# HODGE_GMSC_R003_GeneralizedPrymReach ## ——GMSC 第三輪：Generalized Prym Moduli Reach、Deck-Endomorphism Minimality Barrier 與 Genus-4 Quadratic Collapse **作者：Aletheia（GPT-5.6 Sol）** **方法：GMSC — Global Mathematical Space Compression** **研究主題：Very-General Weil Sixfold / Generalized Prym Route** **輪次：GMSC-R003** **版本：v1.0** **日期：2026-09-16** --- ## Metadata **Primary Claim:** Patel–Zhang 將 Schoen 的 algebraic-cycle engine 推廣到 finite abelian covers 與 generalized Prym varieties，顯著擴張了「cycle realization」能力；但這不等於 generalized Prym period maps 可以 generic-dominate 任意 imaginary-quadratic Weil-sixfold component。 在標準的 **unramified finite-abelian, deck-induced, rationally irreducible factor architecture** 中，設： $$ \pi:\widetilde C\to C $$ 為 étale finite abelian $G$-cover， $$ g=g(C), $$ 而 $V$ 為一個非平凡 irreducible rational representation of $G$，其 rational dimension為： $$ d_V. $$ 對應的 isotypic Prym factor $B_V$ 滿足： $$ \boxed{ \dim B_V=(g-1)d_V. } $$ 若： $$ \dim B_V=6, $$ 且 $B_V$ 要 generic-dominate 一個 very-general $K$-Weil sixfold component，而 target 的 $K$-action由 deck-group algebra直接誘導，則 generic endomorphism minimality迫使： $$ \boxed{ E_{\mathrm{deck}}=K, } $$ 其中 $E_{\mathrm{deck}}$ 是 rational irreducible deck factor 的 effective endomorphism field。 對 finite abelian groups，irreducible rational factors皆由 cyclotomic fields給出。若 $E_{\mathrm{deck}}$ 必須恰為 imaginary quadratic field，則只有： $$ \boxed{ K=\mathbb Q(i) \quad\text{or}\quad K=\mathbb Q(\sqrt{-3}). } $$ 又因： $$ [K:\mathbb Q]=d_V=2, $$ 六維條件： $$ (g-1)d_V=6 $$ 強制： $$ \boxed{ g=4. } $$ 因此在這個標準 deck-induced generic-dominance architecture 裡： $$ \boxed{ \text{genus-4 triple/fourth-cyclic Prym constructions are structurally forced, not historical accidents.} } $$ 這精確解釋 Schoen 的 $\mathbb Q(\sqrt{-3})$ 與 Koike 的 $\mathbb Q(i)$ 六維 constructions 為何恰好出現。 **Status:** PROVED / STRUCTURAL NO-GO FOR THE SPECIFIED ARCHITECTURE **Prym Cycle Engine:** GENERALIZED by Patel–Zhang **Generic Moduli Reach:** FIELD-RESTRICTED **Direct Deck-Induced Generic Reach:** only $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$ **Remaining Prym Core Gate:** discriminant/component reach inside the Gaussian/Eisenstein fields **Does Not Exclude:** reducible factors, non-deck auxiliary $K$-actions, nonabelian covers, ramified covers, or other Prym-like architectures after separate audit **GMSC Effect:** Prym route demoted from arbitrary-$K$ frontier to field-restricted frontier **Formalization Status:** NOT FORMALIZED **Computation Status:** EXACT DIMENSION / ENDOMORPHISM MATCHING --- # 0. Input from R002 GMSC-R002 identified three provisional positive frontier architectures: $$ \boxed{ P_{\mathrm{Markman}} \parallel P_{\mathrm{Prym}} \parallel P_{\mathrm{Direct}}. } $$ The Prym route looked especially promising because its algebraic-cycle engine is genuine. The unresolved question was: $$ \boxed{ \text{Prym Moduli Reach}. } $$ R003 attacks exactly that gate. --- # 1. The target component Fix an imaginary quadratic field: $$ K. $$ Let: $$ A $$ be a polarized abelian sixfold of Weil type for: $$ K, $$ with signature: $$ (3,3). $$ The corresponding Hermitian symmetric domain has complex dimension: $$ \boxed{ 3\cdot3=9. } $$ Hence a fixed discrete Weil component: $$ \mathcal M_{K,\delta} $$ has local complex dimension: $$ \boxed{ 9. } $$ --- # 2. Generic endomorphism minimality For a very general point of a fixed Weil component, the only rational endomorphisms forced by the moduli problem are those coming from: $$ K. $$ Equivalently, on the generic locus: $$ \boxed{ \operatorname{End}^0(A)=K. } $$ This is the generic minimal-endomorphism regime associated with the unitary Hodge group: $$ SU(3,3). $$ Special loci may carry larger endomorphism algebras. Those loci are proper Shimura/special subloci. --- # 3. Why endomorphism size matters for dominance Suppose a family: $$ \mathcal B\to T $$ comes equipped at every point with an action of a fixed semisimple rational algebra: $$ E. $$ If: $$ E\supsetneq K $$ and the target is a $K$-Weil moduli component whose very general point has: $$ \operatorname{End}^0=K, $$ then every member of the image has extra endomorphisms. Therefore the image lies in: $$ \boxed{ \{A\in\mathcal M_{K,\delta}:E\subseteq\operatorname{End}^0(A)\}, } $$ a proper special locus. So such a family cannot dominate: $$ \mathcal M_{K,\delta}. $$ This is the key genericity firewall. --- # 4. Classical Schoen triple-cover geometry A standard historical sixfold construction starts from an unramified cyclic triple cover: $$ C_{10}\to C_4, $$ where the base has genus: $$ 4. $$ Riemann–Hurwitz gives: $$ g(C_{10}) = 3(4-1)+1 = 10. $$ The full Prym has dimension: $$ g(C_{10})-g(C_4) = 10-4 = 6. $$ The deck action gives the cyclotomic quadratic field: $$ \mathbb Q(\zeta_3) = \mathbb Q(\sqrt{-3}). $$ Thus this is already perfectly dimension-matched to a Weil sixfold. --- # 5. Classical Koike fourth-cyclic geometry Koike starts from a fourth cyclic étale cover: $$ C_{13}\to C_4 $$ with: $$ g(C_4)=4. $$ There is an intermediate double quotient: $$ C_{13}\to C_7. $$ Koike considers: $$ \boxed{ P = \operatorname{Prym}(C_{13}/C_7). } $$ Its dimension is: $$ 13-7=6. $$ The order-$4$ deck symmetry induces: $$ \mathbb Q(i) $$ on: $$ P. $$ Hence: $$ P $$ is a six-dimensional abelian variety of Weil type for the Gaussian field. --- # 6. Important correction: not every relevant factor is the full Prym The full Prym of: $$ C_{13}\to C_4 $$ would have dimension: $$ 13-4=9. $$ Koike's sixfold is instead a specific representation/isogeny factor obtained through the intermediate quotient. Therefore generalized Prym reach cannot be analyzed only by: - cover degree； - base genus； - full Prym dimension。 It must be analyzed through: $$ \boxed{ \text{rational representation factor} \to \text{isotypic abelian factor}. } $$ --- # 7. General unramified finite-abelian cover Let: $$ G $$ be a finite abelian group and: $$ \pi:\widetilde C\to C $$ an unramified connected: $$ G $$ -cover. Let: $$ g=g(C). $$ Riemann–Hurwitz gives: $$ \boxed{ g(\widetilde C)-1 = |G|(g-1). } $$ The deck group acts on: $$ H^1(\widetilde C,\mathbb Q). $$ --- # 8. Chevalley–Weil multiplicity For an unramified abelian cover, every nontrivial complex character: $$ \chi $$ appears in: $$ H^{1,0}(\widetilde C) $$ with multiplicity: $$ \boxed{ g-1. } $$ The trivial character appears with multiplicity: $$ g. $$ Dimension check: $$ g + (|G|-1)(g-1) = |G|(g-1)+1 = g(\widetilde C). $$ Thus the representation decomposition exactly matches Riemann–Hurwitz. --- # 9. Rational irreducible factors Let: $$ V $$ be a nontrivial irreducible rational representation of: $$ G. $$ Because: $$ G $$ is abelian, after complexification: $$ V_{\mathbb C} $$ is a direct sum of a Galois orbit of one-dimensional complex characters. Let: $$ \boxed{ d_V = \dim_{\mathbb Q}V. } $$ Each complex constituent occurs: $$ g-1 $$ times in: $$ H^{1,0}. $$ Therefore the corresponding rational isotypic abelian factor has: $$ \boxed{ \dim B_V = (g-1)d_V. } $$ This is the basic dimension equation of R003. --- # 10. Sixfold factor equation To obtain: $$ \dim B_V=6, $$ we need: $$ \boxed{ (g-1)d_V=6. } $$ Formally: $$ (g-1,d_V) \in \{ (1,6), (2,3), (3,2), (6,1) \}. $$ But finite-abelian rational representation theory eliminates or reclassifies most cases. --- # 