# RCHM × Hodge Conjecture — Research Round 15 ## A Bridge Search Compiler: From Residual Hodge Signatures to Geometric Bridge Candidates **Status:** Internal research note + executable bridge-search prototype **Version:** Round 15 / v1.5 **Date:** 2026-09-17 **Methodological base:** RCHM; Typed Relation & Constraint Algebra; X-Layer Mathematical Completeness; Graph-Native Mathematics **Previous rounds:** RCHM × Hodge Conjecture — Rounds 1–14 **Canonical source:** UTF-8 Markdown --- # 0. Round 15 Objective Round 14 validated a third witness architecture: $$ \boxed{ \text{Symmetry-Amplified Bridge}. } $$ Round 15 asks the harder question: > Given only the structural signature of the unresolved Hodge defect, how should an AI search for a bridge before knowing the correct geometry? The proposed architecture is: $$ \boxed{ \text{Residual Hodge Signature} \to \text{Cohomological Bridge Candidates} \to \text{Geometric Bridge Candidates}. } $$ The main principle is: $$ \boxed{ \text{representation/type filtering should precede geometric search}. } $$ --- # 1. Target Residual Defect Let $$ D^p(X) = \operatorname{Hdg}^p(X) / \mathcal A_{\mathrm{easy}}^p(X) $$ be the unresolved Hodge defect after quotienting by classes already known to be algebraic. The bridge search should operate on $$ \boxed{ D^p(X), } $$ not on the full cohomology ring. --- # 2. Bridge Signature Define $$ \boxed{ \operatorname{BridgeSig}(D) = ( w, h^{p,q}, Q, E, \dagger, MT, Mon, \mathbb F, \Lambda, \mu_E ). } $$ The components are: - $w$: Hodge weight; - $h^{p,q}$: Hodge numbers; - $Q$: polarization or bilinear form; - $E$: endomorphism algebra; - $\dagger$: Rosati/adjoint involution when relevant; - $MT$: Mumford–Tate group or bound; - $Mon$: monodromy representation; - $\mathbb F$: field of definition; - $\Lambda$: lattice/discriminant data; - $\mu_E$: minimal number of $E$-module generators. Unknown entries remain typed unknowns. --- # 3. Typed Gate State Every bridge candidate carries gates with three values: $$ \boxed{ \operatorname{PASS}, \qquad \operatorname{UNKNOWN}, \qquad \operatorname{FAIL}. } $$ A candidate with a hard FAIL is rejected. A candidate with only PASS/UNKNOWN remains on the bridge frontier. This is deliberately not a probabilistic score. --- # 4. Stage I — Cohomological Bridge Synthesis A cohomological bridge is a Hodge structure or representation $V$ which can contain, generate, transport or receive the defect. Stage I ignores whether $V$ is already geometrically realized. The first question is: $$ \boxed{ \text{Which Hodge-theoretic objects can even match }D? } $$ Only after this is answered do we ask: $$ \boxed{ \text{Which varieties realize those objects?} } $$ --- # 5. Gate C0 — Weight / Tate Compatibility A correspondence may introduce a Tate twist. A candidate must permit a morphism of the form $$ D \longrightarrow V(r) $$ or an appropriate quotient/subrepresentation. If no integer $r$ makes the weights compatible, reject the bridge. --- # 6. Gate C1 — Hodge-Type Compatibility For an embedding $$ D\hookrightarrow V(r), $$ a necessary condition is $$ h_D^{p,q} \le h_{V(r)}^{p,q}. $$ A mismatch is a hard type error. --- # 7. Gate C2 — Polarization / Quadratic Similarity When the bridge requires an isometry or similarity, seek $$ \psi:D\to V $$ such that $$ Q_V(\psi x,\psi y) = \lambda Q_D(x,y) $$ for some $$ \lambda\in\mathbb Q^\times. $$ Rank, signature and discriminant can reject candidates before geometry is searched. --- # 8. Gate C3 — Endomorphism Compatibility If $$ E\subseteq\operatorname{End}_{\mathrm{Hdg}}(D), $$ the bridge should support the required $E$-action or a compatible transport of it. A necessary target is often $$ E \hookrightarrow \operatorname{End}_{\mathrm{Hdg}}(V). $$ Rosati compatibility can further restrict the bridge. --- # 9. Gate C4 — Mumford–Tate Representation A cohomological bridge requires a nonzero Hodge morphism. In representation language: $$ \boxed{ \operatorname{Hom}_{MT}(D,V(r)) \neq0 } $$ or the analogous condition for a tensor construction. If the relevant irreducible representation does not occur, geometry cannot repair the mismatch. --- # 10. Gate C5 — Level-One Tensor No-Go Kuga–Satake-like bridges try to realize a