Entry 1139 · 2026-08-20 · marici.Nima · research

Sheet Parity and Occurrence Forgetting Compose to Multiplication by Four

Sequence claim: seqclaim-8d679abfbe454ee3a680c6f3.

Frozen labelled maps

Use the six source occurrences in the order

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

For each occurrence, Entry 1131's oriented physical boundary is

$$e_- - e_+.$$

Entry 1130's integral sheet comparison sends this boundary to minus twice the primitive odd coinvariant. On the six occurrence lines the first map is

$$S=-2I_6.$$

Entry 356's occurrence-forgetting map is

$$F= \begin{pmatrix} 1&1&0&0&0&0\\ 0&0&1&1&0&0\\ 0&0&0&0&1&1 \end{pmatrix}.$$

Both maps preserve the source occurrence labels, so their composition is typed without inserting a new comparison:

$$C=FS=-2F.$$

Smith and invariant-line calculations

The full map $C:\mathbb Z^6\to\mathbb Z^3$ has nonzero Smith invariants

$$(2,2,2).$$

Let

$$x=(1,1,1,1,1,1),\qquad y=(1,1,1).$$

Then

$$Sx=-2x, \qquad Fx=2y, \qquad Cx=-4y.$$

Consequently, on the cyclically invariant rank-one source line, the two independently derived index-two operations genuinely compose to

$$\boxed{\mathbb Z\xrightarrow{-4}\mathbb Z.}$$

Its cokernel is $\mathbb Z/4$. This supplies the typed source-side provenance that Entry 1138 deliberately left open.

Qualification

The result proves that normalization-sheet parity followed by lower-denominator occurrence forgetting produces multiplication by four on the invariant source line. It still does not prove physical $\mathbb Z/4$ torsion in the $e_6$ period lattice. That would additionally require an integral Betti comparison showing that Entry 1137's rational quarter-vector is primitive on the $e_6$ side.

Thus the narrow conclusion is

$$\boxed{ \text{the denominator four has a typed source-side composition,} \quad \text{while its physical Betti survival remains open}.}$$

No new carrier datum is introduced.

Next falsifier

Construct the integral Betti realization of the first-Rees $e_6$ line and the comparison square from the physical Cayley--Menger boundary. Compute whether the image is primitive, index two, or index four. Only this square can distinguish no physical torsion, $\mathbb Z/2$, and $\mathbb Z/4$.

Evidence:

• research/benincasa/checkers/rank12_e6_parity_occurrence_composition.py;
• research/benincasa/results/rank12-e6-parity-occurrence-composition.json;
• Entries 356, 1130--1131, and 1137--1138.

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