# RCHM × Hodge Conjecture — Research Round 9 ## Envelope-Relative Factorization Incidence as a Carrier-Free Purity Splitter **Status:** Internal research note + reproducible symbolic experiment **Version:** Round 9 / v0.9 **Date:** 2026-09-17 **Methodological base:** RCHM; Typed Relation & Constraint Algebra; X-Layer Mathematical Completeness; Graph-Native Mathematics **Previous rounds:** RCHM × Hodge Conjecture — Rounds 1–8 **Canonical source:** UTF-8 Markdown --- # 0. Round 9 Objective Round 8 produced an explicit rational-quartic negative control. For the primitive Hodge class $\alpha(C)$ of a smooth rational quartic $C$ in a smooth quartic surface $S$, the literature gives: $$ I_{\alpha(C),2}=I_{C,2}, $$ and: $$ \dim I_{\alpha(C),3}=16. $$ The explicit geometry constructed in Round 8 gave: $$ \dim G_{\alpha(C),3}=14, $$ hence: $$ \operatorname{PD}_3(\alpha(C))=2. $$ The unresolved question was: > Can one detect the geometric part of the cubic signature without already knowing the carrier $C$? Round 9 gives the first positive partial answer. The method is: $$ \boxed{ \text{safe envelope} \to \text{multigraded factor incidence} \to \text{candidate carrier factors} \to \text{geometric signature sector}. } $$ The key idea is that a geometric carrier inside a known envelope does not contribute arbitrary equations. Its equations generate structured multiplication images in the envelope coordinate ring. --- # 1. Safe Envelope from Level 2 For the rational quartic: $$ I_{\alpha(C),2} = \langle q\rangle, $$ where: $$ q=xw-yz. $$ Thus the level-$2$ signature reconstructs the unique smooth quadric envelope: $$ \boxed{ Q=V(q). } $$ This step is carrier-safe but not carrier-identifying. The carrier satisfies: $$ C\subset Q, $$ but: $$ C\neq Q. $$ The strategy of Round 9 is to work **inside $Q$** rather than continue treating all ambient cubic equations equally. --- # 2. Cox Coordinates of the Envelope Use the Segre parametrization: $$ x=ac, \qquad y=ad, \qquad z=bc, \qquad w=bd. $$ Then: $$ Q\simeq\mathbb P^1\times\mathbb P^1. $$ The Cox ring is: $$ \mathbb C[a,b,c,d] $$ with bidegrees: $$ \deg(a)=\deg(b)=(1,0), $$ $$ \deg(c)=\deg(d)=(0,1). $$ An ambient cubic restricts to: $$ H^0(Q,\mathcal O_Q(3,3)). $$ Its dimension is: $$ \boxed{ \dim H^0(Q,\mathcal O_Q(3,3))=16. } $$ Write: $$ W_{3,3} = H^0(Q,\mathcal O_Q(3,3)). $$ --- # 3. The Two Actual Carrier Factors The rational quartic $C$ is defined on $Q$ by: $$ \boxed{ F_C = a^3d-b^3c. } $$ Thus: $$ F_C \in H^0(Q,\mathcal O_Q(3,1)). $$ The residual quartic $D_0$ from: $$ Q\cap S=C\cup D_0 $$ is defined by: $$ \boxed{ F_D = a(c^3+cd^2-d^3) + b(-c^3-c^2d-2cd^2+d^3). } $$ Thus: $$ F_D \in H^0(Q,\mathcal O_Q(1,3)). $$ Round 9 will recover these factors from multiplication incidence rather than assuming them as input. --- # 4. Factorization Incidence Operator Let: $$ A\subseteq W_{3,3} $$ be a signature subspace. For a candidate bidegree: $$ (u,v), $$ define: $$ H_{u,v} = H^0(Q,\mathcal O_Q(u,v)). $$ Let the complementary degree be: $$ (3-u,3-v). $$ Define the operator: $$ \boxed{ \Theta^A_{u,v}: H_{u,v} \to \operatorname{Hom} \left( H_{3-u,3-v}, W_{3,3}/A \right) } $$ by: $$ \boxed{ \Theta^A_{u,v}(F)(G) = FG\bmod A. } $$ Then: $$ F\in\ker\Theta^A_{u,v} $$ if and only if: $$ \boxed{ F\cdot H_{3-u,3-v} \subseteq A. } $$ This is the fundamental carrier-factor test. It asks: > Does the signature subspace