Hodge: source and evidence

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1 \hypertarget{transfer210-carrier-transfer-and-obstructions-for-fermat-hodge-packets}{%
2 \chapter{Carrier transfer and obstructions for Fermat Hodge
3 packets}\label{transfer210-carrier-transfer-and-obstructions-for-fermat-hodge-packets}}
4 
5 \textbf{Komorebi P. --- research block v0.2.0}\\
6 \textbf{Status:} manuscript-level mathematical arguments with exact
7 computational certificates. The geometric argument is not formally
8 verified or peer reviewed. Exact certificates were independently
9 replayed for this edition. The degree-210 algebraicity conclusion uses
10 the degree-70 carrier proposition from v0.1.0 and explicitly identified
11 classical results. No general Hodge-conjecture solution or historical
12 novelty claim is made.
13 
14 \hypertarget{transfer210-main-results-and-proof-boundaries}{%
15 \section{Main results and proof
16 boundaries}\label{transfer210-main-results-and-proof-boundaries}}
17 
18 Let
19 
20 \[
21 X_m^n=\{x_0^m+\cdots+x_{n+1}^m=0\}\subset\mathbf P^{n+1}_{\mathbf C}.
22 \]
23 
24 This block obtains three different kinds of results.
25 
26 \begin{enumerate}
27 \def\labelenumi{\arabic{enumi}.}
28 \tightlist
29 \item
30   \textbf{Algebraicity transfer.} The rational primitive Hodge packet
31   associated to \[
32   b=(2,9,129,142,168,180)\pmod {210}
33   \] is algebraic, using the v0.1.0 degree-70 carrier proposition and
34   classical Fermat cycle correspondences. Its rational dimension is 48.
35   Its full conductor is 210, so it is not a coordinate-power inflation
36   of a lower-conductor packet. A 14-term signed identity supplies the
37   transfer.
38 \item
39   \textbf{Global matrix obstruction.} No nonnegative integral matrix
40   capturing this packet can have positive normalized weights whose sum
41   exceeds one. This is proved for every matrix size-six candidate of the
42   previous criterion, not just for a bounded search. An exact dual
43   certificate has two independent finite verifications, with a uniform
44   argument covering all larger columns.
45 \item
46   \textbf{Structural obstructions.} An infinite balanced family also
47   cannot have such a matrix. A separate holomorphic-character-power
48   criterion excludes all direct diagonal-quotient uniruled carriers for
49   six explicitly checked test packets.
50 \end{enumerate}
51 
52 The positive transfer and negative matrix result are compatible. They
53 prove that the direct matrix method is incomplete even on packets that
54 can be made algebraic by combining a geometric seed with classical cycle
55 operations.
56 
57 The new standalone verifier checks 46 finite assertions. The preserved
58 predecessor verifier was separately rerun and passed all 24 of its
59 checks. Neither count is a formal verification of the geometric
60 theorems.
61 
62 \hypertarget{transfer210-character-conventions}{%
63 \section{Character conventions}\label{transfer210-character-conventions}}
64 
65 A full-support character is an ordered tuple \(a=(a_0,\ldots,a_{n+1})\)
66 of nonzero residues modulo \(m\), with sum zero modulo \(m\). Its
67 eigenspace in primitive cohomology is a complex line. For even \(n\),
68 its rational Galois packet is of middle Hodge type precisely when
69 
70 \begin{equation}\label{eq:transfer210-2.1}
71 \sum_i\langle ta_i\rangle_m=\frac{n+2}{2}m
72 \quad\text{for every }t\in(\mathbf Z/m\mathbf Z)^\times.\end{equation}
73 
74 Here residues are least positive representatives. This is the classical
75 Fermat character decomposition, see \cite{DaSilvaFermat}, Theorem 2.1.
76 Rational orbit dimension means the number of distinct \textbf{ordered}
77 unit conjugates, not the number of representatives after coordinate
78 permutations.
