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1 \hypertarget{capacity_cycles-a-matrix-capacity-obstruction-and-explicit-five-phase-fermat-cycles}{%
2 \chapter{A matrix-capacity obstruction and explicit five-phase Fermat
3 cycles}\label{capacity_cycles-a-matrix-capacity-obstruction-and-explicit-five-phase-fermat-cycles}}
4 
5 Komorebi research continuation, version 0.2.0. Session: 21 September
6 2026, America/Chicago.
7 
8 \hypertarget{capacity_cycles-status-and-scope}{%
9 \section{Status and scope}\label{capacity_cycles-status-and-scope}}
10 
11 This note contains manuscript proofs and replayable exact algebraic
12 certificates. It does not prove the general Hodge conjecture. The exact
13 certificates were independently replayed in October. The geometric
14 arguments are not formally verified or peer reviewed.
15 
16 The positive algebraicity family reconstructed here is the \textbf{known
17 five-phase standard family of Aoki}. The October original-source audit
18 checked Theorem 2-1 of the 1987 paper. Its equations give the same
19 family after the Newton identities below. The construction is a direct
20 recomputation of those known surfaces. The determinant and period are
21 derived explicitly using the checked complete-intersection theorem of
22 Villaflor Loyola. No priority for the surfaces, equations, algebraicity
23 or determinant is claimed.
24 
25 The concrete advances within this research attempt are:
26 
27 \begin{enumerate}
28 \def\labelenumi{\arabic{enumi}.}
29 \tightlist
30 \item
31   An exact global upper bound on the normalized weight sum of every
32   invariant-column matrix candidate, with equality for the earlier
33   degree-70 carrier.
34 \item
35   A uniform obstruction showing that the earlier positive-anticanonical
36   matrix criterion cannot cover all balanced Fermat packets, even with
37   unbounded exponents and coordinate-power lifts.
38 \item
39   Explicit complete-intersection surfaces, a uniform determinant
40   identity, and nonzero exact periods for the infinite standard family
41   missed by that criterion.
42 \end{enumerate}
43 
44 The original v0.1.0 archive is preserved byte for byte in
45 \texttt{predecessor/}. Its original 24-check verifier was replayed
46 successfully before this continuation. The geometric arguments of the
47 predecessor have not been independently formally verified.
48 
49 \hypertarget{capacity_cycles-conventions}{%
50 \section{Conventions}\label{capacity_cycles-conventions}}
51 
52 Let
53 
54 \[
55 X_m=\{F_m=x_0^m+\cdots+x_5^m=0\}\subset\mathbf P^5_{\mathbf C},
56 \qquad G_m=(\mu_m)^6/\mu_m.
57 \]
58 
59 Write \(\langle r\rangle_m\) for the least-positive residue of a nonzero
60 residue class. A full-support character is
61 
62 \[
63 a=(a_0,\ldots,a_5),\qquad 0<a_i<m,\qquad \sum a_i\equiv0\pmod m.
64 \]
65 
66 The Griffiths residue basis of the Fermat hypersurface diagonalizes the
67 action of \(G_m\). The monomial \(x_0^{a_0-1}\cdots x_5^{a_5-1}\) at
68 pole order three gives the character of middle Hodge type when
69 \(\sum a_i=3m\). The entire rational orbit has type \((2,2)\) when
70 
71 \begin{equation}\label{eq:capacity_cycles-1.1}
72 \sum_i\langle ta_i\rangle_m=3m
73 \quad\text{for every }t\in(\mathbf Z/m\mathbf Z)^\times.\end{equation}
74 
75 A rational packet is the rational subrepresentation whose
76 complexification is the sum of these distinct conjugate eigenlines. An
77 individual complex eigenvector is not being identified with an
78 individual rational class.
79 
80 The Jacobian ideal and socle monomial are
81 
82 \begin{equation}\label{eq:capacity_cycles-1.2}
83 J_{F_m}=(x_0^{m-1},\ldots,x_5^{m-1}),\qquad
84 S_m=\prod_{i=0}^5x_i^{m-2}.\end{equation}
85 
86 We use
87 
88 \[
89 \Omega=\sum_{i=0}^5(-1)^ix_i\,dx_0\wedge\cdots\widehat{dx_i}\cdots\wedge dx_5
90 \]
91 
92 and the residue normalization of Villaflor Loyola, Theorems 1.1 and 1.2.
