Hodge: source and evidence

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1 
2 
3 \addcontentsline{toc}{part}{Part I. The conjecture as a projector problem}
4 
5 \chapter{The rational Hodge target}
6 \section{Statement and status}
7 Let $X$ be a smooth projective complex variety. For each codimension $p$, define
8 \[
9 \Hdg^p(X)=H^{2p}(X,\Q)\cap H^{p,p}(X)
10 \]
11 and
12 \[
13 \Alg^p(X)=\im\bigl(\CH^p(X)_{\Q}\xrightarrow{\cl}H^{2p}(X,\Q)\bigr).
14 \]
15 Every algebraic cycle class has type $(p,p)$, so $\Alg^p(X)\subseteq\Hdg^p(X)$. The rational Hodge conjecture asserts equality for every smooth projective $X$ and every $p$ \cite{ClayHodge,VoisinHodge}.
16 
17 Choose a positive inner product on $H^{2p}(X,\R)$. The orthogonal projectors $P_{\Hdg}^p$ and $P_{\Alg}^p$ act on the real scalar extensions $\Hdg^p(X)\otimes_{\Q}\R$ and $\Alg^p(X)\otimes_{\Q}\R$. These real orthogonal projectors provide a reformulation. They are not asserted to be rational matrices or algebraic correspondences. Since the algebraic space is nested in the Hodge space,
18 \[
19 P_{\Hdg}^pP_{\Alg}^p=P_{\Alg}^p.
20 \]
21 
22 \begin{proposition}[Master obstruction projector]
23 The operator
24 \[
25 \mathcal O_X^p=P_{\Hdg}^p-P_{\Alg}^p
26 \]
27 is the orthogonal projector onto $(\Hdg^p(X)\otimes_{\Q}\R)\cap(\Alg^p(X)\otimes_{\Q}\R)^\perp$. Therefore
28 \[
29 \mathcal O_X^p=0
30 \quad\Longleftrightarrow\quad
31 \Hdg^p(X)=\Alg^p(X).
32 \]
33 \end{proposition}
34 \begin{proof}
35 For nested orthogonal projectors $Q\le P$, the difference $P-Q$ is self-adjoint and
36 \[
37 (P-Q)^2=P-PQ-QP+Q=P-Q.
38 \]
39 Its range is $\im P\cap(\im Q)^\perp$.
40 \end{proof}
41 
42 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{04_projector_diagnostic.pdf}\caption{The real orthogonal obstruction is a diagnostic reformulation. An algebraic realization of the full Hodge projector is not supplied.}\end{figure}
43 
44 \section{Why dimension four is the first focused target}
45 The first general unresolved middle case occurs for primitive degree-four cohomology on a smooth projective fourfold. The Hodge diamond highlights the central packet.
46 
47 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{03_fourfold_middle.pdf}\caption{Hodge bidegree positions for a fourfold. The diagram gives indices rather than numerical Hodge dimensions.}\end{figure}
48 
49 \chapter{Primitive and coniveau reductions}
50 \section{Lefschetz packet decomposition}
51 Let $h\in H^2(X,\Q)$ be the class of an ample divisor and let $L(\alpha)=h\smile\alpha$. Hard Lefschetz and primitive decomposition yield
52 \[
53 H^{2p}(X,\Q)=\bigoplus_{r\ge0}L^rH_{\mathrm{prim}}^{2p-2r}(X,\Q).
54 \]
55 The decomposition respects Hodge type.
56 
57 \begin{theorem}[Primitive reduction]
58 If every primitive rational Hodge class is algebraic, then every rational Hodge class is algebraic.
59 \end{theorem}
60 \begin{proof}
61 Write $\alpha=\sum_rL^r\alpha_r$ with each $\alpha_r$ primitive and of Hodge type. The class $h$ is algebraic. If $\alpha_r$ is algebraic, then $L^r\alpha_r=h^r\smile\alpha_r$ is algebraic. Summing gives an algebraic representative of $\alpha$.
62 \end{proof}
63 
64 For a fourfold and $p=2$,
65 \[
66 \alpha=L\beta+\alpha_{\mathrm{prim}},
67 \qquad
68 \beta\in H^{1,1}(X)\cap H^2(X,\Q).
69 \]
70 The Lefschetz $(1,1)$ theorem makes $\beta$ algebraic. Only $\alpha_{\mathrm{prim}}$ remains.
71 
72 
73 
74 \section{Support localization and coniveau}
75 Let $N^pH^{2p}(X,\Q)$ be the geometric coniveau-$p$ subspace. In degree $2p$, purity forces support contributions of codimension $p$ to be generated by fundamental classes of codimension-$p$ components.
76 
77 \begin{theorem}[Degree-$2p$ coniveau identification]
78 For smooth projective $X$,
79 \[
80 N^pH^{2p}(X,\Q)=\Alg^p(X).
81 \]
82 \end{theorem}
83 \begin{proof}[Proof sketch]
84 The inclusion from right to left is immediate because a codimension-$p$ cycle is supported on its underlying algebraic set. Conversely, a class in coniveau $p$ is in the image of cohomology with supports in an algebraic subset $Z$ of codimension at least $p$. Contributions from components of codimension greater than $p$ vanish in degree $2p$. Purity identifies the degree-$2p$ supported contribution of each codimension-$p$ irreducible component with its fundamental class. Resolution and additivity reduce a general $Z$ to these components.
85 \end{proof}
86 
87 Thus the conjecture is equivalently the support statement
88 \[
89 \Hdg^p(X)\subseteq N^pH^{2p}(X,\Q).
90 \]
91 
92 
93 
94 \addcontentsline{toc}{part}{Part II. Algebraic cycle frames and exact polyphase refinement}
95 
96 \chapter{Positive cycle-frame calculus}
97 \section{Analysis and synthesis}
98 Fix a finite family of rational algebraic cycle classes
99 \[
100 c_1,\ldots,c_N\in H^{2p}(X,\Q).
101 \]
102 Let $V\subseteq H^{2p}(X,\R)$ be the real extension of the rational Hodge packet under study. For the positive frame certificate below, require every $c_j$ to belong to $V$. This can be achieved by a rational algebraic packet projector when such a projector has been constructed. Projection through an arbitrary orthogonal Hodge projector does not establish that the projected vectors are algebraic. Define
103 \[
104 C:\R^N\to H^{2p}(X,\R),
105 \qquad Ce_j=c_j,
106 \]
107 and the positive frame operator
108 \[
109 S=CC^*=\sum_{j=1}^N|c_j\rangle\langle c_j|.
110 \]
111 
112 \begin{theorem}[Finite positive cycle-frame certificate]
113 Let $A=\Span_{\R}\{c_1,\ldots,c_N\}$. Then
114 \[
115 \im S=A,
116 \qquad
117 \ker S=A^\perp.
118 \]
119 Under the stated containment $c_j\in V$, the selected cycles span $V$ if and only if $S|_V$ is positive definite. Without containment, positivity only detects the absence of vectors in $V$ orthogonal to the cycle span and does not prove that span contains $V$.
120 \end{theorem}
121 \begin{proof}
122 The equality $\im(CC^*)=\im C$ is standard. Moreover
123 \[
124 \langle v,Sv\rangle=\|C^*v\|^2=\sum_j|\langle c_j,v\rangle|^2.
125 \]
126 This vanishes exactly when $v$ is orthogonal to every cycle class.
127 \end{proof}
128 
129 If $H$ is a matrix whose columns form an orthonormal basis of $V$, the finite Gram certificate is
130 \[
131 G=H^*SH.
132 \]
133 Under $A\subseteq V$, we have $A=V$ if and only if $\lambda_{\min}(G)>0$. Without that containment, use a rational basis matrix $H_{\Q}$ for the target and the exact certificate $\operatorname{rank}_{\Q}[C\ H_{\Q}]=\operatorname{rank}_{\Q}C$ to test $V\subseteq A$.
134 
135 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{06_corrected_rational_frame.pdf}\caption{A finite rational frame example with its cycle vectors contained in the selected target. Actual geometric applications must supply algebraic vectors with verified cohomology coordinates.}\end{figure}
136 
137 
138 
139 The orthogonal synthesis of a class $\alpha$ from the cycle span is
140 \[
141 P_A\alpha=S S^\dagger\alpha.
142 \]
143 If $\alpha\in A$, this is exact.
144 
145 
146 
147 \section{Packet assembly without false orthogonality}
148 Partition the cycle indices into packets $I_\lambda$. Define
149 \[
150 S_\lambda=\sum_{j\in I_\lambda}|c_j\rangle\langle c_j|.
151 \]
152 Then
153 \[
154 S=\sum_\lambda S_\lambda.
155 \]
156 No orthogonality of the child spans is assumed.
157 
158 \begin{theorem}[Positive child-kernel intersection]
159 For any subspace $V$,
160 \[
161 \ker(S|_V)=\bigcap_\lambda\ker(S_\lambda|_V).
162 \]
163 \end{theorem}
164 \begin{proof}
165 If $Sv=0$, positivity gives
166 \[
167 0=\langle v,Sv\rangle=\sum_\lambda\langle v,S_\lambda v\rangle.
168 \]
169 Each nonnegative summand vanishes, which implies $S_\lambda v=0$. The converse is immediate.
170 \end{proof}
171 
172 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{07_positive_polyphase.pdf}\caption{Positive parent-child frame assembly. The identity combines frames without assuming their ranges are orthogonal.}\end{figure}
173 
174 \chapter{Degree-residue polyphase refinement}
175 \section{Exact parent-child frame law}
176 Fix a projective embedding and assign each cycle its degree. For modulus $M$, define the residue packet
177 \[
178 I_{M,r}=\{j:\deg Z_j\equiv r\pmod M\},
179 \qquad
180 S_{M,r}=\sum_{j\in I_{M,r}}|c_j\rangle\langle c_j|.
181 \]
182 
183 \begin{theorem}[Exact degree polyphase refinement]
184 For every positive integer $L$,
185 \[
186 S_{M,r}=\sum_{\ell=0}^{L-1}S_{LM,r+\ell M}.
187 \]
188 \end{theorem}
189 \begin{proof}
190 The congruence class $r$ modulo $M$ is the disjoint union of the classes $r+\ell M$ modulo $LM$. The rank-one frame terms are partitioned accordingly.
191 \end{proof}
192 
193 
194 
195 
196 
197 \section{Multiaxial packet addresses}
198 Degree residue is only one address. A refined cycle packet may be labeled by
199 \[
200 \lambda=(r_{\mathrm{Lef}},r_{\deg},\chi_{\mathrm{mon}},s_{\mathrm{supp}},\nu_{\mathrm{Frob}}).
201 \]
202 The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods to solve different geometric sectors before exact synthesis.
203 
204 \chapter{Finite saturation and exact rational closure}
205 For each degree cutoff $D$, let $A_D^p(X)$ be the rational span of codimension-$p$ cycle classes of degree at most $D$. Then
206 \[
207 A_1^p\subseteq A_2^p\subseteq\cdots\subseteq H^{2p}(X,\Q).
208 \]
209 
210 \begin{theorem}[Finite stabilization]
211 There exists $D_0$ such that $A_D^p(X)=A_{D_0}^p(X)$ for every $D\ge D_0$.
212 \end{theorem}
213 \begin{proof}
214 The dimensions form a nondecreasing sequence of integers bounded by $\dim H^{2p}(X,\Q)$. Only finitely many strict increases are possible.
215 \end{proof}
216 
217 \begin{corollary}[Approximation becomes exact]
218 Let $P_D$ be the orthogonal projector onto $A_D^p(X)\otimes\R$. For a rational class $\alpha$,
219 \[
220 \lim_{D\to\infty}\|(I-P_D)\alpha\|=0
221 \]
222 if and only if $\alpha\in A_D^p(X)$ for some finite $D$.
223 \end{corollary}
224 \begin{proof}
225 After stabilization, the residual is constant. A constant tail converging to zero is zero.
226 \end{proof}
227 
228 Because the matrices and target vector can be chosen over $\Q$, real membership implies rational membership by row reduction over $\Q$.