79 
80 Write \(\nu(a)\) for its integer residue-multiplicity vector. Write
81 \(a\uplus b\) for concatenation regarded as a multiset. Permuting
82 coordinates is an algebraic automorphism, so multiset identities suffice
83 for algebraicity transfer. Write \(-a\) for coordinatewise negation
84 modulo \(m\).
85 
86 A character is called algebraic when its eigenspace is contained in the
87 complex span of algebraic cycle classes. The rational orbit then lies in
88 the rational algebraic-cycle subspace: this subspace and the orbit
89 projector are rational, and the projector is a rational linear
90 combination of graphs of diagonal automorphisms.
91 
92 \hypertarget{transfer210-classical-inputs-stated-separately-from-computation}{%
93 \section{Classical inputs, stated separately from
94 computation}\label{transfer210-classical-inputs-stated-separately-from-computation}}
95 
96 We use the following established operations.
97 
98 \textbf{Concatenation.} Algebraic balanced characters concatenate to an
99 algebraic character on the higher-dimensional Fermat variety.
100 
101 \textbf{Complementary-pair removal.} If a character becomes algebraic
102 after concatenating any finite collection of pairs \((r,m-r)\), then it
103 was algebraic already.
104 
105 These are the Shioda--Ran results recorded as Aoki \cite{Aoki}, Theorem
106 1-4(i),(ii). The equivariant Fermat join construction is also described
107 in \cite{DaSilvaFermat}, Theorem 2.2 and Corollary 2.3. The removal
108 operation is a geometric correspondence theorem, not a formal assumption
109 that arbitrary tensor factors can be cancelled.
110 
111 \textbf{Four-entry blocks.} Every rational balanced four-entry packet is
112 algebraic, by the Lefschetz (1,1) theorem for the corresponding Fermat
113 surface.
114 
115 \textbf{Standard blocks.} For an odd prime \(p\mid m\), put \(d=m/p\).
116 Aoki \cite{Aoki}, Theorem 2-1 constructs an algebraic cycle representing
117 
118 \[
119 \sigma_{p,i}=(i,i+d,\ldots,i+(p-1)d,m-pi)
120 \]
121 
122 under the relevant nonzero-entry and coprimality conditions. We only use
123 \((m,p,i)=(35,5,5)\) and \((35,7,2)\). Here \(\gcd(5,7)=\gcd(2,5)=1\),
124 and all entries are nonzero.
125 
126 \textbf{Power transport.} The finite coordinate-power morphism
127 \(X_{km}^n\to X_m^n\) transports algebraic cycle classes by pullback.
128 The character is multiplied by \(k\).
129 
130 \hypertarget{transfer210-source-access-boundary}{%
131 \subsection{Source-access boundary}\label{transfer210-source-access-boundary}}
132 
133 The October audit obtained the original Aoki article. Theorem 1-4 states
134 concatenation and removal of complementary-pair auxiliary blocks.
135 Theorem 2-1 supplies the standard blocks below under its actual
136 nonzero-entry and coprimality conditions. The September source-access
137 limitation is preserved in the evidence archive and resolved for these
138 specific leaves in this edition. Neither theorem allows cancellation of
139 an arbitrary motive. The Shioda--Ran original proofs were not
140 independently re-audited.
141 
142 \hypertarget{transfer210-the-inherited-geometric-seed}{%
143 \section{The inherited geometric
144 seed}\label{transfer210-the-inherited-geometric-seed}}
145 
146 The authoritative predecessor is the included, unchanged archive
147 
148 \texttt{predecessor/Komorebi\_Hodge\_Carrier70\_v0.1.0.zip}.
149 
150 Its proposition asserts algebraicity of the 24-dimensional rational
151 packet
152 
153 \begin{equation}\label{eq:transfer210-4.1}
154 a=(1,20,24,42,61,62)\pmod {70}.\end{equation}
155 
156 The proof uses a concrete monomial correspondence to a normal weighted
157 degree-30 hypersurface with weights \((10,6,3,3,3,8)\). Its resolution
158 is shown to be uniruled, so the Conte--Murre Hodge theorem for uniruled
159 fourfolds applies. See \cite{ArapuraUniruled}, Corollary 4.2. Boundary
160 vanishing, finite étale descent and lowest-weight lifting then transfer
161 cycles back.