93 The order of polynomial rows in each Jacobian below is explicitly fixed.
94 
95 \hypertarget{capacity_cycles-the-matrix-capacity-and-a-certificate-over-an-infinite-set}{%
96 \section{The matrix capacity and a certificate over an infinite
97 set}\label{capacity_cycles-the-matrix-capacity-and-a-certificate-over-an-infinite-set}}
98 
99 Fix a character \(a\) modulo \(m\). Its invariant exponent columns form
100 
101 \[
102 \mathcal L_a=\{v\in\mathbf Z_{\ge0}^6\setminus\{0\}:a\cdot v\equiv0\pmod m\}.
103 \]
104 
105 Define
106 
107 \begin{equation}\label{eq:capacity_cycles-2.1}
108 C(a)=\sup\left\{\sum_vw_v:
109 \sum_vw_vv=\mathbf1,\quad w_v\ge0,
110 \quad\text{only finitely many }w_v\ne0\right\}.\end{equation}
111 
112 Every nonsingular nonnegative integral matrix \(A\) satisfying
113 \(A^Ta\in m\mathbf Z^6\), with \(w=A^{-1}\mathbf1>0\), gives one
114 feasible point. Hence
115 
116 \begin{equation}\label{eq:capacity_cycles-2.2}
117 \mathbf1^TA^{-1}\mathbf1\le C(a).\end{equation}
118 
119 This is a necessary numerical bound, not a sufficient geometric
120 criterion. A capacity maximizer can use fewer than six columns. A
121 six-column maximizer can fail normality or well-formedness. No claim
122 about a normal carrier follows merely from \(C(a)>1\).
123 
124 \hypertarget{capacity_cycles-lemma-2.1-global-dual-certificate}{%
125 \subsection{Lemma 2.1: global dual
126 certificate}\label{capacity_cycles-lemma-2.1-global-dual-certificate}}
127 
128 Suppose \(y\in\mathbf Q_{>0}^6\) and \(y\cdot v\ge1\) for every
129 \(v\in\mathcal L_a\). Then
130 
131 \[
132 C(a)\le y\cdot\mathbf1.
133 \]
134 
135 Indeed,
136 
137 \[
138 \sum_vw_v\le\sum_vw_v(y\cdot v)
139 =y\cdot\sum_vw_vv=y\cdot\mathbf1.
140 \]
141 
142 The infinitely many inequalities can be certified by a finite residue
143 graph. Write \(y_i=c_i/L\), where \(c_i,L\) are positive integers. On
144 \(\mathbf Z/m\mathbf Z\), an edge \(r\to r+a_i\) has cost \(c_i\). An
145 integer function \(d\) satisfying
146 
147 \begin{equation}\label{eq:capacity_cycles-2.3}
148 d(0)=0,\qquad d(r+a_i)\le d(r)+c_i,
149 \qquad d(-a_i)+c_i\ge L\end{equation}
150 
151 for every \(r,i\) certifies the dual inequality for \textbf{every}
152 invariant column, regardless of its total degree.
153 
154 To prove this, order the multiset described by a nonzero invariant
155 column into a nonempty zero-sum word. Repeated application of the first
156 inequalities bounds the cost of its prefix below by \(d(-a_i)\), where
157 \(i\) is its last letter. Adding the last cost gives at least \(L\).
158 Divide by \(L\).
159 
160 The program uses a shortest-path algorithm to find a potential, but
161 verifies \eqref{eq:capacity_cycles-2.3} directly. The correctness of the global conclusion
162 therefore depends on the displayed finite inequalities and this proof,
163 not on trusting a shortest-path implementation or a search-degree
164 cutoff.
165 
166 \hypertarget{capacity_cycles-proposition-2.2-the-old-degree-70-carrier-is-globally-optimal-in-normalized-weight-sum}{%
167 \subsection{Proposition 2.2: the old degree-70 carrier is globally
168 optimal in normalized weight
169 sum}\label{capacity_cycles-proposition-2.2-the-old-degree-70-carrier-is-globally-optimal-in-normalized-weight-sum}}
170 
171 For
172 
173 \[
174 a=(1,20,24,42,61,62)\pmod{70},
175 \]
176 
177 take
178 
179 \begin{equation}\label{eq:capacity_cycles-2.4}
180 y=\frac1{70}(11,10,19,14,6,17).\end{equation}
181 
182 The exact potential in \texttt{results/capacity\_certificates.json}
183 verifies all 420 edge inequalities in \eqref{eq:capacity_cycles-2.3}, and all six last-edge lower
184 bounds are exactly 70. Thus
185 
186 \[
187 C(a)\le\sum_i y_i=\frac{11}{10}.
188 \]
189 
190 The matrix from v0.1.0 is
191 
192 \[
193 A=\begin{pmatrix}
194 0&0&2&0&0&3\\
195 0&0&1&2&7&0\\
196 1&0&2&2&0&1\\
197 0&5&0&0&0&0\\
198 0&0&0&2&0&3\\
199 3&0&0&0&0&0
200 \end{pmatrix},
201 \quad
202 A^{-1}\mathbf1=(1/3,1/5,1/10,1/10,1/10,4/15).
203 \]
204 
205 It is feasible and its weights sum to \(11/10\). Therefore
206 
207 \begin{equation}\label{eq:capacity_cycles-2.5}
208 \boxed{C(1,20,24,42,61,62\bmod70)=11/10.}\end{equation}
209 
210 This is global within the invariant-column matrix model, not merely
211 optimal among columns of degree at most 18. It is not an assertion that
212 a scale-dependent anticanonical index has an invariant maximum.
213 
214 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{26_capacity_values.pdf}\caption{Seven named capacity certificates and the degree-210 target. These eight specified characters are not a census of their degrees.}\end{figure}
215 
216 \hypertarget{capacity_cycles-a-uniform-obstruction-to-universal-matrix-carrier-coverage}{%
217 \section{A uniform obstruction to universal matrix-carrier
218 coverage}\label{capacity_cycles-a-uniform-obstruction-to-universal-matrix-carrier-coverage}}
219 
220 For \(q\ge2\) and \(1\le b<q\), set
221 
222 \begin{equation}\label{eq:capacity_cycles-3.1}
223 m=5q,\qquad
224  a_{q,b}=(b,b+q,b+2q,b+3q,b+4q,5(q-b)).\end{equation}
225 
226 \hypertarget{capacity_cycles-lemma-3.1-balance}{%
227 \subsection{Lemma 3.1: balance}\label{capacity_cycles-lemma-3.1-balance}}
228 
229 Every character in \eqref{eq:capacity_cycles-3.1} is full support and its rational orbit is of
230 type \((2,2)\).
231 
232 Let \(t\) be a unit modulo \(5q\), and let \(b'=tb\bmod q\), with
233 \(1\le b'<q\). Multiplication by \(t\) permutes the five residues in the
234 fiber over \(b'\), because \(t\) is a unit modulo 5. The first five
235 least-positive residues are therefore, in some order,
236 
237 \[
238 b',b'+q,b'+2q,b'+3q,b'+4q.
239 \]
240 
241 The last residue is \(5(q-b')\). Their sum is \(15q=3m\). This is a
242 proof for all parameters, separate from the finite checks in the bundle.
243 
244 \hypertarget{capacity_cycles-no-positive-anticanonical-invariant-column-matrix}{%
245 \subsection{No positive-anticanonical invariant-column
246 matrix}\label{capacity_cycles-no-positive-anticanonical-invariant-column-matrix}}
247 
248 Assume \(q>5\) and \(\gcd(b,q)=1\). For every nonsingular nonnegative
249 integral matrix \(A\) such that
250 
251 \[
252 A^Ta_{q,b}\in5q\mathbf Z^6,
253 \qquad w=A^{-1}\mathbf1>0,
254 \]
255 
256 one has
257 
258 \begin{equation}\label{eq:capacity_cycles-3.2}
259 \boxed{\sum_j w_j\le1.}\end{equation}
260 
261 In fact \(C(a_{q,b})=1\).
262 
263 \textbf{Proof.} Define
264 
265 \[
266 y_q=\frac1q(1,1,1,1,1,q-5),\qquad y_q\cdot\mathbf1=1.
