Gravity Seven Gaps Edge Tensor Sector Periodic Conformal Log Subspace5 Iff Encod

Two views of one subspace

On a three-dimensional grid with periodic boundaries, a recognition event, a discrete record of a local change, can be described in two coordinate systems. The first assigns a number to each vertex, or corner, of the grid. The second assigns a number to each edge, or connection between two corners. The theorem periodicConformalLogSubspace5_iff_encodedConformal states that a strain pattern on the edges belongs to the special conformal class if and only if the same pattern, written in the encoded edge coordinates, also belongs to the conformal class. In plain terms, the two descriptions pick out exactly the same set of edge strain patterns; neither coordinate system adds or removes a pattern from the class.

The statement is a theorem in the machine-checked library of formal theorems, meaning it is fully proved with no gaps or unstated assumptions. The proof uses the definition of the conformal ansatz, which assigns one scalar per vertex and derives a log-strain on each edge as the average of the two vertex values. The theorem confirms that this vertex-based definition agrees with the edge-based encoded definition on the specific 5 by 5 by 5 periodic torus. This agreement matters because later results in the same file use the encoded coordinates to exhibit concrete examples of strains that are not conformal, such as a rectangle shear pattern that puts +1 on two edges and -1 on two others.

What the theorem does not claim is broader. It does not say that the conformal class fills the entire space of edge perturbations. On the 5 by 5 by 5 torus, the full edge space has dimension 875, while the conformal slice has rank at most 125, one per vertex. The theorem only identifies two descriptions of the same small subspace; it does not establish how that subspace sits inside the larger space, nor does it construct the full orthogonal decomposition of the edge space into conformal, longitudinal, and transverse-traceless parts. That decomposition remains open.

/-- The typed conformal slice corresponds exactly to the encoded conformal
predicate across the canonical edge equivalence. -/
theorem periodicConformalLogSubspace5_iff_encodedConformal
    (c : PeriodicEdgePerturbation5) :
    PeriodicConformalLogSubspace5 c ↔
      IsConformalEdgePerturbation PeriodicTorus5.K
        (periodicToEncodedEdgePerturbation5 c) := by
  constructor
  · rintro ⟨ξ, rfl⟩
    refine ⟨ξ, ?_⟩
    funext i
    simp [periodicToEncodedEdgePerturbation5, encodedToPeriodicEdgePerturbation5]
  · rintro ⟨ξ, hξ⟩
    refine ⟨ξ, ?_⟩
    funext e
    have h := congrFun hξ (PeriodicTorus5.edgeEquiv.symm e)
    simpa [periodicToEncodedEdgePerturbation5,
      encodedToPeriodicEdgePerturbation5] using h

theorem periodicTorus5_conformalRange_finrank_le :
    Module.finrank ℝ
      (LinearMap.range (conformalStrainLinearMap PeriodicTorus5.K)) ≤ 125 := by
  have h := conformalRange_finrank_le_nV PeriodicTorus5.K
  exact h.trans periodicTorus5_nV_eq.le

/-- Computed from the definitions: the `5 × 5 × 5` torus has 125 vertices. -/
theorem periodicTorus5_nV_eq : PeriodicTorus5.K.nV = 125 := by
  show Fintype.card PeriodicVertex5 = 125
  simp [PeriodicVertex5, Vertex]

What this page does not claim

The conformal class fills the entire edge-perturbation space. The full orthogonal decomposition of the edge space is constructed. The theorem applies to grids of any size other than the 5 by 5 by 5 periodic torus.

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/SevenGaps/EdgeTensorSector.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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