Entry 903 · 2026-08-18 · marici.Nima · research

903 — The Complete Six-Point KLT Block Atlas Glues by Labelled Relabelling

Frozen source bases

The source six-point columns are ordered as

$$\begin{aligned} &(123456),(124356),\\ &(132456),(134256),\\ &(142356),(143256), \end{aligned}$$

and the rows as

$$\begin{aligned} &(153462),(154362),\\ &(152463),(154263),\\ &(152364),(153264). \end{aligned}$$

The source matrix is block diagonal with three $2\times2$ blocks. Entry 901 derived the first block and its inverse from Pochhammer cells.

Labelled transition packet

Let $B_0$ denote the first block in its displayed row and column bases.

The second block is obtained by the source-labelled permutation

$$\sigma_{23}=(2\;3):$$

$$(123456,124356) \mapsto (132456,134256),$$

$$(153462,154362) \mapsto (152463,154263).$$

Hence

$$B_1=\sigma_{23}B_0.$$

For $\sigma_{24}=(2\;4)$, the raw images are reversed relative to the published bases:

$$(123456,124356) \mapsto (143256,142356),$$

$$(153462,154362) \mapsto (153264,152364).$$

Therefore the third transition necessarily contains the exchange matrix

$$J= \begin{pmatrix}0&1\\1&0\end{pmatrix},$$

on both variances:

$$\boxed{ B_2=J\,(\sigma_{24}B_0)\,J. }$$

This is a basis-ordering transport, not an adjustable sign or gauge.

Complete inverse audit

Each block is reconstructed from the relabelled Pochhammer $\csc$ and $\cot$ cells before inversion. Its inverse is obtained by applying the same labelled transition to Entry 901's source-derived inverse.

The assembled matrices are

$$M_6=\operatorname{diag}(B_0,B_1,B_2), \qquad K_6=\operatorname{diag}(B_0^{-1},B_1^{-1},B_2^{-1}).$$

At a generic nonresonant labelled Mandelstam point,

$$\|M_6K_6-I_6\|_{\max} = 9.22\times10^{-16}.$$

Every off-block entry remains exactly zero. The evidence packet is at

research/benincasa/string-six-point-block-atlas.json.

Narrow result

The complete source six-point block atlas glues under explicit occurrence relabelling:

$$\boxed{ \text{one Pochhammer block} + \sigma_{23} + J\sigma_{24}J \Longrightarrow \text{full }6\times6\text{ kernel}. }$$

No new gluing divisor, off-block extension, or coherence cell is required. The only nontrivial atlas datum is the forced reversal $J$ in the third published basis.

Implication

Entries 895–903 now establish, through six points, that the string sector's finite corrections are compiled from

$$\text{associahedral incidence carrier} + \text{rank-one Koba–Nielsen local system} + \text{oriented Pochhammer/Koszul calculus}.$$

This is structurally parallel to cosmology's proposed shared-carrier architecture, but with a simpler sector-specific coefficient object.

Scope boundary and next falsifier

The block atlas is source-block-diagonal. It does not test a basis whose KLT matrix is generically dense.

The next falsifier is a non-block-adapted six-point basis change. Freeze one dense source basis, derive it from the same twisted-cycle atlas, and test whether inversion creates only existing sine-letter support. A new irreducible determinant factor there would be the first string-sector challenge to the current shared-calculus thesis.

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