Hilbert 12: analytic Fourier branch

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1 # An Ambient Cyclic Cubic and the Elliptic Gamma Fourier Obstruction
2 
3 **Author:** Komorebi P.
4 
5 **Scope:** The pure cubic Hilbert 12 test field and its fixed elliptic gamma specialization.
6 
7 **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**.
8 
9 ## Abstract
10 
11 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.
12 
13 ## 1. The fixed arithmetic and analytic data
14 
15 Let
16 
17 \[
18 \omega^2+\omega+1=0,\qquad \beta=7^{1/3}\omega^2,\qquad J=2+3\omega,
19 \qquad \overline J=-1-3\omega.
20 \]
21 
22 The chosen embedding is the nonreal one with negative imaginary part. Let
23 
24 \[
25 K=\mathbb Q(\beta),\qquad f(X)=X^3-\beta X^2+1.
26 \]
27 
28 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 \(1\).
29 
30 The fixed analytic periods and fifth-smoothed elliptic gamma are
31 
32 \[
33 \tau=\frac{\beta^2+2\beta}{15},\qquad
34 s=-\frac{\beta+2}{15},\qquad t=-\tau,
35 \]
36 
37 \[
38 F(z)=\frac{\Gamma_{\rm ell}(z;\tau,s)^5}{\Gamma_{\rm ell}(5z;5\tau,5s)},
39 \qquad U_\Gamma=F(1/3),\qquad Y_\Gamma=U_\Gamma+U_\Gamma^{-1}.
40 \]
41 
42 The previously constructed algebraic candidate satisfies \(Y_{\rm alg}=Av^2+Bv+C\) where
43 
44 \[
45 A=4\beta^2+\beta-14,\quad B=11\beta-23,\quad C=-\beta^2+3\beta-3.
46 \]
47 
48 The monic cubic polynomial of this coordinate is
49 
50 \[
51 R(Y)=Y^3+(6\beta^2-14\beta+2)Y^2
52  +(4\beta^2-6\beta-1)Y+94\beta^2-124\beta-107.
53 \]
54 
55 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.
56 
57 ## 2. Explicit projective Galois action
58 
59 **Theorem 1.** The matrix
60 
61 \[
62 G=\begin{pmatrix}5&-3\beta\\ \beta^2&-4\end{pmatrix}
63 \]
64 
65 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
66 
67 \[
68 \Phi(X)=\frac{5X-3\beta}{\beta^2X-4},\qquad
69 \Phi^2(X)=\frac{4X-3\beta}{\beta^2X-5},\qquad \Phi^3(X)=X.
70 \]
71 
72 For \(f(v)=0\), these are exactly the three cyclic conjugates. On these roots the same action has polynomial representatives
73 
74 \[
75 \Phi(v)=\beta^2v^2-3v-\beta,\qquad
76 \Phi^2(v)=-\beta^2v^2+2v+2\beta.
77 \]
78 
79 *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. \(\square\)
80 
81 ## 3. An ambient quotient coordinate and cyclic cubic pencil
82 
83 Define
84 
85 \[
86 \boxed{d(X)=\frac{f(X)}{(\beta^2X-4)(\beta^2X-5)}.}
87 \]
88 
89 **Theorem 2.** The coordinate \(d\) is invariant under \(\Phi\):
90 
91 \[
92 \boxed{d(\Phi(X))=d(X).}
93 \]
94 
95 For generic \(X\), the orbit \(X,\Phi(X),\Phi^2(X)\) has elementary symmetric functions
96 
97 \[
98 \begin{aligned}
99 e_1&=\beta(1+7d),\\
100 e_2&=9\beta^2d,\\
101 e_3&=-1+20d.
102 \end{aligned}
103 \]
104 
105 Consequently, the three orbit elements are roots of
106 
107 \[
108 \boxed{P_d(Z)=Z^3-\beta(1+7d)Z^2+9\beta^2dZ+1-20d.}
109 \]
110 
111 The discriminant is the exact perfect square
112 
113 \[
114 \boxed{\operatorname{Disc}_Z(P_d)=(49d^2-13d+1)^2.}
115 \]
116 
117 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.