267 \]
268 
269 If \(v\in\mathcal L_{a_{q,b}}\), reduction modulo \(q\), followed by
270 cancellation of the unit \(b\), gives
271 
272 \[
273 v_0+v_1+v_2+v_3+v_4-5v_5\equiv0\pmod q.
274 \]
275 
276 Hence
277 
278 \[
279 N=v_0+v_1+v_2+v_3+v_4+(q-5)v_5
280 \]
281 
282 is divisible by \(q\). It is a positive integer because \(v\ne0\) and
283 \(q>5\). Therefore \(N\ge q\), or \(y_q\cdot v\ge1\). Lemma 2.1 proves
284 \(C(a_{q,b})\le1\).
285 
286 Conversely \(v=\mathbf1\) is invariant, because \(\sum a_i=3m\), and
287 putting weight 1 on it is a feasible point of \eqref{eq:capacity_cycles-2.1}. Thus
288 \(C(a_{q,b})=1\), proving \eqref{eq:capacity_cycles-3.2}. \(\square\)
289 
290 The invariant column set is unchanged by multiplying a character by a
291 unit modulo \(m\), or by replacing \((m,a)\) by \((km,ka)\). A
292 coordinate permutation merely permutes the columns' entries and the dual
293 costs. Consequently none of these changes evades the obstruction.
294 
295 For example, at degree 35 the packet is \((1,8,15,22,29,30)\), and the
296 dual is \((1,1,1,1,1,2)/7\). No increase in exponent-search bounds can
297 produce a matrix of the earlier type with normalized weight sum greater
298 than 1.
299 
300 \textbf{Interpretation.} This disproves universal applicability of the
301 specific positive-anticanonical matrix criterion proposed in v0.1.0. It
302 does not disprove the Hodge conjecture, exclude other uniruled carriers
303 or correspondences, or exclude non-anticanonical approaches.
304 
305 \hypertarget{capacity_cycles-an-actual-cycle-for-the-missed-family}{%
306 \section{An actual cycle for the missed
307 family}\label{capacity_cycles-an-actual-cycle-for-the-missed-family}}
308 
309 The construction now bypasses the carrier model entirely.
310 
311 Fix \(q\ge2\), put \(m=5q\), and choose \(\kappa\in\mathbf C\) with
312 
313 \[
314 \kappa^q=-1/5.
315 \]
316 
317 Set
318 
319 \[
320 z=x_5,\quad t_i=x_i^q\ (0\le i\le4),\quad P=x_0x_1x_2x_3x_4,
321 \]
322 
323 and let \(e_j\) be the elementary symmetric polynomial of degree \(j\)
324 in \(t_0,\ldots,t_4\). Define
325 
326 \begin{equation}\label{eq:capacity_cycles-4.1}
327 \boxed{
328 Z_\kappa=\{e_1=0,\ e_2=0,\ P-\kappa z^5=0\}\subset\mathbf P^5.
329 }\end{equation}
330 
331 \hypertarget{capacity_cycles-proposition-4.1-containment-in-the-fermat-fourfold}{%
332 \subsection{Proposition 4.1: containment in the Fermat
333 fourfold}\label{capacity_cycles-proposition-4.1-containment-in-the-fermat-fourfold}}
334 
335 Newton's identity is
336 
337 \begin{equation}\label{eq:capacity_cycles-4.2}
338 p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.\end{equation}
339 
340 Since \(e_5=P^q\), on \(Z_\kappa\) one has
341 
342 \[
343 F_m=p_5+z^{5q}=5P^q+z^{5q}
344 =(5\kappa^q+1)z^{5q}=0.
345 \]
346 
347 Thus \(Z_\kappa\subset X_{5q}\). Its defining equations have degrees
348 \(q,2q,5\).
349 
350 \hypertarget{capacity_cycles-proposition-4.2-the-surface-is-an-integral-complete-intersection}{%
351 \subsection{Proposition 4.2: the surface is an integral complete
352 intersection}\label{capacity_cycles-proposition-4.2-the-surface-is-an-integral-complete-intersection}}
353 
354 Let
355 
356 \[
357 S_q=\{e_1(x_0^q,\ldots,x_4^q)=e_2(x_0^q,\ldots,x_4^q)=0\}\subset\mathbf P^4.
358 \]
359 
360 The coordinate-power map to the \(t\)-space is finite, so the dimension
361 is two. In characteristic zero, on \(e_1=0\) the equations are
362 equivalent to
363 
364 \[
365 \sum x_i^q=0,\qquad\sum x_i^{2q}=0.