162 
163 The predecessor's singular-locus, matrix and character checks have been
164 replayed, see \texttt{results/predecessor\_verification.json}. This does
165 not replace review of the predecessor's mixed-Hodge and birational
166 arguments.
167 
168 Pullback under \(X_{210}^4\to X_{70}^4\) gives the algebraic source
169 packet
170 
171 \begin{equation}\label{eq:transfer210-4.2}
172 s=3a=(3,60,72,126,183,186)\pmod {210}.\end{equation}
173 
174 The source has conductor 70 and 24 distinct ordered unit conjugates. The
175 new target
176 
177 \begin{equation}\label{eq:transfer210-4.3}
178 b=(2,9,129,142,168,180)\pmod {210}\end{equation}
179 
180 has conductor 210 and 48 distinct ordered unit conjugates. Every
181 conjugate of both tuples sums to 630. Their Galois orbits are disjoint.
182 These are exact finite checks.
183 
184 The target also occurs in the even-degree residual table accompanying
185 \cite{Jumagulov2026}. We use that table to select a research target, not
186 as proof that the entire census is correct or that this exact packet was
187 previously unknown in all literature.
188 
189 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{23_transfer210_orbits.pdf}\caption{Source and target ordered unit orbits have dimensions 24 and 48, with conductors 70 and 210.}\end{figure}
190 
191 \hypertarget{transfer210-an-exact-transfer-identity}{%
192 \section{An exact transfer identity}\label{transfer210-an-exact-transfer-identity}}
193 
194 All entries in this section are modulo 210. Define
195 
196 \[
197 P_k=(k,210-k)
198 \]
199 
200 and
201 
202 \[
203 \begin{aligned}
204 Q_3&=(3,105,108,204),\\
205 Q_9&=(9,105,114,192),\\
206 Q_{24}&=(24,105,129,162),\\
207 Q_{27}&=(27,105,132,156),\\
208 R_2&=(2,72,142,204).
209 \end{aligned}
210 \]
211 
212 Every one of these four-entry blocks is balanced, as the exact verifier
213 checks against every unit modulo 210. They are algebraic by the surface
214 case.
215 
216 Define the two longer blocks
217 
218 \[
219 S_5=(30,60,72,114,156,198),
220 \qquad
221 S_7=(12,42,72,102,126,132,162,192).
222 \]
223 
224 Their deflations are
225 
226 \[
227 \begin{aligned}
228 S_5/6&=(5,10,12,19,26,33)=\sigma_{5,5}\quad\text{at degree }35,\\
229 S_7/6&=(2,7,12,17,21,22,27,32)=\sigma_{7,2}\quad\text{at degree }35,
230 \end{aligned}
231 \]
232 
233 up to ordering. The standard-cycle theorem and power pullback make both
234 blocks algebraic. The length-eight block lives on a Fermat sixfold, only
235 its explicitly known standard cycle is needed, not the general Hodge
236 conjecture for that sixfold.
237 
238 The exact identity is
239 
240 \begin{equation}\label{eq:transfer210-5.1}
241 \boxed{
242 \begin{aligned}
243  b\uplus P_{24}\uplus P_{27}\uplus P_{105}\uplus Q_3\uplus S_5\uplus S_7
244  ={}&s\uplus P_{12}\uplus P_{30}\uplus P_{42}\uplus P_{102}\\
245  &\uplus Q_9\uplus Q_{24}\uplus Q_{27}\uplus R_2.
246 \end{aligned}}\end{equation}
247 
248 Both sides contain exactly 30 entries, with equal multiplicity of every
249 residue. Equivalently, \(\nu(b)-\nu(s)\) is the signed sum of the
250 fourteen displayed auxiliary generators, each with coefficient \(+1\) or
251 \(-1\). The certificate is stored in \texttt{results/transfer210.json}.
