PeriodicTorus5
plain-language theorem explainer
The standard 5×5×5 periodic Freudenthal lattice used as the discrete geometry for Track 1.D tensor and shear analysis. Anyone working on edge-length perturbations, conformal slices, or TT residual formulas on the finite torus cites this object. It is a one-line specialization of the canonical encoded periodic construction at side length 5 in each spatial direction.
Claim. Write $T_5$ for the canonical encoded periodic Freudenthal torus on a $5\times 5\times 5$ lattice (each side length strictly greater than 2). This is the fixed discrete 3-torus geometry underlying Track 1.D edge and shear calculations.
background
Track 1.D separates pure edge (tensor/shear) perturbations from the older Track 1.B vertex-conformal ansatz. The conformal ansatz assigns one scalar per vertex and averages endpoints onto edges; that slice cannot represent pure shear or transverse-traceless gravitational-wave modes. The module therefore fixes a concrete periodic lattice and studies independent edge-length variations on it.
The ambient spatial dimension is the RS value $D=3$ forced by the forcing chain (T8). The geometry is the encoded periodic Freudenthal torus: a 3-periodic cubical complex with incidence data packaged so that edge and vertex spaces are finite-dimensional and machine-checkable. The constructor canonicalEncodedPeriodicFreudenthalTorus builds that object from side lengths $N_x,N_y,N_z$ (each $>2$) and the canonical periodic endpoint incidence.
Choosing all three sides equal to 5 yields a small but nontrivial torus: large enough for nontrivial shear and rectangle obstructions, small enough that finite ranks (edge space dimension 875, vertex generators, etc.) remain explicit in later lemmas.
proof idea
Definitional abbreviation only. Instantiate the canonical encoded periodic Freudenthal torus at $(N_x,N_y,N_z)=(5,5,5)$, discharging the three side-length hypotheses $2<5$ by decide. No further proof content.
why it matters
This is the fixed background geometry for the entire Track 1.D tensor/shear scaffold. Downstream edge-sector theorems compute finite ranks and conformal characterizations relative to it (edge-perturbation rank equals the torus edge count; conformal log-subspace matches the encoded conformal predicate). Master-theorem handoff endpoints phrase conformal generator span, TT gauge-generator projector reduction, and Hessian–Lichnerowicz residual formulas as statements about edge perturbations on this same $N=5$ torus.
In the broader RS picture it sits inside the gravity track that must eventually connect discrete Regge-type edge data to continuum TT modes in $D=3$. It does not itself prove shear completeness or residual vanishing; it pins the lattice so those claims become finite linear-algebra statements rather than abstract continuum ones.
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