# RCHM × Hodge Conjecture — Research Round 20 ## Küchle $c5$ on the Novelty Frontier: Ambient Lattice Extraction, OG10 False-Positive Rejection, and Degeneration Bridge Candidates **Status:** Internal research note + exact localization verifier **Version:** Round 20 / v2.0 **Date:** 2026-09-17 **Methodological base:** RCHM; Typed Relation & Constraint Algebra; X-Layer Mathematical Completeness; Graph-Native Mathematics; Bridge Search Compiler **Previous rounds:** RCHM × Hodge Conjecture — Rounds 1–19 **Canonical source:** UTF-8 Markdown --- # 0. Round 20 Objective Round 19 classified Küchle $c5$ as a genuine novelty-frontier target in the current ancestry graph. Round 20 attacks the target directly. The first task is deliberately defensive: > Before searching for a new hyperkähler bridge, remove seductive false positives. The middle Hodge diamond of a Küchle $c5$ fourfold contains: $$ \boxed{ h^{3,1}=1, \qquad h^{2,2}=24, \qquad h^{1,3}=1. } $$ Thus: $$ \boxed{ b_4=26. } $$ Two canonical algebraic codimension-$2$ classes come from the ambient Grassmannian. After removing them, a rank-$24$ K3-type block with Hodge numbers: $$ \boxed{ (1,22,1) } $$ appears. Since an OG10-type hyperkähler manifold has: $$ b_2=24 $$ and Hodge numbers: $$ (1,22,1), $$ this produces a very tempting bridge candidate. Round 20 computes the natural quadratic forms and shows that the direct BB-isometric OG10 template fails. --- # 1. Geometric Model of Küchle $c5$ Let: $$ G=\operatorname{Gr}(3,7). $$ Write: $$ \mathcal U $$ for the tautological rank-$3$ subbundle. Kuznetsov describes the Küchle $c5$ fourfold as the zero locus of a sufficiently general section of the rank-$8$ bundle: $$ \boxed{ \mathcal E = \mathcal U^\perp(1) \oplus \mathcal U(1) \oplus \mathcal O(1). } $$ Thus: $$ \boxed{ X=Z(s)\subset G, \qquad \dim X=4. } $$ Equivalently, after the usual bundle identifications, this is the homogeneous Grassmannian model used in modern Fano-fourfold databases. --- # 2. Ambient Codimension-Two Classes Let: $$ \mathcal Q $$ be the rank-$4$ quotient bundle. Consider: $$ \boxed{ \sigma_2=c_2(\mathcal Q)|_X } $$ and: $$ \boxed{ \sigma_{1,1}=c_2(\mathcal U^\vee)|_X. } $$ Both are algebraic classes of type: $$ (2,2). $$ The question is whether they remain linearly independent after restriction to $X$ and what quadratic form they span. --- # 3. Equivariant Localization Setup Let a diagonal torus act on the ambient $7$-space with distinct weights: $$ t_1,\ldots,t_7. $$ A torus fixed point of: $$ \operatorname{Gr}(3,7) $$ is indexed by a $3$-subset: $$ I\subset\{1,\ldots,7\}. $$ At the fixed point: - $\mathcal U$ has weights $t_i$ for $i\in I$; - $\mathcal Q$ has weights $t_j$ for $j\notin I$; - $\mathcal O(1)=\det\mathcal U^\vee$ has weight: $$ h_I=-\sum_{i\in I}t_i. $$ The roots of the defining bundle are: $$ \boxed{ \{-t_j+h_I:j\notin I\} \cup \{t_i+h_I:i\in I\} \cup \{h_I\}. } $$ The tangent Euler class is: $$ \boxed{ e_T(T_IG) = \prod_{i\in I} \prod_{j\notin I} (t_j-t_i). } $$ The fundamental class of $X$ is: $$ c_8(\mathcal E). $$ Therefore: $$ \int_X\alpha = \sum_{|I|=3} \frac{ \alpha|_I\,c_8(\mathcal E)|_I }{ e_T(T_IG) }. $$ --- # 4. Exact Ambient Intersection Matrix The accompanying exact rational-arithmetic verifier evaluates the localization sum for several independent choices of torus weights. Every choice gives: $$ \boxed{ \int_X\sigma_2^2=20, } $$ $$ \boxed{ \int_X\sigma_2\sigma_{1,1}=16, } $$ and: $$ \boxed{ \int_X\sigma_{1,1}^2=14. } $$ Thus the ambient intersection matrix is: $$ \boxed{ M_{\mathrm{amb}} = \begin{pmatrix} 20&16\\ 16&14 \end{pmatrix}. } $$ Its determinant is: $$ \boxed{ \det M_{\mathrm{amb}} = 280-256 = 24. } $$ Since: $$ 20>0, \qquad \det M_{\mathrm{amb}}>0, $$ the matrix is positive definite. Consequently: $$ \boxed{ \sigma_2,\sigma_{1,1} \text{ are linearly independent in }H^4(X,\mathbb Q). } $$ This is stronger than simply guessing that two ambient Schubert directions survive. --- # 