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1 \hypertarget{carrier70-a-normal-weighted-carrier-at-degree-70}{%
2 \chapter{A normal weighted carrier at degree
3 70}\label{ch:carrier70}\label{carrier70-a-normal-weighted-carrier-at-degree-70}}
4 
5 The September 21 construction, reconciled with the October exact replay
6 and source audit.
7 
8 \textbf{Status:} Manuscript-level special-case proof, with a separately
9 replayable exact algebraic certificate. The geometric proof remains at
10 manuscript level. The October review separately replayed the exact
11 algebra and checked the stated classical leaves. This is not a formal
12 verification of the geometry. This note does not prove the general Hodge
13 conjecture, does not cover every degree-70 packet, and makes no
14 historical novelty claim.
15 
16 \hypertarget{carrier70-precise-statement}{%
17 \section{Precise statement}\label{carrier70-precise-statement}}
18 
19 Write
20 
21 \[
22 X_m=\{x_0^m+\cdots+x_5^m=0\}\subset\mathbf P^5_\mathbf C.
23 \]
24 
25 The diagonal group is \(G_m=(\mu_m)^6/\mu_m\). For a character
26 \(a=(a_0,\ldots,a_5)\) with all entries nonzero modulo \(m\) and sum
27 zero modulo \(m\), let \(W_{[a]}\) denote the rational character-orbit
28 summand of primitive \(H^4(X_m,\mathbf Q)\).
29 
30 \textbf{Proposition.} For
31 
32 \[
33 m=70,\qquad a=(1,20,24,42,61,62),
34 \]
35 
36 all classes in the 24-dimensional rational space \(W_{[a]}\) are
37 rational linear combinations of classes of algebraic surfaces on
38 \(X_{70}\). For every positive integer \(k\), the corresponding
39 pulled-back packet \(W_{[ka]}\) on \(X_{70k}\) is also algebraic.
40 
41 The proof below uses the established rational Hodge theorem for smooth
42 projective uniruled fourfolds. It constructs the carrier and the
43 monomial correspondence explicitly. It does \textbf{not} list equations
44 for a basis of the surfaces supplied by that theorem.
45 
46 \hypertarget{carrier70-hodge-type-of-the-packet}{%
47 \section{Hodge type of the packet}\label{carrier70-hodge-type-of-the-packet}}
48 
49 The primitive Fermat eigenspaces for full-support characters are
50 one-dimensional over the splitting field. The Griffiths residue
51 description assigns type \((2,2)\) when the least-positive residues have
52 sum \(3m\). A rational orbit is of type \((2,2)\) precisely when this
53 holds for all its unit conjugates.
54 
55 For every \(t\in(\mathbf Z/70\mathbf Z)^\times\), exact enumeration
56 gives
57 
58 \[
59 \sum_{i=0}^{5}\langle ta_i\rangle_{70}=210.
60 \]
61 
62 There are 24 conjugates, all distinct because \(a_0=1\). Thus the entire
63 rational orbit, not just a single complex residue vector, has type
64 \((2,2)\). The verifier stores every conjugate. The distinction between
65 a complex residue vector and a rational Hodge class is essential here.
66 
67 The standard residue basis follows from Griffiths' description, a modern
68 source with the formula is Villaflor Loyola, \emph{Small codimension
69 components of the Hodge locus containing the Fermat variety},
70 arXiv:2001.01019, Section 1. For Fermat, the basis monomials diagonalize
71 the group action, giving the character description used here.
72 
73 \hypertarget{carrier70-the-exact-carrier}{%
74 \section{The exact carrier}\label{carrier70-the-exact-carrier}}
75 
76 Let
77 
78 \[
79 A=\begin{pmatrix}
80 0&0&2&0&0&3\\
81 0&0&1&2&7&0\\
82 1&0&2&2&0&1\\
83 0&5&0&0&0&0\\
84 0&0&0&2&0&3\\
85 3&0&0&0&0&0
86 \end{pmatrix}.