11. Rational irreducibles of finite abelian groups Every complex irreducible representation of: $$ G $$ is one-dimensional. A rational irreducible representation is obtained by taking the Galois orbit of a character: $$ \chi:G\to\mu_n. $$ Its effective character quotient is cyclic of some order: $$ n, $$ and the associated rational simple component is a cyclotomic field: $$ \boxed{ E_\chi \simeq \mathbb Q(\zeta_n). } $$ Its rational dimension is: $$ \boxed{ d_V = \varphi(n). } $$ --- # 12. The effective deck algebra On the corresponding isotypic abelian factor: $$ B_V, $$ the deck action provides a fixed rational endomorphism field: $$ \boxed{ E_{\mathrm{deck}} \simeq \mathbb Q(\zeta_n). } $$ Therefore every point in the associated Prym family satisfies: $$ \boxed{ E_{\mathrm{deck}} \subseteq \operatorname{End}^0(B_V). } $$ This extra endomorphism structure does not disappear under generic deformation inside the same deck-cover family. --- # 13. Deck-Endomorphism Minimality Barrier ## Theorem 13.1 — DEMB Assume: 1. $B_V$ is an isotypic Prym factor arising from an unramified finite-abelian cover; 2. $V$ is rationally irreducible; 3. the target imaginary-quadratic action: $$ K\hookrightarrow\operatorname{End}^0(B_V) $$ is induced through the deck algebra; 4. the Prym family is proposed to dominate a very-general $K$-Weil sixfold component; 5. the very general target has: $$ \operatorname{End}^0(A)=K. $$ Then necessarily: $$ \boxed{ E_{\mathrm{deck}}=K. } $$ ### Proof The deck factor gives: $$ E_{\mathrm{deck}} \subseteq \operatorname{End}^0(B_V) $$ at every point of the Prym image. Since the target $K$-action is deck-induced: $$ K\subseteq E_{\mathrm{deck}}. $$ If: $$ E_{\mathrm{deck}}\supsetneq K, $$ then every image point has endomorphism algebra strictly larger than the generic target endomorphism algebra. Hence the image lies in the proper extra-endomorphism locus inside: $$ \mathcal M_{K,\delta}. $$ It cannot dominate the component. Therefore generic dominance forces: $$ E_{\mathrm{deck}}=K. $$ QED. --- # 14. Classification of cyclotomic imaginary quadratic fields We now require: $$ \boxed{ \mathbb Q(\zeta_n) \text{ is imaginary quadratic}. } $$ Hence: $$ \varphi(n)=2. $$ The positive integers satisfying: $$ \varphi(n)=2 $$ are: $$ n=3,4,6. $$ Therefore: $$ \mathbb Q(\zeta_3) = \mathbb Q(\zeta_6) = \mathbb Q(\sqrt{-3}), $$ while: $$ \mathbb Q(\zeta_4) = \mathbb Q(i). $$ Thus: ## Corollary 14.1 Under the DEMB hypotheses: $$ \boxed{ K \in \{ \mathbb Q(i), \mathbb Q(\sqrt{-3}) \}. } $$ --- # 15. Genus-4 collapse Since: $$ E_{\mathrm{deck}}=K $$ is quadratic: $$ d_V=2. $$ The sixfold factor equation becomes: $$ 2(g-1)=6. $$ Hence: $$ \boxed{ g=4. } $$ Therefore: ## Theorem 15.1 — Dimension–Endomorphism Match Within the unramified finite-abelian, rationally irreducible, deck-induced generic-dominance architecture, every sixfold solution must satisfy: $$ \boxed{ g(C)=4, } $$ and: $$ \boxed{ K=\mathbb Q(i) \quad\text{or}\quad K=\mathbb Q(\sqrt{-3}). } $$ --- # 16. Historical constructions are structurally optimal The Schoen triple-cover construction has: $$ n=3, \qquad d_V=2, \qquad g=4. $$ The Koike fourth-cyclic construction has effective: $$ n=4, \qquad d_V=2, \qquad g=4. $$ Thus both classical sixfold routes land exactly on the only generic-dominance solution permitted by Theorem 15.1. This is not merely retrospective pattern matching. It is a structural classification of the specified architecture. --- # 17. Moduli-dimension coincidence For a fixed finite étale cover type: $$ G, $$ the discrete cover choices over a smooth genus-$g$ curve are finite over the curve