contain the entire multiplication image generated by the candidate factor $F$? No carrier is assumed in the definition. --- # 5. Factorization-Incidence Variety Define: $$ \boxed{ \mathfrak P_{u,v}(A) = \mathbb P \left( \ker\Theta^A_{u,v} \right). } $$ A point: $$ [F]\in\mathfrak P_{u,v}(A) $$ is a candidate carrier factor of bidegree $(u,v)$. The candidate is only a hypothesis. It must still pass: - geometric realizability; - containment; - field-of-definition; - class verification; - higher-order Hodge checks. But the search is now finite-dimensional and linear-algebraic. --- # 6. Why the Operator Is Computationally Cheap For fixed $A$, choose a basis of the quotient: $$ W_{3,3}/A. $$ Then multiplication: $$ F\cdot G $$ is bilinear in $(F,G)$. However, when $G$ ranges over a fixed basis of: $$ H_{3-u,3-v}, $$ the condition: $$ FG\in A $$ is linear in the coefficients of $F$. Therefore: $$ \boxed{ \ker\Theta^A_{u,v} } $$ is obtained from a single matrix kernel. This is substantially cheaper than a general Hilbert- or Chow-scheme search. --- # 7. The Explicit Geometric Cubic Signature Round 8 constructed: $$ C, \qquad D_0, \qquad S. $$ Define the geometrically explained ambient cubic space: $$ G_3 = I_{C,3} + I_{D_0,3} + J_{S,3}. $$ Its restriction to $Q$ is: $$ A_{\mathrm{geom}} = G_3|_Q \subset W_{3,3}. $$ Exact symbolic computation gives: $$ \boxed{ \dim A_{\mathrm{geom}}=10. } $$ This agrees with the Round 8 geometric dimension count. --- # 8. Carrier Multiplication Sector for $C$ Because: $$ F_C \in H_{3,1}, $$ multiply by: $$ H_{0,2}. $$ Since: $$ \dim H_{0,2}=3, $$ the multiplication image is: $$ \boxed{ U_C = F_C\cdot H_{0,2}. } $$ Exact computation gives: $$ \boxed{ \dim U_C=3. } $$ Moreover: $$ U_C\subseteq A_{\mathrm{geom}}. $$ --- # 9. Carrier Multiplication Sector for $D_0$ Similarly: $$ F_D \in H_{1,3}. $$ Multiply by: $$ H_{2,0}. $$ Since: $$ \dim H_{2,0}=3, $$ define: $$ \boxed{ U_D = F_D\cdot H_{2,0}. } $$ Exact computation gives: $$ \boxed{ \dim U_D=3. } $$ And: $$ U_D\subseteq A_{\mathrm{geom}}. $$ --- # 10. Jacobian Sector Let: $$ J_Q = J_{S,3}|_Q. $$ Exact computation gives: $$ \boxed{ \dim J_Q=4. } $$ Furthermore: $$ \boxed{ \dim(U_C+U_D)=6, } $$ and: $$ \boxed{ \dim(U_C+U_D+J_Q)=10. } $$ Therefore: $$ \boxed{ A_{\mathrm{geom}} = U_C+U_D+J_Q. } $$ This gives an internal structural decomposition of the entire known geometric signature sector. --- # 11. Factorization-Incidence Recovery on the Geometric Sector Apply: $$ \Theta^{A_{\mathrm{geom}}}_{u,v} $$ for several candidate bidegrees. Exact symbolic computation gives: $$ \boxed{ \dim\ker\Theta^{A_{\mathrm{geom}}}_{3,1}=1. } $$ The unique projective kernel is: $$ \boxed{ [F_C]. } $$ Likewise: $$ \boxed{ \dim\ker\Theta^{A_{\mathrm{geom}}}_{1,3}=1. } $$ and the unique projective kernel is: $$ \boxed{ [F_D]. } $$ Thus the splitter reconstructs both carrier factors. --- # 12. Tested Negative Bidegrees For the same $A_{\mathrm{geom}}$, exact computation gives: $$ \ker\Theta_{2,2}=0, $$ $$ \ker\Theta_{3,0}=0, $$ $$ \ker\Theta_{0,3}=0, $$ $$ \ker\Theta_{2,1}=0, $$ $$ \ker\Theta_{1,2}=0. $$ Therefore, among the tested low-complexity factor types, only the actual geometric carrier types survive. This is a nontrivial inverse identification result. --- # 13. Actual Hodge-Annihilator Restriction Has Dimension 12 The literature theorem gives: $$ \dim I_{\alpha(C),3}=16. $$ The restriction map: $$ R_3 \to H^0(Q,\mathcal O_Q(3,3)) $$ has kernel: $$ qR_1. $$ Since: $$ q\in I_{\alpha(C),2}, $$ we have: $$ qR_1 \subseteq I_{\alpha(C),3}. $$ And: $$ \dim qR_1=4. $$ Therefore the actual Hodge-annihilator signature restricted to $Q$ has dimension: $$ \boxed{ \dim A_\alpha = 16-4 = 12. } $$ Thus: $$ A_{\mathrm{geom}} \subseteq A_\alpha \subseteq W_{3,3}, $$ with: $$ \boxed{ \dim A_{\mathrm{geom}}=10, \qquad \dim A_\alpha=12. } $$ The two-dimensional purity defect remains visible after restriction to the safe envelope. --- # 14. Monotonicity of Factor Candidates If: $$ A\subseteq A', $$ then: $$ \boxed{ \ker\Theta^A_{u,v} \subseteq \ker\Theta^{A'}_{u,v}. } $$ Therefore the true carrier factors remain candidates when the signature space is enlarged by non-geometric directions. Since: $$ A_{\mathrm{geom}}\subseteq A_\alpha, $$ we necessarily have: $$ [F_C] \in \mathfrak P_{3,1}(A_\alpha), $$ and: $$ [F_D] \in \mathfrak P_{1,3}(A_\alpha). $$ So the extra impurity sector cannot erase the genuine factor candidates. It can only create additional aliases. --- # 15. The Remaining Risk: Spurious Factor Candidates Because: $$ \dim A_\alpha=12 $$ is larger than: $$ \dim A_{\mathrm{geom}}=10, $$ it is possible in principle that the extra two-dimensional non-geometric sector creates new factor candidates: $$ \dim\ker\Theta^{A_\alpha}_{u,v} > \dim\ker\Theta^{A_{\mathrm{geom}}}_{u,v}. $$ Without an explicit basis of: $$ I_{\alpha(C),3}, $$ Round 9 does not claim this cannot happen for the actual Hodge annihilator. Instead it tests generic robustness. --- # 16. Generic Robustness Experiment The quotient: $$ W_{3,3}/A_{\mathrm{geom}} $$ has dimension: $$ 16-10=6. $$ A twelve-dimensional extension: $$ A^* \supset A_{\mathrm{geom}} $$ corresponds to choosing a two-dimensional subspace of this six-dimensional quotient. The parameter space is: $$ \operatorname{Gr}(2,6). $$ The accompanying exact computation chooses one explicit extension: $$ \boxed{ A^* = A_{\mathrm{geom}} + \operatorname{span} \{a^3c^3,\;a^3c^2d\}. } $$ It has: $$ \dim A^*=12. $$ For this extension: $$ \boxed{ \dim\ker\Theta^{A^*}_{3,1}=1, } $$ $$ \boxed{ \dim\ker\Theta^{A^*}_{1,3}=1. } $$ The kernels are still exactly: $$ [F_C], \qquad [F_D]. $$ The tested false bidegrees still have zero kernel. --- # 17. Generic Uniqueness Consequence Kernel dimension is upper semicontinuous in algebraic families of matrices. Every twelve-dimensional extension containing $A_{\mathrm{geom}}$ must have: $$ \dim\ker\Theta_{3,1}\ge1 $$ because $F_C$ always remains in the kernel. The explicit extension $A^*$ realizes the minimum possible value: $$ 1. $$ Therefore there is a nonempty Zariski-open subset of: $$ \operatorname{Gr}(2,6) $$ on which: $$ \boxed{ \dim\ker\Theta_{3,1}=1. } $$ Likewise: $$ \boxed{ \dim\ker\Theta_{1,3}=1 } $$ on a nonempty Zariski-open subset. Hence: $$ \boxed{ \text{generic two-dimensional impurity does not destroy unique factor recovery}. } $$ This is a genuine robustness result. It does **not** prove that the actual Hodge impurity sector is generic. --- # 18. Envelope-Relative Purity Splitter Round 9 therefore proposes the following algorithm. Given: $$ I_{\alpha,\le m_0} $$ with a carrier-safe lowest level $m_0$: ### Step 1 — Recover envelope Construct: $$ E=V(I_{\alpha,\le m_0}). $$ ### Step 2 — Identify the envelope coordinate ring If possible, use