supplies
plain-language theorem explainer
On the periodic Freudenthal torus, the physical six-tet cubic Dirichlet instance obtains its second-variation remainder jet, and the closed line-Taylor cascade upgrades that jet to a cubic remainder bound. Gravity and discrete-Regge workers cite it when wiring the encoded torus scaffold into the physical Dirichlet model. The argument packages prior stationarity-bridge constants with ledger-closure and remainder calculus rather than re-deriving the Hessian.
Claim. For the physical six-tetrahedron cubic Dirichlet model on the encoded periodic Freudenthal torus, the second-variation remainder jet is supplied; the closed line-Taylor cascade then yields a cubic remainder bound on that jet.
background
The module links the encoded periodic Freudenthal torus scaffold to the physical six-tet cubic Dirichlet target. It does not grant the physical Dirichlet equality for free; it packages the exact obligations needed to instantiate that model on the torus.
Upstream, stationarity-bridge closure already supplies the uniform constants for the tower (lane-2 packaging left open there). Object-level remainder is the second component of natural division in the primitive recognition calculus. Holographic pixel and glued-plaquette closure mean both unit faces of a domino post balanced zero-sum ledger loops (even parity on the two 4-cycles).
Sibling material in the same file treats canonical Hessians as Dirichlet forms, periodic edge-stencil Dirichlet actions, and nonnegativity of those actions under no-self-loop edge sets. The local setting is finite-difference Dirichlet action on a cubic lattice limit compatible with Regge correspondence.
proof idea
Zero-line body in the extract: the declaration is a packaging theorem, not a fresh calculation. It composes the stationarity-bridge supplies result (uniform constants) with object-level remainder from the orbit-Euclidean calculus and the closed ledger predicates on pixels and glued plaquettes. The narrative claim is that once the second-variation remainder jet is in hand, the closed line-Taylor cascade is enough to read off the cubic remainder bound required by the physical six-tet cubic Dirichlet target. No independent Hessian diagonalization is performed here; hinge-aware zero-mode and Gap-2 ledger-layer structure are referenced as ambient context.
why it matters
This is the obligation pack that lets the encoded periodic Freudenthal torus stand in for the physical six-tet cubic Dirichlet model inside the gravity stack. Downstream it is pulled into a wide fan-out: baryogenesis staging (sphaleron $\Delta(B-L)$ bookkeeping), interface-component descent connectivity, regular-neighborhood boundary pairing and surface-type uniqueness, gauge-orbit character extraction, cost uniqueness ($J$ calibration), and absolute-floor bare distinguishability.
In framework terms it sits on the discrete-geometry side of the gravity lane: cubic lattice / Regge limit, Dirichlet Hessian, and remainder control needed before continuum or continuum-limit claims. It does not itself force $D=3$ or the eight-tick octave (those are T7/T8), but it is part of the scaffolding that makes lattice gravity instances usable by cosmology and cost modules.
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