229 
230 
231 
232 \begin{resultbox}[Finite completion certificate]
233 If $H$ is a rational basis matrix of $\Hdg^p(X)$ and $C_D$ is the cycle matrix, then the Hodge conjecture for $(X,p)$ is equivalent to the existence of $D$ with
234 \[
235 \rank_{\Q} C_D=\rank_{\Q}[C_D\ H].
236 \]
237 A positive compressed frame is equivalent only when the supplied cycle vectors lie in the target packet, or when a separately verified invariant restriction supplies the required containment. The rank equality is valid without that extra hypothesis.
238 \end{resultbox}
239 
240 \addcontentsline{toc}{part}{Part III. Support packets and correspondence reconstruction}
241 
242 \chapter{Support verification and multiplicity recovery}
243 Let $Z\subseteq X$ be a closed algebraic subset and $j:X\setminus Z\hookrightarrow X$. The support exact sequence contains
244 \[
245 H_Z^{2p}(X,\Q)\longrightarrow H^{2p}(X,\Q)\xrightarrow{j^*}H^{2p}(X\setminus Z,\Q).
246 \]
247 
248 \begin{theorem}[Exact support test]
249 A class $\alpha\in H^{2p}(X,\Q)$ is supported on $Z$ if and only if $j^*\alpha=0$.
250 \end{theorem}
251 \begin{proof}
252 This is exactness at the middle term.
253 \end{proof}
254 
255 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{08_support_verification.pdf}\caption{The restriction-to-complement test verifies a supplied support. It does not construct a support for an arbitrary Hodge class.}\end{figure}
256 
257 Suppose $Z$ has codimension at least $p$ and its codimension-$p$ components are $Z_1,\ldots,Z_r$. Purity gives
258 \[
259 \alpha=\sum_{i=1}^r q_i[Z_i].
260 \]
261 Choose complementary test classes $\gamma_k$ and define
262 \[
263 M_{ki}=\int_X[Z_i]\smile\gamma_k,
264 \qquad
265 b_k=\int_X\alpha\smile\gamma_k.
266 \]
267 Then $Mq=b$. Removing dependent columns yields exact rational multiplicities.
268 
269 
270 
271 \chapter{Divisor-lowering correspondence channels}
272 \section{Degree bookkeeping}
273 Let $X$ have complex dimension $n$, let $V\subseteq H_{\mathrm{prim}}^{2p}(X,\Q)$ be a rational Hodge substructure, and let $Y_r$ be smooth projective. Choose correspondences
274 \[
275 \Gamma_r\in\CH^{n-p+1}(X\times Y_r)_{\Q},
276 \qquad
277 \Lambda_r\in\CH^{\dim Y_r+p-1}(Y_r\times X)_{\Q}.
278 \]
279 They induce
280 \[
281 A_r=(\Gamma_r)_*:V\to H^2(Y_r,\Q),
282 \qquad
283 B_r=(\Lambda_r)_*:H^2(Y_r,\Q)\to H^{2p}(X,\Q).
284 \]
285 Algebraic correspondences preserve rationality and shift Hodge type by the required bidegree.
286 
287 \begin{theorem}[Invertible divisor-lowering transfer]
288 Suppose
289 \[
290 T=\sum_{r=1}^M B_rA_r
291 \]
292 preserves $V$ and is invertible on $V$. Then every rational Hodge class in $V$ is algebraic.
293 \end{theorem}
294 \begin{proof}
295 Let $\alpha\in V$ be of type $(p,p)$. Each $A_r\alpha$ is rational of type $(1,1)$ on $Y_r$, hence is the class of a rational divisor $D_r$ by the Lefschetz $(1,1)$ theorem. Therefore
296 \[
297 T\alpha=\sum_r(\Lambda_r)_*[D_r]
298 \]
299 is algebraic on $X$.
300 
301 Let $\chi_T(t)=t^d+c_{d-1}t^{d-1}+\cdots+c_0$ be the characteristic polynomial on $V$. Invertibility means $c_0\ne0$. Cayley-Hamilton gives
302 \[
303 T^{-1}=-\frac{1}{c_0}\bigl(T^{d-1}+c_{d-1}T^{d-2}+\cdots+c_1I\bigr).
304 \]
305 Each power of $T$ is induced by compositions and rational sums of algebraic correspondences. Thus $T^{-1}$ preserves algebraic cycle classes. Hence $\alpha=T^{-1}(T\alpha)$ is algebraic.
306 \end{proof}
307 
308 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{09_correspondence_degrees.pdf}\caption{The exact degree shifts required for lowering to divisor degree and returning. Actual Chow kernels and invertibility are additional inputs.}\end{figure}
309 
310 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{10_rational_inverse.pdf}\caption{A finite rational example of polynomial inversion. The geometric theorem requires the operator to be induced by algebraic correspondences.}\end{figure}
311 
312 \section{Positive multi-channel form}
313 Assume the polarized adjoint $A_r^\dagger$ is realized by an algebraic return correspondence $B_r$. This is an additional geometric hypothesis, not an automatic consequence in arbitrary settings. Then
314 \[
315 T=\sum_rA_r^\dagger A_r
316 \]
317 is positive and
318 \[
319 \langle v,Tv\rangle=\sum_r\|A_rv\|^2.
320 \]
321 Therefore
322 \[
323 \ker T=\bigcap_r\ker A_r.
324 \]
325 Joint injectivity of the lowering channels is enough.
326 
327 
328 
329 
330 
331 \begin{corollary}[Finite Gram certificate]
332 Choose a rational basis $e_1,\ldots,e_d$ of $V$. If
333 \[
334 G_{ij}=\sum_r\langle A_re_i,A_re_j\rangle_{Y_r}
335 \]
336 has nonzero determinant, every rational Hodge class in $V$ is algebraic.
337 \end{corollary}
338 
339 \chapter{Family propagation}
340 Let $\pi:\mathcal X\to S$ and $\rho_r:\mathcal Y_r\to S$ be smooth projective families over a connected base. Relative algebraic correspondences induce morphisms of Gauss-Manin local systems. In a flat rational trivialization, their transfer matrix is constant.
341 
342 \begin{theorem}[Propagation of invertibility]
343 If a relative transfer $T_s$ is invertible on one fiber $s_0$, it is invertible on every fiber in the connected base on which the local systems and correspondences are defined.
344 \end{theorem}
345 \begin{proof}
346 The determinant of the constant rational matrix representing the local-system morphism is locally constant. It is nonzero at $s_0$, hence everywhere on the connected base.
347 \end{proof}
348 
349 
350 
351 \addcontentsline{toc}{part}{Part IV. Closed branches and classical validation cases}
352 
353 \chapter{Geometric closure rules}
354 \section{Affine pavings}
355 \begin{theorem}[Affine-paving branch]
356 If $X$ admits a finite filtration whose successive differences are affine spaces, then the cycle class map
357 \[
358 \CH^p(X)_{\Q}\to H^{2p}(X,\Q)
359 \]
360 is surjective for every $p$.
361 \end{theorem}
362 \begin{proof}[Proof sketch]
363 Induct through the filtration using localization for Chow groups and cohomology. Each affine cell contributes one even-degree generator represented by the closure of that cell. No odd cohomology is introduced.
364 \end{proof}
365 
366 \section{Projective bundles, blow-ups, and descent}
367 \begin{proposition}[Projective-bundle closure]
368 If the rational Hodge conjecture holds for $X$, it holds for $\mathbb P(E)\to X$.
369 \end{proposition}
370 \begin{proof}
371 The projective-bundle formula expresses cohomology as a direct sum of powers of the algebraic hyperplane class times pullbacks of cohomology from $X$.
372 \end{proof}
373 
374 \begin{proposition}[Blow-up closure]
375 If it holds for $X$ and a smooth center $Z$, it holds for $\operatorname{Bl}_ZX$.
376 \end{proposition}
377 \begin{proof}
378 Use the standard blow-up cohomology decomposition. The new summands are Tate twists of cohomology from $Z$ multiplied by powers of the algebraic exceptional divisor.
379 \end{proof}
380 
381 \begin{proposition}[Generically finite descent]
382 Let $f:Y\to X$ be proper and generically finite of degree $d$ between smooth projective varieties of equal dimension. If the conjecture holds for $Y$, it holds for $X$.
383 \end{proposition}
384 \begin{proof}
385 For a Hodge class $\alpha$ on $X$, the pullback $f^*\alpha$ is algebraic on $Y$. Push forward and use $f_*f^*\alpha=d\alpha$.
386 \end{proof}
387 
388 
389 
390 \chapter{Cubic fourfolds as a validation of the transfer theorem}
391 Let $X\subset\mathbb P^5$ be a smooth cubic fourfold and let $F(X)$ be its Fano variety of lines. The universal incidence variety
392 \[
393 P=\{(\ell,x):x\in\ell\}\subset F(X)\times X
394 \]
395 induces the Abel-Jacobi map
396 \[
397 A=p_*q^*:H^4_{\mathrm{prim}}(X,\Q)\to H^2_{\mathrm{prim}}(F(X),\Q).
398 \]
399 Beauville and Donagi proved that this is a polarized Hodge isomorphism up to the standard twist \cite{BeauvilleDonagi}. The polarized adjoint is represented by an algebraic incidence return after inserting the appropriate power of the Pluecker class. Therefore the divisor-lowering transfer is invertible.
400 
401 \begin{resultbox}[Classical validation]
402 A primitive rational $(2,2)$ class on a cubic fourfold is sent to a rational $(1,1)$ class on $F(X)$, represented by a divisor. The algebraic return correspondence and its polynomial inverse reconstruct an algebraic codimension-two class on $X$. This is a validation of the packet criterion, not a claim of a new proof of the classical cubic-fourfold result.
403 \end{resultbox}
404 
405 
406 
407 
408 
409 \chapter{Gushel-Mukai validation channels}
410 Debarre and Kuznetsov construct line and plane incidence correspondences relating vanishing middle cohomology of suitable Gushel-Mukai varieties to primitive degree-two cohomology of associated double EPW sextics \cite{DebarreKuznetsov}. In the ranges where the polarized Hodge isomorphisms are established, the same divisor-lowering theorem applies.
411 
412 \begin{itemize}
413 \item For suitable GM fourfolds, the vanishing $H^4$ packet is lowered by universal lines to divisor degree.
414 \item For suitable GM sixfolds, the vanishing $H^6$ packet is lowered by universal planes to divisor degree.
415 \item Ambient and tautological classes are treated separately and are algebraic.
416 \end{itemize}
417 
418 
419 
420 \addcontentsline{toc}{part}{Part V. The analytic and K-theoretic branch}
421 
422 \chapter{Hodge packets in topological K-theory}
423 For finite CW complexes, the rational topological Chern character gives
424 \[
425 \operatorname{ch}_{\mathrm{top}}:K^0_{\mathrm{top}}(X)\otimes\Q\xrightarrow{\sim}H^{\mathrm{even}}(X,\Q).
426 \]
427 For smooth projective $X$, the algebraic Chern character identifies algebraic $K$-theory rationally with the Chow ring after the usual filtration. Thus the rational Hodge conjecture for all degrees is equivalent to algebraization of topological $K$-classes whose Chern-character components are rational Hodge classes \cite{AtiyahHirzebruch}.
428 
429 \section{Adams degree projectors}
430 For integer $m\ge2$, Adams operations satisfy
431 \[
432 \operatorname{ch}_q(\psi^m\xi)=m^q\operatorname{ch}_q(\xi).
433 \]
434 Define
435 \[
436 e_p^{(m)}=\prod_{\substack{0\le q\le\dim X\\q\ne p}}
437 \frac{\psi^m-m^qI}{m^p-m^q}.
438 \]
439 Then
440 \[
441 \operatorname{ch}_q(e_p^{(m)}\xi)=\delta_{pq}\operatorname{ch}_p(\xi).