a natural multigrading or decomposition of: $$ H^0(E,\mathcal L). $$ ### Step 3 — Restrict higher signature Map: $$ I_{\alpha,m} \to H^0(E,\mathcal L_m). $$ ### Step 4 — Search factor-incidence kernels For candidate divisor types $\tau$ compute: $$ \ker\Theta^A_\tau. $$ ### Step 5 — Generate factorable geometric sectors For every candidate factor $F$ form: $$ F\cdot H_{\mathrm{complement}}. $$ ### Step 6 — Add structural sectors For hypersurfaces include: $$ J_{S,m}|_E. $$ ### Step 7 — Define recovered geometric subspace $$ \boxed{ \widehat G_{\alpha,m} = J_{S,m}|_E + \sum_F F\cdot H_{\mathrm{complement}}. } $$ ### Step 8 — Quarantine residual signature $$ \boxed{ \widehat N_{\alpha,m} = A_\alpha/\widehat G_{\alpha,m}. } $$ This is the proposed purity splitter. --- # 19. Exact Recovery on the Known Geometric Sector For the explicit rational-quartic geometry: $$ \widehat G_3 = U_C+U_D+J_Q. $$ Exact computation gives: $$ \boxed{ \dim\widehat G_3=10. } $$ And: $$ \boxed{ \widehat G_3=A_{\mathrm{geom}}. } $$ Thus, when fed the geometrically generated signature sector, the factorization splitter recovers the entire geometric subspace exactly. --- # 20. Predicted Behavior on the Full Hodge Signature The full envelope-restricted Hodge signature satisfies: $$ \dim A_\alpha=12. $$ If the actual two-dimensional impurity sector lies in the generic open set described above, then factorization incidence would still return only: $$ F_C $$ and: $$ F_D $$ at the tested bidegrees. In that case: $$ \dim\widehat G_{\alpha,3}=10, $$ and: $$ \boxed{ \dim\widehat N_{\alpha,3}=2, } $$ exactly matching the theoretical purity defect. This is a testable prediction for a future period computation of the explicit Round 8 quartic. It is not asserted as already verified. --- # 21. Why This Is Carrier-Free in the Relevant Sense The method does not receive: $$ C $$ or: $$ D_0 $$ as input. It receives: 1. the safe envelope $Q$; 2. a signature subspace $A$; 3. candidate factor bidegrees. The factors are recovered as kernels of linear operators: $$ \Theta^A_{u,v}. $$ Thus the algorithm is carrier-free but **type-guided**. The bidegree search itself can be enumerated or prioritized from: - Hilbert-function constraints; - degree budget; - symmetry; - adjunction; - known envelope geometry. --- # 22. Relation to Common-Factor Incidence Geometry The construction is a multigraded version of a standard algebraic idea: a collection of forms sharing a common factor is described by an incidence condition involving multiplication maps. Round 9 adapts this principle from tuples of forms to an entire Hodge-signature subspace. The object of interest is not merely: $$ F\mid p $$ for one polynomial $p$. It is: $$ \boxed{ F\cdot H_{\mathrm{complement}} \subseteq A. } $$ This stronger condition is what makes the factor a candidate carrier divisor. --- # 23. Why Saturation Alone Is Not Enough The full annihilator ideal: $$ I_\alpha $$ is Artinian modulo the hypersurface/Jacobian structure and is generally much larger than a carrier ideal. Therefore simply saturating: $$ I_\alpha $$ does not isolate the carrier. The safe-envelope strategy first identifies a geometric ambient container at a degree where purity is known, then uses its internal factorization structure at the next degree. Hence: $$ \boxed{ \text{safe envelope} + \text{multigraded multiplication} } $$ is more informative