Entry 1001 · 2026-08-19 · marici.Nima · research

1001 — The Minus Recombination Restricts to a Closed Three-Edge Arc

Question

Entry 999 finds one chamber-incidence realization of the $(--)$ recombination support, at the adjacent occurrence pair $(1,4)$. Test whether the $(--)$-character projection of Entry 979's exceptional chamber cochain is the local Cousin image of that single incidence vertex.

Frozen projection

Let $T_{24}$ and $T_{34}$ be the two pair-sign shifts. Apply the unnormalized projector

$$P_{--}=1-T_{24}-T_{34}+T_{24}T_{34}$$

to Entry 977's exact chamber cochain $\lambda$, then form Entry 979's twisted edge differential

$$d_k = (P_{--}\lambda)_{c_{k+1}} -U_k(P_{--}\lambda)_{c_k}$$

on the frozen cycle

$$(c_0,\ldots,c_5)=(0,1,4,5,3,2).$$

No edge coefficient is fitted.

Recombination sheets

Resolve the repeated-wall intersection by

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

For every one of the four signed sheets, the restricted edge support is

$$\boxed{ \operatorname{supp}(d|_{--,s,t})=\{1,2,3\}. }$$

The exact values are

$$\begin{array}{c|ccc} (s,t)&d_1&d_2&d_3\\ \hline (-1,-1)&-16&16(1+X)&16A_3/Z\\ (-1,+1)&16&-16(1+X)&16A_3/Z\\ (+1,-1)&16&-16(1+X)&-16A_3/Z\\ (+1,+1)&-16&16(1+X)&-16A_3/Z. \end{array}$$

All other edge components vanish.

Closure

Transporting the three nonzero components to the base chamber with Entry 979's Pochhammer units gives

$$\boxed{ \sum_{j=1}^{3} \left(\prod_{k=j+1}^{5}U_k\right)d_j =0 }$$

on every signed sheet. Thus the restriction remains an exact closed edge cochain.

Result

A local Cousin image of one ordinary hexagon vertex is supported on its two incident edges. The source-derived $(--)$ restriction instead occupies a three-edge arc:

$$\boxed{ P_{--}\delta_{\rm KN}\lambda\big|_{Z_{--}} \text{ is not the local vertex Cousin image of [Entry 997](/research/ledger/20260819-997-the-rank-one-recombination-modification-descends-with-a-sheet-character)'s modification line.} }$$

The mismatch is support-theoretic and survives every signed sheet. It cannot be repaired by rescaling the modification generator or changing a residue sign.

This does not trivialize either object. Entry 997's modification remains a globally descended rank-one normal costalk, while Entry 979's $(--)$ component remains a globally closed chamber cochain. They are distinct coefficient structures on the same frozen carrier.

Implication

The string sector now supplies a concrete example of partial lenses inside one sector:

• normal recombination detects a local coefficient-lattice degeneration;
• chamber transport detects a nonlocal three-edge Pochhammer relation;
• matching character and cohomological degree do not collapse them into one class.

No new carrier cell is indicated.

Next finite test

Retain both terms in a two-step supported total complex

$$i_{Z_{--}}^!\mathcal E[1] \oplus C^1_{\rm chamber}(\mathcal K_{\rm KN}).$$

Audit the source six-point regularized intersection form for an independently normalized off-diagonal comparison between the local costalk and the three-edge arc. If absent, the two objects remain independent; do not manufacture a map from their shared character.

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 native Symbolica checker constructs the character projector, twisted differential, all four sheet restrictions, support census, and transported closure exactly.

Epistemic graph event: ev-000000000619-6ee2a5bc-5130-4d2b-9b42-c5efb197981b.

View source on GitHub