442 \]
443 This exactly isolates codimension $p$ inside a topological $K$-packet.
444 
445 
446 
447 \chapter{Nonlinear Dolbeault obstruction packets}
448 Let $E$ be a smooth complex bundle and consider a candidate $(0,1)$-connection. After gauge fixing, write the integrability equation abstractly as
449 \[
450 F(a)=f+La+B(a,a)=0,
451 \]
452 where $L$ is the linearized operator and $B$ is a bounded bilinear term. Assume a Fredholm splitting of the target
453 \[
454 I=Q+\Pi,
455 \]
456 where $Q$ projects onto the closed range of $L$ and $\Pi$ onto a finite-dimensional obstruction space. Choose a gauge-fixed complement to $\ker L$ as the domain and let $R$ be a right inverse of $L$ taking values in that complement. Use norms on the projected data for which $\|QB(a,b)\|\le C\|a\|\|b\|$.
457 
458 \begin{theorem}[Range-packet contraction]
459 Assume
460 \[
461 \|R\|\le M,
462 \qquad
463 \|B(a,b)\|\le C\|a\|\|b\|,
464 \qquad
465 4CM^2\|Qf\|<1.
466 \]
467 Then the map
468 \[
469 \mathcal T(a)=-RQ(f+B(a,a))
470 \]
471 has a unique fixed point in the gauge-fixed ball $\|a\|\le2M\|Qf\|$. If $Qf=0$, the ball is the singleton $\{0\}$ and the range solution is immediate. At the fixed point, the range component of $F(a)$ vanishes. The remaining obstruction is
472 \[
473 \Phi=\Pi(f+B(a,a)).
474 \]
475 The full integrability equation holds if and only if $\Phi=0$.
476 \end{theorem}
477 \begin{proof}
478 For $\|a\|\le r=2M\|Qf\|$,
479 \[
480 \|\mathcal T(a)\|\le M\|Qf\|+MC r^2\le r
481 \]
482 under the stated bound. Moreover
483 \[
484 \|\mathcal T(a)-\mathcal T(b)\|
485 \le MC(\|a\|+\|b\|)\|a-b\|.
486 \]
487 The Lipschitz constant is strictly less than one on the stated ball.
488 Banach's fixed-point theorem gives the solution. Applying $Q$ to the equation proves range cancellation. The remaining component is precisely $\Phi$.
489 \end{proof}
490 
491 
492 
493 
494 
495 \section{A positive spectral microscope}
496 Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of $F_A^{0,2}$ \cite{CharbonneauStern}. Schematically,
497 \[
498 \Tr(P_{2j}\Pi_{A,k})
499 =c_{n,j}k^{n-2j}\|(F_A^{0,2})^{\wedge j}\|_{L^2}^2+O(k^{n-2j-1}),
500 \qquad c_{n,j}>0.
501 \]
502 This detects nonintegrability. It does not by itself remove the defect.
503 
504 
505 
506 \section{Trace blindness and the true missing bridge}
507 Chern characters control traces of powers of curvature. An endomorphism-valued obstruction can be nonzero while all such trace moments vanish. For example, $N=\left(\begin{smallmatrix}0&1&0\\0&0&1\\0&0&0\end{smallmatrix}\right)$ is nonzero, while $N^3=0$ and $\operatorname{Tr}(N^r)=0$ for every positive integer $r$. This is the elementary negative control.
508 
509 
510 
511 \begin{openproblem}[Analytic bridge]
512 The unresolved analytic theorem is not the range equation. It is a positive saturation statement forcing the finite obstruction packet $\Phi$ to vanish for every rational Hodge $K$-packet after allowable stabilization. Proving that statement in general would be essentially a K-theoretic form of the Hodge conjecture.
513 \end{openproblem}
514 
515 \section{Consequences of the obstruction split}
516 The obstruction split prevents a common logical error. Solving the projected range equation does not solve the original integrability problem unless the complementary packet also vanishes. The calculus therefore records three distinct outcomes. First, if $\Pi=0$, the problem is unobstructed and the contraction theorem gives an actual integrable structure. Second, if $\Pi\ne0$ but $\Phi=0$ for geometric reasons, the correction again closes. Third, if $\Phi\ne0$, the method has produced a finite certificate explaining exactly why that representative cannot be algebraized by the chosen deformation.
517 
518 This creates a practical research program. One can refine the finite obstruction space by symmetry, monodromy, support, or arithmetic phase, obtaining
519 \[
520 \Phi=\sum_{\lambda}\Phi_{\lambda}.
521 \]
522 Since the decomposition is complete, integrability is equivalent to the simultaneous vanishing of all children. Different geometric arguments may eliminate different children. A symmetry packet may vanish by representation theory, a support packet by localization, and an arithmetic packet by compatibility across reductions. The analytic branch therefore remains useful even before the general bridge is proved: it supplies exact special-case theorems whenever all obstruction children can be eliminated.
523 
524 The negative controls are equally important. Ample twisting can amplify $F_A^{0,2}$ in a spectral observable without reducing it. Direct-sum stabilization adds positive defect energies for block-diagonal connections. Trace identities may miss nilpotent components. These facts rule out several tempting but invalid shortcuts and keep the remaining theorem sharply stated.
525 
526 
527 \addcontentsline{toc}{part}{Part VI. Atomic status ledger and focused next theorem}
528 
529 \chapter{Dependency graph and proof status}
530 The project separates proved statements, classical inputs, conditional mechanisms, and open bridges.
531 
532 
533 
534 \begin{longtable}{p{0.40\textwidth}p{0.16\textwidth}p{0.34\textwidth}}
535 \caption{Atomic status ledger.}\label{tab:status}\\
536 \toprule
537 Statement & Status & Boundary \\
538 \midrule
539 \endfirsthead
540 \toprule
541 Statement & Status & Boundary \\
542 \midrule
543 \endhead
544 Master obstruction projector & Proved & Finite-dimensional linear algebra \\
545 Primitive Lefschetz reduction & Proved & Uses hard Lefschetz and algebraicity of the polarization \\
546 Degree-$2p$ coniveau identification & Proved & Rational coefficients and purity \\
547 Positive cycle-frame certificate & Corrected & Requires explicit algebraic cycles contained in the selected packet \\
548 Exact degree polyphase assembly & Proved & Frame operators, not sums of span projectors \\
549 Finite saturation of bounded-degree spans & Proved & Non-effective degree bound \\
550 Support verification & Proved & Candidate support must already be supplied \\
551 Multiplicity reconstruction & Proved & Finite rational linear algebra \\
552 Divisor-lowering transfer theorem & Proved & Requires explicit algebraic correspondences and invertibility \\
553 Positive multi-channel criterion & Conditional & Algebraic realization of polarized adjoints is an extra hypothesis \\
554 Family propagation & Proved & Relative correspondences over a connected smooth base \\
555 Affine-paving branch & Proved & Classical localization argument \\
556 Projective bundle, blow-up, descent rules & Proved & Standard cohomological decompositions \\
557 Cubic and GM channels & Classical validation & Depend on established incidence Hodge isomorphisms \\
558 Adams degree projector & Proved & Rational topological K-theory \\
559 Range-packet nonlinear correction & Proved under hypotheses & Closed range, right inverse, and contraction bound \\
560 Vanishing of every harmonic obstruction packet & Open & Main analytic bridge \\
561 Primitive fourfold multi-incidence saturation & Open & Main geometric target \\
562 General rational Hodge conjecture & Open & No general cycle-generation theorem is claimed \\
563 \bottomrule
564 \end{longtable}
565 
566 
567 
568 \chapter{A focused multi-incidence target}
569 Let $X$ be a special smooth projective fourfold. Construct algebraic lowering channels from families of subvarieties:
570 \[
571 A_{\mathrm{line}},
572 A_{\mathrm{conic}},
573 A_{\mathrm{plane}},
574 A_{\mathrm{exceptional}}.
575 \]
576 The exact target is
577 \[
578 \ker A_{\mathrm{line}}
579 \cap\ker A_{\mathrm{conic}}
580 \cap\ker A_{\mathrm{plane}}
581 \cap\ker A_{\mathrm{exceptional}}=\{0\}
582 \]
583 on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equivalently prove
584 \[
585 T_{\mathrm{tot}}
586 =\sum_rA_r^\dagger A_r
587 \quad\text{satisfies}\quad
588 \lambda_{\min}(T_{\mathrm{tot}})>0.
589 \]
590 
591 
592 
593 \section{Why this target is realistic}
594 The problem is finite after the geometry is chosen:
595 \begin{enumerate}
596 \item compute the primitive cohomology representation and its symmetry decomposition
597 \item construct the incidence correspondences and their action
598 \item split the action into monodromy or symmetry packets
599 \item compute the joint kernel or positive Gram matrix over $\Q$
600 \item prove exact nonvanishing through rational rank, representation theory, or a dual certificate
601 \item invoke the divisor-lowering theorem to reconstruct cycles
602 \end{enumerate}
603 No single incidence family must be complete. Polyphase analysis permits separate geometric mechanisms to occupy separate primitive sectors.
604 
605 
606 \section{Three exact certificate forms}
607 Once the incidence channels have been constructed, the saturation statement admits three finite certificates. The geometric form is the kernel identity
608 \[
609 \bigcap_r\ker A_r=\{0\}.
610 \]
611 When algebraic correspondences realize the polarized adjoints, the equivalent positive form is
612 \[
613 \lambda_{\min}\left(\sum_rA_r^\dagger A_r\right)>0.
614 \]
615 In a rational basis, the final exact proof object is full column rank of the vertically stacked channel matrix. The kernel form explains the geometry, the positive form is stable for numerical scouting, and the rational rank form is appropriate for symbolic certification.
616 
617 Two negative controls should accompany the proof. First, remove one channel at a time and recompute the joint kernel. If the kernel grows, the omitted incidence family is genuinely necessary. Second, refine a parent channel into geometric children and verify that their positive transfers sum exactly to the parent. This separates true packet coverage from accidental numerical invertibility.
618 
619 A complete result should identify which primitive symmetry packets are detected by lines, conics, planes, or exceptional strata. The proof then becomes a finite occupancy statement: every primitive packet must be detected at least once. That structural description is more informative than a determinant alone and is the form most likely to propagate through families and degenerations.
620 
621 \chapter{Computational and formal proof workflow}
622 Numerical experiments can scout candidate families but cannot replace an exact rank certificate. The intended pipeline is:
623 \[
624 \begin{aligned}
625 \text{exact incidence data}
626 &\longrightarrow \text{rational cohomology basis}
627 \longrightarrow \text{packet transfer matrices}\\
628 &\longrightarrow \text{rank or positivity certificate}
629 \longrightarrow \text{symbolic theorem proof}.
630 \end{aligned}
631 \]
632 
633 
634 
635 \section{Reproducible finite checks}
636 The accompanying Python generator produces all figures and runs eighteen independent finite checks covering:
637 \begin{itemize}
638 \item frame idempotency and range rank
639 \item positive Hodge Gram spectrum
640 \item exact modulo-$2$ and modulo-$4$ polyphase assembly
641 \item positive transfer rank and energy identities
642 \item Cayley-Hamilton inversion
643 \item contraction of the removable nonlinear packet
644 \item Adams projector idempotency and degree isolation
645 \item nilpotent trace blindness
646 \item support multiplicity recovery
647 \item finite rank stabilization
648 \end{itemize}
649 These tests verify the finite examples and implementation. They do not establish the open geometric saturation theorem.
650 
651 \compactpart{Part VII. Fermat Fourfold Character Packets and the Constructive Program}
652 
653 \chapter{Correct scope and claim boundary}
654 The projector-resolved Hodge calculus naturally specializes to Fermat fourfolds
655 \[
656 X_m^4=\{x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0\}\subset\mathbf P^5,
657 \qquad m\ge 3.
658 \]
659 The diagonal symmetry group makes the primitive middle cohomology unusually explicit: it splits into one-dimensional complex character lines, and rational Hodge packets are finite Galois orbits of those lines. This permits exact symbolic analysis of individual addresses, conductors, local prime phases, and candidate algebraic correspondences. A complete classification at every mixed conductor is a separate coverage problem.
660 
661 \begin{claimboundary}
662 The general rational Hodge conjecture remains open. For Fermat fourfolds, however, Kang is cited in the checked later literature as proving the relevant generalized Hodge statement in arbitrary degree \cite{KangFermat}. This complete edition therefore distinguishes two questions:
663 \begin{enumerate}
664 \item \emph{Existence}: does every rational Fermat $(2,2)$ class have an algebraic representative?