118 
119 *Proof.* Work over \(K(X)\). Write \(D=(\beta^2X-4)(\beta^2X-5)\). Expanding the symmetric functions gives the three exact numerator identities
120 
121 \[
122 \begin{aligned}
123 D(e_1-\beta)&=7\beta f(X),\\
124 De_2&=9\beta^2 f(X),\\
125 D(e_3+1)&=20 f(X).
126 \end{aligned}
127 \]
128 
129 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. \(\square\)
130 
131 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.
132 
133 ## 4. Universal primitive Fourier identities
134 
135 Let
136 
137 \[
138 X_0=X,\quad X_1=\Phi(X),\quad X_2=\Phi^2(X),
139 \]
140 
141 \[
142 L=X_0+\omega X_1+\omega^2X_2,\qquad
143 L'=X_0+\omega^2X_1+\omega X_2.
144 \]
145 
146 Define \(q(d)=1-13d+49d^2\).
147 
148 **Theorem 3.** As identities of rational functions over \(K(\omega)\),
149 
150 \[
151 \boxed{LL'=\beta^2q(d),}
152 \]
153 
154 \[
155 \boxed{L^3=(J^2+49d)q(d),\qquad
156 (L')^3=(\overline J^{\,2}+49d)q(d).}
157 \]
158 
159 *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
160 
161 \[
162 L^3-(L')^3=3(\omega-\omega^2)q(d).
163 \]
164 
165 The universal cubic Fourier identity gives
166 
167 \[
168 L^3+(L')^3=27e_3-e_1^3+3e_1LL'=(98d-13)q(d).
169 \]
170 
171 Adding and subtracting yields the displayed cubes because
172 \(J^2=(-13+3(\omega-\omega^2))/2\) and
173 \(\overline J^{\,2}=(-13-3(\omega-\omega^2))/2\). \(\square\)
174 
175 At \(d=0\) the equations become exactly
176 
177 \[
178 LL'=\beta^2,\qquad L^3=J^2,\qquad (L')^3=\overline J^{\,2}.
179 \]
180 
181 But importantly the same symmetry and deformed Fourier laws exist for every generic \(d\). Therefore the cyclic symmetry alone **cannot** select the arithmetic fiber \(d=0\).
182 
183 ## 5. Canonical inverse to the earlier arithmetic coordinate
184 
185 Set
186 
187 \[
188 \begin{aligned}
189 D_Y(Y)&=A(Y-C)+B^2+\beta AB,\\
190 N_Y(Y)&=(Y-C)(B+\beta A)-A^2,\\
191 c&=3(652\beta^2-1230\beta-33).
192 \end{aligned}
193 \]
194 
195 **Theorem 4.** The following identity holds in \(K[Y]\):
196 
197 \[
198 \boxed{
199 N_Y(Y)^3-\beta N_Y(Y)^2D_Y(Y)+D_Y(Y)^3=cR(Y).
200 }
201 \]
202 
203 Moreover
204 
205 \[
206 \boxed{N_Y'(Y)D_Y(Y)-N_Y(Y)D_Y'(Y)=c\ne0.}
207 \]
208 
209 Hence \(h(Y)=N_Y(Y)/D_Y(Y)\) is a Möbius transformation. On the algebraic cubic it inverts \(Y=Av^2+Bv+C\).
210 
211 *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
212 
213 \[
214 A^2f(v)=D_Y(Y)v-N_Y(Y)
215 \]
216 
217 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. \(\square\)
218 
219 Consequently, the **actual** gamma function defines the meromorphic candidate
220 
221 \[
222 \boxed{
223 v_\Gamma(z)=\frac{N_Y(F(z)+F(z)^{-1})}{D_Y(F(z)+F(z)^{-1})}.
224 }
225 \]
226 
227 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 \(z\).
228 
229 Define its invariant
230 
231 \[
232 \boxed{d_\Gamma(z)=d(v_\Gamma(z)).}
233 \]
234 
235 Substitution of Theorem 4 gives the exact scalar meromorphic formula
236 
237 \[
238 \boxed{
239 d_\Gamma(z)=
240 \frac{c\,R(Y(z))}{D_Y(Y(z))\,[\beta^2N_Y(Y(z))-4D_Y(Y(z))]\,[\beta^2N_Y(Y(z))-5D_Y(Y(z))]},
241 }
242 \]
243 
244 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 \(R(Y(z))=0\).
245 
246 Since \(Q(U)=U^3R(U+U^{-1})\), this is exactly the original reciprocal-sextic membership target, now realized as an ambient cubic invariant.