than blind saturation. --- # 24. Hodge-Annihilator as Artinian Gorenstein Data The literature shows that: $$ A_\alpha=R/I_\alpha $$ is Artinian Gorenstein with a fixed socle degree in the surface case. Therefore $I_\alpha$ is best viewed as an apolar/annihilator object encoding the Hodge functional, not as a direct carrier ideal. Round 9's splitter respects this distinction. It uses: $$ I_\alpha $$ as a **signature source** and extracts geometric multiplication patterns from it, rather than identifying: $$ I_\alpha=I_C. $$ --- # 25. New Principle: Multiplication-Coherence Test A single annihilator equation may be non-geometric. A candidate carrier factor must satisfy a much stronger coherence condition: $$ \boxed{ F\cdot H_{\mathrm{complement}} \subseteq A. } $$ Call this the: $$ \boxed{ \textbf{Multiplication-Coherence Test}. } $$ This transforms carrier detection from equation-level matching to subspace-level algebraic coherence. --- # 26. New Principle: Envelope-Relative Geometry Carrier inference should be performed relative to the strongest safely reconstructed envelope. $$ \boxed{ \text{ambient projective space} \to \text{safe envelope} \to \text{carrier}. } $$ In the rational quartic: $$ \mathbb P^3 \to Q \to C. $$ The intermediate envelope provides the multigrading needed to distinguish carrier types. --- # 27. New Principle: Impurity Cannot Remove Genuine Factors If: $$ A_{\mathrm{geom}}\subseteq A_\alpha, $$ then: $$ \boxed{ \mathfrak P_\tau(A_{\mathrm{geom}}) \subseteq \mathfrak P_\tau(A_\alpha). } $$ Thus added non-geometric signature directions can create aliases, but cannot erase a genuine multiplication-coherent carrier factor. This gives the splitter one-sided robustness. --- # 28. New Principle: Generic Alias Resistance For a fixed geometric signature sector, extra impurity directions form a family of extensions. If one extension realizes the minimum possible kernel dimension, then upper semicontinuity implies a nonempty open family of impurity extensions with the same carrier-identification dimension. Therefore carrier recovery can be: $$ \boxed{ \text{generically robust to bounded non-geometric contamination}. } $$ The explicit rational-quartic experiment proves this generic statement for the two tested carrier types. --- # 29. AMRAL Representation Suggested factor-incidence object: ```text factor_incidence: envelope_ref signature_space_ref target_multidegree complement_multidegree operator_matrix_ref kernel_dimension factor_candidates[] generic_robustness_status ``` Suggested carrier-factor node: ```text carrier_factor_candidate: factor_polynomial multidegree multiplication_image_dimension multiplication_image_ref realizability_status class_verification_status alias_status ``` Suggested purity split: ```text purity_split: signature_dimension jacobian_dimension factorable_geometric_dimension recovered_geometric_dimension residual_dimension residual_status ``` --- # 30. Reproducible Experiment The accompanying Python verifier performs exact rational symbolic linear algebra. It verifies: 1. Round 8 rational quartic and residual factors; 2. $\dim A_{\mathrm{geom}}=10$; 3. $\dim U_C=3$; 4. $\dim U_D=3$; 5. $\dim J_Q=4$; 6. $\dim(U_C+U_D+J_Q)=10$; 7. $\ker\Theta_{3,1}$ is one-dimensional and generated by $F_C$; 8. $\ker\Theta_{1,3}$ is one-dimensional and generated by $F_D$; 9. tested alternative bidegrees have zero