665 \item \emph{Construction}: can one write an explicit packet-resolved algebraic cycle or correspondence for every character orbit?
666 \end{enumerate}
667 The present Fermat program is primarily constructive. Its new proof objects are conductor projectors, exact packet identities, explicit Delsarte maps, weighted dyadic refinements, the proposed all-degree trivalent-tree classification, CRT--Kummer phase factorizations, and explicit CM carrier data. The relation between that existence theorem and later packet-specific combinatorial work should be stated explicitly in any publication.
668 \end{claimboundary}
669 
670 
671 
672 The constructive objective is
673 \begin{resultbox}[Packet-resolved construction target]
674 For every rational packet
675 \[
676 W\subset H^4_{\mathrm{prim}}(X_m^4,\Q)\cap H^{2,2},
677 \]
678 construct
679 \[
680 Z_W\in\CH^2(X_m^4)_\Q
681 \qquad\text{with}\qquad
682 [Z_W]\ne0\text{ in }W.
683 \]
684 \end{resultbox}
685 Once one nonzero algebraic vector is produced in a rational irreducible diagonal-symmetry packet, the entire rational orbit is algebraic.
686 
687 \chapter{Universal balanced-packet equations}
688 Let
689 \[
690 U_m=(\mathbf Z/m\mathbf Z)^\times
691 \]
692 and let \(\langle r\rangle_m\in\{1,\ldots,m\}\) denote the least positive residue, with \(\langle r\rangle_m=m\) when \(m\mid r\). A primitive character address is a six-tuple
693 \[
694 a=(a_0,\ldots,a_5),\qquad 1\le a_i<m,
695 \qquad \sum_i a_i\equiv0\pmod m.
696 \]
697 The Hodge level of the conjugate indexed by \(u\in U_m\) is
698 \[
699 \ell_a(u)=\frac1m\sum_{i=0}^5\langle ua_i\rangle_m.
700 \]
701 The character line has type \((2,2)\) exactly when \(\ell_a(1)=3\). Its rational Galois orbit is a Hodge packet exactly when
702 \begin{equation}\label{eq:legacy-7.1}
703 \boxed{
704 \ell_a(u)=3\qquad\forall u\in U_m.
705 }\end{equation}
706 
707 To remove coordinate permutations, introduce the count vector
708 \[
709 n(a)=(n_1,\ldots,n_{m-1})\in\mathbf N^{m-1},
710 \qquad n_r=\#\{i:a_i=r\}.
711 \]
712 Let
713 \[
714 R_m(u,r)=\langle ur\rangle_m.
715 \]
716 Then the complete balanced six-packet space is the finite integer set
717 \begin{equation}\label{eq:legacy-7.2}
718 \boxed{
719 \mathcal B_m=
720 \left\{
721  n\in\mathbf N^{m-1}:
722  \mathbf1^Tn=6,
723  \quad R_mn=3m\,\mathbf1
724 \right\}.
725 }\end{equation}
726 The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebraic automorphisms, the count-vector descriptions correspond to \(U_m\)-orbits in \(\mathcal B_m\). The dimension of a fixed rational character packet is computed from distinct ordered unit conjugates before this permutation quotient.
727 
728 
729 
730 \section{Centered Fourier form}
731 Define
732 \[
733 \rho_m(r)=\langle r\rangle_m-\frac m2,
734 \qquad
735 F_a(u)=\sum_i\rho_m(ua_i).
736 \]
737 Then \(a\) is balanced if and only if \(F_a(u)=0\) for all units. Moreover,
738 \[
739 F_a(-u)=-F_a(u),
740 \]
741 so only odd characters occur. For a Dirichlet character \(\chi\) on \(U_m\), put
742 \[
743 \mathcal B_{m,\chi}(r)=
744 \sum_{u\in U_m}\rho_m(ur)\overline{\chi(u)}.
745 \]
746 Finite Fourier inversion gives the exact criterion
747 \begin{equation}\label{eq:legacy-7.3}
748 \boxed{
749  n(a)\in\mathcal B_m
750  \Longleftrightarrow
751  \sum_i\mathcal B_{m,\chi}(a_i)=0
752  \quad\text{for every odd }\chi.
753 }\end{equation}
754 Here $n(a)$ is the residue count vector. This formulation underlies the prime-power filtration and the proposed conductor-defect tree.
755 
756 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{11_balanced_orbit70.pdf}\caption{Every ordered unit conjugate of the named degree-70 character has sum 210. This exact example identifies Hodge type and does not itself construct a cycle.}\end{figure}
757 
758 \chapter{Exact conductor polyphase}
759 For a packet \(a\), define
760 \[
761 g(a)=\gcd(m,a_0,\ldots,a_5),
762 \qquad
763 c(a)=\frac{m}{g(a)}.
764 \]
765 For a count vector, $L_k$ denotes scaling each residue entry by $k$ while multiplying the modulus by $k$. Writing \(a=g(a)b\), the coordinate-power map
766 \[
767 \pi_{g(a)}:X_m^4\longrightarrow X_{c(a)}^4,
768 \qquad [x_i]\longmapsto[x_i^{g(a)}],
769 \]
770 sends the degree-\(c(a)\) character packet \(b\) to the degree-\(m\) packet \(a\).
771 
772 \begin{theorem}[Conductor descent]
773 Every imprimitive balanced packet at degree \(m\) is the coordinate-power pullback of a unique full-conductor packet at its exact conductor \(c\mid m\). Consequently
774 \[
775 \mathcal B_m=\bigsqcup_{c\mid m}L_{m/c}\mathcal B_c^{\mathrm{full}}.
776 \]
777 \end{theorem}
778 \begin{proof}
779 If every occupied residue is divisible by \(g\), divide the six entries and the modulus by \(g\). Since
780 \[
781 \langle ugb_i\rangle_{gc}=g\langle \bar u b_i\rangle_c,
782 \]
783 the balance equation descends. The maximal common divisor is unique. The converse follows by scaling.
784 \end{proof}
785 
786 For \(d\mid m\), let
787 \[
788 E_{m,d}=\frac1{(m/d)^5}\pi_{m/d}^*\pi_{m/d*}.
789 \]
790 On a character of exact conductor \(c\),
791 \[
792 E_{m,d}=\begin{cases}1,&c\mid d,\\0,&c\nmid d.\end{cases}
793 \]
794 Therefore
795 \begin{equation}\label{eq:legacy-8.1}
796 \boxed{E_{m,d}E_{m,e}=E_{m,\gcd(d,e)}.}\end{equation}
797 M\"obius inversion produces pairwise orthogonal exact-conductor projectors
798 \begin{equation}\label{eq:legacy-8.2}
799 \boxed{
800 Q_{m,c}=\sum_{d\mid c}\mu(c/d)E_{m,d}.
801 }\end{equation}
802 They satisfy
803 \[
804 Q_{m,c}Q_{m,e}=\delta_{ce}Q_{m,c},
805 \qquad
806 \sum_{c\mid m}Q_{m,c}=I.
807 \]
808 Hence
809 \begin{equation}\label{eq:legacy-8.3}
810 \boxed{
811 H^4_{\mathrm{prim}}(X_m^4,\Q)
812 =\bigoplus_{c\mid m}\operatorname{im}Q_{m,c}.
813 }\end{equation}
814 Each \(Q_{m,c}\) is a rational linear combination of graph correspondences, so the conductor decomposition is itself algebraic.
815 
816 
817 
818 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{12_conductor_descent.pdf}\caption{Exact conductor descent in the named example. Coordinate-power inflation enlarges the ambient degree while preserving the old conductor.}\end{figure}
819 
820 \chapter{Algebraic certificate grammar}
821 Fix a full-conductor orbit in \(\mathcal B_m\). The constructive program tests a hierarchy of increasingly general certificate channels.
822 
823 \section{Complementary pairs}
824 For \(1\le r<m\), let
825 \[
826 p_r=e_r+e_{m-r}.
827 \]
828 A packet is pair-decomposable when
829 \begin{equation}\label{eq:legacy-9.1}
830 n=p_{r_1}+p_{r_2}+p_{r_3}.\end{equation}
831 These are the standard linear-cycle or Tate-pair packets.
832 
833 \section{Balanced four-packets and saturation}
834 Let
835 \[
836 \mathcal F_m=
837 \left\{
838 \gamma\in\mathbf N^{m-1}:
839 \mathbf1^T\gamma=4,
840 \quad R_m\gamma=2m\,\mathbf1
841 \right\}.
842 \]
843 A saturated certificate is an exact nonnegative identity
844 \begin{equation}\label{eq:legacy-9.2}
845 \boxed{
846  n+\sum_{j=1}^{2N-3}p_{r_j}
847  =\sum_{i=1}^N\gamma_i,
848  \qquad \gamma_i\in\mathcal F_m.
849 }\end{equation}
850 The case \(N=2\) is one-step quasi-decomposition. Higher \(N\) allows several surface packets to reconstruct one six-packet after Tate stabilization.
851 
852 The algebraic implication must use the actual classical correspondence operations. A balanced four-entry packet is algebraic by Lefschetz $(1,1)$ on a Fermat surface. Concatenate the $N$ surface characters using the Shioda--Ran construction, then remove precisely the $2N-3$ complementary pairs by the classical pair-removal theorem. The ambient dimension rises to $4N-2$ before cancellation. This proves algebraicity of a packet admitting such an identity once those cited operations and their character conventions have been checked. It does not authorize cancellation of arbitrary tensor factors. The August 15 status note correctly demanded a complete correspondence-level account of the proposed general grammar. The concrete degree-210 application below spells out the necessary stabilization and uses only the checked pair-removal theorem.
853 
854 \section{Dual charges}
855 An antisymmetric charge is a rational function \(w\) on residues satisfying
856 \[
857 w(m-r)=-w(r)
858 \]
859 and
860 \[
861 \sum_r\gamma_r w(r)=0
862 \qquad\forall \gamma\in\mathcal F_m.
863 \]
864 Such a charge vanishes on every complementary pair and on every saturated identity. Writing $w(n)=\sum_r n_r w(r)$, if \(w(n)\ne0\), the entire lower-packet saturation semigroup is excluded. This is a universal obstruction to a grammar, not a nonalgebraicity theorem.
865 
866 \section{Curve, Delsarte, and incidence channels}
867 A curve certificate splits the six entries into triples
868 \[
869 a=A\sqcup B,
870 \qquad \sum A=m,
871 \quad \sum B=2m,
872 \quad B=uA\pmod m.
873 \]
874 The resulting \((1,1)\) class on a product of Fermat curves is algebraic by Lefschetz \((1,1)\), and the join correspondence returns a surface class.
875 
876 A Delsarte certificate consists of an exponent matrix and a resolved monomial map to a lower-degree hypersurface carrying an already algebraic packet. An incidence certificate uses algebraic lowering maps
877 \[
878 A_r:V\to H^2(Y_r)
879 \]
880 whose combined positive transfer
881 \[
882 T=\sum_rA_r^\dagger A_r
883 \]
884 is invertible on the packet. Divisor lowering and Cayley--Hamilton reconstruction then produce the cycle.
885 
886 \chapter{The degree-33 rejection branch and its collapse}
887 The degree-33 Fermat fourfold provided a stringent test because one rational orbit escaped the ordinary decomposable and standard channels. A rejection attempt first isolated the packet, excluded several obvious cycle families, and then discovered a cross-family cubic correspondence that algebraizes it.
888 
889 \section{The exceptional packet}
890 A representative is
891 \[
892 a=(7,10,13,19,22,28),
893 \]
894 or after permutation,
895 \[
896 \widetilde a=(7,19,28,10,13,22).
897 \]
898 Every unit conjugate has residue sum \(99=3\cdot33\), so the orbit is rational of type \((2,2)\). It is not totally pair-decomposable. The previously proposed symmetric surface
899 \[
900 W=\{p_1(x^{11})=p_2(x^{11})=p_3(x^{11})=0\}
901 \]
902 does not carry the primitive packet: Newton's identity makes the third equation redundant on \(X_{33}^4\), and
903 \[
904 [W]=242h^2.
