Explicit Infinity-Gysin Projection and the Rank-Seven Algebraic Kernel

Record

Date: 2026-08-15

Status: conditional exact theorem on the generic fiberwise de Rham locus. The infinity-Gysin map, its rank, and its kernel are explicit. Extension through the discriminant, integral lattice normalization, physical-chain compatibility, and the connection on the algebraic kernel remain open.

This entry closes the generic elliptic embedding problem left open in entries 148 and 149. It does not identify the unpublished rank-one factor (L_1).

Source surface and elliptic boundary

Use the (q_{\mathcal G_{12}})-residue surface and compactification

[ S_E:\ w^2=K_0(a,b),\qquad \overline S_E:\ W^2=\overline K_E(a,b,s), \qquad D_\infty={s=0}. ]

Write

[ x=X_1,\quad y=X_2,\quad z=X_3,\quad E=x+y+z,\quad h=x^2+y^2-z^2. ]

On the chart (b\ne0), put

[ b=s^{-1},\qquad a=t/s,\qquad w=W/s^2. ]

Then

[ \overline K_E(t,1,s)=F(t)+s^2G(t)+s^4H, \qquad F(t)=x^2t^4-ht^2+y^2, ]

with

[ \begin{aligned} G(t)={}& \left[-2x^2(y^2+E^2)+h(x^2+E^2)\right]t^2\ &+h(y^2+E^2)-2y^2(x^2+E^2). \end{aligned} ]

Thus

[ D_\infty:\ W^2=F(t),\qquad \omega_0=\frac{dt}{W},\qquad \omega_2=\frac{t^2dt}{W}. ]

Explicit infinity-Gysin map

The final four source masters are

[ e_6=\phi_{002},\qquad e_7=\phi_{001},\qquad e_8=y_{23}^2\phi_{001},\qquad e_9=y_{31}^2\phi_{001}. ]

In the established (q)-residue normalization, the simple-pole classes are

[ \Omega_7=\frac{da\wedge db}{w},\qquad \Omega_8=\frac{a^2da\wedge db}{w},\qquad \Omega_9=\frac{b^2da\wedge db}{w}. ]

Since

[ da\wedge db=\frac{ds\wedge dt}{s^3},\qquad w=\frac W{s^2}, ]

one obtains

[ \boxed{R_\infty(e_7)=\omega_0.} ]

For (P(a,b)=s^{-2}p(t)), removing the cubic normal pole by an exact form leaves logarithmic coefficient

[ -\frac{p(t)G(t)}{2F(t)^{3/2}}dt. ]

Exact reduction on (D_\infty) gives

[ -\frac{t^2G}{2F^{3/2}}dt \equiv \left[\frac{E^2+y^2}{2}-\frac{E^2+x^2}{2}t^2\right] \frac{dt}{\sqrt F}, ]

[ -\frac{G}{2F^{3/2}}dt \equiv \left[\frac{E^2+x^2}{2} -\frac{x^2(E^2+y^2)}{2y^2}t^2\right]\frac{dt}{\sqrt F}. ]

Consequently,

[ \boxed{ R_\infty(e_8) =\frac{E^2+y^2}{2}\omega_0-\frac{E^2+x^2}{2}\omega_2, } ]

[ \boxed{ R_\infty(e_9) =\frac{E^2+x^2}{2}\omega_0 -\frac{x^2(E^2+y^2)}{2y^2}\omega_2. } ]

In the ordered bases ((e_7,e_8,e_9)) and ((\omega_0,\omega_2)),

[ \boxed{ R_\infty= \begin{pmatrix} 1&\dfrac{E^2+y^2}{2}&\dfrac{E^2+x^2}{2}\[3mm] 0&-\dfrac{E^2+x^2}{2}&-\dfrac{x^2(E^2+y^2)}{2y^2} \end{pmatrix}. } ]

It has rank two at generic nonsoft kinematics.

Double-pole master and algebraic kernel

The double-pole master has residue representative

[ \Omega_6\propto K_1(a,b)\frac{da\wedge db}{K_0^{3/2}}, \qquad K_1=\left.\partial_qK\right|_{q=0}, ]

where (K_1) has degree two. At infinity,

[ K_1=O(s^{-2}),\quad da\wedge db=O(s^{-3}),\quad K_0^{3/2}=O(s^{-6}), ]

so (\Omega_6=O(s)ds\wedge dt) has no logarithmic pole. Hence

[ \boxed{R_\infty(e_6)=0.} ]

The last-three-master kernel is generated by

[ \boxed{ \begin{aligned} v_{\rm alg}={}& (x^2-y^2)(x^2y^2-E^4)e_7\ &+2x^2(E^2+y^2)e_8-2y^2(E^2+x^2)e_9. \end{aligned} } ]

Direct substitution gives (R_\infty(v_{\rm alg})=0). Therefore

[ \boxed{ \ker\left(R_\infty|{\langle e_6,e_7,e_8,e_9\rangle}\right) =\langle e_6,v{\rm alg}\rangle. } ]

The generic quotient is

[ \boxed{ \frac{\langle e_6,e_7,e_8,e_9\rangle} {\langle e_6,v_{\rm alg}\rangle} \simeq H^1(D_\infty)(-1). } ]

This is a de Rham quotient. Under the solution/local-system realization the variance reverses, so it is compatible with the elliptic sub-local-system language associated with (L_3=L_1L_2).