level-two Hodge structure in $$ W\otimes W $$ for a level-one Hodge structure $W$. For K3/hyperkähler type, $$ h^{2,0}=1, $$ the Kuga–Satake construction provides such a bridge. But Varesco–Voisin recall a hard obstruction: if a polarized level-two Hodge structure has maximal Mumford–Tate group $$ MT(V)=\mathrm{SO}(q) $$ and $$ h^{2,0}>1, $$ then it cannot be realized as a Hodge substructure of $W\otimes W$ for any level-one $W$. Thus: $$ \boxed{ h^{2,0}>1 + MT=\mathrm{SO}(q)\text{ maximal} \Rightarrow \text{reject the generic level-one tensor bridge}. } $$ This is a compiled no-go theorem. --- # 11. Gates C6–C7 — Monodromy and Rationality A family bridge must be compatible with monodromy, not merely one fiber. The transported relation must also live over the required rational structure. Track the field chain $$ \mathbb Q \subseteq \mathbb F_{\mathrm{signature}} \subseteq \mathbb F_{\mathrm{bridge}} \subseteq \mathbb C. $$ Galois-stable witness ensembles remain admissible when their rational descent is explicit. --- # 12. Reverse Kuga–Satake Fingerprint For a polarized K3-type Hodge structure $V$ of rational dimension $n$, $$ \dim_{\mathbb Q}\mathrm{Cl}^+(V) = 2^{n-1}. $$ Since this is the dimension of $H^1$ of the Kuga–Satake abelian variety, $$ \boxed{ \dim\mathrm{KS}(V) = 2^{n-2}. } $$ Therefore a suspected identity $$ A^m \sim \mathrm{KS}(V) $$ must satisfy $$ \boxed{ m\dim A = 2^{n-2}. } $$ This converts an abelian-side signature into a discrete candidate rank for the hidden K3-type Hodge bridge. --- # 13. Weil Fourfold Replay Let $A$ be an abelian fourfold of $$ K=\mathbb Q(\sqrt{-d}) $$ Weil type with discriminant one. A theorem recalled by Varesco–Voisin gives $$ \boxed{ A^4 \sim \mathrm{KS}(V) } $$ for a polarized K3-type Hodge structure $V$ of rank $$ \boxed{ 6. } $$ Numerically, $$ \dim A^4 = 4\cdot4 = 16. $$ For rank $6$, $$ \dim_{\mathbb Q}\mathrm{Cl}^+(V) = 2^5 = 32, $$ and therefore $$ \boxed{ \dim\mathrm{KS}(V)=16. } $$ So the target signature itself passes an exact reverse-Kuga–Satake dimension test. --- # 14. Weil Lattice Fingerprint The same theorem identifies the quadratic space $$ \boxed{ V \simeq U^{\oplus2} \oplus \langle-1\rangle \oplus \langle-d\rangle, } $$ where $$ U= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}. $$ Hence $$ \operatorname{rank}V=6 $$ and $$ \boxed{ \det V=d. } $$ Thus the imaginary quadratic field parameter appears in the lattice fingerprint. The bridge search has now produced a **rank-six K3-type cohomological candidate before naming a variety**. --- # 15. Stage II — Geometric Lift Search Once a cohomological candidate survives Stage I, search for geometric realizations in categories such as: - K3 surfaces; - higher-dimensional hyperkähler varieties; - generalized Kummer varieties; - $K3^{[n]}$-type varieties; - OG6-type varieties; - moduli spaces of sheaves; - abelian varieties; - incidence varieties; - singular symplectic varieties and resolutions. The cohomological bridge constrains the geometric search space. --- # 16. Gate G0 — Period / Lattice Realizability For a hyperkähler candidate, test: - Beauville–Bogomolov lattice; - period domain; - polarization; - divisibility; - discriminant; - monodromy orbit. If the required lattice cannot occur, reject the candidate. --- # 17. Gate G1 — Algebraic Correspondence Availability A Hodge-structure match is not enough. Require an algebraic map or correspondence: $$ \Gamma \subset X\times Y. $$ Possible sources include: - actual morphisms; - isogenies; - universal sheaves; - incidence cycles; - Abel–Jacobi correspondences; - known algebraic Kuga–Satake correspondences; - quotient/resolution diagrams. Without such a bridge the status remains UNKNOWN. --- # 18. Gate G2 — Canonical Seed Library Search canonical algebraic seed classes on the bridge: $$ c_i(T_Y), \qquad \operatorname{ch}(E), \qquad [\Delta_Y], \qquad [\text{exceptional locus}], $$ $$ [\text{incidence locus}], \qquad \text{tautological cycles}. $$ Bridge search should try universal classes before arbitrary cycles. --- # 19. Gate G3 — Residual-Hit Test For every seed $$ \beta\in B_{\mathrm{alg}}(Y), $$ test $$ \boxed{ \pi_D(\Gamma_*\beta). } $$ If all available seeds have zero residual projection, the