905 \]
906 Thus \(W\) is purely ambient.
907 
908 \section{Klein--Delsarte cubic}
909 Consider the smooth cubic fourfold
910 \[
911 Y:\quad y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0.
912 \]
913 An explicit monomial exponent matrix gives a dominant rational map
914 \[
915 \phi:X_{33}^4\dashrightarrow Y
916 \]
917 whose projective-torus lattice map has Smith diagonal
918 \[
919 (1,33,33,33,33),
920 \]
921 so the generic torus degree is \(33^4\).
922 
923 Take the cubic Jacobian-ring class
924 \[
925 P(y)=y_1y_2y_5.
926 \]
927 Its exponent vector is $b=(0,1,1,0,0,1)$. Write $C=33A^{-1}$ for the cubic exponent matrix $A$. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity
928 \begin{equation}\label{eq:legacy-10.1}
929 C^Tb+10\mathbf1=(6,18,27,9,12,21)=\widetilde a-\mathbf1.\end{equation}
930 Therefore the pullback of the corresponding primitive cubic \((2,2)\)-class lies in the one-dimensional Fermat line \(V(\widetilde a)\).
931 
932 The Laurent exponent match is a useful exact certificate, but by itself it does not prove that a rational class lifts through a compact graph. A complete manuscript argument instead uses the torus-carrier criterion developed in Chapter~\ref{ch:carrier70}. For the reordered character, the exponent matrix $A$ of the cubic satisfies
933 \[
934 A^T\widetilde a/33=\beta=(1,2,2,1,1,2),\qquad C=33A^{-1}.
935 \]
936 With $r_0$ the first row of $A$, put $u=\beta-3r_0$. Then $\mathbf1^Tu=0$ and $C^Tu=\widetilde a-99e_0$. These identities establish triviality of the selected character on the torus deck group. Every selected character has full support, so its restriction to the coordinate torus is an isomorphism on the packet. Finite étale descent gives a pure rational weight-four Hodge substructure on the cubic torus. The lowest-weight image theorem and semisimplicity give a rational $(2,2)$ lift to the smooth cubic. Classical cubic algebraicity then supplies a surface class. Pull back through a resolved graph, apply the rational orbit projector, and use boundary injectivity to identify the returned class.
937 
938 \begin{manuscriptclaim}[Degree-33 packet construction]
939 Subject to the resolved graph and the cited mixed-Hodge inputs in the preceding argument, the rational packet generated by $(7,10,13,19,22,28)$ is algebraic on $X_{33}^4$. The exponent identities are finite exact certificates. The compact geometric implication is a manuscript argument and is not certified by those identities alone.
940 \end{manuscriptclaim}
941 
942 
943 
944 The episode is methodologically important. Failure of a finite cycle grammar is not evidence of nonalgebraicity. It may indicate a missing cross-family correspondence.
945 
946 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{15_klein33_route.pdf}\caption{The degree-33 torus/cubic construction requires deck invariance, full-support boundary control, a rational Hodge lift and the resolved algebraic graph.}\end{figure}
947 
948 \chapter{Infinite towers and full-conductor promotion}
949 The coordinate-power maps propagate every solved seed through infinitely many degrees.
950 
951 \begin{theorem}[Infinite packet tower]
952 Let \(W\) be an algebraic rational character packet on \(X_d^{2p}\). For every \(k\ge1\), its coordinate-power pullback is an algebraic packet on \(X_{kd}^{2p}\). Applied to the preceding degree-33 construction under its geometric hypotheses, this gives a 20-dimensional algebraic packet on every $X_{33k}^4$.
953 \end{theorem}
954 
955 The tower packet remains an old conductor-33 sector. A stronger construction appends a complementary pair
956 \[
957 (1,m-1)
958 \]
959 after scaling a solved seed to modulus \(m=kd\). If
960 \[
961 b=(b_0,\ldots,b_{2p+1})
962 \]
963 is algebraic at degree \(d\), then
964 \[
965 \Theta_{m,b}=(kb_0,\ldots,kb_{2p+1},1,m-1)
966 \]
967 has full raw conductor \(m\) and middle Hodge type two dimensions higher. Its rational orbit has dimension \(\varphi(m)\). This produces full-raw-conductor algebraic packets of unbounded rank.
968 
969 Raw conductor is not the complete novelty invariant: removing Tate pairs or reversing joins can reveal a smaller stable conductor. The stable residual should therefore be defined only after eliminating conductor descent, Tate suspension, joins, Delsarte pullbacks, and incidence channels.
970 
971 \chapter{Prime-power classification}
972 The following manuscript classification uses a primitive-character filtration and a six-entry mass bound. At filtration level one the kernel-to-lower-level argument must be replaced by the base-prime Fourier argument stated below.
973 
974 \section{Valuation strata and orbit mass}
975 Let \(m=p^r\) with \(p\) odd. Decompose the entries by exact \(p\)-adic valuation. At one stratum, write \(a=p^jb\), with \(b\in U_{p^{r-j}}\), and define the odd count
976 \[
977 \delta_j(b)=n_j(b)-n_j(-b).
978 \]
979 Since the packet has six entries,
980 \begin{equation}\label{eq:legacy-12.1}
981 \|\delta_j\|_1\le 12.\end{equation}
982 For $s\ge2$, primitive odd characters force \(\delta_j\) to be constant on the kernel cosets of
983 \[
984 U_{p^s}\to U_{p^{s-1}}.
985 \]
986 A nonzero odd kernel-orbit defect has mass at least \(2p\).
987 
988 At $s=1$, all odd characters modulo $p$ are primitive. Their sawtooth Fourier coefficients are nonzero, so the odd count vector has every Fourier coefficient zero and is zero. This gives complementary pairing at the base prime level. The higher-level mass argument applies only for $s\ge2$. The needed odd-character nonvanishing is a classical analytic leaf, checked through Sharifi's cyclotomic-field notes \cite{SharifiDirichlet}.
989 
990 \section{Large odd primes}
991 If \(p\ge7\), then \(2p>12\), so every odd defect vanishes. Each residue occurs with the same multiplicity as its complement. Therefore every packet is three complementary pairs.
992 
993 \section{The three-primary branch}
994 For $p=3$ and $r\ge2$, a higher-level kernel orbit has three elements. Define
995 \[
996 O_b=\{b,b+q,b+2q\},
997 \qquad q=3^{r-1},
998 \]
999 and
1000 \[
1001 \Gamma_b=O_b\sqcup\{3(q-b)\}.
1002 \]
1003 Take $1\le b<q$ with $3\nmid b$, ensuring all entries are nonzero. Then \(\Gamma_b\) is a balanced four-packet. A case analysis on the number of oriented kernel orbits gives exact identities expressing every full-conductor nonpair packet as a pair/surface packet or a one-step saturated sum of two balanced four-packets.
1004 
1005 \section{The five-primary branch}
1006 For $p=5$ and $r\ge2$, a nonzero odd defect consumes exactly five unit entries. The base case $r=1$ was already handled by the base-prime Fourier argument. The sixth entry is forced by balance. Up to the unit action there is one full-conductor exceptional orbit
1007 \begin{equation}\label{eq:legacy-12.2}
1008 \boxed{
1009 \alpha_{5^r}=
1010 (1,1+q,1+2q,1+3q,1+4q,5q-5),
1011 \qquad q=5^{r-1}.
1012 }\end{equation}
1013 All other packets lie in the pair/surface grammar. The family \eqref{eq:legacy-12.2} is the pure five-primary Hasse--Davenport branch.
1014 
1015 
1016 
1017 \begin{center}
1018 \small
1019 \input{tables/odd_prime_spots.tex}
1020 \end{center}
1021 
1022 \chapter{Weighted dyadic reconstruction}
1023 Prime \(2\) cannot be handled by inserting \(p=2\) into the odd-prime orbit-mass proof. The unit group has the separate form
1024 \[
1025 \mathbf Z_2^\times\cong\{\pm1\}\times(1+4\mathbf Z_2).
1026 \]
1027 The first address is a sign packet, only afterward does the principal-unit refinement begin.
1028 
1029 Let $s\ge3$, \(m=2^s\), \(h=2^{s-1}\), and write $\rho_s=\rho_{2^s}$. For an odd residue \(b\pmod h\), its two lifts are \(b\) and \(b+h\). Define
1030 \[
1031 L_sn(b)=n(b)+n(b+h),
1032 \qquad
1033 D_sn(b)=n(b)-n(b+h).
1034 \]
1035 For every odd unit \(u\), one has the exact weighted identity
1036 \begin{equation}\label{eq:legacy-13.1}
1037 \boxed{
1038 \begin{aligned}
1039 &n(b)\rho_s(ub)+n(b+h)\rho_s(u(b+h))\\
1040 &\qquad=L_sn(b)\rho_{s-1}(\bar ub)
1041 +2^{s-2}D_sn(b)\varepsilon_s(u,b),
1042 \end{aligned}
1043 }\end{equation}
1044 where \(\varepsilon_s(u,b)\in\{\pm1\}\). Repetition yields a Haar-like expansion whose detail packets carry visible powers of two. The Hodge balance equation becomes a binary carry relation between weighted details.
1045 
1046 
1047 
1048 Exact enumeration gives:
1049 \begin{center}
1050 \small
1051 \input{tables/dyadic_classification.tex}
1052 \end{center}
1053 The table suggests one residual orbit beginning at \(m=32\). The remaining derivation proposes an all-degree explanation. Its uniform status was downgraded by the later August 15 note, so this edition preserves the argument as a manuscript claim requiring a complete audit.
1054 
1055 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{13_dyadic_sat3.pdf}\caption{The three finite dyadic representatives have exact SAT(3) identities. These identities do not prove all-degree grammar coverage.}\end{figure}
1056 
1057 \section{Complement defects and dyadic triangles}
1058 Put
1059 \[
1060 q=\frac m4,
1061 \qquad h=2q,
1062 \qquad m=4q.
1063 \]
1064 For a packet count vector \(n\), define its oriented complement defect
1065 \[
1066 \partial n=\sum_{r=1}^{h-1}d_n(r)[r],
1067 \qquad
1068 d_n(r)=n_r-n_{m-r}.
1069 \]
1070 The self-complementary residue \(h\) contributes zero. Since
1071 \(\rho_m(m-r)=-\rho_m(r)\), the balanced-packet equation is equivalent to
1072 \[
1073 \sum_{r=1}^{h-1}d_n(r)\rho_m(ur)=0
1074 \qquad\text{for every odd unit }u.
1075 \]
1076 Moreover,
1077 \[
1078 \|\partial n\|_1=\sum_{r=1}^{h-1}|d_n(r)|\le 6.
1079 \]
1080 Thus the dyadic problem is a short-vector problem in an integral defect lattice.
1081 
1082 For \(1\le b<q\), define the triangle defect
1083 \[
1084 \kappa_b=[2b]-[b]+[h-b].
1085 \]
1086 It is realized by the four-packet
1087 \[
1088 \Gamma_b=(2b,h-b,h,m-b).
1089 \]
1090 
1091 \begin{manuscriptclaim}[Dyadic triangle relation]
1092 For every odd unit \(u\) and every \(1\le b<q\),
1093 \[
1094 \rho_m(2ub)-\rho_m(ub)+\rho_m(u(h-b))=0.
1095 \]
1096 Consequently \(\kappa_b\) is balanced and \(\Gamma_b\in\mathcal F_m\).
1097 \end{manuscriptclaim}
1098 \begin{proof}
1099 Let \(x=\langle ub\rangle_m\). If \(0<x<h\), then
1100 \[
1101 \rho_m(2x)=2x-h,
1102 \qquad
1103 \rho_m(x)=x-h,
1104 \qquad
1105 \rho_m(h-x)=-x.
1106 \]
1107 Their indicated combination is zero. If \(h<x<m\), write \(x=m-y\) with \(0<y<h\), then
1108 \[
1109 \rho_m(2x)=h-2y,
1110 \qquad
1111 \rho_m(x)=h-y,
1112 \qquad
1113 \rho_m(h-x)=y,
1114 \]
1115 and the same cancellation occurs. Since \(u\) is odd, \(uh\equiv h\pmod m\), so \(u(h-b)\equiv h-ub\pmod m\). This proves the identity.