87 \]
88 
89 Rows are monomial exponents, columns correspond to \(y_0,\ldots,y_5\).
90 Direct rational arithmetic gives
91 
92 \[
93 \det A=-2100,\qquad
94 A^{-1}\mathbf1=(1/3,1/5,1/10,1/10,1/10,4/15),
95 \]
96 
97 \[
98 \mathbf1^TA^{-1}\mathbf1=11/10.
99 \]
100 
101 Set \(W=(10,6,3,3,3,8)\) and \(D=30\), so \(AW=D\mathbf1\). Define
102 
103 \[
104 Y=\{F=0\}\subset\mathbf P(10,6,3,3,3,8),
105 \]
106 
107 \[
108 F=y_0^3+y_1^5+y_2^2y_5^3+y_2y_3^2y_4^7
109   +y_0y_2^2y_3^2y_5+y_3^2y_5^3.
110 \]
111 
112 Every monomial has weighted degree 30. The adjunction coefficient is
113 
114 \[
115 D-\sum W_i=30-33=-3.
116 \]
117 
118 This model is singular and is not quasismooth. The argument therefore
119 cannot invoke a quasismooth residue theorem or assume that its
120 resolution is Fano. Normality and uniruledness are proved separately
121 below.
122 
123 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{19_carrier70_matrix.pdf}\caption{The exact exponent matrix. Rows are monomials and columns are target coordinates.}\end{figure}
124 
125 \hypertarget{carrier70-an-explicit-finite-cover-on-the-tori}{%
126 \section{An explicit finite cover on the
127 tori}\label{carrier70-an-explicit-finite-cover-on-the-tori}}
128 
129 Let \(M=210\), \(\alpha=3a\), and
130 
131 \[
132 B=210A^{-1}=\begin{pmatrix}
133 0&0&0&0&0&70\\
134 0&0&0&42&0&0\\
135 42&0&63&0&-63&-21\\
136 -63&0&63&0&42&-21\\
137 12&30&-27&0&-3&9\\
138 42&0&-42&0&42&14
139 \end{pmatrix}.
140 \]
141 
142 Then
143 
144 \[
145 AB=210I,\qquad B\mathbf1=7W.
146 \]
147 
148 On the projective tori define \(y_i=\prod_jx_j^{B_{ij}}\). Negative
149 exponents are allowed there. The second identity makes this well-defined
150 into the weighted projective torus, the first gives
151 
152 \[
153 F(y(x))=x_0^{210}+\cdots+x_5^{210}.
154 \]
155 
156 The induced map of five-dimensional tori is an isogeny. Consequently its
157 restriction is a finite étale cover
158 
159 \[
160 q:U_{210}\longrightarrow U_Y,
161 \]
162 
163 where \(U_{210}=X_{210}\cap\{\prod x_i\ne0\}\) and
164 \(U_Y=Y\cap\{\prod y_i\ne0\}\).
165 
166 The target torus hypersurface is smooth because its finite étale
167 pullback is smooth. It is irreducible because its Laurent polynomial
168 pulls back under an injective full-rank monomial substitution to an
169 irreducible Fermat Laurent polynomial. Non-monomial factors remain
170 non-monomial under this substitution. The polynomial \(F\) has no
171 coordinate factor, so its projective closure \(Y\) is also irreducible.
172 
173 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{22_carrier70_torus_matrix.pdf}\caption{The exact Laurent exponent matrix, with negative entries permitted on the torus.}\end{figure}
174 
175 \hypertarget{carrier70-normality-and-the-canonical-class}{%
176 \section{Normality and the canonical
177 class}\label{carrier70-normality-and-the-canonical-class}}
178 
179 Put \((u,v,s,t,c,z)=(y_0,y_1,y_2,y_3,y_4,y_5)\), so
180 
181 \[
182 F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.
183 \]
184 
185 Its gradient is
186 
187 \[
188 \begin{aligned}
189 F_u&=3u^2+s^2t^2z, &F_v&=5v^4,\\
190 F_s&=2sz^3+t^2c^7+2ust^2z,
191 &F_t&=2t(z^3+sc^7+us^2z),\\
192 F_c&=7st^2c^6,
193 &F_z&=3z^2(s^2+t^2)+us^2t^2.