5. Operational Hard Block Define the ambient algebraic plane: $$ \boxed{ A_{\mathrm{amb}} = \operatorname{span}_{\mathbb Q} \{\sigma_2,\sigma_{1,1}\}. } $$ Since the form on $A_{\mathrm{amb}}$ is nondegenerate, define the orthogonal hard block: $$ \boxed{ D_{c5} = A_{\mathrm{amb}}^\perp \subset H^4(X,\mathbb Q). } $$ This is an operational residual block. It is not being asserted that $A_{\mathrm{amb}}$ contains every algebraic middle class of every $c5$ fourfold. For a very general target, it is the canonical ambient algebraic sector removed before bridge search. --- # 6. Rank and Hodge Numbers of the Hard Block The full middle Hodge numbers are: $$ (1,24,1). $$ Removing two independent $(2,2)$ directions gives: $$ \boxed{ D_{c5,\mathbb C}: (1,22,1). } $$ Therefore: $$ \boxed{ \dim_{\mathbb Q}D_{c5}=24. } $$ This is the source of the OG10 rank alert. --- # 7. Signature of the Full Middle Intersection Form For a compact Kähler fourfold, the signature of the middle intersection form can be computed from the Hodge decomposition. For $c5$: $$ \operatorname{sign} H^4(X) = -h^{3,1} + h^{2,2} - h^{1,3}. $$ Thus: $$ \boxed{ \operatorname{sign} H^4(X) = -1+24-1 = 22. } $$ Since: $$ b_4=26, $$ the positive and negative indices satisfy: $$ b_+-b_-=22, $$ $$ b_++b_-=26. $$ Hence: $$ \boxed{ (b_+,b_-) = (24,2). } $$ --- # 8. Signature of the Hard Block The ambient plane is positive definite of signature: $$ (2,0). $$ Therefore its orthogonal complement has intersection-form signature: $$ \boxed{ \operatorname{sig}(D_{c5}) = (22,2). } $$ After the conventional sign reversal associated with viewing the Tate-twisted block as a K3-type weight-$2$ structure, the natural comparison form has signature: $$ \boxed{ (2,22). } $$ Even without fixing the sign convention, the unordered pair of indices is: $$ \boxed{ \{22,2\}. } $$ --- # 9. The OG10 Rank Alert For an irreducible hyperkähler manifold of OG10 deformation type: $$ \boxed{ b_2=24. } $$ Its second-cohomology Hodge numbers are therefore: $$ \boxed{ (1,22,1). } $$ At the level of raw rank and Hodge numbers: $$ \boxed{ D_{c5} \quad\text{and}\quad H^2(\mathrm{OG10}) } $$ look identical. This is exactly the kind of candidate a purely pattern-matching AI would over-prioritize. --- # 10. OG10 Beauville--Bogomolov Lattice The Beauville--Bogomolov lattice of OG10 type is: $$ \boxed{ L_{\mathrm{OG10}} = U^{\oplus3} \oplus E_8(-1)^{\oplus2} \oplus A_2(-1). } $$ Its rank is: $$ 3\cdot2+2\cdot8+2=24. $$ Its signature is: $$ \boxed{ (3,21). } $$ Its discriminant has absolute value: $$ \boxed{ 3. } $$ The $U$ and $E_8$ factors are unimodular; the nontrivial discriminant comes from $A_2(-1)$. --- # 11. Direct OG10 Quadratic Gate Failure The $c5$ hard block has natural cup-product signature: $$ (22,2). $$ After sign reversal: $$ (2,22). $$ OG10 has Beauville--Bogomolov signature: $$ (3,21). $$ After global sign reversal: $$ (21,3). $$ Neither choice agrees with: $$ (22,2) $$ or: $$ (2,22). $$ Therefore: $$ \boxed{ D_{c5} \not\simeq H^2(\mathrm{OG10}) } $$ as quadratic spaces preserving the natural forms up to a nonzero scalar. The direct template: ```text c5_hard_block == entire_OG10_H2 preserving_natural_quadratic_form_up_to_scale ``` is rejected. --- # 12. False-Positive Certificate The bridge compiler records: ```text candidate: direct_OG10_BB_bridge C0_weight: PASS C1_rank: PASS c5: 24 OG10: 24 C1_hodge_numbers: PASS c5: [1,22,1] OG10: [1,22,1] C2_natural_quadratic_signature: FAIL c5_hard: [22,2] or sign-reversed [2,22] OG10_BB: [3,21] or sign-reversed [21,3] decision: REJECT_DIRECT_BB_ISOMETRY ``` This is a canonical example of: $$ \boxed{ \text{Hodge-number match} \not\Rightarrow \text{quadratic Hodge bridge}. } $$ --- # 13. What the OG10 Failure Does Not Prove The failure above does **not** prove: > no geometry of OG10 type can ever be related to $c5$. It proves only that the simplest bridge template: $$ \boxed{ D_{c5} \simeq H^2(\mathrm{OG10}) } $$ with natural forms