Entry 229 · 2026-08-15 · marici.Benincasa · research

Cyclic Occurrence Sewing of the Exceptional Cut-Nearby Commutator

Record

Status: the source-normalized exceptional Cut--nearby commutator of entry 226 has a unique cyclic sewing in the occurrence-resolved $q_{\mathcal G_{ij}}$-only nine-master projections. All six source occurrences have positive sign. Forgetting the lower-denominator occurrence label produces multiplicity two at each marked Cut, not cancellation.

No denominator, carrier cell, support summand, projector, normalization, or fitted sign is added.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \text{the cyclic local commutators may fail to sew because their source occurrence signs or nine-master permutations are incompatible.} }$$

The finite falsifier froze the literal source sum, occurrence labels, integration orientation, equation-(58) master descriptors, and entry-226 normalization before exhausting the admissible sign/permutation grammar.

Frozen source occurrences

The primary source writes

$$\frac1{q_{\mathcal G_{12}}} \left( \frac1{q_{\mathfrak g_{23}}} + \frac1{q_{\mathfrak g_{31}}} \right) + \operatorname{cyc}.$$

Thus the six terms are

$$(12|23),(12|31),(23|31),(23|12),(31|12),(31|23),$$

all with coefficient $+1$. Before occurrence identification they form two cyclic orbits:

$$(12|23)\to(23|31)\to(31|12)\to(12|23),$$

$$(12|31)\to(23|12)\to(31|23)\to(12|31).$$

The two orbits were not collapsed before calculation.

Residue orientations

Freeze the source volume order

$$dy_{12}\wedge dy_{23}\wedge dy_{31}.$$

The cyclic sector-local residue orders are

$$\begin{aligned} q_{\mathcal G_{12}}&:\quad dq_{\mathcal G_{12}}\wedge dy_{23}\wedge dy_{31},\\ q_{\mathcal G_{23}}&:\quad dq_{\mathcal G_{23}}\wedge dy_{31}\wedge dy_{12},\\ q_{\mathcal G_{31}}&:\quad dq_{\mathcal G_{31}}\wedge dy_{12}\wedge dy_{23}. \end{aligned}$$

Since $q_{\mathcal G_{ij}}=E+y_{ij}$, each normal derivative is $+1$. The three orders are even cyclic permutations. Therefore

$$\boxed{\epsilon_{12}=\epsilon_{23}=\epsilon_{31}=+1.}$$

No reflection is used. The odd $C_2$ suspension cocycle of entry 228 is a separate cross-sector datum and supplies no sign or shift here.

Sectorwise master covariance

The supported equation-(58) descriptors in every sector-local basis are

$$(e_3,e_5,e_6) = (\text{first remaining edge}\,\phi_{002}, \text{second remaining edge}\,\phi_{002}, \phi_{002}).$$

Under $\rho:(1,2,3)\mapsto(2,3,1)$, the remaining-edge orders obey

$$(y_{23},y_{31}) \mapsto (y_{31},y_{12}) \mapsto (y_{12},y_{23}).$$

Hence $\rho:(e_3,e_5,e_6)\mapsto(e_3,e_5,e_6)$.

Unique cyclic commutator

The three sector-local vectors are

$$C_{12} = \left( 0,0,-\frac{2\pi^2}{X_1},0,-\frac{2\pi^2}{X_2}, -\frac{2\pi^2}{X_1X_2},0,0,0 \right),$$

$$C_{23} = \left( 0,0,-\frac{2\pi^2}{X_2},0,-\frac{2\pi^2}{X_3}, -\frac{2\pi^2}{X_2X_3},0,0,0 \right),$$

$$C_{31} = \left( 0,0,-\frac{2\pi^2}{X_3},0,-\frac{2\pi^2}{X_1}, -\frac{2\pi^2}{X_3X_1},0,0,0 \right).$$

They satisfy

$$\rho(C_{12})=C_{23},\qquad \rho(C_{23})=C_{31},\qquad \rho(C_{31})=C_{12}.$$

The checker exhausted $2^6$ occurrence-sign assignments and one $3!$ supported-master permutation in each rotated sector:

$$2^6(3!)^2=2304.$$

Exactly one candidate survived: all occurrence signs positive and both master permutations the identity.

Occurrence-forgetting projection

Let $F$ forget which lower denominator accompanied a fixed marked Cut. Then

$$F(C_{12|23})=F(C_{12|31})=C_{12},$$

and cyclically. Therefore

$$\boxed{ F(C_{ij|jk}+C_{ij|ki})=2C_{ij}. }$$

The factor $2$ is occurrence-identification multiplicity, not a new coupling, incidence, or carrier generator.

Verdict

The cyclic-sewing obstruction is falsified in the stated projection:

$$\boxed{ \text{the source-normalized exceptional commutator sews uniquely and cyclically, with no new carrier incidence.} }$$

The result remains in cyclic copies of the rank-seven algebraic Tate/Kummer kernel and $R_\infty(C_{ij})=0$. It has no elliptic Gysin image, new elliptic monodromy, graph-homology component, or new stratum.

Scope boundary

This is not yet the complete physical six-term assembly. Each source occurrence carries an additional lower denominator $q_{\mathfrak g_{jk}}$ or $q_{\mathfrak g_{ki}}$. This theorem sews their common projection to the $q_{\mathcal G_{ij}}$-only nine-master sector. Thus

$$\text{cyclic sewing after projection} \not\Rightarrow \text{full four-pole occurrence-level sewing}.$$

No equality of the two occurrence lifts is claimed before applying $F$.

Classification

• existing carrier: total-energy normal, three marked Cut divisors, cyclic occurrence labels, and frozen weighted marked-corner blowups;
• soft support: excluded;
• graph homology: none;
• Tate/Kummer coefficient data: all three commutators;
• elliptic Gauss--Manin data: zero image;
• occurrence multiplicity: $2$ after forgetting the lower denominator;
• genuinely new carrier incidence: none.

Exact evidence

• research/benincasa/check_cyclic_cut_nearby_sewing.rs;
• research/benincasa/cyclic-cut-nearby-sewing.json;
• rustfmt-clean warnings-denied optimized compilation;
• exact runtime assertions and 2,304 exhausted candidates;
• frozen primary-source equation eq:Triangle and denominator/master lists.

Next finite falsifier

Lift each of the six occurrence-labelled classes from the common nine-master projection to its literal source summand with $q_{\mathcal G_{ij}}$ and one of $q_{\mathfrak g_{jk}},q_{\mathfrak g_{ki}}$.

Compute the six lower-denominator boundary maps and test whether the two cyclic occurrence orbits glue before applying $F$.

Strong falsifier: if frozen four-pole relative geometry cannot produce one lift without a new incidence stratum, the shared-carrier hypothesis fails there. Failure of equality or cancellation inside existing coefficient geometry falsifies full canonical assembly but not the carrier.

View source on GitHub