kernel; 10. an explicit twelve-dimensional extension preserves the same unique factor kernels. All computations are exact over $\mathbb Q$. --- # 31. What Has Not Yet Been Computed Round 9 does **not** possess an explicit basis for the true: $$ I_{\alpha(C),3} $$ of the Round 8 quartic. Therefore it does not yet verify: $$ \dim\ker\Theta^{A_\alpha}_{3,1}=1 $$ for the actual Hodge signature. What is proved is: 1. the true factors must remain candidates; 2. the factorization splitter exactly recovers the full known geometric sector; 3. generic two-dimensional contamination preserves uniqueness; 4. the actual Hodge contamination can be tested once the period/annihilator basis is computed. This evidence typing is essential. --- # 32. Immediate Next Experiment The obvious next computational target is now precise. For the explicit Round 8 quartic: ### Step 1 Compute periods or otherwise compute a basis of: $$ I_{\alpha(C),3}. $$ ### Step 2 Restrict to: $$ Q. $$ ### Step 3 Obtain the actual: $$ A_\alpha\subset W_{3,3}. $$ ### Step 4 Run: $$ \Theta^{A_\alpha}_{3,1}, \qquad \Theta^{A_\alpha}_{1,3}. $$ ### Step 5 Check whether: $$ \ker\Theta_{3,1} = \langle F_C\rangle, $$ and: $$ \ker\Theta_{1,3} = \langle F_D\rangle. $$ ### Step 6 If yes, compute: $$ \widehat N_{\alpha,3} = A_\alpha/ (U_C+U_D+J_Q). $$ The prediction is: $$ \boxed{ \dim\widehat N_{\alpha,3}=2. } $$ This would be the first direct intrinsic split of a known non-perfect Hodge annihilator. --- # 33. Round 9 Conclusion Round 9 produces the first operational candidate for the **purity splitter** requested at the end of Round 8. The method is based on: $$ \boxed{ \Theta^A_{u,v}(F)(G)=FG\bmod A. } $$ A carrier factor is detected not because one equation resembles a known carrier equation, but because its entire multiplication image lies inside the signature subspace. For the explicit rational-quartic geometry: $$ \boxed{ \ker\Theta_{3,1} = \langle F_C\rangle, } $$ and: $$ \boxed{ \ker\Theta_{1,3} = \langle F_D\rangle. } $$ The associated multiplication images contribute: $$ 3+3 $$ dimensions. Together with the four-dimensional Jacobian sector: $$ \boxed{ 3+3+4=10, } $$ which exactly reconstructs the known geometric cubic signature space. The actual Hodge signature restricted to the quadric has dimension: $$ 12. $$ Thus the expected unresolved sector remains: $$ \boxed{ 12-10=2. } $$ The new methodological architecture is: $$ \boxed{ \text{safe envelope} \to \text{factorization incidence} \to \text{multiplication-coherent carrier sectors} \to \text{Jacobian sector} \to \text{residual impurity}. } $$ This is the main closure of Round 9. --- # References 1. Maria Gioia Cifani, Gian Pietro Pirola, Enrico Schlesinger, **Reconstructing curves from their Hodge classes**, Rendiconti del Circolo Matematico di Palermo Series 2, 72 (2023), 945--958. arXiv: https://arxiv.org/abs/2104.05576 DOI: https://doi.org/10.1007/s12215-021-00704-9 2. Hossein Movasati, Emre Can Sertöz, **On reconstructing subvarieties from their periods**, Rendiconti del Circolo Matematico di Palermo Series 2, 70 (2021), 1441--1457. arXiv: https://arxiv.org/abs/1908.03221 DOI: https://doi.org/10.1007/s12215-020-00562-x 3. Standard incidence principle for common factors: common-factor loci can be encoded using multiplication maps and projective incidence varieties. Round 9 uses a multigraded subspace version of this principle.