194 \end{aligned}
195 \]
196 
197 The cone singular locus is exactly
198 
199 \[
200 \{y_0=y_1=y_2=y_3=0\}
201 \ \cup\ \{y_0=y_1=y_3=y_5=0\}
202 \ \cup\ \{y_0=y_1=y_4=y_5=0\}.
203 \]
204 
205 Here is a direct verification of exhaustion. At a singular point,
206 \(v=0\) and \(st^2c^6=0\). The cases \(s=0\) or \(t=0\) give the
207 displayed planes immediately from the other derivatives. In the
208 remaining case \(s,t\ne0\), one has \(c=0\). If \(z=0\), then \(u=0\),
209 giving the third plane. If \(z\ne0\), the equations \(F_s=F_t=0\) give
210 
211 \[
212 z^2+ut^2=z^2+us^2=0.
213 \]
214 
215 Hence \(s^2=t^2=R\ne0\) and \(u=-z^2/R\). But then \(F_z=5Rz^2\ne0\), a
216 contradiction.
217 
218 There is also an independent exact ideal certificate. For the gradient
219 ideal \(J\), the verifier proves
220 
221 \[
222 \sqrt J=(y_0,y_1,y_2y_5,y_3y_4,y_3y_5).
223 \]
224 
225 It checks that the gradient vanishes on the three planes and that the
226 powers
227 
228 \[
229 y_0^3,\quad y_1^4,\quad (y_2y_5)^3,\quad
230 (y_3y_4)^{13},\quad (y_3y_5)^3
231 \]
232 
233 belong to \(J\) by Gröbner reduction over \(\mathbf Q\). The ideal on
234 the right is the intersection of the three coordinate prime ideals, so
235 these two inclusions prove equality of radicals.
236 
237 The cone has dimension five and singular locus dimension two. It is a
238 hypersurface, hence Cohen--Macaulay, and is regular in codimension one.
239 Serre's criterion proves normality. Taking the weighted quotient
240 preserves normality.
241 
242 The ambient weighted projective space is well formed. Its largest
243 singular stratum is the \(\mu_3\)-stratum \(y_0=y_5=0\), of dimension
244 three. The restriction of \(F\) there is \(y_1^5+y_2y_3^2y_4^7\), not
245 identically zero. All other ambient singular strata have dimension at
246 most two. Thus \(Y\) meets the ambient singular locus in codimension at
247 least two. The verifier checks all coordinate strata, not just the
248 largest one.
249 
250 On the smooth ambient locus, the weighted Euler sequence gives canonical
251 class \(-33H\). Ordinary hypersurface adjunction gives \(-3H\) on the
252 smooth part of \(Y\) there. Both complements have codimension at least
253 two, so normality extends the equality as a canonical Weil divisor
254 class:
255 
256 \[
257 K_Y\sim_{\mathbf Q}-3H.
258 \]
259 
260 Here \(H\) is the ample rational hyperplane class, not an assertion that
261 \(\mathcal O_Y(1)\) is everywhere a line bundle. This argument
262 establishes the adjunction formula in the present non-quasismooth
263 example directly.
264 
265 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{21_carrier70_singular_locus.pdf}\caption{The certified coordinate singular planes. Normality uses their codimension in the cone.}\end{figure}
266 
267 \hypertarget{carrier70-why-a-resolution-is-uniruled}{%
268 \section{Why a resolution is
269 uniruled}\label{carrier70-why-a-resolution-is-uniruled}}
270 
271 Take a projective resolution \(r:\widetilde Y\to Y\) which is an
272 isomorphism over \(U_Y\). A general complete-intersection curve \(C\)
273 cut out by three sufficiently ample Cartier divisors avoids the singular
274 locus of \(Y\), which has dimension at most two. Its strict transform
275 avoids the exceptional divisors. Therefore
276 
277 \[
278 K_{\widetilde Y}\cdot\widetilde C=K_Y\cdot C=-3H\cdot C<0.
279 \]
280 
281 These curves move in a covering family. Thus \(K_{\widetilde Y}\) is not
282 pseudoeffective. The theorem of Boucksom--Demailly--Păun--Peternell
283 implies that \(\widetilde Y\) is uniruled.
