HEMM Space

A HEMM Space is a two-by-three array of subspaces, each of which is itself a two-by-three matrix of elemental coordinates. One of these subspaces (here chosen as $(M_{22})$) carries the Cosmic Index Factor (CIF (Cosmic Index Factor) 🧱), enabling universe-scale calibrations.

1. Block Structure

We organize HEMM as: $$[ \mathrm{HEMM} = \begin{pmatrix} M_{11} & M_{12} & M_{13} \ M_{21} & M_{22} & M_{23} \end{pmatrix}, ]$$ where each block $(M_{ij}) is a (2\times3)$ matrix of base elements $$(a_{kl}^{(ij)}): [ M_{ij} = \begin{pmatrix} a_{11}^{(ij)} & a_{12}^{(ij)} & a_{13}^{(ij)} \ a_{21}^{(ij)} & a_{22}^{(ij)} & a_{23}^{(ij)} \end{pmatrix}, \quad i\in{1,2},; j\in{1,2,3}. ]$$

1.1 The CIF‐Bearing Subspace

By convention, the central block $(M_{22})$ is endowed with the Cosmic Index Factor: $$[ M_{22} ;\xrightarrow{;\text{CIF};}; M_{22}^{(\mathrm{cif})}, ]$$
linking the local HEMM coordinates to the global cosmic scale via the Cosmic Index Format (CIF) 🧱 definition.

2. LaTeX Representation

$$ \mathrm{HEMM} = \begin{pmatrix} M_{11} & M_{12} & M_{13} \ M_{21} & M_{22} & M_{23} \end{pmatrix}, \quad M_{ij} = \begin{pmatrix} a_{11}^{(ij)} & a_{12}^{(ij)} & a_{13}^{(ij)} \ a_{21}^{(ij)} & a_{22}^{(ij)} & a_{23}^{(ij)} \end{pmatrix}. $$`

And the CIF‐tagged block:

$$ M_{22}^{(\mathrm{cif})} = \mathrm{CIF};\times; \begin{pmatrix} a_{11}^{(22)} & a_{12}^{(22)} & a_{13}^{(22)} \ a_{21}^{(22)} & a_{22}^{(22)} & a_{23}^{(22)} \end{pmatrix}. $$`

3. Conceptual Notes

• Each $(a_{kl}^{(ij)})$ can represent a coordinate axis, physical parameter, or “mode” in that subspace.
• The CIF rescales all entries in $(M_{22})$ to align local physics with the Cosmic Index Frame.
• You can choose any other $(M_{ij})$ for CIF or even support multiple CIF‐bearing blocks for more exotic geometries.