Global rank-seven kernel

The map is equivariant under (C_2^{(a)}\times C_2^{(b)}). The elliptic boundary cohomology occupies the character of the final four-dimensional block. The other source blocks, of ranks (1,2,2), map to zero. Together with the two-dimensional kernel in the final block,

[ 1+2+2+2=7. ]

Thus, on the generic de Rham locus,

[ \boxed{ 0\longrightarrow\mathcal T_7 \longrightarrow\mathcal M_q^{(9)} \xrightarrow{R_\infty}\mathbb V_{\rm ell}(-1) \longrightarrow0. } ]

The kernel character multiplicities are

[ \boxed{(1,2,2,2).} ]

The rank and geometry are those expected from the primitive algebraic (E_7) part of an anticanonical complement of a degree-two del Pezzo surface. This is not yet an integral (E_7)-lattice theorem.

Consequence for (L_3=L_1L_2)

On the last-three-master space,

[ 0\longrightarrow\langle v_{\rm alg}\rangle \longrightarrow\langle e_7,e_8,e_9\rangle \xrightarrow{R_\infty}H^1(D_\infty)(-1) \longrightarrow0. ]

The quotient is the binary-quartic system with published operator (L_2). The source-defined candidate for (L_1) is therefore the rank-one flat subquotient selected by the last-three cyclic module inside the flat saturation of

[ \langle e_6,v_{\rm alg}\rangle. ]

The displayed (v_{\rm alg}) need not itself be horizontal; its derivative may mix with (e_6). Only the induced rank-one connection is invariant.

Status of (\mathcal Q)

The Gysin quotient depends on

[ F(t)=x^2t^4-ht^2+y^2 ]

and its elliptic discriminant (x^2y^2AB). The source quartic

[ \mathcal Q=4AB-(A+B-E^2)^2 ]

does not enter (R_\infty). It must belong to one of

[ \nabla_{\mathcal T_7},\qquad \nabla_{L_1},\qquad \operatorname{Ext}^1(\mathbb V_{\rm ell}(-1),\mathcal T_7). ]

It is not part of the pure elliptic quotient.

Boundary

This entry does not establish:

extension through the discriminant locus;
integral normalization or a canonical (E_7) lattice;
compatibility with the physical relative integration chain;
compatibility with the integrand-level graphical Cut/coaction;
the connection on (\langle e_6,v_{\rm alg}\rangle);
identification of its rank-one subquotient with (L_1);
whether (\mathcal Q) controls (L_1) or the larger extension class.

No new carrier divisor, fitted projector, or post hoc splitting is used.

Next falsifier

Compute the Gauss–Manin connection on

[ \mathcal A_{–}=\langle e_6,v_{\rm alg}\rangle. ]

Identify the invariant rank-one line or quotient selected by the last-three cyclic module, compute its nonintegral support, and test

[ L_1\overset?\sim_{\rm rat}\partial-\frac12d\log(-\mathcal Q). ]

If this fails, determine whether (\mathcal Q) instead controls the extension between (\mathcal T_7) and the elliptic quotient. Failure of the (L_1/\mathcal Q) identification does not falsify the Gysin quotient.

Outcome contract

{
  "claim": "On the generic fiberwise de Rham locus, the source nine-master q-sector has an explicit infinity-Gysin quotient onto the rank-two binary-quartic elliptic system. The kernel has rank seven; inside the final four-dimensional block it is generated by e6 and v_alg.",
  "status": "conditional",
  "assumptions": [
    "The established q-residue normalization and source master representatives are retained.",
    "The compactification and exact boundary reductions are valid on the generic smooth locus.",
    "The character action is the source-defined C2 x C2 action."
  ],
  "factorization_test": {
    "R_infinity_matrix": "explicit",
    "generic_rank": 2,
    "R_infinity_e6": 0,
    "final_block_kernel": "span(e6,v_alg)",
    "global_kernel_rank": 7,
    "kernel_characters": [1, 2, 2, 2],
    "elliptic_quotient": "H1(D_infinity)(-1), with PF operator L2",
    "L1": "uncomputed algebraic flat subquotient",
    "Q_provenance": "confined to the algebraic kernel or extension sector"
  },
  "counterevidence": [
    "No extension through the discriminant or integral lattice theorem is proved.",
    "Physical relative-chain and Cut/coaction compatibilities are untested.",
    "The displayed algebraic kernel vector is not asserted to be horizontal."
  ],
  "next_experiment": "Compute the connection on span(e6,v_alg), extract the invariant rank-one factor, and test its gauge class against one-half dlog(-Q)."
}