periodicRawSplittingOfEncoded5
plain-language theorem explainer
Transports any raw conformal/gauge/TT splitting of edge perturbations from the encoded finite edge index set of the 5×5×5 Freudenthal torus onto the native periodic-edge type. Track 1.D gravity authors use it to move algebraic decompositions across the canonical edge equivalence without re-proving reconstruction. Each component is conjugated by the encode/decode maps; reconstruction is discharged by simp.
Claim. Given a raw additive splitting of real edge perturbations on the finite encoded edge set of the $5\times 5\times 5$ periodic Freudenthal torus into conformal, gauge, and transverse-traceless parts (summing pointwise to the original field), produce the corresponding splitting on the native periodic-edge type by conjugating each component map with the canonical encode/decode equivalence, preserving the reconstruction identity.
background
Track 1.D opens the tensor/shear sector of weak-field gravity in Recognition Science. Track 1.B's conformal ansatz assigns one scalar potential per vertex and induces edge-length changes by averaging endpoints; that scalar slice cannot represent pure shear, so it misses transverse-traceless gravitational-wave modes. This module separates independent edge perturbations from vertex-conformal ones and records the elementary rectangle obstruction for the conformal ansatz.
A raw edge-perturbation splitting is purely algebraic: three maps (conformal, gauge, TT) from edge fields $E\to\mathbb{R}$ to themselves, together with the pointwise identity that their sum recovers the input. It does not yet assert membership in geometric kernels or images; those predicates are the later Track 1.D target, to be supplied by periodic Freudenthal operators.
The ambient geometry is the canonical encoded $5\times 5\times 5$ periodic Freudenthal torus, with native periodic edges PeriodicEdge 5 5 5 and a finite encoded edge index set. Encode/decode maps identify the two edge presentations.
proof idea
Definition by transport along the edge equivalence. For each of the three component maps, push the periodic edge field to the encoded finite index set, apply the corresponding component of the input splitting $D$, then pull back to periodic edges. The reconstruction field is a short tactic proof: introduce the edge field and edge, then simp with the two transport maps and $D$'s reconstruction identity, so the conjugated sum collapses to the original field.
why it matters
Without this bridge, every conformal/gauge/TT decomposition written on the encoded finite edge space would have to be re-stated by hand on native periodic edges. The definition keeps the algebraic splitting datum portable across the two presentations used by Track 1.D payload generators and the Freudenthal torus API.
It sits upstream of the orthogonal conformal/gauge/TT surface (the finite $N=5$ edge-space inner product and the genuine membership predicates). In the broader RS gravity program it supports the move from the scalar conformal ansatz toward a full weak-field metric sector that can carry shear and TT modes, consistent with the $D=3$ spatial forcing (T8) already fixed in the foundation chain. No downstream theorems yet cite it (used_by is empty); it is scaffolding infrastructure for the tensor/shear track rather than a closed physical claim.
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