# RCHM × Hodge Conjecture — Research Round 10 ## Impurity Alias Discriminants and a Codimension-Three Robustness Bound **Status:** Internal research note + reproducible exact algebra experiment **Version:** Round 10 / v1.0 **Date:** 2026-09-17 **Methodological base:** RCHM; Typed Relation & Constraint Algebra; X-Layer Mathematical Completeness; Graph-Native Mathematics **Previous rounds:** RCHM × Hodge Conjecture — Rounds 1–9 **Canonical source:** UTF-8 Markdown --- # 0. Round 10 Objective Round 9 introduced the envelope-relative factorization operator $$ \Theta^A_\tau $$ and showed that, for the explicit rational-quartic geometry, $$ \ker\Theta^{A_{\mathrm{geom}}}_{3,1} = \langle F_C\rangle, $$ and $$ \ker\Theta^{A_{\mathrm{geom}}}_{1,3} = \langle F_D\rangle. $$ The actual Hodge-annihilator signature on the quadric envelope has dimension $$ 12, $$ while the geometrically explained signature has dimension $$ 10. $$ Thus the unknown non-geometric contamination is a two-dimensional extension. Round 10 asks: > Without knowing the actual two-dimensional impurity basis, how exceptional must that impurity be in order to create a false carrier factor? The answer is controlled by residual multiplication rank and Grassmannian incidence. For the two genuine carrier types $(3,1)$ and $(1,3)$, the bad impurity set has codimension at least $$ \boxed{3} $$ inside the impurity Grassmannian $$ \operatorname{Gr}(2,6). $$ Thus generic two-dimensional impurity preserves unique carrier recovery. --- # 1. Ambient Signature Spaces Let $$ W = H^0(Q,\mathcal O_Q(3,3)). $$ For the quadric envelope: $$ \dim W=16. $$ Let $$ G = A_{\mathrm{geom}} \subset W $$ be the known geometric signature sector. Round 9 established: $$ \dim G=10. $$ Define the quotient: $$ \boxed{ U=W/G. } $$ Then: $$ \boxed{ \dim U=6. } $$ The actual Hodge signature is an extension: $$ A_\alpha = G+\widetilde N, $$ where its image in $U$ is a two-plane: $$ \boxed{ N\in\operatorname{Gr}(2,U) \simeq \operatorname{Gr}(2,6). } $$ The impurity parameter space therefore has dimension: $$ \boxed{ \dim\operatorname{Gr}(2,6)=8. } $$ --- # 2. Residual Multiplication Image Fix a carrier type $$ \tau=(u,v). $$ Let: $$ F\in H^0(Q,\mathcal O_Q(u,v)). $$ Let the complementary space be: $$ H_{\bar\tau} = H^0(Q,\mathcal O_Q(3-u,3-v)). $$ Multiplication gives: $$ \mu_F: H_{\bar\tau} \to W. $$ Project to the quotient: $$ \pi_G: W\to U=W/G. $$ Define the **residual multiplication map**: $$ \boxed{ \overline\mu_F = \pi_G\circ\mu_F: H_{\bar\tau} \to U. } $$ Define its image: $$ \boxed{ R_G(F) = \operatorname{Im}\overline\mu_F \subseteq U. } $$ and residual rank: $$ \boxed{ r_G(F) = \dim R_G(F). } $$ --- # 3. Exact Alias Criterion Let: $$ N\subseteq U $$ be a $p$-dimensional impurity space. The enlarged signature is: $$ A_N = G+\pi_G^{-1}(N). $$ A factor $F$ becomes a carrier candidate exactly when: $$ F\cdot H_{\bar\tau} \subseteq A_N. $$ After quotienting by $G$, this is equivalent to: $$ \boxed{ R_G(F)\subseteq N. } $$ Therefore: $$ \boxed{ F\text{ is an impurity-induced alias} \iff R_G(F)\subseteq N. } $$ This is the central exact criterion of Round 10. --- # 4. Residual-Rank Barrier If: $$ \dim N=p, $$ then: $$ R_G(F)\subseteq N $$ is impossible whenever: $$ r_G(F)>p. $$ Hence: $$ \boxed{ r_G(F)>p \Rightarrow F\text{ cannot be created as an alias by }p\text{-dimensional impurity}. } $$ For the rational-quartic problem: $$ p=2. $$ Therefore every false factor with: $$ r_G(F)\ge3 $$ is automatically safe against the unknown two-dimensional impurity. --- # 5. Fixed-Factor Schubert Cost Let: $$ \dim U=d. $$ Fix a factor $F$ with: $$ r_G(F)=r\le p. $$ The set of impurity $p$-planes containing the fixed residual $r$-plane is: $$ \boxed{ \Sigma_{R_G(F)} = \left\{ N\in\operatorname{Gr}(p,U): R_G(F)\subseteq N \right\}. } $$ It is isomorphic to: $$ \operatorname{Gr}(p-r,d-r). $$ Hence: $$ \dim\Sigma_{R_G(F)} = (p-r)(d-p). $$ Since: $$ \dim\operatorname{Gr}(p,d) = p(d-p), $$ the codimension is: $$ \boxed{ \operatorname{codim} \Sigma_{R_G(F)} = r(d-p). } $$ Call this the **Schubert Alias Cost**. --- # 6. Moving-Factor Alias Incidence Let: $$ Z_r^\tau = \left\{ [F]: r_G(F)=r \right\} $$ be the projective rank-$r$ factor stratum. Define the incidence: $$ \boxed{ \mathfrak B_r^\tau = \left\{ ([F],N): [F]\in Z_r^\tau, \; R_G(F)\subseteq N \right\}. } $$ Then: $$ \dim\mathfrak B_r^\tau \le \dim Z_r^\tau + (p-r)(d-p). $$ Its image in: $$ \operatorname{Gr}(p,d) $$ is the impurity set capable of generating at least one rank-$r$ false alias. Therefore: $$ \boxed{ \operatorname{codim} B_r^\tau \ge r(d-p) - \dim Z_r^\tau. } $$ This is the **Family-Discounted Alias Bound**. --- # 7. Exact Residual Matrix for Type $(3,1)$ For: $$ \tau=(3,1), $$ the factor space has dimension: $$ (3+1)(1+1)=8. $$ Thus factors are parametrized projectively by: $$ \mathbb P^7. $$ The complementary multiplication space is: $$ H^0(Q,\mathcal O_Q(0,2)), $$ of dimension: $$ 3. $$ Therefore: $$ \overline\mu_F $$ is represented by a: $$ 6\times3 $$ matrix linear in eight factor coefficients. The accompanying exact SymPy computation constructs this matrix directly from the Round 8--9 explicit geometry. --- # 8. Rank Stratification for Type $(3,1)$ Let: $$ Z_{\le r}^{3,1} = \left\{ [F]\in\mathbb P^7: r_G(F)\le r \right\}. $$ The exact determinantal ideals give: $$ \boxed{ \dim Z_{\le2}^{3,1}=3, } $$ $$ \boxed{ \dim Z_{\le1}^{3,1}=1, } $$ and: $$ \boxed{ \dim Z_0^{3,1}=0. } $$ The rank-zero locus is precisely the genuine carrier factor: $$ [F_C]. $$ Thus: - generic factor rank is $3$; - rank-$2$ false factors form a projective $3$-dimensional locus; - rank-$1$ false factors form a projective $1$-dimensional locus; - rank-$0$ consists of the genuine factor. --- # 9. How the Dimensions Were Certified The rank-$\le2$ locus is defined by all: $$ 3\times3 $$ minors of the residual: $$ 6\times3 $$ matrix. A Gröbner basis over: $$ \mathbb Q $$ has initial monomial ideal with radical height: $$ 4. $$ Therefore its affine cone has dimension: $$ 8-4=4, $$ and projective dimension: $$ 3. $$ Similarly, the rank-$\le1$ locus is defined by: $$ 2\times2 $$ minors. Its initial monomial ideal has radical height: $$ 6. $$ Thus its affine cone dimension is: $$ 8-6=2, $$ and projective dimension: $$ 1. $$ All calculations are exact. --- # 10. Alias Codimension for Type $(3,1)$ Now use: $$ d=6, \qquad p=2. $$ ## Rank-one aliases For: $$ r=1, $$ the fixed-factor Schubert cost is: $$ r(d-p) = 1\cdot4 = 4. $$ The false rank-one factor locus has dimension: $$ 1. $$ Therefore: $$ \boxed{ \operatorname{codim} B_1^{3,1} \ge 4-1 = 3. } $$ --- ## Rank-two aliases For: $$ r=2, $$ the fixed-factor Schubert cost is: $$ 2\cdot4=8. $$ The rank-two factor locus has dimension: $$ 3. $$ Therefore: $$ \boxed{ \operatorname{codim} B_2^{3,1} \ge 8-3 = 5. } $$ --- ## Rank-three factors They cannot be absorbed by a two-dimensional impurity: $$ \boxed{ B_3^{3,1}=\varnothing. } $$ Hence the total same-type bad impurity locus satisfies: $$ \boxed{ \operatorname{codim} B^{3,1} \ge3. } $$ --- # 11. Exact Rank Stratification for Type $(1,3)$ The residual operator for: $$ \tau=(1,3) $$ is again a: $$ 6\times3 $$ linear matrix on a factor space: $$ \mathbb P^7. $$ The exact Gröbner computation independently gives: $$ \boxed{ \dim Z_{\le2}^{1,3}=3, } $$ $$ \boxed{ \dim Z_{\le1}^{1,3}=1, } $$ $$ \boxed{ \dim Z_0^{1,3}=0. } $$ The rank-zero point is the genuine residual carrier factor: $$ [F_D]. $$ Therefore the same incidence calculation gives: $$ \boxed{ \operatorname{codim} B^{1,3} \ge3. } $$ --- # 12. Simultaneous Robustness Let: $$ B^{\mathrm{true}} = B^{3,1}\cup B^{1,3}. $$ A finite union of closed/proper incidence images of codimension at least $3$ remains a