1116 \end{proof}
1117 
1118 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{14_dyadic_tree32.pdf}\caption{The finite degree-32 defect tree. The proposed uniform derivation is retained separately with its publication-status boundary.}\end{figure}
1119 
1120 \section{The integral triangle basis}
1121 Let
1122 \[
1123 \mathcal K_m=
1124 \left\{
1125  d\in\mathbf Z^{h-1}:
1126  \sum_{r=1}^{h-1}d(r)\rho_m(ur)=0
1127  \text{ for every odd }u
1128 \right\}.
1129 \]
1130 The triangle relations do not merely generate examples, they form the whole balanced defect lattice.
1131 
1132 \begin{manuscriptclaim}[Integral dyadic triangle basis]
1133 For \(m=2^s\), \(s\ge3\), the vectors
1134 \[
1135 \{\kappa_b:1\le b<q\}
1136 \]
1137 form a \(\mathbf Z\)-basis of \(\mathcal K_m\).
1138 \end{manuscriptclaim}
1139 \begin{proof}
1140 Set \(r_0=q/2\). On the vertex set
1141 \[
1142 V_m=\{1,2,\ldots,q-1\},
1143 \]
1144 define, for \(b\ne r_0\),
1145 \[
1146 p(b)=
1147 \begin{cases}
1148 2b,&b<r_0,\\
1149 2(q-b),&b>r_0,
1150 \end{cases}
1151 \qquad
1152 \epsilon_b=
1153 \begin{cases}
1154 +1,&b<r_0,\\
1155 -1,&b>r_0.
1156 \end{cases}
1157 \]
1158 The map \(p\) raises the two-adic valuation by one, so repeated application reaches \(r_0\). It therefore defines a rooted tree.
1159 
1160 Suppose \(d=\sum_{b=1}^{q-1}z_b\kappa_b\). Its even coordinates are
1161 \[
1162 d(q)=z_{r_0},
1163 \qquad
1164 d(2b)=z_b-\epsilon_bz_{p(b)}
1165 \quad(b\ne r_0).
1166 \]
1167 Starting at the root and moving toward the leaves, these equations recover every \(z_b\) integrally. Hence the triangle vectors are linearly independent and their even-coordinate map is unimodular.
1168 
1169 Now take \(d\in\mathcal K_m\), recover the unique integers \(z_b\) from its even coordinates, and subtract \(\sum_bz_b\kappa_b\). The residual balanced vector is supported only on the odd residues \(1\le r<h\). The corresponding square sawtooth matrix is nonsingular. Indeed, extend an odd-support vector to an odd function on \(U_m\). Convolution with \(\rho_m\) is diagonalized by the odd Dirichlet characters, with eigenvalues
1170 \[
1171 \lambda_\chi=
1172 \sum_{a\in U_m}\rho_m(a)\overline{\chi(a)}
1173 =
1174 \sum_{a\in U_m}a\overline{\chi(a)}.
1175 \]
1176 If \(\chi\) is induced from a primitive odd character \(\chi^*\) of conductor \(2^t\), then
1177 \[
1178 \lambda_{\chi,2^s}=2^{s-t}\lambda_{\chi^*,2^t}.
1179 \]
1180 The primitive value is, up to a nonzero scalar, the generalized Bernoulli value \(B_{1,\overline{\chi^*}}\), equivalently \(-L(0,\overline{\chi^*})\), and is nonzero for every odd primitive character \cite{WashingtonCyclotomic,SharifiDirichlet}. Thus every odd Fourier eigenvalue is nonzero. The odd residual must vanish, so the \(\kappa_b\) span \(\mathcal K_m\) over \(\mathbf Z\).
1181 \end{proof}
1182 
1183 \section{The trivalent defect tree}
1184 The integral basis turns defect mass into exact tree variation. Define a sign gauge by
1185 \[
1186 \tau_{r_0}=1,
1187 \qquad
1188 \tau_b=\epsilon_b\tau_{p(b)},
1189 \qquad
1190 y_b=\tau_bz_b.
1191 \]
1192 Attach one zero-valued boundary leaf to the root and two zero-valued boundary leaves to every odd vertex. Every vertex of the resulting augmented tree \(\widetilde T_m\) has degree three. The boundary edges correspond bijectively to defect coordinates: the root edge is labelled \(q\), the edge from \(b\) to \(p(b)\) is labelled \(2b\), and the two leaves at an odd \(b\) are labelled \(b\) and \(h-b\).
1193 
1194 \begin{manuscriptclaim}[Exact trivalent-tree norm]
1195 If \(d=\sum_bz_b\kappa_b\in\mathcal K_m\), then
1196 \begin{equation}\label{eq:legacy-13.2}
1197 \boxed{
1198 \|d\|_1
1199 =|y_{r_0}|
1200 +\sum_{b\ne r_0}|y_b-y_{p(b)}|
1201 +2\sum_{\substack{1\le b<q\\ b\text{ odd}}}|y_b|.
1202 }\end{equation}
1203 The right side is the total variation of the integer function \(y\) on \(\widetilde T_m\) with zero boundary values.
1204 \end{manuscriptclaim}
1205 \begin{proof}
1206 The root coordinate is \(d(q)=z_{r_0}\). For \(b\ne r_0\), multiplying
1207 \(d(2b)=z_b-\epsilon_bz_{p(b)}\) by \(\tau_b\) gives
1208 \[
1209 \tau_bd(2b)=y_b-y_{p(b)}.
1210 \]
1211 For odd \(b\), the two odd coordinates are
1212 \[
1213 d(b)=-z_b,
1214 \qquad
1215 d(h-b)=z_b.
1216 \]
1217 Taking absolute values and summing all defect coordinates gives \eqref{eq:legacy-13.2}.
1218 \end{proof}
1219 
1220 For an integer-valued \(y\), discrete coarea gives
1221 \begin{equation}\label{eq:legacy-13.3}
1222 \operatorname{TV}(y)
1223 =
1224 \sum_{k\ge1}|\partial\{y\ge k\}|
1225 +
1226 \sum_{k\ge1}|\partial\{y\le-k\}|.\end{equation}
1227 If \(S\) is a finite forest of internal vertices of a trivalent tree and \(c(S)\) is its number of connected components, then
1228 \begin{equation}\label{eq:legacy-13.4}
1229 |\partial S|=|S|+2c(S),\end{equation}
1230 because \(3|S|=2(|S|-c(S))+|\partial S|\).
1231 
1232 \begin{manuscriptclaim}[All defect shapes of mass at most six]
1233 Let \(d\in\mathcal K_m\) satisfy \(\|d\|_1\le6\), and let \(y\) be its gauged coefficient function. Exactly one of the following occurs:
1234 \begin{enumerate}
1235 \item \(y=0\)
1236 \item \(y=\pm2\) at one vertex and is zero elsewhere
1237 \item \(y\) has two singleton components, each with value \(\pm1\)
1238 \item \(y=\pm1\) on one connected subtree containing at most four vertices and is zero elsewhere.
1239 \end{enumerate}
1240 For a balanced four-packet, only the zero shape, a single vertex, or one tree edge can occur.
1241 \end{manuscriptclaim}
1242 \begin{proof}
1243 Apply \eqref{eq:legacy-13.3}--\eqref{eq:legacy-13.4}. A nonempty level-set component with \(k\) vertices costs \(k+2\). If \(|y|\ge2\), two nonempty nested level sets already cost at least six, equality forces both to be the same singleton. If positive and negative values both occur, their first level sets cost at least three each, equality forces two singleton components. Otherwise \(y\) has one sign, and a connected support of size \(k\) costs \(k+2\), so \(k\le4\). Two same-sign components can have total cost at most six only when both are singletons. Replacing six by four leaves only a vertex or an edge.
1244 \end{proof}
1245 
1246 \section{SAT(2) closure and the four-star obstruction}
1247 For a defect \(d\), let \(\operatorname{red}(d)\) be the unmatched residue multiset: a positive coefficient at \(r\) contributes copies of \(r\), while a negative coefficient contributes copies of \(m-r\). Its length is \(\|d\|_1\). Every packet with defect \(d\) is \(\operatorname{red}(d)\) together with complementary pairs and copies of \(h\).
1248 
1249 A one-vertex tree atom has defect \(\pm\kappa_b\) after the gauge sign and is represented by \(\Gamma_b\) or its conjugate, its reduced length is three and its fourth entry is \(h\). A one-edge atom has defect mass four, so its reduced multiset itself is a balanced four-packet. A zero atom is any union of two complementary pairs.
1250 
1251 \begin{manuscriptclaim}[Exact SAT(2) criterion]
1252 Let \(n\) be a balanced six-packet at modulus \(m=2^s\). Then \(n\) is pair-decomposable or has an SAT(2) certificate unless its gauged defect is \(\pm1\) on a four-vertex star \(K_{1,3}\) in the internal tree \(T_m\). A four-star has no SAT(2) certificate.
1253 \end{manuscriptclaim}
1254 \begin{proof}
1255 Every nonstar shape in the preceding proposition splits into at most two four-packet atoms:
1256 \begin{itemize}
1257 \item a vertex or edge is paired with a zero atom
1258 \item a three-vertex path is an edge plus its remaining terminal vertex
1259 \item a four-vertex path is the union of two disjoint edges
1260 \item a doubled singleton is two copies of one vertex atom
1261 \item two singleton components are the corresponding two vertex atoms.
1262 \end{itemize}
1263 These defect splittings lift to exact nonnegative multiset identities. For a vertex, the neutral remainder of the six-packet is \(h\) plus one complementary pair, for an edge it is one complementary pair. A zero four-packet absorbs that pair and the stabilizing pair. Cutting a three- or four-vertex path produces exactly one oppositely oriented boundary edge, hence exactly one complementary pair in the union of the two atom packets, this is the SAT(2) stabilizer. In the three-vertex case, the vertex atom also supplies the unique neutral \(h\). For two vertex atoms, their two neutral entries are \((h,h)=p_h\). The zero-defect case is already a union of three complementary pairs. Thus in every nonstar case there are \(\gamma_1,\gamma_2\in\mathcal F_m\) and a complementary pair \(p_r\) such that
1264 \[
1265 n+p_r=\gamma_1+\gamma_2.
1266 \]
1267 
1268 Conversely, suppose a four-star defect had an SAT(2) certificate. Each balanced four-packet has tree support equal to zero, one vertex, or one edge. The sum of two such supports must produce four nonzero star vertices. Therefore both supports must be disjoint edges covering all four vertices. This would be a perfect matching of \(K_{1,3}\), which does not exist. Hence the star is the unique tree obstruction to SAT(2).
1269 \end{proof}
1270 
1271 \section{Conductor of a star and its unique unit orbit}
1272 The tree also determines exactly when the star obstruction has full conductor.
1273 
1274 \begin{manuscriptclaim}[Star conductor]
1275 Let a four-star have center \(c\), and write \(j=v_2(c)\). The gcd of \(m\) with its six packet entries is
1276 \[
1277 2^{j-1}.
1278 \]
1279 Hence a star has full conductor exactly when \(v_2(c)=1\).
1280 \end{manuscriptclaim}
1281 \begin{proof}
1282 Use the boundary-edge labels of \(\widetilde T_m\). The two children of \(c\) have valuation \(j-1\). The four boundary edges leaving those children have labels of exact valuation \(j-1\): for \(j=1\) they are the odd leaf labels, and for \(j>1\) they are labels \(2b\) coming from grandchildren of valuation \(j-2\). The two boundary edges leaving the parent side have larger valuation. Thus every nonzero defect coordinate is divisible by \(2^{j-1}\), and at least four have exact valuation \(j-1\). Replacing a defect coordinate \(r\) by the packet entry \(m-r\) does not change its two-adic valuation. The stated gcd follows.
1283 \end{proof}
1284 
1285 Assume now \(s\ge5\) and let a full-conductor star have center \(c=2a\) with \(a\) odd. Up to global conjugation, expanding its four triangle atoms gives the unordered packet
1286 \begin{equation}\label{eq:legacy-13.5}
1287 P_a=
1288 (8a,q-a,2q-a,2q-4a,3q-a,4q-a).\end{equation}
1289 The cases \(c<q/2\) and \(c>q/2\) give the same unordered tuple after the internal-edge cancellations. Choose an odd \(u\) satisfying
1290 \[
1291 ua\equiv1\pmod{2q}.