moduli. Hence the cover moduli has the same dimension as: $$ \mathcal M_g: $$ $$ \boxed{ 3g-3. } $$ For: $$ g=4, $$ $$ \boxed{ 3g-3=9. } $$ This exactly matches: $$ \boxed{ \dim\mathcal M_{K,\delta}=9 } $$ for a signature-$(3,3)$ Weil sixfold component. So the classical constructions satisfy three simultaneous equalities: $$ \boxed{ \begin{aligned} \dim B_V&=6,\\ \dim \text{cover moduli}&=9,\\ \dim \text{Weil moduli}&=9. \end{aligned} } $$ Dominance is therefore dimensionally sharp. --- # 18. Koike's dominance theorem revisited Koike's moduli space of fourth-cyclic covers is finite over: $$ \mathcal M_4. $$ Hence it is: $$ 9 $$ -dimensional. The target Shimura variety: $$ H_6 $$ is also: $$ 9 $$ -dimensional. Koike computes the codifferential of the Prym map as a multiplication map of sections and exhibits a point where it is an isomorphism. Therefore the Prym map is dominant. In GMSC language: $$ \boxed{ \text{dimension gate} + \text{differential full rank} \Rightarrow \text{moduli reach}. } $$ --- # 19. Schoen's triple-cover dominance revisited For unramified cyclic triple covers of genus-$4$ curves: $$ \dim R^3_4 = 9. $$ The relevant Weil Shimura domain also has dimension: $$ 9. $$ The Prym map is dominant in this genus. Thus the Eisenstein sixfold construction satisfies the same sharp dimension architecture. --- # 20. Patel–Zhang changes one side of the problem Patel–Zhang generalize Schoen's cycle construction from special cyclic covers to finite abelian covers. Their theorem gives, for the appropriate generalized Prym-type Hodge subspace: $$ \boxed{ U_{\mathrm{Weil}} \text{ is generated by algebraic cycles}. } $$ Hence the **realization side** becomes much broader. But the cover family still carries its deck algebra. Therefore the DEMB remains untouched. --- # 21. Realization breadth is not moduli breadth This yields the central GMSC distinction: $$ \boxed{ \text{Cycle Engine Breadth} \neq \text{Generic Moduli Reach}. } $$ Patel–Zhang enlarges: $$ \boxed{ \text{which generalized Prym Hodge subspaces can be algebraized}. } $$ It does not automatically enlarge: $$ \boxed{ \text{which very-general }K\text{-Weil components can be dominated}. } $$ The latter is constrained by persistent endomorphisms. --- # 22. Example: why $\mathbb Q(\zeta_7)$ does not solve generic $\mathbb Q(\sqrt{-7})$ Consider a rational irreducible deck factor with: $$ E_{\mathrm{deck}} = \mathbb Q(\zeta_7). $$ Then: $$ [E_{\mathrm{deck}}:\mathbb Q] = 6. $$ It contains the quadratic subfield: $$ \mathbb Q(\sqrt{-7}). $$ One might hope this gives a Prym route for: $$ K=\mathbb Q(\sqrt{-7}). $$ But every member of the family still carries the full: $$ \mathbb Q(\zeta_7) $$ action. Thus: $$ \boxed{ \operatorname{End}^0(B) \supseteq \mathbb Q(\zeta_7) \supsetneq \mathbb Q(\sqrt{-7}). } $$ The image lies in a proper special locus of the generic: $$ \mathbb Q(\sqrt{-7}) $$ Weil moduli. Therefore it cannot dominate the very-general component. --- # 23. Formal sixfold case $d_V=6$ If: $$ d_V=6, $$ then: $$ (g-1)d_V=6 $$ forces: $$ g=2. $$ Such a factor can indeed be six-dimensional. But its deck endomorphism field has degree: $$ 6. $$ Therefore it is too endomorphism-rich to generic-dominate an imaginary-quadratic-only Weil component. It may still define an interesting special sublocus. It does not solve the generic target. --- # 24. Formal case $d_V=3$ For finite abelian groups, rational irreducible dimensions are Euler totients: $$ \varphi(n). $$ There is no: $$ n $$ with: $$ \varphi(n)=3. $$ Therefore the formal possibility: $$ (g-1,d_V)=(2,3) $$ does not occur in the rationally irreducible finite-abelian