247 
248 ## 6. A further reduction: two chart orbit laws plus one trace identity
249 
250 The transported order three action on \(Y\) is the exact Möbius map
251 
252 \[
253 \Psi=h^{-1}\circ\Phi\circ h,
254 \]
255 
256 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.
257 
258 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
259 
260 \[
261 \boxed{
262 Y_0+Y_1+Y_2
263 =-6\beta^2+14\beta-2
264 +\frac{\kappa d}{1+\theta d},
265 }
266 \]
267 
268 where
269 
270 \[
271 \kappa=-\frac{1300\beta^2-3486\beta+2723}{41},
272 \qquad
273 \theta=-\frac{14(2\beta^2+7\beta+45)}{41}.
274 \]
275 
276 Both formulas were checked by polynomial coefficient comparison. In particular \(\kappa\ne0\). For a finite orbit and nonzero denominator, the single trace normalization
277 
278 \[
279 \boxed{Y_0+Y_1+Y_2=-6\beta^2+14\beta-2}
280 \]
281 
282 forces \(d=0\), **provided the three values really belong to the same \(\Psi\) orbit**.
283 
284 Our three existing BCG gamma charts are
285 
286 \[
287 (t_0,s_0)=(t,s),\quad
288 (t_{10},s_{10})=((t-1)/10,(s+3)/10),\quad
289 (t_{23},s_{23})=((t-5)/23,(s-2)/23).
290 \]
291 
292 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
293 
294 \[
295 \boxed{Y_{23}=\Psi(Y_0),\qquad Y_{10}=\Psi^2(Y_0),}
296 \]
297 
298 and
299 
300 \[
301 \boxed{Y_0+Y_{10}+Y_{23}=-6\beta^2+14\beta-2.}
302 \]
303 
304 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.
305 
306 ## 7. Numerical experiment and independent negative control
307 
308 The verifier evaluates each smoothed gamma using the convergent, fixed-branch logarithmic series
309 
310 \[
311 \log F_{t,s}(z)=-5\sum_{\substack{n\ge1\\5\nmid n}}
312 \frac{e^{2\pi in(t+z)}-e^{2\pi in(s-z)}}
313 {n(1-e^{2\pi int})(1-e^{2\pi ins})}.
314 \]
315 
316 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
317 
318 \[
319 |\text{tail}|\le\frac{10r^{N+1}}{(N+1)(1-r)^2(1-q)}.
320 \]
321 
322 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.
323 
324 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 \(1.4\times10^{-42}\).
325 
326 At \(z=1/4\), the corresponding invariant magnitudes were approximately
327 
328 \[
329 0.1280579,\quad 0.1428319,\quad 0.1430370.
330 \]
331 
332 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.
333 
334 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.
335 
336 ## 8. An attempted theta route and its remaining obstruction
337 
338 The previous polyphase proof supplies the exact distribution identity
339 
340 \[
341 F_{(t-a)/n,(s+b)/n}(z)
342 =\prod_{u=1-n}^0\prod_{v=0}^{n-1}
343 F_{t,s}\left(z+u\frac{t-a}{n}+v\frac{s+b}{n}\right).
344 \]
345 
346 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.
347 
348 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.
349 
350 The present exact Möbius orbit and scalar trace reduction makes the next research task precise:
351 
352 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.
353 2. **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 \(d_\Gamma(1/3)=0\).
354 3. **Branch step:** Apply the already retained analytic enclosure or another rigorous branch selector to identify the intended root of the reciprocal sextic.
355 
356 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.
357 
358 ## 9. Verification boundaries
359 
360 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.
361 
362 ### Research foundations inspected
363 
364 - *Cubic Divisor Transport and Reconstruction*, Komorebi P., 8 October 2026.
365 - *An ambient dual slice for the pure cubic Hilbert 12 experiment*, Komorebi P.
366 - *Hilbert's Twelfth Problem: A Phase-Closing Attempt*, Komorebi P.
367 - *Komorebi Hilbert 12: Complete Markdown Working Archive*, 2 October 2026.
368 - *Holomorphic descent, complementary gerbe torsion, and the oriented Gauss character*, Komorebi P., 3 October 2026.
369 
370 The external literature and prior manuscripts used in those foundations are not rechecked as part of this new exact algebraic continuation.

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