366 \]
367 
368 At a hypothetical failure of Jacobian rank, there would be constants
369 \(\lambda,\mu\), not both zero, such that for each nonzero coordinate
370 
371 \[
372 \lambda+2\mu x_i^q=0.
373 \]
374 
375 The case \(\mu=0\) is impossible. Otherwise all nonzero \(x_i^q\) equal
376 the same nonzero constant. Their sum cannot vanish in characteristic
377 zero. Thus \(S_q\) is smooth. It is a connected complete intersection of
378 positive-dimensional ample divisors, hence irreducible.
379 
380 The same argument with four coordinates shows that \(S_q\cap\{x_i=0\}\)
381 is a smooth reduced complete-intersection curve. Distinct such curves
382 share no irreducible component, since imposing two coordinate zeros
383 gives a zero-dimensional intersection. In particular the divisor of
384 \(P\) has valuation one at the generic point of each coordinate curve.
385 
386 Forgetting \(z\) defines a finite flat cover \(Z_\kappa\to S_q\) of
387 degree five. On a chart trivializing \(\mathcal O_{S_q}(1)\), its
388 equation is \(z^5=P/\kappa\), with the corresponding trivializations
389 understood. At a generic coordinate curve its right side has valuation
390 one. The degree-five polynomial is Eisenstein over that discrete
391 valuation ring and is therefore irreducible over the function field of
392 \(S_q\). Finite flatness makes the cover generically integral without
393 vertical or embedded components, locally its monic algebra injects into
394 the generic field extension. Hence the cover is integral.
395 
396 The two equations of \(S_q\), followed by the monic equation in \(z\),
397 form a homogeneous regular sequence. The resulting projective ideal is
398 saturated and defines the integral complete intersection \eqref{eq:capacity_cycles-4.1}.
399 Therefore
400 
401 \[
402 \dim Z_\kappa=2,\qquad\deg Z_\kappa=q(2q)5=10q^2.
403 \]
404 
405 The surface need not be smooth. Villaflor Loyola's complete-intersection
406 theorem does not impose smoothness of \(Z\), its proof explicitly
407 reduces to a smooth perturbation. We use this version, not an unverified
408 smooth-cycle variant.
409 
410 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{30_explicit_fivephase_surface.pdf}\caption{A direct reconstruction of the known Aoki standard surface. Containment follows from exact Newton identities.}\end{figure}
411 
412 \hypertarget{capacity_cycles-the-exact-jacobian-determinant}{%
413 \section{The exact Jacobian
414 determinant}\label{capacity_cycles-the-exact-jacobian-determinant}}
415 
416 Define
417 
418 \[
419 \begin{aligned}
420 f_1&=e_1,&g_1&=e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4,\\
421 f_2&=e_2,&g_2&=-5e_3,\\
422 f_3&=P-\kappa z^5,&g_3&=5\sum_{j=0}^{q-1}P^{q-1-j}\kappa^jz^{5j}.
423 \end{aligned}
424 \]
425 
426 The geometric-series identity and \eqref{eq:capacity_cycles-4.2} give
427 
428 \begin{equation}\label{eq:capacity_cycles-5.1}
429 F_m=f_1g_1+f_2g_2+f_3g_3.\end{equation}
430 
431 Let
432 
433 \[
434 H=(f_1,g_1,f_2,g_2,f_3,g_3),\qquad
435 D_\kappa=\det\operatorname{Jac}_{(x_0,\ldots,x_4,z)}H,
436 \]
437 
438 and
439 
440 \[
441 \Delta(t)=\prod_{0\le i<j\le4}(t_i-t_j).
442 \]
443 
444 \hypertarget{capacity_cycles-theorem-5.1-uniform-determinant-formula}{%
445 \subsection{Theorem 5.1: uniform determinant
446 formula}\label{capacity_cycles-theorem-5.1-uniform-determinant-formula}}
447 
448 \begin{equation}\label{eq:capacity_cycles-5.2}
449 \boxed{
450 D_\kappa=625q^5\Delta(x_0^q,\ldots,x_4^q)
451 \sum_{b=1}^{q-1}\kappa^{q-b}P^{b-1}z^{5(q-b)-1}.
