Gravity Analysis Regge Hinge4 Dflat Kernel Seed Orbit Assembly Support Projectio

The support projection

In Regge calculus, spacetime is approximated by flat four-dimensional blocks glued together. The framework's library of formal theorems contains a machine-checked map called the support projection. It takes any assignment of numbers to the fifteen possible edge classes in a four-dimensional cell and sets to zero the numbers on edges that cannot participate in a particular geometric structure. The structure in question is a seed hinge: a triangle with vertices at the origin, the first coordinate axis, and the sum of the first two axes.

The map is built from a proved combinatorial fact. Exactly two of the twenty-four four-simplices in the Freudenthal cell contain that seed triangle. Those two simplices are the permutations that begin with the first two axes in order. The support projection keeps only the edge classes that appear in those two simplices. Three edge classes are absent from both, so the projection sends them to zero. These are called decoy classes. The three edge classes on the boundary of the hinge each appear in both simplices, giving them multiplicity two.

The projection is not an approximation or a guess. Its behavior is proved by exhaustive case analysis, which the machine checks. The incidence support is nonempty, it is invariant under swapping the third and fourth axes, and the three decoy classes are explicitly identified. The sum of the incidence counts over all fifteen classes is twenty, which matches the two simplices times their ten edges each.

In Recognition Science, this projection is part of a larger assembly skeleton for a flat Hessian of the four-dimensional Regge action. The skeleton contracts per-hinge area weights against per-hinge deficit kernels, and the projection forces the result to vanish off the incidence support. The true per-hinge kernels remain open: the dihedral and Cayley-Menger calculus that would supply numeric class weights is not evaluated. The projection is the combinatorial factor those kernels must contract against.

What the projection does not do is as important as what it does. It does not complete the flat Hessian of the four-dimensional Regge action. It does not prove that the recognition-science action converges to the Einstein-Hilbert action in four dimensions. It does not recover the gap action. It does not reverse-engineer weights from Einstein-Hilbert. The projection is a proved combinatorial support map, not a physical derivation.

/-- THEOREM: exactly two of the 24 Freudenthal simplices contain the
seed hinge. -/
theorem seedHinge_simplex_count :
    (Finset.univ.filter (fun s : Fin 24 => containsSeedHinge s = true)).card =
      2 := by
  decide

/-- THEOREM (decoy): three explicit edge classes have zero seed-hinge
incidence. -/
theorem seedHingeIncidence_decoy_zero :
    seedHingeIncidenceNat decoyClass4 = 0 ∧
      seedHingeIncidenceNat decoyClass8 = 0 ∧
        seedHingeIncidenceNat decoyClass12 = 0 := by
  decide

/-- THEOREM: every seed-hinge boundary class has positive incidence. -/
theorem hingeBoundary_incidence_pos (i : Fin 3) :
    0 < seedHingeIncidenceNat (hingeBoundaryClass i) := by
  fin_cases i <;> decide

/-- THEOREM (nonvacuity): class `0` (hinge-boundary edge `e₀`) has
multiplicity `2`. -/
theorem seedHingeIncidence_nonvacuous :
    seedHingeIncidenceNat (0 : Fin 15) = 2 ∧
      seedHingeIncidenceNat (0 : Fin 15) ≠ 0 := by
  decide

/-- THEOREM (symmetry): seed-hinge incidence is invariant under the
lattice symmetry that swaps axes `2` and `3` and fixes the hinge. -/
theorem seedHingeIncidence_swap23 (d : Fin 15) :
    seedHingeIncidenceNat (swap23Class d) = seedHingeIncidenceNat d := by
  fin_cases d <;> decide

/-- THEOREM (decoy): three explicit edge classes have zero seed-hinge
incidence. -/
theorem seedHingeIncidence_decoy_zero :
    seedHingeIncidenceNat decoyClass4 = 0 ∧
      seedHingeIncidenceNat decoyClass8 = 0 ∧
        seedHingeIncidenceNat decoyClass12 = 0 := by
  decide

/-- Total incidence mass on the seed hinge (= 2 simplices × 10 edges). -/
theorem sum_seedHingeIncidenceNat :
    (∑ d : Fin 15, seedHingeIncidenceNat d) = 20 := by
  unfold seedHingeIncidenceNat simplex0Classes simplex1Classes
  decide

What this page does not claim

The projection does not complete the flat Hessian of the four-dimensional Regge action. The projection does not prove convergence to the Einstein-Hilbert action in four dimensions. The projection does not recover the gap action or reverse-engineer weights from Einstein-Hilbert.

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/Analysis/ReggeHinge4DFlatKernel.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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