bridge does not attack the unresolved Hodge defect. --- # 20. Gate G4 — Symmetry Closure Let $\mathcal E$ act algebraically on $D$. Compute $$ \operatorname{SymCl}_{\mathcal E}(I_\Gamma). $$ Success requires $$ \boxed{ \operatorname{SymCl}_{\mathcal E}(I_\Gamma)=D. } $$ The number of independent bridge seeds needed is controlled by $$ \boxed{ \mu_{\mathcal E}(D), } $$ not by $\dim_{\mathbb Q}D$ alone. --- # 21. Gates G5–G6 — Family Closure and Carrier Synthesis A bridge may solve the very general fiber but not the whole moduli locus. Family closure must therefore track deformation, specialization and monodromy. If an explicit cycle is required, a final synthesis stage may use: - zero loci of bundle sections; - degeneracy loci; - pushforwards or pullbacks; - moduli cycles; - effective representatives. Explicit carrier synthesis is the last stage, not the first. --- # 22. 2026 Hyperkähler Replay The Stage-I reverse fingerprint points toward K3-type/hyperkähler geometry. Van Geemen–Rapagnetta construct a map from the Weil-fourfold setting to a hyperkähler sixfold $X$ of $K3^{[3]}$ type. They use the canonical algebraic seed $$ \boxed{ c_2(T_X). } $$ Its pullback is an algebraic codimension-two class that is not generated by divisor intersections and therefore hits the hard Hodge-Weil sector. This passes the geometric correspondence, canonical seed and residual-hit gates. --- # 23. Symmetry Closure of the Weil Defect The Hodge-Weil residual sector is two-dimensional over $\mathbb Q$ but one-dimensional as a module over $K$. Hence $$ \boxed{ \mu_K(D)=1. } $$ One nonzero algebraic bridge seed closes the entire residual sector under the $K$-action. Thus the search path is $$ \boxed{ \text{Weil signature} \to \text{rank-6 K3-type bridge} \to \text{hyperkähler category} \to c_2(T_X) \to K\text{-symmetry closure}. } $$ --- # 24. Why Hyperkähler Becomes a Natural Candidate Family Several independent cues point to hyperkähler geometry: 1. reverse Kuga–Satake produces a K3-type weight-two structure; 2. K3-type structures naturally occur in hyperkähler $H^2$; 3. the Kuga–Satake correspondence is known to be algebraic for generalized Kummer deformation type; 4. hyperkähler varieties have canonical algebraic Chern classes; 5. generalized Kummer geometry is related to Weil-type abelian fourfolds through intermediate Jacobians and algebraic correspondences. No single cue proves the final bridge. Together they justify early exploration of the hyperkähler category. --- # 25. Hard Negative Replay Consider a polarized level-two Hodge structure with $$ h^{2,0}>1 $$ and maximal Mumford–Tate group $$ MT=\mathrm{SO}(q). $$ The level-one tensor bridge is rejected immediately by the known no-go theorem. The compiler should emit: ```text candidate: generic_level_one_tensor_bridge status: FAIL reason: maximal_MT_level_two_no_go ``` No expensive geometric search should follow. --- # 26. Bridge Frontier Maintain $$ \boxed{ \mathcal F_{\mathrm{bridge}} = \{ \mathcal B: \text{no hard FAIL} \}. } $$ For every surviving candidate, record the first UNKNOWN gate. That gate determines the next research domain. Examples: - lattice UNKNOWN $\to$ lattice/period theory; - geometric realization UNKNOWN $\to$ deformation theory; - correspondence UNKNOWN $\to$ moduli/incidence/derived geometry; - seed UNKNOWN $\to$ characteristic and tautological cycles; - closure UNKNOWN $\to$ symmetry and monodromy. This turns bridge search into a directed research graph. --- # 27. Cost-Aware Gate Ordering A useful default order is $$ \boxed{ \text{Hodge type} \to \text{representation} \to \text{lattice} \to \text{geometry} \to \text{correspondence} \to \text{seed} \to \text{carrier}. } $$ Cheap exact rejection should happen before expensive geometric construction. --- # 28. Candidate Certificate A bridge candidate should emit: ```text bridge_candidate: target_defect cohomological_type weight_tate_gate hodge_number_gate polarization_gate endomorphism_gate mumford_tate_gate no_go_gate monodromy_gate lattice_gate geometric_realization correspondence_status canonical_seed_candidates raw_bridge_rank symmetry_closed_rank family_closure_status carrier_synthesis_status ``` This certificate lets another AI resume