252 
253 \hypertarget{transfer210-algebraicity-of-the-degree-210-packet}{%
254 \subsection{Algebraicity of the degree-210
255 packet}\label{transfer210-algebraicity-of-the-degree-210-packet}}
256 
257 Assuming the inherited degree-70 carrier proposition, every rational
258 class in \(W_{[b]}\) is algebraic.
259 
260 \textbf{Proof.} Let \(L\) be the auxiliary multiset to the left of \eqref{eq:transfer210-5.1}
261 and \(R\) the auxiliary multiset to its right. Thus
262 
263 \[
264 b\uplus L=s\uplus R,
265 \qquad |L|=|R|=24.
266 \]
267 
268 All component packets of \(L\) and \(R\) are algebraic. Negation
269 preserves their algebraicity, either by the rational Galois packet
270 statement or by conjugating the corresponding cycle class. Append \(-L\)
271 to both sides:
272 
273 \begin{equation}\label{eq:transfer210-5.2}
274 b\uplus L\uplus(-L)=s\uplus R\uplus(-L).\end{equation}
275 
276 The right-hand side is algebraic by concatenation, because the source
277 and every auxiliary component are algebraic. The left-hand auxiliary
278 part \(L\uplus(-L)\) is a multiset of exactly 24 complementary pairs.
279 The full character has 54 entries and is a character on
280 \(X_{210}^{52}\). Complementary-pair removal therefore gives
281 algebraicity of \(b\) on \(X_{210}^4\).
282 
283 The character projector and its rational Galois orbit transfer this
284 statement to every class in \(W_{[b]}\). Its dimension is 48. This
285 proves the proposition. \hfill\ensuremath{\square}
286 
287 For every \(k\ge1\), coordinate-power pullback gives the associated
288 packet on \(X_{210k}^4\). It does not establish the Hodge conjecture for
289 all packets on any of these fourfolds.
290 
291 This proof does not subtract unknown algebraic cycles, assert
292 cancellation of arbitrary motives, or assume a cycle-existence theorem
293 for the new packet. Every negative coefficient in the incidence identity
294 is removed through explicitly algebraic auxiliary blocks and the
295 classical complementary-pair theorem.
296 
297 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{24_transfer210_incidence.pdf}\caption{An exact signed multiplicity equality. Its algebraic implication also uses the stated classical correspondence operations.}\end{figure}
298 
299 \hypertarget{transfer210-a-global-obstruction-to-the-direct-matrix-route}{%
300 \section{A global obstruction to the direct matrix
301 route}\label{transfer210-a-global-obstruction-to-the-direct-matrix-route}}
302 
303 The previous search criterion required a nonsingular nonnegative
304 integral matrix \(A\), with columns \(v_j\), satisfying
305 
306 \begin{equation}\label{eq:transfer210-6.1}
307 b\cdot v_j\equiv0\pmod {210},\qquad
308 w=A^{-1}\mathbf1>0,\qquad
309 \sum_jw_j>1.\end{equation}
310 
311 The following exact certificate rules this out for the new target.
312 
313 Set
314 
315 \begin{equation}\label{eq:transfer210-6.2}
316 \eta=\frac1{420}(31,87,57,101,84,60).\end{equation}
317 
318 Its coordinates are positive and sum to one. For every nonzero
319 \(v\in\mathbf Z_{\ge0}^6\) satisfying \(b\cdot v\equiv0\pmod {210}\),
320 
321 \begin{equation}\label{eq:transfer210-6.3}
322 \boxed{\eta\cdot v\ge1.}\end{equation}
323 
324 \hypertarget{transfer210-proof-of-the-universal-inequality}{%
325 \subsection{Proof of the universal
326 inequality}\label{transfer210-proof-of-the-universal-inequality}}
327 
328 If \(|v|\ge14\), the numerator of \(\eta\cdot v\) is at least
329 \(14\cdot31=434>420\). It therefore suffices to check \(1\le |v|\le13\).