matched up to scalar cannot hold. Possible more complicated relations could involve: - a proper subquotient; - extra algebraic directions; - another cohomological degree; - a correspondence not identifying the natural quadratic forms; - a degeneration or lattice-polarized construction. Those are separate templates and must pass separate gates. --- # 14. Unknown Integral Hard Lattice The ambient plane has determinant: $$ 24. $$ But this does not by itself determine the integral discriminant of: $$ D_{c5}. $$ One still needs: - the integral middle lattice; - the saturation index of the ambient Schubert sublattice; - the gluing data between ambient and orthogonal pieces. Therefore the next lattice task is: $$ \boxed{ \text{compute the saturated integral lattice }D_{c5,\mathbb Z}. } $$ This is a high-priority exact problem. --- # 15. Abstract Kuga--Satake Fallback The hard block has K3-type Hodge numbers: $$ (1,22,1) $$ and rank: $$ 24. $$ At the Hodge-theoretic level the Kuga--Satake construction is available. For rank: $$ n=24, $$ the associated abelian dimension is: $$ \boxed{ 2^{n-2} = 2^{22} = 4,194,304. } $$ Thus it is a formally available but extraordinarily expensive fallback. The algebraicity of the required Kuga--Satake correspondence remains a separate theorem obligation. --- # 16. Historical Twisted-Cubic Candidate Long before the present compiler, the $c5$ geometry was already compared with cubic-fourfold geometry. An ICM-era discussion suggested that a natural candidate for an associated hyperkähler space could be a moduli/Hilbert space of twisted cubic curves. The status must remain: $$ \boxed{ \text{HISTORICAL / CONJECTURAL CANDIDATE}. } $$ The current source set does not provide a completed hyperkähler construction from twisted cubics on a general $c5$ fourfold. Nevertheless this candidate is structurally aligned with the compiler's canonical-incidence strategy and should remain on the frontier. --- # 17. Lagrangian-Grassmannian Birational Scaffold Kuznetsov proves that the $c5$ fivefold is birational to a hyperplane section of: $$ \operatorname{LGr}(3,6). $$ For the fourfold hyperplane section, he obtains a birational model as a singular quadratic section of that Lagrangian-Grassmannian hyperplane section, with singularities along a curve. Therefore the compiler records: ```text candidate: LGr_birational_scaffold geometry_status: PASS hard_Hodge_transfer: UNKNOWN bridge_to_solved_layer: UNKNOWN ``` This is valuable ancestry/birational geometry but not yet a closed Hodge bridge. --- # 18. Derived-K3 Frontier Kuznetsov's original motivation was to find a rectangular Lefschetz decomposition of the $c5$ fivefold whose hyperplane section would produce a noncommutative K3 category. The 2016 paper explicitly states that this construction had not been completed there. Later literature continues to treat the $c5$ K3 category as a genuinely new/conjectural case. Therefore: $$ \boxed{ \text{noncommutative-K3 branch} = \text{ACTIVE / NOT CLOSED}. } $$ This branch is likely important for any future moduli-of-objects hyperkähler construction. --- # 19. 2026 Rational Degenerations Federico Tufo's 2026 thesis studies rational degenerations of the general Küchle $c5$ fourfold. The chapter constructs several singular degenerations together with small resolutions and birational models. Among the explicit structures are conic and quadric fibrations. For example, one model is a conic fibration over a smooth quadric threefold and is shown rational by exhibiting a rational section. Another model is a quadric fibration over: $$ \mathbb P^1\times\mathbb P^1 $$ and is likewise treated through rational sections. Thus the geometry supplies concrete degeneration pathways into the same broad conic/quadric-bundle world that has produced Hodge bridges for cubic and GM fourfolds. --- # 20. 