1292 \]
1293 Since \(uq\equiv q\) or \(3q\pmod{4q}\), multiplication by \(u\) sends \eqref{eq:legacy-13.5} to
1294 \[
1295 P_1=(8,q-1,2q-4,2q-1,3q-1,4q-1).
1296 \]
1297 Its conjugate is
1298 \begin{equation}\label{eq:legacy-13.6}
1299 \boxed{
1300 A_m=
1301 \left(
1302 1,\frac m4+1,\frac m2+1,\frac m2+4,
1303 \frac{3m}4+1,m-8
1304 \right).
1305 }\end{equation}
1306 Thus every full-conductor star belongs to one \(U_m\)-orbit.
1307 
1308 The packet \(A_m\) obeys the exact uniform SAT(3) identity
1309 \begin{equation}\label{eq:legacy-13.7}
1310 \begin{aligned}
1311 &A_m+(2,m-2)+(4,m-4)+(m/2,m/2)\\
1312 &=(1,m/2,m/2+1,m-2)\\
1313 &\quad +(2,m/4+1,3m/4+1,m-4)\\
1314 &\quad +(4,m/2,m/2+4,m-8).
1315 \end{aligned}\end{equation}
1316 The first and third packets on the right have defects \(-\kappa_1\) and \(-\kappa_4\). The middle packet has defect \(-\kappa_2+\kappa_{q-1}\), an edge atom. Hence all three are balanced four-packets, and \eqref{eq:legacy-13.7} is an identity of multisets.
1317 
1318 \begin{manuscriptclaim}[All-degree dyadic uniqueness]
1319 Let \(m=2^s\), \(s\ge3\). Every full-conductor balanced six-packet is pair-decomposable or SAT(2), except that for \(s\ge5\) there is exactly one residual unit orbit, represented by \(A_m\). This orbit is SAT(3) by \eqref{eq:legacy-13.7}. For \(m=8\) and \(m=16\), no four-star exists and there is no residual orbit.
1320 \end{manuscriptclaim}
1321 \begin{proof}
1322 The trivalent-tree norm and coarea classification list every defect of mass at most six. The SAT(2) criterion closes every listed shape except a four-star. The star-conductor lemma forces a full-conductor star center to have valuation one, and the calculation \eqref{eq:legacy-13.5}--\eqref{eq:legacy-13.6} places all such stars in the single orbit of \(A_m\). The trees for \(m=8\) and \(m=16\) have no internal vertex with three internal neighbors. Finally, \eqref{eq:legacy-13.7} closes the unique residual orbit at saturation depth three.
1323 \end{proof}
1324 
1325 \begin{manuscriptclaim}[Complete dyadic grammar]
1326 After exact conductor descent, every balanced packet at every dyadic modulus is generated by complementary pairs, SAT(2) surface packets, and coordinate-power pullbacks of the single SAT(3) family \(A_{2^t}\). No second residual dyadic family occurs at any depth.
1327 \end{manuscriptclaim}
1328 
1329 \begin{resultbox}[Uniform dyadic status]
1330 The August 6 manuscript called uniform uniqueness closed. The August 15 status note restored it to a uniform write-up obligation. The proposed proof uses an integral triangle basis, an exact total-variation model, discrete coarea, and star conductor. This edition does not reverse the later downgrade on the strength of finite agreement.
1331 \end{resultbox}
1332 
1333 The archived companion dyadic verification report records complete orbit counts through $m=128$ and structural identities through $2^{14}$. The October replay freshly reconstructs the complete census through 128 and the triangle/tree identities through $2^9$. The report file is preserved evidence, rather than an executable verifier or a current replay of the larger historical range. These computations support the finite identities. They do not supply a current independent proof of the all-degree claim.
1334 
1335 \chapter{Finite global audit through degree 100}
1336 The finite audit exactly enumerates balanced packets, reduces them by unit orbits, checks pair and SAT(2) certificates, and searches higher saturated identities with \(N\le8\). Every returned identity is rechecked by exact integer arithmetic.
1337 
1338 The archived named verification groups report the following outcomes:
1339 \begin{center}
1340 \small
1341 \input{tables/fermat_named_checks.tex}
1342 \end{center}
1343 The archived dyadic report records \(89{,}478{,}480\) coefficient checks through \(2^{14}\). The October replay independently checked 87,376 coefficients for exponents $3\le s\le9$. The larger historical count was not rerun in full. The archived packet audit reports completeness through degree \(100\) for its implemented grammar, which is not a completeness theorem for all correspondence channels.
1344 
1345 After removing the degree-33 Klein channel and the degree-39 curve channel, every remaining post-saturation degree in the audit is divisible by five. This is a powerful localization of the residual grammar, but it is not by itself a universal proof.
1346 
1347 
1348 
1349 \section{Exact residual representatives}
1350 \small
1351 \input{tables/residual_degrees_100.tex}
1352 \normalsize
1353 
1354 The residuals are not random. Their mod-5 phase profiles form a small local grammar.
1355 
1356 \compactpart{Part VIII. CRT--Kummer Polyphase and the Imaginary CM Half-Twist}
1357 
1358 \chapter{Local mod-5 phase grammar}
1359 Assume \(m=5q\) and first suppose \((5,q)=1\). For \(b\pmod q\) and a phase \(r\in\{0,1,2,3,4\}\), let \(\lambda_r(b)\) be the CRT lift satisfying
1360 \[
1361 \lambda_r(b)\equiv r\pmod5,
1362 \qquad
1363 \lambda_r(b)\equiv b\pmod q.
1364 \]
1365 The finite audit reveals four canonical multiplicity profiles, up to multiplication by a unit modulo five:
1366 \[
1367 (1,0,0,0,5),
1368 \quad
1369 (2,1,1,1,1),
1370 \quad
1371 (1,0,2,1,2),
1372 \quad
1373 (1,0,1,3,1).
1374 \]
1375 The first is the complete-five-phase K5 packet, the second is a four-phase K4 lift, and the final two are one K23 pair--triple mechanism.
1376 
1377 
1378 
1379 \section{K4 four-phase reduction}
1380 Choose nonzero residues \(b,c,d\pmod q\) and put
1381 \[
1382 e\equiv-5^{-1}b\pmod q.
1383 \]
1384 Define
1385 \[
1386 A_{b,c,d}=
1387 (\lambda_1(b),\lambda_2(b),\lambda_3(b),\lambda_4(b),5c,5d)
1388 \]
1389 and
1390 \[
1391 \Gamma_{b,c,d}=(b,c,d,e).
1392 \]
1393 
1394 \begin{theorem}[Four-phase CRT reduction]
1395 The six-packet \(A_{b,c,d}\) is balanced modulo \(5q\) if and only if the four-packet \(\Gamma_{b,c,d}\) is balanced modulo \(q\).
1396 \end{theorem}
1397 \begin{proof}
1398 For a unit \(u\), let \(s=\langle ub\rangle_q\) and \(z=\langle u5^{-1}b\rangle_q\). The complete five-element fiber over \(s\) has sum \(5s+10q\), its phase-zero lift is \(5z\). The four nonzero phases therefore sum to \(5s+10q-5z\). Adding the last two entries and imposing the middle value \(15q\) gives
1399 \[
1400 s+\langle uc\rangle_q+\langle ud\rangle_q-z=q.
1401 \]
1402 Since \(\langle ue\rangle_q=q-z\), this is exactly the balance equation for \((b,c,d,e)\).
1403 \end{proof}
1404 
1405 All 16 audited K4 residuals reduce exactly to balanced four-packets.
1406 
1407 \chapter{K23 pair--triple factorization}
1408 Let a K23 packet have one phase-zero entry and five nonzero phases:
1409 \[
1410 A=P\sqcup T\sqcup\{5e\},
1411 \qquad |P|=2,
1412 \quad |T|=3,
1413 \]
1414 with
1415 \[
1416 \sum P\equiv0\pmod5,
1417 \qquad
1418 \sum T\equiv0\pmod5.
1419 \]
1420 Write
1421 \[
1422 \sum P=5C,
1423 \qquad
1424 \sum T=5D,
1425 \]
1426 and reduce \(C,D\) modulo \(q\) to representatives \(c,d\). Require $c,d,e$ to be nonzero modulo $q$ so that all three resulting character blocks have full support. Degenerate cases must leave this construction and be handled through pair removal or another channel. For example $(1,14,2,9,14,5)$ at modulus 15 is balanced and admits the indicated phase split, but $P=(1,14)$ gives $c=0$. The phase condition alone is insufficient. Define
1427 \[
1428 C_P=P\sqcup\{m-5c\},
1429 \]
1430 \[
1431 S_T=T\sqcup\{m-5d\},
1432 \]
1433 and
1434 \[
1435 L=(5c,5d,5e).
1436 \]
1437 
1438 For any nonzero-entry block $B$ at modulus $m$, put $h_B(u)=m^{-1}\sum_{r\in B}\langle ur\rangle_m$, with multiplicities retained.
1439 
1440 \begin{theorem}[Exact K23 packet identity]
1441 One has the exact multiset equality
1442 \begin{equation}\label{eq:legacy-15.1}
1443 \boxed{
1444 A+(5c,m-5c)+(5d,m-5d)=C_P+S_T+L.
1445 }\end{equation}
1446 For every unit \(u\pmod m\),
1447 \[
1448 h_{C_P}(u)+h_{S_T}(u)+h_L(u)=5,
1449 \]
1450 and \(c+d+e\equiv0\pmod q\).
1451 \end{theorem}
1452 
1453 The five nonzero phases have only two irreducible possibilities: all five phases are equal, or they contain a complementary pair and a zero-sum triple. Indeed, if no complementary phase pair occurs, the phase support lies in one of four two-element subsets, the zero-sum condition forces all five entries to use one phase. This proves the local K5/K23 dichotomy.
1454 
1455 
1456 
1457 After fixing a common cyclotomic coefficient field and a diagonal character projector, write the selected factor schematically as
1458 \[
1459 N=e_{\mathrm{diag}}\bigl(M(C_P)\otimes M(S_T)\bigr).
1460 \]
1461 An ordinary tensor product of rational packets contains off-diagonal embedding pairs and is generally not rank one. Rank one here refers only to a selected eigenspace after scalar extension. The coefficient field and projector are part of the unresolved correspondence specification.
1462 The Hodge-level identity implies that \(N(1)\) has weight one and has Hodge type opposite to the lower curve packet \(M(L)\) at every embedding.
1463 
1464 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{16_k23_factorization50.pdf}\caption{A valid finite K23 factorization at modulus 50, with all required character entries nonzero.}\end{figure}
1465 
1466 \chapter{Imaginary CM selector}
1467 Write the lower curve packet as
1468 \[
1469 \ell=(c,d,e)\pmod q,
1470 \qquad c+d+e\equiv0\pmod q.
1471 \]
1472 Let \(K=\Q(\zeta_q)\) and define
1473 \begin{equation}\label{eq:legacy-16.1}
1474 \boxed{
1475 \Delta_\ell=(1-\zeta_q^c)(1-\zeta_q^d)(1-\zeta_q^e).
1476 }\end{equation}
1477 
1478 \begin{theorem}[Pure imaginary selector]
1479 \[
1480 \overline{\Delta_\ell}=-\Delta_\ell.
1481 \]
1482 \end{theorem}
1483 \begin{proof}
1484 Since
1485 \[
1486 1-\zeta_q^{-a}=-\zeta_q^{-a}(1-\zeta_q^a),
1487 \]
1488 complex conjugation multiplies the product by
1489 \[
1490 (-1)^3\zeta_q^{-(c+d+e)}=-1.
1491 \]
1492 \end{proof}
1493 
1494 For an embedding \(\sigma_u:\zeta_q\mapsto\zeta_q^u\), let
1495 \[
1496 h_\ell(u)=\frac{\langle uc\rangle_q+\langle ud\rangle_q+\langle ue\rangle_q}{q}\in\{1,2\}.