330 
331 The exact verifier enumerates all 27,131 such nonnegative six-tuples.
332 Exactly 143 satisfy the invariant-column congruence. Each has numerator
333 at least 420, and the minimum is exactly 420. The enumerator uses
334 integer arithmetic only and includes all compositions in the specified
335 range. Together with the large-degree bound, this proves \eqref{eq:transfer210-6.3} for every
336 column, not just those within a search cutoff.
337 
338 As an independent implementation, place residue states in
339 \(\mathbf Z/210\) and give the six transitions \(r\mapsto r+b_i\)
340 integer costs \((31,87,57,101,84,60)\). A nonempty zero-sum column is a
341 return walk. Exact shortest-path computation finds minimum nonempty
342 return cost 420. All costs are positive. This verifies the same
343 inequality by a different method.
344 
345 \hypertarget{transfer210-no-direct-positive-weight-matrix-carrier}{%
346 \subsection{No direct positive-weight matrix
347 carrier}\label{transfer210-no-direct-positive-weight-matrix-carrier}}
348 
349 If a matrix in \eqref{eq:transfer210-6.1} existed, then
350 
351 \[
352 1=\eta^T\mathbf1=\eta^TAw
353  =\sum_jw_j(\eta\cdot v_j)\ge\sum_jw_j.
354 \]
355 
356 This contradicts \(\sum_jw_j>1\). \hfill\ensuremath{\square}
357 
358 The single invariant column \(v=\mathbf1\), with coefficient one,
359 supplies a matching value-one witness for the relaxed column linear
360 program. Thus its global optimum is exactly one. This is not a claim
361 that a nonsingular six-column positive solution attains that optimum.
362 
363 The invariant-column condition is unchanged by unit-conjugating \(b\),
364 permuting its coordinates with the corresponding dual permutation, or
365 inflating \(b\) and the modulus together. The obstruction persists under
366 those operations.
367 
368 the algebraicity-transfer proposition above and the direct-matrix
369 obstruction corollary above together show a definite limitation of the
370 direct carrier criterion: a packet can be algebraic by a certified
371 transfer while every direct matrix of this type is excluded. This does
372 not rule out all geometric carriers or non-diagonal correspondences.
373 
374 \begin{figure}[tbp]\centering\includegraphics[width=0.98\linewidth]{27_global_dual210.pdf}\caption{The exact global dual combines a complete finite premise with a uniform large-degree inequality.}\end{figure}
375 
376 \hypertarget{transfer210-a-uniform-infinite-family-excluded-by-the-same-criterion}{%
377 \section{A uniform infinite family excluded by the same
378 criterion}\label{transfer210-a-uniform-infinite-family-excluded-by-the-same-criterion}}
379 
380 For any integer \(q>5\), let
381 
382 \begin{equation}\label{eq:transfer210-7.1}
383 a_q=(1,1+q,1+2q,1+3q,1+4q,5q-5)\pmod {5q}.\end{equation}
384 
385 \hypertarget{transfer210-theorem-7.2}{%
386 \subsection{Theorem 7.2}\label{transfer210-theorem-7.2}}
387 
388 Every \(a_q\) is a full-support middle-balanced character. No matrix
389 satisfying the previous direct positive-weight criterion exists for
390 \(a_q\).
391 
392 \textbf{Proof of balance.} Let \(t\) be a unit modulo \(5q\), and let
393 \(s\) be its least positive residue modulo \(q\). The first five
394 multiplied residues permute
395 
396 \[
397 s,s+q,s+2q,s+3q,s+4q.
398 \]
399 
400 Their sum is \(5s+10q\). The last multiplied residue is \(5(q-s)\), so
401 the total is \(15q=3(5q)\).
402 
403 \textbf{Proof of the obstruction.} Set
404 
405 \[
406 \eta_q=\frac1q(1,1,1,1,1,q-5).
