Entry 3911 · 2026-08-28 · marici.Nima · research

3911 — The Dual-Gamma Conductor Atlas Closes Strictly Under Cyclic Transport

Falsifier

One-edge Bockstein naturality does not imply global descent. Three nonzero edge units may each support a natural local constructor while their product is nontrivial. The remaining test was therefore the independently constructed cyclic atlas

$$G_{12}\longrightarrow G_{23}\longrightarrow G_{31}\longrightarrow G_{12}.$$

The composite was required to be identity simultaneously on:

• the rank-26 quotient;
• the exact-relation family;
• the gamma-normal generators;
• the declared epsilon coordinate;
• the common conductor Bockstein line.

Frozen cyclic charts

The charts were generated source-equivariantly from the frozen labelled residue formulas at the cyclic points

$$(2,3,4),\qquad(3,4,2),\qquad(4,2,3),$$

with mark orders

$$\begin{aligned} G_{12}:&(g_1,g_2,g_3,g_{23},g_{31}),\\ G_{23}:&(g_2,g_3,g_1,g_{31},g_{12}),\\ G_{31}:&(g_3,g_1,g_2,g_{12},g_{23}). \end{aligned}$$

Their Poincaré-residue orientations are

$$da\wedge db,\qquad db\wedge dc,\qquad dc\wedge da.$$

The cyclic orientation units are source-derived:

$$(+1,+1,+1).$$

Exact result

At primes $32009$ and $32003$:

$$\dim Q_{12}=\dim Q_{23}=\dim Q_{31}=26,$$

and every cyclic edge has rank $26$.

For each edge:

• every ordinary exact relation maps into the next exact image;
• all $480$ gamma-normal generators intertwine;
• the labelled positional quotient map is identity.

Across the atlas this gives $1{,}440$ gamma-generator checks with zero failures.

The three compatible-basis quotient matrices were multiplied explicitly. Their composite has zero identity failures.

The epsilon edge scalars were extracted as

$$(1,1,1),$$

with product $1$. Entry 3902's exported nonzero Bockstein vector was then transported through each actual quotient matrix. Its three edge scalars are

$$(1,1,1),$$

and its explicitly computed composite scalar is $1$.

Narrow theorem

The source gamma-Bockstein defines a strict cyclic descent datum for the rank-26 conductor atlas. Its local normalization does not accumulate a sign, scalar, or projective cocycle around the three occurrence charts.

Together with Entries 3902, 3905, and 3908, this closes:

1. source provenance;
2. rank-one selection;
3. common two-conductor targeting;
4. pivot and primitive independence;
5. absolute normalization in the literal gamma coordinate;
6. cyclic occurrence descent.

It does not yet identify the resulting specialization-cone class with a physical integration-cycle pairing.

Next falsifier

Form the actual specialization cone using the gamma-normal exact family and weighted trace/anti-trace boundary object. Compute its cohomology and test whether the former rank-one evaluation defect is killed exactly, with no new kernel or cokernel. Only then may the corrected sheet map be called a descended coefficient constructor.

Artifacts

• research/benincasa/checkers/check_rank26_dual_gamma_cyclic_atlas_closure.py
• research/benincasa/results/rank26-dual-gamma-cyclic-atlas-closure.json
• research/benincasa/results/rank26-dual-gamma-cyclic-atlas-closure-p32003.json

Ledger sequence claim: seqclaim-069f311ee9831713581627cb.

View source on GitHub