452 }\end{equation}
453 
454 \textbf{Proof.} First,
455 
456 \begin{equation}\label{eq:capacity_cycles-5.3}
457 df_1\wedge dg_1\wedge df_2\wedge dg_2
458 =25\,de_1\wedge de_2\wedge de_3\wedge de_4.\end{equation}
459 
460 Only the term \(-5de_4\) in \(dg_1\) survives after wedging with
461 \(de_1,de_2,de_3\), the row permutation has positive sign.
462 
463 Next, direct differentiation yields
464 
465 \begin{equation}\label{eq:capacity_cycles-5.4}
466 df_3\wedge dg_3=
467 25q\sum_{j=1}^{q-1}\kappa^jP^{q-1-j}z^{5j-1}\,dP\wedge dz.\end{equation}
468 
469 For each interior index \(j\), the two contributing coefficients add to
470 \(j+(q-j)=q\). The endpoint terms have zero derivative coefficient. Thus
471 \eqref{eq:capacity_cycles-5.4} is an identity for every positive integer \(q\), not an
472 interpolation from checked values.
473 
474 Finally,
475 
476 \[
477 de_1\wedge\cdots\wedge de_5
478 =\Delta(t)\,dt_0\wedge\cdots\wedge dt_4.
479 \]
480 
481 Use \(e_5=P^q\) and
482 \(dt_0\wedge\cdots\wedge dt_4=q^5P^{q-1}\,dx_0\wedge\cdots\wedge dx_4\).
483 Cancellation in the polynomial ring gives
484 
485 \begin{equation}\label{eq:capacity_cycles-5.5}
486 de_1\wedge\cdots\wedge de_4\wedge dP
487 =q^4\Delta(x^q)\,dx_0\wedge\cdots\wedge dx_4.\end{equation}
488 
489 Combining \eqref{eq:capacity_cycles-5.3}, \eqref{eq:capacity_cycles-5.4}, and \eqref{eq:capacity_cycles-5.5}, and putting \(b=q-j\), proves \eqref{eq:capacity_cycles-5.2}.
490 The identity extends across \(P=0\) because it is a polynomial identity.
491 \(\square\)
492 
493 The degree of every monomial in \eqref{eq:capacity_cycles-5.2} is \(15q-6=3m-6\), as required for
494 a middle residue numerator.
495 
496 \hypertarget{capacity_cycles-nonzero-periods-not-just-a-formal-character-match}{%
497 \section{Nonzero periods, not just a formal character
498 match}\label{capacity_cycles-nonzero-periods-not-just-a-formal-character-match}}
499 
500 Fix \(1\le b<q\). In the order used in \eqref{eq:capacity_cycles-3.1}, the coefficient of
501 
502 \begin{equation}\label{eq:capacity_cycles-6.1}
503 x_0^{b-1}x_1^{b+q-1}x_2^{b+2q-1}x_3^{b+3q-1}x_4^{b+4q-1}
504  z^{5(q-b)-1}\end{equation}
505 
506 in \(D_\kappa\) is
507 
508 \begin{equation}\label{eq:capacity_cycles-6.2}
509 625q^5\kappa^{q-b}\ne0.\end{equation}
510 
511 All permutations of the first five exponents have the same nonzero
512 magnitude, with the Vandermonde sign. Different values of \(b\) and
513 different permutations give different monomials. Every exponent lies
514 between zero and \(m-2\), so no term is killed by the Fermat Jacobian
515 ideal.
516 
517 Define the dual numerator
518 
519 \begin{equation}\label{eq:capacity_cycles-6.3}
520 Q_b=\prod_{i=0}^5x_i^{m-a_{q,b,i}-1}.\end{equation}
521 
522 It has degree \(3m-6\). Because the product \(Q_bD_\kappa\) has socle
523 degree, a surviving monomial modulo \(J_{F_m}\) must have all six
524 exponents equal to \(m-2\). Formula \eqref{eq:capacity_cycles-6.1} is the unique term with this
525 complement. Therefore
526 
527 \begin{equation}\label{eq:capacity_cycles-6.4}
528 Q_bD_\kappa\equiv625q^5\kappa^{q-b}S_m\pmod{J_{F_m}}.\end{equation}
529 
530 For the Fermat polynomial,
531 
532 \[
533 \det\operatorname{Hess}(F_m)=[m(m-1)]^6S_m.