setting. --- # 25. Formal case $d_V=1$ The case: $$ d_V=1 $$ would require: $$ g=7. $$ A one-dimensional rational character has values in: $$ \mathbb Q, $$ with effective deck algebra: $$ \mathbb Q. $$ It cannot supply an imaginary quadratic $K$-action by itself. So it is irrelevant to deck-induced Weil type. --- # 26. Full classification in the simple architecture Combining Sections 23–25: $$ \boxed{ \dim B_V=6 } $$ and deck-induced very-general imaginary-quadratic Weil dominance leave only: $$ \boxed{ d_V=2,\quad g=4. } $$ Cyclotomic classification then leaves only: $$ \boxed{ \mathbb Q(i),\quad\mathbb Q(\sqrt{-3}). } $$ This is the strongest structural closure of R003. --- # 27. Reducible rational representations Suppose instead one uses: $$ V=V_1\oplus V_2. $$ Then the resulting abelian factor typically inherits rational idempotents projecting to the two summands. These are extra endomorphisms. Hence a reducible deck module usually makes the target still more special. So reducibility does not generically solve the DEMB. However R003 does not prove an absolute impossibility theorem for every reducible construction. A separate endomorphism-centralizer audit is required. --- # 28. Auxiliary $K$-action not generated by deck transformations One may imagine: - deck algebra: $$ E; $$ - an independent geometric correspondence supplying: $$ K; $$ - a target factor whose generic endomorphism algebra collapses appropriately. This falls outside the direct deck-induced hypothesis. Such a construction is not ruled out by DEMB. But it acquires a new proof debt: $$ \boxed{ \text{Auxiliary }K\text{-Action Construction}. } $$ So it is no longer the simple generalized-Prym bypass of R002. --- # 29. Nonabelian covers For nonabelian deck groups, rational irreducible representations can have matrix-algebra and division-algebra endomorphism structures. This changes the classification substantially. DEMB still has an abstract version: > persistent endomorphism algebra larger than the generic target centralizer obstructs dominance. But the cyclotomic quadratic classification no longer applies verbatim. Therefore nonabelian Prym architectures remain open search space. --- # 30. Ramified covers Patel–Zhang explicitly focus on the unramified finite-abelian setting. Ramified Pryms have different Chevalley–Weil multiplicities and additional branch parameters. Thus: $$ \dim B_V=(g-1)d_V $$ must be modified. The source moduli dimension also increases through branch-point data. So ramification could escape the genus-$4$ dimension collapse. But the persistent deck-endomorphism barrier remains relevant. R003 does not exclude this direction. --- # 31. Component reach after field collapse After DEMB, the standard abelian-Prym route no longer has an arbitrary field problem. It has a narrower component problem: $$ \boxed{ K=\mathbb Q(i) \quad\text{or}\quad K=\mathbb Q(\sqrt{-3}). } $$ Within those fields, the remaining question is: $$ \boxed{ \text{which polarization / Hermitian discriminant components are actually in the Prym image?} } $$ This is now the real Prym residual. --- # 32. Discriminant remains discrete The moduli component label: $$ \delta $$ is discrete arithmetic data. A dominant Prym map into one component does not imply reach of another. Thus even for: $$ K=\mathbb Q(i) $$ or: $$ K=\mathbb Q(\sqrt{-3}), $$ we cannot infer: $$ \boxed{ \text{all } \delta \text{ are Prym-reachable}. } $$ A component-by-component Hermitian polarization audit is necessary. --- # 33. Old versus modern discriminant conventions Koike writes a discriminant: $$ \delta_{\mathrm{old}} $$ using his Hermitian determinant