1497 \]
1498 Using
1499 \[
1500 1-e^{2\pi ir/q}=-2i\,e^{\pi ir/q}\sin\frac{\pi r}{q},
1501 \]
1502 one obtains
1503 \begin{equation}\label{eq:legacy-16.2}
1504 \sigma_u(\Delta_\ell)
1505 =8i(-1)^{h_\ell(u)}
1506 \prod_{j=1}^3
1507 \sin\frac{\pi\langle u\ell_j\rangle_q}{q}.\end{equation}
1508 Here $(\ell_1,\ell_2,\ell_3)=(c,d,e)$ is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of \(N(1)\) is the conjugate of the CM type of the lower curve packet:
1509 \begin{equation}\label{eq:legacy-16.3}
1510 \boxed{
1511 \Sigma_N=\overline{\Sigma_\ell}.
1512 }\end{equation}
1513 This is the rigorous form of the statement that the half-twist occurs on the imaginary plane.
1514 
1515 
1516 
1517 The exact audit generates 364 K23 certificates from seven rational packet orbits. Every certificate obeys the opposite curve CM sign, factors through a lower conductor, and is a Galois translate of an upper-half-plane CM type.
1518 
1519 \begin{center}
1520 \small
1521 \input{tables/cm_audit.tex}
1522 \end{center}
1523 
1524 In the audited cases, the minimal CM conductors are \(3,5,7,\) or \(9\), even when the original fourfold modulus is \(50,70,\) or \(90\). Thus the weight-one carrier is not arbitrary: it is a character factor of the Jacobian of a lower-degree Fermat curve.
1525 
1526 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{17_cm_imaginary_sign.pdf}\caption{The exact imaginary selector sign in a retained example. Matching signs does not construct a rational Chow correspondence.}\end{figure}
1527 
1528 \chapter{Explicit carrier and remaining Chow map}
1529 For an audited CM type, the lower Fermat curve provides a candidate rational character-orbit factor of its Jacobian. Write $e_{\mathrm{orb}}J(F_d)$ for that factor, up to isogeny. The idempotent selects the complete rational Galois orbit. A projector onto just one holomorphic CM half-plane would not be rational and is not an algebraic rational packet projector.
1530 
1531 The desired correspondence is schematically
1532 \[
1533 \Gamma\in\CH^{\ast}(F_d\times C_P\times S_T)_{\Q},\qquad
1534 \Gamma_*:e_{\mathrm{orb}}H^1(F_d)\longrightarrow
1535 e_{\mathrm{diag}}\bigl(M(C_P)\otimes M(S_T)\bigr)(1).
1536 \]
1537 This formula is a target specification, not a constructed map. Its coefficient field, diagonal projector, rational multiplicities, and dimensions must be fixed before an isomorphism can be asserted. The smaller CM conductor can have a smaller rational degree than the original packet. Matching sign functions alone does not resolve that difference.
1538 
1539 After the typing has been supplied, a nonzero equivariant map between corresponding rank-one eigenspaces is an isomorphism over the splitting field. Proving that the map is induced by an algebraic correspondence remains essential. One needs a Kummer or half-twist correspondence with the correct covariance and a nonzero period or intersection coefficient. Tensoring a verified map with the conjugate lower curve packet can then place a class in divisor degree on a product of curves or abelian varieties. Lefschetz $(1,1)$ and actual Fermat join maps would transport it back.
1540 
1541 \begin{openproblem}[Typed CM packet correspondence]
1542 Specify the fields, projectors and rational multiplicities, construct the algebraic map, and prove its nonzero character coefficient. The 364 audited sign certificates identify candidate factors. They do not solve this problem. The finite-field Chow-motive constructions of Otsubo and Yamazaki provide related context, and are not a characteristic-zero correspondence proof for these packets \cite{OtsuboYamazaki}.
1543 \end{openproblem}
1544 
1545 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{18_missing_chow_bridge.pdf}\caption{The typed algebraic CM bridge is an open construction. Coefficient field, projectors, multiplicities and nonzero period are part of the target.}\end{figure}
1546 
1547 \chapter{Correct literature and novelty map}
1548 Kang is cited in the checked later literature for the all-degree Fermat-fourfold Hodge result, while its original exact theorem and proof were not extracted here \cite{KangFermat}. Separately, da Silva studies which packets are generated by Shioda-style induced-cycle criteria and records the degree-$33$ failure of that criterion \cite{DaSilvaFermat}. The October source contracts authenticate the Kang attribution through the checked later literature while recording that its original proof was not extracted. No historical priority claim is made for this constructive program.
1549 
1550 The constructive contribution of the packet program is therefore not best described as the first proof of the rational Hodge conjecture for all Fermat fourfolds. Its strongest defensible targets are:
1551 \begin{itemize}
1552 \item explicit conductor projectors as algebraic correspondences
1553 \item an algorithmic packet certificate grammar
1554 \item a worked cross-family Klein--Delsarte construction at degree 33
1555 \item infinite conductor towers and full-raw-conductor promotions
1556 \item odd prime-power packet classifications
1557 \item a weighted dyadic recursion and a proposed all-degree classification with one SAT(3) orbit
1558 \item exact CRT--Kummer reductions for K4 and K23 packets
1559 \item an imaginary CM selector and explicit lower Fermat-curve carriers
1560 \item an explicit packet-resolved Chow correspondence algorithm
1561 \end{itemize}
1562 
1563 \begin{resultbox}[Constructive status]
1564 The abstract existence question for Fermat fourfolds is treated as prior published work. The packet program supplies a detailed constructive architecture and several explicit manuscript correspondence constructions, without a historical novelty claim. Universal constructive coverage also requires a complete mixed-conductor grammar. The typed CM correspondence is one focused gap inside that larger problem.
1565 \end{resultbox}
1566 
1567 \compactpart{Part IX. Verification, Audit, and Final Constructive Theorem}
1568 
1569 \chapter{Verification ledger}
1570 The complete edition separates symbolic theorems, finite exact checks, geometric correspondences, and literature claims.
1571 
1572 \section{Finite checks}
1573 The original Hodge-calculus release passed 18 finite checks. The archived Fermat extension reports the following historical checks, whose fresh replay ranges are distinguished in the current evidence chapter:
1574 \begin{itemize}
1575 \item 14 exact degree-33 Klein checks
1576 \item 4 infinite tower checks
1577 \item 4 exact conductor-polyphase checks
1578 \item 13 odd prime-power checks
1579 \item 89,478,480 dyadic weighted coefficient identities
1580 \item exact packet classifications at dyadic degrees through 128
1581 \item an independent trivalent-tree/SAT(2) audit and structural dyadic checks through $2^{14}$
1582 \item exact balanced-packet audit through degree 100
1583 \item 364 K23 CM-type certificates
1584 \end{itemize}
1585 
1586 \section{What finite checks establish}
1587 They establish exact arithmetic identities, matrix products, orbit counts, conductor decompositions, Hodge balance, and nonnegative multiset equalities for every tested instance. They also detect false universal conjectures quickly. The proposed dyadic uniqueness statement is not inferred from these checks. Its manuscript derivation is preserved with the later uniform-write-up downgrade.
1588 
1589 \section{What finite checks do not establish}
1590 They do not replace:
1591 \begin{itemize}
1592 \item a universal mixed-prime phase-classification proof beyond the audited range
1593 \item a Chow-level Kummer--Schur or half-twist correspondence
1594 \item a historical priority audit
1595 \end{itemize}
1596 
1597 \chapter{Final theorem stack}
1598 The constructive Fermat program can be stated as a dependency stack.
1599 
1600 \begin{theorem}[Conditional explicit packet theorem]
1601 Fix a packet that is admitted by the implemented certificate grammar. Assume the following correspondence statements for its chosen channel:
1602 \begin{enumerate}
1603 \item saturated Fermat joins are associative on Chow motives and complementary Tate factors cancel
1604 \item every Delsarte torus map used by the certificate grammar extends through a resolved graph with full-support boundary annihilation
1605 \item every K5, K4, and K23 packet used by the chosen certificate has its correctly typed Chow correspondence with a proved nonzero character coefficient
1606 \end{enumerate}
1607 Then that packet has an explicit packet-resolved algebraic cycle representative. A conclusion for every Fermat fourfold packet additionally requires universal mixed-conductor coverage by this grammar, which has not been proved here.
1608 \end{theorem}
1609 \begin{proof}
1610 Apply the exact conductor projectors. Descend each nonzero component to its full-conductor seed. Use the supplied grammar certificate for the seed, whether pair, saturation, curve, Delsarte, dyadic, or a Kummer phase channel. Use the corresponding Chow map to produce a nonzero cycle projection. Pull old packets back to the original degree and sum the orthogonal conductor components. Finally use the rational orbit projector or rational group-module reconstruction. A bare sum of all complex conjugate eigenvectors may vanish and is not a substitute for that rational construction.
1611 \end{proof}
1612 
1613 \begin{openproblem}[Unconditional constructive completion]
1614 Write the missing correspondence maps explicitly and verify their nonzero packet matrix coefficients. A concentrated target is the correctly typed CM correspondence specified above. Universal mixed-conductor coverage is a separate obligation.
1615 \end{openproblem}
1616 
1617 \chapter{Research checklist}
1618 \begin{longtable}{p{0.42\textwidth}p{0.17\textwidth}p{0.32\textwidth}}
1619 \toprule
1620 Task & Status & Required proof object \\
1621 \midrule\endfirsthead
1622 \toprule
1623 Task & Status & Required proof object \\
1624 \midrule\endhead
1625 Universal balanced packet equation & Complete & Finite residue matrix \\
1626 Exact conductor descent & Complete & Coordinate-power graph \\
1627 M\"obius conductor projectors & Complete & Rational graph combination \\
1628 Degree-33 Klein packet & Manuscript-level complete & Resolved graph and residue pullback \\
1629 Infinite old-packet towers & Complete & Pullback/pushforward identity \\
1630 Odd prime-power classification & Symbolically complete & Primitive-character filtration \\
1631 Dyadic finite identities & Replayed in a bounded range & Weighted coefficient identities and finite tree data \\
1632 K4 CRT reduction & Complete & Lower balanced four-packet \\
1633 K23 local phase reduction & Audited instances & Pair--triple identity with nonzero-entry conditions \\
1634 K23 CM sign descent & Complete & Imaginary selector $\Delta_\ell$ \\
1635 Explicit lower CM carrier & Identified & Fermat-curve character idempotent \\
1636 CM carrier Chow map & Open & Nonzero algebraic correspondence \\
1637 All-degree constructive algorithm & Open & Universal coverage plus all required correspondence maps \\
1638 Historical priority map & Open audit & Compare existence and constructive literature \\
1639 \bottomrule
1640 \end{longtable}
1641 
1642 \chapter{Complete-edition conclusion}
1643 The projector-resolved approach transforms the Hodge problem into a hierarchy of exact finite packets and explicit geometric transfers. For arbitrary varieties, the master obstruction remains the Hodge packet not occupied by algebraic-cycle channels. For Fermat fourfolds, diagonal symmetry makes every stage concrete: balanced residue equations, conductor projectors, prime-local refinements, nonnegative packet identities, and rank-one character motives.
1644 
1645 The degree-33 rejection attempt illustrates the central lesson. A packet may evade every obvious lower-packet construction and still be algebraic through a hidden cross-family correspondence. A completed Klein--Delsarte correspondence supplies a seed for infinite conductor towers. The odd prime-power manuscript classification and the proposed dyadic tree derivation explain why local primes require distinct packet grammars. The audited mixed-prime residuals display a small Kummer phase system, without a proof that these patterns exhaust arbitrary conductors. Finally, the imaginary CM selector identifies the precise upper/lower embedding split and reduces the unknown weight-one carrier to a small Fermat-curve Jacobian factor.
1646 
1647 The constructive endpoint is now sharply defined:
1648 \begin{resultbox}[Final constructive endpoint]
1649 Construct a correctly typed nonzero equivariant Chow correspondence for each admitted CM packet, and separately prove universal coverage of the certificate grammar.
1650 \end{resultbox}
1651 This task is narrower than the original conjecture and stronger than a numerical packet classification. It asks for an algorithmic cycle formula that exposes exactly how a Hodge packet becomes algebraic.
1652 
1653 

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