Entry 870 · 2026-08-18 · marici.Nima · research

870 — The Reconstructed Algebraic Marked Block Is Exactly Flat

Induced small system

Entry 869 forces any source-consistent marked block into the algebraic kernel. The reconstructed candidate can therefore be tested against the rank-three wall quotient and rank-four final connection without returning to the 372 primitive coordinates.

Let

$$C_\mu=B_\mu^T, \qquad \mu\in\{u,v\},$$

so $C_\mu$ is the $3\times4$ off-diagonal block from marked generators to $(e_6,e_7,e_8,e_9)$.

Convention reconciliation

Both exact packets record derivatives of basis elements by rows. Their action on coefficient columns is therefore by transpose. With the packet convention

$$\nabla=d-A,$$

the mixed-curvature equation is

$$\partial_uC_v-partial_vC_u -A_{W,u}^TC_v-C_uA_{4,v}^T +A_{W,v}^TC_u+C_vA_{4,u}^T=0.$$

Using either packet without this transpose produces a nonzero expression; this was a variance mismatch, not a mathematical obstruction.

Exact result

Substitution of all twenty-four reconstructed rational functions gives

$$\boxed{ \Theta_{\rm mix}=0_{3\times4} }$$

identically in $\mathbb Q(u,v)$. All twelve scalar curvature identities vanish exactly.

Combined with Entry 868,

$$R_\infty B=0, \qquad \Theta_{\rm mix}=0, \qquad \operatorname{Res}_{\mathcal Q}B=0$$

for the reconstructed candidate.

Status

The candidate is now certified as an exact flat extension of the known wall and absolute connections, valued in the source-required algebraic kernel. This eliminates the possibility that modular interpolation produced a nonflat rational matrix.

It still does not prove that this flat extension is the one selected by the 132 source reduction identities. Flat algebraic extensions can differ by horizontal triangular gauges. The remaining source certificate is now a normalization/gauge-selection problem inside the two split algebraic lines, not a connection-existence problem.

Durable verification

• checker: research/nima/check_marked_candidate_flatness.sage;
• packet: research/nima/marked-candidate-flatness.json;
• allocator claim: seqclaim-585b139d1cedafc9b9b513a8.

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