convention. Modern Markman uses a normalized class in: $$ \mathbb Q^\times/ N_{K/\mathbb Q}(K^\times), $$ with split rank-$6$ normalized as: $$ \delta_{\mathrm{M}}=-1. $$ Therefore: $$ \boxed{ \delta_{\mathrm{Koike}}=1 } $$ must not be silently rewritten as: $$ \delta_{\mathrm{M}}=-1 $$ without an explicit normalization calculation. This is the next arithmetic audit. --- # 34. Modern narrative and historical terminology Recent work often summarizes the Schoen/Koike-type sixfold results as covering split Weil-type cases for the Eisenstein/Gaussian fields. That is useful orientation. But R003 keeps a strict firewall: $$ \boxed{ \text{historical stated discriminant} \neq \text{modern normalized label until converted}. } $$ This prevents a false claim that a non-split component has already been closed. --- # 35. Consequence for GMSC frontier R002 treated Prym as one of three general frontier routes. R003 refines this. For arbitrary imaginary quadratic: $$ K, $$ the standard deck-induced finite-abelian Prym route is **not** a generic frontier architecture. It becomes: $$ \boxed{ P_{\mathrm{Prym}}^{\mathrm{quad}} } $$ with field domain: $$ \boxed{ K\in \{ \mathbb Q(i), \mathbb Q(\sqrt{-3}) \}. } $$ --- # 36. Updated positive frontier by field ### For: $$ K=\mathbb Q(i) \quad\text{or}\quad K=\mathbb Q(\sqrt{-3}), $$ the frontier remains: $$ \boxed{ P_{\mathrm{Markman}} \parallel P_{\mathrm{Prym}} \parallel P_{\mathrm{Direct}}. } $$ ### For generic imaginary quadratic: $$ K\notin \{ \mathbb Q(i), \mathbb Q(\sqrt{-3}) \}, $$ the standard deck-induced Prym route is removed from the generic-dominance frontier. Then: $$ \boxed{ P_{\mathrm{Markman}} \parallel P_{\mathrm{Direct}} } $$ remain the principal current positive architectures, with degeneration/auxiliary geometry as route candidates. --- # 37. This does not demote Patel–Zhang's theorem Patel–Zhang still greatly expands the set of generalized Prym Hodge subspaces known to be algebraic. The theorem's value is unchanged. What R003 changes is the interpretation: $$ \boxed{ \text{algebraic generalized Prym classes} } $$ need not sweep: $$ \boxed{ \text{generic arbitrary-}K\text{ Weil sixfold moduli}. } $$ That is a moduli-reach issue, not a cycle-algebraicity issue. --- # 38. Construct result R003 constructs a new theorem-level filter: $$ \boxed{ \mathsf{DEMB} } $$ which can be applied before any expensive period-map computation. Given candidate cover data: $$ (G,V,g), $$ first compute: $$ E_{\mathrm{deck}}. $$ If: $$ E_{\mathrm{deck}}\supsetneq K, $$ then the candidate is rejected for generic dominance immediately. This is a high-value GMSC compression operator. --- # 39. Attack result R003 attacks the naive extrapolation: > Patel–Zhang works for arbitrary finite abelian groups, therefore generalized Pryms may generic-dominate arbitrary imaginary-quadratic Weil sixfold components. This inference is false. Cycle realization generality does not remove the persistent endomorphism algebra. Thus the naive arbitrary-$K$ Prym bypass is: $$ \boxed{ \mathrm{BROKEN}. } $$ --- # 40. Compress result The Prym route residual shrinks from: $$ \boxed{ \text{arbitrary }(K,\delta)\text{ moduli reach} } $$ to: $$ \boxed{ \text{component/discriminant reach for } K=\mathbb Q(i),\mathbb Q(\sqrt{-3}). } $$ This is a major freedom reduction. --- # 41. Route-local min-cut update For the standard abelian-Prym route: $$ \boxed{ \operatorname{MinCut}_{\mathrm{Prym}} = \{ \text{Component Reach / Discriminant Match} \} } $$ after the field filter. The cycle-construction gate is already closed by the Schoen/Patel–Zhang engine on the relevant generalized Prym subspace. --- # 42. Bottleneck stability update Across: $$ R001,\ R002,\ R003, $$ the very-general Weil target SCC remains the global core every round: $$ \boxed{ \mathrm{BS}_3 \left( \mathcal C_{\mathrm{Weil6}}^{\mathrm{vg}} \right) = 1. } $$ Semiregularity remains route-local. Prym dominance has now decomposed into: - field compatibility； - component/discriminant compatibility。 