the
plain-language theorem explainer
Packages the displacement class that periodic Freudenthal geometry assigns to each local tetrahedral edge slot, independent of cell. Gravity and Regge-calculus workers cite it when assembling hinge and edge-diagonal blocks over the seven displacement classes. It is a pure definitional class, fixed by the cell-independent local-edge map on the torus.
Claim. For each tetrahedron type $t$ and local edge slot $f$, the class records the displacement class that the actual periodic geometry assigns to that slot in every cell of every torus. Cell-independence follows from the construction of the local-to-global edge map.
background
This module sits in the QG full-theory campaign (Paper C / Pillar 1, Lane C) and closes the hinge-aware zero-mode gate for Regge TT analysis. The ambient geometry is the periodic Freudenthal triangulation of the torus: each cell carries six tetrahedron types, each with six local edge slots. A local-to-global edge map sends every (cell, tet, slot) triple to a periodic edge; by construction the displacement part of that edge does not depend on the cell.
Upstream, the cube triangulation and two-cube strip supply the base local-edge tables, and the periodic torus lifts them by translating the cube-edge base and displacement. The seven displacement classes are the discrete labels over which hinge and stencil blocks are later summed. RS-independence of coupled axes is in scope only as a primitive-separation side condition, not as a dynamical input here.
The sympy diagnostic context (not a proof) found that the stencil-only constant block fails to vanish under a reported TT witness, while the assembled hinge-minus-stencil block does vanish. That assembly is built from per-class edge coefficients, so a stable, cell-independent slot-to-class assignment is a prerequisite.
proof idea
No proof body: this is a definitional class (def/abbrev style). It names the displacement class read off the periodic local-edge map for slot $f$ of tetrahedron type $t$. Cell-independence is inherited from the torus local-edge construction, which factors the edge into a translated base and a cube-edge displacement that does not depend on the cell. No algebraic reduction or tactic script is involved.
why it matters
Without a cell-independent slot-to-class assignment, the hinge edge-diagonal block and the assembled constant block cannot be written as clean sums over the seven displacement classes. Those objects are the content of Gate C-A3: the hinge block is the assembled O(1) sum with edge-class coefficients from the symbol program; the headline zero-mode theorem states that the Schlaefli-reduced assembled constant block vanishes at $k=0$ under TT.
In the broader Recognition gravity stack this is scaffolding for the Regge TT kernel on the Freudenthal torus, not a forcing-chain landmark (T5–T8) or an RCL identity. It does not touch mass ladders, $\alpha$, or the eight-tick octave. Downstream consumers in-module are the hinge block, the witness residual cancellation, and the assembled zero-mode statement; cross-module edges on the bare name are mostly name-collision noise.
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