284 
285 No assertion of log terminal singularities, rational connectedness, or
286 Fano positivity on the resolution is used.
287 
288 The rational Hodge conjecture for smooth projective uniruled fourfolds
289 is a proved theorem of Conte--Murre, also stated as Corollary 4.2 of
290 Arapura, arXiv:0706.2506. Therefore every rational \((2,2)\)-class on
291 \(\widetilde Y\) is algebraic.
292 
293 For context, this theorem does not use the unknown general fourfold
294 case: a uniruled fourfold is dominated generically finitely by a
295 resolution of \(T\times\mathbf P^1\), with \(\dim T=3\). The relevant
296 Hodge classes on the product are algebraic by the threefold case and
297 Künneth, blowup centers have dimension at most two, push-pull transfers
298 algebraicity.
299 
300 \hypertarget{carrier70-the-packet-is-actually-carried-by-this-cover}{%
301 \section{The packet is actually carried by this
302 cover}\label{carrier70-the-packet-is-actually-carried-by-this-cover}}
303 
304 It remains to verify that the target is not merely a convenient variety
305 unrelated to the desired character.
306 
307 Direct computation gives
308 
309 \[
310 A^Ta=70\beta,\qquad \beta=(3,3,1,3,2,3),
311 \]
312 
313 and hence \(B^T\beta=\alpha\). To obtain a genuine character of the
314 projective torus, let \(r_0=(0,0,2,0,0,3)\) be the first row of \(A\)
315 and set
316 
317 \[
318 u=\beta-3r_0=(3,3,-5,3,2,-6).
319 \]
320 
321 Then
322 
323 \[
324 W\cdot u=0,\qquad B^Tu=\alpha-630e_0.
325 \]
326 
327 Thus \(u\) is a character of the weighted projective torus. Its pullback
328 restricts on \(G_{210}\) to the Fermat character \(\alpha\), since
329 \(630=3\cdot210\).
330 
331 Let \(K\) be the kernel of the torus isogeny. The relation \(AB=210I\)
332 shows that \(K\) embeds in \(G_{210}\): an element in the kernel has
333 representatives whose 210th powers are all the same, and a common scalar
334 makes them roots of unity. The displayed character identity proves
335 \(\chi_\alpha|_K=1\). The same holds for every rational conjugate.
336 Therefore the whole rational packet is \(K\)-invariant.
337 
338 \hypertarget{carrier70-boundary-control-and-hodge-lifting}{%
339 \section{Boundary control and Hodge
340 lifting}\label{carrier70-boundary-control-and-hodge-lifting}}
341 
342 \hypertarget{carrier70-no-full-support-packet-is-lost-at-the-fermat-boundary}{%
343 \subsection{No full-support packet is lost at the Fermat
344 boundary}\label{carrier70-no-full-support-packet-is-lost-at-the-fermat-boundary}}
345 
346 The coordinate boundary of \(X_{210}\) is a simple normal crossing
347 divisor. Every nonempty coordinate stratum is fixed pointwise by at
348 least one coordinate scaling subgroup \(\mu_{210}\). Our full-support
349 character is nontrivial on that subgroup. The same is true for every
350 character in its rational orbit.
351 
352 In the boundary Gysin spectral sequence, the corresponding character
353 projector therefore kills every boundary term. The normal
354 orientation/Tate factors have trivial coordinate-scaling action.
355 Consequently restriction induces an isomorphism on the packet:
356 
357 \[
358 e_{[\alpha]}H^4(X_{210},\mathbf Q)
359 \xrightarrow{\sim}
360 e_{[\alpha]}H^4(U_{210},\mathbf Q).
361 \]
362 
363 This argument is independent of any asserted torus-residue extension for
364 singular weighted hypersurfaces.
365 
366 \hypertarget{carrier70-descent-to-the-open-carrier}{%
367 \subsection{Descent to the open
368 carrier}\label{carrier70-descent-to-the-open-carrier}}
369 
370 Because \(q\) is finite étale with deck group \(K\),
371 
372 \[
373 q^*:H^4(U_Y,\mathbf Q)
374 \xrightarrow{\sim}H^4(U_{210},\mathbf Q)^K.
