An Ambient Cyclic Cubic and the Elliptic Gamma Fourier Obstruction
Author: Komorebi P.
Scope: The pure cubic Hilbert 12 test field and its fixed elliptic gamma specialization.
Status: The ambient cyclic cubic, rational Fourier identities, analytic candidate, and trace reduction are proved by elementary algebra. Exact symbolic checks and an independent numerical evaluation of the gamma series accompany the arguments. The elliptic gamma and class field equality is not proved.
Abstract
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 \(d_\Gamma(1/3)=0\) remains open.
1. The fixed arithmetic and analytic data
Let
The chosen embedding is the nonreal one with negative imaginary part. Let
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 \(1\).
The fixed analytic periods and fifth-smoothed elliptic gamma are
The previously constructed algebraic candidate satisfies \(Y_{\rm alg}=Av^2+Bv+C\) where
The monic cubic polynomial of this coordinate is
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.
2. Explicit projective Galois action
Theorem 1. The matrix
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
For \(f(v)=0\), these are exactly the three cyclic conjugates. On these roots the same action has polynomial representatives
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. \(\square\)
3. An ambient quotient coordinate and cyclic cubic pencil
Define
Theorem 2. The coordinate \(d\) is invariant under \(\Phi\):
For generic \(X\), the orbit \(X,\Phi(X),\Phi^2(X)\) has elementary symmetric functions
Consequently, the three orbit elements are roots of
The discriminant is the exact perfect square
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
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. \(\square\)
This is a realization of a classical order three Möbius quotient, not by itself a new Hilbert 12 solution or a claim of novelty for cyclic cubic theory.
4. Universal primitive Fourier identities
Let
Define \(q(d)=1-13d+49d^2\).
Theorem 3. As identities of rational functions over \(K(\omega)\),
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
The universal cubic Fourier identity gives
Adding and subtracting yields the displayed cubes because \(J^2=(-13+3(\omega-\omega^2))/2\) and \(\overline J^{\,2}=(-13-3(\omega-\omega^2))/2\). \(\square\)
At \(d=0\) the equations become exactly
But importantly the same symmetry and deformed Fourier laws exist for every generic \(d\). Therefore the cyclic symmetry alone cannot select the arithmetic fiber \(d=0\).
5. Canonical inverse to the earlier arithmetic coordinate
Set
Theorem 4. The following identity holds in \(K[Y]\):
Moreover
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
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. \(\square\)
Consequently, the actual gamma function defines the meromorphic candidate
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 \(z\).
Define its invariant
Substitution of Theorem 4 gives the exact scalar meromorphic formula
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.
6. A further reduction: two chart orbit laws plus one trace identity
The transported order three action on \(Y\) is the exact Möbius map
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
where
Both formulas were checked by polynomial coefficient comparison. In particular \(\kappa\ne0\). For a finite orbit and nonzero denominator, the single trace normalization
forces \(d=0\), provided the three values really belong to the same \(\Psi\) orbit.
Our three existing BCG gamma charts are
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
and
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.
7. Numerical experiment and independent negative control
The verifier evaluates each smoothed gamma using the convergent, fixed-branch logarithmic series
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
This is an analytic tail inequality. The implementation uses arbitrary-precision floating point rather than outward-rounded interval evaluation, so its displayed numeric errors are diagnostic, not certified enclosures.
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 \(1.4\times10^{-42}\).
At \(z=1/4\), the corresponding invariant magnitudes were approximately
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.
8. An attempted theta route and its remaining obstruction
The previous polyphase proof supplies the exact distribution identity
This puts the index 10 and index 23 gamma values inside one base function. Their finite grids are invariant under the cubic unit. The index 23 grid consists of one fixed label and a single orbit of length 528. However, evaluating that orbit through the gamma function requires the normalized analytic cocycle, including its theta carries and the scalar left over after descent.
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.
The present exact Möbius orbit and scalar trace reduction makes the next research task precise:
- 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.
- 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 \(d_\Gamma(1/3)=0\).
- 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.
9. Verification boundaries
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.
Research foundations inspected
- Cubic Divisor Transport and Reconstruction, Komorebi P., 8 October 2026.
- An ambient dual slice for the pure cubic Hilbert 12 experiment, Komorebi P.
- Hilbert's Twelfth Problem: A Phase-Closing Attempt, Komorebi P.
- Komorebi Hilbert 12: Complete Markdown Working Archive, 2 October 2026.
- Holomorphic descent, complementary gerbe torsion, and the oriented Gauss character, Komorebi P., 3 October 2026.
The external literature and prior manuscripts used in those foundations are not rechecked as part of this new exact algebraic continuation.