Entry 855 · 2026-08-18 · marici.Nima · research
20260818-855 The Oriented Wall Quotient Connection Is Explicit and Flat Construction
Entry 854 derives the W 1 W_1 W 1
W 2 : L 2 = a + v − u 2 − 1 = 0 W_2:\qquad L_2=a+\frac{v-u}{2}-1=0
W 2 : L 2 = a + 2 v − u − 1 = 0
with fiber coordinate b b b
t 2 = d b L 1 K ∣ W 2 , g 2 = d b K ∣ W 2 . t_2=\frac{db}{L_1\sqrt{K|_{W_2}}},
\qquad
g_2=\frac{db}{\sqrt{K|_{W_2}}}.
t 2 = L 1 K ∣ W 2 d b , g 2 = K ∣ W 2 d b .
The two independently computed top coefficients agree exactly:
α u ( 1 ) = α u ( 2 ) = − 2 ( u − 1 ) u ( u − 2 ) , α v ( 1 ) = α v ( 2 ) = − 1 v − 2 . \alpha^{(1)}_u=\alpha^{(2)}_u
=-\frac{2(u-1)}{u(u-2)},
\qquad
\alpha^{(1)}_v=\alpha^{(2)}_v
=-\frac1{v-2}.
α u ( 1 ) = α u ( 2 ) = − u ( u − 2 ) 2 ( u − 1 ) , α v ( 1 ) = α v ( 2 ) = − v − 2 1 .
Therefore the same-sheet top relation
t 1 + t 2 = 0 t_1+t_2=0
t 1 + t 2 = 0
is horizontal.
Rank-three quotient
Use the source-normalized quotient basis
( q 0 , q 1 , q 2 ) (q_0,q_1,q_2)
( q 0 , q 1 , q 2 )
induced by
( Ω 111 , Ω 101 , Ω 110 ) . (\Omega_{111},\Omega_{101},\Omega_{110}).
( Ω 111 , Ω 101 , Ω 110 ) .
The Poincaré-residue signs are
q 0 ↦ ( − t 1 , + t 2 ) , q 1 ↦ − g 1 , q 2 ↦ + g 2 . q_0\mapsto(-t_1,+t_2),
\qquad
q_1\mapsto-g_1,
\qquad
q_2\mapsto+g_2.
q 0 ↦ ( − t 1 , + t 2 ) , q 1 ↦ − g 1 , q 2 ↦ + g 2 .
Both base connections are therefore
A 3 , μ = ( α μ 0 0 β 1 , μ γ 1 , μ 0 β 2 , μ 0 γ 2 , μ ) . A_{3,\mu}=
\begin{pmatrix}
\alpha_\mu&0&0\\
\beta_{1,\mu}&\gamma_{1,\mu}&0\\
\beta_{2,\mu}&0&\gamma_{2,\mu}
\end{pmatrix}.
A 3 , μ = α μ β 1 , μ β 2 , μ 0 γ 1 , μ 0 0 0 γ 2 , μ .
All ten characteristic-zero coefficient functions are exported in the durable packet. They are obtained from the two wall reductions, not from the finite-field reconstruction table of Entry 853 .
Flatness
The checker verifies the three diagonal curvature identities
∂ u a v − ∂ v a u = 0 \partial_u a_v-\partial_v a_u=0
∂ u a v − ∂ v a u = 0
and both extension identities
∂ u β i , v − ∂ v β i , u + β i , u α v + γ i , u β i , v − β i , v α u − γ i , v β i , u = 0. \partial_u\beta_{i,v}-\partial_v\beta_{i,u}
+\beta_{i,u}\alpha_v+\gamma_{i,u}\beta_{i,v}
-\beta_{i,v}\alpha_u-\gamma_{i,v}\beta_{i,u}=0.
∂ u β i , v − ∂ v β i , u + β i , u α v + γ i , u β i , v − β i , v α u − γ i , v β i , u = 0.
Hence
∂ u A 3 , v − ∂ v A 3 , u + [ A 3 , u , A 3 , v ] = 0. \partial_uA_{3,v}-\partial_vA_{3,u}
+[A_{3,u},A_{3,v}]=0.
∂ u A 3 , v − ∂ v A 3 , u + [ A 3 , u , A 3 , v ] = 0.
Consequence
The wall quotient connection required by Entry 852 is now source-derived, explicit, and flat. The rank-twelve problem is reduced to deriving
B u , B v ∈ Mat 9 × 3 B_u,B_v\in\operatorname{Mat}_{9\times3}
B u , B v ∈ Mat 9 × 3
with the frozen A 9 A_9 A 9 A 3 A_3 A 3
This does not yet establish the extension blocks, their triangular-gauge class, infinity-Gysin compatibility, or intrinsic Q \mathcal Q Q
Durable artifacts
research/benincasa/marici-gm/src/bin/marked_wall_laurent_reduction.rsresearch/benincasa/marked-wall-quotient-connection.jsonEpistemic event ev-000000000470-b3be6f01-9bca-463f-8ad2-5f2e7a8e75f3
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