375 \]
376 
377 This is an isomorphism of mixed Hodge structures onto its image.
378 Normalized transfer supplies the inverse. The rational pure
379 \((2,2)\)-packet therefore descends to a pure weight-four Hodge
380 substructure of \(H^4(U_Y,\mathbf Q)\).
381 
382 \hypertarget{carrier70-lifting-is-justified-by-the-lowest-weight-theorem}{%
383 \subsection{Lifting is justified by the lowest-weight
384 theorem}\label{carrier70-lifting-is-justified-by-the-lowest-weight-theorem}}
385 
386 Deligne's Corollary 3.2.17 in \emph{Théorie de Hodge II} gives
387 
388 \[
389 W_4H^4(U_Y,\mathbf Q)
390 =\operatorname{im}\bigl(H^4(\widetilde Y,\mathbf Q)
391 \to H^4(U_Y,\mathbf Q)\bigr).
392 \]
393 
394 Here \(W_4\) denotes the weight filtration, not the Fermat packet
395 notation. The descended classes have weight four, so belong to this
396 image. The category of polarizable pure rational Hodge structures is
397 semisimple. The surjection from \(H^4(\widetilde Y)\) onto its image
398 thus has a Hodge-theoretic section. Every descended rational
399 \((2,2)\)-class has a rational \((2,2)\)-lift on \(\widetilde Y\).
400 
401 This lifting is not being inferred from an arbitrary complex residue
402 numerator. Rationality, purity, and the surjection of pure Hodge
403 structures have all been checked separately.
404 
405 \hypertarget{carrier70-producing-the-algebraic-classes-and-returning-to-degree-70}{%
406 \section{Producing the algebraic classes and returning to degree
407 70}\label{carrier70-producing-the-algebraic-classes-and-returning-to-degree-70}}
408 
409 By the uniruled-fourfold theorem, each lifted class on \(\widetilde Y\)
410 is a rational combination of algebraic surface classes. The resolved
411 graph of the monomial rational map
412 
413 \[
414 X_{210}\dashrightarrow\widetilde Y
415 \]
416 
417 defines an algebraic correspondence. Its pullback on cohomology agrees
418 with \(q^*\) on the torus. Apply the rational orbit projector
419 \(e_{[\alpha]}\), which is a rational linear combination of graphs of
420 diagonal automorphisms. Therefore it preserves the span of algebraic
421 cycle classes.
422 
423 The resulting class and the prescribed class agree on \(U_{210}\), their
424 difference lies in the packet and restricts to zero. The preceding
425 boundary isomorphism proves that difference is zero. This establishes
426 algebraicity of the entire \(W_{[\alpha]}\) packet on \(X_{210}\).
427 
428 Finally, use the finite coordinate-cube map
429 
430 \[
431 \pi:X_{210}\longrightarrow X_{70},\qquad
432 [x_i]\longmapsto[x_i^3].
433 \]
434 
435 Its pullback identifies \(W_{[a]}\) with \(W_{[3a]}\). The identity
436 
437 \[
438 \pi_*\pi^*=(\deg\pi)\operatorname{id}
439 \]
440 
441 then transports the algebraic classes down to \(X_{70}\). Division by
442 the positive degree is legitimate because all cycle coefficients are
443 rational.
444 
445 For arbitrary \(k\ge1\), the coordinate-power map \(X_{70k}\to X_{70}\)
446 pulls the same packet back algebraically. This proves the stated
447 infinite family of packets. It does not prove the full Hodge conjecture
448 for those varieties.
449 
450 \hypertarget{carrier70-reusable-criterion-and-the-remaining-universal-problem}{%
451 \section{Reusable criterion and the remaining universal
452 problem}\label{carrier70-reusable-criterion-and-the-remaining-universal-problem}}
453 
454 The argument proves this criterion:
455 
456 \textbf{Torus-carrier criterion.} Let \(W\) be a full-support rational
457 middle Hodge packet on a Fermat fourfold. Suppose a finite torus isogeny
458 identifies its open Fermat variety with the inverse image of a smooth
459 torus hypersurface, the deck group acts trivially on \(W\), and that
460 torus hypersurface has a smooth projective compactification for which
461 the rational degree-four Hodge conjecture is known. Then \(W\) is
462 algebraic.
463 
464 A normal, well-formed weighted hypersurface with negative canonical
465 hyperplane coefficient supplies one such compactification after
466 resolution: the resolution is uniruled by the uniruledness argument
467 above.
468 
469 For searching, choose nonnegative integral columns \(v_j\) satisfying
470 \(a\cdot v_j\equiv0\pmod m\). Seek a nonsingular six-column matrix \(A\)
471 with
472 
473 \[
474 A^{-1}\mathbf1>0,\qquad \mathbf1^TA^{-1}\mathbf1>1.
475 \]
476 
477 These inequalities discover candidate weights only. Normality and
478 codimension of the ambient singular intersections still require
479 geometric certificates.
480 
481 \textbf{Matrix-carrier corollary.} Let \(a\) be a full-support balanced
482 rational Hodge character modulo \(m\). Suppose \(A\) is a nonsingular
483 nonnegative integral \(6\times6\) matrix such that
484 
485 \[
486 A^Ta\in m\mathbf Z^6,\qquad A^{-1}\mathbf1>0,
487 \qquad\mathbf1^TA^{-1}\mathbf1>1.
488 \]
489 
490 Let \(W/D=A^{-1}\mathbf1\), with primitive positive integral weights
491 \(W\), and assume the row-monomial hypersurface is integral, normal, and
492 well formed in \(\mathbf P(W)\). Then the rational Fermat packet
493 \(W_{[a]}\) is algebraic.
494 
495 Indeed, choose \(M\) divisible by \(m\) and all denominators of
496 \(A^{-1}\). Put \(B=MA^{-1}\), \(\alpha=(M/m)a\), and \(\beta=A^Ta/m\).
497 For any row \(r_i\) of \(A\),
498 
499 \[
500 u=\beta-3r_i,\qquad W\cdot u=0,\qquad
501 B^Tu=\alpha-3Me_i.
502 \]
503 
504 Here \(W\cdot\beta=3D\) follows from \(\sum a_j=3m\). Thus deck
505 invariance follows uniformly from the matrix congruence, it is not an
506 extra conjecture. The positive weight sum gives negative canonical
507 coefficient, normality and well-formedness justify the canonical
508 computation, and the uniruledness, descent, lifting and return arguments
509 above finish the proof. This corollary replaces an assumed cycle seed
510 with a finite exponent matrix and checkable geometric hypotheses.
511 
512 There is no proof here that such carriers exist for every balanced
513 packet. Still less is there a reduction of arbitrary smooth projective
514 varieties to these diagonal Fermat packets. Those are the missing
515 universal coverage statements, they are not consequences of the
516 successful example.
517 
518 The earlier proposed fixed-degree central-defect realization lemma
519 remains unproved and is not used anywhere in this proof. Likewise, this
520 construction does not establish the general CM half-twist Chow
521 correspondence listed as open in the pre-existing Komorebi manuscript.
522 
523 \hypertarget{carrier70-computation-audit}{%
524 \section{Computation audit}\label{carrier70-computation-audit}}
525 
526 The narrower atomic chain/loop search considered 6,600 labeled graphs
527 and 24 unit conjugates, for 158,400 pairs. Its maximum normalized weight
528 sum was \(296/315<1\), it found no Fano candidate in that implemented
529 family. A broader functional-graph search also failed to find one. These
530 failures are not evidence against the Hodge conjecture.
531 
532 The unrestricted column search enumerated 1,955 invariant columns of
533 total degree at most 18. A floating-point linear program selected a
534 six-column support. Its result was then replaced by exact rational
535 matrix arithmetic. The actual certificate and all conclusions about the
536 selected matrix are independent of floating-point tolerances.
537 
538 The final verifier checks 24 assertions, including every unit conjugate,
539 matrix identities, all ambient coordinate singular strata, and the exact
540 radical-gradient ideal certificate. It does not compute algebraic
541 cycles, formally prove mixed Hodge theory, or independently authenticate
542 the cited geometric theorems.

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