2026 Even-Nodal Direction Current 2026 seminar announcements by Fatighenti and Tufo describe an analogy with the classical construction: $$ \text{blow-up of cubic along a line} \to \text{conic bundle} \to \text{quintic discriminant surface with 16 nodes}. $$ The announcements state that an analogous result is obtained through a degeneration of Küchle $c5$ fourfolds. This is ongoing/current work and must not be upgraded beyond the public abstract. Its compiler status is: ```text degeneration_geometry: ACTIVE conic_bundle_route: PASS_ON_SPECIAL_DEGENERATION even_nodal_analogy: WORK_IN_PROGRESS / ANNOUNCED general_fiber_Hodge_bridge: UNKNOWN smoothing_transport: UNKNOWN ``` --- # 21. Why the Degeneration Route Is Promising A 2024 program on even nodal surfaces of K3 type shows that conic-bundle Fano fourfolds can be linked directly to hyperkähler varieties, and that the associated nodal surfaces generalize Kummer quartic surfaces. The c5 2026 degeneration announcement explicitly positions itself as an analogous construction. Therefore this is not arbitrary pattern matching. The bridge compiler recognizes a previously successful motif: $$ \boxed{ \text{K3-type fourfold} \to \text{conic bundle} \to \text{even nodal discriminant} \to \text{hyperkähler geometry}. } $$ But for general $c5$ the transport back from the singular degeneration remains an unresolved gate. --- # 22. Degeneration Bridge Architecture Define a degeneration bridge candidate: $$ \boxed{ \mathcal X \to \Delta } $$ with: - general fiber $X_t$; - special fiber $X_0$; - resolution or birational model $\widetilde X_0$; - conic/quadric fibration: $$ \widetilde X_0\to B; $$ - discriminant surface $\Delta_B$; - possible double cover / nodal-surface / hyperkähler structure. The desired path is: $$ \boxed{ D(X_t) \rightsquigarrow D(X_0) \to D(\Delta_B) \to H^2(Y_{\mathrm{HK}}). } $$ The first arrow is the dangerous one. --- # 23. Degeneration Gates A degeneration bridge must pass: ### D0 — Flatness / Family The degeneration is controlled by a flat family. ### D1 — Special-Fiber Geometry The conic/quadric bundle or other bridge geometry is verified on $X_0$. ### D2 — Monodromy / Invariant Cycle Determine which part of the general-fiber hard Hodge structure specializes. ### D3 — Rank Preservation Check that the candidate special-fiber bridge sees the full required rank. ### D4 — Correspondence Construct the algebraic correspondence on the special fiber. ### D5 — Smoothing Transport Prove the algebraic/Hodge bridge extends or controls the general fiber. ### D6 — Witness Closure Verify that the transported bridge closes the desired Hodge defect. For current $c5$ work: $$ D1 $$ has substantial evidence. The later gates remain research obligations. --- # 24. Rationality Is Not Hodge Transport The 2026 thesis proves rationality of several singular degenerations. This is geometrically valuable. But: $$ \boxed{ X_0\text{ rational} \not\Rightarrow D(X_t)\text{ is solved}. } $$ A singular degeneration may change: - cohomology rank; - monodromy; - vanishing cycles; - Hodge filtration; - algebraic-cycle groups. Therefore rationality is not itself a bridge-closure certificate. This distinction is important for avoiding another type-erasure error. --- # 25. Current $c5$ Bridge Frontier After the OG10 rejection, the active frontier is: ## F1 — Twisted-Cubic Moduli ```text canonical_incidence: plausible historical_candidate: yes hyperkahler_proof: unknown Hodge_correspondence: unknown ``` ## F2 — Degeneration / Even-Nodal Route ```text special_fiber_geometry: active and partially explicit conic_quadric_fibration: yes on degenerations even_nodal_analogy: current 2026 work general_fiber_transport: unknown ``` ## F3 — Lagrangian-Grassmannian Scaffold ```text birational_geometry: pass singular_quadratic_model: pass direct_Hodge_bridge: unknown ``` ## F4 — Noncommutative K3 / Moduli Route ```text K3_category_expected: yes rectangular_Lefschetz_closure: not established in cited source moduli_hyperkahler: unknown ``` ## F5 — Abstract Kuga--Satake ```text cohomological_template: pass dimension: 4194304 algebraic_correspondence: unknown priority: fallback ``` The direct full-$H^2$ OG10 template is removed. --- # 26. Frontier Priority The compiler should prioritize by the first missing high-value gate. ### Priority A — Integral hard lattice Compute: $$ D_{c5,\mathbb Z} $$ including saturation and discriminant form. This will reject or identify additional hyperkähler lattice templates. ### Priority B — Twisted-cubic Hilbert component Compute: - expected dimension; - smoothness; - symplectic candidate form; - deformation type; - Abel--Jacobi map to $D_{c5}$. ### Priority C — Degeneration invariant cycles For the 2026 conic/quadric degeneration, compute: - limiting mixed Hodge structure; - monodromy; - invariant part of $D_{c5}$; - discriminant-surface correspondence. ### Priority D — Derived numerical lattice Finish or import the numerical Grothendieck lattice of the putative K3 category and compare its Mukai lattice with: $$ D_{c5}. $$ ### Priority E — Lagrangian-Grassmannian singular model Analyze whether the singular quadratic section carries a natural resolution/moduli object whose $H^2$ can receive the hard block. --- # 27. OG10 as a Compiler Regression Test The OG10 alert should remain permanently in the AMRAL benchmark. A weak model might reason: ```text c5 hard: rank 24 Hodge (1,22,1) OG10: b2 24 Hodge (1,22,1) therefore likely bridge ``` The correct graph reasoning is: ```text rank gate: PASS Hodge-number gate: PASS quadratic-form gate: FAIL direct bridge: REJECT ``` This is an ideal X-layer test because the rejected relation remains stored with its failure reason. --- # 28. New Principle — Quadratic False-Positive Rejection For a proposed Hodge bridge between polarized quadratic Hodge structures: $$ (V,Q_V), \qquad (W,Q_W), $$ rank and Hodge-number compatibility are only preliminary gates. A direct similarity bridge requires: $$ \boxed{ Q_W(\phi x,\phi y) = \lambda Q_V(x,y) } $$ for some: $$ \lambda\in\mathbb Q^\times. $$ If real signatures are incompatible even after global sign: $$ \boxed{ \text{reject before geometric search}. } $$ This is exactly what happens for the direct c5--OG10 template. --- # 29. New Principle — Degeneration Frontier, Not Degeneration Proof A special fiber may expose a powerful bridge architecture that is invisible on the general fiber. Therefore degenerations should be used to **generate bridge candidates**. But the candidate remains on the frontier until monodromy and smoothing gates are closed. Formally: $$ \boxed{ \text{special-fiber bridge} \not\Rightarrow \text{general-fiber bridge}. } $$ --- # 30. AMRAL Representation Suggested false-positive record: ```text bridge_false_positive: target: Kuchle_c5_hard_block candidate: OG10_H2 rank_gate: PASS hodge_gate: PASS target_signature: [22,2] candidate_signature: [3,21] quadratic_gate: FAIL decision: reject_direct_isometric_bridge ``` Suggested degeneration candidate: ```text degeneration_bridge: target: general_Kuchle_c5 special_model: c5_rational_degeneration fibration_type: conic_or_quadric discriminant_geometry: even_nodal_candidate special_geometry_status: PASS_OR_ACTIVE monodromy_status: UNKNOWN invariant_cycle_status: UNKNOWN general_fiber_transport: UNKNOWN ``` --- # 31. Exact Verifier The accompanying Python verifier performs: 1. exact equivariant localization on $\operatorname{Gr}(3,7)$; 2. three independent torus-weight evaluations; 3. reconstruction of: $$ M_{\mathrm{amb}} = \begin{pmatrix} 20&16\\ 16&14 \end{pmatrix}; $$ 4. determinant and positivity checks; 5. full and hard-block signature arithmetic; 6. OG10 lattice rank/signature/discriminant checks; 7. direct quadratic-gate rejection; 8. Kuga--Satake fallback dimension. All numerical computations are exact rational/integer arithmetic. --- # 32. Evidence Typing This round separates: ## Published geometry - definition and Hodge diamond of Küchle $c5$; - Lagrangian-Grassmannian birational scaffold; - historical