407 \]
408 
409 This vector is positive and has sum one. For a nonzero invariant column
410 \(v\ge0\), put \(S=v_0+\cdots+v_4\). Reducing
411 \(a_q\cdot v\equiv0\pmod {5q}\) modulo \(q\) gives
412 \(S-5v_5\equiv0\pmod q\). Therefore
413 
414 \[
415 \eta_q\cdot v=\frac{S+(q-5)v_5}{q}
416 =\frac{S-5v_5}{q}+v_5
417 \]
418 
419 is a positive integer, and hence at least one. Apply the same matrix
420 inequality as in the preceding dual argument. \hfill\ensuremath{\square}
421 
422 This is a proof for every \(q>5\), not an extrapolation from finite
423 runs. The verifier's checks for \(6\le q\le80\) are benchmarks only.
424 These packets are themselves the standard family \(\sigma_{5,1}\), so
425 the obstruction is to the method, not to algebraicity.
426 
427 \hypertarget{transfer210-a-stronger-obstruction-holomorphic-powers-of-a-hodge-character}{%
428 \section{A stronger obstruction: holomorphic powers of a Hodge
429 character}\label{transfer210-a-stronger-obstruction-holomorphic-powers-of-a-hodge-character}}
430 
431 \hypertarget{transfer210-proposition-8.1}{%
432 \subsection{Proposition 8.1}\label{transfer210-proposition-8.1}}
433 
434 Let \(a\) be a full-support Fermat fourfold character. Suppose some
435 integer multiple
436 
437 \[
438 c=ka\pmod m
439 \]
440 
441 has all entries nonzero and least-positive residue sum \(m\). A direct
442 quotient by a subgroup of the diagonal automorphism group that captures
443 \(a\) cannot have a smooth projective uniruled compactification.
444 
445 \textbf{Proof.} If \(K\) is a diagonal deck subgroup and
446 \(\chi_a|_K=1\), then
447 
448 \[
449 \chi_c|_K=\chi_a^k|_K=1.
450 \]
451 
452 The residue description places the \(c\)-line in \(H^{4,0}\) under the
453 direct-action convention (or its conjugate under the inverse-character
454 convention). Its rational Galois span contains a nonzero \((4,0)\) part
455 and is also \(K\)-invariant.
456 
457 Full support makes restriction to the open coordinate torus injective on
458 this character space. As in the predecessor, each boundary stratum is
459 fixed by a coordinate scaling subgroup on which the character is
460 nontrivial, so the relevant Gysin terms vanish. Finite étale descent
461 gives the invariant pure weight-four space on the open quotient.
462 
463 For any smooth projective compactification, the lowest-weight image
464 theorem and the semisimplicity of polarizable pure Hodge structures lift
465 this space from open cohomology to compact cohomology. In particular the
466 compactification has a nonzero \(H^{4,0}\). These are the same classical
467 mixed-Hodge inputs as in the predecessor, see \cite{ArapuraUniruled},
468 \cite{DeligneHodgeII}.
469 
470 A smooth projective uniruled fourfold has no nonzero holomorphic top
471 form: a generically finite dominant parametrization from a resolution of
472 a threefold times \(\mathbf P^1\) would pull such a form back
473 injectively, whereas the latter has none. This is a contradiction. \hfill\ensuremath{\square}
474 
475 The assertion concerns this direct diagonal-quotient construction, not
476 arbitrary algebraic correspondences. It persists after coordinate-power
477 inflation of the source.
478 
479 \Needspace{10\baselineskip}
480 
481 \hypertarget{transfer210-exact-witnesses}{%
482 \subsection{Exact witnesses}\label{transfer210-exact-witnesses}}
483 
484 {\small\begin{longtable}[]{@{}
485   >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2308}}
486   >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2308}}
487   >{\raggedleft\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.3077}}
488   >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2308}}@{}}
489 \toprule\noalign{}
490 \begin{minipage}[b]{\linewidth}\raggedright
491 Modulus
492 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
493 Middle-balanced character a
494 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
495 k
496 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
497 Full-support character ka with sum m
498 \end{minipage} \\
499 \midrule\noalign{}
500 \endhead
501 \bottomrule\noalign{}
502 \endlastfoot
503 110 & (1,24,62,71,81,91) & 11 & (11,44,22,11,11,11) \\
504 110 & (1,31,55,71,81,91) & 11 & (11,11,55,11,11,11) \\
505 114 & (1,13,43,72,103,110) & 27 & (27,9,21,6,45,6) \\
506 114 & (1,13,43,80,102,103) & 9 & (9,3,45,36,6,15) \\
507 114 & (1,7,78,79,86,91) & 3 & (3,21,6,9,30,45) \\
508 220 & (1,62,111,142,162,182) & 11 & (11,22,121,22,22,22) \\
509 \end{longtable}}
510 
511 Each source's middle balance and each product's nonzero entries and
512 residue sum are checked exactly. The multipliers are nonunits, so there
513 is no contradiction with the middle balance of all \textbf{unit}
514 conjugates of the source character.