534 \]
535 
536 Apply Villaflor Loyola's Theorem 1.2 with dimension four and the
537 ordering fixed above. Its constant is
538 
539 \[
540 c=\frac{625q^5\kappa^{q-b}}{[m(m-1)]^6}.
541 \]
542 
543 The exact period is consequently
544 
545 \begin{equation}\label{eq:capacity_cycles-6.5}
546 \boxed{
547 \int_{Z_\kappa}\operatorname{res}\left(\frac{Q_b\Omega}{F_m^3}\right)
548 =\frac{(2\pi i)^2}{50q}\,\kappa^{q-b}\ne0.
549 }\end{equation}
550 
551 The normalization follows from \(625q^5/[2(5q)^6]=1/(50q)\). In
552 conventions using a differently normalized residue map, the displayed
553 constant changes with that convention, nonvanishing does not.
554 
555 \hypertarget{capacity_cycles-rational-algebraicity-of-whole-packets}{%
556 \subsection{Rational algebraicity of whole
557 packets}\label{capacity_cycles-rational-algebraicity-of-whole-packets}}
558 
559 Let \(h\) denote the hyperplane class. Since \(\deg Z_\kappa/m=2q\), the
560 primitive part
561 
562 \[
563 \xi_\kappa=[Z_\kappa]-2q h^2
564 \]
565 
566 is rational and algebraic. Villaflor Loyola's Theorem 1.1 identifies its
567 primitive residue numerator with \(-2D_\kappa/m\). Thus its complex
568 eigenspace support consists of exactly the \(120(q-1)\) characters
569 obtained from \eqref{eq:capacity_cycles-3.1} by permuting the first five entries and varying
570 \(b\).
571 
572 That set is closed under multiplication by units modulo \(m\), by Lemma
573 3.1. Each of its eigenspaces has dimension one. The
574 \(\mathbf Q[G_m]\)-submodule generated by \(\xi_\kappa\) therefore has
575 complexification equal to the entire sum of those eigenlines: applying
576 the complex character idempotents isolates each nonzero component. Since
577 the generator and the group action are rational, this is the
578 complexification of the claimed rational submodule.
579 
580 Every vector in that rational submodule is a rational linear combination
581 of classes of diagonal translates of \(Z_\kappa\) and of \(h^2\). Hence
582 every rational packet in the family is algebraic. Its rational algebraic
583 subspace has dimension
584 
585 \[
586 120(q-1)
587 \]
588 
589 for this fixed choice of the last coordinate. This is not the dimension
590 of all Hodge classes on \(X_{5q}\).
591 
592 The conclusion recovers Aoki's standard family, including the elementary
593 decomposable midpoint case at even levels. It does not close arbitrary
594 nonstandard five-primary packets.
595 
596 \hypertarget{capacity_cycles-a-concrete-example-at-degree-35}{%
597 \section{A concrete example at degree
598 35}\label{capacity_cycles-a-concrete-example-at-degree-35}}
599 
600 Take \(q=7\), \(b=1\), \(\kappa^7=-1/5\). Then
601 
602 \[
603 a=(1,8,15,22,29,30),\quad
604 Z_\kappa=\left\{
605 \begin{array}{l}
606 \sum_{i=0}^4x_i^7=0,\\
607 \sum_{0\le i<j\le4}x_i^7x_j^7=0,\\
608 x_0x_1x_2x_3x_4=\kappa x_5^5
609 \end{array}\right\}\subset X_{35}.
610 \]
611 
612 The cycle has degree 490. The rational packet has dimension
613 \(\varphi(35)=24\), since its first character entry is 1. The
614 determinant coefficient is
615 
616 \[
617 10,504,375\,\kappa^6.
