalization alone cannot force \\(d=0\\). The missing input must be analytic and arithmetic normalization, not another abstract finite-group identity.","The attached Python verifier checks the exact rational function identities by polynomial reduction over \\(\\mathbb Q[\\beta,\\omega]\\) with \\(\\beta^3=7\\) and \\(\\omega^2+\\omega+1=0\\). Its numerical gamma evaluation is separate and diagnostic. No Lean source was compiled for this continuation. No numerical residual was treated as a proof of exact algebraicity. The result does not solve Hilbert's twelfth problem or establish the conjectured elliptic gamma special-value equality."]}