K3-category / curve-moduli expectations; - 2026 rational degenerations; - 2026 c5/even-nodal ongoing direction; - OG10 Beauville--Bogomolov lattice. ## New exact computation in this round - ambient intersection matrix; - determinant $24$; - positivity of the ambient plane; - rank-$24$ hard block; - hard-form signature; - direct OG10 quadratic-form rejection. ## Research proposals - integral hard-lattice computation; - twisted-cubic Hilbert bridge; - degeneration invariant-cycle bridge; - derived Mukai-lattice comparison. No new hyperkähler construction for general $c5$ is claimed. --- # 33. Round 20 Conclusion Küchle $c5$ remains on the novelty frontier, but the bridge search is now substantially narrower. The exact ambient Schubert calculation gives: $$ \boxed{ M_{\mathrm{amb}} = \begin{pmatrix} 20&16\\ 16&14 \end{pmatrix}, \qquad \det=24. } $$ Thus two independent positive algebraic directions can be removed from: $$ H^4(X). $$ The resulting hard block has: $$ \boxed{ \operatorname{rank}=24, \qquad h=(1,22,1), \qquad \operatorname{signature}=(22,2) } $$ for the cup-product form. This initially mimics OG10. But the OG10 Beauville--Bogomolov lattice has: $$ \boxed{ \operatorname{signature}=(3,21). } $$ Hence the most obvious direct OG10 bridge is a false positive: $$ \boxed{ \text{rank PASS} + \text{Hodge PASS} + \text{quadratic gate FAIL}. } $$ The frontier that survives is genuinely geometric: $$ \boxed{ \begin{array}{c} \text{twisted-cubic moduli candidate}\\ \text{2026 degeneration / even-nodal route}\\ \text{Lagrangian-Grassmannian birational scaffold}\\ \text{noncommutative-K3 / moduli route}\\ \text{abstract Kuga--Satake fallback} \end{array} } $$ The most important immediate task is now: $$ \boxed{ \text{compute the saturated integral hard lattice of }c5. } $$ The most promising geometric task is: $$ \boxed{ \text{extract the invariant Hodge block through the 2026 conic/quadric degeneration}. } $$ This is the main closure of Round 20. --- # References 1. Alexander Kuznetsov, **Küchle fivefolds of type c5**, Mathematische Zeitschrift 284 (2016), 1245--1278. DOI: https://doi.org/10.1007/s00209-016-1707-9 Relevant material: definition of the rank-$8$ bundle, Hodge diamond, birational relation to a Lagrangian-Grassmannian hyperplane section, singular quadratic model for the fourfold, and the unresolved rectangular-Lefschetz/K3-category program. 2. Fano Fourfold Database, **Fano fourfold #321: Küchle c5**. https://fano-fourfolds.hyperkaehler.info/rho/1 3. Federico Tufo, **On the geometry of some Fano and hyperkähler varieties via homogeneous spaces**, PhD thesis, University of Bologna, 2026. DOI: https://doi.org/10.48676/unibo/amsdottorato/13069 Relevant material: Chapter 4, rational degenerations of the general Küchle c5 fourfold, including conic/quadric fibration models. 4. Enrico Fatighenti, **Fano 4-folds of type c5 and even nodal surfaces**, 2026 seminar abstracts, joint work with Federico Tufo. Public abstract states that an analogue of the cubic/GM conic-bundle even-nodal construction is obtained via a degeneration of c5. 5. Marcello Bernardara, Enrico Fatighenti, Grzegorz Kapustka, Michał Kapustka, Laurent Manivel, Giovanni Mongardi, Fabio Tanturri, **Even nodal surfaces of K3 type**, arXiv:2402.08528. Relevant motif: conic-bundle K3-type Fano fourfolds linked geometrically to hyperkähler varieties through even nodal discriminant surfaces. 6. Marcello Bernardara, Enrico Fatighenti, Laurent Manivel, Fabio Tanturri, **Fano fourfolds of K3 type**, Annales de l'Institut Fourier, online first 2026. DOI: https://doi.org/10.5802/aif.3761 7. OG10 lattice: the Beauville--Bogomolov lattice is $$U^3\oplus E_8(-1)^2\oplus A_2(-1),$$ of rank $24$, signature $(3,21)$ and discriminant $3$. 8. Historical c5 hyperkähler candidate: ICM-era discussions proposed moduli/Hilbert spaces of twisted cubic curves as a natural object to investigate. This remains a candidate in this report, not a theorem.