Canonical blocks and reports

The cell dictionary in Appendix app:coding has 55 entries, so six location bits suffice. At the source boundary, only valid incoming labels are admitted.

Fix integers

The clock stores $s=j{2}^{-p_{\mathrm{clk}}}$. A source input has the exact integer

representing $s=-1/2$. No event decision is inferred from a rounded length report.

For each of the 24 finite slots, store

inactive slots have report zero. All finite lengths are less than 16. Indeed, compact endpoint spatial norms are at most 4:

The developed spatial distance between two endpoints is at most 8, and a spacelike edge's induced length is no larger than its spatial length. Hence $p_{\mathrm{rep}}+4$ report bits suffice.

metadata consists of five fixed one-bit identifiers and one six-bit particle-cell label.

Fix injective encodings of ${\mathcal{T}}_q$ into the index fields and of $j\in [-2^{p_{\mathrm{clk}}},2^{p_{\mathrm{clk}}}]$ into the clock field. The stated widths suffice. All admitted metadata values are fixed.

A canonical geometric input $x$ has stage 0, incoming incidence mode, the exact source clock, indices $m_{\mathrm{old}},m_{\mathrm{src}}\in {\mathcal{T}}_q$, permission $1\le {\nu }_{\mathrm{src}}\le 2^{w_{\mathrm{src}}}-1$, fixed metadata, and the canonical reports for ${\mu }_{m_{\mathrm{old}}}$.

A second $B$-bit block stores complete geometric input history. Each original matter name has six location bits and six name-history bits, reserved separately.

The decoder reconstructs the entire family from its packet index, clock, and fixed symbolic formulas. Reports are consistency data; they are not the underlying exact geometry. The decoder ignores implementation history and returns the matter state unchanged.