periodicExternalDispCoordNat5
plain-language theorem explainer
Maps each of the seven nonzero positive cube displacements on the 5-periodic torus to a real coordinate triple by reading the three Boolean direction bits as 0/1. Returns the selected spatial component (or 0 for out-of-range indices). Downstream TT normal-equation generators cite it to build external-order displacement columns. The body is a direct match on component after calling the displacement-bit decoder.
Claim. For a displacement index $d \in \{0,\ldots,6\}$ encoding one of the seven nonzero positive cube steps and a component index $c \in \mathbb{N}$, the external-order displacement coordinate is $1$ if $c \in \{0,1,2\}$ and the corresponding direction bit of $d$ is set, and $0$ otherwise.
background
Track 1.D opens the tensor/shear sector of the weak-field metric. The conformal ansatz of Track 1.B assigns one scalar potential per vertex and cannot represent pure shear, so it misses transverse-traceless gravitational-wave modes. This module therefore separates independent edge-length perturbations from vertex-conformal ones.
The ambient geometry is the 5-periodic Freudenthal torus. Its seven nonzero positive cube displacements are encoded by dispBits, which sends each index in $\mathrm{Fin},7$ to a triple of Booleans, one per spatial axis (e.g. index 0 is $(1,0,0)$, index 3 is $(1,1,0)$). Those bits are the only upstream data this definition needs.
External order means the coordinate frame used by the numerical TT normal-equation generator: rows are external edges, and selected columns carry longitudinal vertex-component generators built from these displacement coordinates.
proof idea
Pure definitional match. Decode the displacement via the bit map into a Boolean triple, then on component $0,1,2$ return $1$ or $0$ according as the corresponding bit is true or false; every other natural component returns $0$. No lemmas are applied.
why it matters
Supplies the coordinate readout that the external TT normal-equation generator matrix and its sparse row-dot product both call when filling longitudinal vertex-component columns (indices $125$–$499$). Without a stable $0/1$ embedding of the seven cube steps, the Python payloads that assemble the discrete TT system cannot distinguish pure shear generators from the conformal vertex slice.
In the broader Recognition scaffold this is infrastructure for the tensor sector that must eventually sit beside the conformal ansatz, so that the weak-field metric can carry transverse-traceless modes rather than only scalar potentials. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is a coordinate helper inside the gravity geometry layer.
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