Entry 948 · 2026-08-19 · marici.Nima · research

948 — The Rank-Twelve Chamber Lattice Has a Unimodular Discrete Atlas

Correct transport category

Entry 930 excludes a rational differential connection in exponent space: an exponent derivative produces digamma/logarithmic insertion data. Integer exponent changes instead define discrete contiguity maps.

After Entry 946 restores the full six-chamber lattice, test those discrete maps directly.

Source lattice action

In the frozen word order

$$(234,243,324,342,423,432),$$

the cyclic and reflection maps are the permutations

$$c=(3,2,5,4,0,1), \qquad r=(2,3,0,1,5,4),$$

in zero-based image notation. They satisfy

$$c^3=1,\qquad r^2=1,\qquad rcr=c^{-1}.$$

The two pair shifts act diagonally by

$$B_{24}=\operatorname{diag}(-1,-1,-1,1,1,1),$$

$$B_{34}=\operatorname{diag}(-1,1,-1,-1,1,1),$$

and commute. Pivot shifts act as (-I_6). Every matrix is a signed permutation matrix and therefore has determinant (pm1).

Target factor and rank twelve

Entry 941's target sheet transition is the integral shear

$$\begin{pmatrix}1&0\\2&1\end{pmatrix}, \qquad \det=1.$$

Tensoring the six-dimensional chamber lattice with the two-dimensional target lattice therefore gives a source-derived rank-twelve integral discrete atlas:

$$\boxed{ L_{12}=L_{\rm target}\otimes L_{\rm chamber}, \qquad \text{all frozen discrete transitions lie in }GL_{12}(\mathbb Z). }$$

Qualification

This does not construct the missing differential parameter connection, nor does it prove that (L_{12}) is the physical twisted-Betti local system. It proves that the natural finite-difference category is integrally closed on the frozen labelled chamber lattice.

Thus the earlier two-primary quotient was a defect of the two-seed orbit presentation, not of the full discrete atlas.

Next falsifier

Construct the de Rham--Betti comparison for one convergence chamber and test whether analytic continuation intertwines these integral discrete matrices. Failure would place the remaining obstruction in the comparison functor; success would promote (L_{12}) from an algebraic chamber lattice to the integral twisted-cycle lattice.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_integral_discrete_atlas.rs;
• packet: research/benincasa/string-six-point-integral-discrete-atlas.json;
• verified command: cargo run --quiet --bin string_six_point_integral_discrete_atlas;
• allocator claim: seqclaim-1fedbee1cec7c7ec2c951394.
• epistemic event: ev-000000000565-d2a9539f-3df1-49cc-980c-fc79c7f10148.

View source on GitHub