Gravity Physical Six Tet Cubic Dirichlet Instance Canonical Periodic Full Regge

The periodic Dirichlet bridge

A Dirichlet problem asks for a function that takes prescribed values on a boundary and satisfies a governing equation in the interior. In Recognition Science, the framework that derives physical structure from a ledger of recognition events, this declaration assembles the exact obligations needed to instantiate a physical model on a periodic Freudenthal torus scaffold. The torus is a repeating three-dimensional grid of tetrahedra, and the declaration packages the theorem obligations that would connect this discrete scaffold to a finite-difference Dirichlet operator.

The declaration itself does not prove the physical Dirichlet equality. The pack states this plainly: the finite-difference operator is a placeholder, separated from the abstract canonical graph Dirichlet energy, and a later proof should replace it with the actual six-tet cubic stencil expression. What the declaration does establish is a collection of supporting theorems and definitions. One theorem, periodicMatchingBaseCell_unique, proves that for any target vertex and any of the eight tetrahedral positions, the base cell that maps to it is unique. Another theorem, freudenthalLocalThreeAngleIdentityTarget, proves a local identity about sums of angles in the Freudenthal realization, and a third theorem, FreudenthalAxisDisp0LocalFiberSumEqAuditTarget_holds, verifies a specific numerical sum of paired angle contributions.

These theorems are not idle bookkeeping. They are the load-bearing pieces that a future proof would need to show that the periodic lattice obeys a Dirichlet-like condition. The uniqueness theorem guarantees the matching between the abstract lattice and the physical vertices is well-defined. The angle-sum theorems provide the local geometric identities that a discrete Laplacian or energy functional would rely on. The declaration also includes witness values, such as a grid size of 5 in each direction, and explicit counts of stationary points and symmetry reductions, which serve as concrete checks that the structures behave as intended on a small example.

What the declaration does not claim is the physical content. It does not assert that gravity, as described by the Einstein field equations, emerges from this lattice. It does not claim that the placeholder finite-difference operator is the correct physical stencil. The bridge from the abstract recognition ledger to the physical six-tet cubic Dirichlet model remains open, and this declaration is the scaffolding on which that bridge would be built, not the bridge itself.

theorem periodicMatchingBaseCell_unique
    {Nx Ny Nz : ℕ} [NeZero Nx] [NeZero Ny] [NeZero Nz]
    (a : Fin 8) (target : Vertex Nx Ny Nz)
    {cell : Vertex Nx Ny Nz}
    (h : target = addVertexBits cell a) :
    cell = periodicMatchingBaseCell a target :=
  (Classical.choose_spec (existsUnique_addVertexBits_eq a target)).2 cell h

/-- A supplied continuum Einstein-Hilbert integral value for the canonical
periodic Freudenthal finite-to-integral interface.  It is intentionally just a
real number here: the analytic work lives in the Riemann-sum hypothesis that
identifies finite EH/Dirichlet aggregates with this value. -/
abbrev CanonicalPeriodicContinuumEHIntegral := ℝ

What this page does not claim

The declaration does not prove the physical Dirichlet equality for the six-tet cubic model. The declaration does not assert that gravity emerges from the periodic Freudenthal torus scaffold. The declaration does not provide the actual physical stencil expression for the finite-difference operator.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/PhysicalSixTetCubicDirichletInstance.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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