Gravity Physical Six Tet Cubic Dirichlet Instance

The six-tet cubic instance

The Dirichlet problem asks for a function that satisfies a fixed boundary condition and minimizes a given energy. In this framework, the ledger (a discrete record of recognition events) is laid out on a cubic grid with periodic boundaries, and the module builds the formal scaffolding for a six-tet cubic instance of that problem. The name "six-tet" refers to the six tetrahedral cells that meet at each interior vertex of the cubic lattice, the natural local unit for a discrete geometry.

The module does not assert the physical Dirichlet equality for free. Its own documentation states this plainly: it packages the exact obligations needed to instantiate the physical model on a periodic Freudenthal torus. A Freudenthal torus is a specific periodic triangulation of three-dimensional space, and the module connects that encoded scaffold to the PhysicalSixTetCubicDirichletModel target. The physical finite-difference Dirichlet operator is a placeholder, separated from the abstract canonical graph Dirichlet energy; a later proof should replace it with the actual six-tet cubic stencil expression.

What the module does establish is a set of structural facts. It proves periodicMatchingBaseCell_unique, a uniqueness result: given a target vertex and one of the eight corner offsets of a cube, there is exactly one base cell that maps to that target. It proves periodicDispSqEdge_pos, that every periodic displacement edge has positive squared length, and freudenthalLocalThreeAngleIdentityTarget, a local angle identity on the Freudenthal scaffold. It also constructs a witness at grid size 5, AxisDisp0EndpointUnitWitness5, with explicit coordinates and a computed audit sum of -4, verifying that the local fiber sum matches the expected endpoint contribution.

In plain language, the module is a carefully checked blueprint. It shows that the discrete geometry is consistent enough to support the statement of a physical Dirichlet problem, and it records the exact conditions a future proof must satisfy. The payoff is not a physical law; it is a precise, machine-checked map of what remains to be shown. The actual equality between the abstract graph energy and the physical six-tet stencil is left as a target, not claimed as a result.

/-- 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

/-- Flat-deficit zero target for the Regge flat background.  This is the exact
geometric input needed to normalize the flat Regge action to zero; the
remaining periodic-Freudenthal task is to prove this target from the canonical
flat geometry. -/
def FlatDeficitZeroTarget (K : Triangulation3D) : Prop :=
  ∀ e : Fin K.nE, deficitAngle K (zeroPotential K) e = 0

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

theorem periodicDispSqEdge_pos (d : Fin 7) : 0 < periodicDispSqEdge d := by
  fin_cases d <;> simp [periodicDispSqEdge]

/-- Audit-mirrored per-pair contributions for axis class `0` at endpoint-unit data
(base `(1,0,0)`, `N = 5`).  Matches `scripts/freudenthal_explicit_fiber_endpoint_analysis.py`. -/
def freudenthalAxisDisp0EndpointUnitAuditSum : ℝ :=
  (-1 / 2 : ℝ) + (-1 / 2) + (-1) + (-1 / 2) + (-1) + (-1 / 2)

What this page does not claim

The physical Dirichlet equality is not proved; the module only states the obligations. The six-tet cubic stencil expression is not derived; the operator remains a placeholder. No claim is made that the Freudenthal torus is the unique or preferred discretization of space. The module does not connect the Dirichlet instance to the forcing chain that derives physical constants.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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