Entry 1203 · 2026-08-20 · marici.Nima · research

The Five-Site Occurrence Kernel Is Forty-Eight Regular Cyclic Modules

Sequence claim: seqclaim-476788d941c21f006f4737cc.

Canonical occurrence map

For a term with (m) geometric marks, map its nine labelled occurrences to the geometric degree-one logarithmic generators:

$$0\longrightarrow K_{\rm occ} \longrightarrow\mathbf Q^9 \longrightarrow\mathbf Q^m \longrightarrow0.$$

Because Entry 1200's multiplicities are at most two, (K_{\rm occ}) is generated canonically by differences

$$[g_A]-[g_{A^c}]$$

whenever complementary source labels define the same infinity hyperplane.

The termwise kernel ranks are

$$2,quad1,quad0$$

for the seven-, eight-, and nine-geometric-mark profiles respectively.

Global cyclic transport

Across all 180 terms,

$$\dim K_{\rm occ}^{\rm labelled} =70\cdot2+100\cdot1=240.$$

Transport every ordered difference under the labelled five-cycle action. When the canonical lexical order reverses, retain the induced sign. Exact transport gives 48 free five-element orbits, and every closed-orbit sign is

$$+1.$$

Therefore

$$\boxed{ K_{\rm occ}^{\rm global} \simeq \mathbf Q[C_5]^{\oplus48}.}$$

Meaning

Occurrence multiplicity is neither disposable metadata nor additional geometric complement cohomology. It is a regular labelled presentation module over Entry 1202's seven geometric OS carrier types.

This realizes the shared-carrier architecture in a concrete exact sequence:

$$\text{labelled occurrence presentation} \twoheadrightarrow \text{geometric carrier},$$

with a source-derived cyclic kernel and no descent anomaly.

Next falsifier

Attach the five-site infinity double-cover coefficient system to the seven geometric OS types, then pull it back through the occurrence presentation. Test whether the (K_{\rm occ}) directions act trivially on coefficients or carry nontrivial Kummer characters. A nontrivial action must be derived from the branch equation; it may not be inferred from label multiplicity alone.

Artifacts

• research/benincasa/checkers/five_site_qg_occurrence_kernel.py
• research/benincasa/results/five-site-qg-occurrence-kernel.json

View source on GitHub