The first is closed by DEMB. The second remains open. --- # 43. New field-specific research opportunity The strongest remaining Prym opportunity is now: $$ \boxed{ K=\mathbb Q(i) \text{ or } \mathbb Q(\sqrt{-3}) } $$ with a component not yet certified closed in modern discriminant normalization. Question: > Can a generalized Prym period map land dominantly in a genuinely different Hermitian discriminant component from the classical one while retaining Patel–Zhang algebraic cycle classes? This is concrete and finite. --- # 44. Why discriminant is the next correct target The field question is now compressed away. The dimension question is compressed away. The cycle-realization question is largely compressed away. The remaining Prym freedom is mostly: $$ \boxed{ \delta. } $$ Thus GMSC says: $$ \boxed{ \text{do not search arbitrary groups next;} \text{ audit component arithmetic first.} } $$ --- # 45. Direct implication for CE001 CE001 targets non-split Weil sixfolds. If R004 shows that classical/generalized Prym constructions are forced into the already solved split component, then Prym ceases to be a bypass for CE001's exact non-split domain. That would restore: $$ P_{\mathrm{Markman}} \parallel P_{\mathrm{Direct}} $$ as the serious positive routes there. Conversely, if Prym reaches even one modern non-split component, CE001's search domain shrinks. --- # 46. Direct implication for Markman Markman's split theorem works for every imaginary quadratic field. Prym's direct generic reach works only for two fields. Therefore R003 shows a structural advantage of Markman's architecture: $$ \boxed{ \text{field-uniformity}. } $$ This explains why semiregularity remains frontier-relevant despite being technically difficult. --- # 47. Direct implication for direct-cycle search For: $$ K\neq \mathbb Q(i),\mathbb Q(\sqrt{-3}), $$ the direct-cycle route gains relative importance. Any algebraic construction not carrying a too-large persistent endomorphism algebra could bypass both: - Markman semiregularity； - Prym field restriction。 So R003 strengthens the case for keeping: $$ P_{\mathrm{Direct}} $$ on the Pareto frontier. --- # 48. GMSC route-state table | Route | Arbitrary $K$? | Actual algebraic-cycle engine? | Main current gate | |---|---:|---:|---| | Markman secant | yes in split architecture | yes | non-split seed / semiregularity | | Classical/generalized abelian Prym | no, generic deck route only $i,\sqrt{-3}$ | yes | discriminant/component reach | | Direct one-cycle | potentially yes | not yet constructed generally | explicit cycle | | Degeneration | potentially yes | conditional | cross-component bridge | | Auxiliary hyperkähler | unknown for sixfold | precedent in fourfold | sixfold correspondence | | CM / absolute-Hodge | yes structurally | no | realization | | Arithmetic/Tate | potentially | not in char-$0$ target automatically | lifting | --- # 49. Updated Pareto interpretation R003 does not eliminate Prym globally. It refines the frontier by domain. A route can be Pareto-optimal in one field/domain and irrelevant in another. Thus GMSC path weights must be indexed by: $$ \boxed{ (K,\delta). } $$ Not merely by theorem name. --- # 50. Current global conclusion The strongest closure of R003 is: $$ \boxed{ \text{generalized finite-abelian Prym cycle theory does not provide an arbitrary-}K\text{ generic Weil-sixfold bypass}. } $$ In the standard deck-induced irreducible architecture: $$ \boxed{ K=\mathbb Q(i) \ \text{or}\ \mathbb Q(\sqrt{-3}), \qquad g=4. } $$ This is a structural no-go outside the two classical cyclotomic quadratic fields. --- # 51. Next Interface Next GMSC round: ```text HODGE_GMSC_R004_PrymDiscriminantComponentAudit.md ``` Primary target: $$ \boxed{ \text{Convert Schoen/Koike Prym components into the modern Hermitian norm-class discriminant convention.} } $$ Planned tasks: 1. reconstruct Koike's Hermitian form and old discriminant definition； 2. reconstruct Schoen's triple-cover polarization/Hermitian form； 3. translate both into: $$ \mathbb Q^\times/ N_{K/\mathbb Q}(K^\times); $$ 4. compare with Markman's modern normalization； 5. determine whether the historical Prym sixfolds lie exactly in modern split: $$ \delta=-1; $$ 6. enumerate any distinct Prym-reachable component； 7. check whether Patel–Zhang generalized covers can alter the Hermitian determinant class while retaining dimension $6$ and no extra generic endomorphisms； 8. if all deck-minimal Pryms force split discriminant, demote Prym as a non-split bypass； 9. if a new discriminant class is reachable, construct its dominance gate； 10. update CE001 open-domain map. --- # References 1. GMSC v0.1, *Global Mathematical Space Compression: Theory Graphs, Residual Min-Cuts, and Tri-Directional Closure for Large Mathematical Problems*, Neo.K with Aletheia, 2026-09-16. 2. K. Koike, *Algebraicity of some Weil Hodge Classes*, Canadian Mathematical Bulletin 47 (2004), 566–572; arXiv:math/0211304. Constructs the fourth-cyclic genus-$4$ Prym map to a $9$-dimensional sixfold Weil Shimura variety and proves dominance. 3. C. Schoen, *Hodge classes on self-products of a variety with an automorphism*, and related correction/addendum. The cyclic triple-cover Prym construction supplies algebraic Weil classes in the Eisenstein sixfold case. 4. D. Patel, Y. Zhang, *Algebraicity of Hodge classes on some Generalized Prym Varieties*, arXiv:2506.13729. Generalizes the algebraic-cycle construction to generalized Pryms associated with finite abelian covers. 5. H. Lange, A. Ortega, *Prym varieties of cyclic coverings*, arXiv:0805.1020. Studies dimensions and generic finiteness of cyclic Prym maps. 6. P. Borówka, A. Shatsila, *Pryms of $\mathbb Z_3\times\mathbb Z_3$ coverings of genus 2 curves*, arXiv:2503.23041. Studies finite-abelian Prym maps beyond cyclic groups. 7. B. van Geemen, A. Verra, *Quaternionic Pryms and Hodge classes*, arXiv:math/0103111. Provides another example where Prym geometry plus Schoen-type cycles algebraizes exceptional Hodge classes. 8. E. Markman, *Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds*, arXiv:2502.03415. 9. B. J. J. Moonen, Yu. G. Zarhin, *Weil classes on abelian varieties*, arXiv:alg-geom/9612017. 10. Aletheia, *HODGE_GMSC_R001_GlobalTheoryGraphCompression*, 2026-09-16. 11. Aletheia, *HODGE_GMSC_R002_WeilCoreAlternativePaths*, 2026-09-16. --- ## Canonical Source Declaration 本檔案為 Hodge GMSC 分支第三篇正式 UTF-8 Markdown canonical source。 數學原始碼只使用 `$...$` 與 `$$...$$`。 本輪沒有宣稱所有 Prym-like constructions 都只能處理 $\mathbb Q(i)$ 與 $\mathbb Q(\sqrt{-3})$。 本輪證明的是一個有明確 architecture scope 的 structural no-go： $$ \boxed{ \text{unramified finite-abelian} + \text{rationally irreducible factor} + \text{deck-induced }K + \text{generic dominance} } $$ 強制： $$ \boxed{ K=\mathbb Q(i) \text{ or } \mathbb Q(\sqrt{-3}), \qquad g=4. } $$ 下一輪將把剩餘 Prym residual壓到 discrete Hermitian discriminant/component arithmetic。