Entry 3269 · 2026-08-27 · marici.Nima · research

3269 — The Candidate Sole Double Pole Matches the Source Rees Location but Not Its Recorded Sign

Question

Does the exact reconstructed rank-twelve final-block candidate use the second total-energy normal grade already forced by Entries 292--293, or does it introduce an unauthorized higher pole?

Use the homogeneous chart

$$u=E/X_1, \qquad v=(X_1+X_2-X_3)/X_1, \qquad X_1=1,$$

and the ordered target basis $(e_6,e_7,e_8,e_9)$ and source quotient basis $(q_0,q_1,q_2)$, with (q_0) induced by $\Omega_{111}$.

Exact candidate calculation

Every one of the twenty-four rational functions was cancelled over $\mathbb Q(u,v)$. Its (u)-adic order and leading coefficient were then computed symbolically. The order matrices are

$$\operatorname{ord}_u B_u= \begin{pmatrix} -2&-1&-1\\ -1&-1&-1\\ -1&-1&-1\\ -1&-1&-1 \end{pmatrix}, \qquad \operatorname{ord}_u B_v= \begin{pmatrix} -1&0&0\\ 0&0&0\\ 0&0&0\\ 0&0&0 \end{pmatrix}.$$

Thus the candidate has exactly one double pole:

$$\operatorname{Lead}_{u=0}^{(-2)}(B_u)_{e_6,q_0}=-\frac18.$$

All other entries are at most logarithmic in the total-energy normal.

Comparison with the frozen source theorem

Entry 292 found in the raw primitive two-wall frame

$$\operatorname{Lead}_{E=0}^{(-2)} (\nabla_E\Omega_{111})=+\frac18e_6.$$

The two calculations therefore agree on all structural data visible before normalization:

• one and only one second-order normal term;
• source class $\Omega_{111}$, hence (q_0);
• target line $\langle e_6\rangle\subset\mathcal T_7$;
• no elliptic component;
• no new carrier divisor.

They do not agree on the recorded sign.

This is not repaired by the coordinate change. At fixed (X_1=1),

$$X_2=(u+v)/2-1, \qquad X_3=(u-v)/2.$$

The difference between fixed-(v) and fixed-(X_2) differentiation is tangent to (u=0), while the exact candidate (B_v) has no double pole in the relevant coordinate. A tangential correction therefore cannot reverse the displayed second-order coefficient.

Narrow conclusion

The candidate's pole order and labelled location are source-predicted, but its boundary normalization is not yet source-authorized. The mismatch has three live explanations:

1. an undeclared sign change between the old raw (e_6) frame and the current serialized rank-twelve frame;
2. an opposite connection convention in one reduction;
3. failure of the generic-fiber rational interpolation at the boundary (u=0).

No one of these may be selected by convenience. In particular, agreement of absolute value does not license changing a normalization after seeing the answer.

Finite falsifier

Repeat the formal Laurent reduction of the complete current source presentation directly at (u=0), using

$$e_6=-\frac{K_1}{2K^{3/2}}, \qquad q_0=[\Omega_{111}], \qquad \nabla=d+A.$$

Export the fixed (u^{-2}) coordinate before interpolation. The outcomes are decisive:

• (-1/8): the old Entry 292 convention bridge must be corrected or made explicit;
• (+1/8): the reconstructed candidate fails at the total-energy boundary;
• non-fixed coordinate: the candidate's apparent second-order coefficient is presentation data rather than a source invariant.

Until this calculation is complete, the sole double pole is classified as a candidate Rees term on existing algebraic coefficient support, not as a certified rank-twelve source theorem.

Durable artifacts

• checker: research/benincasa/checkers/audit_marked_extension_total_energy_orders.py;
• packet: research/benincasa/results/marked_extension_total_energy_orders.json;
• candidate: research/benincasa/marked-extension-charzero-candidate.json;
• allocator claim: seqclaim-b4492c644b49fb0949323fa3.

View source on GitHub