proper subset. Thus: $$ \boxed{ \operatorname{codim} B^{\mathrm{true}} \ge3. } $$ Consequently there exists a nonempty Zariski-open subset: $$ \boxed{ \mathcal U_{\mathrm{safe}} \subset \operatorname{Gr}(2,6) } $$ such that every: $$ N\in\mathcal U_{\mathrm{safe}} $$ preserves uniqueness of both genuine factor types. This upgrades Round 9's one explicit safe extension to a quantified generic robustness theorem for the explicit geometry. --- # 13. Interpretation The unknown two-dimensional Hodge impurity is not automatically dangerous. To manufacture a false $(3,1)$ or $(1,3)$ carrier factor, it must align with the residual multiplication image of a nongenuine low-rank factor. Most false factors have: $$ r_G(F)=3 $$ and are completely immune to a two-dimensional impurity. The exceptional false factors with rank $1$ or $2$ occupy determinantal subvarieties. The impurity must then additionally lie in the appropriate Schubert incidence condition. Thus aliasing requires **two simultaneous degeneracies**: $$ \boxed{ \text{low residual rank} + \text{impurity alignment}. } $$ --- # 14. Double-Degeneracy Principle Round 10 therefore proposes: $$ \boxed{ \textbf{Double-Degeneracy Principle} } $$ A non-geometric signature sector can create a false carrier only if: 1. the candidate factor already has unusually low residual multiplication rank relative to the geometric signature; 2. the impurity subspace aligns with that residual image. This explains why generic contamination may leave the true carrier inference unchanged. --- # 15. Alias Discriminant Define the full type-$\tau$ alias discriminant: $$ \boxed{ \mathcal D_{\mathrm{alias}}^\tau = \left\{ N\in\operatorname{Gr}(p,U): \exists [F]\notin\mathfrak P_\tau(G), \; R_G(F)\subseteq N \right\}. } $$ It is the bad parameter locus for the impurity. For the explicit rational-quartic geometry and true types: $$ \boxed{ \operatorname{codim} \mathcal D_{\mathrm{alias}}^{3,1} \ge3, } $$ and: $$ \boxed{ \operatorname{codim} \mathcal D_{\mathrm{alias}}^{1,3} \ge3. } $$ --- # 16. Alias Risk Index For a rank-$r$ factor family of projective dimension: $$ z_r, $$ define the lower-bound risk codimension: $$ \boxed{ \rho_r = r(d-p)-z_r. } $$ Interpretation: - large positive $\rho_r$: alias generation is geometrically expensive; - $\rho_r=1$: alias lies on a divisor-like bad locus; - $\rho_r\le0$: dimension counting alone gives no generic safety guarantee. For the explicit true carrier types: ### Rank one $$ \rho_1 = 4-1 = 3. $$ ### Rank two $$ \rho_2 = 8-3 = 5. $$ Therefore the controlling risk is: $$ \boxed{ \rho_{\min}=3. } $$ --- # 17. Robustness Margin Against Impurity Dimension The same framework varies with impurity dimension $p$. For fixed quotient dimension: $$ d=6, $$ the Schubert alias cost is: $$ r(6-p). $$ Thus increasing impurity dimension makes aliases easier. For example: ### $p=1$ Only rank-$1$ factors can alias. A fixed rank-$1$ factor requires the unique impurity line to equal its residual line. ### $p=2$ Rank $1$ and rank $2$ factors can alias. This is the current Hodge case. ### $p=3$ Rank-$3$ generic factors become potentially absorbable. At that point the residual-rank barrier no longer excludes generic factors of the $(3,1)$ and $(1,3)$ families. Thus: $$ \boxed{ p=3 } $$ is a qualitative robustness threshold for these carrier types. --- # 18. Carrier-Type Security Radius Let: $$ r_{\mathrm{gen}}^\tau $$ be the generic residual rank for type $\tau$. Define the **impurity security radius**: $$ \boxed{ p_{\mathrm{sec}}^\tau = r_{\mathrm{gen}}^\tau-1. } $$ If: $$ p\le p_{\mathrm{sec}}^\tau, $$ generic factors cannot become aliases solely by arbitrary impurity. For: $$ (3,1) $$ and: $$ (1,3), $$ we have: $$ r_{\mathrm{gen}}=3, $$ so: $$ \boxed{ p_{\mathrm{sec}}=2. } $$ The actual Hodge impurity has exactly dimension: $$ 2. $$ Thus the rational-quartic problem lies at the maximal impurity dimension for which generic factors are still protected by the residual-rank barrier. --- # 19. Relation to the Actual Period Computation The true impurity plane: $$ N_\alpha \in \operatorname{Gr}(2,6) $$ is determined by the actual Hodge periods. The numerical-period algorithm of Movasati--Sertöz computes the annihilator spaces by: 1. computing the period isomorphism; 2. reducing polynomial products in the Jacobian ring; 3. forming the Hodge pairing matrix; 4. taking a left kernel. Therefore once the actual: $$ I_{\alpha(C),3} $$ basis is available, its impurity point: $$ N_\alpha $$ can be located in the Grassmannian and checked against: $$ \mathcal D_{\mathrm{alias}}. $$ The present round proves that a generic impurity point is safe before that expensive computation is carried out. --- # 20. Why This Matters for AMRAL AMRAL should not store only: ```text factor_candidate = true ``` It should store: ```text factor_candidate: carrier_type residual_rank impurity_dimension schubert_alias_cost factor_stratum_dimension alias_codimension_lower_bound robustness_status ``` For the current factors: ```text (3,1): generic_residual_rank: 3 impurity_dimension: 2 rank1_alias_codim_ge: 3 rank2_alias_codim_ge: 5 ``` and identically for: ```text (1,3). ``` This records **why** a candidate is robust. --- # 21. Evidence Type This round uses only: - exact symbolic construction of the Round 8--9 geometry; - exact rational linear algebra; - exact determinantal minors; - exact Gröbner bases; - standard Grassmannian dimension formulas. It does **not** assume the unknown actual impurity basis. Therefore the codimension-three bound is independent of the unresolved numerical period computation. --- # 22. Reproducible Verification The accompanying Python script reconstructs: 1. the explicit rational quartic; 2. the residual carrier; 3. the geometric signature $G$; 4. the six-dimensional quotient $U$; 5. the symbolic residual matrices for $(3,1)$ and $(1,3)$; 6. the rank-$\le2$ determinantal ideals; 7. the rank-$\le1$ determinantal ideals; 8. their Gröbner bases; 9. projective dimensions from the initial monomial ideals; 10. the alias codimension bounds. Expected output: ```text type (3,1): factor projective dimension = 7 generic residual rank = 3 dim Z_rank<=2 = 3 dim Z_rank<=1 = 1 dim Z_rank=0 = 0 rank1 bad impurity codim >= 3 rank2 bad impurity codim >= 5 total bad impurity codim >= 3 type (1,3): same dimensions and bounds ``` --- # 23. Limits of the Result This round establishes robustness against **same-type extra aliases** for the two genuine carrier types. It does not yet prove that a special impurity cannot generate a false carrier of some completely different bidegree or geometric family. Round 9 tested several alternative bidegrees on: $$ G $$ and on one explicit $12$-dimensional extension, but a full cross-type alias discriminant requires separate determinantal analysis for each carrier family. Thus: $$ \boxed{ \text{same-type robustness} \neq \text{universal carrier-type robustness}. } $$ The next natural extension is a cross-type alias atlas. --- # 24. New Principles ## Principle A — Residual-Rank Barrier $$ \boxed{ r_G(F)>p \Rightarrow p\text{-dimensional impurity cannot create }F. } $$ --- ## Principle B — Schubert Alias Cost For a fixed residual rank-$r$ factor: $$ \boxed{ \operatorname{codim} = r(d-p) } $$ inside the impurity Grassmannian. --- ## Principle C — Family-Discounted Alias Bound For a moving rank-$r$ factor family of dimension $z_r$: $$ \boxed{ \operatorname{codim}B_r \ge r(d-p)-z_r. } $$ --- ## Principle D — Double Degeneracy False carrier generation requires both: $$ \boxed{ \text{factor rank degeneration} + \text{impurity alignment}. } $$ --- ## Principle E — Security Radius If the generic residual rank is: $$ r_{\mathrm{gen}}, $$ then impurity dimension: $$ p