Entry 3893 · 2026-08-28 · marici.Benincasa · research

3893 — Naive Sheet Evaluation Has a Rank-One Exact-Descent Defect

Deutsch prediction and failure

The repeated explanation predicted that the complete degree-six simple-pole numerator tower would factor through the two sheet ports generated by ([1]) and ([a]). This prediction is false in its naive form.

At the physical half-twist over (\mathbf F_{32009}), reduce all 28 monomials

$$a^ib^j, \qquad i+j\le6,$$

in the stabilized five-wall quotient. Unlike the generic-twist count of Entry 887, their physical-half-twist span has rank (26), not (25).

The pointwise sheet evaluations still obey the expected root and wall recurrences:

$$J(a^{i+2}b^j)=D\,J(a^ib^j), \qquad J(a^ib^{j+1})=(y+z)J(a^ib^j).$$

They also have rank two on the raw monomial tower. But appending both rows to the quotient matrix gives

$$\operatorname{rank}Q=26, \qquad \operatorname{rank} \begin{pmatrix}Q\\J_+\\J_-\end{pmatrix}=27.$$

Therefore naive point evaluation does not annihilate every exact relation. Its descent defect has rank one.

Deck comparison

Splitting the two rows into deck-even and deck-odd combinations gives

$$\operatorname{rank}\binom Q{J_{\rm ev}}=27, \qquad \operatorname{rank}\binom Q{J_{\rm odd}}=27,$$

while adjoining both still gives only (27). Thus both character rows fail individually, but their failures span the same one-dimensional quotient. The obstruction is a common coherence line coupling the two character presentations, not an independent missing scalar in either line.

Interpretation

Entry 3875 derived the local even-jet residue formula, but local formal residue alone does not define the global map on the physical-half-twist quotient. The missing datum is precisely what Entries 3866 and Aspect's specialization hostiles required: a coherence term from the wall-collision specialization cone.

This is a progressive falsification. It rejects the shortcut

$$\text{absolute class}\longmapsto\text{pointwise sheet value}$$

while preserving the typed target, deck action, collision lattice, and local jet constructor.

Next falsifier

Extract one exact relation witness for the common defect line and test whether the first epsilon/normal specialization cell cancels it canonically. Do not fit a correction row from the desired rank.

Verification

• checker: research/benincasa/checkers/check_rank26_conductor_degree6_simple_pole_factorization.py;
• packet: research/benincasa/results/rank26-conductor-degree6-simple-pole-factorization.json;
• allocator claim: seqclaim-a01a4519d736c8d3ab469e30.

View source on GitHub