the research without rereading every source. --- # 29. Prototype Compiler The accompanying Python prototype implements a small typed gate engine. It verifies: 1. the reverse Kuga–Satake dimension identity; 2. the rank-six Weil lattice determinant; 3. the two-dimensional symmetry closure from one $K$-seed; 4. a surviving reverse-KS cohomological candidate; 5. a hyperkähler geometric candidate with unresolved gates; 6. a hard-rejected maximal-Mumford–Tate tensor candidate; 7. a replay of the published 2026 bridge as a nearly closed candidate. The prototype does not discover new geometry. It verifies that the search logic is executable. --- # 30. Exact Fingerprint Checks The prototype confirms $$ \dim_{\mathbb Q}V=6, $$ $$ \dim_{\mathbb Q}\mathrm{Cl}^+(V)=32, $$ $$ \dim\mathrm{KS}(V)=16, $$ and $$ \dim A^4=16. $$ It also confirms $$ \det \left( U^{\oplus2} \oplus \langle-1\rangle \oplus \langle-d\rangle \right) = \boxed{d}. $$ --- # 31. Discovery vs. Proof The compiler outputs $$ \boxed{ \text{structurally admissible bridge candidate}. } $$ It does not automatically prove: - existence of a realizing variety; - algebraicity of a correspondence; - nonvanishing of the transported seed; - family closure. Thus $$ \boxed{ \text{bridge discovery} \neq \text{bridge proof}. } $$ This separation is mandatory. --- # 32. New Principles ## Principle A — Cohomology Before Geometry Search the representation/Hodge structure first, then search for a variety realizing it. ## Principle B — Reverse-Construction Fingerprints A construction such as Kuga–Satake can be inverted at the level of dimensions, lattices and endomorphisms to generate candidate bridges. ## Principle C — No-Go Compilation A theorem excluding an entire architecture should be compiled into an early hard gate. ## Principle D — Module-Generator Search The number of algebraic bridge seeds required is controlled by $$ \mu_E(D), $$ not only by $\dim_{\mathbb Q}D$. ## Principle E — Frontier-Directed Research The first unresolved gate determines the next mathematical domain the AI should search. --- # 33. Round 15 Conclusion Round 15 turns bridge discovery into a typed compiler: $$ \boxed{ D \to \operatorname{BridgeSig}(D) \to \text{cohomological candidates} \to \text{hard gates} \to \text{geometric lifts} \to \text{correspondences} \to \text{canonical seeds} \to \text{symmetry closure}. } $$ For a discriminant-one Weil fourfold, the target structure itself generates a reverse Kuga–Satake fingerprint: $$ A^4 \sim \mathrm{KS}(V), \qquad \dim_{\mathbb Q}V=6. $$ The Clifford-dimension check gives $$ 2^{6-1}=32, $$ hence $$ \dim\mathrm{KS}(V)=16=\dim A^4. $$ The lattice $$ U^{\oplus2} \oplus \langle-1\rangle \oplus \langle-d\rangle $$ also preserves the Weil field parameter. These independent constraints push the search toward K3-type/hyperkähler bridge geometry before the 2026 geometric bridge is supplied. Conversely, maximal Mumford–Tate level-two structures with $h^{2,0}>1$ are rejected from generic level-one tensor bridges before geometric search begins. The result is not an automatic Hodge-conjecture solver. It is a mathematically pruned bridge-discovery search space: $$ \boxed{ \text{candidate generation} + \text{hard rejection} + \text{frontier-directed research}. } $$ This is the main closure of Round 15. --- # References 1. M. Varesco, C. Voisin, **Hyper-Kähler Manifolds of Generalized Kummer Type and the Kuga-Satake Correspondence**, Milan Journal of Mathematics 90 (2022), 433–444. DOI: https://doi.org/10.1007/s00032-022-00369-8 2. Mauro Varesco, **Hodge similarities, algebraic classes, and Kuga-Satake varieties**, Mathematische Zeitschrift 305, 69 (2023). DOI: https://doi.org/10.1007/s00209-023-03390-8 3. Bert van Geemen, Antonio Rapagnetta, **Hyperkähler sixfolds, abelian fourfolds of Weil type and a Hodge class**, arXiv:2607.18341 (2026). 4. Salvatore Floccari, Mauro Varesco, **Algebraic cycles on hyper-Kähler varieties of generalized Kummer type**, Mathematische Annalen 391 (2025), 4443–4453. DOI: https://doi.org/10.1007/s00208-024-03027-z 5. Bert van Geemen, **Fourfolds of Weil type and the spinor map**, Expositiones Mathematicae 41 (2023), 418–447. DOI: https://doi.org/10.1016/j.exmath.2023.04.006