Adiv:=spanQ(cl1(CH1(Y))cl1(CH1(Y)))A_{\mathrm{div}}\;:=\;\operatorname{span}_{\mathbb Q}\big(\operatorname{cl}_1(\mathrm{CH}^1(Y))\otimes\operatorname{cl}_1(\mathrm{CH}^1(Y))\big)Adiv:=spanQ(cl1(CH1(Y))cl1(CH1(Y)))
has dimension at most 400400400 when (Y)=20\rho(Y)=20(Y)=20. The difference
:=TX/Adiv\Delta \;:=\; T_X \;/\; A_{\mathrm{div}}:=TX/Adiv
is a four-dimensional rational vector space isomorphic to T(Y)QT(Y)T(Y)\otimes_{\mathbb Q} T(Y)T(Y)QT(Y) (where T(Y)T(Y)T(Y) is the transcendental lattice of YYY). In CAS-6 language this is precisely a failure of closure between the topological layer (L,C)(L,C)(L,C) and the algebraic layer (W,P)(W,P)(W,P): four topological "nodes" (cohomological configurations) remain without algebraic weights and so produce no geometric outputs.
Below we analyze this incomplete closure more formally, describe how it manifests within the CAS-6 components, and explain the mathematical and heuristic consequences.
1. Precise formulation of incomplete closure
Adopt the shorthand mapping from CAS-6 to Hodge theory:
(L,C) H2p(X,Q)Hp,p(X),(W,P) clp(CHp(X)Q).(L,C)\ \longleftrightarrow\ H^{2p}(X,\mathbb Q)\cap H^{p,p}(X),\qquad (W,P)\ \longleftrightarrow\ \operatorname{cl}_p(\mathrm{CH}^p(X)\otimes\mathbb Q).(L,C) H2p(X,Q)Hp,p(X),(W,P) clp(CHp(X)Q).
For X=YYX=Y\times YX=YY and p=2p=2p=2 we write
TX=(L,C)XandA:=(W,P)X.T_X = (L,C)_X \quad\text{and}\quad A := (W,P)_X.TX=(L,C)XandA:=(W,P)X.
Closure of the system at this level is the statement that cl2\operatorname{cl}_2cl2 is surjective onto TXT_XTX:
