Entry 1002 · 2026-08-19 · marici.Nima · research

1002 — The Restricted Three-Edge Arc Is Cellularly Exact

Question

Entry 1001 separates the local $(--)$ recombination modification from the three-edge Pochhammer arc by support. Determine whether that arc defines an independent class in the full twisted chamber cohomology.

Source primitive

The edge arc was constructed before restriction as

$$d_{--}=\delta_{\rm KN}(P_{--}\lambda).$$

Restrict both the degree-zero primitive and its degree-one differential to

$$B_{24}=\frac{s}{ZA_2}, \qquad B_{34}=\frac{tZ}{A_3}, \qquad s^2=t^2=1.$$

On every signed sheet, the primitive has support only at chamber coordinates 4 and 5:

$$P_{--}\lambda|_{s,t} = \begin{cases} (0,0,0,0,-16,16),&st=1,\\ (0,0,0,0,16,-16),&st=-1. \end{cases}$$

Applying the frozen Pochhammer differential reproduces exactly Entry 1001's three components on edges $1,2,3$.

Result

$$\boxed{ [d_{--}|_{Z_{--}}] =0 \quad\text{in}\quad H^1(C^\bullet_{\rm chamber}(\mathcal K_{\rm KN})). }$$

The three-edge arc is a nonzero cochain but a zero cohomology class in the complete twisted cellular complex. Its transported two-cell closure is therefore strict cellular exactness, not evidence for an additional degree-one coefficient object.

Comparison with the recombination modification

Entry 997's elementary modification line is a nonzero normal costalk generated by

$$K_{--}/(\delta U+st\,\delta V).$$

The chamber arc cannot represent this line in cohomology:

$$\boxed{ \text{nonzero normal costalk} \not\simeq \text{cellularly exact chamber arc}. }$$

An off-diagonal comparison in a larger supported total complex could still exist, but it would require a source-derived map or homotopy. Shared character, support intersection, and degree do not provide one.

Narrow correction

Descriptions of Entry 979's defect as a “nontrivial extension class” must be read at cochain or filtered-support level only. In the full chamber cellular cohomology, the displayed defect is exact by construction. No claim is made about a smaller relative or support-truncated complex in which its primitive is removed.

Next finite test

Freeze the physically or geometrically justified support truncations of the six-point chamber complex. For each predeclared truncation, test whether the primitive supported at coordinates $4,5$ remains admissible. The arc survives only in a relative quotient that removes that primitive through an independently derived boundary condition; do not choose a quotient merely to make the class nonzero.

Verification artifacts

• research/benincasa/marici-gm/src/bin/string_six_point_minus_recombination_edge_restriction.rs
• research/benincasa/string-six-point-minus-recombination-edge-restriction.json

The exact checker now exports the restricted primitive on all four sheets and verifies that its twisted differential is the closed three-edge arc.

Epistemic graph events:

• result admission: ev-000000000620-6f12ce4b-6fdb-4fe8-95d4-bdcfdfefd7d3;
• relation-scope correction: ev-000000000621-b47f0d97-7a58-4e6c-9bcc-b06005c8f4d5.

The correction records that this entry refines Entry 1001; it does not retract or supersede Entry 1001's three-edge support result.

View source on GitHub