515 
516 These witnesses explain why merely increasing the direct diagonal
517 carrier search cannot address these targets. They do not show that the
518 targets are nonalgebraic.
519 
520 \hypertarget{transfer210-computation-contract-failures-and-remaining-work}{%
521 \section{Computation contract, failures and remaining
522 work}\label{transfer210-computation-contract-failures-and-remaining-work}}
523 
524 \hypertarget{transfer210-exact-final-replay}{%
525 \subsection{Exact final replay}\label{transfer210-exact-final-replay}}
526 
527 \texttt{python\ checks/verify.py} needs Python 3.10 or later and only
528 its standard library. It runs without network access, optimization
529 software or floating-point arithmetic. It verifies the frozen identity,
530 all named character conditions, the standard-block deflations, the
531 54-entry stabilization, the dual's complete finite premise and degree
532 cutoff, an independent exact shortest-cycle check, six holomorphic-power
533 witnesses and 75 infinite-family benchmarks.
534 
535 The code does not construct equations for a basis of representing
536 surfaces, verify the external theorems, or compile a proof in Lean. The
537 standard-block and carrier constructions give existence through known
538 algebraic correspondences, not a list of all representing surfaces.
539 
540 \hypertarget{transfer210-discovery-distinct-from-proof}{%
541 \subsection{Discovery, distinct from
542 proof}\label{transfer210-discovery-distinct-from-proof}}
543 
544 The transfer was found by a bounded mixed-integer search among
545 complementary pairs and standard blocks at degree 210. The solver uses
546 floating-point internals. The stored result has fourteen nonzero
547 coefficients and was accepted only after exact integer replay. Its
548 reported optimizer optimum is not a claim that fourteen is globally
549 minimal among all possible geometric identities.
550 
551 An earlier Hermite-normal-form membership attempt exceeded its 20-second
552 execution allowance and was stopped without a conclusion. Direct
553 integer-programming discovery replaced it successfully. Bounded
554 degree-18 matrix scans were exploratory and never used as global
555 impossibility proofs. The exact dual arguments in the two dual
556 obstruction arguments above replace those bounded observations for the
557 targets they cover.
558 
559 Original publisher PDF and some ancillary-log fetches failed. No missing
560 log was used as evidence, the transfer and its verification were
561 produced locally. The source-access limit for the classical Aoki theorem
562 is resolved in the source-access discussion above.
563 
564 \hypertarget{transfer210-what-remains}{%
565 \subsection{What remains}\label{transfer210-what-remains}}
566 
567 The general Hodge conjecture is not established. This block does not
568 cover all packets on \(X_{210}^4\), establish universal closure of the
569 Fermat packet calculus, or reduce arbitrary projective varieties to
570 diagonal hypersurfaces. It does not complete the earlier central-defect
571 realization lemma or a general CM half-twist Chow correspondence.
572 
573 The direct matrix criterion cannot be the universal bridge: the two dual
574 obstruction arguments above disprove that assertion. Direct diagonal
575 uniruled quotients cannot capture the holomorphic-power examples either.
576 A further positive argument must therefore use stabilization and
577 transfer, a different known-algebraic carrier class, or genuinely
578 non-diagonal correspondences. That is a mathematical constraint on the
579 next construction, not a conjectured failure of Hodge.

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