618 \]
619 
620 For
621 
622 \[
623 Q_1=x_0^{33}x_1^{26}x_2^{19}x_3^{12}x_4^5x_5^4,
624 \]
625 
626 one obtains
627 
628 \[
629 \boxed{
630 \int_{Z_\kappa}\operatorname{res}\left(\frac{Q_1\Omega}{F_{35}^3}\right)
631 =\frac{(2\pi i)^2\kappa^6}{350}\ne0.
632 }
633 \]
634 
635 At the same time, the uniform matrix-obstruction theorem above proves
636 that this packet has capacity exactly 1 and cannot use the
637 positive-anticanonical matrix route. Thus this is an explicit example
638 where failure of that constructor is compatible with, and bypassed by,
639 an actual algebraic cycle.
640 
641 At degree 70 the standard packet \((1,15,29,43,57,65)\) obtained by
642 setting \(q=14\) is \textbf{different} from the earlier packet
643 \((1,20,24,42,61,62)\). Their capacities are respectively 1 and
644 \(11/10\). They must not be conflated.
645 
646 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{31_socle_pairing35.pdf}\caption{At degree 35 the exact socle coefficient gives a nonzero period and a genuine packet component.}\end{figure}
647 
648 \hypertarget{capacity_cycles-exact-computations-and-limitations}{%
649 \section{Exact computations and
650 limitations}\label{capacity_cycles-exact-computations-and-limitations}}
651 
652 The final verifier uses integers, rational numbers, and symbolic
653 polynomial identities. It uses no floating-point arithmetic, random
654 sampling, network request, or optimization routine.
655 
656 Its 137 grouped assertions include:
657 
658 \begin{itemize}
659 \tightlist
660 \item
661   balance and exact primal/global-dual certificates for seven specified
662   packets,
663 \item
664   the predecessor archive checksum,
665 \item
666   Newton's identity and the elementary-symmetric Jacobian identity,
667 \item
668   exact six-by-six intermediate determinants for \(q=2,3,4,5,7,11,17\),
669 \item
670   independent socle-pairing checks for every \(b\) at \(q=2,7,11\),
671 \item
672   all \(b\), all units, and all first-five permutations for
673   \(2\le q\le64\), totaling 241,920 distinct support monomials and
674   218,836 balance/conjugacy instances.
675 \end{itemize}
676 
677 The seven globally certified capacity values are:
678 
679 \begin{longtable}[]{@{}lll@{}}
680 \toprule\noalign{}
681 Modulus & Character & Capacity \\
682 \midrule\noalign{}
683 \endhead
684 \bottomrule\noalign{}
685 \endlastfoot
686 25 & (1,6,11,16,20,21) & 6/5 \\
687 33 & (1,4,16,22,25,31) & 2 \\
688 35 & (1,8,15,22,29,30) & 1 \\
689 39 & (1,7,16,22,34,37) & 2 \\
690 40 & (1,9,17,25,33,35) & 1 \\
691 55 & (1,12,23,34,45,50) & 1 \\
692 70 & (1,20,24,42,61,62) & 11/10 \\
693 \end{longtable}
694 
695 These are seven specified inputs, not a census of all characters at
696 those levels. Capacities 2 in the table do not themselves supply a
697 six-column normal carrier.
698 
699 The finite identities support the implementations. The all-parameter
700 assertions come from the separate mathematical proofs in the balance,
701 cycle and period arguments above, conditional only on the stated
702 standard algebraic-geometric and residue theorems. These proofs remain
703 at manuscript level.
704 
705 \hypertarget{capacity_cycles-unclosed-targets}{%
706 \subsection{Unclosed targets}\label{capacity_cycles-unclosed-targets}}
707 
708 No proof is given of the general Hodge conjecture, of coverage of every
709 Fermat packet, of the full CM half-twist Chow correspondence, or of the
710 earlier central-defect realization lemma. The universal
711 positive-anticanonical invariant-matrix conjecture is false by the
712 uniform matrix-obstruction theorem above and should not remain in a gap
713 ledger as an unproved plausible universal lemma.
714 
715 The usable research architecture is now a collection of separately
716 certified constructors: carrier transfer where its hypotheses hold,
717 explicit standard surfaces where the capacity barrier is 1, and other
718 correspondences for packets not handled by either route. A proof that
719 such constructors exhaust all Hodge classes is still missing.

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